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Deciphering Precision Cosmic Microwave Background Data

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Dechiphering Precision Cosmic Microwave Background Data
By
MARIUS MILLEA
M.S. (University of California, Davis) 2015
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in
Physics
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Approved:
Prof. Lloyd Knox, Chair
Prof. Andreas Albrecht
Prof. Chris Fassnacht
2015
i
Pro Q ue st Num b e r: 3723682
All rig hts re se rve d
INFO RMATIO N TO ALL USERS
The q ua lity o f this re p ro d uc tio n is d e p e nd e nt up o n the q ua lity o f the c o p y sub m itte d .
In the unlike ly e ve nt tha t the a utho r d id no t se nd a c o m p le te m a nusc rip t
a nd the re a re m issing p a g e s, the se will b e no te d . Also , if m a te ria l ha d to b e re m o ve d ,
a no te will ind ic a te the d e le tio n.
Pro Q ue st 3723682
Pub lishe d b y Pro Q ue st LLC (2015). Co p yrig ht o f the Disse rta tio n is he ld b y the Autho r.
All rig hts re se rve d .
This wo rk is p ro te c te d a g a inst una utho rize d c o p ying und e r Title 17, Unite d Sta te s Co d e
Mic ro fo rm Ed itio n © Pro Q ue st LLC.
Pro Q ue st LLC.
789 Ea st Eise nho we r Pa rkwa y
P.O . Bo x 1346
Ann Arb o r, MI 48106 - 1346
Abstract
This thesis centers around work which was performed in the lead up to and during analysis of
high resolution cosmic microwave background (CMB) data coming from the South Pole Telescope
and from the Planck satellite. Analysis of such data necessitates modeling of extra-galactic foregrounds, and here we develop a model for these foregrounds for use with current data. This model is
shown to be both necessary and sufficient for unbiased estimation of cosmological parameters. Additionally, we develop a novel method for using auxiliary data in the form of galaxy number counts
to put priors on certain components of this model. Finally, we perform an analysis of CMB data,
taking into account these extra-galactic foregrounds, to calculate new constraints on hypothetical
particles like axions or axion-like particles. We find new data from Planck and from measurements
of primordial deuterium newly rule out some theoretically interesting regions of parameter space.
In addition, we show that in a more speculative scenario which includes the presence of both axions
and other dark radiation, some of these regions are again allowed and could explain hints of high
primordial helium fraction, or could allow one or two thermalized sterile neutrinos to exist.
ii
Acknowledgments
Thanks to Lloyd Knox, who was the best advisor anyone could ask for. I couldn’t be where I
am today without your help, and I hope to make you proud with where I am going. Thanks to
my friends and family who supported me throughout this journey and made sacrifices so I could
be here. Finally, thanks to the South Pole Telescope and Planck collaborations for allowing me to
join and work with such cutting edge data, and for the chance to give many talks on these results.
iii
Contents
Abstract
ii
Acknowledgments
iii
List of Figures
vii
List of Tables
xii
Chapter 1.
1.1.
1
Theoretical Background
Chapter 2.
2.1.
Introduction
3
Modeling Extragalactic Foregrounds
Modeling
2.1.1.
6
9
Emission from External Galaxies
9
2.1.1.1.
Radio Galaxies
10
2.1.1.2.
Dusty Star-Forming Galaxies
10
2.1.1.3.
Polarization
12
2.1.2.
Thermal SZ Effect
12
2.1.3.
Kinetic SZ Effect
16
2.1.3.1.
Ostriker-Vishniac Effect
16
2.1.3.2.
Patchy Reionization
18
2.1.4.
tSZ-DSFG Correlation
19
2.1.5.
Galactic Foregrounds
20
2.1.6.
CMB
20
2.2.
Fiducial Model and Current Constraints
21
2.3.
Survey Properties
22
2.4.
Forecasting Methodology
24
2.4.1.
2.5.
Non-Gaussianity
26
Compression to a CMB Power Spectrum Estimate
iv
28
2.5.1.
Splitting the power spectra into CMB–free and a CMB estimate
28
2.5.2.
Modeling the foreground residuals with principal components
31
2.5.3.
Discussion of linear combination analysis
31
2.6.
Results
32
2.6.1.
Importance of the Different Components
33
2.6.2.
Modeling Sufficiency
35
2.6.3.
Statistical Error Increase with and without Auxiliary Data
37
2.7.
Conclusions
Chapter 3.
38
Predicting Poisson Foreground Amplitudes
40
3.1.
Methodology
40
3.2.
Source Count Modeling
41
3.3.
Results
43
3.4.
Conclusions
44
Chapter 4.
New Constriants on Axions and Axion-like Particles
48
4.1.
The Scenario
49
4.2.
Constraints
55
4.2.1.
Cosmic Microwave Background
55
4.2.1.1.
Frequency Spectrum
55
4.2.1.2.
Angular Power Spectrum
60
4.2.2.
Primordial Abundance Inferences
61
4.2.3.
Laboratory
66
4.2.4.
Globular Clusters and SN1987A
66
4.3.
Discussion
67
4.3.1.
The MeV-ALP Window
67
4.3.2.
A Loophole in the Presence of Extra Radiation
70
4.3.3.
A Simple Expression for Exclusion Bounds
71
4.3.4.
Forecasts
72
Conclusion
73
4.4.
Chapter 5.
Conclusion
75
v
Appendix A.
Principal Component Analysis for Power Spectra
76
Appendix B.
CMB Linear Combination Generalization to Off-Diagonal Correlations
77
Bibliography
79
vi
List of Figures
2.1 Three model DSFG clustering auto-spectra at 217 GHz (black), and approximations to
them with our parameterized model (solid, red), all normalized (with one exception) at
= 3000. Our fiducial model is the thickest curve. Also plotted are estimates of the
clustering power from Planck [Planck Collaboration, 2011r] and SPT [Hall et al., 2010].
For both sets of data points we have subtracted estimates of the Poisson power from
the reported total CIB power. The lowest amplitude solid (red) curve is the result of a
“by-hand” adjustment of our model parameters to fit the Planck and SPT data.
13
2.2 Comparison between recent models and simulations of the tSZ effect (black lines) and fits
of our PCA model to each (thin red lines). The thickest red line shows the fiducial tSZ
power spectrum used in this work. All results are plotted at 146 GHz and are scaled to
σ8 = 0.8. The blue arrow shows the SPT 95% confidence upper limit on thermal SZ power
at = 3000 [Shirokoff et al., 2010].
15
2.3 Comparison between recent simulations of the kSZ effect (black lines) and our model.
We consider contributions from the post-reionization kSZ effect (solid red line) and from
patchy reionization (dashed red line). Note that the simulations plotted here assume
homogeneous reionzation and thus do not include a patchy contribution. The SPT 95%
confidence upper limit for the kSZ power at = 3000 is 6.5 μK2 .
16
2.4 All 36 power spectra which can be formed from Planck 70 GHz–217 GHz temperature and
E-mode polarization, and the prediction of our fiducial model for the CMB and foreground
power in each of them (with the exception of the tSZ-DSFG correlation which is shown at
30% instead of its fiducial value of 0%). The black dashed line shows the errors bars for
-bins of width Δ = 256. Dotted lines indicate negative power.
2.5 Top: The -dependent weightings which form the CMB linear combination (Eq. 2.29). All
possible auto/cross spectra from Planck channels in Table 2.2 were considered. Dashed
lines indicate negative weight. Middle: The mean foreground contribution to the CMB
vii
25
linear combination for our fiducial model. Note for example that tSZ (purple) is not present
at high because only 217 GHz is used there. The dominant non-Poisson component for
the -range where Planck is most sensitive is the DFSG clustering. Bottom: Principal
components of foreground residuals (constrained by the CMB–free linear combinations)
with amplitudes set to 1-σ. Note that we only need two principal component amplitudes
to be accurate to > 1μK2 . (The errors in bin widths of Δ = 256 for both the CMB linear
combination and for 217 GHz alone are plotted as dashed lines in the bottom two plots.)
29
2.6 68% (and 95% in the bottom panel) confidence contours for a suite of test cases examining
the effect of neglecting to model different foregrounds. Unless explicitly stated above,
other parameters were included in the data at their fiducial values listed in Table 2.3 and
were marginalized over in the analysis. N corresponds to the maximum number of -bins
per power spectrum one could use and still detect the error in modeling at 3 sigma (see
Sec. 2.6.1 for further discussion).
34
2.7 The effect on cosmological parameters from trying to fit our model to simulated data which
includes (orange) the Battaglia et al. [2010] tSZ template and (black) the Amblard and
Cooray [2007] clustering template. These two models are the most dissimilar to ours, and
thus show our model can protect against biases of a few percent up to Planck sensitivity.
The inclusion of Ground data necessitates more detailed modeling of only the clustering.
36
3.1 The blue points with error bars show the M13 source count data, and the line drawn
through them is our best fit model integrated down to individual frequencies. At 150 GHz
we also show the dusty sources in green, as these are used in calculating the dusty
Poisson contribution there (see text). The lines at the bottom of each plot show the
fractional contribution from each logarithmic flux bin to the Planck radio Poisson power.
Dotted/dashed/dot-dashed lines are for the contribution at 100, 143, and 217 GHz. For
example, the dotted line in the 220 GHz plot corresponds to the contribution to the 100 GHz
Poisson power from sources with those particular 220 GHz fluxes. The normalization is
arbitrarily. The best-fit model is used to calculate these curves.
3.2 Posteriors on radio Poisson amplitudes at 100 and 143 GHz from LCDM chains when the
data used is (blue) Planck2013+WP (green) Planck2013+WP+highL. These posteriors
are nearly unchanged for extended models. The prior from this procedure is in (black). To
viii
45
arrive at the radio contribution in the 2013 chains, 143 GHz has had the expected dusty
contribution of (8 ± 2)μK2 subtracted from the total Poisson amplitude (see text).
45
3.3 (Top row) Slices through the joint incompleteness function, I(S100 , S143 , S217 ). Each
column shows the incompleteness for S143 vs. S217 , at a fixed value of S100 , given in
the label above the plot. (Bottom row) The same slices but approximating the joint
incompleteness as independent, I(S100 , S143 , S217 ) ≈ I(S100 )I(S143 )(S217 ). Doing the
integral in Eq. 3.2 with this approximation leads to about a 3% difference, thus we use
the more accurate joint incompleteness. The top plots shows visible shot noise from the
simulations used to compute the incompleteness function. We have split these simulations
in two and using either half gives the same answer to within 1%.
46
3.4 Constraints on the six parameter source count model from (blue) integrating the model
down to individual frequencies and fitting the M13 differential source count measurements
at 90, 150, and 220 GHz and (orange) also including the M13 distribution of spectral
indices as a direct constraint on the spectral index and scatter parameters.
47
4.1 Key regions and contours in the mass-lifetime parameter space according to the analytic
approximations in Sec. 4.1. As in other plots in this paper, dashed lines correspond to the
temperature at neutrino decoupling, dot-dashed lines the start of BBN, and dotted lines
the end of BBN. Blue lines show contours of constant Primakoff freeze-out temperature,
Tfo , and black lines show contours of constant two-photon re-equilibration temperature,
Tre . The line Tre = mφ divides two regions A and B. Region A is vertically hatched and
corresponds to out-of-equilibrium decays. Region B is cross hatched and corresponds to
in-equilibrium decays. Constant decay-time contours in region A are Tre = const whereas
they are mφ = const in region B. Region C has no hatching and corresponds to decays
before neutrino decoupling, where ALPs leave no cosmologically observable traces. The
line Tfo =QCD leaves a sharp feature on cosmological constraints as g∗ changes suddenly
during this phase transition.
52
4.2 The evolution of the energy densities in the various components of the universe for different
scenarios which have similar decay time. The temperature of the photons today is held
fixed and the y-axis units are such that the final value of the neutrino line is the value
CMB . As in other plots in this paper, dashed lines correspond to the scale factor at
of Neff
ix
neutrino decoupling, dot-dashed lines the start of BBN, and dotted lines the end of BBN.
The dashed red line is not actually a component, but is shown for illustrative purposes; it
is the equilibrium ALP energy density (that is, the energy density APLs would have if they
were in chemical and kinetic equilibrium with the photons). As per Eqn. 4.5, interactions
serve to always drive the ALP energy density towards equilibrium. The plots labeled A
and B correspond to the same regions in Fig. 4.1.
56
4.3 Exclusion regions in the ALP mass-lifetime parameter space. The dashed and dotted lines
labeled “ν dec” (neutrino decoupling), and “BBN start/end” correspond to particles which
decay at these particular times (with decay here arbitrarily defined as when maximum
energy injection occurs). The two thick dashed lines are the consistency relations for two
particular axion models (see Sec. 4.3.1). The CMB, D/H, and Yp regions are excluded at
3σ, the Collider and Beam Dump regions are excluded at 2σ, and the SN1987a and HB
Stars regions are less formal, rough bounds (see Sec. 4.2.4).
57
4.4 A comparison of exclusion regions from previous works (left panel) and those presented
here (right panel). The right panel is identical to Fig. 4.3.
58
4.5 The colored contours show the prediction for each of the labeled quantities as a function
of different values of ALP mass and lifetime. The dotted/dashed/solid lines give 1/2/3 σ
contours given the measurements for these quantities discussed in Sec. 4.2. No lines are
visible on the lithium plot because the entire parameter space is excluded at > 3σ (our
CMB
scenario does not alleviate the lithium problem). We do not give contours for the Neff
plot because the CMB constraint is highly degenerate with Yp . For the D/H panel, the
colored contours are calculated assuming a best-fit η from the CMB, and uncertainties
in η and nuclear reaction rates are taken into account in producing the σ contours (see
Sec. 4.2.2 for discussion).
59
CMB and Y from
4.6 The contours show the 1- and 2-σ confidence regions for Neff
p
Planck +WP+highL. The dotted lines give the 1-σ constraint on Yp from Aver
et al. [2013]. The dashed line is the relation if standard BBN is assumed, and the dot
CMB = 3.046. Colored points show
along this line corresponds to the standard value of Neff
CMB and Y arising from ALP masses and lifetimes sampled from a grid over
values of Neff
p
√
the entire region shown in Fig. 4.3. They are colored by mφ τφγ which is an important
x
quantity for the CMB constraint since it controls the fractional energy injected into the
CMB ≈ 2.44 gives
photons (Eqn. 4.10). The maximum value for in-equilibrium decays of Neff
the sharp cutoff visible above. Points along the standard BBN consistency line arise from
decays happening between neutrino decoupling and the beginning of BBN, and correspond
to the island of low helium visible in Fig. 4.5.
61
4.7 Parameter constraints in the MeV-ALP region of parameter space when we also allow
extra radiation present besides neutrinos and the ALP. Our likelihood includes all of the
constraints shown in Fig. 4.3. In the ALP case (top panel) the lifetime is marginalized
over whereas in the DFSZ-EN2 case (bottom panel) it is fixed by the consistency relation
(Eqn. 4.28). The vertical dashed line is a forecast for SUPER-KEKB, showing that it could
close the remaining allowed parameter window or detect a particle there. We show Neff
evaluated both prior to BBN when neutrinos, extra radiation, and the ∼MeV ALP (which
here adds 4/7 to Neff ) contribute, and at the CMB epoch after the ALP has decayed.
69
CMB and Y from
4.8 (Top) The contours show the 1- and 2-σ confidence regions for Neff
p
Planck +WP+highL. The dotted lines give the 1-σ constraint on Yp from Aver et al. [2013].
The dashed line is the relation if standard BBN is assumed, and the dot along this line
CMB = 3.046. Colored points show values of N CMB
corresponds to the standard value of Neff
eff
and Yp arising from ALP masses and lifetimes taken from the ΛCDM+ΔNeff +ALP chain
described in Sec. 4.3.2. They are colored by mφ which controls decay time and can alter
CMB . (Bottom) Same as the top panel, but with
BBN but otherwise does not affect Neff
105 D/H shown on the x-axis. The vertical dotted lines give the 1-σ constraint from Cooke
preBBN
. Sufficiently tight constraints around
et al. [2014]. Points are colored instead by Neff
the standard value (black dot) could rule out the ALP scenario even in the presence of
extra radiation.
