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Large-scale structure and microwave background anisotropies in cosmological models and stellar photometry techniques with the wide field/planetary camera of the Hubble space telescope

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Order N um ber 9003285
L arge sca le stru ctu re and m icrow ave background an isotrop ies in
co sm ological m od els and stella r p h o to m etry techniq ues w ith th e
w id e fie ld /p la n e ta r y cam era o f th e H u b b le space telesco p e
Holtzman, Jon Andrew, Ph.D.
University of California, Santa Cruz, 1989
UMI
300 N . Zeeb Rd.
Ann Arbor, MI 48106
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UNIVERSITY OF CALIFORNIA
SANTA CRUZ
LARGE SCALE STRUCTURE AND
MICROWAVE BACKGROUND ANISOTROPIES
IN COSMOLOGICAL MODELS
and
STELLAR PHOTOMETRY TECHNIQUES WITH THE
WIDE FIELD/PLANETARY CAMERA OF THE
HUBBLE SPACE TELESCOPE
A dissertation subm itted in partial satisfaction of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
in
ASTRONOMY AND ASTROPHYSICS
Jon A. Holtzman
June 1989
The dissertation of Jon A. Holtzman
is approved:
M . - F t& r
Dean of G raduate Studies and Research
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T able O f C ontents
A bstract
vi
P art A . Large Scale S tru ctu re and M icrow ave B ack grou nd
B ack grou n d A n iso tro p ies in C osm ological M o d els
I. I n tr o d u c ti o n ........................................................................................................
2
II. C a lc u la tio n s........................................................................................................
6
III. R e s u lts .................................................................................................................
12
a. Evolution w ith t i m e ..................................................................................
12
b. Fluctuation spectrum of baryons at present
19
c. £(R) and ^
..................................................
25
d. Bulk v e l o c i t i e s ..............................................
26
28
IV. D is c u s s io n ..........................................
a. Normalization
.............................
A rF
35
35
b. Secondary effects on Op- . . . .
37
c. Bulk v e l o c i t i e s .............................
38
d. Comparison w ith observations
39
V. C o n c lu sio n s ..........................................
44
Appendix A. Derivation of Equations
.
45
Appendix B. Com putational Techniques
63
R e fe re n c e s ..................................................
65
T a b le s ...........................................................
68
Figures
77
......................................................
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P a r t B . S te lla r P h o to m e tr y T e c h n iq u es W ith T h e W id e F ie ld /
P la n e ta r y C a m e ra o f th e H u b b le S p a c e T e lesco p e
I. I n tr o d u c tio n ...............................................................................................
95
II. The W ide Field/P lanetary C a m e r a ......................................................................... 97
III. Stellar Photom etry T e c h n iq u e s ............................................................................. 99
IV. Simulated D a t a ........................................................................................................ 104
V. Optimization of Techniques For W FC D a t a .......................................................108
a. Param eters for Stax Detection
....................................................................... 108
b. Param eters for aperture photom etry and PSF d e f in itio n ..........................109
c. Pixel weighting sch em e ........................................................................................110
d. Choice of fitting radius
................................................................................... 114
e. S p e e d .....................................................................................................................115
f. C onclusions............................................................................................................ 117
VI. PSF Representation For Undersampled, Isolated S t a r s .................................. 119
a. High signal-to-noise s t a r s ....................................................................................121
b. Different S / N ........................................................................................................ 125
c. Conclusions
........................................................................................................ 127
d. The Real W o rld .................................................................................................... 127
VII. Variations in the P S F ...................................................................................... '.
130
a. PSF Variations w ith Stellar C o l o r ................................................................... 131
b. PSF Variations with P o s itio n ........................................................................... 132
c. PSF Variations w ith T i m e ............................................................................... 134
d. The Real W o rld .................................................................................................... 135
VIII. Crowding in the W F / P C ....................................................................................137
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V
a. New fitting and grouping te c h n iq u e s............................................................... 139
b. Effects of PS F m is re p re s e n ta tio n ................................................................... 139
c. Sky d e te rm in a tio n ................................................................................................141
d. Quality of crowded field p h o to m e try ...............................................................142
IX. C onclusions...............................................................................................................145
R e fe re n c e s .........................................................................................................................146
T a b le s ................................................................................................................................. 147
F i g u r e s ............................................................................................................................. 153
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A b stract
This dissertation consists of two separate parts. T he first presents cal­
culations of microwave background anisotropies at various angular scales and of
expected large scale bulk velocities and mass correlation functions for a variety of
models which include baryons, radiation, cold dark m atter (CDM), and massive
and massless neutrinos. Free parameters include 0,, H q, the mass fractions of each
component, and th e initial conditions; nearly 100 different models are considered.
Open and flat models w ith both adiabatic and isocurvature initial conditions are
calculated for models without massive neutrinos. A set of flat models w ith both
massive neutrinos and CDM w ith adiabatic initial conditions is also considered.
Fitting functions for the mass transfer function and small angle radiation correla­
tion function are provided for all of the models. A discussion of the evolution of the
perturbations is presented. Results are compared with some recent observations
of large scale velocities and limits on microwave background anisotropies. CDM
and baryon models have difficulty satisfying observational limits, although they are
not completely ruled out. Hybrid models with massive neutrinos and CDM satisfy
current observational data.
The second part of the dissertation is a discussion of stellar photom etry re­
duction techniques for d ata to be obtained w ith the Wide F ield/ Planetary Camera
(W F /P C ) of the Hubble Space Telescope (HST). Detailed simulations are used to
determine optimum techniques to use and to assess the expected accuracy of such
techniques.
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P art A . Large S cale S tru ctu re and M icrow ave B ack grou nd
B ack grou nd A n iso tro p ies in C osm ological M od els
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I. In tro d u ctio n
In “standard” models where structure in the universe forms as a result
of gravitational amplification of initially small perturbations in the mass distribu­
tion, definite predictions can be made for the amplitudes of microwave background
anisotropies and large scale structure in the m atter distribution. Recent observa­
tions of large scale structure combined w ith increasingly more stringent limits on
anisotropies in the microwave background can constrain cosmological param eters in
such models. In particular, the standard cold dark m atter (CDM) scenario, while
successful in many other respects, may fail to predict sufficient large scale power to
m atch current observations. Large scale structure in CDM models can be enhanced
in several ways: for example, by increasing the baryon fraction a n d /o r by allowing
an open universe (Blumenthal, Dekel, and Primack, 1987). It also may be possible
to get large scale power by allowing for a small neutrino mass. In the search for
a combination of components which will produce the structures observed, however,
one finds th a t the addition of large scale power may generate microwave background
fluctuations above current observational limits. The advantage of studying struc­
ture in the large scale m atter distribution and the microwave background is th at
fluctuations in these components are found to be linear, so theoretical predictions
for their amplitudes are relatively straightforward.
New d ata on both large scale structure and the background anisotropy are
becoming available. Several new experiments for measuring A T J T at a variety of
angular scales are in progress (Readhead et al. 1987, RELIK T), and a possible
detection of anisotropy on the scale of a few degrees has been reported (Davies et
al. 1987). The detection of very large structures in the galaxy distribution and the
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measurement of large scale bulk flows of m atter (e.g. Lynden-Bell et al. 1988) are
providing evidence of substantial power on very big scales.
This paper presents a catalog of the expected large scale velocities and
microwave background anisotropies for a wide variety of cosmological models th at
include baryons, photons, CDM, and massive and massless neutrinos, in which
fluctuations evolve as a result of gravitational interaction.
In such models, the
param eters include the relative densities of each component, the Hubble param eter h
(= # o /(1 0 0 km /sec/M pc)), the cosmological constant A (or vacuum energy density
£ l Va c
= A/3-Hq), and the initial conditions. The density in radiation is well known
from the tem perature of the microwave background (taken to be 2.7K here). The
number density of neutrinos depends on the number of neutrino species and is
proportional to the radiation density; in this paper I use three neutrino species. The
predictions are obtained from detailed numerical solutions for the transfer function
and radiation distribution function in different cosmological models.
Nearly 100 different models are considered. These axe best divided into two
classes, namely models w ith and without massive neutrinos. For the models without
massive neutrinos, both flat (J2 = 1 and
Q,
= 0.2 , Q , V a c = 0.8) and open (fl = 0.2)
models are considered, m otivated by theoretical prejudice for the flat cases and
observational evidence for the open cases. In particular, the effect of varying the
baryon content from purely baryonic models to models in which JIcdm ^
studied in detail. Several choices for initial conditions are considered including the
standard scale-invariant adiabatic and CDM isocurvature modes. For the models
with massive neutrinos, the param eter space is larger and the calculations take
longer, so only flat models are considered. Two different baryon fractions are used
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(fij = 0.01 and fij = 0.1), and a range of massive neutrino fractions from
to
= 0.1
= 0.99 are considered, with the remaining mass in CDM. For each massive
neutrino fraction two cases are considered, one w ith the entire neutrino mass in
one neutrino species and the other w ith the mass evenly divided between all three
species. An index of the models considered here is given in Table 1.
Many of the results for models without massive neutrinos have been cal­
culated previously by other authors (Peebles and Yu, 1970; Peebles, 1981; W il­
son and Silk, 1981; Wilson, 1983; Primack and Blum enthal, 1983; Bond and Efstathiou, 1984; Blumenthal et al, 1984; Vittorio and Silk, 1985; Efstathiou and Bond,
1986,1987; Bardeen, Bond, and Efstathiou, 1987). Here, I extend these results to
cover more fully the cosmological param eter space for CDM models (particularly
w ith respect to varying the baryon content and introducing a cosmological con­
stant in the open universes) and concisely summarize all the results. W here overlap
between the current and past results occurs, the agreement is generally good.
In addition, I present new calculations for massive neutrino models and
hybrid models with massive neutrinos and CDM. Some of the results for the mass
distribution in such models have been presented by Bond and Szalay (1983), Valdam ini and Bonometto (1985), and Achilli et. al. (1985). Small angle microwave
background anisotropies in some massive neutrino models have been considered by
Bond and Efstathiou (1984). Here I consider more models and include calculations
for A T / T in hybrid models.
The m ajor intent of this paper is to provide a reference catalog of model
predictions. In this spirit, some models are included th at do not satisfy conventional
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ideas about the baryon content of the universe or current particle physics ideas
about the generation of initial fluctuations. In addition, a discussion of th e physical
mechanisms involved in generating large scale structure and microwave anisotropies
is provided. This discussion is intended for observers, non-specialists, and students
who wish to understand the predictions of different models. Finally, a comparison
of the model predictions w ith recent observational results is presented.
Section II describes how the calculations are performed. In Section III, I
present results. A description of the evolution of perturbations in given in Section
Ilia; the more knowledgeable reader may wish to bypass this. Section Illb and IIIc
present results for the baryon transfer function and power spectrum , and the mass
correlation functions. Section Illd presents results for bulk velocities on large scales.
The A T / T results are given in Ille. Finally, Section IV discusses some uncertainties
in the calculations and presents comparisons w ith recent observations.
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II. C alculations
The calculations here include radiation, baryonic m atter, CDM, and three
species of neutrinos and describe the evolution of linear density perturbations. In
general, the fluctuation amplitude is a function of b o th position and momentum.
For massless neutrinos and photons, the momentum dependence corresponds to a
directional dependence only; for massive neutrinos there is also a velocity depen­
dence. The CDM momentum dependence can be ignored because the velocities of
the CDM are small, so the evolution of CDM perturbations can be reduced to the
fluid equations describing the conservation of energy-momentum; here, perturba­
tions are a function of position only. The baryons can also be treated as a fluid.
Before recombination, the baryons and photons are coupled by Thomson scattering;
the anisotropic scattering is included in the calculations.
The spatial dependence of all the components is handled by solving for the
evolution of the Fourier components (<5(k, t)) of the perturbations:
6( x, t ) = (F /(2 tr )3)
J
d ? k 6(k ,t)eik'x
(1 )
where V represents a large volume in which the universe is assumed to be periodic.
This normalization constant is determined by scaling th e model predictions to m atch
current observed features of th e m atter distribution (see discussion below).
The plane wave expansion used here is strictly valid only for flat models;
for universes w ith curvature, this is inadequate for scales comparable to and larger
than th a t of the cur vat m e, given by:
L c = cH q 1(1 - ft)-1 / 2 « 3000h- 1 ( l - f t ) " 1/ 2 Mpc
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(2).
For the open models considered here, however, perturbations on these scales con­
tribute only to microwave background fluctuations on angular scales > 10 degrees.
The angular distribution of the radiation is handled by expanding the
Fourier components of the radiation perturbation in Legendre polynomials:
6rad(k, q» <) = S
S*(k >
(3)
A
where fi = k • q. Generally, the angular scale of the variations decreases w ith time,
so a larger number of orders is needed (oc kc J d r/a ) as the perturbations evolve. A
sufficient number of orders (up to 800) is used to insure th a t the angular distribu­
tion is well described until the end of the calculation. For the current calculations,
the equations are integrated all the way to the present for scales which cross the
horizon around or after recombination, for comoving wavenumbers k < 0.03M pc- 1 .
For smaller wavelengths, it is computationally expensive to integrate the radiation
equations numerically after recombination. On these scales, the small angle ap­
proximation of Bond and Efstathiou (1984) and Wilson and Silk (1981) is used to
extend the results from recombination to the present. This approxim ation ignores
the Sachs-Wolfe (1967) effect (the gravitational influence of the m atter perturba­
tions on the radiation), but this is im portant only on the larger scales. Several
overlap cases were computed for each model to confirm th a t the approximation is
valid at the crossover point. This technique provides angular correlation functions
and predictions for A T / T which are valid on all angular scales for the flat models.
The results for models with curvature are accurate only for angle < 10 degrees
owing to the limitations of the plane wave expansion.
The angular distribution of the massless neutrinos is calculated by evolving
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the perturbations in each of 20 different directions. A finer grid in angles (or an
expansion in Legendre polynomials) for the neutrinos is not necessary because they
are gravitationally im portant only when they are reasonably isotropic and we have
no need to know about their angular distribution in detail at later times.
The massive neutrinos are computationally more difficult because of the
dependence on both velocity and direction. Here, I adopt the strategy of Bond and
Szalay (1983), in which only the first two moments (with respect to direction) of
the neutrino distribution function are calculated. This is sufficient because only
these moments enter as source terms in the gravitational equations and because we
do not wish to predict the angular dependence of the neutrino distribution. The
velocity dependence is handled by solving for the evolution of perturbations at 15
different comoving velocities.
For all the equations, the expansion scale factor, a(t), is used as the inde­
pendent variable and is normalized in such a way th a t o(now) = 1. The evolution of
perturbations with comoving wavenumbers from 3 x 10-4 to 0.6Mpc~* is computed
in detail, along w ith some larger wavenumbers for models without massive neutri­
nos. The recombination physics of Peebles (1968) is used, allowing for a helium
content of 25 percent by mass. The equations are in the synchronous gauge; results
should be independent of the gauge so long as only physical (gauge invariant) modes
of perturbation are used for initial conditions. Derivations of the relevant equations
are presented in Appendix A. A brief discussion of th e numerical results is given in
Appendix B. The numerical solutions are expected to be accurate to a few percent,
w ith most of the error arising from the finite number of wavenumbers considered.
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T he uncertain physics in the calculation is the initial conditions to be used.
C urrent theories of the very early universe do not predict am plitudes of density fluc­
tuations and even th e perturbation mode (the relative amplitudes of perturbations
in the different components) is not firmly predicted. For a multicomponent universe
there are many modes of solutions. Only those which are gauge invariant can be
considered to be physical modes, b u t this still leaves several modes as well as all
linear combinations of these modes. In this paper I consider prim arily the fastest
growing mode, which is adiabatic:
f>rad(ty = $v(k) = ~6i,ar(k ) = -<5cdm(&)o
o
(4)
For models w ithout massive neutrinos, I also consider two isocurvature modes. In
models w ith CDM, the isocurvature mode has:
Srad(k) = 6v{k) = ! ^ ar(fc) = - S cdm/ A
(5)
^ Pba,r(tjnit) _j_ A _|_ Pv^init) A Pradi^init)
4 PCDM^inzt)
\
Prad(tinit)) PCDM^im'i)
The constant A for the isocurvature case is determ ined from the requirement that
the total curvature is zero initially. In the purely baryonic models, the isocurvature
mode considered has
c _
®bar ~
c (Pradjtinit) ~b Pvjtinit))
°rad
n
\
Pbar\zinit)
'
The adiabatic mode is well m otivated by the predictions of most inflationary sce­
narios. The CDM isocurvature mode could be generated if axions are the CDM
particles (Efstathiou and Bond, 1986). The baryon isocurvature mode is not ex­
pected from any known mechanism. Isocurvature modes could also exist in models
w ith massive neutrinos but these are not considered here.
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10
The initial spectrum is assumed to be of the form:
« *"
(?)
Inflation generally predicts the initial spectrum to be of the scale-invariant form,
w ith n = 1 for the adiabatic mode, or n = —3 for the isocurvature mode; these
spectra guarantee th a t the power in perturbations when they cross the horizon be
independent of size. Most of the results here are presented for a scale-invariant
initial spectrum , but results for other initial spectra can be computed from the
inform ation given.
Although the shape of the initial spectrum of perturbations may be pre­
dicted by inflationary theories, the am plitude is not. To get the correct amplitude,
unnormalized calculations for all the models are performed and then scaled to match
some current observation of large scale structure. Consequently, the results for var­
ious models differ not only because the different physics in each model causes a
different relative amount of growth for each wavenumber, but also because the nor­
malization factor is different for each model. The results presented here are normal­
ized so th a t (A M / M ) rms = 1 at a scale of 8h~^Mpc, which comes from an analysis
of the CfA catalog (Peebles, 1982, Davis and Peebles, 1983). The normalization is
uncertain for several reasons:
1.
The am plitude of observed large scale structure is not known accurately, and
th e normalization from A M / M = 1 is not identical w ith th a t which would be
obtained from other large scale observational results, such as the correlation
function, its moments, or large scale velocities (see section IVa).
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11
2.
Linear theory is not strictly accurate up to the point where A M / M = 1.
From analysis of Zeldovich approximations, Hoffmann (1987) has calculated
th at the normalization used here could be in error by a factor of roughly
5/3, in the sense th at the results here for A T /T , correlation functions, and
bulk rms velocities should be reduced by this factor. The exact value is
somewhat uncertain, however, since Zeldovich approximations may not be
more accurate th an linear calculations for many of the models presented here.
3.
The observations actually measure the light distribution rather th an the mass
distribution, so if light does not trace mass, the normalization will be in error.
In several models, in particular if 0 = 1, observations may require th a t light
does not trace mass (Davis et al., 1985). The uncertainty in the extent to
which light traces mass is param etrized by a biasing factor b, defined as
62 = igg/tpp, where £,gg and ( pp are the correlation functions of galaxies
(light) and mass, respectively (see Bardeen et al. 1986 for a more detailed
discussion). If b > 1 the light distribution is more clumped th an the mass
distribution. If non-unit biasing is used, the results for A T /T , £(R), and
vbulk scale as b~^. Values in the range 1 < b < 3 are currently favored.
Because of the uncertainties in the normalization, one must exercise caution
when comparing the calculations w ith observations; there is possibly freedom to
adjust the theoretical results by up to a factor of three.
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12
III. R esu lts
a. Evolution with time
In these models, there are several im portant epochs. For CDM models,
a critical time is when the energy density in nonrelativistic particles (baryons and
CDM) becomes comparable to th at in relativistic ones (photons and massless neu­
trinos). This occurs at
aeq « 2.4 x 10_5(ftft2)- 1 ( l + .227JV„)
(8)
where N v is the number of massless neutrino species. For the radiation perturba­
tions, an im portant epoch is when recombination occurs at
arec ^ 0.001.
(9)
Another im portant time occurs when the horizon becomes large enough to encom­
pass one half of a wavelength of the perturbation; for flat models,
ahor ~ 1
X
10
kComoving
(^ )
before aeq and
ah„ » 2.5 x 1(T6 ( a h 2) k ^ mins
(11)
well after aeq, for k in Mpc. Finally, for models w ith massive neutrinos, a relevant
tim e is when the neutrinos become nonrelativistic, given by
anr
5 x 10 4
,
for (mj/C2) in eV.
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(12)
13
i. Growth outside the horizon
T he growth of perturbations outside the horizon is different for the adi­
abatic and isocurvature modes. The relevant equations and solutions in the syn­
chronous gauge are given by Meszaros (1980) and Efstathiou and Bond (1986). In
either mode, the quantity S = (3 /4 6rad — ^cdm) is constant outside th e horizon. In
the adiabatic mode, perturbations in all components before horizon crossing grow
as S oc a 2 dining the radiation dom inated era and as 6 oc a during th e m atter domi­
nated era. In the isocurvature mode, the sign of the CDM perturbations is opposite
to th a t of the photon perturbations initially. The photon perturbations are small
initially but grow to keep the to tal energy density near zero, and become comparable
in size to the CDM perturbations around aeq■ This growth of th e photon p ertu r­
bation means th a t the CDM perturbations m ust decrease in am plitude to keep S
constant; this decrease becomes noticeable compared w ith the initial perturbation
after aeq. The decrease continues until th e perturbations cross th e horizon. Well
after the perturbations cross the horizon th e growth is different for each component
but the same for the two modes considered here; a description of th e evolution of
each component follows.
ii. Baryons
If the baryon perturbations cross the horizon after recom bination (true for
the largest scales), they grow because of self gravity. If they cross th e horizon before
recombination, however, they are coupled to the photons by Thom son scattering.
This interaction provides a pressure to counteract gravitational attractio n , so these
perturbations oscillate if they are smaller th an th e Jeans mass, which is comparable
to the horizon size before recombination. On the smallest scales which undergo
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14
many oscillations, the fluctuations are essentially wiped out by photons diffusing out
of the perturbation before recombination (Silk damping). This is very im portant for
pure baryon models because there is no mechanism for regenerating perturbations
on small scales; consequently, in such models, there is significantly more power on
large scales than on small scales at present. It is less im portant for models with
CDM, since the CDM perturbations, which are not affected by Silk damping, act
as seeds to regenerate the dam ped baryon perturbations after recombination.
Once recombination occurs, the Jeans mass drops abruptly since there
is essentially no pressure to prevent collapse; gas pressure stops collapse only on
small scales corresponding to a present mass of ~ 1O6M0 . Any baryon fluctuations
larger th a n this which have survived Silk damping begin to grow again. If CDM
fluctuations already exist at recombination, the baryons fall into these perturbations
and then both components grow together.
f^CDM ^
In the standard CDM model where
the baryons fall into the CDM perturbations very quickly without
affecting the growth of the CDM perturbations, but in a model where ftcDM ~
the growth in the CDM perturbations is slowed by the presence of the smoother
component of baryons. Once the CDM and baryon perturbation amplitudes are
equal, both grow at the standard rates for the appropriate 0 and Slvac- In all
models considered here, the baryon and CDM perturbations become equal before
the present.
Analytic derivations for the grow th laws after recombination are given by
Peebles (1980, 1984), and several examples are shown in Figure 1. For a spatially
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flat universe w ith zero cosmological constant, perturbations grow as
6 oc. a oc
(13)
If there is nonzero curvature or a cosmological constant, the evolution of p erturba­
tions is the same until the expansion rate differs from th a t of a flat universe. In all
the models considered here, this does not occur until after recombination. Negative
curvature and positive cosmological constant increase the expansion rate by
/ q\ 2
H ^
8 tcG
( —) = —-— (pj + pcDM + Pr + pv) + QvcicH q d
V CLJ
O
(1 —ft — £lVac)
(14)
Cl
An increased expansion rate slows the growth of perturbations. Note th at in the
models considered here, the most growth between recombination and the present
occurs in the flat models, followed by the open models with positive cosmological
constant, then the open models with zero cosmological constant, as seen in Figure
1. The effect of negative curvature is less abrupt th an th at of a positive cosmologi­
cal constant; w ith a cosmological constant, growth is almost completely suppressed
soon after the cosmological constant term becomes dominant in driving the ex­
pansion. For the models considered here (ft = 0.2,), however, the curvature term
becomes im portant around recombination, much earlier th an the epoch at which
the cosmological constant term (for ft = 0.2, £lVac = 0.8) becomes im portant, which
is just shortly before the present.
Hi. CDM
For CDM perturbations, the critical scale is th at crossing the horizon at
deq '•
hor,eq ~ 0.3 (ftft2) M p c ~ l
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(15)
16
On larger scales which cross the horizon after aeq, the CDM perturbations grow
w ithout being affected by the presence of other components. O n smaller scales
for which horizon crossing occurs before aeq, the CDM perturbations cannot grow
substantially because radiation is the dominant component gravitationally and the
coupling of the baryons to the radiation by Thomson scattering causes the pertur­
bations in the baryon-radiation fluid to oscillate. W hen the CDM energy density
becomes comparable to th a t of the radiation around aeq, the CDM perturbations
begin to grow. If ficDM
&b the CDM perturbations grow w ith the standard
growth laws. If baryons axe significant gravitationally (flcDM ~
the growth
of the CDM perturbations is slowed until after recombination because of the oscil­
lation of the baryon perturbations; after recombination, the baryons fall into the
perturbation seeds of the CDM after which the perturbations in b o th components
grow together.
iv. Radiation
Before recombination, the radiation is coupled to the baryons and the per­
turbations oscillate as discussed above for scales smaller th an the horizon. During
this time, the perturbations remain reasonably isotropic because of the small mean
free p ath of the photons. W hen recombination occurs, perturbations in the de­
coupled baryons begin to grow. The last scattering of photons off the collapsing
baryons can introduce some radiation perturbations. Once recombination is com­
plete, the photons are relativistic collisionless particles and they free stream out of
the perturbations; at any point, the amplitude decreases as a result of canceling
contributions coming from different directions which originate at different locations
in the wave. On the largest scales which cross the horizon around or after recombi­
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17
nation, perturbations in the photons are gravitationally induced by baryon, CDM,
and massive neutrino perturbations; this is the Sachs-Wolfe (1967) effect.
v. Massless neutrinos
Massless neutrinos axe relativistic collisionless particles. They grow before
horizon crossing, but free stream once they cross the horizon. Well after horizon
crossing, they are gravitationally unim portant since the am plitude of the neutrino
perturbations is small. The prim ary effect of the massless neutrinos in the present
calculations is to set the epoch aeq.
vi. Massive neutrinos
For the massive neutrino cases considered here ( l e V < m v<? < lOOeF),
the massive neutrinos are relativistic at the start of th e calculation and become
nonrelativistic by recombination. After horizon crossing, the neutrino p erturba­
tions will decay if they are relativistic, owing to the directional dispersion discussed
above for the radiation and massless neutrinos. If they are semirelativistic, they
dam p because of the smearing of the perturbations by particles moving at differ­
ent velocities. The largest scales, which cross the horizon when nonrelativistic,
exhibit continuous growth like CDM. Smaller scales undergo a period of damping
until the neutrinos become nonrelativistic, after which perturbations grow. Con­
sequently, the critical scale for massive neutrinos is the size crossing the horizon
when the neutrinos begin to become nonrelativistic. All power on scales smaller
th a n k « 6.2 x 10- ^(m t,c2)M pc-1 is significantly reduced. If massive neutrinos
dominate, then there is much more power at larger scales th an at smaller ones at
present. In hybrid models, the massive neutrino perturbations grow to m atch the
CDM perturbations after the neutrinos become nonrelativistic. For all neutrino
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18
masses considered here, the neutrino, CDM, and baryon perturbations reach equal
amplitudes by the present time. The growth of th e CDM perturbations on scales
smaller than the neutrino damping scale is slowed until the neutrino perturbations
grow to m atch the CDM perturbations. Consequently, there is less power a t smaller
scales relative to larger ones in hybrid models th an in models without massive neu­
trinos. For a more detailed discussion of massive neutrino evolution, see Bond and
Szalay (1983) or Valdarnini and Bonometto (1985) and references therein.
vii. A n example
An example of the evolution of the CDM aivii baryon perturbations on
several different size scales for an
= 1, flcDM 3>
and f^cDM
model is
presented in Figures 2a and 2b for the adiabatic and isocurvature modes. In these
models, logae? ft* —3.8. Solid lines represent the baryon perturbations, dashed
lines the CDM perturbations and dotted lines the massive neutrinos; the evolution
for four different scales is shown. In these figures, perturbations on all scales are
normalized to have the same initial amplitudes, so the relative growth of different
wavelengths can be seen. The largest scale shown (smallest k ) crosses the horizon
after recombination at log a ~ —2.5, the interm ediate scales cross the horizon before
arec but after aeq, and the smallest scale shown crosses the horizon before aeq. The
neutrino mass in Figure 2a is m ^c2
ZeV.
In the adiabatic mode (Figure 2a), the CDM, baryon, and massive neutrino
perturbations on the largest scale grow identically, as 6 oc a 2 before aeq and as 8 cx a
after aeq. The massive neutrino perturbation is 4 /3 th a t of the baryons and CDM
initially, but grows to m atch th e CDM perturbation after horizon crossing since
CDM dominates in this model. On the interm ediate scales, the CDM grows as
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on the larger scale, but the baryons start to oscillate before recombination. On
the smallest scale shown, the CDM perturbation grows only slowly from the time
of horizon crossing until aeq, after which it begins to grow as 6 oc a.
For the
baryon perturbation on the smallest scale, Silk dam ping is evident, but the baryon
fluctuation is regenerated by the CDM perturbation after recombination. For the
massive neutrinos, the two largest scales shown cross the horizon when the neutrinos
are nonrelativistic, and damping is evident on the smaller scales.
In the isocurvature mode (Figure 2b), the CDM perturbations are much
larger and of opposite sign compared with the baryon and photon perturbations
initially, to satisfy the isocurvature initial conditions.
The CDM perturbations
outside th e horizon remain approximately constant (actually they are decreasing,
but by an amount small compared with the initial value) before a eq, and decrease
noticeably after a eq until they cross the horizon. After they cross the horizon they
grow as 6 oc a. On large scales which cross the horizon after recombination, the
baryon perturbations grow w ith the radiation perturbations initially, bu t quickly fall
into the CDM perturbations after horizon crossing. On smaller scales, the baryons
oscillate before recombination, then grow to m atch the CDM perturbations as in
the adiabatic case. Note th a t the relative growth factor for th e different scales is
very different from th a t for the adiabatic case because of the different behavior
before horizon crossing; in the isocurvature case, smaller scales grow more than
larger scales, opposite to the adiabatic case.
b. Fluctuation spectrum of baryon fluctuations at present
The results for the present mass spectrum are given in several forms. In
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20
Tables 2a and 2b, I present five param eter fitting functions for the normalized
baryon transfer function:
, _ Sb(k,a)
T (k, a) — ^ n/2
T
n
_
,
.