69
xi
List of Tables
2.1 SZ Cosmological Scaling
18
2.2 Survey Properties
23
2.3 Summary of Model Parameters
24
2.4 Statistical Error Degradation
37
4.1 Best-fit parameters
71
4.2 Best-fit χ2
71
xii
CHAPTER 1
Introduction
The era of precision cosmology is currently upon us, and observations of the cosmic microwave
background (CMB) are at the forefront of driving current and future progress. The CMB is a
remarkable cosmological probe because of the wealth of physics to which it sensitive; this ranges
from the conditions at the very beginning of time during the inflationary era, to the formation of
the light elements during big bang nucelosynthesis, to the evolution of structure across orders of
magnitude of cosmic time. In addition, the properties of the CMB can be predicted accurately
to first order in linear perturbation theory, meaning precise theoretical predictions can be readily
calculated and compared to observations. For these reasons, the CMB is a truly powerful probe of
physics.
It can be said that the precision era was ushered in by the results from the WMAP satellite
[Bennett et al., 2003, Spergel et al., 2003]. By the time of the final results from nine years of
observation were published [Hinshaw et al., 2013], the temperature fluctuations in the CMB at scales
larger than about a third of a degree, or multipole moments smaller than about ∼ 180◦ /(1/3◦ ) ∼
540, were now measured to the best possible theoretical limit, the cosmic variance limit. The
picture of universe which emerged was a parametrically simple but physically rich “standard model
of cosmology”, ΛCDM, which took the universe to be made up mostly of dark matter and dark
energy, with a small but important contribution from known atoms, with stars which reionized the
interstellar medium between redshifts z ∼ 7 − 14, and with initial conditions consistent with those
predicted from the simplest models of inflation. Falsifying this model, or alternatively confirming
and refining its predictions, has continued to drive the field of cosmology.
Further progress on the CMB side necessitated better measurements of the temperature fluctuations at small scales, and better measurements of the CMB polarization at all scales. The
latter is outside of the scope of this thesis, while the former is the main topic. The first results
following WMAP pertaining to small scale temperature fluctuations were published by the ACT
and SPT collaborations [Dunkley et al., 2011, Keisler et al., 2011, respectively], which were, at
least in certain regions of the sky, now at the cosmic variance limit up to and above ∼ 2500.
1
This essentially completed the measurement of the CMB temperature fluctuations in these regions
since above this scale almost no cosmological information can be extracted due to irreducible foregrounds and confusion from gravitational lensing. While some hints of failures of ΛCDM did arise,
[Dunkley et al., 2010, Hou et al., 2013, 2014], none were of convincing statistical significance. Thus
in the lead up to the results from the Planck satellite, the post-WMAP picture of the universe had
remained qualitatively very similar [Calabrese et al., 2013], albeit with more and more alternatives
to ΛCDM ruled out. The release of the first Planck results [Collaboration et al., 2013a] marked the
completion of measurements of the temperature fluctuations up to ∼ 2500 to the cosmic variance,
now across the entire sky.
Measurements by Planck, ACT, and SPT, probe a qualitatively new regime of the CMB power
spectrum called the damping tail. This region corresponds to scales small enough that they entered the horizon deep in the radiation dominated epoch of the universe, and additionally whose
amplitude has been damped significantly due to photon diffusion in the last scattering surface.
This allows these measurements to probe the physics of known or hypothetical relativistic particles
which contribute to the radiation energy density, or which can affect this photon diffusion.
Additionally, by virtue of being at small scales, extraction of cosmological information from
these modes now necessitates taking into account new foreground components which are otherwise
negligible at large scales. These foregrounds, called extra-galactic foregrounds, arise from emission at microwave frequencies or distortion of the CMB spectrum by sources outside of our own
galaxy such as other galaxies and galaxy clusters. A large part of this thesis centers around these
foregrounds, and contains work which was performed by the author in the lead up and during the
analysis of SPT and Planck data. Ch. 2 follows from Millea et al. [2012] and develops a realistic
model for such contamination. Since Millea et al. [2012] was written before the release of the Planck
results, it uses a set of Planck -like simulations to show that subtracting this model is crucial for
such an analysis. Ch. 3 follows from work soon to be published as part of a Planck publication,
and gives a novel method for constraining particular components of this foreground model with
auxiliary data.
Finally, in Ch. 4, which follows from Millea et al. [2014], we use CMB observations to constrain
a model of a hypothetical scalar particle called the axion, and a more general class of so called
axion-like particles.
2
1.1. Theoretical Background
We begin with a very brief theoretical introduction to the power spectrum of the temperature
fluctuations in the CMB, a quantity which will be central to this thesis. The temperature of
the primary1 CMB as a function of position on the sky, T (n̂), is modeled as a homogeneous and
isotropic Gaussian random field. This means that the correlation between different spots on the sky
is independent of their location and orientation, and thus can only be dependent on the distance
between them,
(1.1)
T (n̂)T (m̂) = ξ(|n̂ − m̂|)
where ξ is known as the correlation function. Here, the brackets denote the “ensemble average”,
taken over many hypothetical realizations of our universe. One can show that in terms of the
spherical harmonic coefficients of the same map,
(1.2)
am =
dn̂ Ym (n̂)T (n̂)
Eqn. 1.1 is equivalent to,
(1.3)
a∗m a m = δ δmm C
where C is referred to as the power spectrum. Higher order correlations, i.e. ones with more than
two am ’s appearing such as am a m a m can be calculated using Wick’s theorem. It states that
such correlations can be written as the sum over all permutations of products of correlations of
one and two am ’s. Using am = 0 and Eqn. 1.3, it is clear that all such higher order correlations
can themselves only depend on C and various numerical factors, thus the entirety of the statistical
information contained in the T (n̂) field is contained in its power spectrum, C . For this reason,
using the power spectrum is a form of loss-less data compression, one which is also very convenient
as it is most directly predicted by theory.
The fluctuations in the temperature of the CMB are sourced by quantum fluctuations which
become classical perturbations during inflation, then causally evolve according to classical physics
until today. Their power spectrum can be written as an integral over spatial Fourier modes k, taking
the product of the initial perturbation set up during inflation, P(k), and a transfer function, Δ2 (k),
1We will use “primary” to refer to the CMB before gravitational lensing
3
which describes the subsequent causal evolution and projects each mode onto the last scattering
surface,
(1.4)
C =
d ln k P(k)Δ2 (k)
Here P(k) is related to fluctuations in the inflaton field, P(k) = |φ(k)|2 (evaluated long after each
mode has left the horizon). The transfer function, Δ2 (k), comes from solving the coupled set of
Einstein and Boltzmann equations which track how perturbations in the various components of the
universe and also in the background metric evolve.
While constraining both P(k) and Δ2 (k) are of interest in cosmology, in this thesis we will
focus only on the transfer function. In particular, in Ch. 4, we discuss how the addition of an axion
or an axion-like particle to the coupled set of Boltzmann and Einstein equations affect the transfer
function and how CMB and other cosmological observations can place constraints on these types
of hypothetical particles.
Maps at microwave frequencies contain not just CMB fluctuations, but also contamination
from extra-galactic foregrounds. Here, objects along the line of sight to the CMB last-scattering
surface emit microwaves, or distort the CMB spectrum. The observed temperature, T will then be
T = T CMB + T FG with,
(1.5)
T FG (n̂) = f N (n̂)
where we have assumed that each one of N objects along the line of sight in direction n̂ contributes
an amount t to the temperature (we will shortly generalize to the realistic case where objects
contribute different amounts). The distribution of object locations is modeled as being randomly
sampled from some smooth underlying distribution (which in our case is a biased traced of the
total matter distribution, although in general it does not need to be). Thus the field N (n̂) arises
from a Cox Poisson process, and the actual number density in any direction (for our realization of
the universe) will be drawn from a Poisson distribution,
(1.6)
N (n̂) ∼ P(N̄ (n̂)) ≈ N (N̄ (n̂), N̄ (n̂))
where N̄ is the underlying distribution and P(λ) represents the Poisson distribution with mean
λ. As will become important in a moment, in the limit of large N , this is approximately equal
to a Gaussian distribution N (μ, σ 2 ) with mean μ and variance σ 2 both equal to N̄ (n̂). The field
4
N̄ is itself a Gaussian random field, whose distribution is most cleanly expressed in terms of its
harmonic coefficients,
N̄m ∼ N (0, N̄ )
(1.7)
where N is the power spectrum of the smooth underlying distribution as predicted by theory.
One observation which helps understand the resulting power spectrum of N (n̂) is that, under
the Gaussian approximation to the Poisson distribution, the field can be written as,
(1.8)
N (n̂) = N̄ (n̂) + ΔN (n̂)
N̄ (n̂)
with
(1.9)
ΔN (n̂) ∼ N (0, 1)
where we have separated out the mean contribution, and written the remainder explicitly as a
uniform white noise term multiplied by a term which spatially scales its variance.
In this form, it is more obvious that the power spectrum N (n̂) will have three terms, one auto
correlation for each of the two terms in Eqn. 1.8, and one cross correlation. The auto power of N̄ (n̂)
is by definition N , which is the quantity predicted by theory. The auto power of the second term
is the mean object density, N̄ (which is independent of direction), also as predicted by theory. To
arrive at this result one can simply plug in Eqn. 1.8 into Eqns. 1.2 and 1.3. Using the same method,
one can also see that the cross term is zero. Therefore, the power spectrum of the foregrounds is,
(1.10)
CFG = f 2 (N + N̄ )
The first term is known as the “clustering” term, because it depends on how the objects trace the
underlying distribution, and hence how they cluster. The second term is known as the Poisson term
because it is independent of the underlying distribution and only depends the Poisson fluctuations
around the mean number of objects.
Ch. 2 will give theoretical predictions for both of these terms for a number of different objects.
It will turn out that the clustering term is only important for dusty galaxies, whereas for radio
galaxies, and for distortion of the CMB spectrum by galaxy clusters via the tSZ and kSZ effect,
only the Poisson term is important.
5
CHAPTER 2
Modeling Extragalactic Foregrounds
At angular scales smaller than a tenth of a degree, extragalactic foregrounds1 become important for three reasons: 1) the CMB power spectrum is dropping in amplitude, 2) cosmic variance
is smaller and 3) foregrounds are growing in amplitude. At sufficiently small angular scales, foregrounds become the dominant signal at all CMB frequencies. Furthermore, unlike galactic foregrounds, they are statistically isotropic and thus cannot be avoided by masking regions of higher
contamination. Their modeling is an unavoidable necessity.
In this chapter we present a parameterized, physically-motivated, phenomenological model for
the extragalactic foregrounds and consider it in the context of extracting cosmological parameters
from the primary CMB anisotropy. We demonstrate that for an analysis of Planck data, such
modeling is necessary to avoid significant biases in cosmological parameter estimates, but that
marginalization over even a very rich foreground model is essentially “for free”; the foregrounds are
sufficiently orthogonal to the primary CMB that the statistical errors on cosmological parameters
are degraded by at most 20% for ns and less than 10% for other parameters. With the addition
of higher resolution ground-based data or non-CMB Planck bands to clean the foregrounds, the
degradation is reduced to a few percent for all parameters.
The importance of extragalactic foregrounds for CMB analysis has been recognized for a long
time [Tegmark and Efstathiou, 1996, Bouchet and Gispert, 1999, Knox, 1999, Tegmark et al., 2000,
Leach et al., 2008, Cardoso et al., 2008, Dunkley et al., 2011]. Potential biases from extragalactic
contaminants have been pointed out previously by Knox et al. [1998], Santos et al. [2003], Zahn et al.
[2005], Serra et al. [2008] and Taburet et al. [2009]. Distinguishing our work is the simultaneous
consideration of all foreground components necessary for an analysis of Planck data, and physical
modeling of these components informed from recent measurements beyond the damping tail by SPT
[Hall et al., 2010, Vieira et al., 2010, Shirokoff et al., 2010] and ACT [Dunkley et al., 2011]. In this
chapter, we will consider the foreground power contributions from shot noise due to radio galaxies
1We will henceforth refer to both extragalactic foreground contaminants and secondary anisotropies as just “fore-
grounds” since they cannot be modeled from first principles like the primary CMB.
6
and dusty star forming galaxies (DSFGs), the clustering of the DSFGs, the thermal and kinetic
Sunyaev-Zeldovich effects (tSZ and kSZ), and correlation between the tSZ and DSFG components.
We now turn to summarizing recent developments in both modeling and measurements of these
extragalactic foregrounds.
Our understanding of the power spectrum due to DSFGs at frequencies relevant for CMB
analysis has been rapidly improving. We demonstrate here that for analysis of Planck data, the
effects of DSFG clustering are the most important of the foregrounds to model. Although it is
the most important effect, it has been almost entirely ignored by previous cosmological parameter
error forecasting work. To date, the only papers to consider the impact of DSFG clustering on
cosmological parameter estimates are Dunkley et al. [2011] and Serra et al. [2008].
DSFG clustering power was first detected at CMB frequencies by the SPT [Hall et al., 2010],
with subsequent confirmation and improved constraints from ACT [Dunkley et al., 2011] and SPT
[Shirokoff et al., 2010]. The recent suite of early Planck papers [Planck Collaboration, 2011r,
in particular] have also provided significant constraints on both the amplitude and shape of the
clustering power. The Planck measurements rule out many otherwise viable models which generally
predict higher power (on the scales relevant for analysis of the primary CMB power spectrum) than
observed.
Radio galaxy source counts from high-resolution ground-based data are particularly useful for
Planck since they are sensitive to the decade in brightness below Planck ’s flux cut. The radio
sources in this brightness range create the dominant source of shot noise power in most of the
Planck frequencies which contain significant CMB information. SPT measurements of point source
populations [Vieira et al., 2010] have offered valuable information about the amplitude of Poisson
power, as well as the coherence of these shot-noise fluctuations from frequency to frequency.
Recent data, as well as recent theoretical developments, inform our modeling of the power
spectrum of the tSZ effect—a spectral distortion that arises due to inverse Compton scattering of
CMB photons off the hot electrons in groups and clusters. The magnitude of the tSZ signal is
proportional to the thermal pressure of the intra-cluster medium (ICM) integrated along the line
of sight. Upper limits on the amplitude of the tSZ power (set by Lueker et al. [2010], confirmed by
Dunkley et al. [2011] and further tightened by Shirokoff et al. [2010]) were found to be surprisingly
low compared to predictions from halo model calculations [Komatsu and Seljak, 2002] and nonradiative hydrodynamical simulations [White et al., 2002]. Recent work has demonstrated that the
7
inclusion of a significant non-thermal contribution to the total gas pressure in groups and clusters in
analytic models can significantly reduce the predicted amplitude of the tSZ power spectrum [Shaw
et al., 2010, Trac et al., 2010]. Non-thermal pressure, sourced by bulk gas motions and turbulence,
reduces the thermal pressure required to support the ICM against gravitational collapse and thus
the amplitude of the tSZ signal. Similarly, Battaglia et al. [2010] demonstrated that the inclusion
of radiative cooling, star formation and AGN feedback in hydrodynamical simulations substantially
lowers the tSZ power compared to simulations that omit these processes. Current predictions for tSZ
power from models and simulations are consistent with the upper limits derived from observations.
These recent modeling developments are supported by data from Planck ; when the models are
used to extrapolate from X-ray measurements to a predicted tSZ signal, the predictions agree with
Planck SZ observations. Agreement is seen both in observations of single galaxy clusters [Planck
Collaboration, 2011h,i] and via a stacking analysis over a broad range in X-ray luminosity down to
masses as small as M500 ∼ 5 × 1013 M [Planck Collaboration, 2011j].
Current data provide no direct lower limits to the amplitude of tSZ power due to a degeneracy
with the kinetic SZ power spectrum [Lueker et al., 2010]. The kinetic SZ effect arises due to
the Doppler Thomson scattering of CMB photons off of regions of ionized gas with bulk peculiar
velocities. Upper limits on kSZ power set by Lueker et al. [2010] and now substantially tightened by
Shirokoff et al. [2010], are ruling out some models of patchy reionization. It is useful to decompose
the kSZ power into contributions arising from an inhomogeneous transition from a neutral to ionized
inter-galactic medium, so called “patchy reionization,” and those from the post-reionization era,
the “Ostriker-Vishniac” (OV) effect. The former is much more uncertain than the latter, and our
best knowledge of its amplitude comes directly from the upper limits in Shirokoff et al. [2010]. The
OV power level has a current theoretical uncertainty that we estimate to be about a factor of 2.
Despite its low levels, kSZ power is a worrisome source of potential bias of cosmological parameters
since its spectral dependence is the same as the primary CMB temperature anisotropies.
We expect that the only potentially significant extragalactic contributions to polarization anisotropy are Poisson power from radio sources and DSFGs. A polarization analog for DSFG clustering
could only arise due to (unexpected) correlations between galaxies in the polarization orientations
of their emission. Polarization signals arise from scattering off of electrons in clusters and groups
[Sazonov and Sunyaev, 1999, Carlstrom et al., 2002, Amblard and White, 2005] and in reionized
patches [Knox et al., 1998, Santos et al., 2003], but these are also expected to be negligibly small.
8
In addition to developing and exploring the implications of an extragalactic foreground model
that takes into account recent developments, we introduce a new approach to analyzing the multifrequency data. We show how the complexities of our modeling can be reduced to a fairly simple
description of the contamination of the estimates of CMB power spectra. The contamination can
be described by just a few principal components whose amplitudes are constrained by CMB–free
linear combinations of the auto and cross-frequency power spectra.
The outline of this chapter is as follows. In Sec. 2.1 we describe our foreground models before
describing our fiducial models and surveys in Sec. 2.2 and 2.3 respectively. In Sec. 2.4 we describe
our general methodology before detailing our principal component approach in Sec. 2.5. We finally
present our results in Sec. 2.6 and discuss them in Sec. 2.7.
2.1. Modeling
2.1.1. Emission from External Galaxies. In the frequency range in which Planck is most
sensitive to the CMB (roughly 70 GHz to 217 GHz), external galaxies are well approximated by
power-law intensities Iν ∝ ν α , and divide fairly cleanly into those with spectral indices α < 1
(radio galaxies) and those with α > 1 (DSFGs) [Vieira et al., 2010]. We assume all sources have
no spatial extent, an approximation which might be worrisome for radio sources because of long
relativistic jets. However, if we extrapolate from 1.4 GHz up to our frequency range, we find 99%
of sources have a major axis FWHM less than 30 arc-seconds [Hodge et al., 2011], too small to be
detectable with the 1.5 arc-minute beams of typical ground-based experiments.
External galaxies lead to anisotropy via their discreteness, usually modeled with a Poisson
distribution, and also via correlations due to their tracing of the large-scale structure. The Poisson
fluctuations are important for both radio galaxies and DSFGs, while clustering is only significant
for the dusty galaxies [Hall et al., 2010].
The Poisson contribution depends on the brightness function, dN/dS, via
(2.1)
C =
Sc
0
dS S 2
dN
dS
where Sc is the flux cut; map pixels with sources with S > Sc are masked.
9
Clustering power, in contrast, scales approximately with the square of the mean intensity, Iν2 ,
with
Iν =
(2.2)
Sc
dS S
0
dN
.
dS
Although radio sources do cluster, their mean intensity at the relevant frequencies is much smaller
than for the DSFGs; sufficiently smaller that their clustering power is negligible.
2.1.1.1. Radio Galaxies. From Vieira et al. [2010] we know the radio galaxies at 150 GHz and
220 GHz and at flux densities below 100 mJy are described quite well by the de Zotti et al. [2005]
model2. This model has a brightness function that is approximately a power-law SdN/dS ∝ S γR .
This translates into Poisson power which depends on the flux cut via C ∝ ScγR +2 . Due to the
inhomogeneity of the Planck sky coverage, Sc will vary significantly across the sky. So that these
angular variations can be taken into account, we chose to model the radio galaxies in terms of
dN/dS rather than C .
For frequency dependence, we assume the spectral indices of the source population form a
Gaussian distribution with mean α and width δα2 = σ 2 (uncorrelated from source to source).
With these assumptions our power spectra from radio sources are given by3
2
γR + 2 αR + ln(νν /ν02 )σR
/2
Sc
νν
(2.3)
CR = C R,0
S0
ν02
where C R,0 is some overall normalization factor.
2.1.1.2. Dusty Star-Forming Galaxies. Due the shape of DSFG brightness function, the integrals in Equations 2.1 and 2.2 are nearly independent of the upper bound [Hall et al., 2010], thus
dusty power is nearly independent of flux cut and we choose to build our model in C rather than
dN/dS. In that case, the DSFG Poisson contribution is given simply by
(2.4)
CD = C D,0
νν ν02
2
αD + ln(νν /ν02 )σD
/2
and again C D,0 is an overall normalization.
2We also know from recent Planck results [Planck Collaboration, 2011m] that at brighter flux densities the deZotti
model significantly over predicts the number counts.
3In deriving this form we have used the identity that for a zero-mean Gaussian random variable x, exp(−x) =
exp(x2 /2). This identity and its applicability in this context, was pointed out to us by Challinor, Gratton and
Migliaccio.
10
A number of authors have considered the clustering of the infrared background, starting with
Bond et al. [1986, 1991]. Further theoretical investigation [Scott and White, 1999, Haiman and
Knox, 2000] was stimulated by the detection of the infrared background in COBE data [Puget et al.,
1996, Fixsen et al., 1998], and the detection of bright “sub-millimeter” galaxies in SCUBA data
[Hughes et al., 1998]. Subsequently, the clustering has been detected at 160 microns [Lagache et al.,
2007], at 250, 350 and 500 microns by the Balloon-borne Large Aperture Submillimeter Telescope
[Viero et al., 2009, BLAST] and at 217 GHz [Hall et al., 2010, Dunkley et al., 2011]. Recent Planck
measurements of the Cosmic Infrared Background [CIB; Planck Collaboration, 2011r] have extended
to much larger angular scales than before at 217 GHz, 353 GHz, 545 GHz, and 857 GHz and recent
Herschel measurements [Amblard et al., 2011] have tightened up the BLAST measurements and
extended them to smaller angular scales. The field is rapidly evolving.
For the clustering, we assume the same model as in Hall et al. [2010], extended to phenomenologically include the consequences of non-linear clustering by including a multiplicative factor which
is a power-law in for > 1500. This extension is able to fit many different models in the literature and allows us to explore the theoretical uncertainty in a statistical manner by marginalizing
over the value of the multiplicative factor. We neglect one aspect of the Hall et al. [2010] model
because it leads to corrections of only about 1% across the relevant frequency range; we ignore the
-dependent spectral index. Thus, the DSFG clustering power spectra are given by
(2.5)
CC = C C,0 ΦH10
⎧
α C ⎪
⎨ 1
n C
⎪
ν02
⎩
1500
νν < 1500
> 1500
is the Hall et al. [2010] clustering template.
where ΦH10
Though the same sources generate both the Poisson power and the clustering power, they are
weighted differently, thus for our baseline model we conservatively assume no relationship between
the clustering spectral index and the Poisson spectral index.
To gain some idea of the range of possible shapes of the DSFG clustering power spectrum,
we show a sampling of power spectra from models in the literature in Figure 2.1. They are all
normalized at = 3000 to highlight similarities/differences in shape. The models are the fiducial
model from Righi et al. [2008], the β = 0.6 model from Amblard and Cooray [2007] and a non-linear
version of the model by Haiman and Knox [2000], hereafter HK00. Righi et al. [2008] associate
11
the sources of infrared light with starbursts triggered by mergers. Amblard and Cooray [2007]
incorporate nonlinearities using a halo model. For the ‘HK00nonlin’ curve, we used the luminosity
densities for the fiducial model of HK00, assumed light is a biased tracer of mass, and calculated
the non-linear mass power spectrum using the prescription by Peacock and Dodds [1996]. Though
these template arise from very different modeling assumptions, they have similar shapes in the linear
regime at large scales, then turn to a power-law behavior at small scales. It is this observation which
informed our phenomenological model. We will also show our model to be sufficient for reproducing
these shapes with enough accuracy for Planck cosmological parameter estimation.
One result of the Planck measurements, available only after our calculations for this paper
were completed, is that the CIB power spectrum uncertainty at < 2000 is now much smaller than
before. At least two of the three models shown in Figure 2.1 that guided our understanding of
the range of possible amplitudes have shapes that are inconsistent with the combined Planck and
SPT data. That range of possible amplitudes is now given by the Planck CIB power spectrum
measurement uncertainty.
2.1.1.3. Polarization. We expect polarized emission from the sources we consider to be very
small and uncorrelated from source to source. For a collection of sources with polarization fraction
f , contributing a Poisson temperature power spectrum of CT T,P , we have
(2.6)
CEE = CBB =
f 2 CT T,P
CT E =
f CT T,P
We parameterize both radio source and DSFG contributions with the above forms, with f = fD
for DSFGs and f = fR for radio sources.
2.1.2. Thermal SZ Effect. The thermal SZ effect is a distortion of the CMB caused by
inverse Compton scattering of CMB photons off electrons in the high temperature plasma within
galaxy clusters. To first order, the temperature change of the CMB at frequency ν is given by
ΔT /TCMB (xν ) = f (xν )y, where f (xν ) = xν (coth(xν /2) − 4), xν = hν/kB TCMB , and y is the
dimensionless Compton-y parameter
(2.7)
y=
kB σT
me c2
ne (l)Te (l)dl ,
12
Figure 2.1 Three model DSFG clustering auto-spectra at 217 GHz (black), and approximations to
them with our parameterized model (solid, red), all normalized (with one exception) at = 3000.
Our fiducial model is the thickest curve. Also plotted are estimates of the clustering power from
Planck [Planck Collaboration, 2011r] and SPT [Hall et al., 2010]. For both sets of data points we
have subtracted estimates of the Poisson power from the reported total CIB power. The lowest
amplitude solid (red) curve is the result of a “by-hand” adjustment of our model parameters to fit
the Planck and SPT data.
where the integral is along the line of sight. TCMB is the CMB temperature, ne and Te are the
number density and electron temperature of the ICM, respectively.
The thermal SZ power spectrum can be calculated by simply summing up the squared, Fourierspace SZ profiles, ỹ, of all clusters:
(2.8)
CtSZ
= f (xν )
2
dV
dz
dz
d ln M
dn(M, z) 2
ỹ (M, z, )
d ln M
where V(z) is the comoving volume per steradian and n(M, z) is the number density of objects of
mass M at redshift z. For the latter we use the fitting function of Tinker et al. [2008]. y(M, z, r)
is the projected radial SZ profile for a cluster of mass M and redshift z. Note that this calculation
assumes that halos are not spatially correlated; Komatsu and Kitayama [1999] demonstrated that
for > 1000 the two-halo (or clustered) contribution to the tSZ power spectrum is several orders
of magnitude smaller than the Poisson contribution given by Eq. 2.8.
13
To calculate the thermal SZ signal we adopt the analytic intra-cluster gas model presented in
Shaw et al. [2010]. This model provides a prescription for calculating the compton-y (or equivalently,
thermal pressure) profiles of hot gas in groups and clusters. The model assumes that gas resides
in hydrostatic equilibrium in the potential well of dark matter halos with a polytropic equation of
state. The dark matter potential is modeled by a Navarro-Frenk-White profile [Navarro et al., 1997]
using the halo mass - concentration relation of Duffy et al. [2008]. The model includes parameters to
account for gas heating via energy feedback (from AGN or supernovae) plus dynamical heating via
mergers. The stellar component of the baryon fraction in groups/clusters is determined using the
stellar mass fraction - total mass relation observed by [Giodini et al., 2009]. A radially-dependent
non-thermal pressure component of the gas is incorporated by calibrating off the non-thermal
pressure profiles measured in hydrodynamical simulations [Lau et al., 2009]. In total the model has
four free parameters relating to astrophysical processes in groups and clusters. Shaw et al. [2010]
explored the range in which these parameters reproduce radial profiles and scaling relations derived
from X-ray observations of nearby groups and clusters.
To allow the astrophysical uncertainty to be marginalized over quickly in our MCMC chains,
we perform a principal component analysis (PCA) described in Appendix A. A suite of 10,000
simulated power spectra were created, each time randomly sampling from the input astrophysical
parameter distribution (with the cosmological parameters fixed to their fiducial values described
in Sec. 2.2). We find that two principal components are sufficient to achieve 1% accuracy out to
= 10, 000 on the model power spectra.
In Figure 2.2 we plot the thermal SZ power spectrum predicted by a number of recent simulations (black lines) as well as a fit to each with our PCA model (red lines). The dotted line
represents the thermal SZ power spectrum measured from the Mare-Nostrum simulation – a nonradiative simulation run using the smoothed-particle hydrodynamics code, Gadget-2. The black
solid line shows the results of the non-radiative simulation of Battaglia et al. [2010] and the black
dashed line the results of a rerun of this simulation including radiative cooling, star-formation and
energy feedback. The dot-dashed line shows the ‘standard’ tSZ model from the simulations ofTrac
et al. [2010]. The thickest red lines represents our fiducial thermal SZ model in this work. The
blue point with errorbars show the recent SPT constraint on the amplitude of thermal SZ power
at = 3000. All models are plotted at 146 Ghz and have been scaled to our fiducial cosmology.
14
16
14
M B
B
T t D [K 2 ]
12
1
8
6
4
2
2
4
6
8
1
Figure 2.2 Comparison between recent models and simulations of the tSZ effect (black lines) and
fits of our PCA model to each (thin red lines). The thickest red line shows the fiducial tSZ power
spectrum used in this work. All results are plotted at 146 GHz and are scaled to σ8 = 0.8. The
blue arrow shows the SPT 95% confidence upper limit on thermal SZ power at = 3000 [Shirokoff
et al., 2010].
Our PCA model can accurately reproduce all the simulations in Fig 2.2 other than the nonradiative simulation of Battaglia et al. [2010] which peaks at much smaller angular scales than the
other simulations. We note that the Shaw et al. [2010] model inherently assumes that some fraction
of cluster gas has been converted to stars, whereas this simulation did not include these processes.
Turning off star formation produces a power spectrum that peaks at smaller scales. In Section
2.6.2 we investigate the bias on measured cosmological parameters when the Battaglia et al. [2010]
non-radiative template is used for the tSZ signal. We find that the PCA will adapt sufficiently to
prevent a bias in the measured cosmological parameters.
The final step is to determine the cosmological scaling of the power spectrum of our fiducial
model so that the amplitude can be scaled accordingly in our analysis. We find that the tSZ power
spectrum is principally sensitive to Ωm , Ωb , ns , and σ8 , with a particularly strong dependence on
the latter. To determine the scaling we simply evaluate the Shaw et al. [2010] model varying each
cosmological parameter in the range ±25% of its fiducial value while holding the other three fixed
(at their fiducial value). We then fit to the resulting power spectra, with our results summarized
in Table 2.1.
15
T M B t t t
40
D [K 2 ]
4
30
3
20
2
10
1
0
2
4
6
8
1
Figure 2.3 Comparison between recent simulations of the kSZ effect (black lines) and our model.
We consider contributions from the post-reionization kSZ effect (solid red line) and from patchy
reionization (dashed red line). Note that the simulations plotted here assume homogeneous reionzation and thus do not include a patchy contribution. The SPT 95% confidence upper limit for the
kSZ power at = 3000 is 6.5 μK2 .
2.1.3. Kinetic SZ Effect. The kinetic SZ effect is a temperature anisotropy that arises from
the Compton scattering of CMB photons off of electrons that have been given a line-of-sight peculiar
velocity by density inhomogeneities in the matter field. We break up the kSZ into contributions
from the post-reionization period and from a period of inhomogeneous “patchy” reionization.
2.1.3.1. Ostriker-Vishniac Effect. When the density fluctuations which source electron velocities are in the linear regime the effect is known as the Ostriker-Vishniac (OV) effect, as derived
in Ostriker and Vishniac [1986] and Vishniac [1987]. The post-reionization kSZ effect can then be
modeled as the nonlinear extension of the OV effect as described below.