(16)
where n is defined by equation (7). The normalization constants given here are valid
for scale-invariant initial spectra using units of inverse Mpc\ for other initial spectra,
the transfer functions must be renormalized. The normalization is determined by
setting (A M /M )rm s =
rm s
1
at 8 /i
/A M
)
1
Mpc, using
2
poo
= /
k 2dkknT (k ) W (k R )
(17)
JO
where W ( k R ) represents the window function of a sphere in k space (Peebles, 1980).
The fitting functions given in the notes to Tables 2a and 2b were chosen to give the
correct asymptotic behavior; different functions are used for the adiabatic (Table
2a) and isocurvature (Table 2b) cases. Also, note th a t a slightly different fitting
function is used for massive neutrino dominated models (see notes to Table 2a).
The fitting functions are less accurate for models with a significant baryon fraction
th an for CDM dominated models, but never off by more th an 5-10 percent. No
results are given for pure baryon models because the transfer function for these
models oscillates and is difficult to fit.
The baryon fluctuation spectrum, defined by P 2 (fc,a) = &3 J(& ,a)2, gives
the contribution of each wave to the variance of the mass density per logarithmic
increment of k; the peak of P (k ) indicates which perturbations will collapse first.
The relative amplitudes of different size perturbations at present is fully determined
for each model from the growth laws discussed above w ith some choice for the initial
fluctuation spectrum.
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Figures 3a-c present the baryon fluctuation spectra for several representa­
tive models. The scale-invariant adiabatic cases are shown in Figures 3a and 3b
for models w ithout and w ith massive neutrinos, respectively. Figure 3c presents
the fluctuation spectra for scale-invariant isocurvature models. In Figures 3a and
3c, the left and right panels have
fraction varies from
=
0 .0 1
= 1 and Q, = 0.2, respectively, and the baryon
in the top graphs to purely baryonic models in the
bottom graphs. For the massive neutrino models (Figure 3b), the baryon fraction
is always fij = 0.01; th e massive neutrino fraction varies from €lv = 0.3 in the top
panels to
= 0.99 in the bottom panels. The graphs on the left have the neutrino
mass evenly divided between the three neutrino species, whereas those on the right
have only one massive neutrino species. These graphs show only a subset of all the
models for which results are presented in Table 2.
i. Adiabatic Mode
First I consider the CDM models. If JIcdm ^
the initial spectrum is
preserved for large scales which cross the horizon during the m atter dom inated era,
since all these scales grow at the same rate at all times. On smaller scales which
cross the horizon eaxlier, the spectrum is depressed relative to the larger scales
because these perturbations undergo a phase of suppressed growth, when they have
crossed the horizon but cannot grow because radiation is dominant gravitationally.
The amount by which small scales are depressed relative to larger scales is given
by T (k) oc k ~
2
for scales which cross the horizon well before aeq if the slow growth
of the CDM before aeq is neglected. This is because the small scale fluctuations
rem ain approximately constant from horizon crossing to aeq, while the larger ones
are growing as 8 oc a2; since the scale factor when the small scales cross the horizon
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22
is proportional to A:- 1 , the am plitude of the small scale relative to the large scale
fluctuations is reduced by A:2. The transition from T (k ) « constant to T{k) oc A; - 2
occurs over a large range of wavenumbers, centered on A;^or e 9 « 0.3 (fl/i2) M pc~
Consequently, for a scale-invariant initial spectrum (Sinit oc A:1/ 2), P (k ) increases
as A;2 for sizes crossing the horizon after aeq, then remains approxim ately constant
for smaller scales which cross the horizon before aeq. Actually, since the CDM
perturbations do grow slightly between horizon crossing and aeq, P (k ) in all CDM
models always increases for smaller scales; this leads to hierarchical clustering with
small scales collapsing first.
If flcDM ~
then the initial spectrum is preserved only on scales which
cross the horizon after recombination, kjl0r rec ~ 0.05 (fi/i2) * ^ M p c 1. The growth
of the CDM perturbations which cross the horizon between aeq and arec is reduced
(relative to the pure CDM case) by the presence of the sm oother component of
baryons which oscillate until arec. On these intermediate scales, the am plitude of
the baryon perturbations at arec oscillates as a function of the wavenumber; these
oscillations reflect the phase of the photon-baryon oscillation when recombination
occurs. Consequently, the fluctuation spectrum in any model w ith a significant
baryon fraction will show some wiggles on scales th a t cross th e horizon before re­
combination because of the gravitational influence of the baryon perturbations after
recombination. If CDM dominates, the baryons quickly fall into the CDM potential
wells and the wiggles are erased.
In a purely baryonic universe, the perturbations on small scales are de­
stroyed by Silk damping, so the power spectrum drops on scales smaller th an the
damping scale. In this case, there can be significantly more power on larger scales
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23
th an on smaller ones, so large structures axe expected to form first. This is evident
in the bottom panel of Figure 3a.
Qualitatively, the effects of the presence of massive neutrinos are similar
to those of baryons. Purely massive neutrino models have the prim ordial spectrum
at large scales with almost no power at scales smaller th an the horizon-crossing
scale when the neutrinos become nonrelativistic; this is seen in the bottom panels
of Figure 3b. This scale depends on the mass of the neutrino, and consequently on
both the total neutrino density €tv and the number of massive neutrino species.
In hybrid models w ith CDM and massive neutrinos, the CDM p ertu rb a­
tions act as seeds to regenerate neutrino perturbations after the neutrinos become
nonrelativistic. If ficDM ^ ^ j/> the neutrinos fall quickly into the CDM p ertu rb a­
tions and the power spectrum is very similar to th at for pure CDM. As the neutrino
fraction or the number of massive neutrino species is increased, the presence of the
smooth neutrino perturbations after damping begins to influence the growth of the
CDM perturbations. Consequently, the fluctuation spectra in hybrid models are
flattened below the neutrino damping scale relative to the pure CDM case, with
the degree of flattening depending on the relative proportions of CDM and massive
neutrinos. If CDM dominates, the fluctuation spectrum continues to rise below
the damping scale. If massive neutrinos dominate, the spectrum drops below the
damping scale.
ii. Isocurvature mode
If flcDM ^ ^bi the transfer function for the isocurvature mode is approx­
imately constant on small scales which cross the horizon before a eq, because all
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24
these perturbations remain almost constant until aeq, then grow as 6 oc a. Larger
scales which cross the horizon between aeq and the present have reduced am plitudes
because of the decay period between aeq and horizon crossing for the CDM p ertu r­
bations. Thus, the isocurvature transfer function scales as T ( k ) oc
&2
for small k
and as T (k ) « constant for large k. As in the adiabatic case, the transition occurs
over a wide range of wavenumbers. On the largest scales which are yet to cross
the horizon or have ju st done so, the baryons have not yet fallen into th e CDM
perturbations. Since they are of different signs initially, the transfer function of the
baryons has a zero at very large scales. The transfer function of the CDM has the
primordial shape at these small wavenumbers.
If the baryon fraction is increased, the CDM perturbations which cross the
horizon between aeq and arec cannot grow as much before recombination as in the
CDM dom inated case because the oscillating baryon perturbation is gravitationally
im portant. Just as in the adiabatic case, the power at these scales is reduced relative
to the pure CDM case, although the effect is less marked since the isocurvature
perturbations grow only after horizon crossing.
For an isocurvature scale-invariant initial spectrum (8ina oc &- ^/2), P ( k )
for the isocurvature mode turns out to be similar to th a t for the adiabatic mode;
it increases as fc2 for the largest scales and remains approximately constant or
increases slightly on small scales. This similarity is illustrated in Figure 3c, which
shows the scale-invariant isocurvature fluctuation spectra for the same models as
those shown in Figure 3a. The break in the spectra appears at slightly larger scales
in the isocurvature mode, but the spectra are otherwise similar.
Hi. Origin o f large scale power
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25
The relative amounts of power in large scale perturbations is primarily
influenced by the relative normalization of the models at 8h~^Mpc. The normal­
ization differences manifest themselves in several ways:
1.
As ft/i 2 increases, aeq is earlier, so th e critical scale th a t crosses the horizon
at aeq becomes smaller, and the break in the m atter fluctuation spectrum
moves to larger wavenumbers for CDM models. Since the spectrum increases
toward smaller scales for all CDM models, there is less power at large scales
relative to the scales contributing to the norm alization as Q,h? increases. This
effect is clearly seen in Figures 3a-c in the relative power at large scales at
present.
2.
The normalization scale ( 8 /i- 1 Mpc) decreases as h increases. This further
reduces the amplitude of large scale power relative to th at of the normal­
ization scale as h increases, since the power spectrum rises towards smaller
scales in CDM models.
3.
As tlb/Q or
increases, the spectrum is increasingly flattened at smaller
scales owing to the growing gravitational effect of a smoother component of
baryons or massive neutrinos on the CDM perturbations, suppressing their
growth. Consequently, there is more power on large scales relative to the
normalization scale as
or
increases.
c. f ( i? ) a n d A M / M
Although the fluctuation spectrum , P (k ), best indicates the relative
strength of various size fluctuations and which scales will collapse first, it is not
directly observable. Predictions for observables such as the two point correlation
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26
function and the rms overdensity in spheres can be made using:
f(J2) = f ° ° k 2d k k nT
JO
(
k
(18)
kR
(Peebles, 1981) and equation (17).
Results for £(i2) and (A M / M ) rms are shown for only the scale-invariant
adiabatic models w ithout massive neutrinos in Figures 4 and 5. The results for the
other models are qualitatively similar and can be produced from the d ata in Table
2.
The different relative amounts of large scale power are very evident in the
plots of the correlation function. These plots clearly dem onstrate the increase in
large scale structure as £lh2 decreases and as
increases.
The correlation function shown here is valid only for £(i2) < 1, since these
are ju st linear calculations. The d ata in this regime are still uncertain, so meaningful
comparisons w ith observations are difficult. Also, these are the correlation functions
of the mass, not the light, distribution. If the light distribution is different from th at
of the mass, as may be the case for galaxy clusters or galaxies if there is biasing,
the am plitude of the correlation function would be modified.
d. Bulk Velocities
The rms bulk velocity of a sphere of radius R is given by
( " L t ( R ) ) rms = / k n T ( k f w \ k R ) d k
(19)
where W ( k R ) is the window function. For comparison w ith observations, it is not
clear w hat is the most appropriate window function to use, since the observations
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27
are incomplete, are subject to selection effects, and may be composed of a biased
sample, for example, if only one type of galaxy is observed. Here I present results
for w hat would be expected if a sphere in space were completely sampled; this
introduces some uncertainties for comparison of observations w ith models and ne­
cessitates deciding on a typical size of a given set of observations, but at least the
numbers presented have a simple physical interpretation. In Figures
6
a-c, I give
the expected rms velocities in the adiabatic and isocurvature models for spheres
of 2000, 3000, and 5000 km /sec in radius; this should bracket the expected range
for recent observations of large scale flows, in particular for the velocities obtained
by Lynden-Bell et al. (1988). The models predict rms velocities; for a gaussian
distribution in 3 dimensions, there is a 90% chance th a t an observed velocity will
lie between 0.34urms and 1.6tVms- The vertical bars in Figures
6
a-c show these
limits for some typical model velocities. Plotted in these figures is the dependence
of the bulk velocity on baryon (Figure 6 a and 6 c) or neutrino (Figure 6 b) fraction
for models with different O, h and O-vac- As in all figures, the results are normalized
to A M / M at
8
h~^M pc with no biasing.
Several trends are evident from these figures, namely th a t the large scale
velocities increase as Oh? decreases and as 0 ^ / 0 or 0 u/ 0 increases. In general
the bulk velocities are larger in scale-invariant isocurvature models th an in scaleinvariant adiabatic models, although they are comparable for purely baryonic uni­
verses. These trends exist because the scales contributing most of the power for
the bulk velocities are larger than the normalization scale. The window function
for a sphere cuts off sharply for k > 1 /R , so the dominant contribution to v^if.
comes from scales k ~ 1 /R if 6(k) increases w ith k, which is the case for all models
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28
considered here on large scales. Consequently, only large scale power contributes to
vbulk f°r large volumes, and the scalings of the results are as discussed above.
e. A T / T
The final amplitudes of the radiation perturbations can be used to deter­
mine the angular correlation function (C(8)) of the radiation. The equations have
been presented by Bond and Efstathiou (1987) for the Q, =
1
cases and for the small
angle approximation, and by Efstathiou and Bond (1987) for the open cases.
The angular correlation function gives the expected average value of the
product of the tem perature perturbation in two directions separated by 8, and can
be used to make predictions for A T / T at a given angle. The exact expression
depends on the specific experimental setup, for example, the beam sizeand the
switching p attern. For an ideal two beam experiment with infinitely narrow beams:
C(0) =
cW=
(^j
(20)
( « i )
where Ji ■% — cos 8. Consequently,
M
' ATV
n 2
V
=, // (T
( ('h
T f) e- )1 - T ( r f \
T ) rm! \
T§
2(C(0)_ C W )
(21)
For real experiments, the effects of beam smearing must be accounted for; the
relevant equations are given by Bond and Efstathiou (1984) and Silk and Wilson
(1980). Power at all scales at recombination contribute to C (0), but only scales
larger th an the size associated w ith the angle 8 contribute to C(8). The values of
A T / T calculated here are prim ary anisotropies; any electromagnetic interactions
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29
between the photons and baryons which might occur after recombination (if the
universe reionizes, for example) are not included (see Discussion).
The results
presented below have been normalized w ithout any biasing; if biasing is desired (or
required) in any model, th e predictions for A T / T will be reduced as 6 - 1 .
The results for C(0) and a fitting function for C (0) —C{6) are presented
in Tables 3a and 3b. The fitting function used here is valid only for angles smaller
th a n 10-12 degrees. Also included in the tables are values of the am plitude of the
quadrupole moment, <Z2 , which is the average of the square of the spherical harmonic
coefficients C2m- Since
depends only on rather large scales for which the evolution
of perturbations is reasonably simple, analytic results can be computed; see Bond
and Efstathiou (1987) for derivations. The results in Tables 3a and 3b are numerical,
but they agree closely w ith analytic estimates. Results are not given for C(0) and
02
for the models with curvature because these are affected by perturbations on
scales comparable to the curvature, which are not calculated accurately w ith the
plane wave expansion used here.
Several graphical examples of A T / T for an ideal two beam experiment are
presented as a function of angle Figures 7a-b for the adiabatic and isocurvature
scale-invariant models. T he results of the calculations for A T / T at 4.5 arcmin, 7.5
arcmin, and
6
in Figures
a-c. These plots are similar to Figures 6 a-c for the velocities and use
8
degrees for some particular experimental setups and
are presented
the same symbols.
The 4.5 minute calculations are calculated for a triple beam switching
experiment w ith a beam width of 1.5 arcmin so th at they can be compared to the
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30
Uson and Wilkinson (1984) upper limit of 5 x 10- 5 (Kaiser and Lasenby, 1988). The
7.15 m inute results are calculated for the setup used by Readhead et al. (1987) (see
also Bond, 1987), which is a triple beam switching experiment w ith a beamwidth of
1.8 arcmin. This group has reported a tentative upper limit at the 95% confidence
level of A T / T < 1.5 x 10- 5 . T he expected results at
6
degrees axe given for the
double beam switching experiment with a beamwidth of 2.2 degrees of Melchiorri
et al.(1981), who give an upper limit th a t corresponds to 4.8 x 10- ^ at the 95%
confidence level. Davies et al. (1987) have reported a possible detection of A T / T =
3.7 x 10- ® at ~
8
degrees. Their setup and sky coverage makes model predictions
somewhat complicated, but they have attem pted to extract y /C (0) from their data,
finding a value of 5.7
X
10~®, although to do this extraction they assume th a t C{6)
is a gaussian, which is not the case for most of the models considered here; it is
not clear to what extent using the true form of C(0) would change their result.
The predictions for y /C ( 0) are included in Figures
8
a-c. The am plitude of the
quadrupole moment, a<i is also shown; the observational limits comes from the
study of Fixsen et al. (1983) (see also Efstathiou and Bond, 1987). All of the
observational limits are shown in Figures 8 a-c as horizontal dashed lines. I postpone
the comparison of the observational limits with the model predictions until section
IV. Predictions for other experimental setups may be determined using the fitting
functions for C(ff) given in Table 3.
Several trends are evident from these figures:
1.
A T / T increases as 9 increases. This is best seen in Figures 7a and 7b. Note
th a t different beam smearings for the different experiments mask this effect
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31
in Figures 8 a-c.
2.
A T / T increases as
3.
A T / T increases as f 'lb? decreases.
4.
The large angle anisotropies are significantly larger for the CDM isocurvature
increases.
models than for the corresponding adiabatic ones; small angle anisotropies
are comparable.
To understand these results, it is necessary to know what size scales con­
tribute most to the background anisotropy. On the smallest scales, perturbations are
smeared out if they are smaller than the width of the last scattering surface, which
is computed from the recombination history to be ~ 7h~^ Mpc, and corresponds
to an angular scale of several arc minutes. Consequently, anisotropies decrease for
smaller angular scales, as seen in Figures 7a and 7b. As (fijh 2) increases (more
baryons), the mean free p a th of the photons decreases, so the anisotropies at small
angles become larger.
The size of the horizon at recombination corresponds to roughly one degree
on the sky today or several hundred Mpc (comoving) at recombination. On angular
scales smaller than this but larger th an the w idth of the last scattering surface,
the background anisotropy is dominated by anisotropies generated by the radiation
scattering off the baryons as they begin to collapse when recombination occurs.
Consequently, the amplitude of fluctuations on scales from arcminutes to about
a degree is directly related to the power spectrum of the baryon fluctuations at
recombination at relatively large scales. Since this is similar at recombination for
the adiabatic and isocurvature modes for scales which have already crossed the
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32
horizon, the adiabatic and isocurvature fluctuations on these scales are similar.
On scales larger than the horizon at recombination, corresponding to an­
gular scales of several degrees or larger, the amplitude of the perturbations reflects
the initial amplitude and growth of the fluctuations outside the horizon. For scaleinvariant adiabatic initial conditions, the large scale perturbations are much smaller
than those generated at recombination on sizes smaller th an the horizon. For isocur­
vature scale-invariant initial conditions, however, there is much more power at large
scales th an in the corresponding adiabatic case because on these scales the initial
spectrum, which varies according to 6 oc
^ /2
(as opposed to S oc fc1/ 2 for the adia­
batic case), has a significant amount of large scale power. This source of large scale
fluctuations is the isocurvature effect described by Efstathiou and Bond (1987).
There is an additional contribution to large scale fluctuations from the Sachs-Wolfe
(1967) effect where the m atter perturbations after recombination generate radia­
tion perturbations gravitationally. It is the dominant source of large scale angular
fluctuations in the adiabatic models, but is small compared w ith the isocurvature
effect for scale-invariant isocurvature models.
The amplitude of A T / T fluctuations at a given angle 9 at present in the
different cosmological models is set by two factors: the am plitude of the p ertu r­
bations on various scales at recombination, and the scale at recombination th at
corresponds to an angular separation of 6 today. The former varies for different
models because of intrinsic differences between the models (different growth laws,
etc.) and also because of normalization differences between the models. The latter
differs for each model because of the different angle-distance relation for flat, open,
and cosmological constant models.
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33
The normalization of th e power spectrum has the greatest effect on the
amount of large scale power for the different models. Since the normalization scale
(~
8
/i- 1 Mpc) is smaller than the scales which dominate the contribution to A T /T
at all angular scales, models w ith more large scale power will have increased A T /T .
Similarly, the growth rate of perturbations since recombination affects the
final am plitude of A T /T ; since the models are normalized a t the present time,
those models w ith less growth since recombination will have a larger amplitude
at recombination th an those in which perturbations have grown more. If there
is negative spatial curvature, the growth of perturbations after recombination is
reduced relative to th a t in a flat universe (see Figure 1). Consequently to match
the current observed amplitude requires a larger amplitude at recombination in a
model w ith negative curvature; the power at all scales at recombination increases
over th a t in a flat model so A T /T increases. This factor alone increases A T /T by
about a factor of three in the
for similar
=
0 .2
for the same h, the
models over the corresponding
0
=
0 .2
0
=
1
models
results axe even larger because of the
normalization differences. For th e models with cosmological constant considered
here, the growth factor from recombination to the present is larger than in the
corresponding open model w ithout
reduced in the O =
0 .2
, Q ,v a c =
0 .8
Q Va c
(see Figure 1). Consequently, A T /T is
model relative to the 0, — 0 .2 , f l v a c =
0 .0
model
by about a factor of two. It is still enhanced over th a t in the flat model. This effect
accounts for about half the difference in A T /T between the open models w ith and
without a cosmological constant at small angular scales. The rest of the difference
comes from the different angle-distance relation in the two models.
For a given angle, the proper distance at recombination is larger for a flat
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34
universe with
0 = 1
than for an open universe or one with a positive cosmological
constant. Since the fluctuations increase as the angle increases, this causes the
anisotropies in the open and cosmological constant to be reduced over those in the
flat models.
Although the normalization effects explain most of the trends seen in Fig­
ures
8
a-c, other effects also play a role. This is dem onstrated in Figure 9, which
shows the intrinsic differences between the adiabatic models at an angular scale of
1 degree. Here all the models are normalized to have the same initial amplitude,
so differences in the growth laws rather than in the normalization are shown; the
absolute scale is arbitrary here. There are clearly still some differences between the
models, but they are smaller than those seen in Figure 8 a, showing the dominance of
the norm alization effects on A T / T . The intrinsic differences reflect several factors.
The most im portant of these is the difference in m ean free p ath of the photons at
recombination. As the baryon fraction is increased, the mean free p a th decreases,
so the expected anisotropy increases. O ther factors include the difference in size of
the horizon at recombination, the different' am plitude of the perturbations at aeq
due to the different values of aeq, and the angle-distance effect.
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35
IV . D iscu ssio n
It is im portant to understand the limitations and uncertainties in the mod­
els and observations before attem pting to use them to rule out possible cosmological
models. In particular, three issues need to be considered for the comparison of ob­
servations of A T / T and Vbulki namely the uncertainties in the norm alization of the
theoretical results, the importance of secondary effects on A T / T , and the extent to
which the bulk velocity predictions can be compared w ith the observations.
a. Normalization
As noted in the Introduction, the normalization used could be in error
because of observational uncertainties or because the linear calculations used here
may not be valid at the normalization scale. By normalizing to A M / M at
Mpc and not applying any nonlinear correction, the results for A T / T and
8
h- 1
if
in error, are likely to be overestimated rather th an underestim ated, except for the
pure baryon or massive neutrino dom inated models (see below). Normalizing to a
different observational result generally tends to lower the predictions for A T / T and
vbulk• For example, normalizing to J^(R), the second moment of the correlation
function, at R =
Mpc lowers the predicted value for A T / T and Vbuik by
25 ± 10% for all the models considered here; normalizing directly to the correla­
tion function where it equals unity gives similar results to the normalization from
A M / M . The nonlinear correction factor suggested by Hoffmann would also work
in the direction of reducing A T / T and Vbuik- Even in the worst case, however, they
are not likely to be off by more th an a factor of two. Consequently, if the model
predictions for A T / T exceed the observational limits by more th an this amount,
the models probably have some problems. If the models have trouble giving large
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36
enough bulk velocities, revising the normalization is likely only to make th e m atter
worse.
In addition to this, the theoretical values for A T / T and
will also be
lowered if the galaxy distribution is biased w ith respect to the mass distribution. If
0 =
1,
biasing seems to be required from dynamical measurements of ft. Also, the
formation of galaxies in a CDM universe may have a b uilt-in biasing mechanism
(W hite et al., 1987). Note that the results presented in Figures 6 a-c and
normalized for no biasing. E biasing exists, the predictions for A T / T and
be reduced by a factor of
6
8
a-c are
will
1, where b is the biasing factor introduced by Davis et
al.(1985); b is generally expected to be in the range 1 ----- 3.
The normalization to A M / M at 8 h~^ Mpc is more uncertain for the purely
baryonic or for the massive neutrino dominated models th an for those w ith CDM,
because in these models large structures form first and may influence the evolution
of structure on smaller scales. An alternative m ethod of norm alization for these
models is to require th a t the first scales to go nonlinear must do so at an early
enough time to account for observations of objects at high redshift.
Using the
A M / M normalization, I find that the first scales go nonlinear at about the present
epoch (see Figures 3a-c). To form objects earlier, the amplitude must be larger; if
we wish to form objects by z = 3, then the amplitude of the fluctuations in the pure
baryon models must be increased by a factor of four in the flat ft =
8 oc a oc
(1
1
models (since
+ z)- 1 ) and by more th a n this in the open and cosmological constant
models. Consequently the estim ates for A T / T and
for the purely baryonic
and massive neutrino dominated models in Figures 6 a-c and
8 a-c
are likely to be
too low by a significant factor. This makes it very difficult to reconcile these models
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with lim its on A T / T .
b. Secondary effects on A T / T
The results presented for A T / T assume standard recombination and th a t
the photons free stream from recombination to the present w ithout interacting with
the baryons. If an early generation of luminous objects were to reionize the universe
after recombination, the amplitude of the fluctuations would be affected, especially
if reionization takes place fairly soon after recombination (or if th e universe never re­
combines). The ionization history depends on when the ionizing objects are formed.
This is more likely to happen early if the power spectrum of the m atter fluctuations
is steep, because in such a case, various sized objects form over a long interval in
time. Consequently, the prim ary anisotropies presented here for the scale-invariant
models axe probably not affected by secondary effects, since the power spectra are
reasonably flat for these models, especially on the small scales which are expected
to collapse first. This may not be the case for models w ith other initial spectra.
If reionization did occur, it is not completely clear w hat its effects would
be. Conventionally, it has been assumed th a t reionization would reduce the fluctu­
ation am plitude because of the increase in the size of the surface of last scattering.
Vishniac (1987), however, has shown th a t reionization can actually increase the ex­
pected A T / T by the scattering off of moving electrons, particularly on small angular
scales. Unless the universe never recombines or reionizes soon after recombination,
it is unlikely th a t the large angular fluctuations would be significantly affected.
A nother possible secondary effect has been suggested by Kashlinsky (1988),
in which multiple gravitational lensings by early forming objects smear out mi­
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38
crowave background anisotropies. This should not be a large effect on large angular
scales, but it may be im portant on smaller scales. For models w ith a late epoch of
galaxy formation such as most of the scale-invariant models presented here, there
are probably not enough condensed objects to reduce the anisotropy significantly
even on the scale of several arcminutes, although a detailed calculation needs to be
performed to confirm this.
c. Bulk Velocities
Several problems arise when one wishes to compare bulk velocity predic­
tions with the observations. Besides the completeness problems and difficulty with
assigning a typical shape and size to a set of observations mentioned in Hid, there
is the troublesome problem of comparing an rms prediction w ith a single obser­
vation. The extent to which our location in space is typical is difficult to decide.
Perhaps more im portant, it is difficult to understand how to deal w ith th e fact th a t
smaller clumps w ith large stream ing velocities inside a larger sample affect the bulk
m otion solution for the entire sample, particularly when all galaxies do not enter
the calculation with equal weights owing to observational uncertainties. Because of
smaller scale motions and dum piness, the bulk velocity solution can be sensitive to
the exact velocity cutoff of the sample and the type of galaxies of which the sample
is comprised.
A detailed analysis of these effects is beyond the scope of this paper. For the
comparison w ith the theoretical predictions below, I use the updated bulk velocities
for a combined sample of elliptical galaxies (Lynden-Bell et al., 1988) and spiral
galaxies of Aaronson et al.
(1982) and B othun et al.
(1984) as com puted by
S.M.Faber (1988, private communication). The most likely bulk velocity solution
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39
gives 457, 421, and 398 km /sec for galaxies w ith spheres of 2000, 3000, and 5000
km/sec; these are the velocities th at are shown in Figures
6 a-c
as dashed lines.
These solutions are obtained without subtracting any flow model; this should be
appropriate to use for comparison with theoretical predictions of bulk streaming
velocities. The velocities used here differ from those published by Lynden-Bell et
al. because of a revised weighting scheme and the inclusion of the spiral galaxy
sample, and the fact th a t no Virgo infall is subtracted off first.
The values of bulk velocities for the largest sphere may not be truly rep­
resentative of bulk motions on this scale, since the more distant galaxies enter the
bulk m otion solution w ith less weight than nearby galaxies. Consequently, the com­
parisons w ith model results are less conclusive at this scale th an for smaller spheres.
d. Comparison with Observations
The observational d ata (mean bulk velocity and upper limits for A T / T )
are plotted on Figures
6 a-c
and 8 a-c as horizontal lines; the A T / T results come
from the experiments listed in Section Hie. In addition to the constraints from
these data, one might wish to consider the conventional nucleosynthesis constraint
th at f If, be less than 0.035/i~2 (Yang et al., 1984).
i. Scale-invariant adiabatic models without massive neutrinos
These are shown in Figures 6 a and
8
a. At face value, the 0 =
1
CDM
dominated models appear to be compatible with observations. These models, how­
ever, generally require biasing of the light distribution w ith respect to the mass
distribution to agree w ith observations. A biasing factor of two or smaller would
still be marginally consistent w ith the observations if the extent of the bulk motion
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40
is small, but otherwise is ruled out a t greater th an the 90% confidence level. Also,
recall th a t if the normalization is in error, the predicted velocities would probably
be lowered.
The £2 = 1 models w ith equal parts baryons and CDM satisfy most of the
constraints although they are marginally ruled out by the results a t 7.15 arcmin and
possibly by the Davies et al. detection. Both these differences could be salvaged w ith
some biasing or favorable errors in normalization. The models w ith a large baryon
content, however, have generally been ruled out by nucleosynthesis constraints.
The pure baryon models predict too large a A T / T at all angular scales,
especially considering th a t the results shown should be increased by a factor of
four or more, as discussed above. Reionization seems unlikely in these models and
would probably not be able to erase fluctuations at all angular scales in any case.
In addition, these models violate the nucleosynthesis constraint.
The open £1 = 0.2 CDM models with a very small baryon fraction satisfy
all constraints except the 7.15 arcm in limit. This seems to be a m ajor problem,
especially for a small Hubble constant. These models m atch observations without
biasing, so they are likely to be able to provide sufficient large scale velocities.
Introducing a cosmological constant might solve the A T / T problem for a sufficiently
small baryon fraction, though this may seem like a fine tuning of param eters.
The open models with comparable amounts of baryons and CDM have
been investigated by Blumenthal, Dekel, and Primack (1987) who find th a t they
produce sufficient large scale power to account for observed large scale structure, in
agreement w ith the results here. As they note, however, these models are difficult
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41
to reconcile w ith small and interm ediate angle limits on A T / T . Even with a cos­
mological constant the model predictions are a factor of two to five above the 7.15
arcmin limit.