We follow the analytic prescription given in Hu [2000] which describes the angular power spectrum of the linear Vishniac effect as
(2.9)
π2
C = 5
2
3
dχDA
Ġ
g
G
2
Δ4δb IV .
where χ is the conformal time, G is the cosmological growth function, DA is the comoving angular
diameter distance, g is the visibility function, Δ2δb is the linear theory baryon density power spectrum
16
and IV represents the mode coupling of the linear density and velocity fields:
(2.10)
IV =
∞
0
dy1
1
−1
dμ
2
2
(1 − μ2 )(1 − 2μy1 ) Δδb (ky1 ) Δδb (ky2 )
,
y13 y25
Δ2δb (k) Δ2δb (k)
with
μ = k̂ · k̂1
y1 = k1 /k
(2.11)
y2 = k2 /k =
1 − 2μy1 + y12 .
Due to an incomplete treatment of the effects of pressure feedback from baryons we slightly overpredict the power on very small scales. As described in Hu [2000], in this formulation we can consider
the kSZ effect to be the nonlinear extension of the linear Vishniac effect. This approximation
requires replacing the linear density power spectrum in Eq. 2.9 with its nonlinear extension while
leaving the contribution from the velocity power spectrum unchanged:
(2.12)
CkSZ,OV
π2
= 5
2
3
dχDA
Ġ
g
G
2
2(N L)
Δ δb
Δ2δb IV .
For the nonlinear power spectra (NL) we utilize the HALOFIT [Smith et al., 2003] model. In this
calculation of the kSZ effect we assume that the nonlinear density fluctuations are uncorrelated
with the bulk velocity field in which they lie. Zhang et al. [2004] argue that this approximation
may not hold in highly nonlinear regimes where contributions from the curl of the nonlinear velocity
field may become important however we neglect these corrections here.
As in the previous section, we find a power-law approximation for estimating the kSZ power
as a function of cosmological parameters. The kSZ angular power spectrum was calculated under
the full analytic formulation for a large suite of WMAP7-allowed LCDM cosmologies. An MCMC
was then performed in the 6 dimensional fitting-function parameter space and best-fit marginalized
values were found and are listed in Table 2.1
In Figure 2.3 we compare our calculation of the kSZ power spectrum (solid red line) with that
measured from recent simulations (black lines). As in Figure 2.2, we plot the power spectrum
predicted by the Mare-Nostrum simulation (dotted), the ‘standard’ model of Trac et al. [2010]
(dot-dashed) and the non-radiative hydrodynamical simulations of Battaglia et al. [2010] (solid).
17
Table 2.1 SZ Cosmological Scaling
AOV [μK2 ]
500
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
1.18
1.81
2.64
3.06
3.33
3.53
3.67
3.78
3.87
3.94
4.00
AtSZ [μK2 ]
500
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
2.06
3.59
4.86
5.04
4.82
4.44
4.03
3.62
3.24
2.89
2.59
ns
Ωb
Ωc
σ8
τ
0.96 0.045 0.22 0.8 0.09
-1.44 1.83 -1.06 4.36 0.25
-1.36 1.91 -1.13 4.82 0.24
-0.94 1.96 -1.12 5.26 0.22
-0.45 1.94 -1.03 5.38 0.20
-0.22 1.96 -1.04 5.54 0.17
-0.03 1.98 -1.05 5.66 0.15
0.13 2.00 -1.06 5.75 0.13
0.27 2.01 -1.07 5.83 0.12
0.38 2.02 -1.08 5.89 0.10
0.49 2.03 -1.09 5.94 0.09
0.58 2.04 -1.10 5.99 0.08
ns
Ωb
Ωm
σ8
h
0.96 0.045 0.265 0.8 0.71
-1.01 2.41 0.69 8.57 1.30
-0.75 2.45 0.63 8.49 1.43
-0.36 2.52 0.53 8.40 1.65
-0.08 2.57 0.48 8.36 1.73
0.15 2.62 0.44 8.34 1.88
0.31 2.66 0.42 8.33 2.03
0.49 2.70 0.39 8.32 2.13
0.58 2.73 0.38 8.32 2.18
0.77 2.78 0.36 8.32 2.27
0.87 2.80 0.35 8.33 2.32
0.96 2.83 0.34 8.33 2.37
Note. — The -dependent power-law cosmological scalings for the Ostriker-Vishniac effect and the thermal SZ
effect. The numbers immediately below the cosmological parameters are the pivot points for the power law, and
the numbers in the table are the power-law indices. For example, the top row says that for the OV effect, D500 =
1.18μK2 (ns /0.96)−1.44 (Ωb /0.045)1.83 ...
The red dashed line shows our model for the contribution to the kSZ signal from inhomogeneous
reionization (not included in the other lines), as described in the following section.
2.1.3.2. Patchy Reionization. We also consider the contribution to the kSZ power from inhomogeneous reionization [Gruzinov and Hu, 1998, Knox et al., 1998, Hu, 2000, Zahn et al., 2005,
McQuinn et al., 2005, Iliev et al., 2007]. Simulations and analytic models of HII bubble formation both indicate that the first galaxies and quasars were highly clustered and led to gradual
reionization in “bubbles” that quickly grew to sizes of several Mpc [Zahn et al., 2010].
18
In our estimates we use the analytic Monte-Carlo “FFRT” model of Zahn et al. [2010]. It has
been shown to agree well with the most sophisticated radiative transfer simulations on scales of 100
comoving Mpc/h, while having the added advantage of allowing the modeling of arbitrarily large
volumes (the analytic scheme is about 4 orders of magnitude faster, at a given dynamic range, than
radiative transfer). This is especially important for kSZ, since large scale velocity streams lead to
the bulk of the signal. Our particular template (dashed red line in Figure 2.3) was calculated in
a 1.5 Gpc/h cosmological volume where x- and y-axes correspond to roughly 15 degrees on a side
and z-axis corresponds to redshift, with a median redshift of 8. We shift this template left-right
logarithmically by a “patchy shift” parameter RP ; that is, the power spectrum for a given shift is
related to the fiducial RP = 1 spectrum by,
f id
C = CR
P ×
(2.13)
RP is to be thought of as scaling the size of the bubbles, and is, to good approximation, proportional
to the duration of the patchy phase. The timing of reionization has a secondary small effect on the
shape and amplitude, which we neglect here.
2.1.4. tSZ-DSFG Correlation. It is reasonable to expect some correlation between the
DSFG clustering and tSZ components since they both trace the same underlying dark matter distribution, with significant overlap in redshift. Simulations which associate emission with individual
cluster member galaxies predict anti-correlations—DSFGs fill in SZ decrements at frequencies below 217 GHz considered here—on the order of tens of percent, with a correlation coefficient nearly
independent of scale [Sehgal et al., 2010]. This effect was explored in Shirokoff et al. [2010] which
found correlation consistent with zero but with significant uncertainty due to degeneracies with the
tSZ and kSZ components.
Assuming a fixed correlation rtSZ,C , the total power spectrum isn’t simply the sum of the tSZ
and DSFG clustering terms, but must also include a term given by,
(2.14)
C,νν = rtSZ,C
C C tSZ +
C,νν
,ν ν tSZ C C
C,νν
,ν ν
Note that this effect can be larger than either component individually in cross spectra between
frequencies in which each component is large. For example, even with only moderate levels of
19
correlation, the 217 × 70 GHz foreground contribution would be dominated by this correction
because of the large DSFG power at 217 GHz and tSZ power at 70 GHz.
2.1.5. Galactic Foregrounds. Though galactic foreground cleaning represents a key challenge for Planck , the aim of this chapter is to understand the impact of the extragalactic foregrounds rather than to provide the most accurate Planck forecast possible. For treatments of
galactic foregrounds see, for example, Tegmark et al. [2000], Gold et al. [2011]. The impact of
ignoring the galactic foregrounds on the temperature power spectrum is minimal provided sufficiently conservative masking and template cleaning. Our neglect of galactic foregrounds will be a
bad approximation at low- polarization and leaves our forecasts for r and τ overly optimistic; we
don’t report these forecasts.
2.1.6. CMB. For the primary CMB signal itself, we use an 8-parameter model which includes
the baryon density Ωb h2 , the density of cold dark matter Ωc h2 , the optical depth to recombination
τ , the angular size of the sound horizon at last scattering Θ, the amplitude of the primordial density
fluctuations ln[1010 As ], the scalar spectral index ns , the dark energy equation of state parameter w,
and the tensor-to-scalar ratio r. Freeing w opens up the “geometric degeneracy” which is typically
broken by adding an external dataset such as supernovae data. Rather than do this, we simply
put a ±0.3 Gaussian prior on w to reasonably constrain the chain while allowing it to explore the
parameter space.
Because we are interested in the simplest description of how the foregrounds affect cosmological
parameters, we do not consider extensions to our model such as a running spectral index, non-flat
universes, non-standard effective number of neutrino species, or a difference in primordial helium
from standard big bang nucleosynthesis. Due to the small angular scales where they affect the CMB
anisotropy, it is possible that such parameters could be even more degenerate with the foregrounds
than the “vanilla” set we consider.
For quick and highly accurate CMB calculations during our MCMC chains, we use a PICO
[Fendt and Wandelt, 2007] interpolation of a training set generated by CAMB [Lewis et al., 2000].
PICO was trained using the June 2008 version of CAMB which uses a now-outdated recombination
code. Though we use the older code, we do it in a self-consistent manner and don’t expect any
impact on our forecasting. Additionally, the training set includes the option of a non-linear lensing
contribution described in Challinor and Lewis [2005] which we use.
20
2.2. Fiducial Model and Current Constraints
For our forecasting, we create simulated power spectra (henceforth the “simulated data”) using
the model described in the previous sections. We pick one single set of model parameter values,
called the “fiducial values” or the “fiducial model” in general, which is the baseline for the different
cases of simulated data which we consider. The model used to analyze the simulated data, which
generally contains small changes relative to the fiducial model, will be called the “analysis model.”
In Table 2.3 we summarize all of the parameters in our model, the naming convention, and their
fiducial values. The fiducial values are chosen to be consistent with current cosmological constrains
from WMAP7 [Komatsu et al., 2010], and with constraints on the foreground components from
ground-based data such as SPT and ACT. In the following paragraphs, we describe the method
used to arrive at our fiducial model.
Because the expected SZ power depends strongly on cosmology, special care was taken so
that our fiducial SZ power and cosmology agree. To achieve this, we use the constraint from
Lueker et al. [2010] on the linear combination DtSZ + .46 × DkSZ = 4.2 ± 1.5 μK2 (at = 3000
and 153 GHz) along with the cosmological scalings in Table 2.1. We then importance sample the
WMAP7 ΛCDM+TENS4 chain by calculating at each step the expected SZ (kinetic and thermal)
power assuming no theory uncertainty, then applying the prior from Lueker et al. [2010]. The new
best fit point in the post-processed chain mainly shifts SZ power up relative to best fit SPT value
and σ8 down relative to the best fit WMAP7 value. All other cosmological parameters are also
affected (at a smaller level), and their new mean values form our fiducial cosmology, which remains
1-σ consistent across all parameters with WMAP7.
For the radio sources, the tightest constraints on the expected Planck power come from the
Vieira et al. [2010] catalog which contains sources in the decade of brightness just below the Planck
flux cut. Fitting a de Zotti et al. [2005] model to the data yields the values listed in Table 2.3,
notably radio Poisson power of 53 μK2 at 143 GHz assuming a 330 mJy flux cut.
Since the DSFG Poisson contribution is nearly independent of flux cut, we expect the same
Poisson power in Planck maps as in SPT maps, adjusting only for bandpass differences. We get
our fiducial value for Planck Poisson power at 143 GHz by extrapolating in frequency from the
best-fit value of the SPT 150 GHz power as given in Shirokoff et al. [2010]. Our fiducial values for
αD and αC also come from the best-fit values in Shirokoff et al. [2010]. We set σD to 0.4 following
4Available at http://lambda.gsfc.nasa.gov/
21
the arguments in Knox et al. [2004], although it is not yet well constrained by observations. We
adopt a clustering tilt nC = 1 so that it (roughly) has the shape expected at small scales due to the
the observed clustering properties of high-redshift galaxies. As argued by Scott and White [1999],
the observed clustering properties of z ∼ 3 Lyman break galaxies, namely an angular correlation
function proportional to θ−0.9 [Giavalisco et al., 1998], correspond to D ∝ 1.1 . Since we multiply
the power-law by the linear theory template, this is similar (at > 1500) to the power-law only
D ∝ 0.8 shape used as baseline models in both Dunkley et al. [2011] and Shirokoff et al. [2010].
Although ruled out by the Planck data, our fiducial model is at least closer to the measurements
than all of the other models plotted in Figure 2.1. The agreement is sufficient for our purposes
here, though we will certainly be updating our CIB modeling in the near future.
Following Battye et al. [2011] which found a mean fractional polarization of 4.5% at 86 GHz (and
varying weakly with frequency) for the WMAP point source catalog [Wright et al., 2009], we adopt
a fiducial value of fR = 0.05. For DSFGs we expect an even smaller level of average polarization
fraction. Polarized dust emission arises due to alignment of grains in interstellar magnetic fields.
We somewhat arbitrarily set fD = 0.01 for our fiducial model which is consistent with the finding
that, in our own galaxy, the coherence length for magnetic fields is much smaller than the extent
of the dust emission [Prunet et al., 1998].
Figure 2.4 shows the fiducial CMB and foreground contribution to Planck TT, TE, and EE
power spectra (with the exception of tSZ-DSFG correlation which is plotted at 30% rather than its
fiducial value of 0%).
2.3. Survey Properties
We consider simulated Planck data in the four bands between 70 GHz and 217 GHz. These are
chosen because they contain nearly all of the significant CMB information. Though the neglected
channels place little extra constraints on the CMB, they are crucial for understanding and cleaning
the foregrounds. We consider their effect implicitly by testing limits such as lowered Poisson
power amplitudes, or fixed DSFG clustering shapes. Additionally, we also consider the benefit of
higher resolution ground-based data, which we model after SPT 90 GHz, 150 GHz, and 220 GHz
channels. We divide the data into two fields: a 100 deg2 “deep” field and a 1000 deg2 “wide” field.
We henceforth refer to these two datasets as Ground-deep and Ground-wide. The depths, sky
coverage, and flux cuts used in our forecasting are summarized in Table 2.2.
22
Table 2.2 Survey Properties
Band
T (E/B)
Beam
Notes
(μK-arcmin) (arcmin)
(GHz)
Planck
70
177 (253)
14
fsky = 70%
61 (98)
10
Sc = 330 mJy
100
42 (80)
7.1
143
217
64 (132)
5
Ground-Deep
90
53
1.6
fsky = 100 deg2
150
13
1.15
Sc = 6.4 mJy
35
1.05
220
Ground-Wide
90
53
1.6
fsky = 1000 deg2
18
1.15
Sc = 6.4 mJy
150
80
1.05
220
Note. — Instrument properties used to generate simulated power spectra. The beam width is given as a full-widthhalf-max. Sc refers to the flux cut above which brighter sources are masked out.
Our simulated data take the form of auto and cross spectra from as many bands as are present
for a given patch of sky. The four Planck frequency channels form 10 TT, EE, and BB and 16
TE power spectra, with an additional 18 TT power spectra from the three extra frequencies in
regions of Ground overlap5. We do not assume overlap between Ground deep and wide, nor do
we form cross spectra between Ground temperature and Planck polarization as these are expected
to be a very small contribution to the CMB and foreground information. Planck BB polarization
is also ignored except in one test case where we find its impact is minimal on our cosmological
parameterization.
We simulate power spectrum assuming a uniform masking threshold across the sky. The only
exception is in the case of Planck and Ground overlap. For such patches of sky, we assume Planck
maps can be masked using a point source mask from the higher resolution Ground data. Thus, for
the overlap areas, even the Planck auto spectra will have greatly reduced radio Poisson power.