Similar conclusions for the models w ithout massive neutrinos have been
reached by Bardeen, Bond, and Efstathiou (1987).
ii. Scale-invariant adiabatic models with massive neutrinos
The results for adiabatic scale-invariant massive neutrino and hybrid m od­
els are presented in Figures
6b
and
8
b. These models were considered because a
component of massive neutrinos helps to boost the large scale power, while a small
baryon fraction can keep the microwave background anisotropy on small angular
scales from becoming too large. Figure
6b
dem onstrates that bulk velocities are
indeed increased over the CDM models, so th a t nearly all of the hybrid models are
compatible w ith velocity observations, even w ith biasing (although models with a
large Hubble constant are marginal). Figure 8 b shows th a t the anisotropy limit at
7.15 arcmin is the only significant constraint on these models. Even this constraint
is satisfied w ith biasing or if models w ith a small baryon fraction ( f lj/f l as 0 .0 1 ) are
considered, especially for hybrid models in which CDM dominates.
Consequently, these models are very promising if one does not object to
having two different species of dark m atter. A measurement of a small neutrino mass
with le V < (m „c2) < 30eV would greatly increase the interest of such models.
H ybrid models are appealing because they may satisfy observational constraints
w ithout resorting to peculiar initial spectra or non-standard models. They are
also testable in th a t both a neutrino mass and a CDM particle should be able to
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42
be detected eventually. A disadvantage is th a t they require two species of dark
m atter to exist in the universe w ith roughly compaxable densities (within an order
of magnitude).
Hi. Scale-invariant isocurvature models
These are shown in Figures 6 c and 8 c. All of these models probably pro­
duce sufficient large scale velocities but seem generally ruled out by limits on A T / T ,
particularly at large angular scales; similar conclusions have been reached by Efstathiou and Bond (1986). CDM dom inated
= 1 models are only marginally ruled
out by the large angle fluctuations; this might be avoided if there was biasing and
a large Hubble constant.
iv. Non scale-invariant models
There are no compelling theoretical justifications for adopting non-scaleinvariant spectra. One might consider, however, if models with an initial spectrum
w ith more large scale power would be consistent with observations. For example, an
adiabatic model w ith n =
0
has increased large scale power over th e scale-invariant
adiabatic case. Consequently, larger bulk velocities are produced. Predictions for
A T / T are also increased particularly at large angular scales. For this initial spec­
trum an fl = 1 CDM universe, especially w ith biasing, would be consistent with the
observations considered here. Models with a substantial baryon fraction are ruled
out by the limits on A T / T .
Peebles (1987) has suggested th a t an isocurvature initial spectrum with n =
0 or n = —1 in a purely baryonic open model could m atch current observations. Such
an initial spectrum would produce sufficiently large velocities in a baryon dominated
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43
model, but not in a CDM model. Both Q, = 1 and
= 0.2 purely baryonic models
give large enough velocities for n = —1. These models violate the small angle
anisotropy limits. Since the fluctuation spectra are much steeper in these models
than for the scale-invariant case, however, the first objects to go nonlinear do so at
a much earlier epoch than in the scale-invariant case. Consequently, reionization
might occur and resolve the A T / T problem. Efstathiou and Bond (1987) have
considered various ionization histories in such models including scattering off moving
electrons and find th at isocurvature models with n — —1 are probably compatible
with the small angle limits at 4.5 arcmin; the 7.15 arcmin limits may be more
difficult to satisfy.
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44
V . C o n clu sio n
Calculations of large scale bulk velocities and microwave background
anisotropies have been presented for a wide variety of models in which structures
form as a result of gravitational amplification of initially small, gaussian p ertur­
bations. The general difficulty of having such models be consistent w ith obser­
vations arises because the scales th at generate large scale velocities also generate
microwave background anisotropies. Standard CDM and baryon models have nearly
been squeezed out between these observational constraints, although more needs to
be understood about the velocity observations and biasing to seal their fate firmly.
Massive neutrino models predict too large background anisotropies. H ybrid models
w ith both massive neutrinos and CDM, however, appear to satisfy both observa­
tional constraints.
I would like to thank Sandy Faber, George Blumenthal, and Joel Primack
for encouragement and help they provided throughout the completion of this work.
The computations presented here were done on a Microvax supported by a ?? grant
to S.M. Faber.
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45
A p p e n d ix A . D e riv a tio n o f e q u a tio n s
The goal is to calculate the linear evolution of an assumed initial spectrum
of fluctuations through the era of recombination in a m ulti-com ponent universe for
various cosmologies. T he solutions can be decomposed into complex plane waves,
S (x, t) =
where each mode (k) evolves independently. We will calculate
the linear transfer function, T ( k ) = Sf. (present) /6f. (initial).
In this calculation the universe can include up to five components: ordinary
m atter, radiation, massless neutrinos, massive neutrinos, and some kind of massive
weakly interacting particle (CDM). The differential equations to be solved are the
conservation equations for each component as well as Einstein’s gravitational field
equations. Each component may have a different transfer function, but in most cos­
mological models of interest, the transfer functions for all components are identical
at the present time.
The conservation equations take one of two forms. For components which
interact electromagnetically, an isotropic velocity distribution exists at each point in
space, so the relevant equations are just the fluid equations of mass and momentum
conservation; here, the perturbations are a function of position and tim e only. For
particles which interact only via gravity, however, the perturbations are also a
function of direction, since at a given point there may be an anisotropic velocity
distribution. In this case, the relevant equation is the Boltzmann equation (or if the
particles are collisionless, the Vlasov equation); this assures th a t the to tal number
of particles is conserved.
The components which behave as a fluid are the ordinary m atter and the
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46
CDM. The CDM can be treated as a fluid even though th e paxticles are noninteract­
ing because the random velocities of the particles are very small (since the particles
are massive). The neutrinos are collisionless and the evolution of their perturbations
are thus described by the Vlasov equation. The evolution of radiation fluctuations
is calculated using the Boltzmann equation to account for the Com pton interaction
w ith the baryonic m atter.
i. Coordinates and metric
Throughout this calculation we use time orthogonal coordinates and the
synchronous gauge, where:
000
=
1
0 00
90a = 0
9a0 = ~ a i^afi
=
1
g0a= 0
9 ^ ~
(x ?*0)
~ ~ ^2
(A l)
a/3 d" ha/3 (x, t))
In this and all future equations, latin indices range from 0-3, while greek indices
range from 1-3. The determ inant of the metric gij is:
|<7 | ~ —a (1 —h)
(A2)
where h = ^2 hiiii. Stress-energy tensors
a. Perfect fluid:
The stress energy tensor for a perfect fluid is given by:
T ij = (p +
b V - gijp
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(AZ)
47
t Q0 =
P°2
T 00 = pc2
Too = - a 2 (Saf - h a f ) ^
Tap
T ° “ = ( p + $ ) cu«
= a* (Sa-, - /.«,) («v -
hv )
ni
r * ? = (/> + $ ) ^
P
For the fluids that we axe interested in (m atter and CDM) pc2 > p so the
pressure terms can be neglected, but I’ll leave them in for now anyway.
b. Stress-energy tensor for massive collisionless particles
By definition:
T tj =
j
y ~ jY j 2 2S ( 9ljPiPj ~ m 2 c4) p V /
(A4)
where f is the distribution function (in spacetime):
/ = /(x ,p ) = f( t , x i , X 2 , x 3,p 0,p 1,p2,P3).
The delta function appears because the four p ’s are not all independent:
gVpiPj = m 2c4.
(Ah)
Following Peebles (1980), we will express the distribution function as a function of
several other variables, namely particle energy pg, direction angles to convey the
direction of photon motion (these 2 angles are expressed in term s of the 3 direction
cosines j i = sin 6 cos < ^ ,7 2 = sin 0 sin ^ , 7 3 = cos 6 and the constituency relation
XX7 a)^ =
1 )?
plus an auxiliary variable e which gives the degree of freedom for the
normalization of the 4-m om entum (which we will proceed to integrate out):
/ = /(*, ®1, ®2, S3,P0,0, <f>,e) = /(< ,s i , 3 2, x 3 ,p 0, 71,72,73, e)
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48
pa = - p a t J Q
P = (P o~ ™2) 2
(-46)
In these variables:
9ljPiPj ~ m 2 C4 = Po ~
“ 2
(*W + >*<*/?) P<*P(3 ~ ™2cA
(A7)
=
p
2
1
- e2
(1
+ 7 a 7 /?^a/?)]
so the integral for stress energy tensor is:
Ti =
J
^
^
2
6 (p 2 ( l - e2 ( 1 +
P V f M
( ^ 8)
where J is the Jacobian of the coordinate transformation:
p i = —pae sin 6 cos <^>,P2 = —pae sin 6 sin <j>,P3 = —pae cos 9
\J\ = —p^a?e2 sin 6
(^49)
Using this, the integral for the stress energy tensor is:
T tj =
J
6 (p 2 ( l - e 2 ( l + /»a /?7 a 7 /?))) P V / ( ~ p 3 a 3 e2 sint?) (A 1 0 )
Doing the integral over the delta function, we have:
T* = f ifJ dU\ 2p V / (-pV sm9) f
•/ o 3 ( l - | )
V
+
\
^ \-2 p 2 ( 1 + ^ 7 ^ ) V
So finally, to first order in h:
Tij =
(l + | )
J d p d Q ^ ^ -f
( l -
\ h apl a ip
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(A12)
49
In term s of components, this expression, w ith / =
0- 0 :
00
+
<C
is:
J P0P2dPd^ (f ~\fbKl3l<xl(^j
T00= (l +
r
f , f \, f\ fi
= j p o p 2dpdti ( f + f b ( j £ ~ ^ f b K p l a l p ^ ,
but / dQ.'ya'yp = %g-8ap so:
T00=
j pop2dpdtif =pic2(1+6(x,<))
(.413)
0 -ct:
*=(i +1) / dpdaljff (i - f ^ )
.0
_
P ‘ = PO
P ° = s " ‘l>,
= JJ (<<,3 + Ao/j) P “ 'I 3 ( l - 5 /1037073 )
Since 7 a is first order, this becomes:
=~ j P^dpdSlfja
T0a
(A14)
a -0 :
^ = (X+1) /
(l -
= [ d p d Q .^ -^ J a l3 +
J
POO
/
4f
dPd^
f h
~ I ( 2TaT^ + /laT7T (T/?+ 7a^“
5
\
2 hpalpl<TlctlP )
using:
/ d^7a7/? = i f W
and:
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50
J d ^ l j a j a j y l s — H ) {^a/3^j6 + ^ ay 3/36 + &ab^(3y)
we have:
T ap
=Y + J
{6ap
haj3)
j
+
dp& ±
dpdnp 4 { f ~ fb \ a yp
(-415)
c. Stress-energy tensor fo r massless collisionless particles
For massless particles, p — po, so the stress energy tensor becomes:
j
T®a =~J
J ^ +J
T 00 =
p Zd p d S l f
P3d p d Q f j a
Taf
=y
(Safi + hafi)
p Zdp
pZd p d ( l ^ - ^ - 7a^
The mean energy density is given by:
J
P6C 2 / 4 t t =
p 3d p f b
(416)
and A(0, <j>), the fractional perturbation to the brightness, is defined as follows:
p bc 2 (1 + A ( 6 , 4 ) )
/4 tt =
J p Zd p f
6=j dfiA/47r
J
J
fa —
7
(418)
a A d fi/ 4 ?r
(419)
« 7 /?Adf2 / 4 7 r
(420)
7
Va/3 =
(417)
Then
T00 = Pb<? (1 + 5)
Toa = -aPbc2f a
rp
___
2 2
= Pbc a
o (faff
(A 2 1 )
haft) + 'Ha(3
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51
Hi. Perturbation equations, perfect fluid, no interactions
The equations of mass and momentum conservation expressed in covariant form
axe:
= ( l - s l 1/ 2 2 ? )
(422)
- \ \- 9 \1/2g j k j T * = o
In the linear approximation:
J
P = Pb (1 + £) i P = Pb + <%Pbt, c« =
< c, 6 < 1
(423)
The time component of the conservation equation is:
d
dt
3
1 pc
5
a3 I
dxa
2
1
5 )(' + 3 ) “ "
(424)
Keeping terms only to first order:
d_
pbc2a 3
dt
1
^
2a
(1
udh
a
"dt
+ 8- |
n
+ a 3 ( Pb +
) cu“a
h
2 • /
2
r
° a \ pf>+ CsPb° - nPb
(425)
The unperturbed p art gives:
| («cV)
=
- t o
(426)
apb
Substituting this in the perturbation equation gives:
—
dt
_
4.
2
d t\
JEL.
P—6 C2
P6
cs
Pbc 2
c2
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(4 2 7 )
52
The spatial component of the conservation equation is:
(x- 1)
a3 [
.- 5
1
° 3
(*
- 1) m , ° T ik =
(^2S)
0
To first order:
(A29)
The two conservation equations have plane wave solutions of the form
S = £ £jfee*k'x , u = u ke ^ 'x . If we choose one coordinate axis to lie along k, we then
get:
d
dt
dh _
dt
1
2
dh /
h
dt V
~ « 5
( Pb + ^ t ) uc] + ikaZph<?s8k =
A
1 + ( h
/>6 C /
\
n v *“ ‘
Pbc /
c =
(A30)
0
* f ( pbc2
c2
U 31)
iv. Expression for Compton interaction term:
The interaction term enters the conservation equations for m atter as a force term:
T l] = gl
Peebles and Yu derive a covariant expression for
(A32)
by noting th a t in the m atter
rest frame:
<7 ° = 0
(A33)
ga = a n eT%daZ
(A34)
since T ^ d gives the radiation flux density, thus crneT®®d gives the net volume force
(the a 3 enters to account for volume changes in time).
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53
= c ,u a — 0),
The covariant expression th a t gives this in the m atter rest frame
is:
* =
[T 'iju j -
(A 3 5 )
We need g{, not gl, for the way the conservation equations were expressed in Section
III:
9i
=
9ik9k
so = g» = [ c l * +
=
0
- c (t„“ + T% ,ua +
1
^ / ^
) ]
„3
(.436)
to first order
Using the results from the last section, we get:
ga = - a Za n epbc2 ( a c fa +
(A37)
Using this expression in the spatial component of th e conservation equation gives:
~dt
^ 2 ) U c + a Pm,bcs~Q^a ~ a a n zPr,bc
cfoe +
ua ^
(A38)
where (m) refers to m atter, (r) to radiation.
If we neglect pressure term s (which is a good approximation for ordinary m atter)
and use v a = aua = ^ u a , the equation becomes:
^
(^P m ,b v<Xc) = aZ<Tnepr>bc2 ^ a c fa +
= aAa n eprbc2 ( c f a -
W ithout pressure term s, th e unperturbed p art of th e tim e component equation
gives:
d t ( a3^m’6) = °
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54
so:
d
3
.
n \
4
2 (
r
Pm’bd t ^av C' = a anep^ hC \ cf a ~
^ • + V
&
= ^ ^ ( c / „ - V
Pm,b
a
\
3
3
U
)
(A 39)
)
v. Perturbation equations fo r collisionless particles
The Vlasov equation expresses the conservation of the to ta l num ber of particles:
f=0.
(AiO)
As in Section II, the distribution function is expressed as:
/ = f(t,x i,X 2 ,x z ,p o ,'Y a )
so the Vlasov equation is:
df
d f dxa
d f dpQ
d f dja
Ht + d ^ ~ d T + d ^ 1 F + d ^ ~ d T ~
^
^
In this expression, both of the factors in the last term are first order in the p ertu r­
bation, so it can be dropped in the linear calculation. In the second term:
Since
dxa
pa
pja
dt
p®
pqci
is first order in the perturbation, the Vlasov equation becomes (to first
order):
df
df
p ja
d f dpo
,
idtt + pqci d x a + 7T—
1T = °dpQ dt
(A 43
The last term is calculated from the geodesic equation:
dpQ
^
pip^
1
=
a
1 d h ajj
\
p 22
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<^>
.
55
Thus, the Vlasov equation becomes:
df
p ja d f
d f fa
dpo \ a
1
dhla/3
) d dt ) PO
(A45)
a. Massless particles
For massless particles, p = po> so:
df
ya d f
a dxa
d f fa
dp \ a
)p -
1
<^2 a^P Qt
(A46)
The fractional perturbation to the density is defined by by:
/
PZf d p =
(A47)
(! + A )
Multiply the Vlasov equation by j? and integrate over p to get:
d_
dt
(1 + A )) +
H
*Pb (1 + A) = 0
+ (5 -
(A48)
The unperturbed part gives:
(A49)
and the first order part is:
dpiA t qapt 3A
dt + “ T
~ 2w
^ - a r + V
6A = 0
(A50)
so:
dA ,
7
« dA
¥
i s ?
d h ap
’
7 “ ' l''? _ a T =
(451)
This equation has a plane wave solution of the form A = A ^eJ^ x . If we chose the
x ® axis to lie in the direction of k, then by symmetry h \\ = h%2 and h \ 2 =
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^13
=
56
^23 =
0
, and:
. 2 n dhl l ,
2 n dh23
™ i r - § r = sm * -& + cos e~ d T
(1
—^ 2) oft
2
dt
/ I —l**
V
2
‘A dh22
J dt
(A52)
and y = cos 9
(A53)
y
2
where:
y =
^1
—y 2^ ^
— ( l —3ju2^
so the Vlasov equation becomes:
dA
iky, .
- + - f A = !,
.
(.454)
b. Massive particles
Here we axe interested in calculating the perturbation equations for a massive
particle which may be relativistic at early times. The m ain candidate here is the
massive neutrino, so we take the unperturbed distribution function to be the FermiDirac distribution:
/0=
(A55)
where we note th a t -Jpp is independent of time (since p and T both scale the same
way). The perturbed part of the Vlasov equation is:
d fl
dt
, P Jadf]_
, dfodp _ n
p 0a d x a + d p dt
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(
^
57
•
•
•
•
•
_/
Using
the expression
for the unperturbed distribution
and m
T = -£■:
/ ca
dfodp
dp dt
pc da \
dp
J
dt
(ei? t + l ) 2 ^ T° + kTQ 9 p
pc
(A57)
c
ekT
(e #
+ l)
f
kT<>\
dp
,\
*
Since Pq = pz + m?c^, podpQ == pdp, the geodesic equation becomes:
dp
fa
I
d h ap \
d t = - p { a - 2 7a7e - d T )
(A58)
so th e Vlasov equation is:
£C_
ekT
+^
d f l + P lo t d f i
dt
p0a d x a
2
d h aj3
cap
lo ti0dt
2k T
=
0.
(A59)
If we define:
9=
(A60)
then:
dg , p i a dg
1
dhap
— + ------------- = - l a i p - ------dt
pqo d x a
2
dt
(A 61)
ression for g = gke^ x and the expression for 7 a 7 /?
W ith the plane wave expression
given in Section Va, this reduces to:
dg
ik y p
y
H
at '---------a pq9 ~ 74
(AQ2)
vi. Compton interaction for photons
We wish to calucate the change in th e photon distribution function from
collisions w ith electrons. In the m atter rest frame (denoted by a prime), the scat-
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58
tering from point
1
changes the distribution function at point
2
such that:
/ ( 2 ) ' - / ( l ) ' = crnec « , ( / + ( p J | ) - / ')
(.463)
where:
OX1 dt
OX1PQ
PQ
(-464)
and:
f+ M =
if the
j
f 0 4 7«)
(-465)
scattering is isotropic in the m atter rest frame (which Peebles thinks is a
reasonable assumption). To get an expression for p 'q, note th at p iu 1is an invariant,
so:
+
p'Qu Q'
P q = PQU ^
=
p'a u a t = p q u Q + p a u a
+
PaU a
m atter rest frame
—a j a U
pq
(A66)
01^
to first order
(-467)
= Pq (u ° - j a v a^
For massless particles where
pq
=
dp
p,
=
this gives:
d p 1 (c
—J a v 01) -
(A 6 8 )
^
The Boltzmann equation for the photons is:
|
= ™ e(/+ - /)
(-469)
Since
(/+ - / ) is first order, we can neglect the
since it equals 1 to zeroth
order.
Asin the previous section, we will multiply by p ^ and integrate over p. The
background energy density of the radiation in the m atter rest frame is:
Pb,rc2 = J PS' f ' dP'
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(-470)
59
Define the energy scattered into the radiation:
J p 3U d p = ^ A +
(A71)
The total energy density at any point is:
Pb,r (1 + A +) = J f'p ^d p d tt'
= J f'p 3' (1
=
(1
- 7a«“ )-4
dp'dn1
(472)
+ 4 j a va ) p'b
To first order:
P'b,rc =Pb,rC2 { l + S r)
(A73)
This gives:
(1 + A +) =
(1
+ 4ya va ) ( 1 + Sr) ^ 1 + 4ya va + 6r
(474)
So the Boltzmann equation for radiation becomes (using the results for the left hand
side derived in the last section):
dA
j a dA
n
aT + V & ? “ ^
d h af3
- g r = <m‘ (A + " A)
(475)
= a n e (Sr + 4 j a v a - A)
Using the plane wave expression for the perturbation gives:
dA
iku
+ - ^ - A —y = a n e (Sr + 4pv —A)
vii.
Equations fo r the
(476)
components
a. CDM
The CDM is treated as a pressureless perfect fluid. From (A31),we have:
^cdm _ 1 d h
dt
~ 2 dt
ik v c u M
a
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( A77\
{
}
60
Prom (A30), we get:
<focDM
a
—j — = — vcdm
u t
s ana\
(A78)
CL
b. Baryons
Baryons are similar to the CDM, but Compton interactions must be in­
cluded. Prom (A31):
dSbar
dt ~
1
2
dh
ikVhgjd t~
a
( d'7Q\
1
}
From (A39):
dt>cDM
7,
—
dt
a
, crne/9rad ( c
^CDM ~F
( C/rad
a
Pm t \
4
\
o^bcir I
3
/
(A80)
where / rad is defined by (A19). c. Radiation
Prom (A76) we have:
—
— ( l - 7<2) ^
- ( l - 3/(2j
+ <roe ( 6 rad 4- 4/zt>bar) - A ri,(i
(>181)
We expand the angular dependence of the radiation perturbation in Legendre poly­
nomials:
Arad = ^
=
(A82)
^
^
= 1
(483)
Taking the first few moments, we get:
d60
dt
S\
dt
d6
2
d<
_
- i k c Si , Adh
a 3
3 dt
^ j _j_ a n ^c (4 ^ ^ — Si)
ifcc
a \5
(I*2+*°)
ctt
32xGa
—ike / 3 C , 2 c \\ , 4 d/i
327rGa
(^3
+
j
+
*
-------a V7
2
3^ 1
3
(A85)
f
< " * .< * 2
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(^ )
8 6
61
dSj __ dt =
+ 2 T 1 61- 1) - ° n ‘ c>'
1> 2
(A87)
where:
F = ipiadfrad "f" i Pisf IS ”1“ ipbax^bai
ipVCDM
(A 8 8 )
d. Massless neutrinos
From (A59) we have:
<*»>
In these calculations, we just compute A v for 20 different values of p. and integrate
numerically to get the source terms for th e gravity equations.
e. Massive neutrinos
Equation (A62) gives:
dAv
ik p p
_ n 0(l ~
-f---------iSj/ —
dt
a q
4
)p
As noted by Bond and Szalay (1982), this equation has the solution:
A „(f) = A(<0)exp (—ikp(z(p, t) - z ( p ,t0)))
+ j [ d t ’exp (ikcp{z{p, t ’) - z(p , t)))
^ n0 > +
One can take angular moments of the above equation to obtain the source term s in
th e gravity equations. The relevant equations are presented in Bond and Szalay.
/. Gravitational field equations
Finally, we need the equations which relate the gravitational source terms,
h = ^
and
A 33
=
to the perturbations in the components. These are derived
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62
from the field equation:
R ij = S tcG ( Tij - ± g ijT )
(A92)
where R ij is the Ricci tensor, which can be calculated from the m etric tensor, and
T is the trace of the to tal stress-energy tensor. The first order equations, when the
unperturbed parts are subtracted, yield (Peebles 1980):
Z at + ~a k = 47rG ( 2 T 0 0 - T ~ P b - 3p&)
(^493)
A3 3 = h — 8itGT q3
(A94)
where pi and p i refer to the background, unperturbed, density and pressure. The
stress-energy tensor in these equations refer to the sum of the stress-energy tensors
of each component.
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63
A p p en d ix B . N um erical tech n iq u es
The general problem is to solve for the evolution of the perturbations in
each component from some assumed initial conditions.
For the CDM and the
baryons, we just need to solve for the overdensities, <5>cdm and £bar, and the pe­
culiar velocities, for a range of different wavelengths. For the radiation and the
massless neutrinos, the overdensities are a function of direction as well as posi­
tion, so we must solve for overdensities at a variety of angles. For the radiation,
th e perturbations are expanded in Legendre polynomials as discussed in Appendix
A. For the massless neutrino perturbations, solutions are obtained in 20 different
directions. For massive neutrinos, we also m ust include a momentum dependence.
Initially, perturbations are all components are nearly isotropic.
Conse­
quently, the massless neutrino angular distribution is well represented by the
20
different zones, and the radiation angular distribution can easily be described by
ju st a few Legendre polynomials. After horizon crossing, however, the angular de­
pendence of the perturbations can get very fine. W hen this happens, however,
th e integral over the angular distribution, which is the gravitational source term ,
generally becomes small. Consequently, by the time the massless neutrino angular
distribution is no longer well represented by
20
zones, they are totally unim portant
gravitationally, and they axe manually damped. We cannot do the same for the
radiation, since we are interested in the fine scale angular structure of the radiation
perturbation at present. Consequently, higher orders of Legendre polynomials are
added as they are needed as the calculations proceed. By shortly after recombi­
nation, several hundred orders may be required. As discussed in Section II, the
calculations are stopped at this time, and the small angle approximation is used to
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64
extend the results to the present.
Since the evolution must be calculated for several components through
several different regimes (before and after horizon crossing, during the radiation and
m atter dominated era, before and after recombination), it is not possible to solve the
equation analytically, so numerical techniques are used. The numerical integration
is a bit tricky, as the radiation and baryon perturbations m ust be followed closely
as they oscillate before recombination. The equation have been integrated using
the DG EAR subroutine of the IMSL m ath library. This routine uses a variety of
techniques. Before recombination, a backward differentiation technique developed
by Gear has been used. After recombination, standard techniques for integrating
differential equations can be used.
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65
R eferen ces
Aaronson, M. et al. 1982, Ap. J. Supp. 50, 241.
Achilli, S., Occhionero, F., and Scaramella, R, 1985, Ap. J. 299, 577.
Bardeen,J.M ., B ond,J.R ., and Efstathiou,G. 1987, Ap. J. 321, 28.
Bardeen,J.M ., B ond,J.R ., Kaiser,N., and Szalay,A.S. 1986 Ap. J. 304, 15.
Blumenthal,G.R., Dekel,A., and Prim ack,J.R., 1987, Ap. J. 326, 539.
Blumenthal,G.R., Faber,S.M., Prim ack,J.R., and Rees,M.J., 1984, Nature 311, 517.
Bond,J.R. and Szalay,A.S., 1983, Ap. J. 274, 443.
Bond,J.R. and Efstathiou,G ., 1984, Ap. J. Lett. 285, L45.
Bond,J.R. and Efstathiou,G ., 1987, M .N .R .A .S 226, 655.
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Universe, ed. J. Audouze and A. Szalay (Dordrecht: Reidel), p. 93.
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Davis,M. and Peebles,P.J.E., 1983, Ap. J 267, 465.
Efstathiou,G. and B ond,J.R ., 1986, M .N .R .A .S. 218, 103.
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Efstathiou,G . and Bond,J.R., 1987, M .N .R .A .S. 227, 33P.
Fixsen,D.J., Cheng,E.S., and W ilkinson,D.T. 1983, Phys. Rev. Lett. 50, 620.
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Kaiser, N. and Lasenby,A.N. 1988, Nature, in press.
Kashlinsky, A., 1988, Ap. J. Lett. 331, LI.
Lynden-Bell,D., Faber,S.M., Burstein,D., Davies,R.L., Dressier,A., Terlevich,R.J.,
and Wegner,G. 1988, Ap. J. 326, 19.
Melchiorri,F., Melchiorri,B.O., Ceccaxelli,C., and Pietranera,L., 1981, Ap. J. Lett.
250, LI.
Meszaxos, P., 1980, Ap. J. 238, 781.
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J.
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Peebles,P.J.E., 1984, Ap.
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284, 439.
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Peebles,P.J.E., 1981, Ap.
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Prim ack,J.R. and Blumenthal,G.R., 1983, in Fourth Workshop on Grand Uni­
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H.A. Weldon, P. Laugacker, and P.J. Steinhardt (Boston:
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Audouze and A. Szalay (Dordrecht: Reidel), p. 37.
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281, 493.
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T able 1
Cosmological Param eters of Models Calculated
ft
h
ft
ft
ftiiac
Adiabatic Models:
1 .0
1.0, 0.5
0 .0
1.0, 0.5, 0.3, 0.1
0.05, 0.03, 0.01
0 .0
0 .2
1.0, 0.5
0 .0
1.0, 0.5, 0.3, 0.1
0.05, 0.03, 0.01
0 .0
0 .2
1.0, 0.5
0 .0
1.0, 0.5, 0.3, 0.1
0.05, 0.03, 0.01
0 .8
1 .0
1.0, 0.5
0.1, 0.3, 0.6
0.8, 0.9
0 .1
0 .0
1 .0
1.0, 0.5
0.1, 0.3, 0.6
0.8, 0.99
0 .0 1
0 .0
Isocurvature Models:
1 .0
1.0, 0.5
0 .0
1.0, 0.5, 0.03
0 .0
0 .2
1.0, 0.5
0 .0
1.0, 0.5, 0.03
0 .0
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
Reproduced
with permission
T a bl e 2A
of the copyright owner.
Further reproduction
prohibited
without permission.
IP
»3
FITTING FUNCTIONS FOR TRANSFER FUNCTIONS: ADIABATIC MODELS®
Tr
N
b
iy mv
^vac
h
ic
h
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0
0
0
0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0
0
0
0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0
0
0
0
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0
0
0
0
Q
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1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.01
0.03
0.05
0.10
0.30
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1.00
1.0
1.0
1.0
1.0
1.0
1.0
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7.279E+01
7.585E+01
7.896E+01
8.717E+01
1.305E+02
2.010E+02
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-4.306E-01
-5.131E-01
-6.020E-01
-8.282E-01
-9.952E-01
1.503E+00
6.505E+00
7.189E+00
7.845E+00
9.184E+00
6.299E+00
-2.407E+01
4.040E+00
3.925E+00
3.912E+00
4.806E+00
2.906E+01
1.231E+02
3.337E+00
3.706E+00
4.063E+00
4.666E+00
1.799E+00
-1.475E+01
3.09
3.09
3.09
3.09
3.09
3.09
0.50
0.50
0.50
0.50
0.50
0.50
0.50
5.519E+02
5.707E+02
5.899E+02
6.415E+02
9.139E+02
1.409E+03
-6.789E-01
-8.402E-01
-9.876E-01
-1.323E+00
-2.311E+00
1.214E-01
2.313E+01
2.480E+01
2.627E+01
2.912E+01
3.111E+01
-2.450E+01
4.153E+01
4.229E+01
4.351 E+01
5.029E+01
1.208E+02
4.359E+02
4.413E+01
4.719E+01
5.045E+01
5.739E+01
8.795E+01
8.759E+01
3.09
3.09
3.09
3.09
3.09
3.09
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
3.960E+02
4.109E+02
4.268E+02
4.691E+02
7.173E+02
1.165E+03
...
-7.277E-01
-9.133E-01
-1.075E+00
-1.473E+00
-2.428E+00
-1.225E+00
•••
2.846E+01
3.032E+01
3.182E+01
3.538E+01
3.500E+01
-5.983E+00
6.037E+01
6.174E+01
6.444E+01
7.237E+01
1.741E+02
4.633E+02
6.709E+01
7.140E+01
7.532E+01
8.608E+01
1.259E+02
1.990E+02
3.09
3.09
3.09
3.09
3.09
3.09
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.50
0.50
0.50
0.50
0.50
0.50
0.50
3.955E+03
4.076E+03
4.189E+03
4.494E+03
6.181E+03
9.361E+03
6.420E-01
1.085E-01
-2.983E-01
-1.062E+00
-2.306E+00
8.744E-01
7.733E+01
7.969E+01
8.293E+01
8.904E+01
6.969E+01
-7.928E+01
6.472E+02
6.812E+02
7.036E+02
7.617E+02
1.294E+03
2.850E+03
8.955E+02
9.083E+02
9.377E+02
1.041E+03
1.497E+03
2.116E+03
3.09
3.09
3.09
3.09
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R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
Reproduced
with permission
T a ble 2 A ( c o n t .)
p
of the copyright owner.
Q,
IP
FITTING FUNCTIONS FOR TRANSFER FUNCTIONS: ADIABATIC MODELS®
su
Tf
ft
Further reproduction
N b
^ vac
h
/c
h
h
h
^5
kfcma
d x
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.126E+02
1.033E+02
1.475E+02
1.111E+02
1.664E+02
1.065E+02
1.789E+02
1.082E+02
1.766E+02
1.070E+02
-4.664E-01
-2.456E-01
2.885E-01
-4.369E-01
-2.935E+00
-2.340E+00
4.700E+00
4.121E+00
-1.862E+00
4.062E+00
2.764E+00
1.607E+00
-5.315E+00
6.065E+00
4.720E+01
3.152E+01
-4.568E+00
4.569E+01
1.869E+02
4.703E+01
3.353E+01
3.008E+01
4.321E+01
2.749E-01
-2.286E+02
-1.039E+02
6.061E+02
-1.678E+01
-5.993E+02
-2.854E+01
-9.280E+00
-6.979E+00
3.136E+01
4.337E+01
4.435E+02
1.631E+02
6.846E+03
6.776E+02
4.399E+04
7.726E+02
0.62
0.62
0.62
0.62
0.62
0.62
0.49
0.49
0.49
0.49
0.50
0.50
0.60
0.50
0.50
0.50
0.50
0.50
0.50
0.50
8.741E+02
8.181E+02
8.969E+02
1.021E+03
1.801E+03
9.827E+02
2.346E+03
1.129E+03
2.332E+03
1.115E+03
-1.866E+00
-1.155E+00
1.348E+00
-4.678E-01
-1.330E+01
-1.245E+01
3.891E+01
1.817E+01
3.729E+00
1.305E+01
2.925E+01
2.010E+01
-2.709E+01
1.695E+01
3.394E+02
2.752E+02
-2.142E+03
3.771E+02
1.371E+03
7.513E+02
1.155E+02
1.165E+02
2.091E+02
1.203E+01
-2.809E+03
-1.780E+03
7.584E+04
3.255E+03
-2.733E+03
-2.795E+03
8.358E+01
9.562E+01
4.067E+02
6.521E+02
8.598E+03
4.234E+03
2.529E+06
1.325E+06
1.134E+08
3.919E+06
0.62
0.62
0.62
0.62
0.62
0.62
0.19
0.19
0.10
0.10
1.0
1.0
1.0
1.0
1.0
1.0
*1.0
*1.0
*1.0
*1.0
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.80
0.80
0.60
0.60
0.30
0.30
0.10
0.10
0.00
0.00
0.10
0.10
0.30
0.30
0.60
0.60
0.80
0.80
0.90
0.90
3
3
1
3
1
3
1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.0
1.0
1.0
1.0
1.0
1.0
*1.0
*1.0
*1.0
*1.0
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.80
0.80
0.60
0.60
0.30
0.30
0.10
0.10
0.00
0.00
0.10
0.10
0.30
0.30
0.60
0.60
0.80
0.80
0.90
0.90
3
1
3
1
3
1
3
1
3
1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
prohibited
a
T he fitting form ula used here is T
l
3
1
'
(
a n asterisk. These fall off steeper th a n 1 /fc2 a t large
f
k,
c
)
=
f
5^^) for all th e m odels except for those m odels which are a nnotated by
so th e fitting form ula for these is T ( f c ) = f x /( l.+ f 2 f c + f 3 f c 2 + t4 fc ^ + fs fc 6 ) . T h e values for the
without permission.
constants above are for k in M p c ~*. T he fitting function is accurate to a few percent in th e worst cases. The original results of th e calculations are available
up o n request. Note th a t the fitting function should n o t b e used for k>kmax.
^ Nmu is the num ber of massive neutrino species. If N mv is greater them one, the neutrino m ass is equally divided betw een th e num ber of species
listed.
^ T h e norm alization constant given here is valid only for scale invariant initial spectra, where the models have been norm alized to A M / M at
8h ~ 1 M p c - 1
(see text).
^ T he value of kmax given here is the m axim um value (in Mpc~ *) used to determ ine the fitting param eters. T he results should probably n o t be
e x trapolated to larger wavenumbers.
* Models w ith an asterisk use a different fitting function than other models. See note (a).
Reproduced
with permission
T a bl e 2B
FITTING FUNCTIONS FOR TRANSFER FUNCTIONS: ISOCURVATURE MODELS®
of the copyright owner.
Further reproduction
TF
N m v
S lv a c
h
th
h
0.97
0.50
0.00
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
1.00
1.00
1.00
1.226E+02
2.709E+02
0.03
0.50
1.00
0.97
0.50
0.00
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
0.50
0.50
0.50
0.2
0.2
0.2
0.03
0.50
0.97
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0.2
0.2
0.2
0.03
0.50
1.00
0.97
0.50
0.00
0.00
0.00
0
0
0
SI
ft
1.0
1.0
1.0
0.03
0.50
1.00
1.0
1.0
1.0
1.00
0.00
® T he fitting form ula used here is
<3
*4
^5
1cc
/vm a x
-2.927E+00
5.259E+00
2.093E+01
-4.642E+01
-1.039E+01
1.401E+02
3.368E+01
9.988E+01
3.09
3.09
6.856E+02
1.058E+03
-1.009E+01
-7.901E+00
1.007E+02
4.936E+01
-1.948E+02
-4.262E+01
4.321E+02
8.587E+02
3.09
3.09
0.00
1.00
1.00
1.00
5.931E+02
1.078E+03
-1.111E+01
-7.845E+00
1.188E+02
4.966E+01
-2.360E+02
-2.170E+01
6.113E+02
1.196E+03
3.09
3.09
0.00
0.00
0.00
0.50
0.50
0.50
7.027E+03
-1.549E+01
3.128E+02
-6.899E+02
7.256E+03
3.09
* **
•••
•••
•••
T(Je)=t\k^ / ( \
. •■
•. .
h
...
■• .
■■•
. ■.
•••
••.
...
•. .
...
•. •
...
for all th e models. No fitting functions are given for th e pure baryon
m odels since they cannot be fit by this function. T he values for the constants above are for
k
in
Mpc~L
^ T he norm alization constant given here is valid only for scale invariant initial spectra, where the models have been norm alized to A M / M at
prohibited
S h ^ M p c - 1 (see
text).
C T he value of kmax given here is the m axim um value (in
extrapolated to larger wavenumbers.
Mp c ~ *) used to determ ine the fitting param eters. T he results should probably not be
to
without permission.
Reproduced
with permission
T a b l e 3A
F IT T IN G F U N C T IO N S F O R R A D IA TIO N C O R R E L A T IO N FU N C TIO N S: A D IA B A TIC M ODELS 0
of the copyright owner.
ft
ft
ft
■
Lymi/
N b
Further reproduction
ftvac
h
\A?(<0
02
ic
l l
0.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.197E-05
1.480E-05
1.801E-05
2.730E-05
6.975E-05
1.282E-04
3.041E-04
4.179E-06
4.364E-06
4.551E-06
5.050E-06
7.615E-06
1.132E-05
2.104E-05
1.064E-09
1.817E-09
4.468E-09
1.064E-08
1.210E-07
5.306E-07
3.078E-06
-1.630E+00
-2.707E+00
2.938E-01
2.231E+00
9.391E+00
1.481E+01
1.495E+01
1.295E+00
7.376E+00
4.603E+00
-2.896E+00
-2.290E+01
-3.880E+01
-4.301E+01
6.839E+00
-1.654E+00
-1.919E+00
1.694E+00
1.587E+01
2.865E+01
3.382E+01
6.367E+00
9.194E+00
1.528E+01
1.473E+01
2.217E+01
2.668E+01
2.613E+01
0.50
0.50
0.50
0.50
0.50
0.50
0.50
2.617E-05
2.937E-05
3.160E-05
3.710E-05
6.847E-05
1.243E-04
3.611E-04
7.924E-06
8.216E-06
8.511E-06
9.304E-06
1.351E-05
2.038E-05
4.469E-05
6.856E-09
1.105E-08
1.311E-08
1.597E-08
1.973E-07
3.159E-06
6.832E-06
1.025E+00
-1.109E+00
-2.316E+00
-3.677E+00
1.767E+01
1.538E+02
3.596E+01
-4.317E+00
1.638E+00
5.667E+00
1.101E+01
-1.764E+01
-2.793E+02
-8.121E+01
8.684E+00
4.357E+00
5.063E-01
-6.010E+00
4.654E+00
1.674E+02
5.599E+01
8.870E+00
1.266E+01
1.381E+01
1.367E+01
4.474E+01
1.828E+02
4.236E+01
*3
h
^5
prohibited
without permission.
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0
0
0
0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0
0
0
0
0
0
0
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
. 0.70
0.50
0.00
0.00
0.00
0
0
0
0
0
0
0
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
9.781E-08
1.923E-07
3.143E-07
3.907E-06
1.092E+01
6.650E-02
7.165E-02
4.120E+00
3.934E+00
5.767E+00
9.842E+01
1.258E+08
3.145E+05
2.554E+04
-1.711E+01
-1.439E+01
-1.589E+01
-1.928E+02
-2.616E+08
-8.497E+05
-7.984E+04
3.171E+01
3.280E+01
3.764E+01
3.054E+02
2.240E+08
7.525E+05
7.535E+04
3.821E+01
6.408E+01
9.280E+01
8.559E+02
6.802E+08
1.048E+06
7.137E+04
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.01
0.03
0.05
0.10
0.30
0.50
1.00
0.99
0.97
0.95
0.90
0.70
0.50
0.00
0.00
0
0
0
0
0.00
0.00
0.50
0.50
0.50
0.50
0.50
0.50
0.50
1.073E-06
1.660E-06
2.292E-06
4.293E-06
2.624E-03
1.845E+00
1.460E+03
3.610E+00
2.597E+00
2.477E+00
3.581E+00
2.623E+03
-8.641E+05
7.075E+08
-1.403E+01
-9.436E+00
-7.863E+00
-8.000E+00
-5.262E+03
1.756E+06
-2.220E+09
2.344E+01
2.005E+01
1.955E+01
2.198E+01
6.974E+03
-1.939E+06
1.641E+09
4.644E+01
6.264E+01
7.735E+01
1.145E+02
3.030E+04
-8.700E+06
-1.689E+09
o.oo
0.00
0.00
0.00
0.00
0.00
0.00
o:oo
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
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0
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission
Reproduced
of the copyright owner.
ft
p
FITTING FUNCTIONS FOR RADIATION CORRELATION FUNCTIONS: ADIABATIC MODELS®
IP
with permission
T a b l e 3A ( c o n t .)
Tf
%
N b
Further reproduction
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.80
0.80
0.60
0.60
0.30
0.30
0.10
0.10
0.00
0.00
0.10
0.10
0.30
0.30
0.60
0.60
0.80
0.80
0.90
0.90
3
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.80
0.80
0.60
0.60
0.30
0.30
0.10
0.10
0.00
0.00
0.10
0.10
0.30
0.30
0.60
0.60
0.80
0.80
0.90
0.90
3
l
3
1
3
1
3
1
1
3
1
3
1
3
1
3
1
3
1
& va c
h
% /c ( o )
a2
ic
h
*2
<3
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
3.447E-05
3.176E-05
4.390E-05
3.435E-05
5.165E-05
3.405E-05
5.340E-05
3.336E-05
5.355E-05
3.300E-05
6.489E-06
5.922E-06
8.411E-06
6.380E-06
9.867E-06
6.286E-06
1.015E-05
6.145E-06
1.016E-05
6.077E-06
1.440E-08
1.275E-08
2.270E-08
1.553E-08
3.092E-08
1.534E-08
3.249E-08
1.438E-08
2.956E-08
1.344E-08
1.476E+00
1.597E+00
2.197E+00
2.120E+00
2.699E+00
2.099E+00
2.705E+00
1.877E+00
1.977E+00
1.482E+00
-1.301E+00
-1.710E+00
-2.899E+00
-2.959E+00
-4.466E+00
-3.081E+00
-4.794E+00
-2.728E+00
-3.634E+00
-1.999E+00
4.169E-01
8.119E-01
1.532E+00
1.780E+00
2.818E+00
1.947E+00
3.172E+00
1.730E+00
2.460E+00
1.219E+00
1.277E+01
1.321E+01
1.221E+01
1.354E+01
1.171E+01
1.355E+01
1.141E+01
1.328E+01
1.040E+01
1.276E+01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.50
0.50
0.60
0.50
0.50
0.50
0.50
0.50
0.50
0.50
5.171E-05
4.762E-05
7.312E-05
5.854E-05
1.178E-04
6.383E-05
1.310E-04
6.364E-05
1.333E-04
6.305E-05
1.276E-05
1.183E-05
1.823E-05
1.463E-05
2.927E-05
1.600E-05
3.272E-05
1.597E-05
3.337E-05
1.585E-05
3.192E-08
2.735E-08
5.977E-08
4.218E-08
1.814E-07
5.050E-08
2.235E-07
4.992E-08
2.321E-07
4.870E-08
-3.478E+00
-3.611E+00
-3.972E+00
-3.607E+00
-3.258E+00
-3.701E+00
-3.330E+00
-3.754E+00
-3.346E+00
-3.734E+00
1.084E+01
1.092E+01
1.166E+01
1.087E+01
9.976E+00
1.113E+01
1.001E+01
1.123E+01
1.001E+01
1.117E+01
-6.151E+00
-5.915E+00
-6.389E+00
-5.669E+00
-4.858E+00
-5.778E+00
-4.721E+00
-5.845E+00
-4.651E+00
-5.804E+00
1.408E+01
1.413E+01
1.324E+01
1.433E+01
1.498E+01
1.444E+01
1.488E+01
1.438E+01
1.490E+01
1.430E+01
h
^5
prohibited without permission.
0 The fitting formula used here is for C ( 0 ) —C ( 0 ) = f i 0 ^ / ( l . + < 2 ^ ''’+ f3 ^"H 4 $ * ‘'*+f50^)- This expression is valid only for angular scales less than
10—15 degrees. The values for the constants above are for 6 in degrees. Also given is \ / C { 0) and the amplitude of the quadrupole moment d 2 -
N m v is the number of massive neutrino species. If N m v is greater than one, the neutrino mass is equally divided between the number of species
listed.
C The normalization constant given here is valid only for scale invariant initial spectra, where the models have been normalized to A M /M at
8 /t_ 1 M p c - 1 (see text).
Reproduced
with permission
T a ble 3B
F IT T IN G FU N C T IO N S F O R R A D IA T IO N C O R R EL A TIO N F U N C T IO N S : ISOCU RV A TU RE MODELS®
of the copyright owner.
Further reproduction
ik
Tf
N mu
ftu a c
h
% /C (0 )
02
thb
0.97
0.50
0.00
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
1.00
1.00
1.00
9.704E-05
1.997E-04
2.890E-04
4.795E-05
8.041E-05
1.035E-04
6.361E-09
6.171 E-07
3.385E-06
0.03
0.50
1.00
0.97
0.50
0.00
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
0.50
0.50
0.50
1.765E-04
2.965E-04
4.281E-04
9.641E-05
1.568E-04
2.185E-04
0.2
0.2
0.2
0.03
0.50
1.00
0.97
0.50
0.00
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
1.00
1.00
1.00
• ••
0.2
0.2
0.2
0.03
0.50
1.00
0.97
0.50
0.00
0.00
0.00
0.00
0
0
0
0.00
0.00
0.00
0.50
0.50
0.50
ft
ft
1.0
1.0
1.0
0.03
0.50
1.00
1.0
1.0
1.0
a The fitting formula used here is for C(Qi)—C[6)=t\6^ /
. . .
h
f4
^5
-1.269E+00
1.758E+01
3.309E+01
2.544E+00
-6.145E+01
-1.172E+02
1.125E+00
7.402E+01
1.376E+02
3.769E-01
1.009E+00
1.417E+01
1.478E-08
7.566E-08
3.328E-07
-1.391E+00
-1.616E+00
-3.986^01
3.523E+00
9.168E-01
-4.193E+00
-2.003E-01
3.743E+00
1.071E+01
4.616E-01
-9.001E-04
-4.577E-01
2.281E-07
9.996E-07
4.500E-06
-2.482E+00
-4.218E+00
-4.654E+00
6.066E+00
8.552E+00
8.576E+00
-3.789E-01
-9.275E-01
8.975E-01
1.503E+00
1.848E+00
2.219E+00
9.219E-07
2.371E-06
4.508E-06
-2.530E+00
-3.234E+00
-3.966E+00
6.979E+00
9.503E+00
1.109E+01
-2.835E+00
-4.046E+00
-5.005E+00
1.448E+00
1.954E+00
2.168E+00
Thi s expression is valid only for angular scales less than
10—15 degrees. The values for the constants above are for 0 in degrees. Also given is y/C ( 0) and the amplitude of the quadrupole moment Cl2-
prohibited
b The normalization constant given here is valid only for scale invariant initial spectra, where the models have been normalized to A M /M at
8 h - 1 M p c - 1 (see text).
oj
without permission.
77
Figure 1.
Growth Laws From Recombination to the Present
/
2.5
0 = 1.0
log
8
0 = 0 .2 O vac=0.8
-
2
-
1
0
1
lo g s c a l e fa c to r (a )
Figure 1. Evolution of perturbations in C DM from recombination to the present
for several cosmological models. All models are normalized to have the same
amplitude fluctuations around recombination.
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78
Figure 2a. Growth of Adiabatic Perturbations Through Recombination
8
log 5
log 8
6
4
2
8
k=3x10'' Mpc
log 8
log 8
6
4
2
-6
-4
log a
-2
-6
-4
log a
-2
Figure 2a. Evolution of adiabatic C D M and baryons from early times through
recombinatiion for
«
£ 2 ^ and £2V«
£ 2 ^ . Solid
lines represent the baryon perturbations, dashed lines the C D M perturbations
and dotted lines the massive neutrino perturbations. Perturbations on four
size scales are shown; each scale is normalized to have the sam e initial
amplitude.
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79
Figure 2b.
i
r
|
i
i
i
1 |—
—
i— i—
i— ri—
i—
|—
i—
r
Growth of Isocurvature CDM and Baryon Perturbations
Through Recombination
, k=3 Mpc
log |S|
///; k = 3 x 1 0 '1 Mpc'1
k=3x10'2 M pc1
0
—
k=3x10'3 Mpc'1
-1
—
-2
J
-6
-4
i
L
-2
log a
Figure 2b. Sam e as Figure 2 a but for isocurvature initial conditions.
No massive neutrinos are included here.
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0
80
Figure 3a. P(k), sc a le invariant adiabatic m odels
1£2=0.2:1£2,/£2=0.01
I T
I£2=1.0:I£2j/£2=0.01_^ss
f I I
/ /
/ /
/
/ /
/ /
/ ✓
/ /
/ /
—/ /
/ /
/ /
” /
■'
II
II
II
“
/
/
—
""
---- h=0.5
- - h=1.0
II
/
/ /
// / ✓
/ /
/ /
- - h=1.0
I1
1I
log k3^ !
1 I 1
0.5
I
1
£2,/£2=1 .0
1I
1
I
1
I
I
I
- - h=1.0
II
I
- 2 - 1 0
log k (Mpc1)
1
-4
-3
I
n„/£2
I.
—
-2
—
-4
I
i | , | , | ,
-3
II
1
-4
I
---- h=0.5
- i S
I
-4
—
/V
"
—
“
/ /
/ /
/ /
/ ✓
/ ✓
/ /
/ /
S '
/
---- h=0.5
I
1
-2
---- h=0.5
—
/ /
1
1
—
0
/ ✓
/ /
- / /
/ /
—/
/
—
- - h=1.0
£2,/£2=0.5
-
0
X /
/
/
/ /
/
/
II
1
-2 -1
0
log k (Mpc'1)
Figure 3a. Matter fluctuation spectra P(k)=k3C8(k) at present for models
with scale invariant adiabatic initial conditions and no massive neutrinos.
Left panels have £2=1, right panels £2=0.2. Top plots are C DM dominated,
middle plots have £2CDM=£2b, lower plots are purely baryonic models.
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0
-2
-4
81
Figure 3b. P(k), sca le invariant adiabatic m odels, m assive v ’s
log k3/2|8jj
I
I
I
£2=1.0 £2(=0.01:
I
I
£2V=0.3
I
I
I
n=i.oi\=o.oi:
I
T
^ = 0 .3
-
“
/ /
/ /
/
/
/ /
/ / ✓/
/ /
/ /
/ /
/ /
”_
- / f
— h=0.5 mvc2=2.4eV
/ f
--- h=0.5 mvc2=7.3eV- / /
- - h=1.0 m c2=9.7eV- - / /
- - h=1.0 mvc2=29eV'/ / /
/ /
/
/
- 'I I
II
'
II I I
II
II
II
II
II I I
£2V=0.6 £2,=0.6
__
_—
_
/ /
// //
/ /
/ /
/ ✓
/ /
_
/ /
- / /
— h=0.5 mvc2=4.9eV
— h=0.5 mv
// / ' '
vc2=14eV-V / '
- - h=1.0 mvc2=20eV- - / / /
h=1.0 mvcz=59eV' /
/
/
“✓
/
'
II -----1I
I
I
I
•'
II
II
I
I
I
II
II
II
£2V=0.99 £2V=0.99
/
X
- / /
/- / / /
Y
-4
i
-3
/
7
'
/ \ \
\ \
' ‘
— h=0.5 mvca=8.0eV
// /'
- - h=1.0 mvc2=32eV- ~
i . i
i
- 2 - 1 0
log k (Mpc'1)
i
1
/ l
-4 -3
— h=0.5 m„c2
u =24eV- ” h=1.0mvc2=96eVI
I
I
- 2 - 1 0
log k (M pc1)
I
1
Figure 3b. S am e as Figure 3 a but for models with massive neutrinos. Top plots
have £2CDM>£2V, middle plots have £2V>Q CDM, and
bottom plots have only massive neutrinos. Left panels have neutrino mass
equally divided between all three neutrino species; right panels have
all the mass in one species.
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82
Figure 3c. P(k), sca le invariant isocurvature m odels
0
0
■2
-2
4
h=0.5
h=0.5
h=1.0
h=1.0
-4
0
0
2
-2
C\J
O)
O
-4
h=0.5
h=0.5
h=1.0
h=1.0
-4
0
0
2
-2
4
-4
-3
h=0.5
h=0.5
h=1.0
h=1.0
- 2 - 1 0
log k (Mpc"1)
1
-4
-3
- 2 - 1 0
log k (Mpc"1)
1
Figure 3c. Sam e as Figure 3a for scale invariant isocurvature initial conditions.
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-4
83
Figure 4. £(R), sca le invariant adiabatic m odels
1
1
.8
.8
.6
h=0.5
.4
--h = 1 .0
h=0.5
.6
- -h=1.0 —
.4
.2
.2
0
0
1
1
.8
DC
1 XJ'
.8
.6
h=0.5
.4
--h = 1 .0
.6
h=0.5
-
-h=1.0 —
.4
.2
.2
0
0
1
1
.8
.8
.6
h=0.5
.4
--h = 1 .0
h=0.5
.6
- -h=1.0 —
.4
.2
.2
0
0
0
50
100
150
R(Mpc)
200
50
100
150
R(Mpc)
200
Figure 4. Two point mass correlation functions for the same models shown in
Figure 3a.
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84
Figure 5. (AM/M)^, sc a le invariant adiabatic m odels
I
I
I
0
N. N
\, X l.
>*
N.
-1
T
I
I
0
----- h=0.5
- ------- h=1.0
' ' X
I
I
-1
'x .
I
1
I
I
I
I
—
I
I
£2b/£2=0.5
log (AM/M),
I
“
'^ X .
I
I
I
>, ■sx
V. ^
N X
s X^
NS \ X.
------- h=1.0
I
I
I
£2=0.2: £2b/£2=0.01
**
------- h=0.5
-2
I
£2=1.0: £2„/£2=0.01
-
I
I
I
I
-2
I
I
£2b/£2=0.5
0
0
N‘ \ ^N.X\
-1
-1
\ ^
------- h=0.5
''J X
------h=0.5
s
------- h=1.0
I
I
------- h=1.0
I
I
I
I
I
I
I
I
I
I
—
I
I
£2b/£2=1.0
I
I
I
I
I
I
I
I
£2b/£2=1.0
0
-1
------- h=0.5
-2
X
------h=0.5
------- h=1.0
I
15
------- h=1.0
I
I
I
16 17 18
log M ass (MJ
I
I
19
15
—
I
I
16 17 18
log Mass (MJ
19
Figure 5. RMS overdensities in spheres of various sizes for the sam e models
as in Figure 3a. The mass in the perturbation is related to the size of
the perturbation by M(R)=4/37tRA3p0.
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-2
85
Figure 6a. vbulk, scale invariant adiabatic models with C D M + baryons, bias=1
-
Bulk Velocity a t 2 0 0 0 km /sec
Bulk V elocity a t 3 0 0 0 k m /sec
1500
1500
1000
1000
500
b u lk
500
>
-
Bulk V elocity a t 5 0 0 0 km /sec
1500
#£ 2 = 1 .0
h = 0 .5
Qw=0.0
n=1
x £2=1.0
h = 1 .0
£2„=0.0
n=1
O £2=0.2
h = 0 .5
£2„=0.0
n=1
□ £2=0.2
h = 1 .0
£2U= 0 .0
n=1
# £ 2 = 0 .2
h = 0 .5
£2„=0.8
n=1
X £2=0.2
h = 1 .0
£2U= 0 .8
n=1
1000
500
0
.2
.4
.6
£2b/£2
.8
1
0
.2
.4
.6
£2b/£2
.8
1
Figure 6a. Predicted RM S bulk velocities in spheres of radii 2000, 3000, and
5000 km/sec for the scale invariant adiabatic initial conditions for models
without massive neutrinos. Skeletal points represent flat £2=1 models,
open points open £2=0.2 models, and stellated points the flat £2=0.2,
£2vac=0.8 models. Vertical bars located at various points represent
the range of velocities within the 9 0 % confidence level for some typical
R M S velocities. The horizontal line represents observed values (see text).
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Figure 6b. vbulk, scale invariant adiabatic models with massive vs + CDM + baryons, bias=1
-
Bulk V elocity a t 2 0 0 0 km /sec
Bulk V elocity a t 3 0 0 0 km /sec
1500
1500
1000
1000
500
b u lk
500
>
-
Bulk V elocity a t 5 0 0 0 km /sec
1500
#£2=1.0
h=0.5
£2^0.10
n=1
x £2=1.0
h=1.0
£2b=0.10
n=1
O £2=1.0
h=0.5
£2b=0.01
n=1
□ £2=1.0
h=1.0
£2b=0.01
n=1
1000
—
3 massive v ’s, 0 massless v ’s
500
- - - 1 massive v, 2 massless v ’s
0
.2
.4
.6
.8
1
0
.2
Q.JQ.
.4
.6
.8
1
n jn
Figure 6b. Sam e as Figure 6a, but for models with massive neutrinos. Here,
skeletal points have £2b= 0 .1 , while open points have £2b= 0 .0 1 . All
models have £2,M= 1 . Points connected by a solid line have the neutrino
mass evenly divided between the three neutrino species; those connected by
a dotted line have all the mass in one species.
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87
Figure 6c. vbulk, scale invariant isocurvature models with C D M + baryons, bias=1
1
1
1
1
I
1
Bulk V elocity a t 2 0 0 0 km /sec
I
I
I
I
Bulk V elocity a t 3 0 0 0 km /sec
1500
1500
1000
1000
500
500
n = -3
o
n = -3
# £ 2 = 1 .0
h = 0 .5
X £2=1.0
h = 1 .0
O £2=0.2
h = 0 .5
£2V= 0 .0
n = -3
□ £ 2 = 0 .2
h = 1 .0
n = -3
# Q = 0 .2
h = 0 .5
Q v= 0 .8
n =-3
X £2=0.2
h = 1 .0
<
II
O
>
n
1000
L
o
o
1500
I
a
Bulk Velocity a t 5 0 0 0 km /sec
I
<O
II
o
b
J
-
<
II
o
bo
500
n = -3
J____ L
.6
£2b/£2
.8
1
.2
.4
.6
.8
£2b/£2
Figure 6c. Sam e as Figure 6a, but for scale invariant isocurvature initial
conditions.