The non-zero width of frequency bandpasses creates a different effective frequency for each
component in each band. For components with uncertain spectral shapes, the variation in effective
frequency leads to percent level corrections which can be neglected. In this chapter, values quoted
from ground-based experiments are normalized at, and explicitly cite, the corresponding effective
5In general N frequency channels can be used to create N (N + 1)/2 power spectra of type TT, EE, and BB, and N 2
of type TE
23
Table 2.3 Summary of Model Parameters
Parameter
Cosmological
Ω b h2
Ω c h2
Θ
τ
w
ns
ln(1010 As )
r
Dusty Poisson
αD
σD
DD
fD
Radio Poisson
αR
σR
DR
γR
fR
Dusty Clustered
αC
DC
nC
SZ Effects
DtSZ
DkSZ,OV
DkSZ,P
RP
Correlations
rtSZ,C
Fiducial Value
Current Constraints (1σ)
Definition
.022565
.10709
.010376
.0799
-1
.9669
3.1462
.13
.00073
.0063
.000029
.015
.13
.014
.045
<.36 (95%)
Baryon density
Cold dark matter density
Angular size of the sound horizon at last scattering
Optical depth to reionization
Dark energy equation of state parameter
Scalar spectral index
Scalar amplitude
Tensor-to-scalar ratio
3.8
.4
5.9 μK2
.01
0.35
Spectral index
Spectral index intrinsic spread
Amplitude at = 3000, ν = 143 GHz
Dusty polarization fraction
-.5
.1
53 μK2
-.8
.05
0.1
< 0.6 (95%)
10 μK2
0.1
Spectral index
Spectral index intrinsic spread
Amplitude at = 3000, ν = 143 GHz, Sc = 330 mJy
Brightness function power law index
Polarization fraction
3.8
3.9 μK2
1
0.4
1.2 μK2
Spectral index
Amplitude at = 3000, ν = 143 GHz
Nonlinear tilt
4.3 μK2
2.7 μK2
1.5 μK2
1
< 6.8 μK2 (95%)
< 6.5 μK2 (95%)
< 6.5 μK2 (95%)
tSZ amplitude at = 3000, ν = 143 GHz
OV amplitude at = 3000
Patchy amplitude at = 3000
Patchy shift
0.8
Correlation between tSZ and DSFGs at = 3000
0
Note. — A summary of the parameters in our model. The fiducial values generate our simulated data. The current
constraints column gives the 1σ constraints on our model given WMAP power spectra and radio source counts, SPT
power spectra and radio/DSFG source counts, and ACT power spectra. Note that due to the process by which the
fiducial values were chosen (Sec. 2.2) they are not necessarily the most likely values given current data; they are,
however, totally consistent with the most likely value to within 1σ.
frequency. For Planck , it is sufficient for our forecasting purposes to ignore this and use the nominal
band centers for all components.
2.4. Forecasting Methodology
The analysis of the simulated data assumes perfectly known Gaussian beams, no calibration
uncertainty, isotropic noise, and ignores the effects of mode-mode coupling on the cut sky. While
these assumptions are not sufficient for modeling real data, we expect them to be adequate for
our purpose of modeling the extragalactic foregrounds, and understanding their importance on
cosmological parameter biases and statistical errors.
24
T
1000
z
100
H
G
10
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z
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100
7
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000
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 6
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000
000
1000
000
000
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h__ `bc f g
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]???
qh^ `bc f g
Figure 2.4 All 36 power spectra which can be formed from Planck 70 GHz–217 GHz temperature
and E-mode polarization, and the prediction of our fiducial model for the CMB and foreground
power in each of them (with the exception of the tSZ-DSFG correlation which is shown at 30%
instead of its fiducial value of 0%). The black dashed line shows the errors bars for -bins of width
Δ = 256. Dotted lines indicate negative power.
25
Under these assumptions, the so called “pseudo power spectrum” which includes both signal
and noise is given by,
(2.15)
S
+ δij w−1 exp(2 σb2 )
Cij, = Cij,
Here i and j each label one of the maps, and the noise is parameterized by the weight per solid
angle w, and the beam width in radians σb . Assuming the Gaussianity (the validity of which we
discuss in Sec. 2.4.1), the covariance on our estimate of the signal power spectra is,
Σ(ij)(kl) ≡
(2.16)
=
S
S
ij
S − C S )
(C
− Cij
)(C
kl
kl
1
(Cil Cjk + Cik Cjl )
(2l + 1)fsky
where we have suppressed the dependence for notational simplicity. We use the covariance to
form the likelihood as a function of parameters θ,
(2.17)
S
S
S
ij
S
(θ) − C
(θ)
−
C
Σ−1
C
−2 ln L(θ) = Cij
kl
kl
(ij)(kl)
Note we have neglected the normalization term since it does not vary with θ.
Our simulated data are the mean expected power spectra; i.e., they do not include a sample of
the errors from the bandpower covariance matrix. Leaving out these fluctuations has the benefit
of making the best fit χ2 equal to exactly 0 (as long as our analysis model and simulation model
are the same) and has no affect on our forecasting abilities.
Tests we performed showed that a Gaussian propagation of uncertainty from the C ’s to the
model parameters can be insufficiently accurate due to the highly non-Gaussian shape of the foreground parameter posterior likelihoods. Instead, we run a full Markov-Chain Monte-Carlo (MCMC)
analysis using a custom multi-frequency extension to CosmoMC [Lewis and Bridle, 2002].
2.4.1. Non-Gaussianity. In the previous section, we make assumptions that the signal and
noise are a Gaussian random field (Equation 2.16) and that the likelihood itself is Gaussian (Equation 2.17). The latter is a good approximation for examining the foregrounds since they are relevant
at high- where averaging over many alm ’s drives the likelihood to Gaussianity. The former, however, can be a very bad approximation for the radio/dusty Poisson and tSZ (which is just the
one-halo term) since they are sourced by Poisson number fluctuations. In general, we expect the
tSZ to suffer most from non-Gaussian effects since the fluctuation power is most heavily weighted
26
towards a small number of bright objects; the dusty Poisson component by contrast receives most
of the power from well below the flux-cut where there are many sources.
Non-Gaussianity can impact our likelihood both by producing long tails in the C probability
distribution and by the C variance differing from that expected from sample variance. The first
problem is alleviated by large sky coverage. We can use the tSZ as a worst-case scenario, where
calculations by Zhang and Sheth [2007] and simulations by Peel et al. [2009] suggest for 1000 deg2
one would expect a skewness of less than 0.1 in the distribution of C ’s. The second problem,
however, does not average away with fsky , and notably becomes worse when binning in . The full
expression for the C covariance is,
(2.18)
ΔC ΔC =
−1
fsky
2C 2 δ
T
+
(2 + 1)Δ
4π
where the first term is the sample variance for a Gaussian field, and the second is the non-Gaussian
tri-spectrum contribution [Komatsu and Seljak, 2002]. To examine the impact of ignoring the trispectrum in our analysis, we use the following method. Since binning in brings out the effect
of a non-zero tri-spectrum, we consider making one giant -bin which is inverse variance weighted
across the entire angular range. If the variance in that bin still receives a negligible contribution
from the tri-spectrum, then we can safely ignore its effect (including -to- correlations) in our more
reasonably binned analysis.
The primary contribution to the tSZ tri-spectrum comes from galaxy cluster shot noise [Komatsu and Seljak, 2002]. Shaw et al. [2009] calculate this contribution, and we use their tri-spectrum
to apply the binning procedure described above. Using only Planck 143 GHz data and forming the
single -bin, we find the error bar is increased by 1%. For Ground 150 GHz, it is increased by 30%,
although we expect this number to decrease for an analysis which does not ignore mode-coupling.
To calculate the radio tri-spectrum, we create a set of 1000 full sky radio realizations by Poisson
sampling our fiducial brightness function. We find, somewhat surprisingly, that the radio power
spectrum with a Planck flux cut is even more non-Gaussian than the tSZ, dropping to a similar
level for a Ground flux cut. Applying the binning procedure yields an error increase of 6% for a
Planck flux cut and Planck 143 GHz data, and 4% for a Ground flux cut and Ground 150 GHz.
Given the small error increase in all cases, we conclude it is safe to ignore the foreground nonGaussianities in our analysis. We do note, however, the possibility that non-Gaussianities could
27
be relevant to other analyses, for example, component separation methods which produce heavily
-binned foreground-only maps, or lensing reconstructions which rely on off-diagonal correlations.
2.5. Compression to a CMB Power Spectrum Estimate
Before getting to our results, it is useful to explore the foreground contamination in a more
model-independent manner, motivated by two drawbacks of our procedure. First, there is a large
amount of data one must work with—our bandpower covariance matrix at each is 46×46 for
Planck and 64×64 for Planck +Ground. Second, 17 foreground parameters must be marginalized
over, and if one wanted to examine constraints on a new cosmological model, the whole procedure
would have to be repeated. Here we present a procedure for compressing all the power spectra to
1) a single CMB estimate and 2) a low dimensional parametrization of the residual foregrounds in
this estimate. We describe the procedure here for temperature-only power spectra with errors that
are uncorrelated from multipole to multipole. The generalization to include to correlations and
polarization is in Appendix B.
Given N power spectra (for example the 10 TT spectra we consider for Planck ), we would like
to split our data up into N − 1 linear combinations of power spectra that have no sensitivity to
the CMB and then find the remaining linear combination that contains CMB and whose errors are
uncorrelated with those of the N − 1. With this split made, we can then derive our foreground
constraints using the CMB–free linear combinations. Doing so means foreground constraints can
be made independent of our modeling of the CMB (other than the assumed frequency dependence).
We use the N − 1 CMB–free linear combinations to find the constraints they place on our foreground model parameters via MCMC. For each point in the chain we can determine the contribution
to the CMB linear combination. We sample over all these contributions to find the mean contribution and fluctuations about that mean. We find a low-dimensional description of the fluctuations
via a principal component decomposition.
2.5.1. Splitting the power spectra into CMB–free and a CMB estimate. Let us begin
by first considering arbitrary linear combination of the power spectra,
(2.19)
C̃ μ =
wiμ Ci
i
where Ci are the i = 1 to N power spectra. The weightings we consider will be -dependent; the
lack of any labeling by is solely for notational simplicity. Here, μ is merely a label to distinguish
28
1.00
‰
0.50
0.20
t
h
ig 0.10
e
W
0.02
4‰
‰
4‰4
4‰
CMB
usy Cusrd
usy Posson
Rado Posson
1000
‰4
‰
0.05
SZ
kSZ
H op
aL
H 7 GzL
100
K
m@
l
D
10
1
CMB
kSZ
H op
aL
H 7 GzL
Rsdua PC's
1000
100
K
m@
D
l
10
1
1000
1500
2000
2500
3000
3500
Figure 2.5 Top: The -dependent weightings which form the CMB linear combination (Eq. 2.29).
All possible auto/cross spectra from Planck channels in Table 2.2 were considered. Dashed lines
indicate negative weight. Middle: The mean foreground contribution to the CMB linear combination for our fiducial model. Note for example that tSZ (purple) is not present at high because
only 217 GHz is used there. The dominant non-Poisson component for the -range where Planck
is most sensitive is the DFSG clustering. Bottom: Principal components of foreground residuals
(constrained by the CMB–free linear combinations) with amplitudes set to 1-σ. Note that we only
need two principal component amplitudes to be accurate to > 1μK2 . (The errors in bin widths of
Δ = 256 for both the CMB linear combination and for 217 GHz alone are plotted as dashed lines
in the bottom two plots.)
different weightings; the C̃ μ are a linear combination of the old power spectra with weight wiμ . Note
that if a weighting satisfies,
(2.20)
wiμ = 0
i
it is not sensitive to the CMB.
29
We would first like to find the CMB weighting wCMB which will be statistically orthogonal to
the N − 1 linear combinations which satisfy the CMB–free condition (Eq. 2.20). We would also
like this weighting to be properly normalized so that,
(2.21)
wiCMB = 1.
i
To satisfy the orthogonality condition it helps to work in a primed space defined by a linear
transformation via
wαμ =
(2.22)
Lαi wiμ
i
where L is the Cholesky decomposition of the bandpower error covariance matrix, Σ = LLT . The
advantage of the primed space is that the basis vectors in the primed space correspond to power
spectra whose errors are statistically orthogonal; i.e., with the weightings set so that wαμ = δμα
(now setting μ = 1...N ) the errors in the corresponding power spectra satisfy
δ C̃ μ δ C̃ ν = δμν .
(2.23)
The primed weights that satisfy the CMB–free condition satisfy
(2.24)
L−1
iα wα = 0.
α,i
Thus any power spectrum with primed weighting proportional to
wαCMB =
(2.25)
L−1
iα
i
is perpendicular to any vector satisfying the CMB–free condition, as one can easily verify. To find
the CMB weighting in the unprimed space we perform the inverse transform and normalize to
satisfy the normalization condition (Eq. 2.21)
(2.26)
wkCMB =
⎡
−1 ⎣
L−1
kα Li,α
i,α
(2.27)
i
30
⎤−1
−1 ⎦
L−1
kα Liα
i,k,α
=
⎡
⎣
Σ−1
ik
i,k
⎤−1
⎦
Σ−1
ik
.
Note that this is the expression for inverse-variance weighting.
Our remaining task is to construct the N − 1 CMB–free weightings in a manner that leaves
them all statistically orthogonal to the CMB weighting. We do so by applying the Gramm-Schmidt
procedure in the primed space. This gives us N − 1 orthogonal vectors that are all orthogonal to
the CMB direction as well, that we will call vαμ for μ = 2, N . The weightings in the unprimed space
are then given by
μ
viμ = L−1
iα vα .
(2.28)
We now define the matrix W so that
(2.29)
W1i =
wiCMB
Wμi =
viμ {for μ = 2...N }.
This matrix defines the linear combinations of the power spectra that have all the properties we
desire. The first row is the optimal CMB weighting and subsequent rows give the N − 1 CMB–free
linear combinations. All the linear combinations are statistically orthogonal; i.e., the covariance
matrix for the new power spectra, W T ΣW , is diagonal. Furthermore, W is non-singular so we have
not lost any information through this re-weighting.
2.5.2. Modeling the foreground residuals with principal components. With the weight
matrix W defined, we can constrain the foreground power in the N − 1 CMB–free power spectra by
running an MCMC chain. Despite the large number of parameters and power spectra, this analysis
is fast in practice because the foreground model consists of simple analytic forms and precomputed
templates, and does not depend on any costly Einstein-Boltzmann solver or lensing models. For the
set of foreground parameters at each step in this chain, we calculate the corresponding foreground
contribution to the CMB linear combination. These -dependent contributions form the columns of
the Y matrix in a principal component analysis (see Appendix A). Following the PCA procedure,
we have a few principal components and priors on their amplitudes which must be marginalized
over in a separate chain which uses only the CMB linear combination.
2.5.3. Discussion of linear combination analysis. The weightings which make up the
CMB and CMB–free linear combinations depend on the bandpower covariance matrix, and thus
on the noise properties of the instrument, any filtering which is performed, and on the true power
31
spectrum on the sky. The principal components for the foreground residuals also depend on the
choice of foreground model. For a Planck temperature-only forecast and for our fiducial model, we
present the results of a linear combination analysis.
In the top panel of Figure 2.5 we plot the weights for the CMB linear combination as a function
of . At high where the measurement is noise dominated, nearly all of the CMB information is
contained in the 217 GHz map which is the least noisy. At lower where we become dominated
by cosmic variance, the CMB information comes from the channels with the lowest foreground
contamination.
Given these weights, we plot in the middle panel of Figure 2.5 the foreground contribution to
the CMB linear combination and the error bars on this new power spectrum. Also shown are the
error bars for the 217 GHz channel alone for comparison. The maximum improvement is at = 2000
where the error bars tighten by a factor of 1.4. We also see that the dominant contribution to the
foreground power in the range where Planck is most sensitive to the CMB is the radio Poisson,
followed by the DSFG clustering.
Finally, we perform a PCA on the foreground residuals in the CMB linear combination. The
first several principal components are shown in the bottom panel of Figure 2.5. We find that all
of the variation > 1 μK2 can be described by two amplitude parameters, as compared to the 14
parameters which govern these foregrounds. Another way to put this is that using the CMB–free
combinations we can clean out almost 140 μK2 of foregrounds (at = 3000), leaving only tens of
μK2 of residual uncertainty, modeled with the two principal components.
2.6. Results
With the model and forecasting tools in place, we are ready to present the results of our main
analysis. We want to find which components can potentially cause large biases in an analysis of
Planck data, so that we can model them with sufficient care. We would also like to know how
much constraining power is reduced due to foreground confusion. Could significant improvements
in cosmological parameter constraints be achieved by using additional data and/or modeling? To
answer these questions, we run a suite of forecasting analyses aimed at singling out the effects of
each foreground contribution.
The next sub-sections are organized as follows. First we examine the importance of each
component by turning it on or off in the analysis. For the components which we deem important,
32
we check whether our modeling is sufficient to protect the cosmological parameters from biases,
both at the Planck and Planck +Ground sensitivity levels. We then examine the degradation in
statistical errors from the need to marginalize over the foregrounds, and finally we explore the
impact of adding in ground-based data.
2.6.1. Importance of the Different Components. Figure 2.6 shows the effect of removing
four foreground components—the DSFG clustering, tSZ, kSZ, and tSZ-DSFG correlation—one at
a time from the analysis model, while they are actually present in the simulated data at their
fiducial value. We present the results by plotting likelihood contours in the ns and Ωc h2 plane,
since changes in those two parameters affect the primary CMB at the smallest scales and are the
most susceptible to foreground biases. We also show the amplitudes of the clustering and SZ effects
as their -space shapes make them most degenerate with cosmological parameters. All of the chains
in this section include only Planck power spectra in the simulated data.
We expect the DFSG clustering to be extremely important to model since it is the second
largest foreground contribution to the CMB linear combination in the -range where Planck is
most sensitive. When marginalized over, this contribution is constrained to be 10.5 ± 0.6 μK2 at
= 1500, so setting it to zero is about an 18σ change. The dot-dashed green contours in Figure
2.6 show that this is compensated by a systematic bias of 7σ is ns and 11σ in Ωc h2 , along with an
increased kSZ power to about 30 μK2 . Using the middle panel of Figure 2.5 as a visual guide, we
can examine how this happens. Though the kSZ increases to compensate for the missing clustering
power at high-, its shape is flatter than the DSFG contribution to the CMB linear combination,
so ns decreases to roughly remove the extra power at a low-.
At 217 Ghz, the kSZ power in our fiducial model is a factor of 20 times smaller than the
DSFG clustering and is therefore (in terms of cosmological parameter estimation) less troublesome.
However, due to its identical frequency dependence to the CMB, we do expect the amplitude of the
kSZ signal to be degenerate with cosmological parameters. The solid black contours in Figure 2.6
demonstrate the bias introduced if the kSZ component is omitted from the foreground modeling.
We find a roughly 0.5σ bias in ns as it increases to try to fill in the missing 4.2 μK2 of kSZ power.
The DSFG and tSZ plane (bottom right panel of Figure 2.6) shows that the other foregrounds are
largely unaffected.
The thermal SZ component is neither frequency independent, nor does it contribute as much
power as the DSFG clustering, so we do not expect a bias as large as in either previous case. It
33
0.98
0.97
0.96
0.95
Fiducial
No kSZ
No tSZ
No tSZ-CIB correlation
No DSFG Clustering
0.94
0.106
0.108
0.110
W h
0.112
0.114
0.116
D
3 D @ mK No
No
No
No
D
D
D @ mK D @ mK Data
Analysis Δχ2 N
tSZ
DtSZ = 4.3
DtSZ = 0
43
6
DkSZ = 4.2 DkSZ = 0
2
1
kSZ
DC = 3.9
DC = 0
791 1930
DSFG Clustering
3
1
tSZ-DSFG Corr. rtSZ,C = −.3 rtSZ,C = 0
Figure 2.6 68% (and 95% in the bottom panel) confidence contours for a suite of test cases examining the effect of neglecting to model different foregrounds. Unless explicitly stated above,
other parameters were included in the data at their fiducial values listed in Table 2.3 and were
marginalized over in the analysis. N corresponds to the maximum number of -bins per power
spectrum one could use and still detect the error in modeling at 3 sigma (see Sec. 2.6.1 for further
discussion).
34
does, however, project into the CMB linear combination to an -shape very similar to the CMB
itself (see the middle panel of Figure 2.5), making it more likely to be degenerate with cosmological
parameters. From the results, we see about 0.3σ biases in each of ns , Ωc h2 , and Ωb h2 .
Finally, we consider neglecting a 30% tSZ-DSFG correlation, a value on the high end of expected
correlation, but still consistent with Shirokoff et al. [2010]. We expect this to have the smallest
effect on the cosmological parameters since the power contribution is sub-dominant to all of the
other foreground components at all frequencies which appear in the CMB linear combination at
> 1%. While the measured tSZ amplitude is biased at a few sigma as it raises to compensate for
the missing power, the effect is not large enough to significantly impact any of the cosmological
parameters.
One question is whether any of these analysis errors would be caught by a goodness-of-fit
test. To address this question we present Δχ2 values in the table in Fig. 2.7. We can expect rms
√
fluctuations in χ2 to be Nb where Nb is the total number of bandpowers which is roughly equal to
the number of degrees of freedom. If one is searching for signs of a contaminant that is very slowly
varying in , then one would bin coarsely to reduce the statistical fluctuations in χ2 , to make a
more stringent goodness-of-fit test.
We define N to be the number of -space bins such that the absolute Δχ2 from the fit is 99.7%
inconsistent with random fluctuations. Thus we have N = (Δχ2 )2 /(9 × Nspec ) where Nspec is the
number of power spectra (36 here). We see that binning would not have to be coarse at all to detect
the poor fit caused by neglecting clustering. We also see that for the other entries in the table,
binning would have to be extremely coarse for the fits to be noticeably poor. Indeed, the binning
would have to be coarser than is practical since the signals of interest, as well as the contaminants,
would vary significantly across a bin. We conclude that only the “no clustering” case would produce
a noticeably bad fit for Planck only.
2.6.2. Modeling Sufficiency. Given the demonstrated importance of the foreground components, we would now like to see if our modeling is sufficient to protect the cosmological parameters
from biases if we have modeled the components, but modeled them incorrectly. In this section we
consider the DSFG clustering and the tSZ effect.
For the DSFG clustering, we turn to the models plotted in Figure 2.1. Our parameterization
should have the most trouble reproducing the Amblard and Cooray [2007] model, which switches
to a power-law (as a consequence of non-linear clustering) at ≈ 2500 rather than at = 1500 as
35
0.970
0.965
0.960
0.955
Fiducial
Battaglia
Battaglia
Amblard
Amblard
0.106
tSZ H P nk L
tSZ H P nk +Groud L
Clusterig H P nk L
Clusterig H P nk +Groud L
0.107
W h
0.108
2
0.109
Figure 2.7 The effect on cosmological parameters from trying to fit our model to simulated data
which includes (orange) the Battaglia et al. [2010] tSZ template and (black) the Amblard and
Cooray [2007] clustering template. These two models are the most dissimilar to ours, and thus
show our model can protect against biases of a few percent up to Planck sensitivity. The inclusion
of Ground data necessitates more detailed modeling of only the clustering.
in our fiducial model. The orange contours in Figure 2.7 show the results obtained when fitting our
model to simulated power spectra that assume the Amblard and Cooray [2007] clustering template.
As we had hoped, for the case of Planck only (solid lines), there is no significant biasing.
We also explore the ability of our tSZ principal component model [based on the analytic model
of Shaw et al., 2010] to encompass the variations in the tSZ models shown in Figure 2.2. We
elect the Battaglia et al. [2010] one as the most dissimilar, since it lacks the effects of radiative
cooling, and should be the most difficult for the Shaw et al. [2010] model to reproduce. Despite
these differences, Figure 2.7 shows that for Planck the model is sufficient to encompass the shape
uncertainty and protect cosmological parameters.
When we add in Ground (dashed lines), the requirements on the modeling accuracy are more
stringent. For the clustering case, we see an almost 1σ bias from using our fiducial model when the
true model is the Amblard and Cooray [2007] clustering template. Analyses with current data can
tolerate much more discrepant clustering shapes [Dunkley et al., 2011]. For future Planck +Ground
analyses, the clustering shape will need to be modeled more accurately. For tSZ the modeling
appears to be more robust; tSZ-induced biases are small even in the Planck +Ground case.
36
Table 2.4 Statistical Error Degradation
Planck (fgs fixed)
Planck (fgs marginalized)
Planck +Ground
Planck (Clean DSFG)
104 Ωb h2
1.1
1.2
1.1
1.1
103 Ωc h2
1.0
1.0
1.0
1.0
104 Θ
2.6
2.6
2.6
2.6
103 ns
3.0
3.6
3.3
3.3
ln(1010 As )
1.3
1.4
1.3
1.3
DD
–
3.4
0.3
2.0
DR
–
6.0
3.0
6.0
DC
–
1.3
0.5
DtSZ
–
1.0
0.6
1.0
DkSZ
–
4.4
2.6
4.4
Note. — Entries are 1-σ constraints. Dashes indicate the parameter was fixed, while blanks mean the parameter
is not applicable to that case. The normalization parameters Dx are in units of μK2 . The different cases correspond
to: (fgs fixed) Fixing all of the foregrounds at their fiducial values. (fgs marginalized) Marginalization over our full
foreground model. (+Ground) Also including Ground auto and cross spectra in the simulated data. (Clean DSFG)
Assuming 90% reduced clustering power due to cleaning from higher frequencies.
2.6.3. Statistical Error Increase with and without Auxiliary Data. We have demonstrated the possibility of σ-level biases in cosmological parameters arising from failure to model
foregrounds. To prevent these biases, the foregrounds need to be jointly estimated or marginalized
over. We now turn to two questions: 1) How much do the cosmological parameter statistical errors
degrade due to foreground uncertainty? and 2) How much can be gained from using other data to
constrain foregrounds and thereby reduce that degradation?
The top two rows of Table 2.4 show the effect of marginalizing over our entire foreground model
as opposed to fixing it at fiducial values. In each row, the difference from 100% is the percent
degradation due to foreground marginalization. The second row shows the degradation is limited
to 20% for ns and 10% for As and Ωb h2 . We see no degradation in τ and r since they are mainly
constrained by large scales where the extragalactic foregrounds we consider are negligible. The
dark energy equation of state w is unaffected because it is mainly constrained by our ±0.3 prior.
Ground data can help reduce this degradation by better constraining the foregrounds using
auto and cross spectra that are more sensitive at small scales. The improvement from adding these
to the simulated data is shown in the row labeled Planck +Ground. The measurement of DSFG
shot noise is improved ten-fold, with the clustering and SZ effects also tightened by a factor of two.
The radio amplitude is improved through constraints on the spectral dependence, and could be
further improved though a prior on γR from Ground source counts. The effect on the cosmological
parameters is to remove essentially all of the degradation we incurred from marginalizing over the
foreground model.
Above about 300 GHz, the DSFGs are the dominant source of anisotropy power on all scales.
Correlations with maps at these higher frequencies, for example maps from Planck bands above
37
217 GHz or Herschel, can be used to place tight constraints on the DSFG components, at the price
of requiring more sophisticated modeling for the spectral dependence and shape. Even with such
modeling, the correlations are no longer fully coherent across frequencies so there is a limit to how
much of the DSFG power can be “cleaned out” of the lower frequency maps. Following results in
Knox et al. [2001], which assumes a redshift dependent grey-body emissivity density tracing the
linear matter power spectrum, we assume that we could clean out 90% of DSFG clustering power
at the lower frequencies. As in the previous case of adding in Ground data, this again is enough to
eliminate nearly all of the degradation on cosmological parameters.
2.7. Conclusions
To make full use of Planck ’s very small statistical error on CMB power spectra out to ∼ 2500,
without introducing significant bias in the cosmological parameters, we must include contributions
from extragalactic foregrounds and secondaries in our model of the data. Here we have presented
a model of these contaminants, based on the latest data and modeling developments, and demonstrated its ability to remove biases in an 8-parameter cosmological model. The foreground model
has 17 parameters – many more than any extragalactic foreground model used in analysis of CMB
data to date. Despite the large number of nuisance parameters, marginalizing over all of them only
increases statistical uncertainties in the cosmological parameters by, at most 10 to 20%. Almost all
of this degradation can be avoided by inclusion of ground-based data or higher frequency Planck
bands.
Our model includes Poisson components from both radio galaxies and DSFGs, a clustering
component due to DSFGs, contributions to kSZ power from patchy reionization, as well as after
reionization is complete, and tSZ power. If kSZ power and tSZ power are at our fiducial values
(slightly higher than the preferred values given current high-resolution ground-based data) then
ignoring them in an analysis of Planck data would produce small, almost negligible biases, to
cosmological parameter estimates. On the other hand, ignoring the clustering of DSFGs, would
lead to a very large bias in cosmological parameters.
To avoid having to marginalize over these 17 parameters every time a new cosmological model
is analyzed, we broke our procedure up into a two-step process, with the first step independent
of the model of the primary CMB power spectra. The second step is an analysis of the CMB
power spectra estimated in the first step, with a small number of foreground template amplitude
38
parameters to marginalize over. The shapes of these templates, and priors on their amplitudes, are
also outputs of the first step. Only the second step needs to be repeated in order to get constraints
on the parameters of a new model of the primary CMB power spectra.
Looking toward the near future, the model will definitely evolve, increasing the faithfulness
with which it represents reality, as we gain more information from the CMB-dominated channels in
Planck , higher-frequency Planck channels, higher-resolution ground-based data (SPT, SPTpol and
ACTpol) and higher-resolution, higher-frequency space-based data (Herschel). One could easily use
our foreground model to study potential biases in extensions of the primary cosmological model,
to include, for example, departure of the Helium mass fraction from the predicition of Big Bang
Nucleosynthesis, or a difference in the number of effective neutrino species from the standard model
value.
39
CHAPTER 3
Predicting Poisson Foreground Amplitudes
In the previous chapter, we showed that a realistic model of extra-galactic foregrouds must
include a term due to Poisson fluctuations of dusty and radio galaxies. Including and marginalizing
over such a term degrades constraints on cosmological parameters, so it is of interst to try and put
priors on it. Alternatively, we would like to check that the resulting constraints on the Poisson
amplitude are in general agreement with other sources of knowledge about these galaxies.
In this section, we show how we can use auxilliary data in the form of number counts of such
galaxies to put tight priors on the expected Poisson amplitudes for Planck. The method will center
around using source catalogs from SPT, which resolves many more such galaxies than Planck due
to its better angular resolution. Additionally, because of the non-trivial way in which sources are
masked in the Planck analysis, we develop a novel method based on simulations for calculating the
Poisson amplitude.
3.1. Methodology
In the previous chapter, and in general in the literature, the Poisson power is calculated as,
C =
(3.1)
Scut
0
dS S 2
dN
,
dS
where dN/dS are the differential number counts, Scut is some effective flux cut above which which
sources are masked, and the integral is performed independently at each frequency of interest.
Although satisfactory for rough consistency checks, Eq. 3.1 ignores that Planck point source mask
are built from a union of sources detected at different frequencies, that the Planck flux cut varies
across the sky, and the effect of Eddington bias. In order to accurately account for all of these
effects, the method we have developed calculates the Poisson power as
(3.2)
Cij
=
∞
0
dS1 ...dSn Si Sj
dN (S1 , ..., Sn )
I(S1 , ..., Sn ),
dS1 ...dSn
40
where 1...n are a set of frequencies, the differential source count model, dN/dS, is now jointly a
function of the flux at each frequency, and I(S1 , ..., Sn ) is the joint “incompleteness” of our catalogue
for the particular cut that was used to build the point source mask.