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Figure 7a. AT/T(0), sc a le invariant adiabatic m odels
n=0 .2 :£y£2=0.01
-3
h=0.5
-4
— h=1.0 —
h=0.5—
-5
- - h = 1 .0 -
log (AT/T)
-3
-4
h=0.5 —
h=0.5—
- - h = 1 .0
-5
— h=1.0 -
-3
-4
-5 —
h=0.5 —
h=0.5—
- - h=1.0 -
— h=1.0
1
2
log 0 (arcmin)
1
2
log 0 (arcmin)
Figure 7a. Predicted AT/T for an ideal two beam experim ent a s a function of
angle for the sa m e adiabatic m odels show n in Figure 3a.
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-5
89
Figure 7b. AT/T(0), sca le invariant isocurvature m odels
I
—
1
I
1
n = o. 2 : ___________________ —
fi= 1.0:0,/0=0.01
-4
^ / ✓^
—
I
I
_
------ h=0.5 - ------h=0.5—
----- h=1.0
I
I
-3
I
I
-5
----- h=1.0 I
I
_________
-3
£ y « = 0 .5
log (AT/T)
—
-4
------h=0.5—
I
I
—
----- h=1.0
I
I
I
I
-5
----- h=1.0 I
I
_________ —
-3
fl,/O=1.0
-4
—
-
----- h=0.5 !
r
1
2
log 0 (arcmin)
-w
.
------h=0.5—
I
- - - h = 1 .0 -
1
2
log 0 (arcmin)
Figure 7b. Predicted AT/T for an ideal two beam experim ent a s a function of
an gle for the sam e isocurvature m odels show n in Figure 3c.
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-5
90
Figure 8a. (AT/T), scale invariant adiabatic models with CDM + baryons, bias=1
I
I
log(AT/T) a t T.
_ lo g (A T/T ) a t 415
log(A T/T) a t 6
log A T /T
log C (0 )
-
# fi= 1 .0
h = 0 .5
a .= 0 .0
n=1
x Q = 1 .0
h = 1 .0
Q .,= 0 .0
n=1
O Q = 0 .2
h = 0 .5
Q v= 0 .0
n=1
□ Q = 0 .2
h = 1 .0
Q v= 0 .0
n=1
# Q = 0 .2
h = 0 .5
Q v= 0 .8
n=1
X Q = 0 .2
h = 1 .0
Q v= 0 .8
n=1
J
Qb/f>
.8
1
I
I
I
I
.6
Qb/Q
.8
L
Figure 8a. Predicted A T/T for several experimental setups (see text) and
predicted quadrupoie moment for the scale invariant adiabatic models without
massive neutrinos. Symbols are the same as in Figure 6a. Horizontal lines
are the observed upper limits for the various experiments.
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Figure 8b. (AT/T), scale invariant adiabatic models with massive vs + CDM + baryons, bias=1
I
-3 .5
I
I
I
I
I
_ log(A T/T) a t 4 ’. 5
1
1
1
1
I
1
_ lo g (A T/T) a t 7115
_
-4
-4
-4 .5
-4 .5
-5
jjjj
-5
£ 0
-----tJ............. t i — --Q------ □
-5 .5
log A T /T
-3
-5 .5
i
i
i
_ log(A T/T) a t 6 ‘
i
i
i
I
I
I
I
I
I
_ log C (0 )1/z
I
I
!
T
I
I
_
-3 .5 —
—
-4 —
-3
-3 .5
-4
-4 .5
_ e - -----------5 -------0 —
Q_
-5
-4 .5
-5
i
-3 .5
-3 .5
_ log a ,
i
i
i
i
i
#£2=1.0
h=0.5
£2b=0.10
n=1
x £2=1.0
h=1.0
£2b= 0 .1 0
n=1
O £2=1.0
h=0.5
£2b=0.01
n=1
0 £2=1.0
h=1.0
£2b=0.01
n=1
-4
-4 .5
-5
—
3 massive v ’s, 0 massless v ’s
-5 .5
—
1 massive v, 2 massless v ’s
1
1
1
0
.2
.4
Q ja
.
1.
.6
1 ......... 1
.8
1
1
0
I
I
I
I
I
.2
.4
.6
.8
1
Q.JQ.
Figure 8b. Sam e as Figure 8a, but for models with massive neutrinos.
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Figure 8c. (AT/T), scale invariant isocurvature models with C D M + baryons, bias=1
_3 5
__
_ lo g (A T/T) a t 4 ’. 5
log(A T/T) a t 7115
-3 .5
-4 .5
-4 .5
-5 .5
-5 .5
_ !og(A T/T) a t 6 ‘
-3 .5
-3 .5
-4 .5
-4 .5
*1
O
o)
-4 .5
-5 .5
0
.2
.4
.6
£2b/£2
.8
1
# £ 2 = 1 .0
h = 0 .5
Qw=0.0
n = -3
x £ 2 = 1 .0
h = 1 .0
£2..=0.0
n = -3
O £2=0.2
h = 0 .5
£2U= 0 .0
n = -3
□ £2=0.2
h = 1 .0
£2w=0.0
n = -3
# £ 2 = 0 .2
h = 0 .5
£2 = 0 .8
n = -3
x £2=0.2
h = 1 .0
£2U= 0 .8
n = -3
0
.2
.4
.6
£2b/£2
.8
1
Figure 8c. Sam e as Figure 8a, but for scale invariant isocurvature initial
conditions.
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93
Figure 9.
I
1
1
1
I
Relative log AT/T (1 deg)
Intrinsic differences at 1 degree
£2=1.0 h=0.5
X
I
O £2=0.2 h=0.5
□
-8
£2=1.0 h=1.0
—
£2=0.2 h=1.0
£2=0.2 h=0.5 £2V=0.8
X £2=0.2 h=1.0 £2V=0.8
_
I I I
i i i
.2
.4
I
.6
i i i
J
I
L
.8
£2 ^ 0
Figure 9. Predicted difference in AT/T at one degree for different adiabatic
models if all models are normalized to have the sam e initial amplitude.
T h e ordinate scale is arbitrary here.
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P a rt B . S tellar P h o to m e tr y T echniques W ith T he W id e F ie ld /
P la n e ta ry C am era o f th e H ub ble Space T elescop e
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95
I. In tro d u ctio n
The Hubble Space Telescope (HST) will provide tremendous new oppor­
tunities to study astronomy because of its capability to obtain observations at very
high spatial resolution and of very faint objects. The HST will outperform ground
based telescopes because the atmosphere of the earth acts to blur the light from
astronomical objects, which sets the limiting factor on angular resolution obtain­
able from the ground, at least for conventional telescopes, and because the level
of the background signal is greatly reduced for HST w ith respect to ground based
telescopes. These capabilites will allow a wide variety of scientific problems to be
studied.
Some substantial subset of these problems will require the measurement
of the brightnesses of individual stars from HST observations. Several examples
are the measurement of the periods of Cepheid variables in nearby galaxies, which
is expected to reduce the current uncertainty in the distance scale of galaxies and
which may be considered one of the prime goals of the entire HST project, as well as
the study of stellar populations in both our own and nearby galaxies, which should
provide information about the ages and compositions of these stellar systems. Many
of these stellar photom etry observations will be made with the prim ary camera of
the HST, the W ide Field / Planetary Cam era (W F /P C ).
The measurement of stellar brightnesses from images taken w ith the
W F /P C , however, will not be a simple m atter. The optical design of the instrum ent
introduces optical distortions across the field of view, the nature of the detector
(CCD’s) introduce sampling effects, and to exploit the unique capabilities of HST,
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96
the scientifically interesting observations will be of faint objects in very crowded
fields.
T he purpose of this paper is to determine the best techniques to use for
stellar photom etry with the W F /P C , and to assess the accuracy and limitations of
these techniques. We are concerned here with stellar photom etry reduction after
basic image reduction (dark subtraction, flat fielding, etc.) has been performed; an
analysis of the basic reduction procedure has been presented by Lauer (1989).
The organization of the paper is as follows. First I describe the W F /P C
and discuss the expectations of the appearance and characteristics of stellar im­
ages. Next I discuss the basic techniques of stellar photom etry (which have been
developed for ground based data) and the differences of W F /P C d ata from ground
based data, and briefly describe the philosophy of this study. Next, I present the
simulations to be used, and I discuss ways to optimize stellar photom etry techniques
for the W F /P C . Finally, I focus on each of what I consider to be the three m ajor
complications of W F /P C data, namely the undersampling of the stellar images by
the detector, the variations of the point spread function (PSF) of stellar images as
a function of color and of position on the detector, and the extreme crowding of
stellar images expected on W F /P C pictures.
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II. T h e W id e F ield / P la n e ta ry C am era (W F P C )
The W ide Field / Planetary Cam era is to be located at the Cassegrain
focus of HST, and can be operated in either of two modes, th e W ide Field mode
(f/13), which gives a 160 arcsec field of view, and the Planetary Cam era mode
(f/30), with a 70 arcsec field. In both modes, the field is split into four sections,
each of which is imaged through separate optics onto an 800 x 800 CCD. These
chips have 15// pixels, corresponding to 0.10 arcsec/pixel for the W ide Field Cam era
(W FC) and 0.043 arcsec/pixel for the Planetary Camera (PC ). More details about
the instrum ent design can be found in the instrum ent handbook (Griffiths 1985).
Since there is essentially no atmospheric seeing from the orbit of HST, the
PSFs for the W F /P C are expected to be determined by a combination of diffraction
and th e aberrations introduced by the cam era and the imperfections of the optics.
Since the W F /P C is located on the optical axis of HST, the telescope images are
almost entirely set by diffraction, at least at optical wavelengths. The diffraction
p a tte rn for a circular aperture is the Airy p a tte rn of a bright central disk surrounded
by concentric dark and bright rings. The radius of the first dark ring is given
by r = 1.22AF , where F is the focal ratio.
Consequently, Airy disk diameters
range from 12.7// at 4000A to 25// at 8000A for the WFC; these diameters are 2.3
times larger for the PC. Thus it is clear th a t diffraction limited images are severely
undersampled by the 15// CCD pixels, particularly in the W FC.
The optics in the W F /P C camera introduce some low order aberrations to
the telescope images, predominantly astigm atism , which has the effect of elongating
stellar images as the distance from the optical axis of the camera increases. Because
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98
of the larger field size, the effect is larger in the W FC th an in the PC . Since the
beam is split into four sections, the cam era optical axes are located at the centers
of the four CCDs.
Detailed software to compute expected point spread functions has been
developed by Chris Burrows and Hashima Hasan of the Space Telescope Science
Institute (STScI). This software uses ray trace d ata of the optical system to predict
the aberrations, and also includes realistic modelling of m irror defects, the presence
of dust on the mirror, and telescope jitter. Some images of com puted PSFs are
presented in Figures 1A to ID. These are magnified views, with the location of
the PSF on the page representative of the the location on a CCD which has the
optical telescope axis off to the lower left. Figure 1 A show PSFs for th e WFC for
a broad V filter (F555W), computed a t the center and four comers of a CCD. In
this and subsequent images, the grey scale is logarithmic, w ith each star normalized
to have 1 x 104 counts. The tick m arks show the sizes of 15/i pixels and the color
b ar shows the greyscale intensities. T he optical distortions across the field and the
undersampling of the images by the CCD pixels are very evident. Figure IB shows
the same d a ta for the wide I filter (F785LP), and Figures 1C and ID repeat the
d a ta for the PC. In the PC , both the optical distortions and the undersampling
problem are less severe th a n for the W FC. Note th a t these images are computed
on a 1.5(i grid; when HST actually flies, we will never directly see the detail on this
scale, but only as it is integrated over the 15\i pixels.
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III. S tellar P h o to m e tr y T echniques
In principle, measuring the brightnesses of stars from an image taken with
a linear digital detector is trivial. One merely needs to add up the counts in all
pixels in which light from the star falls, estim ate th e background contribution in
these pixels from nearby pixels which do hot contain any stars, and subtract the
background from the total counts to get the stellar brightness. This is customarily
known as aperture photometry.
This technique fails, however, for stars which have neighbors th a t con­
tribute light into the aperture of the star for which the brightness is desired. The
strategy usually adopted here is to determine the shape of the PSF from some
other stars with no (or few) neighbors, and then, w ith the assum ption th a t the
stars of interest have the same PSF, perform a least squares fit of the overlapping
stars simultaneously to get their brightnesses relative to the stars th a t were used
to determine the PSF. In addition, one may choose to include the positions of the
stars an d /o r the sky background as free param eters in the fit. The whole set of
relative brightnesses can be transformed to absolute brightnesses by doing aperture
photom etry on the PSF stars, or by using standard stars.
PSF photom etry is also preferable to aperture photom etry for faint stars,
even when they are isolated, because b etter signal-to-noise can be obtained (King
1983).
Determining the PSF from observations of a few stars, however, is not nec­
essarily easy. Since the values of the PSF integrated over pixels vary for different
stars of the same total brightness because of different pixel centerings, some inter­
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polation scheme must be used to predict the PSF for an arbitrary pixel centering
from d ata from a few stars w ith given pixel centerings. It is possible to directly
interpolate an observed PSF if the PSF is well sampled, b u t very difficult in the
case of undersampled d ata without some a priori idea of the exact form of the true
underlying PSF. This will be discussed extensively in section VI.
In very crowded fields, no star finding routine is able to locate all of the
stars, as some are buried under the wings of brighter stars. In such cases, the
reduction is usually repeated several times; the first pass fits the most easily found
stars, these fits are used to subtract these stars from the frame and then the star
finder can find some previously hidden stars. These are included w ith the originally
found stars, and the entire frame is rereduced w ith the augmented list.
Although the technique of PS F photom etry is reasonably straightforward,
there are many styles w ith which it can be performed. These differ with respect
to the interpolation technique for the PSF, the number of pixels included in the
fit, weighting schemes, star detection and rejection techniques, least squares fitting
techniques, etc. One goal of this study is to determine the optimum param eters
and techniques to use for W F /P C data.
Several software packages have been developed for doing this type of reduc­
tion, including DAOPHOT (Stetson 1987), ROM APHOT (Buonanno et al. 1979),
HAOPHOT (Gilliland and Brown 1988), W OLF (Lupton and Gunn 1986), and
others. The approach adopted here has been to take DAOPHOT, one of the most
widespread of the photom etry packages, as a starting point and to make changes
as desired. For this study, DAOPHOT was incorporated into a general image pro­
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101
cessing system (VISTA), which greatly facilitated the process of doing stellar pho­
tom etry reductions by allowing easy interaction with a display and the capability
to write command procedures for standard DAOPHOT tasks. As will be discussed
below, some p arts of DAOPHOT have been modified, particularly the PSF and
star fitting routines. The star finding and aperture photom etry routines have been
relatively untouched, although they have been modified to ru n under environments
other than VMS. An effort was m ade to make most of the modifications through
the addition of new param eters (DAOPHOT OPTIONS), which can be changed at
run tim e and which default to values which reproduce the standard DAOPHOT
techniques.
Approach of this study
The basic approach of this study is to use simulated images in a series of
controlled experiments to determine optim um param eters and techniques to use for
W F /P C stellar photometry. I start w ith very simple images and progress towards
more complicated and more realistic images. After optimizing the techniques, the
most realistic frames are used to attem p t to estim ate the expected errors for W F /P C
photometry.
I have chosen to use the simulated PSFs from the STScI as the input
d a ta for the simulations. One must continually keep in mind, however, th at we
want to determine the best techniques to use for actual HST data, not simulated
data. In the best of all worlds, the true d ata will closely resemble the simulations;
more realistically, the simulations will probably be correct qualitatively, but not
quantitatively. Consquently, we m ust consider not only the best techniques and
param eters to use, but also how robust these choices are to small variations in the
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PSF. The experiments ought to provide guidelines for how to determine the best
techniques once we actually see some W F /P C data, as well as to determine the best
techniques for our expectations of w hat W F /P C d ata will look like. But we also
need to consider w hat we will have to do in the real world.
To consider such issues, I include a section subtitled “The Real World” at
the end of some of the following discussions of W F /P C complications. These sec­
tions discuss w hat d ata will be available after launch and a procedure for analyzing
it to determine optimum methods to use for d ata reduction.
For the purposes of this study, it was impossible to work w ith all the model
PSFs which are available, for all the different field locations and for all possible
W F /P C filters. It is im portant to work with some variety of PSFs, however, to
determine the sensitivity to minor changes in the PSF. Consequently, I have chosen
to use four different PSFs, for two different filters, F555W and F785LP, and at two
different field locations, the center and the far corner of the CCD (although some
interpolation between the PSFs is used on the section of PSF variability across
the frame). All of these are for the W FC. Some subsequent figures and tables are
labelled Figure N-I, where N is the figure number, and I runs from 1 to 4 and refers
to the PS F used in the construction of the figure; the first two in the set refer to
the F555W PSF at the center and far comer of the CCD, and the last two refer to
the F785LP PSFs.
I have chosen not to do any experiments w ith the PC for the time being,
as PC images will be significantly easier to deal w ith for the undersampling and
variable PSF problems th a n the WFC images, and the crowding problem should
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be similar for the two cameras. F555W and F785LP were chosen because these
are the wide V and wide I filters th a t will be used for a large fraction of W F /P C
stellar observations because of their high throughput. I have not considered a PSF
at shorter wavelengths yet since it is even more undersampled than the V image,
which seems difficult enough.
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IV . Sim ulated D a ta
In this section, I summarize the characteristics of the simulated images
which will be used throughout the rest of the paper. These fall into two m ajor
categories: images with isolated stars only (usually placed on a grid), and images
w ith stars placed at random locations at various degrees of crowding.
There are two sets of isolated star images, hereafter denoted as GRID and
FIELD images. The GRID images consist of 400 stars of equal brightness, each
placed at a different location within a pixel. The stars are ordered such th a t star
ista r has pixel centering (.row, .col), where row = m o d (is ta r /10) * 0.05 and col =
(in t(is ta r /10) + 1) * 0.05; star 1 has pixel centering (.05,.05), star 2 (.10,.05), etc.
The coordinate system has (.00,.00) corresponding to a pixel center and (.50,.50)
to a pixel comer. The same PSF was used for all 400 stars, the only difference
being the pixel centering. GRID images were constructed at 5 different signal-tonoise levels, corresponding to 10000, 1000, 200, 100, and 50 counts per star. In
all images, a background of 100 counts was added (a bright background for HST).
A gain of 7.5 photons/count was used w ith a readout noise of 13 electrons (Lauer
1989). Consequently, these images have approxim ate signal-to-noise ratios of 270,
73, 22, 12, and 6 , and will be referred to as GRID-SN270, GRID-SN73, etc. Since
the GRID images just use a single PSF, there is a different set of GRID images for
each PSF (e.g, one set for F555W in the center of the CCD, another for F555W
in the far comer, etc.). Some of the stars in the SN270 frames would be starting
to saturate the 12 bit A /D converter, depending on the pixel centering, but these
roughly correspond to the brightest stars observable w ith the W F /P C detectors..
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105
The FIELD images were created to test the effects of PSF variations across
the frame. In these images, a 19 x 19 grid of stars was placed on an 800 x 800 frame.
Here, different PSFs were used for each star, where th e PSFs were determ ined by
interpolating between the PSFs calculated by the STScI software (for the center and
four corners of the frame). The stars are separated by 40 pixels and also sample
a variety of pixel centerings on a regular grid. The interpolation scheme used to
create th e variable PSFs was:
p s f ( x ,y ) = b + | ( c - a) + x 2
^
- e) + V2 ( ~ y ~ ~
where th e (x,y) coordinate system is oriented at a 45 degree angle to the pixel
coordinate system, w ith the origin at the center of the CCD. Consequently, the five
known PSFs are at a : (—1 ,0), b : (0,0), c : (1,0), d : (0,1), e : (0, —1); b is the PSF
a t the center and the others are the PSFs at the four corners. This interpolation
scheme was designed to be accurate in the presence of low order aberrations. The
FIELD images were created at the same signal-to-noise ratios as the GRID images.
In addition, to the images of isolated stars, two sets of images w ith random
star placements were created. The first set consists of 50 x 50 images where the same
P S F was used across the image. These were created at various degrees of crowding
and will be referred to as the CROWD images. In all cases, th e distribution of stellar
brightnesses was taken to be roughly th a t of a globular cluster luminosity function
in V, w ith the exception th at the stars were created in discrete m agnitude bins,
rath er th a n continuously distributed in brightness. This enables us to do statistics
in each bin to get an idea of the errors as a function of m agnitude without the
complications of stars leaking from one bin into another. The fields were designed
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106
to simulate observations at different locations in M31, using a distance modulus
of 24 magnitudes; the number of counts used for the stars were calculated from
the W F /P C sensitivities given in the July 1988 STScI Newsletter for a half-orbit
exposure (2500s). A background of 33 counts, corresponding to 23 magnitudes per
square arcsec, was added to the images. These images were created at four degrees
of crowding, corresponding to equivalent surface brightnesses of 25, 24, 23, and
22 magnitudes per square arcsec; these are referred to as CROWD25, CROWD24,
etc. An equivalent surface brightness of 25 m ag/square arcsec corresponds to a
reasonably sparse field; a field w ith
22
m ag/square arcsec is extremely crowded,
probably more so than any sane astronomer would care to attem pt. As w ith the
GRID images, a different set of CROWD images exists for each different PSF.
Finally, a set of images designed to be more realistic were created. These
images, referred to as the REAL images, are 800 x 800 images created w ith the
variable PSF used for the FIELD images. They have random star placements and
stellar brightness distributions like the CROWD images.
These have also been
created at various degrees of crowding, and are denoted by REAL25, REAL24, etc.
Since we wish to do controlled experiments to test stellar photom etry tech­
niques, no image defects (flat fielding errors, cosmic rays, bad pixels, saturated stars,
etc.) were added to any of these images. The presence of these will undoubtedly
complicate the reduction process and degrade the quality of the results somewhat,
but th e basic techniques are best developed w ithout worrying about them.
Samples of some of these images for F555W are shown in Figures 2 and 3.
Figure 2 shows a set of CROWD images for the center of the CCD, and Figure 3
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107
shows the REAL23 image. The image characteristics axe summarized in Table
1
,
which also gives the text sections in which they are referred to and th e figures in
which the d ata from each image is shown.
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V . O p tim ization o f T echniques for W F C D a ta
Before proceeding to test for the effects of undersampling, a variable PSF,
and crowding, stellar photom etry techniques need to be optimized for W FC data.
Primarily, we need to determine the correct DAOPHOT param eters to use during
the star detection, aperture photometry, and PSF fitting procedures. In addition
to this, we also discuss ways to speed up the process of PSF photometry. This is
essential for photom etry in very crowded fields.
a. Parameters for star detection
The DAOPHOT FIND routine locates stars by going through the image
pixel by pixel, fitting a gaussian to each location, and determining from the quality
of the fit whether there is likely to be a star neaxby. The candidates located in
this m anner are then filtered for bad pixels, cosmic rays, and galaxies by computing
two statistics, called SHARP and ROUND, for each detection (see Stetson 1987 for
more details). For this procedure, the user needs to specify several param eters: an
approximate FWHM to use for the gaussian, and upper and lower limits for b o th
SHARP and ROUND to be used in the filtering procedure.
To determine the optimum FWHM, we used the GRID-SN 6 field of very
faint stars in the center of the CCD for F555W. In this field, stars have 50 counts
each, w ith a background of 100 counts. The stars centered in a pixel have about
25 counts over background, corresponding to nearly a 7 sigma sky peak; the stars
located at a pixel corner have about
10
counts over background, corresponding to a
2.7 sigma sky peak. Consequently, we expect these stars to be at about the limit of
detectability, since we use a threshold of 3.5 sigma (this value detects some, but not
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109
too many, noise peaks, most of which are filtered out during the fitting procedure
- if anything, a slightly larger value should be used). We attem pted to find stars
w ith five choices of the FW HM param eter: 0.7, 1.0, 1.5, 2.Q, and 3.0. In all cases
the same threshold was used (apart from the relative error factor for the different
FW HM; see DAOPHOT manual). The resulting fractions of stars detected and
num ber of spurious detections, before any filtering with SHARP or ROUND, are
shown in Table 2. These d a ta suggest th at the FWHM be set to about 1.5 pixels
for the maximum number of true detections, w ith a minimum of spurious ones.
To determine appropriate limits to filter detections by SHARP and
ROUND, we ran FIND on all the GRID frames of various signal-to-noise and ta b ­
ulated the extreme observed values of SHARP and ROUND for true detections
as a function of signal-to-noise. These d ata are shown in Table 3. Note th a t the
standard DAOPHOT cutoffs are 0.2 < SHARP < 1.0 and —1.0 < ROUND < 1.0.
Table 2 suggests th a t these cutoffs be modified somewhat for W FC d a ta at F555W,
perhaps to 0.4 < SHARP < 1.2 and —1.25 < ROUND < 1.25.
We have chosen to do these experiments in F555W only, because we expect
th a t most of the star detection will be done on the frames with a sharper PSF. The
final list of stars determined after photom etry of this frame can be used as the
input list for photom etry a t longer wavelengths. Of course, similar tests for other
wavelengths can easily be performed if some situation calls for it.
b. Parameters fo r aperture photometry and P SF definition
During the aperture photom etry and PSF definition stages, we need to
know how large an aperture is needed to contain essentially all of the light from
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the star. The PSF radius is used to determine the size of the area used to subtract
stars, and will also be used as the radius to which aperture corrections need to made.
From aperture photom etry of the model PSFs, we have found th a t an aperture of
radius 5 pixels contains at least 99.5% of the light for all the PSF considered here.
This radius should probably not be used as the prim ary DAOPHOT ra­
dius during aperture photometry.
The prim ary radius is the aperture used by
DAOPHOT as a first guess in the PSF fitting stage. For this, we have found th at a
radius smaller th an the PSF radius is desirable, as the larger radius will tend to in­
clude other stars and throw the initial guess off, causing the solution to take longer
to converge. For all the PSFs considered, a radius of 2.5 pixels includes at least 90%
percent of the light, so we suggest using this value as the prim ary aperture size. A
range of larger apertures would then be used to determine the aperture corrections
once the fitting photom etry is completed.
c. Pixel weighting scheme (fitting techniques)
To test the adequacy of the standard DAOPHOT fitting scheme for WFC
data, the GRID-SN270 field was reduced using an unmodified DAOPHOT. For
this signal-to-noise, a scatter of a = 0.004 magnitudes is expected. For this test,
a fitting radius of 3 pixels was used, and the fitting routine was supplied w ith a
perfect P S F model. The results are shown in Figures 4-Na ( recall, N goes from 1
to 4 and represents results for 4 differents PSFs). In these figures and subsequent
ones, magnitudes are plotted as a function of star number. The magnitudes are
given in the standard DAOPHOT magnitude system which has a magnitude 25
star corresponding to a star w ith
1
count; consequently, stars w ith
1 0 ,0 0 0
counts
have m = 15.
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Ill
Clearly the photom etry shown in Figures 4-Na is much worse th a n expected
from photon statistics alone, even given a perfect representation of th e PSF. After
some experim entation, the source of the additional scatter was determ ined to come
from the scheme by which NSTAR weights the pixels in the fit. Normally, the
weight for each pixel is given by:
weight'- 1 = a 2 = (rn 2) + (D {fG ) + (0.0075£>;)2 + ( 0.027-—*- y
\
°x*y
A
)
where D{ is the data value (in counts) at pixel i, G is the gain in photons per count,
rn the readout noise in electrons, and crx and ay are the standard deviations of the
Gaussian first approximation to the PSF. The first two term s are th e usual ones
for photon statistics, the third term is included to account for minor flat fielding
errors, and the final term is included as some estim ate of the interpolation errors.
In the experiments here, no flat fielding errors exist, but since the term for
such errors is generally small compared w ith the other term s, we have included it
anyway; it causes no apparent problems. For actual W F /P C data, we will want to
estim ate the m agnitude of flat fielding errors and use an appropriate value in the
weighting scheme.
It is the final term for interpolation errors th a t causes the problems for
W F /P C -like data. The constant (0.027) used by DAOPHOT was determ ined from
analysis of high quality ground based data. This expression greatly overestimates
the interpolation errors for the W F /P C PSF. T he same test field used above was
reduced without using this term , and the results are shown in Figures 4-Nb. Now,
we obtain photom etry at the precision expected by photon statistics.
This is not an entirely fair test because a perfect PSF was supplied to
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112
the fitting routine, so there really were no interpolation errors. However, the same
test was performed again, after determining the PSF from five stars of different
pixel centerings using standard DAOPHOT techniques. The results are shown in
Figures 4-Nc when the interpolation term in included in the weighting of pixels,
and in Figures 4-Nd when the term is set to zero. Here, it is clear th a t even when
interpolation errors do exist, the weighting scheme used by DAOPHOT degrades
the accuracy of the result. It is also clear th at there are additional systematic
effects when the PSF is not known exactly, since, although Figures 4-Nd shows
b etter results th an Figure 4-Nc, they are not nearly as good as those shown in
Figure 4-Nb.
The dominant source of the error seen in Figures 4-Nd was found, after yet
more experimentation, to arise from the DAOPHOT scheme of reweighting pixels
during the fitting procedure.
Originally, the weight of the pixels in the fit was
reduced as the 8 th power of the ratio of the observed error over the expected error
in each pixel after the first iteration. This is implemented to reduce the effect of bad
pixels. However, in the case of W F /P C data, when the PSF is not known exactly
or if th e initial guess for the m agnitude is off, the residuals from the bright central
pixels can be much larger than expected from the noise estim ate during the first
few iterations. DAOPHOT was reweighting these pixels before the fit had a chance
to converge. Since some scheme of reweighting is highly desirable for real detectors
which do have bad pixels and cosmic ray events, DAOPHOT was modified to retain
the reweighting scheme, but only to reduce the weights by the
2 nd
power of the
above ratio, and more importantly, only after at least 5 iterations have taken place.
W ith this new technique, I obtained the results shown in Figures 4-Ne. The result
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113
begins to approach the level of photon noise.
Since DAOPHOT provides a statistic, called CHI, which gives an estimate
of th e ratio of the observed noise to the expected noise, this can be used to determine
an appropriate constant to use for the term representing interpolation errors. One
finds th a t if the standard DAOPHOT constant 0.027 is used, the interpolation
errors are overestimated, so CHI comes out significantly less th an one.
I have
found th a t a value approximately 0.1 times the standard DAOPHOT value gives
CHI on the order of unity, as desired. The results from using this value in the
weighting scheme are shown in Figures 4-Nf. It is still evident, however, th at this
gives worse photom etry than if the term is ignored entirely, as in Figure 4-Ne.
This is probably because the interpolation term always reduces the weight of the
central pixels; since these contain the bulk of th e signal for undersampled stars, the
photom etry cannot be accurate if these pixels have reduced weights. In any case,
it is not really correct to include an error estim ate for interpolation, because the
errors arising from interpolation are not random errors; rather, they are correlated
w ith the pixel centering of the stars. Consequently, in all subsequent experiments,
the interpolation term in the error estim ate is set to zero.