The joint incompleteness was determined by injecting simulated point sources into the Planck
sky maps, using the procedure described in Planck Collaboration et al. [2015]. The same point
source detection pipelines that were used to produce the second Planck catalogue of compact sources
(PCCS2) were run on the injected maps, producing an ensemble of simulated Planck sky catalogues
with realistic detection characteristics. The joint incompleteness is defined as the probability that
a source would not be included in the mask as a function of the source flux, given the specific
masking thresholds being considered. The raw incompleteness is a function of sky location, as the
Planck noise varies across the sky. The incompleteness which appears in Eq. 3.2 is integrated over
the region of the sky used in the analysis; the injection pipeline estimates exactly this quantity by
injecting sources only in these regions.
Because computing the N-dimensional incompleteness I(S1 , ..., Sn ) can be costly, we have explored whether it can be approximated as independent, i.e.,
(3.3)
I(S1 , ..., Sn ) = I(S1 )...I(Sn )
For a set of realistic model parameters, performing the integral in Eq. 3.2 with this approximation
given the same answer to within about 3%. This is to say that the chance to detect a source at
each frequency is (nearly) independent; the appearance of some small correlation comes the fact
that part of the noise that contributes to point source detection comes from CMB and galaxy
fluctuations, which are correlated frequency-to-frequency. However, as 3% is on the order of the
statistical uncertainty, we use the more accurate joint incompleteness in our calculations.
3.2. Source Count Modeling
The auxilliary data we use comes from [Mocanu et al., 2013, hereafter M13], which gives a
catalog of detected sources at 90 GHz, 150 GHz, and 220 GHz. These source counts are shown in
Fig. 3.1. To model these source counts, we take a simple phenomenological fit to the M13 data
which works well in the range relevant for Planck Poisson power. That is, we take the scaling of
the differential number counts with flux to be power-law, and we take the frequency dependence
of individual sources to be power-law with a spectral index which is scattered around some mean,
41
with a break in this mean and scatter at 150 GHz. With a convenient choice of normalization, this
is,
(3.4)
(α(S2 , S3 ) − ᾱ32 )2
A(S1 S2 S3 )γ−1
(α(S1 , S2 ) − ᾱ12 )2
dN (S1 , S2 , S3 )
exp −
=
exp −
2
2
dS1 dS2 dS3
2πσ12 σ23
2σ12
2σ32
with,
ln(Sj /Si )
ln(νj /νi )
α(Si , Sj ) =
(3.5)
and 1, 2, and 3 representing the SPT 90, 150, and 220 GHz bands.
This form conveniently integrates down to single frequencies analytically, which we give here
for reference,
(3.6)
dN (S1 )
=
dS1
∞ ∞
0
0
AS13γ−1
(3.7)
=
(3.8)
dN (S2 )
= AS23γ−1
dS2
(3.9)
dN (S3 )
= AS33γ−1
dS3
dN (S1 , S2 , S3 )
dS1 dS2 dS3
2 ln ν3 γα23 + 1 γ 2 σ23
dS2 dS3
ν3
ν2
ν2
ν1
ν2
ν1
2
ν2
2 ln ν2
−γα12 + 1 γ 2 σ12
2
ν1
2 ln ν2
−γα12 + 1 γ 2 σ12
2
ν1
ν2
1
2 2
ν2 2γα12 + 2 (2γ) σ12 ln ν1 ν3 ν2
ln ln
ν1
ν2 ν1
2 ln ν3
γα32 + 1 γ 2 σ32
2
ν2
ν3
ν3 ν2
ln ln
ν2
ν2 ν1
2 ln ν3
−2γα32 + 1 (2γ)2 σ32
2
ν2
ν3
ν3 ν2
ln ln
ν2
ν2 ν1
Finally, we will need to evaluate this model at Planck frequencies. Under our assumption that
sources have power-law frequency dependence (with a break at 150 GHz), it is straight forward to
write down the relation between Planck and SPT fluxes for a given source,
(3.10)
S100 = S100 (S90 , S150 )
(3.11)
S143 = S143 (S90 , S150 )
(3.12)
S217 = S217 (S150 , S220 )
Then we simply need to invert this set of equations, substitute it into Eq. 3.4 and apply the Jacobian
determinant factor. The computation is straight-forward, albeit messy, so we give the expression
42
with the Planck and SPT frequencies numerically evaluated,
(3.13)
S10.203
dN (S1 , S2 , S3 )
A(S11.203 S20.898 S30.899 )γ−1
=1.506 0.101
dS1 dS2 dS3
2πσ12 σ23
S2 S30.102
(α(S2 , S3 ) − ᾱ32 )2
(α(S1 , S2 ) − ᾱ12 )2
exp −
× exp −
2
2
2σ12
2σ32
where now 1, 2, and 3 refer to Planck 100, 143, and 217 GHz.
3.3. Results
We fit the source count model to the M13 data using an MCMC chain. In constructing the
likelihood, if M13 had provided a three-dimensional measured dN/dS in a grid of flux bins, as well
a joint covariance between all these bins, we could just construct a Gaussian likelihood directly.
As is standard, however, only single frequency dN/dS and covariance are provided, with no crosscorrelation between different frequencies. M13 do, however, provide estimates of the spectral index
of each source, and error bars on this quantity. Thus, to fit model we perform two steps. First
we integrate the multifrequency model down the model to individual frequencies (Eqs. 3.6) and
construct a Gaussian likelihood with the single frequency covariances, ignoring cross correlations.
These correlation should be small and dominated by calibration uncertainties which are taken into
account. Note that some information is lost by not using the full three-dimensional source count
data, however this is inevitable without further processing of the source count data. We then use
the distribution of spectral indices to constrain {ᾱ12 , ᾱ23 , σ12 , σ23 } directly. M13 give asymmetric
error bars on the spectral index of each source, which we use in the likelihood by “gluing” together
two half-normal distribution in such a way that the mean and 68% intervals correspond to those
quoted in M13.
After running the chain, we can also apply Eq. 3.2 to calculate the predicted Poisson power in
each Planck band and post process this quantity into the chain. The posteriors on these parameters
are shown in Fig. 3.2 along with posteriors from Planck chains.
At 143 GHz we must also take into account a sub-dominant contribution to the Poisson amplitude from dusty sources. The dusty Poisson amplitude, unlike the radio, is largely insensitive to
increasing the flux-cut from SPT to Planck levels. The constraint from George et al. [2015] scaled
(only in frequency) to 143 GHz gives (6.3 ± 0.3) μK2 and will be a large part of the total Planck
dusty Poisson. We can estimate the remaining contribution by integrating a power-law fit to the
43
measured dusty sources in between flux cuts, shown in the middle panel of Fig. 3.1. A careful treatment of this contribution à la Eq. 3.2 is not necessary as the contribution is very small. We find
0.3 μK2 , and if we flatten the best-fit power-law index from −2.5 to −1.5, since we expect lensed
dusty sources at higher fluxes to boost the number counts, the power increases to only 1.3 μK2 .
We will take (8 ± 2) μK2 , having artificially inflated the error bar to account for the possibility of
lensed sources and our rough treatment of the integration. However we stress that the uncertainty
in this number is still sub-dominant in our analysis. This would not be the case at 217 GHz, hence
why we do not attempt to give a prior there.
3.4. Conclusions
As shown in Fig. 3.2, our main conclusion is that we find excellent agreement between the
Poisson priors and the 100 and 143 GHz posteriors in all cases except chains with highL. The fits
from the non-highL chains give us physically very believable Poisson amplitudes. The tension with
highL will have to be examined further. We note the highL constraint on Poisson amplitudes is
indirect, via constrains on SZ and CIB clustering, and interaction with data and cosmological model.
We find that our priors lead to smaller amplitudes than what was calculated in the Collaboration
et al. [2013b]. Using our procedure we conclude that those numbers have overestimated the power
by neglecting the effect of the union mask; e.g., a significant number of sources at 100 GHz are
masked because they were detected only with 143 and 217 GHz and this can be shown directly by
setting the joint incompleteness I(S1 , S2 , S3 ) to be independent of S2 and S3 . We see marginal
improvement in constraints on extended parameters, about 10% on Yp , Neff , mν , and nrun . The
prior improves the constraint in Planck +highL chains more than in Planck -only chains, suggesting
that once the tension is resolved between the two datasets, they could be used to jointly break a
degeneracy that neither alone can.
44
90 GHz
150 GHz
220 GHz
1.0
S dN/dS [deg−2 ]
0.8
0.6
0.4
0.2
10
-2
10
-1
S [Jy]
10
0
10
-2
10
-1
S [Jy]
10
0
10
-2
10
-1
10
0
0.0
S [Jy]
Figure 3.1 The blue points with error bars show the M13 source count data, and the line drawn
through them is our best fit model integrated down to individual frequencies. At 150 GHz we also
show the dusty sources in green, as these are used in calculating the dusty Poisson contribution
there (see text). The lines at the bottom of each plot show the fractional contribution from each
logarithmic flux bin to the Planck radio Poisson power. Dotted/dashed/dot-dashed lines are for
the contribution at 100, 143, and 217 GHz. For example, the dotted line in the 220 GHz plot
corresponds to the contribution to the 100 GHz Poisson power from sources with those particular
220 GHz fluxes. The normalization is arbitrarily. The best-fit model is used to calculate these
curves.
Figure 3.2 Posteriors on radio Poisson amplitudes at 100 and 143 GHz from LCDM chains when the
data used is (blue) Planck2013+WP (green) Planck2013+WP+highL. These posteriors are nearly
unchanged for extended models. The prior from this procedure is in (black). To arrive at the radio
contribution in the 2013 chains, 143 GHz has had the expected dusty contribution of (8 ± 2)μK2
subtracted from the total Poisson amplitude (see text).
45
A
γ
-0.35
-0.375
-0.4
α12
0
-0.5
-1
α23
0
-0.5
-1
σ12
0.6
0.4
0.2
σ23
0.6
0.4
σ23
0.6
0.4
0.2
σ12
0.6
0.4
0.2
α23
0
-0.5
-1
α12
0
-1
-0.5
γ
-0.35
-0.375
-0.4
A
0.04
0.02
0
0.2
Figure 3.4 Constraints on the six parameter source count model from (blue) integrating the model
down to individual frequencies and fitting the M13 differential source count measurements at 90,
150, and 220 GHz and (orange) also including the M13 distribution of spectral indices as a direct
constraint on the spectral index and scatter parameters.
47
CHAPTER 4
New Constriants on Axions and Axion-like Particles
With a number of analysis methods laid out in the previous two chapters, we now turn to using
CMB data to constrain intersting models. Recent improvements in both CMB measurements as well
as inferences of primordial elemental abundances formed during Big Bang Nucleosynthesis (BBN)
motivate us to reconsider bounds on hypothetical scenarios which alter the expansion rate or inject
energy into the plasma around these two epochs. One of the simplest scenarios involves a radiatively
decaying particle, for which axions or so called “axion-like particles” (ALPs) provide a theoretically
well motivated candidate. Axions arise from perhaps the most elegant solution to the strong CP
problem as the pseudo Nambu-Goldstone boson of a new spontaneously broken symmetry [Peccei
and Quinn, 1977a,b, Wilczek, 1978, Weinberg, 1978]. Axions have a mass and standard model
couplings controlled by a single parameter: the energy scale of the symmetry breaking. ALPs form
a more general class of particles where the mass and couplings are independent. Such models can
arise from other new symmetries which are spontaneously broken [Chikashige et al., 1981, Gelmini
and Roncadelli, 1981], and in string theory [Arvanitaki et al., 2009].
In the parameter space of interest here, ALPs are weakly interacting, making them difficult
to detect in the laboratory. Cosmological constraints serve as a natural complement as the weak
coupling generally leads to later decays allowing the particles to become non-relativistic and pick
up energy compared to the plasma, leading to observable consequences upon their decay. Astrophysical bounds are also important as ALPs provide a new method for energy release from stars
and supernovae. Some early calculations and compilation of cosmological, astrophysical, and laboratory bounds on ALPs include those from Massó and Toldrà [1995, 1997]. Cosmological bounds
were recently updated by Cadamuro et al. [2011] which considered only axions, and Cadamuro and
Redondo [2012] who extended this more generally to ALPs. Among other significant advances,
Cadamuro and Redondo [2012] used newer data, treated out-of-equilibrium decays more carefully,
and performed precise calculations of the implications for BBN. These and other known ALP
bounds are tabulated in Hewett et al. [2012], Essig et al. [2013], Olive and Group [2014]. Our
work updates these by 1) using the latest inferences of primordial element abundances, 2) using the
48
latest measurements of the CMB power spectrum measurements from Planck, 3) having improved
calculations of CMB spectral distortions.
We also highlight the importance of a region in parameter space near m = 1 MeV and τ = 100 ms
which we term the MeV-ALP window. This region is interesting because it previously has evaded
all known constraints (also noted in Hewett et al. [2012], Mimasu and Sanz [2014]) and, as we
show, can correspond to a particular axion model we will call the DFSZ-EN2. Additionally, in
this mass window the symmetry breaking scale is much lower than the often-considered “invisible”
axion models. A main conclusion of this paper is that the possibility of the DFSZ-EN2 or any other
ALP hiding in plain sight in the MeV-ALP window is ruled out by the newer bounds presented
here. We show, however, that these bounds are model dependent and the MeV-ALP window can
be reopened if there is other exotic radiation present.
The paper is structured as follows. In Sec. 4.1 we discuss in more detail the scenario and its
implications for cosmology. Sec. 4.2 describes the new and tabulated constraints which we use. In
Sec. 4.3.1 we further discuss the MeV-ALP window and in Sec. 4.3.4 we give forecasts for future
probes.
4.1. The Scenario
We begin by discussing the cosmological impact of ALPs, which we define as any particle with
a mass mφ and two-photo coupling gφγ . Following standard conventions in the ALP literature, the
effective Lagrangian is
(4.1)
gφγ
1
1
φFμν F̃ μν ,
L = (∂μ φ)(∂ μ φ) − m2φ φ2 −
2
2
4
where F is the electromagnetic field strength tensor, F̃ its dual, and φ the ALP field. We often
describe the two dimensional parameter space with mφ and, in place of gφγ , the lifetime for decay
into photons
(4.2)
τφγ ≡ Γ−1
φγ =
64π
2 .
m3φ gφγ
Two processes drive the cosmological evolution of the ALP energy density. The first is the
Primakoff interaction which allows for conversion between photons and ALPs in the presence of a
charged particle q, via γq ↔ φq. Because it is a four-point interaction, the scattering rate for the
Primakoff process will depend on the density of scatterers (in this case charged particles). At early
49
times, this density grows faster than the Hubble rate, meaning the Primakoff process will always
begin in equilibrium and freeze out at later times. The second process is the direct two-photon
interaction, γγ ↔ φ. Conversely, this three-point interaction has no dependence on scatterers, and
will always begin out of equilibrium then re-equilibrate at later times.
Qualitatively, the ALP scenario depends strongly on the time ordering in which ALPs 1) freezeout from the Primakoff interaction, 2) become non-relativistic, and 3) recouple via the two-photon
interaction. The details of how these events are controlled by the two free parameters forbids certain
orderings, and in fact the vast majority of solutions fall into one of just two cases. If there is a gap
between freeze-out and recoupling during which the ALPs become non-relativistic, meaning Tfo >
mφ > Tre , there is an out-of-equilibrium decay. In this case, upon becoming non-relativistic ALPs
cease to track their equilibrium abundance and instead increase in energy density relative to the
plasma. Decay happens when the two-photon interaction becomes effective, which here is controlled
by only the ALP lifetime, independent of the mass. On the other hand, if ALPs become nonrelativistic only after recoupling, Tre > mφ , they will track their equilibrium abundance throughout
decay, a scenario which might better be called a “Boltzmann suppression.” This suppression occurs
when the temperature reaches the mass, independent of the lifetime. Although some subtleties can
occur if other events reheat the plasma while the ALPs are decoupled, to a large degree only these
two in- and out-of-equilibrium scenarios are important.
In discussing the cosmological impact of ALPs, it is useful to define two quantities. The first
is the effective number of relativistic species, Neff . As usual, this is taken so that in the limit of
complete neutrino decoupling prior to electron-positron annihilation, each neutrino species (with
antineutrinos) contributes 1 to Neff , making the total relativistic energy density,
(4.3)
ρrel = ργ
7
1 + Neff
8
4
11
4 3
.