This has the unfortunate consequence th a t the statistic CHI is no longer
a direct estim ate of the errors of bright stars for which interpolation errors are one
of th e largest sources of error; for these stars, CHI always comes out larger than
unity w ithout the inclusion of the interpolation term (although the photom etry is
excellent). For fainter stars, CHI is still a useful error estim ate, since for these
stars, the interpolation errors are usually small compared w ith Poisson noise. The
new CHI w ithout the interpolation term , however, is very useful as a diagnostic of
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114
how well the PSF is modelled as applied to bright stars; it gives the mean ratio of
observed deviations from the predicted PSF to the expected deviations on the basis
of photon statistics alone.
d. Choice o f fitting radius
The fitting radius determines how many pixels are included in the PSF
fitting process. We have chosen to use a fitting radius of 3 pixels, as this radius
contains about 95% of the total light from a star. This radius is large enough to
allow both the star and the sky background to be accurately fit, even for bright
stars. If the sky background is determined from the aperture stage, not during the
fitting process, as it is in standard DAOPHOT, the fitting radius could be smaller
(see section VIII for further discussion of determining the background level). A
smaller fitting radius has the advantage th a t it requires fewer evaluations of the
PSF during the fitting process.
This can lead to a substantial time savings in
standard DAOPHOT. As discussed in the next section, however, the entire process
can be sped up by calculating the PSF before the fitting process. If this is done,
the tim e taken to evaluate the P S F during the fitting stage is small compared with
the time required to solve the equations, so the time savings for a smaller fitting
radius become less significant. For isolated stars w ith the new technique, using a
fitting radius of 2 takes 25% less time th an a fitting radius of 3, but this factor is
drastically reduced in the realistic case where stars come in groups.
Consequently, we have decided to use a fitting radius of 3 pixels throughout
this study.
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e. Speed!
One of the largest problems expected w ith W F /P C d a ta is the tim e needed
to do the stellar photom etry reductions. Each W F /P C image will consist of four 800
x 800 frames, so there will be a great quantity of information. In addition, photom­
etry in crowded fields can take extremely large quantities of tim e to simultaneously
fit large numbers of stars. Consequently, we felt th a t speeding up the reduction
process as much as possible is highly desirable. We have discovered several ways to
do so, as discussed in the next few paragraphs.
The first m ajor change, as mentioned above, is to integrate the PSF on
a grid of pixel spacings separated by
0 .1
pixels b e fo re doing the profile fitting,
and to obtain values during the fitting stage by bilinear interpolation w ithin the
stored table of the integrated PSF. This results in a speed gain of a factor of two
for isolated stars where the PSF com putation dominates the to tal CPU time. For
larger groups, the tim e savings is reduced as more and more tim e is required for the
solution of the set of equations at each iteration compared w ith the tim e required
for the PSF computation; for groups of 60, we found the library technique resulted
in ~ 25% reduction in CPU time. In all cases, the photom etry is nearly identical,
rarely differing by more than
0 .0 0 1
magnitudes.
An additional time saving can be made by changing the m ethod used to
solve for the brightnesses and positions during the profile fitting stage. For each
group of stars with N members, there are at least 3N param eters to be fit (a bright­
ness and x and y positions for each star). In addition there may be param eters for
the background level, although DAOPHOT in its standard form just uses the aver-
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116
age of the aperture photom etry measurements of the sky for all the group members.
Since the expression for the brightness in each pixel is given in p art by an integral
over a gaussian, it is nonlinear in the param eters which give the stellar positions.
Consequently, an iterative technique is used during the fitting process. At each step
in the iteration, a set of linear equations must be solved to provide corrections to
the input parameters; when the corrections become small, th e fit is judged to have
converged and the next group is reduced. Usually something like 5 to 30 iterations
are required before convergence; if 50 iterations are reached the fitting process is
autom atically term inated. The bulk of the tim e is taken by the routine th a t solves
the linear set of equations at each step. In its standard form, DAOPHOT solves this
set of equations by a m atrix inversion, using a G auss-Jordan technique. The time
required goes roughly as the third power of the number of param eters. There are
quicker ways to solve the equations, and we have instituted a bi-conjugate gradient
m ethod. This takes about the same time for small groups as standard DAOPHOT,
but is several times faster for larger groups. It has the disadvantage th a t it solves
the set of equations w ithout inverting the m atrix, and thus it does not retu rn es­
tim ates of errors in the values of the parameters. Having the errors, especially for
the magnitudes, is essential, so we have instituted a scheme in which we do the
standard m atrix inversion every fourth iteration, and once again after the fit has
converged, so th a t the errors are computed and can be used in the process of de­
termining convergence and weeding out bogus star identifications during the fitting
process. Significant time savings are still realized.
We have also investigated using a different philosophy for the grouping of
the stars. In standard DAOPHOT, groups are built by taking the neighbors of
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117
a given star, then adding in the neighbors of these neighbors, etc., until no more
stars are found. In very crowded fields this can easily continue until almost the
whole frame is included in one group. W hen this happen, the neighbor criterion
m ust be relaxed until groups have more reasonble sizes, possibly at the expense
of photom etric accuracy. This scheme seems inefficient as one would not expect
the neighbor of a neighbor of a neighbor of a neighbor (for example) to affect the
photom etry of a given star very much. Consequently, we have instituted a scheme in
which each sta r is reduced separately using only its neighbors and their neighbors.
This results in many more groups of stars which need to be fit and many stars
which are reduced more than once as members of several groups, b u t the groups are
significantly smaller. It also involves a little tricky bookeeping during the grouping
and fitting procedures. We have found th a t this technique can increase the speed
of th e fitting procedure by a factor of two or more. Some results will be presented
in section VIII.
Between these various techniques, we feel th a t we can speed up the fitting
process in DAOPHOT by a factor of 3 to 10 for crowded fields. This could possi­
bly be increased still further with more modifications, for example, by demanding
convergence of only the prim ary star in the new grouping scheme. Some demonstra­
tions of the relative accuracy of the photom etry and CPU requirements for these
various modifications will be presented in section VIII for fields of various degrees
of crowding.
/. Conclusions
In all subsequent reductions, we adopted the following DAOPHOT param e­
ters: FW HM =1.5, FITTIN G RADIUS=3, LOW SHARP=0.4, HIGH S H A R P= 1 .2 ,
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LOW ROUND=-1.25, HIGH ROUND=1.25, and PSF RADIUS=5. During the fit­
ting stage (NSTAR), no term was included for interpolation errors, reweighting of
pixels was done proportional to the 2 nd power of the mean relative error after at
least 5 iterations, and the library technique for getting values of the PSF was used.
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119
V I. P S F R ep resen ta tio n For U n d ersa m p led , Iso la ted Stars
To be able to do PSF photometry, we must determine the PS F from some
stars to be used as a model to fit others. Specifically, we need to determine the
convolution of the underlying PSF with the pixels of the detectors, because this is
w hat is actually observed. The values of the convolved PSF depend on the exact
location of the star w ithin the pixel. The general problem is to determine the
convolved PSF for arbitrary pixel centering given observations of isolated stars at
some subset of pixel centerings.
This is easy if the PSF is well sampled by the CCD pixels. In this case,
an observed PSF at one pixel centering can be directly interpolated to give a PSF
at any other pixel centering. For undersampled data, however, the convolved PSF
varies over scales less th an the size of a pixel, so direct interpolation schemes will
introduce systematic errors.
The scheme adopted by most stellar photom etry packages for undersampled
d a ta is to fit some analytic function to the observed PSF. This function can represent
either the true underlying PSF or the convolved PSF; if the true PSF is being
fit, then this function is integrated over pixels to provide predictions for the PSF
for arbitrary pixel centering. Given observations of ju st a few stars, much b etter
accuracy is obtained for undersampled images by fitting to the tru e PSF rather
than the convolved one (Buonanno and Iannicola 1989).
Unfortunately, it is difficult to find an analytic formula w ith relatively few
param eters th at can accurately model the entire PSF. Typical choices for analytic
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120
functions are Gaussians (DAOPHOT) or Moffat functions (ROM APHOT), where:
I = p i * exp [ - (p 4 (x - p2f + P5 (y - P z f + P6 (x ~ P2) {v ~ Pz)) ] » Gaussian;
J=,1+f1+( ^ +(^|3)!yP6,Moffat,
\
p%
Pz
)
where the pf axe the param eters of the function to be fit to the data. Because of the
difficulty of getting a function to fit the entire extent of the PSF, it is also possible
to fit an analytic function to only the inner parts of the PSF, where the underlying
PSF is varying most rapidly. The residuals of the PSF stars are stored in a lookup
table, and in the final run, the PSF is a combination of the integral of the Gaussian
plus the residuals of the PSF stars interpolated to the desired pixel centering. The
wings of the PSF are generally smoother than the core, so the interpolation is
usually sufficiently accurate.
This is the scheme used by DAOPHOT, which uses a Gaussian w ith prin­
cipal axes along the directions of the pixels as the analytic function for the core.
In its standard form, DAOPHOT performs the fit of the Gaussian only to the first
PSF star chosen, and stores both the residuals and the aperture m agnitude of the
first PSF star. Subsequent PSF stars are fit with the model from the previous PSF
stars, have the Gaussian subtracted, and the new residuals are added to the old
ones to build up the signal to noise in the array of residuals. The m agnitude of the
PSF is increased correspondingly by the factor attained during the fit. The table
of residuals is stored on a half-pixel grid; residuals from each star are interpolated
by cubic interpolation to the nearest grid location before being stored.
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121
a. High signal-to-noise stars
To test w hether this scheme is adequate for the undersampling of W F /P C
data, we can determine whether there are systematic errors for different pixel cen­
terings for a given P S F representation, and also w hether the results differ depending
on which stars are used to construct the PSF.
Figures 5-N show the results of using the standard DAOPHOT represen­
tation of the PSF on the GRID-SN270 field. Each image is reduced several times.
In all cases, the same 5 stars were used to construct the PSF (centerings (.05,.05),
(.25,.25), (.45,.45), (.60,.90), and (.80,.70) ), but in each case, a different one was
chosen as the first star. Recall th a t DAOPHOT places special emphasis on the first
star because this is the one to which the Gaussian is fit. Figures 5-Na have the
prim ary PSF star near a pixel center and reproduce the results from Figures 4-Ne.
Figures 5-Nb-e show the same field reduced using other prim ary PSF stars; for ex­
ample, Figure 5-Nc shows the results using a star near the corner of a pixel. Also
shown on the figures is the m ean offset in the photometry, the observed standard
deviation, and the average value of CHI for the set of stars.
It is apparent th at there are systematic effects depending on the choice
of the initial PSF star, both in increased scatter an d offsets. Using a star near a
pixel center seems to give the best results. W hen the scatter increases, the offset
will also increase because of the increased error in the fit of the secondary PSF
stars to the prim ary one during the process of building up the PSF. If one uses the
aperture magnitudes to give the relative brightness of the PSF stars instead of the
fit brightness, the offsets usually go away, although still not for the star falling at the
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122
pixel comer. This information is provided w ith the two m ean offsets shown on the
figure; the first mean, labelled m ean(fit), gives th e mean offset if the fit magnitude
is used during the PSF creation to update th e m agnitude of the PSF, while the
second one, labelled m ean(aper), gives the mean offset if th e aperture magnitudes
of th e PS F stars are used to determine the to tal PSF m agnitude. W hen the two
numbers agree and are close to zero, there are no systematic effects introduced by
the choice of prim ary PSF star, and even in th e case where they disagree, good
results can be attained if aperture photom etry can be done on the PS F stars.
Consequently, it seems th a t reasonably accurate photom etry can be ob­
tained for isolated stars, as long as stars near pixel comers are avoided as prim ary
P S F stars. Unfortunately, this is not the whole story. Really, we are concerned
w ith more than just getting the correct fitted m agnitude to the star. It is possible
to get an accurate fit without matching the observed profile exactly. This is fine
for isolated stars, but for crowded images, we also want to m atch the exact profile
as closely as possible so we can subtract the star out cleanly and find additional
neighbors and fit other stars accurately. The residuals from any fit will always come
out noisier than the background because of the increased am plitude of the photon
noise in the location of stars, but any errors in the PSF representation will lead to
extra peaks and valleys in the residual image. The degree to which the PSF repre­
sentation is consistent with the noise expected in the images is given by the statistic
CHI which, as discussed above, gives the mean ratio of expected to observed noise
in each star. A n ideal PSF representation will have CHI for all stars near unity.
The worse the PSF representation, the larger the value of CHI. As shown on Figures
5-N, the mean value of CHI is never less than three, even for the PS F constructed
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123
w ith the well centered star as the prim ary PSF star.
The extent to which this causes problems in more crowded fields will be
discussed in section VIII. Here we concentrate on improving results for single stars,
keeping in mind th a t smaller mean values of CHI are desirable.
In an effort to bypass the system atic errors and to avoid having to hand
pick stars depending on their pixel centering, I tried a PSF fitting scheme where
the analytic function is fit to all of the PSF stars simultaneously. After this fit is
performed, it is subtracted from the first PSF star, and subsequent stars are added
in as they were previously. The results are shown in Figures 5-Nf. This technique
seems to work reasonably well, although still not as good as using a well centered
prim ary PSF star.
To try to improve the PSF representation, different functions were used to
fit the core of the PSF. In particular, I tried a two-component Gaussian fit w ith
principal axes aligned w ith the pixels, a Gaussian fit w ith arbitrary orientation, and
a fit using a Moffat function. A function th a t b etter fits the true PSF will work
b etter because the residuals will be smaller, so errors arising from interpolation will
be smaller. Results will be summarized below.
For a given function, the results can be improved if we can reduce the
interpolation errors. To illustrate the problems w ith interpolation, Figure
6
shows
a cross section of a true convolved PSF (bold line), a best fitting convolved Gaussian
from a well centered star (light line), and the residuals, tabulated on a grid spaced
at 0.1 pixels. An observed PSF would be obtained by taking points spaced by
one pixel, with the zero point determined by the pixel centering. One sees th a t the
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124
gaussian is fit only to the core. In the wings of the stars, the residuals are reasonably
smooth, and interpolating between every ten th point will be reasonably accurate.
In the core, however, the residuals clearly cannot be accurately interpolated if only
every ten th point is sampled. The amplitude of the residuals can be nearly ten
percent of the amplitude of the PSF, so errors in the interpolation cannot be totally
ignored.
The situation can be improved if we have several PSF stars of high signalto-noise at different pixel centerings. In this case, we can store values of the con­
volved PSF at different pixel centerings w ithout interpolating them first to the near­
est half-pixel grid location. The extended residual table can then be interpolated
to get values at arbitrary pixel centering. High signal-to-noise stars are required
because we will not be averaging residuals from different PSF stars. The residuals
from each PSF star will be stored in the appropriate slot in a residual grid tabulated
at 0.1 pixel spacing without any interpolation, rath er th an interpolating them to
the nearest half-pixel grid location, as in standard DAOPHOT. The only trick is
how to fill in the missing spots in the residual grid after as many stars of different
centerings are included; the known values will not necessarily be spaced in a regular
manner. We also wish to know how many spots need to be filled in to give reason­
ably decent results. The scheme used here is to fill in all missing spots by finding the
three nearest points which enclose the point in question, and then using the plane
th at passes through these three points to give the value a t the desired location. A
related technique has been used by Gilliland and Brown (1988) in HAOPHOT, and
they too find th at results are improved over the standard DAOPHOT technique.
Results for several choices of fitting functions and for the new interpola­
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125
tion technique are summarized in Tables 4-N. The mean offsets, scatter, and mean
values of CHI axe given for each fitting technique for several different choices for the
prim ary PSF star; in addition, results are given when the function is simultaneously
fit to 5 PSF stars and to 9 PSF stars spaced on a 0.33 pixel grid.
As noted above, the standard DAOPHOT technique gives good photom etry
if stars near pixel corners are avoided, but still has values of CHI around three for
F555W. Using two Gaussians or a Gaussian w ith arbitrary orientation does not
improve the situation much. A Moffat function gives slightly lower values of CHI
but still has problems for stars near pixel comers. Using the new interpolation
technique gives the best results, w ith no dependence on the choice of prim ary PSF
star. If a 3x3 grid of observations at different pixel spacings for a high S/N star is
used define the PSF, excellent results are obtained, w ith CHI around two.
The quality of the results does depend on the details of the PSF; m od­
elling the PSF a t F785LP, for example, gives consistently worse results than those
obtained at F555W (e.g., note th a t the mean values of CHI are larger in Table 5-3
than in Table 5-1). Generally, however, the techniques which work b etter with one
PSF also work b etter with the other PSFs, so the modified techniques seem robust
against small variations in the PSF. Using the new interpolation technique gives
values of CHI around two for all of the PSFs considered.
b. Different S /N
I next wanted to test whether there are systematic effects for stars of
different signal-to-noise (S/N ), and to see determine whether stars of lower S/N
can be used to define the PSF. To do so, I reduced the GRID-SN73 and the GRID-
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126
SN12 fields. The expected scatter is about 0.015 magnitudes for the GRID-SN73
field and 0.20 magnitudes for the GRID-SN12 field; the correct magnitudes should
be 17.5 and 20.0.
First consider the results of using high S/N PSF stars to reduce fainter
stars. Three cases are presented in Figure 7 for stars of SN73: using a perfect
PSF, a PSF representation from 5 SN270 stars using 1 Gaussian, and a PSF rep­
resentation from 9 SN270 stars using a Moffat function and th e new interpolation
technique. Results for F555W are shown oh the left side, and those for F785LP on
the right; for brevity, only results for the center of the CCD are shown. One can
see th a t the results are sensitive to the details of the PSF. In F555W, either PSF
representation seems to do adequately. For F785LP, however, there are system­
atic errors and increased scatter when the Gaussian w ith old interpolation is used.
The new technique performs quite well. T h at the F785LP results are worse than
the F555W results is slightly difficult to explain, as the undersampling is worse at
F555W than at F785LP. Since the mean values of CHI were higher in the SN270
field for F785LP than for F555W, however, this result was not entirely unexpected.
The corresponding results for the SN12 field axe presented in Figure 8.
Here interpolation errors appear to be unim portant, w ith the same scatter resulting
from either PSF representation as from the perfect PSF, in both colors. There is
a systematic effect, however, such th at the faint stars are always measured to be
slightly too bright. This most likely results from the effect of noise peaks on the
fit the PSF to the star. The offset does not appear to be extremely large, however,
and the systematic errors could be detected by artificially adding faint stars to the
observed frame and reducing them.
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127
Next we consider whether the PSF can be determ ined from stars of lower
S/N . The bottom lines of Tables 4-N show the results for photom etry of SN270
stars using SN73 stars to determine the PSF. For the fainter P S F stars, the old
DAOPHOT technique works b etter than the new interpolation scheme in F555W,
but the new technique is b etter in F785LP. The new technique does not average the
residuals from different stars, so we expected to have more problems w ith fainter
stars. Although the DAOPHOT technique may give b e tte r results, the systematics
w ith pixel centering are still apparent, and they cannot be avoided by using aper­
ture photom etry since very accurate aperture photom etry is not possible for fainter
stars. Consequently, it seems very desirable to have the brightest stars for the de­
term ination of the PSF. The new technique is probably preferable in all colors for
stars of S /N > 100.
c. Conclusions
Generally, it seems th at the effects of undersampling can be kept reasonably
small. If bright stars are available to determine the PSF, photom etry at nearly the
level of photon statistics for isolated stars should be attainable. Even using the
standard DAOPHOT technique and determining a PSF from stars on each frame
is likely to give acceptable results, although care must be taken to see th a t there
are no systematic errors like those seen in the tests above for F785LP. The new
interpolation technique looks very promising if high S/N d ata at a variety of pixel
centerings is available to construct the PSF. In Section VIII, we will investigate the
relative errors from these two techniques in more crowded fields.
d. The Real World
Before discussing how to approach real data, it is im portant to note one of
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128
the prime uncertainties about W F /P C data. This concerns the stability of the PSF
as a function of time. This is crucial for stellar photometry, as we need to know how
often we will need to determine the characteristics of the PSF. If the PSF varies
stochastically from frame to frame, as it does for ground based observations, we will
have to determine the PSF from some small number of stars on each frame, which
may be very difficult. If, on the other hand, the PSF is found to remain stable
over time scales of weeks, then some effort could be put into fully mapping the PSF
characteristics at one time for use in reducing observations at other times. If the
PSF changes gradually, we may be able to use some kind of perturbation technique
to determine the PSF on a given frame on the basis of a detailed mapping of the
PSF at another time.
Consequently, the first thing th at needs to be done after launch is to mon­
itor the characteristics of the PSF over time. Virtually all observations will have
some stars in the field which can be studied in an effort to characterize the PSF.
In the W FC, it is not clear th a t single observations of stars will be sufficient to
characterize the PSF and to detect small variations because of the undersampling;
we know from the experiments above th at a simple fit of gaussian will give differ­
ent results for stars of different pixel centerings, and the same will likely be true
for moments of the light distribution or other similar statistics. I expect th a t an
accurate study of PSF variations in the W FC may require multiple exposures of
stars w ith different pixel centerings. Of course, if the PSF varies significantly from
frame to frame, we cannot use observations even from consecutive frames. The most
promising technique may be to look for PSF variations in the PC where the PSF is
b etter sampled. This should give some idea of the expected variability of the PSF
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in b o th cameras.
After PSF variations, the critical questions will be to understand what type
of PS F representation to use and what d ata this technique requires, and to assess
the expected accuracy of stellar photometry. The most useful d a ta for answering
these questions is a series of exposures of an uncrowded field w ith each exposure
shifted by some fraction of a pixel. The Fine Guidance Sensors of HST are expected
to be able to position to w ithin some small fraction of a pixel, so this is technically
feasible. Such a test is planned as a part of the W F /P C Science Verification tasks
and is referred to as the PSF dither test. Of course, if the PSF varies from frame
to frame, this test is not useful; if this is the case, we are in a bit of trouble and
most likely, all we will do is to determine the PSF as best we can from stars in each
frame, and use repeat measurements of stars to assess the errors in the photometry.
Provided the PSF is constant over the tim e th at the dither test is made, a
PSF representation can be built from observations of a single star at different pixel
centerings. This representation can then be used to do photom etry on the set of
observations at all pixel centerings, and the resulting spread in m agnitudes can be
used as a diagnostic of the suitability of the PSF model. At this stage, one would
also look at D AOPHOT’s CHI statistic (without any term for interpolation errors).
The different techniques (different fitting functions, interpolation techniques) can
be tried, and the one giving the best photom etry and the lowest values of CHI can
be adopted as the technique to use.
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130
V II. V ariations in th e P S F
In this section we are concerned w ith the effects of a PSF which is variable
in color, position on the frame, and possibly w ith time. The general question here
is to determine how far off the P S F used to fit stars on a fram e can be from the
actual PS F of the stars being fit. As w ith the PS F representation, th e answer
depends on the degree of crowding in the field. Here I discuss the systematics
of misrepresentations of the PS F for isolated stars and present a statistic (CHI)
which should give some estim ate of the relative errors of getting th e PS F wrong on
photom etry in more crowded fields.
As noted above, however, the PSF representation is never perfect, as in­
terpolation errors cause the residuals to be larger th an those expected from photon
statistics, even when the photom etry for isolated stars is accurate; this causes val­
ues of CHI to be larger than unity. The top and middle panels of Figure 9A show
graphically the results of the photom etry for a perfect PSF and for a PS F respresentation using a Moffat function and the new interpolation technique, for F555W
and F785LP; judging from the accuracy of the photom etry alone, the PSF repre­
sentation is nearly as good as the perfect representation. The corresponding plots
for CHI are shown in Figure 9B for these cases. For the perfect P S F representation,
the values of CHI are scattered around 1, as expected. For the model PSF, however,
CHI depends on th e pixel centering and is always larger th an unity. As shown in
Tables 4-N, all other PSF representations give even larger values of CHI th an the
new interpolation scheme. The level to which the m isrepresentation of th e PS F af­
fects photom etry in more crowded fields will be discussed in Section VIII. Here we
will use this information as a basis of comparison to judge the relative im portance
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131
of other sources of PSF misrepresentation, namely the effect of PSF variability with
stellar color and position on the frame.
a. P SF Variations With Stellar Color
Since stellar images are nearly diffraction limited in the W F /P C , the PSF
varies significantly w ith color. Consequently, in a broad band filter, the PSF as
integrated over the filter bandpass will vary w ith the slope of the stellar spectrum;
blue stars will have a narrower PSF th an red ones. This must be considered for the
W F /P C , since some very broad filters (> 2000A) will be used for a large num ber of
observations because of their large throughput. All of the test frames used in the
previous section were made for stars w ith (B-V)=0.
To consider the magnitude of this effect, the GRID-SN270 field was reduced
using a PSF appropriate for stars with (B-V)=2.0. The results for the photom etry
are shown in the bottom panel of Figure 9A and the values of CHI are shown in
Figure 9B. Figure 9A shows th a t very accurate photom etry is attained for isolated
stars even for the color mismatch between stars of (B-V)=0.0 and (B-V)=2.0 for
F555W. For F785LP, there is a systematic offset, b u t th e scatter is still quite small.
Such an offset would not be a problem if different transform ation coefficients were
derived for each frame. It would probably be easier, however, ju st to check after
reducing the d ata th a t the PSF stars are not drastically different in color from most
of the other stars. Figure 9B shows th a t, although the residuals are larger than
expected when the PSF of the wrong color is used, the difference is not extremely
significant, and the errors arising from this are likely to be much smaller th an errors
arising from interpolation for even the best PSF representation (since th e values of
CHI in th e lower panel of Figure 9B are significantly smaller th an those seen in the
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132
middle panel).
We conclude th at the variation of the PSF w ith stellar color is not likely
to cause great difficulties; errors from these variations are likely to be swamped by
errors in th e PSF representation. Where possible, of course, the PSF should be
determined from stars of similar color to those being reduced.
b. P SF Variations With Position
Because of the presence of aberrations in the camera optics, W F /P C images
are expected to vary significantly across the frame, as shown in Figures 1A-D. In
addition to the large scale variations from aberrations, there may also be variations
arising from the imperfections of the optics and waviness in the CCDs.
It is im portant to understand the effects of using a PSF determ ined from
stars on one part of the frame to fit stars at other locations because we need to
know how many different PSFs we will need to do photom etry on the whole field. If
we need to construct PSF representations in advance, e.g. from a P S F dither test,
we need to know how many locations they need to be computed at. If it is sufficient
to use one PSF for the entire field, then it may be feasible to determine the PSF on
each frame from a few reasonably isolated stars even in moderately crowded fields.
If we need to use a different PSF every 100 pixels, however, then it will probably
be impossible to determine the PSF on all but the least crowded frames.
We may also b s uiblv to use some interpolation scheme to predict the PSF
at a given spot from observations at other locations. Unfortunately, we cannot test
this in advance as we do not know exactly how the PSF will vary. The simulated
PSFs we are working with are computed for the center and four corners of the CCD.
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133
We have used an interpolation scheme which creates a variable PSF across the 800
x 800 CCD th a t is consistent w ith the presence of low order aberrations, but we
cannot test the adequacy of this interpolation scheme for recovering correct data
w ithout independent simulations at other field locations.
W hat we have chosen to study here are the effects of using a PSF from
one region to do photom etry on stars across the whole field. The FIELD-SN270
frame was reduced using the correct varying PSF, the PSF from th e center of the
field, and an average of the center with the four corner PSFs.
In all cases, a
perfect PSF representation (not a DAOPHOT model) was used, so th a t any errors
from PS F representation will add to the errors from these tests. The results of
the photom tetry are shown in Figure 10A. Once again, for isolated stars, accurate
photom etry is attained even when using an inaccurate PSF to do the fitting. Figure
10B shows th e values of CHI for this field. W hen the correct PSF is supplied, the
values of CHI axe clustered around 1. W hen the PSF from the center is used, the
dependence on the position in the frame is very evident. The oscillations shown
appear w ith a period of 19 stars since the stars axe on a 19 x 19 grid and are
num bered consecutively across columns.
Even the crudest approximation of using the central PSF or an average PSF
to reduce the entire frame does reasonably well. The PSF inaccuracies resulting from
this axe comparable in magnitude to those resulting from errors in the DAOPHOT
P S F representation. Judging from the observed values of CHI, it seems th at fairly
good results would be attained if the PSF were modelled at a reasonably small
num ber of points on the CCD, perhaps on a 3 ' x 3 or a 5 x 5 grid. If the PSF
varies smoothly from one region to another, an interpolation scheme like the one
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134
presented above should be easy to implement.
There may also be variations of the PSF over smaller scales on the CCD
which we are not modelling here. If such variations occur, an interpolation scheme
may be difficult to implement. In this case, it is comforting to know th a t as long as
the th e variations are not too severe, they are not likely to introduce a large amount
of scatter into the photometry.
c. PSF Variations With Time
The next question is to try to assess the effects of PSF,variation in time.
The level of such variations is not well known, and we have chosen to study two
cases: variations caused by changes in the amount of guiding jitte r and by changes
in focus.
All of the PSFs used so far have been created with 7mas RMS guiding jitte r,
appropriate for the fine lock mode of the Fine Guidance Sensors (FGS). We have also
created PSFs w ith 20mas jitte r and used them to reduce the 7mas images. Results
are shown in the middle panels of Figures 11A and 11B. The errors introduced by
this m isrepresentation are comparable to those introduced by a DAOPHOT PSF
representation.
It is expected th a t the focus of HST will vary with time. To get some idea
of the effect on PSF photometry, we have created PSFs w ith an extra 1/10 wave
of focus added to the results from the ray tracing. The results of using this PSF
are shown in the bottom panels of Figures 11A and 11B. Once again, the errors
introduced are of approximately the same magnitude as those caused by the PSF
representation. The photom etry for isolated stars is very accurate.
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135
These results are all encouraging in th a t it seems th a t reasonably accurate
photom etry is attainable in the presence of m oderate variations even if they are
ignored. Unfortunately, these test cases may be too easy, as in reality the changes
may be larger and may involve changes of shape as well as changes of scale. For ex­
ample, jitte r errors m ay result in more elongated images for longer exposures. This
might cause trouble if a PSF were determined from shorter exposures of reasonably
bright stars and used to reduce frames of longer exposures.
d. The Real World
W hen HST flies, we will need to assess the level of variations in th e PSF.
This will be difficult for direct observations of single stars because of the undersam ­
pling. Using a good PSF representation, however, it should be reasonably easy to
look for variations. The PSF representations can be used to construct well sampled
images, which can be directly compared to each other to look for variations. As
m entioned above, the first test will be to look for variations w ith time. If the PSF
varies from frame to frame, we will probably just obtain the best PSF we can from
whatever isolated stars are available on the frame. In this case, we will probably
not get a very good representation, but the tests of this section show th a t we still
may be able to get reasonable photometry.
To sample variations across th e frame, we m ust be able to create indepen­
dent PS F representations of a variety of field locations. Once again, the PS F dither
test will be extremely useful, as it will provide observations of a variety of stars
across the field at a variety of pixel centerings. Once the variations are detected,
we can see whether it is possible to implement some kind of interpolation scheme
to account for PSF variations across the frame. There will also be some variety of
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136
stellar colors in the dither test, so color effects can also be examined.
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137
V III. C row ding in th e W F /P C
Many of the interesting scientific problems to be studied w ith the W F /P C
involve stellar photom etry in very crowded fields. Doing accurate photom etry in
such fields requires simultaneous fitting of large numbers of stars, a good technique
for star finding, and a good method for determining the level of the background. The
scheme usually adopted is as follows. F irst, as m any stars as possible axe found on
the frame. These are then grouped by some kind of neighbors of neighbors scheme,
and a first pass at profile fitting is made. These results are used to subtract stars
from the frame, then more stars axe found in the subtracted frame which were
hidden before. These are added to the original list of stars and the frame is reduced
again. In very crowded fields this may be repeated yet again.
Primarily, we view the crowding problem as a com putational one. In its
standard form, DAOPHOT is limited to groups of 60 or smaller because even groups
of 60 can take many hours of CPU tim e to reduce. As faster machines become
available, this could be increased, but for the laxge volume of d ata th a t we are
talking about (each W F /P C frame is a 1600 x 1600 image, and there will be a lo t
of them ), using the standard technique is likely to take a very long time.
In this section, I present results of photom etry on the CROWD fields of
various surface brightnesses for a variety of techniques and PSF representations.
First I present the results using the new fitting and grouping techniques discussed
in Section V. Then I present the effects of various degrees of PSF misrepresentation
on photom etry in crowded fields. I briefly discuss th e difficult m atter of background
determ ination. Finally, I discuss the expected accuracy of photom etry in crowded
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fields with the WFC.
The results are shown in Figures 12A to 12D for the four degrees of crowd­
ing in F555W, and in Figures 13A to 13D for F785LP; in both cases, the PSF
in the center of the CCD was used. Shown in the figures are the observed errors
in magnitude (true - observed) as a function of true V magnitude; positive differ­
ences mean the star is measured brighter th an it really is. Recall, for these fields a
globular cluster luminosity function was used, b u t with all stars placed at discrete
m agnitude steps. The total number of stars placed in each magnitude bin is shown
at th e top of the figures. For each plot, the number of stars detected in each bin
is shown along w ith the median and standard deviation of the magnitude errors in
the bins. The lines show expected 2cr error bars. All these results were obtained by
taking two passes through the d ata of finding and fitting stars.
The CPU tim e shown is the time in seconds taken by the second fitting
procedure only; for the crowded fields, this is where most of the tim e is spent.
The amount of CPU time depends largely on the group sizes th a t need to be fit.
For th e small fields considered here (each CROWD image is only 50x50 pixels),
there are statistical effects in the group sizes. For example, the CROWD23 field,
although less crowded than the CROWD22 field, takes longer to reduce because the
stars happen to fall such th a t the largest group is larger in the CROWD23 field.
Similarly, the addition of just a few extra spurious detections in the subtracted
frame can increase the CPU time dram atically by linking two smaller groups into a
larger one. Consequently, one cannot take any one CPU tim e too literally for such
a small field, but the times given are generally indicative of the expected relative
C PU times for the different techniques.
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139
a. New fitting and grouping techniques
As discussed in Section V, the speed of the fitting procedure can be in­
creased in crowded fields by using a new equation solver an d /o r a new grouping
technique. The results using the standard techniques are presented in panel (a) of
Figures 12A to 12D, the results using the new equation solver in panel (b), and the
results w ith the new grouping technique in panel (c). In all of these cases, a perfect
PSF was supplied to the fitting routine. Results for cases (b) and (c) were only
obtained for F555W and do not appear on Figures 13A to 13D.
By comparing the upper two panels, one can see th a t the new equation
solver gives essentially identical results as the old technique. The CPU tim e savings
is negligible for sparse fields but can be nearly a factor of three for more crowded
fields.
The new grouping technique, shown in panel (c), is significantly faster in
crowded fields (more than a factor of 7 for CROWD23). Recall, the new technique
just fits a star along with its neighbors and their neighbors, rath er th an continuing
to take neighbors until no more stars are found. The photom etry seems to be ju st as
good as th a t obtained w ith the old technique. The new technique, however, appears
to cause the fitting routine to drop some of the faintest stars. For magnitudes fainter
than V ~ 27, the new technique seems to end up with ~ 10% fewer stars th an the old
technique. For most applications, this is probably acceptable, since the magnitudes
for these stars are very uncertain anyhow.
b. Effects o f P SF misrepresentation
Panels (d), (e), and (f) in Figures 12A-D and 13A-D show results using
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140
several imperfect representations of the PSF. Panel (d) shows photom etry for the
standard DAOPHOT technique using a well centered star as the prim ary PSF star.
Panel (e) uses a PSF representation made using a Moffat function and the new
interpolation scheme discussed in Section V ia. Finally, panel (f) gives the results
when a perfect PSF for a star of a different color (B — V = 2) is used to reduce
the frame. These three PSF representations give m ean values of CHI of about 3,
2
, and 1.4, respectively, for F555W; consequently, they range from fair to good
PSF representations. The DAOPHOT representation utilizes a well centered star,
so worse PSF representations are certainly possible. All three cases gave excellent
photom etry for isolated stars in F555W; the DAOPHOT representation had some
small problems in F785LP for stars of interm ediate brightness. Here, we wish to
assess how photom etry in more crowded fields is affected by errors in the PSF
representation.
It is encouraging to note th at reasonably accurate photom etry can be ob­
tained with any of the PSF representations. In detail, however, one can see the
difference between the representations. This is most evident for the brighter stars,
where the observed scatter increases as the PSF representation gets worse from the
level expected from photon statistics to a scatter of a few percent for the worst PSF
representations. A better PSF representation also usually requires less CPU time
during the fitting process, although this is not always true in the experiments shown
here, probably because of the stochastic effects in small fields mentioned above.
Consequently, we conclude th at it is worth some effort to get a good PSF
representation. Recall th at the DAOPHOT PSF used here is a best case, m ade from
a well centered, isolated, bright star; for a representation much worse th a n this, we
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141
expect more serious problems to occur. We expect, however, th at even if we had
to construct a PSF from stars on each frame, as we do for ground based data, we
could probably m anage to obtain accurate photom etry at the level of a few percent
if some care was taken to define the PSF. The mean values of CHI for isolated stars
seem to be an excellent diagnostic of th e quality of photom etry in more crowded
fields, so this can be used as a tool to determine what PSF representation is best.
c. Sky determination
Determining a background level can be difficult in crowded fields where
there are many faint stars. As discussed by Stetson (1987), we wish to determine
for each star the am ount of fight contributed by the true background plus the fight
from undetected faint stars. In standard DAOPHOT and for all the experiments
presented so far, a background level has been determined for each star by taking
the mode in an annulus from
6
to 10 pixels around each star. During the fitting
process, the average of the values for all stars in each group is used as the sky
background for th a t group. We might expect this technique to have problems in
crowded fields where many faint stars fall in this annulus. Consequently, we have
also tried to do photom etry by fitting for the background in each group. The results
are shown in the bottom panel of Figures 12A to 12D. A close exam ination shows
essentially no difference until a surface brightness of V ~ 23 m ag/square arcsec.
Even for the more crowded fields, there are not very large differences between the
two techniques, although fitting for the sky might give slightly b etter results for the
bright stars in crowded fields.
Consequently, we feel th at the choice of sky determ ination techniques is
probably a m atter of taste. In the most crowded fields, errors introduced by unre­
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142
solved doubles, as discussed in the next section, probably swamp any errors intro­
duced by background errors anyway.
d. Quality o f crowded field photometry
Here we wish to assess the expected accuracy of photom etry in crowded
fields. In particular, we wish to consider the expected scatter, any system atic errors,
and the expected star detection efficiency, all as a function of m agnitude and degree
of crowding.
For sparse fields we seem to be able to do very well. Figures 12A and 13A
show th at nearly all stars in such fields can be measured to nearly the accuracy
expected from photon statistics down to V ~ 28. At the faintest magnitudes,
there is a bias towards observing stars to be too bright because only the stars with
positive noise peaks get detected w ith the threshold used. For this field, all stars
w ith V < 28 are detected. Figures 12B and 13B show th a t the situation is nearly
as good at an equivalent surface brightness of 24 m agnitudes/square arcsec. At
V ~ 27 we begin to see the effect of crowding, as these stars tend to be measured
slightly too bright. This is caused by unresolved blends w ith fainter stars which are
not detected.
For more crowded fields, the effects of crowding become more noticeable.
Figures 12C, 12D, 13C and 13D show th at nearly all stars are measured system­
atically too bright because of unresolved blends. Also, we begin to miss some of
the stars of interm ediate brightness; in the CROWD23 field, we are complete to
V ~ 26, but only to V ~ 24 in the CROWD22 field.
The crowding effects are slightly larger in F785LP than in F555W, which is
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143
expected because the PSF is sharper at F555W. The main effect is th a t fewer fainter
stars are detected in F785LP. Because of this, we expect th a t the most promising
technique for multicolor photom etry will be to reduce the shorter wavelength first,
where more stars will be detected. This list can then be used as the input list for
reductions at longer wavelengths. After a first pass, the list can be modified to
get rid of spurious detections at th e first wavelength and to add any very red stars
th a t might show up more strongly on the redder frame. This technique also makes
bookeeping much easier, as there is ju st one m aster list of stars, and no complicated
matching routines need to be used to merge results from different frames.
A crucial question in crowded fields is to determine if there is any way
we can recognize which stars are measured inaccurately; if we had some way to
weed these out, we could obtain good photom etry even in crowded fields.
We
have checked for correlations between the estim ated errors and observed values or
observed values of CHI and the observed errors, but find th a t neither quantity
is a good indicator of when inaccurate photom etry is done. Basically, we need a
technique th a t can recognize close blends in undersampled data. Unfortunately, at
close separations with this degree of undersampling, two blended stars look very
similar to one brighter star.
Consequently, the only way to assess errors in crowded fields, for b o th
photom etry and luminosity functions, may be to artificially place stars of known
brightness randomly in the image and rereduce them to assess the errors.
We
note th at using the new grouping technique would make such a task reasonably
straightforward, as we can reduce each artificial star reasonably quickly since we
only include its nearest neighbors. W ith the old technique, if we wish to reduce the
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144
artificial stars in the same way as the real one, we will essentially have to rereduce
the whole frame in crowded fields.
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145
IX . C onclusions
A wide variety of tests on simulated d a ta have been carried out to de­
term ine optim um techniques to use for W F /P C stellar photom etry and to assess
the expected accuracy of such techniques. In particular, I have concentrated on
the problems of undersampling and variations in the PS F and used a variety of
techniques to reduce some test fields a t various degrees of crowding to assess their
accuracy.
Generally, the experiments on the simulated d a ta have been encouraging.
Photom etry on reasonably sparse fields should be attainable to nearly the level of
photon statistics. For more crowded fields, the quality of the results will depend on
the characteristics of the W F /P C PSFs during flight. If these are reasonably stable
in tim e, we should be able to get an accurate PSF representation which should give
good results even in crowded fields. If they are not, th e situation is worse, but
photom etry down to the level of several percent can still probably be obtained.
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146
R eferen ces
Buonnano, R., Corsi, C.E., De Biase, G.A., and Ferraro, I., 1979, in Proceedings
of the International Workshop on Image Processing in Astronom y, eds. G.
Sedmak, M.Capaccioli and R .J. Allen, Trieste, Italy, p. 354.
Buonnano, R. and Iannicola, G., 1989, P.A. S.P. 101, 294.
Gilliland, R.L. and Brown, T.M ., 1988. P.A .S.P .
1 0 0
, 754.
Griffiths, R,, 1985. Wide Field and Planetary Camera Instrum ent Handbook, STScI
publication.
King, I.R., 1983. P .A .S.P . 95, 163.
Lauer, T.R., 1989. P .A .S.P . 101, 445.
Lupton, R.H. and Gunn, J.E ., 1986. A .J. 91, 317.
Stetson. P.B., 1987. P .A .S.P . 99, 191.
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Reproduced
with permission
T able 1
INDEX OF SIMULATIONS0
of the copyright owner.
Star Locations
Image
GRID-SN270
GRID-SN73
GRID-SN12
GRID-SN6
FIELD-SN270
grid
grid
grid
grid
of 400
of 400
of 400
of 400
different
different
different
different
pixel
pixel
pixel
pixel
Star brightnesses
centerings
centerings
centerings
centerings
10000 cnts/star
1000 cnts/star
100 cnts/star
50 cnts/star
grid of 361 different field locations
10000 cnts/star
PSF
Text Sections
Figures
V, VI, VII
VI
VI
V
4-N, 5-N
7
8
variable across image
vn
10
constant
constant
constant
constant
across
across
across
across
image0
image0
image0
image0
Further reproduction
CROWD-25
random star distribution in 50x50 field
25 mag/sq.arcsec^
constant across image0
VIII
2, 12A, 13A
CROWD-24
random star distribution in 50x50 field
24 mag/sq.arcsec^
constant across image0
vin
2, 12B, 13B
CROWD-23
random star distribution in 50x50 field
23 mag/sq.arcsec^
constant across image0
VIII
2, 12C, 13C
CROWD-22
random star distribution in 50x50 field
22 mag/sq.arcsec^
constant across image0
vin
2, 12D, 13D
REAL-25
random star distribution in 800x800 field
25 mag/sq.arcsec^
variable across image
3
REAL-24
random star distribution in 800x800 field
24 mag/sq.arcsec^
variable across image
3
REAL-23
random star distribution in 800x800 field
23 mag/sq.arcsec^
variable across image
3
REAL-22
random star distribution in 800x800 field
22 mag/sq.arcsec^
variable across image
3
prohibited without permission.
0 All of these above simulations exist for two colors, F555W and F785LP. The GRID and CROWD images exist for various locations in the field as well.
The CROWD and REAL images have stellar brightness distributions approximately matching a population II luminosity function. The counts are
determined for a 2500s exposure at the distance of M31, and the toted number of stars is determined from the equivalent surface brightness.
148
T a b le 2
ST A R D E T E C T IO N E F F IC IE N C Y F O R V A RIO U S F W H M “
FWHM
Percent Detected
Spurious detections
0.7
1.0
1.5
2.0
3.0
83.5
85.8
91.5
90.5
83.8
105
104
104
110
109
a These are detection efficiencies for the GRID-SN6 field of 400 stars at various pixel
centerings, where each star has 50 counts, with a background level of 100 counts/pixel.
T able 3
SH A R P and R O U N D L imits fo r various S /N
S/N
SHARP LOW
SHARP HIGH
ROUND LOW
ROUND HIGH
270
73
22
12
6
0.75
0.74
0.68
0.60
0.50
1.00
1.01
1.05
1.10
1.25
-0.45
-0.50
-0.65
-1.00
-1.90
0.55
0.60
0.70
1.20
1.30
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149
T a b l e 4-1
P h o t o m e t r y f o r GEID-SN260 s t a r s , F555W, C e n t e r o f CCD
PSF technique/Centering o f primary PSF star
Mean (fit)
Mean (aper)
a
Mean CHI
0.001
0.000
0.005
0.962
0.005
-0.002
-0.084
-0.062
-0.040
-0.004
0.006
0.002
-0.001
-0.025
-0.018
-0.009
-0.005
-0.010
0.007
0.007
0.012
0.010
0.009
0.008
0.009
3.716
3.824
5.325
4.874
4.423
4.086
3.917
0.004
-0.012
-0.105
-0.071
-0.050
-0.008
0.006
0.001
-0.005
-0.032
-0.021
-0.012
-0.010
-0.010
0.007
0.008
0.013
0.010
0.010
0.009
0.009
3.604
3.789
5.769
5.044
4.516
4.328
3.866
0.004
-0.004
-0.097
-0.065
-0.041
-0.005
0.012
0.002
-0.002
-0.029
-0.019
-0.009
-0.005
-0.010
0.006
0.007
0.012
0.010
0.009
0.008
0.009
4.320
3.891
5.613
4.858
4.430
4.095
3.947
0.001
-0.001
-0.079
-0.051
-0.032
-0.008
0.002
0.000
-0.002
-0.023
-0.014
-0.007
-0.005
-0.009
0.007
0.007
0.012
0.010
0.009
0.007
0.008
3.013
3.089
4.588
3.970
3.531
3.275
3.106
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.008
0.009
0.008
0.007
0.008
0.008
0.008
2.776
3.100
2.979
2.840
2.945
2.904
2.006
0.000
0.009
5.019
‘0.018
5.858
P erfect P S F :
O ne G au ssian align ed w ith pixels:
(.05r05)
(.25,.25)
(.45, .45)
(.60, .90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
T w o G aussians align ed w ith pixels:
(.05,.05)
(.25, .25)
(.45,.45)
(.60, .90)
(.80, .70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
O ne G aussian w ith arbitrary orientation:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80, .70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat function:
(.05,.05)
(.25,.25)
(.45,.45)
(.60, .90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat fu n ctio n w ith n ew in terpolation:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
0.002
-0.002
-0.007
-0.003
0.000
0.001
0.003
P S F from 5 S / N 75 stars, O ne gaussian:
0.089
P S F from 9 S / N 75 stars, M offat + new interpolation:
0.008
0.000
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
T a b l e 4 -2
P h o t o m e t r y f o r GRID-SN260 s t a r s , F555W , F a r c o r n e r o f CCD
PSF technique/Centering of primary PSF star
Mean (fit)
Mean (aper)
a
Mean CHI
0.001
0.000
0.005
0.953
-0.004
-0.008
-0.068
-0.062
-0.020
0.003
-0.017
0.003
0.001
-0.017
-0.014
-0.005
-0.001
-0.008
0.007
0.008
0.011
0.010
0.009
0.008
0.009
4.258
4.339
5.414
5.201
4.718
4.607
4.303
-0.006
-0.025
-0.083
-0.070
-0.032
0.003
-0.015
0.001
-0.006
-0.022
-0.017
-0.009
-0.002
-0.012
0.007
0.009
0.011
0.010
0.009
0.008
0.010
4.043
4.501
5.781
5.412
4.950
4.577
4.500
0.008
-0.010
-0.073
-0.052
-0.031
0.007
0.005
0.007
0.000
-0.018
-0.012
-0.008
-0.001
-0.007
0.007
0.008
0.011
0.010
0.010
0.008
0.010
4.148
4.357
5.042
4.479
4.456
3.997
3.920
-0.007
-0.006
-0.051
-0.046
-0.015
-0.001
-0.022
0.001
0.000
-0.011
-0.009
-0.004
0.000
-0.007
0.007
0.007
0.009
0.009
0.008
0.007
0.008
3.999
4.033
4.603
4.476
4.217
4.140
3.884
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.008
0.008
0.008
0.008
0.008
0.008
0.007
2.872
2.944
2.976
2.925
2.901
2.880
1.809
0.012
•
new interpolation: »
-0.022
0.000
0.026
6.281
P e rfect P SF :
O ne G au ssian aligned w ith pixels:
(.05,.05)
(.25,.25)
(.45, .45)
(.60, .90)
(.80, .70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
T w o G aussians align ed w ith pixels:
(.05,.05)
(.25,.25)
(.45,.45)
(.60, .90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
O ne G au ssian w ith arbitrary orientation:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat function:
(.05,-05)
(.25,.25)
(.45,.45)
(.60, .90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat fu n ction w ith n ew interpolation:
(.05..05)
(-25,-25)
(.45,.45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
-0.001
-0.002
-0.005
-0.005
-0.003
-0.001
-0.004
P S F from 5 S / N 75 stars, One gaussian:
-0.050
P S F from 9 S / N 75 sta rs, M offat +
0.000
7.224
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
151
T a b l e 4 -3
P h o t o m e t r y f o r GRID-SN260 s t a r s , F785LP, C e n t e r o f CCD
PSF technique/Centering of primary PSF star
Mean (fit)
Mean (aper)
a
Mean CHI
0.001
0.000
0.005
0.939
0.029
-0.007
-0.071
-0.064
-0.058
0.065
0.025
0.015
0.007
-0.014
-0.011
-0.010
0.002
-0.004
0.008
0.012
0.016
0.015
0.015
0.013
0.014
6.831
7.043
7.710
7.709
7.747
7.182
7.544
0.027
-0.016
-0.078
-0.065
-0.058
0.064
0.025
0.014
0.003
-0.017
-0.011
-0.012
-0.001
-0.006
0.008
0.012
0.016
0.015
0.015
0.013
0.014
6.344
6.536
6.855
7.238
6.703
7.266
6.784
0.009
-0.008
-0.079
-0.066
-0.061
0.065
0.026
0.014
0.006
-0.017
-0.012
-0.011
0.002
-0.004
0.012
0.012
0.016
0.015
0.015
0.013
0.014
7.848
7.045
7.804
7.728
7.779
7.198
7.588
0.026
-0.009
-0.064
-0.055
-0.046
0.057
0.024
0.014
0.006
-0.012
-0.008
-0.008
0.002
-0.003
0.008
0.011
0.016
0.014
0.014
0.012
0.013
5.894
5.916
6.647
6.522
6.523
5.961
6.203
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.008
0.007
0.007
0.007
0.007
0.007
0.006
3.049
2.939
2.839
2.940
2.875
2.805
1.965
0.013
•
n ew interpolation: •
0.030
0.027
0.000
8.141
P erfect P S F :
O ne G aussian align ed w ith pixels:
(.05,.05)
(.25,.25)
(.45, .45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
T w o G aussians align ed w ith pixels:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80, .70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
O ne G aussian w ith arbitrary orien tation :
(.05,.05)
(.25,.25)
(.45,.45)
(.60, .90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat function:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat fu n ction w ith n ew in terp olation :
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80, .70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
0.004
-0.001
-0.008
-0.006
-0.006
0.000
-0.002
P S F from 5 S / N 7 5 sta rs, O ne gaussian:
0.047
P S F from 9 S / N 7 5 sta rs, M offat +
0.000
4.708
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
152
T able 4-4
P h otom etry f o r G R ID -SN 260 stars , F 7 8 5 L P , Fa r co rn er o f CCD
PSF technique/Centering of primary PSF star
Mean (fit)
Mean (aper)
a
Mean CHI
0.001
0.000
0.005
0.953
0.008
-0.038
-0.064
-0.019
-0.001
-0.027
0.081
0.007
-0.009
-0.023
-0.003
0.002
-0.002
0.001
0.008
0.011
0.013
0.010
0.009
0.009
0.010
6.881
7.071
6.878
7.037
6.985
7.091
7.193
0.005
-0.044
-0.062
-0.027
-0.009
-0.026
0.072
0.005
-0.013
-0.022
-0.007
-0.002
-0.004
-0.001
0.008
0.011
0.013
0.011
0.010
0.009
0.010
6.118
5.898
6.843
6.027
6.144
6.181
6.155
0.014
-0.042
-0.062
-0.020
-0.013
-0.028
0.086
0.007
-0.011
-0.022
-0.003
0.001
-0.002
0.001
0.009
0.011
0.013
0.010
0.010
0.010
0.011
7.380
7.099
6.882
6.981
6.985
7.061
7.202
0.004
-0.030
-0.049
-0.016
0.005
-0.008
-0.017
-0.003
0.001
-0.002
0.001
0.007
0.010
0.011
0.009
0.008
0.008
0.009
5.875
5.920
6.097
5.810
5.772
5.791
5.917
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.007
0.007
0.008
0.007
0.007
0.007
0.005
3.032
2.805
2.777
2.833
2J35
2.935
1.796
0.014
-0.007
0.000
P S F from 9 S / N 75 stars, M offat + new interpolation:
-0.022
0.018
0.000
7.806
P e rfect PSF:
O ne G aussian aligned w ith pixels:
(.05,.05)
(.25,.25)
(.45,.45)
(.60, .90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
T w o G aussians aligned w ith pixels:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
O ne G aussian w ith arbitrary orientation:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80, .70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
M offat function:
(.05,.05)
(.25,.25)
(.45,.45)
(.60,.90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
0.000
-0.018
0.063
M offat fu n ction w ith new in terpolation:
(.05,.05)
(.25,.25)
(.45,.45)
(.60..90)
(.80,.70)
all 5 stars above
9 stars on a grid spaced by 0.33 pixels
0.001
-0.005
-0.009
-n.nns
o.o’oo
0.000
0.000
P S F from 5 S /N 75 stars, O ne gaussian:
6.668
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
153
Figure 1A. Wide Field Camera, F555W
Magnified PSFs, Logarithmic Display, OTA Axis Towards Lower Left
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154
Figure 1B. Wide Field Camera, F785LP
Magnified PSFs, Logarithmic Display, OTA Axis Towards Lower Left
m mmm-
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
155
Figure 1C. Planetary Camera, F555W
Magnified PSFs, Logarithmic Display, OTA Axis Towards Lower Left
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156
Figure 1D. Planetary Camera, F785LP
Magnified PSFs, Logarithmic Display, OTA Axis Towards Lower Left
",
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R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
157
Figure 2. CROWD fields
Fields at various surface brightnesses: 25,24,23,22 mag/sq.arcse
1 1
IP
25
80
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
158
Figure 3. REAL23
Simulated WFC field at 23 mag/sq. arcsec
■ m f
25
80
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
159
Figure 4-1. F555W; PSF In center of GGD
i i i i | i i i i - | - r r i i | i i i i m I I '! ) | I I I I |"f I I I | I I I I j-
15.2
I
a) Standard DAOPHOT, perfect PSF
b) No weighting for interpolation errors, perfect PSF I
magnitude
15.1
15
14.9
mean (fit) = 0.007
a = 0.029
mean (lit) = 0.001
mean CHI = 0.180
14.8
^ | - H - h | - M -1 | - | - h
| | | | | I I j:j_ I I M
cr = 0.005
_____
mean CHI = 0.962
| I i I I-1 H
I I | I I I Ij
15.2
I
c) Standard DAOPHOT, measured PSF
d) No weighting for interpolation, m easured PSF,
old reweighting
magnitude
15.1
15
14.9
mean (fit) = 0.009
14.8 l_
15.2
a = 0.030
mean CHI = 0.360
if l-l-H - l—|-h- |- |--|-|—H H -
1- 1
mean (fit) = -0.018
mean CHI = 3.573
i- l -l - 1
I e) No weighting for interpolation, measured PSF,
new reweighting
15.1 I_
magnitude
a = 0.017
■H —H - | - I I I I I I I I I I I I I I j
f) Weighting at 0.1 standard, measured PSF,
new reweighting
15
14.9
mean (fit) = 0.006
14.8
a = 0.007
mean CHI = 3.728
mean (fit) = 0.012
' i ' i I i i i i I i ' i ' I i i '
100
200
Star number
300
cr= 0.019
mean CHI = 1.605
i i ' i I i i i i I i i i i iI i i i i
4000
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
160
Figure 4-2. F555W, PSF in far corner of CCD
15.2
^ I " I T"l | I I I I | I I I I | I I I I
I
~l i i I | ' l i i i | i l i i | i l l i j-
a) Standard DAOPHOT, perfect PSF
b) No weighting for interpolation errors, perfect PSF
magnitude
15.1
15
14.9
mean (fit) = 0.005
14.8
15.2
o = 0.028
mean CHI = 0.184
i
I I II | i I I I | I I I I | I I
I
c) Standard DAOPHOT, measured PSF
11
a = 0.005
mean CHI = 0.953
H I
Ip
15.1
magnitude
mean (fit) = 0.001
I | - 1 I I I | I I I I | I I I l j:
d) No weighting for interpolation, measured PSF,
old reweighting
15
14.9
mean (fit) = 0.002
14.8
15.2 t
a = 0.031
mean CHI = 0.395
mean (fit) = -0.048
mean CHI = 4.107
I I I I I I I I | I I I I | I I I I i t I I I I | I I I I | I I I I | I I I I j:
I e) No weighting for interpolation, measured PSF,
new reweighting
15.1 I_
magnitude
o = 0.024
I) Weighting at 0.1 standard, m easured PSF,
new reweightin
15
14.9
14.8
I I I
'
I i
100
cr= 0.009
mean CHI = 4.263
■ I i i r i I J I
200
300
Star number
mean (fit) = -0.054
mean (fit) =-0.004
1
L
a = 0.030
mean CHI = 1.722
M . l i l t L-i_J—i-i-L
I I I I I I I
4000
100
200
i i i i
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
161
]Figure 4-3. t-785LP, PSF in center of CCD
iiii| ii
15.2
I
-|-t.pr1rit
I
a) Standard DAOPHOT, perfect PSF
I
I
I
|
I
I
I
I
|
1 I' l " T ' | T
I
I 'I J .
b) No weighting for interpolation errors, perfect PSF
magnitude
15.1
15
14.9
mean (fit) = 0.005
14.8
o = 0.024
mean CHI = 0.214
-I -I-I I | I 'I I I | -| | | |
15.2
I
mean (fit) =» 0.000
it I I I I | I I I I | I I I I | I I I I J
d) No weighting for interpolation, measured PSF,
old reweighting
c) Standard DAOPHOT, measured PSF
15.1
magnitude
c =■ 0.005
mean CHI = 0.938
I
15
14.9
mean (fit)
14.8
15.2
mean (fit) = 0.016
I I I I I I I I I I I I I I I I I I I T f I I I I I I I I l I l"l"H —[-H
f) Weighting at 0.1 standard, measured PSF,
new reweighting
I e) No weighting for interpolation, measured PSF,
new reweighting
15.1 I_
magnitude
a = 0.010
mean CHI = 6.744
mean CHI = 0.502
15
14.9
mean (fit) = 0.030
14.8
'
i
'
I
100
■
i i
o = 0.008
mean CHI = 6.821
'
I i ■i i I i
200
Star number
300
4000
100
32.999
^
300
400
'i I i i i i I-
I I I t .I—l"
200
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
162
Figure 4-4. F785LP, PSF in far corner of CCD
15.2
^i i i i
| 11
r i i | i—
rrrp —
iii
a) Standard DAOPHOT, perfect PSF
I I" I
" | 'l F T T T * J ' " r " l ' 'I
I ~ | "I
I
I
I J-
b) No weighting for interpolation errors, perfect PSF I
magnitude
15.1
15
14.9
mean (fit) = 0.008
14.8
15.2
c = 0.027
m ean CHI = 0.219
]_\ I --i—t—|—I—f I I | I I I I | I I
)[Standard DAOPHOT, m easured PS
magnitude
15.1
mean (fit) = 0.000
a = 0.005
mean CHI = 0.953
I
M i l l
d) No weighting for interpolation, measured PSF,
old reweighting
15
14.9
= 0.074
mean CHI = 0.760
mean (fit)
14.8
magnitude
a = 0.015
mean CHI = 6.796
i I I I | I I I i | I I I I I I I I I
I I I I j f - l- l
e) No weighting for interpolation, measured PSF,
new reweighting
f) Weighting at 0.1 standard, measured PSF,
new reweighting
15.2
15.1 Z_
mean (fit) = -0.008
I I | I I I if
15
14.9
mean (fit) = 0.009
14.8
o = 0.008
mean CHI = 6.894
i i i I i i i i I i i i i
HI = 3.916
l i i i i ■i ■iT
400
Star number
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
163
Figure 5-1. F555W, PSF in center of CCD, standard DAOPHOT
■j I I I I | I I I T j
I I I I | I I I I |-(-|
15.2
I
a) Primary PSF star: (.05..05)
l l 'i l | l l l l [—i—i "i" I | I I I I j.
b) Primary PSF star: (.25,.25)
magnitude
15.1
15
14.9
14.8 I_
15.2
mean (fit) = 0.005 a = 0.007
mean (aper) = 0.002 mean CHI = 3.716
j_l I I I | I I I I | I I I I | I I I
I
c) Primary PSF star: (.45,.45)
mean (fit) = -0.002
mean (aper) = -0.001
■ it
II
a = 0.007
mean CHI = 3.824
I I I I | I I I I | I I I I | li
+ + i
d) Primary PSF star: (.60,.90)
magnitude
15.1
15
14.9
mean (fit) = -0.084
mean (aper) = -0.025
14.8
15.2
I
a = 0.012
mean CHI = 5.325
mean (fit) = -0.062
mean (aper) = -0.018
c = 0.010
mean CHI = 4.874
I I I I | I I H~| T I I I | I I I
M il
e) Primary PSF star: (.80,.70)
f) Simultaneous fit to all 5 PSF stars
| I I I I | I I I I | I I
magnitude
15.1
15
14.9
14.8
mean (fit) = -0.040
mean (aper) = -0.009
a = 0.009
mean CHI = 4.423
i i i i I i i i i I i i i i I i i i
100
200
Star number
300
mean (fit) = -0.004 a = 0.008
mean (aper) = -0.005 mean CHI = 4.086
i
■ i i » I ■» ' i I i ' i i I ' i
4000
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
i i
400
164
Figure 5-2. F555W, PSF in far corner of CCD, standard DAOPHOT
1 1 1
15.2
I
"I"
I I I I
1 1 1 1
I 'I 1‘ I'
II
a) Primary PSF star: (.05,.05)
i i i i
T '- r - f - r
i i i i
iiit
b) Primary PSF star: (.25,.25)
magnitude
15.1
15
14.9
14.8 I_
15.2
mean (fit) = -0.004
mean (aper) = 0.003
mean (fit) = -0.008
mean (aper) = 0.001
o = 0.007
mean CHI = 4.258
1 I I i l l I- Hi 1 11 I I I I 1
j j I I I | I I M | I M- 1-| - K
I
II
c) Primary PSF star: (.45,.45)
a = 0.008
mean CHI = 4.339
d) Primary PSF star: (.60,.90)
magnitude
15.1
15
14.9
mean (fit) = -0.068
mean (aper) = -0.017
14.8
15.2 E
I
mean (fit) = -0.062
mean (aper) = -0.014
a = 0.011
mean CHI = 5.414
l l l l
I I I I | -M l I | I I I I | I I I
II
e) Primary PSF star: (.80,.70)
a = 0.010
mean CHI = 5.201
I I I I
I I l I
I I I
f) Simultaneous fit to all 5 PSF stars
magnitude
15.1
15
14.9
14.8 _
mean (fit) = -0.020
mean (aper) = -0.005
mean (fit) = 0.003
mean (aper) = -0.001
a = 0.009
mean CHI = 4.718
a = 0.008
mean CHI = 4.607
T i i i i I i i i i I i i i i I i i i i T T i i i V T , i i i I ................... ■ ,~n
100
200
Star number
300
4000
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
165
Figure 5-3. F785LP, PSF in center, standard DAOPHOT
15.2
-| i i~T t 1
I
i i i 1 j 1 1 1 1 |
1 1 1 1 |- U^-| I T T
a) Primary PSF star: (.05,.05)
I I
p
r r i
| i i i i | i i "i r - j .
b) Primary PSF star: (.25,.25)
I
magnitude
15.1
------------------ - -
15
14.9
I
m ean (fit) = 0.029
14.8 !_ mean (aper) = 0.015
-I i i i i 1 t i i i
n
1 1 1
t i l 1
15.2
I
o = 0.008
m ean CHI = 6.831
1 i i i i 1 i i i i
| t 1 1 1 | I I 1 r
c) Primary PSF star: (.45,.45)
I I
' '
T- i
t J
I I
mean (fit) =-0.007 a - 0.012
I
mean (aper) = 0.007 mean CHI = 7.043
_I
11 11 11 11 1I I 1 11 1I I 1 1I 11 11 11 11 1I I 1 11 11 11 (•
U
d) Primary PSF star: (.60 ,.90)
I
magnitude
15.1
15
14.9
I
mean (fit) =-0.071
mean (aper) = -0.014
14.8 "
-I i i i i I i i i i
-j i 1 1 1 | 1 1 1 1
15.2
I
a = 0.016
m ean CHI = 7.710
I i i i i I i i i i
| 1 1 1 1 II I 1 t
e) Primary PSF star: (.80,.70)
II
mean (fit) =-0.064 o = 0.015
I
mean (aper) = -0.011 mean CHI = 7.709
*
r
- '■ I1 1I I1 I1 1I I I I1 1I I1 1 II I1 I1 I I 1 I1 I1 I1 I1 U
I I
f) Simultaneous fit to all 5 PSF stars
I
magnitude
15.1
15
14.9
mean (fit) = 0.065 a = 0.013
I
mean (fit) =-0.058 c = 0.015
II
“
mean
(aper)
0.002
mean
CHI
=
7.182
_I
"
mean
(aper)
=
-0.010
mean
CHI
=
7.747
~
14.8
T t i i i I i i i r I i i t i I i i i i T A 1 I ! 1 1 1 1 1 ! 1 1 ! 1 1 1 1 1 1 1 1100
200
300
400
100
200
300
4000
Star number
Star number
I
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
166
Figure 5-4. F785LP, PSF in far corner of CCD, standard DAOPHOT
15.2
0TI I I I I I I !
1
| I I I I I I I I I
|- H
a) Primary PSF star: (.05,.05)
I
I
I
I
|
I
I
I" I
[~ I
I
I
I
|
I
I
I
I
b) Primary PSF star: (.25,.25)
magnitude
15.1
15
14.9
14.8 L.
15.2
mean (fit) = 0.008
mean (aper) = 0.007
c> = 0.008
mean CHI = 6.881
j j I I I | I -I-I' l"| i l l - I |-i i
c) Primary PSF star: (.45,.45)
11
_ r_
mean (fit) = -0.038
mean (aper) = -0.009
o = 0.011
mean CHI = 7.071
-jjjj l- i -I I |- l l |-+ -|- | | l | | | | | | j:
II
d) Primary PSF star: (.60,.90)
magnitude
15.1
15
14.9
mean (fit) = -0.064
mean (aper) = -0.023
14.8
| | | | I
a = 0.013
mean CHI = 6.878
I I I I | I I I I | I I I I | I I I IJ
15.2
I
mean (fit) = -0.019 a = 0.010
mean (aper) = -0.003 mean CHI = 7.037
e) Primary PSF star: (.80,.70)
0 Simultaneous fit to all 5 PSF stars
magnitude
15.1
15
14.9
14.8
mean (fit) = -0.001
mean (aper) = 0.002
o = 0.009
mean CHI = 6.985
i i i I i i i i I i ' i i I t i i
100
200
Star number
300
mean (fit) = -0.027 a = 0.009
mean (aper) = -0.002 mean CHI = 7.091
iM . i i t i I < i ' i I ' i i i I i i
300
4000
100
200
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
i
JE
400
167
Figure 6. C ro ss sectio n of convolved P S F
25000 —
2 0 0 0 0
J 5000 —
'1 0 0 0 0
5000
-4
-2
0
2
D istan ce in pixels
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
168
Figure 7. SN73 stars reduced with SN270 PSF, F555W (left) and F785LP (right)
-| I I I I j I I t I "' [
f ) t - 1 "I I | |
-I I I I | I I I I ' l i l t I I l l l
17.7
F555W, Perfect PSF
F785LP, Perfect PSF
magnitude
17.6
17.5
17.4
mean (fit) = 0.003
17.3
17.7
t
mean (fit) = 0.002
a = 0.017
a = 0.020
mean CHI = 0.986
mean CHI = 0.973
I I I I | I I I I | I I I I | I I I I it - | |
F555W, PSF from 5 SN260 stars, 1 gaussian
11
|.
| | | | | f;
F785LP, PSF from 5 SN260 stars, 1 gaussian
magnitude
17.6
17.5
17.4
mean (fit) =-0.001
17.3
17.7
mean (fit) - C.060
a = 0.018
mean CHI = 1.349
a = 0.040
mean CHI = 1.671
j - l -l-l I | I I I I |-I-1 I- I | I I I-1F555W, PSF from 9 SN260 stars, Moffat + new interf£ I
I I I I I I I I -I •I -i- l
F785LP, PSF from 9 SN260 stars, Moffat + new interp
magnitude
17.6
17.5
17.4
mean (fit) = 0.006
17.3
mean (fit) = -0.008
c = 0.020
mean CHI = 1.058
a = 0.021
mean CHI = 1.044
i i - i i I' i i i i I i i i i I i i t i M . i i i i I i i i i I i i i i I i i i iJ
100
200
Star number
300
4000
100
200
Star number
300
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
169
Figure 8. SN12 stars reduced with SN270 PSF, F555W (left) and F785LP (right)
J I I I I | I I IT i r i i - r r~ r f t n r i • i i i ••[ i i i i | i r f t 1l " l" T T~
magnitude
20.5
20
mean (fit) = 0.032
19.5
magnitude
20.5
mean (fit) = 0.038
a = 0.110
I
mean CHI = 0.982
]
1
I
1
I | I I I I | | H -|- | - h -
F555W, PSF from 5 SN260 stars, 1 gaussian
-f] i i i i | i i i
_J_
a = 0.132
mean CHI = 0.977
i - | - i-
1" H - | - h -
F785LP, PSF from 5 SN260 sta^p, 1 gaussian
20
mean (fit) = 0.030
19.5
mean (fit) = 0.094
a = 0.110
mean CHI = 0.995
+
3 -+ +
20.5
magnitude
F785LP, Perfect PSF
F555W, Perfect PSF
II
I ' ' [}
F555W, PSF from 9 SN260 stars, Moffat + new interp_ _
c = 0.132
mean CHI
___
.005
H
i I | I I I I | I
F785LP, PSF ffibm 9 SN260 stars, Moffat + new interp
20
19.5
mean (fit) = 0.035
I I I
I l -l....I
100
mean (fit) = 0.024
a = 0.111
rnean CHI = 0.984
_1_
i i i I i ''i M
200
Star number
300
4000
or = 0.132
mean CHI = 0.980
i i i i I i i i i I i i i i I i i i i
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
170
15.2
Figure 9A. Photometry of GRID-SN270 with various PSFs
I I I I "I I I I I | ! I I I | I I I I
£ i i i i | i i i i | i r' i i | i i i i
: I
F555W, Perfect PSF
F785LP, Perfect PSF
magnitude
15.1
15
m
m
m
14.9
mean (fit) = 0.001
jr
<j = 0.005
mean CHI = 0.939
mean CHI = 0.962
14.8
15.2
mean (fit) = 0.001
a = 0.005
I I I I | l-f I I | I I I I | I I I I Jj
l-t- l I I II I- I I I I I l l
F555W, PSF from 9 SN260 stare, Moffat + new inteijL I
F785LP, PSF from 9 SN260 stare, Moffat + new interp
magnitude
15.1
15
14.9
mean (fit) = 0.003
14.8
15.2
mean (fit) = -0.002
o = 0.008
mean CHI = 2.006
mean CHI = 1.965
I I I | l- H - l I I I I I I I I
f I l i -l | I I I I | I I I I | I I ■HF555W, PSF from B-V=2.0 for B-V=0.0 stars
a = 0.006
: :
F785LP, PSF from B-V=2.0 for B-V=0.0 stars
magnitude
15.1
15
m m m r
14.9
mean (fit) = 0.005
14.8
200
Star number
300
a = 0.005
mean CHI = 1.233
i i i I i i i i I i i i i I i i i iT
i i i i
i ' i i I i i i i I ' ' i '
100
mean (fit) = 0.045
a = 0.005
mean CHI = 1.284
4000
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
171
Figure 9B. CHI for photometry of GRID-SN270 with various PSFs
j
i i i i I i i i t | I i i I | i I I r
I
F555W, Perfect PSF
LT l I i i | i i i r“ | i i t i | i i—i—p
magnitude
F785LP, Perfect PSF
iTTTTTTnTTTTiHiTi^
} 1 1 I I | I I I I | 1 I i i- | h -h
i
.
F785LP, PSF from 9 SN260 stars, Moffat + new interp
magnitude
F555W, PSF from 9 SN260 stars, Moffat + new interp..
} l l I I | I i I I | 1 I II | I I I l f
F555W, PSF from B-V=2.0 for B-V=0.0 stars
I I I I
I I I I
I I I I ■■
F785LP, PSF from B-V=2.0 for B-V=0.0 stars
magnitude
I
I I I
WMw
“
“ —© C3 v CjCj 13*4
Hi til | OHffl
i i i i i i i i t i i i i i i i i i i i rti i 1 1 1 1 1 1 1 1 1 1 1 1 1
. . r
100
200
Star number
300
4000
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
172
Figure 10A. Photometry of FIELD-SN270 with various PSFs
15.2
jpi i i i [ i i i i | i i i t | i i i r j jj~i i i i | i i i i | i i i i | i i i i j.
I
F555W, Perfect PSF
F785LP, Perfect PSF
magnitude
15.1
15
14.9
mean (fit) = 0.001
14.8 _
15.2
jH"
I
1 1 1
a = 0.005
mean CHI = 0.952
mean (fit) = 0.000
I I I I I | I I I I | I I I I
F555W, PSF from center of frame
it
11
a = 0.004
mean CHI = 0.951
- h f - l- H - H -
-i
F785LP, PSF from center of frame
magnitude
15.1
15
14.9
mean (fit) = 0.003
a = 0.006
If
mean (fit) = 0.004
mean CHI = 2.225
14.8
-l- l I -1 i I I I I I I I
15.2
I
mean CHi = 2.029
i t
F555W, average PSF
a = 0.006
11
I I [ I I M | I I I I | I I I I]
F785LP, average PSF
magnitude
15.1
15
14.9
mean (fit) = -0.006
14.8
a = 0.007
mean CHI = 2.828
| |
mean (fit) = -0.005 a = 0.009
mean CHI = 2.795
i ' i i I ' i i i I i i i i I i i i i M-l i i i iI i i i i I i i i i I i i i i I100
200
Star number
300
4000
100
200
Star number
300
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
173
Figure 10B. CHI for photometry of FIELD-SN270 with various PSFs
i
i
i
j
i
l
I
l
|
I
I
I
l
|
l
I
I
l
_
F555W, Perfect PSF
F785LP, Perfect PSF
1
| | | | | I
Ii IIIi III| Ii i
J555W , PSF from center of frame
F785LP, PSF from center of frame
magnitude
magnitude
i
i i t i I i i i i I i i i i I i t i i
j
} l M il I
I I
F785LP, average PSF
F555W, average PSF
magnitude
_
1
100
200
Star number
300
100
200
Star number
300
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
174
Figure 11 A. Photometry of GP.ID-SN270 with various PSFs
-| I I I I | I I I T J T 'I I I |
I I I I
-i
i
i
i
| 1r
i " t - i— j- H
If
- r " i
t l
15.2
F555W, Perfect PSF
F785LP, Perfect PSF
magnitude
15.1
15
14.9
mean (fit) = 0.001
14.8
15.2
mean (fit) = 0.001
a = 0.005
mean CHI = 0.962
tI I I I | I I I I | I I I I | I I I I
I
F555W, Using jitter=20mas PSF for 7mas stars
a = 0.005
mean CHI = 0.939
T
M i l l I l- l - ■I I -M | I I I I [
11
F785LP, Using jit!er=20mas PSF for 7mas stars
magnitude
15.1
15
14.9
mean (fit) = -0.015
14.8
15.2
E
I
mean (fit) = -0.015
a = 0.010
mean CHI = 3.265
H - H - |—H -1
1■| I I I- l- f - H H -
F555W, Using off focus PSF
a = 0.011
mean CHI = 2.671
I I I | I I I I
3F
11
H - H - Tf
F785LP, Using off focus PSF
magnitude
15.1
15
i'm m
14.9
mean (fit) = 0.007
14.8
mean (fit) = 0.004
a = 0.006
a = 0.005
mean CHI = 2.486
mean CHI = 4.646
i i i i I i i i ' I i i i ■ I i i i i M . i ' ' i I i i i i I i i ' M i i i i I100
200
Star number
300
4000
100
200
Star number
300
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
400
175
Figure 11B. CHI for photometry of GRID-SN270 with various PSFs
j i i i i | i i i "i |~ f l i t *|""T t ~i~r UJ~i—i~r i | i i t i | i
F555W, Perfect PSF
'i
'i 1 1 1 L
F785LP, Perfect PSF
magnitude
I
itt
} I I I I | I l- i I | I I
I | I I I I
F555W, Using JI20 PSF
H -h + i
. _
F785LP, Using Ji20 PSF
magnitude
.
1
] - F+ + + -|- H - l"l | I I I I | l- l -1- 4 } I I I
F555W, Using wfcb2 PSF
|...| | | l | | | -f
F785LP, Using wfcb2 PSF
magnitude
.
t-j —
j—|—h —
1 i i i i I i i i i I i i i i I i i i i ITI i i i
100
200
Star number
300
4000
t I i i i i I » i i r I i i t
100
200
300
Star number
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
. ud
400
176
Figure 12A. Photometry of CROWD25 field, F555W
001
002
001
003
009
029
046 n(ref)
n— i— i—
i— 1i— i— r■ — 1— ■i— 1— Ni ^
1
,gl old
p N fitting
f ttf in /t
a) Perfect PSFjold grouping!
.5
—T
1
CPU: 113 =
0
-.5
001
-1
0.000
001
-0.017
-0.004
0.012
003
0.061
0.050
0.000
009
-0.025
0.113
0.078
0.350
pj*4RM<id)
40 o
erfect PSF, old grouping', new fitting
.5
CPU: 105
0
-.5
001
-1
0.000
0.012
001
-0.017
002
.
-0.005
003
0.060
0.047
0.
0.
0.087
0356
0348 o
f
E
c) Perfect PSF, new grouping, old fitting
.5
0.025
0.113
CPU:
74
0
-.5
.5
002
001
-1 r-j
E
I
|
0.012
0.000
|
-0.004
0.004
003
0.061
0.Q50
001
|
-0.018
0.000
009
-0.025
0.115
.
0.080
0351
^U.d4<M4O« n hed)
d) centered DAOPHOT PS f ! old grouping ar
and fitting
CPU:
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<
66 :
0
-.5
001
0.015
.5
001
-0.019
o.qoo ^
o.qoo
-1
003
-0.009
',UU9
0-144
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e)lew iinterpolation PSF, old grouping and fitting
009
-0.059
0.111
0.087
. O.T~'
0.340 0 .
—
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0
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-1
I
|
001
0.013
0.000
002
|
-0.009
0.013
|
001
-0.013
0.000
|
003
0.013
‘ 171
0.1
009
-0.019
0.129
,
0.088
0324
) Perfect PSF for B-V=2, ola grouping ana fitting
o l s miffed):
0.391 a 1
|
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i
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0.021
0.000
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0.007
0.001
001
-0.003
0.
003
0.012
0.123
-0.012
0.121
02
0.088
0.306
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0
001
0.014
o.qoo
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-0.003
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0.106
0.129
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0-195 0 ,__
26
magnitude
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
177
Figure 12B. Photometry of CROWD24 field, F555W
001
.5
0 c-
-.5
-1
.5
002
006
004
007
023
073_______ 116 n(rel)
T
a) Perfect PSF, old grouping1, old fitting
CPU: 2152=
-st­
ool
0.014
o.qoo
0.000
|.
002
0.017
o.qi9
0.019
|
006
0.008
0.032
0.q32
|
004
-0.005
0.031
|
007
-0.007
0.044
0.q44
022
|
0.021
0.175
|
0.086
OS294
o ls fM e d )!
" 224 o ,
b) Perfect PSF, old grouping! new fitting
CPU: 1023:
0
-.5
-1
.5
001
0.014
o.qoo
0.000
002
|,
0.016
o .q is
0.018
|
006
0.008
0.032
0.q32
|
004
-0.005
o.q3i
0.031
^|
007
-0.008
o.q43
0.04:
022
0.023
0.177
|
0.085
0.3289
o la R M f)!
220 a |
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c) Perfect PSF.'new grouping, old fitting
CPU: 391 :
0
-.5
-1
.5
001
0.014
o.ijoo
002
,
0.015
o .q
.018
is
|
006
0.008
0.026
0 .q26
|
004
-0.006
0.031
o.q3i
|
021
0.008
007
-0.009
0.044
189
O.T
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0075
0.280
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si--------- 1-----H
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-1
.5
001
-0.025
0072
-1
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0
-.5
-1
.5
,
007
-0.029
0.q65
022
0.014
0.166
doo O |
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e)kew interpolation PSF, old grouping and fitting
B
0
-.5
004
-0.009
00 3 0
006
-0.030
o.q
002
0.013
o.qoo
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001
0.020
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002
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0.019
0056
|
006
0.003
0031
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004
0.001
0.031
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007
|
0.001
' q48
0048
j
022
0.001
)254
|| 6 at ,t C
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0.174
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for B-V=2, old grouping ana fitting
o
H
001
0.015
o.qoo
0.000
j —.
002
|,
0.025
o.qi6
0016
|
006
0.019
00023
023
CPU: 2145:
B~
j
004
0.009
0031
0.031
|
007
0.009
0.04'
0044
022
0.043
0.171
0.097
0.300
ollsRWri'
v0.214
o_|_
g) Perfect PSF, fitting for sky, old grouping and fitting
CPU: 2525:
■. ■- a —
0
-.5
-1
001
0.008
o.qoo
22
002
,
0.014
o.qi3
23
006
,
0.010
0.Q23
24
,
004
-0.001
o.q29
007
-0.007
Q.q44
022
0.029
0.173
0.088
0.285
28
25
0 IJ 4 M
s.0.323 o ,
29
magnitude
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
30
178
Figure 12C. Photometry of CROWD23 field, F555W
292 n(ref)
a) Perfect PSF,'old grouping, old fitting
CPU: 6392=
002
0.022
0.013
.
005
0.001
0.024
0.152
046
0.084
0.234
0
0.232
0.335
0 598 »
b/ C
hed)
erfect PSF,'old groupin
Amag
CPU: 2119
S
002
0.022
0.013
|
005
0.000
0.024
015
0.016
0.095
016
0.074
0.182
008
0.019
0.069
044
0.078
0.231
0
0.273
0.327
0.091 c |
c) Perfect PSF,'new grouping, old fitting
□
CPU: 871
Amag
I
002
0.025
0.017
.
005
-0.001
0.021
015
0.012
0.091
008
0.025
0.072
.
017
0.046
0.160
□
041
0.097
0.
0.307
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ntered DAOPHOT PSF&old grouping and fitting
Amag
CPU: 1150(7
015
0.007
0.169
—r -----1------ 7------ 1----- T----1—
046
0.078
0.248
0.254 \
0.327 -
0.596 frfeSan
a
'
n .0.122
---------T
e) new interpolation PSF, old grouping and fitting
□
Amag
CPU: 5046:
002
0.113
0.113
.
004
0.002
0.008
015
0.014
0.123
008
0.033
017
0.059
0.161
0.
044
0.113
0.219
07
0.264
0.
f) Perfect PSF for B-V=2, old grouping ana fitting
Amag
CPU: 6304:
□
005
0.016
002
0.029
0.016
0.Q19
015
0.028
0.095
008
0.027
0.075
017
0.060
0.186
□
046
0.111
0.253
0.242
30 a |
g) Perfect PSF,'fitting for sky, old grouping and fitting
CPU: 6703:
Amag
D
002
0.012
0.q25
005
,
-
0.001
0.qi3
017
0.055
0.166
015
0.000
0.q40
25
047
0.042
0.346
0
0.225
0.314
011
0.
0.200 a
26
magnitude
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
179
Figure 12D. Photometry of CROWD22 field, F555W
I
----- 1--------
1--£
a) Perfect PSF,'old grouplngL old fitting
□
a
g
o
CPU: 2207=
B
006
0.053
0.161
012
0.039
0.119
016
0.083
0.a04
0.081
0.138
cm
0.124
0.352
.
0J492
0.610
..
b) Perfect PSF, old grouping^ new fitting
O)
CPU: 1795=
<
006
0.053
0.161
.
012
0.045
0.114
016
0.079
0.195
0.077
0.125
,
cm
0.093
0.254
0.346
0.607
c) Perfect PSF.new grouping, old fitting
006
0.045
n 117
□
H
012
0.045
n in o
034
0.061
n i^ c
a
CPU: 1416
0.092
0.195
f t O C vl
056
ft0.460
io n
0.567
d) centered DAOPHOT PSF&old grouping and fitting
CPU: 4086=
0
004
0.070
0.055
|
012
0.036
0.048
|
035
0.046
0.144
|
035
0.156
016
0.059
0.222
0
086
0.478
0.545
m matched) z
median . =
e) new interpolation PSF, o la grouping aigi fitti
CPU: 4055=
006
0.042
0.117
,
012
0.039
0.031
,
035
0.074
0.130
,
019
0.107
0.ai9
01517
0.657
32
f) Perfect PSF ror B-V=2, oliggrouping aria fitting
□
006
0.061
0.140
012
0.055
0.025
□
□
0.100
0.182
018
0.124
0.ai7
CPU: 1182=
034
0.150
0.a04
058
0.505
0.302
01
0.
0.344
nfmatched)
median . '
a
g) Perfect PSF/fitting for sky, old grouping and fitting
CPU: 4198=
006
0.010
0.q65
,
012
0.006
0.q25
035
0.015
0J05
018
0.009
0.323
25
033
0.012
0.184
-
0.317
0.514
0-389
26
magnitude
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
180
Figure 13A. Photometry of CROWD25 field, F785LP
001
002
001
003
009
029
046 n(ref)
003
0.083
0.164
009
018\
007
0.020
0.113
0 .200
\0 .2 2 6 o
a) Perfect PSF, old grouping, old fitting
S’
CO
E
<
001
0.017
o.qoo
002
0.007
o.qo6
001
0.012
o.qoo
0.181
d) centered DAOPHOT PSF, old grouping and fitting
CD
E
<
001
002
001
003
0.004
o.qoo
0.020
-0.004
0.122
009
-0.015
007
0.q48
o.qoo
0.152
0.182
0.201
009
018
0.027
v0.238o
e) new interpolation PSF, old grouping and fitting
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E
<
-.5
0.001
001
0.001
o.qoi
o.qoo
24
25
002
001
0.004
o.qoo
22
23
-
003
007
n(m atchet)
0.198
26
27
28
29
magnitude
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
30
181
Figure 13B. Photometry of CROWD24 field, F785LP
001
OCS
006
004
007
023
073
116 n(ref)
007
-0.031
0.q79
020
009
011
a) Perfect PSF, old grouping, old fitting
n?
E
<1
-.5
001
0.001
o.qoo
002
0.017
0.Q30
006
0.019
004
0.015
o.q is
o.qi4
n(matche«)
,0.269a
0.141
d) centered DAOPHOT PSF, old grouping and fitting
o>
(O
E
<
-.5
001
002
006
-0.007
-
0.001
0.012
o.qoo
o.q3i
0.q33
004
-
0.011
0.q25
007
-0.047
0.q75
0.032
0TO\ 010
0.185\ 0.64sfH
3.302 \0.236a
020
039
020
e) new interpolation PSF, old grouping and fitting
O
)
ra
E
<
-.5
001
002
-0.004
o.qoo
0.007
0.q25
o.qi3
o.qis
22
23
24
25
006
0.009
004
-
0.012
007
-0.047
0.q83
26
magnitude
ledian
0.138
27
012
|7 2 a
28
29
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
30
182
Figure 13C. Photometry of CROWD23 field, F785LP
002
005
015
009
018
058
184
292 n(ref)
a) Perfect PSF, old grouping, old fitting
-.5
002
005
-0.007
0.Q37
0.034
0.019
015
008
003
017
0.000
0.053
0.074
0.397
p .1 6 9 o
d) centered DAOPHOT PSF, old grouping gnd fitting
(o>
U
E
<3
-.5
002
004
0.147
0.135
0.012
0.023
015
0.018
0.167
008
0.066
0.092
016
0.069
0.187
001
039
0.279
0.340
041
0.158
0.281
04 6 \
0.428
0.354
27
28
n (m a tc h « )
\0.q00o
e) new interpolation PSF, old grouping ancyitting
(o
E
<
-.5
002
0.016
0.018
22
005
0.007
015
-0.007
0.^39
0 .q82
23
24
25
-
0.020
008
0.004
017
0.047
0.195
26
magnitude
n(matche<)
median _
29
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
30
183
Figure 13D. Photometry of CROWD22 field, F785LP
006
■r ~ r
o ft
037
023
046
463
035
0d23
0.373
0.518
0.209
n(ref)
'— I— '— I— '— I— 1— nr
a) Perfect PSF, old gripping, otctfitting
.5
Amag
147
□
I
Q
012
0.042
0.103
103
033
0.095
0.192
016
0.103
0.202
0.2
1
0
.5
005
0.051
0.389
-1
I
'
I
'
0.339
n(matchet)
median I
I
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Amag
.5
□
0
.5
006
0.068
0.137
1-137
-1
012
0.065
0.125
0.125
I' T ' 1
032
0.087
0.209
0.2I
016
030
0.158
0.
n(matche<)
median _
0.228
o.:
a
+
—
e) new interpolation PSF, old grouping ancfcfitting
0
.5
-1
—
□
□
B
=
1
=
□
005
0.027
° f 7
22
012
0.027
,
° T
23
,
033
0.052
°--p
24
017
0.118
(
i P
00
; °
Amag
.5
25
029
OdOO
0.^17
n(matchec)
median
0.232
o
_L
26
magnitude
R ep ro d u ced with p erm ission of the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
30
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