The other quantity is the baryon-to-photon ratio η, or equivalently the energy density in baryons
Ω b h2 ,
(4.4)
η=
nb
= 2.74 × 10−8 Ωb h2 .
nγ
We sometimes superscript these quantities with BBN or CMB depending on the epoch at which
they are evaluated, although we note there is no exact definition as they can change with time in
this scenario.
50
CMB . If the decays occur
ALP decays can only reduce or leave unchanged the value of Neff
CMB does not change. Later decays increase the
while the neutrinos are still fully coupled then Neft
CMB . The value of N BBN
temperature of the plasma relative to the neutrinos and thus decrease Neff
eff
can also be decreased in the same manner if decay happens before BBN, although this region in
parameter space is fairly small. A much larger region of parameter space corresponds to ALPs
BBN by simply contributing to the relativistic energy density
decaying after BBN and increasing Neff
during BBN. ALP decays affect η similarly to Neff by causing its value to be less after the decay.
The difference is that with η CMB held constant, η before the decay is now increased. Assuming
BBN serve
that the physics of BBN is unchanged, both an increase in ηBBN and an increase in Neff
to increase the amount of primordial helium produced. For deuterium, however, there is partial
cancellation, leaving the primordial abundance only slightly reduced. This cancellation is key to
allowing the ALP scenario at all in light of the very tight constraints on primordial deuterium (see
CMB moves along a
Sec. 4.2.2). The increase in primordial helium coupled with the decrease in Neff
degeneracy direction for the CMB constraints, and is hence also generally allowed.
More quantitatively, we must track the evolution of the phase space distribution function fφ ,
which is governed by the Boltzmann equation
dfφ
= (Cq + Cγ )(fφeq − fφ ),
dt
(4.5)
where the f ’s are comoving, fφeq is a Bose-Einstein distribution, and a dependence on the comoving
momentum p is omitted for brevity. When electrons and positrons are the only charged particles
present, the scattering rate for the Primakoff interaction, Cq , is given by
(4.6)
Cq ≈
2
nep gφγ
α
[4E(me + 3T )]2
log 1 + 2 2
16
mγ [me + (me + 3T )2 ]
!
,
where nep is the number density of electrons and positrons, E is the ALP energy, and mγ = eT /3
is the plasmon mass in an electron-positron plasma [Bolz et al., 2001]. As shown by Cadamuro and
Redondo [2012], an accounting of other charged particles gives an approximation for the Primakoff
freeze-out temperature of
(4.7)
"
2
g∗ (Tfo ) 10−9 GeV−1
GeV,
Tfo ≈ 123
gq (Tfo )
gφγ
51
In practice, we begin evolving the Boltzmann equations at a temperature T0 when electrons and
positrons are the only remaining charged particles and we can thus use Eqn. 4.6 for the Primakoff
scattering rate. If the Primakoff process has yet to freeze out at T0 , as given by Eqn. 4.7, we take
take as the initial conditions for fφ a Bose-Einstein distribution at temperature T0 . If freeze-out is
earlier, we use the conservation of comoving entropy to calculate the increase in photon temperature
after Primamkoff freeze-out. The result is that ALPs are now at a reduced relative temperature of
T0 (g∗ (T0 )/g∗ (Tfo ))1/3 .
We now describe key regions in the ALP parameter space using Fig. 4.1 as a guide. An
important quantity governing the ALP evolution is the temperature, Tre , at which the two-photon
interaction re-equilibrates. Eqn. 4.5 shows that the ALP distribution fφ at a given momentum
p becomes equal to its equilibrium value fφeq when the scattering rate, Cγ (p), is on the order of
the Hubble constant. We consider only those momenta which contribute dominantly to the total
energy density, since this is the quantity we are interested in. For mφ T , these are p ≈ 0. In
this case, Cγ vastly simplifies to 1/τφγ and thus recoupling occurs when the Hubble time reaches
the lifetime. For mφ T , the important momenta are instead E ≈ p ≈ T . Ignoring the term in
brackets in Eqn. 4.8 which can be shown to depend only logarithmically on temperature for these
√
momenta, and using H = 1.66 g∗ T 2 , we arrive at (in Planck units),
(4.9)
⎧
1/2
⎪
1
⎪
⎪
⎪
⎨ τφγ 1.66√g∗
Tre = 1/3
⎪
⎪
mφ
1
⎪
⎪
√
⎩
τφγ 1.66 g∗
mφ > Tre
mφ < Tre
Fig. 4.1 shows constant Tre contours in black for three values of Tre corresponding to neutrino
decoupling, and the start and end of BBN. Here and throughout this paper, we take these temperatures to be 1 MeV, 200 KeV, and 20 KeV respectively. Fig. 4.1 also shows constant mass contours
at these three values. The line implicitly formed by Tre = mφ divides regions A and B where the
ALP becomes non-relativistic before and after the two-photon interaction re-equilibrates, respectively. Note that in region A we have mφ > Tre , which is only part of the requirement for an
out-of-equilibrium decay. The other is that Tfo > mφ , however it turns out that this is always
satisfied for any combination of mass and lifetime in region A; that is to say, ALPs can never decay
by the Primakoff process alone.
53
The evolution of the energy densities in the relevant components of the plasma for a typical inequilibrium decay (region B) is shown in the third panel of Fig. 4.2. If ALPs decay after neutrinos
are decoupled, conservation of comoving entropy implies that the temperature of the neutrinos
relative to the photons after the ALP decay and after electrons and positrons have annihilated
CMB will be
is (4/13)1/3 , as compared to (4/11)1/3 in the standard scenario. This means that Neff
reduced by (13/11)4/3 and, at fixed η CMB , the value of η prior to the decay is increased by a factor of
CMB ≈ 2.44. Because the exact timing with respect
13/11. Assuming three neutrinos, this gives Neff
to electron-positron annihilation is unimportant for the final temperature, the entirety of region B
CMB . The BBN data, however, are sensitive to the exact time of decay
shares this same value for Neff
BBN and η
via sensitivity to Neff
BBN . Since decay time when in equilibrium is controlled only by mφ ,
we find characteristic constant-mass contours in this region in the BBN constraints in Figs. 4.3,
4.4, and 4.5.
The second panel of Fig. 4.2 shows instead a typical out-of-equilibrium decay (region A). In this
region an important quantity is the fractional increase in the energy of the photons (or equivalently
the decrease in Neff ) once the ALP decays at t ∼ τφγ . Because the ALP here is non-relativistic,
this is,
(4.10)
#
#
ρφ ##
mφ a−3 ##
1
√
∼
∼
∼ mφ τφγ
#
#
−4
Neff
ργ t=τφγ
a
t=τφγ
where we have assumed that the ALP does not remain non-relativistic for too long before decay
√
hence the universe is radiation dominated and a(t) ∼ t. This assumption is true for models right
√
on the edge of the allowed region, thus we find characteristic contours of constant mφ τφγ for
cosmological constraints in region A as seen in Figs. 4.3, 4.4, and 4.5.
In both regions A and B there are characteristic contours arising from phenomena which depend on the fractional energy injection after a certain reaction has frozen out, i.e. after a certain
temperature. For example, CMB and BBN constraints are only sensitive to energy injected after
neutrino decoupling, as any earlier injection is “invisible” because it is rethermalized among all
components. We will also consider bounds from CMB spectral distortions, which depend on the
energy injected only after the freeze-out of reactions which can bring the CMB spectrum back into
chemical equilibrium.
In region B, fractional energy injection is independent of either mass or lifetime, and the amount
after a certain temperature is controlled only by the mass. On the non-equilibrium side the story
54
is slightly more subtle. The fractional energy increase at a certain time or scale factor depends on
the mass, as per Eqn. 4.10. However, we cannot simply assume a ∼ 1/T because the ALP decay
alters this relation. In particular, the Friedmann acceleration equation shows that it does so in a
way which exactly cancels the mφ dependence, leaving only sensitivity to τφγ . Essentially, more
massive ALPs lead to more total energy injection, but delay the time it takes to reach a certain
temperature, leaving the same amount of energy injected after that temperature. Thus, contours
of constant fractional energy injection are m = const in region B and τ = const in region A. We
note that this is the same as a contour of constant decay time. The final key region in Fig. 4.1,
region C, is delineated by such a contour, and corresponds to zero energy injection after neutrino
decoupling and thus no cosmologically observable imprints.
The Primakoff freeze-out temperature plays a smaller role in the ALP evolution than the recoupling temperature, although we briefly mention two effects stemming from events happening in
the gap between freeze-out and recoupling (in regions of parameter space where the gap exists).
We first note that if no reheating of the plasma occurs in this gap and ALPs are relativistic, they
still track their equilibrium abundance even though they are decoupled. One possible reheating
occurring during the gap is the QCD phase transition which imprints a sharp feature our constraints. During this transition the temperature of the ALPs roughly doubles along with the rest
of the plasma if they are still coupled, leading to an energy density roughly twenty times larger.
Another possibility is reheating from electron-positron annihilation, a case in which the ALPs can
re-equilibrate relativistically thereafter. Such a scenario happens near the top-left part of Fig. 4.1
and is shown in the fourth panel of Fig.4.2. For deuterium and helium constraints shown in Fig. 4.5,
near this region we can find both lines of constant Tre and Tfo .
4.2. Constraints
4.2.1. Cosmic Microwave Background.
4.2.1.1. Frequency Spectrum. The measurement of the CMB frequency spectrum by COBE/FIRAS
places very tight bounds on spectral distortions away from a black-body spectrum [Fixsen et al.,
1996], limiting possible energy injection into the plasma [Wright et al., 1994].
The effects of energy injection depend crucially on when it occurs, with the time-line roughly
divided into three eras. In the earliest era, reactions that change photon number are fast and
any injection of photon energy is quickly rethermalized. This leads to only an adjustment of the
55
[2012] under a further approximation of Eqn. 4.11 where DC and BR are taken to be infinitely fast
until they instantaneously freeze-out at T ≈ 750 eV. In the “previous constraints” panel of Fig. 4.4
we reproduce their result, showing that qualitatively this is a very good approximation. The use of
Eqn. 4.11 becomes more important, however, for constraining scenarios with even smaller chemical
potentials generated deeper into what is currently called the T -era. Two future missions which are
predicted to reach such sensitivity are PIXIE and COrE [Kogut et al., 2011, Collaboration et al.,
2011], for which we give forecasts in Sec. 4.3.4.
4.2.1.2. Angular Power Spectrum. Measurements of CMB anisotropies have been recently improving, both from the ground [Keisler et al., 2011, Das et al., 2011, Story et al., 2013, Das et al.,
2014] and from space [Collaboration et al., 2013b, collaboration et al., 2013]. Better angular resolution and lower noise have tightened up small-scale constraints where the CMB is most sensitive
to changes in Neff and Yp , both of which are altered by the decay of ALPs.
A fully general treatment would include ALPs in the set of Boltzmann equations for calculating
the CMB power spectrum, but it turns out this is not necessary for the scales that are currently
well measured. For these scales, all of the physical effects of ALPs are in fact identical to changes
in Neff and Yp , as long as we assume adiabatic initial conditions. This is essentially because decays
must happen early enough, as enforced by the spectral distortion bound discussed in the previous
section, which requires the decay happen by T ≈ 750 eV or equivalently z ≈ 3 × 106 . The angular
scales well constrained by CMB measurements, roughly 3000, correspond to physical scales
which do not begin to enter the horizon until about z ≈ 3 × 105 . At this point two things are
different in the ALP scenario as opposed to the standard case 1) the amplitude of neutrino density
perturbations upon horizon entry is reduced relative to the photons and 2) the expansion rate is
different. However, both are exactly captured in the standard scenario by changing Neff . No other
scale-dependent changes to spatial perturbations are possible because the relevant scales are still
outside of the horizon by the time of the decay. Finally, we note that the altered helium abundance
is taken as an input to the CMB spectrum calculation, and in the ALP scenario it is now just at a
different value.
In their work, Cadamuro and Redondo [2012] took as a CMB constraint a lower bound on Neff
as given by WMAP7. The Planck data tighten this constraint and are also sensitive to Yp . We use
the joint constraint on these two parameters given by the combination of Planck+WP+highL from
Collaboration et al. [2013b], approximating the likelihood as Gaussian and taking just the mean
60
element abundances thus probe new physics at play during this epoch [e.g., Pospelov and Pradler,
2010]. For our case of an ALP, light element production is affected by changes to the cosmic
expansion rate during BBN and to the extrapolation of η back from the CMB epoch. For the
majority of the parameter space where ALPs decay after neutrino decoupling, they are still present
BBN and the expansion rate. Additionally, because the decay
during BBN and hence increase Neff
decreases η, fixing ηCMB to the observed value generally leads to an increased ηBBN . Both effects
BBN increases D/H
serve to increase Yp , but partially cancel for D/H and 7 Li. The increase in Neff
while the increase in η has the opposite effect, with the latter about twice as large, leaving an
overall reduced D/H. For 7 Li it is instead the former which wins out. The light-element trends in
the mass-lifetime planes of Fig. 4.5 bear out these expectations, as we now see in detail.
In practice we have modified the AlterBBN code of Arbey [2012] to include changes to the
expansion history and η due to ALPs. Our code assumes the photon spectrum is instantaneously
rethermalized, in effect ignoring the possibility that high energy photons from the decay can break
apart already-formed nuclei. Bounds due to this phenomenon constitute so called “photo-erosion”
bounds, discussed in e.g. Cyburt et al. [2009] and references therein. We will consider them
separately at the end of this section.
Deuterium is observable at z ∼ 3 in QSO absorption systems, via the ∼ 82 km/s isotope shift
between D and H Lyman absorption lines. Recent D/H measurements have been reported Cooke
et al. [2014],
(4.16)
D
= (2.53 ± 0.04) × 105 ,
H
which represents a factor ∼ 3 improvement in precision. These bounds are now so tight as to place
the measurement errors on level footing with uncertainties associated with nuclear reaction rates
and with a determination of η from the CMB. To account for these uncertainties, we first consider
the joint likelihood for the CMB, D/H, and nuclear reaction rate measurements, which can be
written as
DH(mφ , τφγ , η, αi ) − DH
− log L =
2
2σMEAS
+ log LCMB (η, Neff , Ω)
(4.17)
+ log LNUCL (αi , Ω )
62
2
where DH ± σMEAS = (2.53 ± 0.04) × 105 as per Eqn. 4.16, η and Neff are evaluated at the CMB
epoch but we omit the label for brevity, Neff = Neff (mφ , τφγ ) is uniquely set by the mass and
lifetime, αi are parameters describing the nuclear reaction rates, and Ω and Ω are any remaining
cosmological and nuisance parameters. Neff is important and appears explicitly because it is both
dependent on the ALP parameters and its measurement from the CMB is significantly degenerate
with η. We next analytically marginalize over all parameters other than mφ and τφγ under the
assumption that these other parameters have Gaussian posterior likelihoods and that D/H depends
linearly on η. This gives
(4.18)
−Neff
DH(mφ , τφγ , η̄ + rση Neff
σNeff , αi ) − DH
2
− log L =
2
2
2 σMEAS
+ σNUCL
+ σETA
2
with
(4.19)
2
σETA
=
dDH
ση
dη
2
(1 − r2 )
where η̄ ± ση and Neff ± σNeff are the mean and standard deviation of the posterior likelihoods from
the CMB with all other parameters marginalized over, and r is the correlation coefficient between
η and Neff . The presence of r in this equation can be understood by considering the r = 1 case,
which would imply that CMB measurements could turn a fixed Neff into a perfect determination
of η; the quantity above at which the D/H prediction is evaluated, η̄ + rση (Neff − Neff )/σNeff , is
the mean of this determination. Because in our case Neff is fixed by the mass and lifetime, it
would mean η is also fixed, leading to no extra uncertainty in D/H. In reality, we find r ≈ 0.4 from
the Planck measurements. We make one further approximation which is that neither the D/H
derivative nor the nuclear reaction rate uncertainty depends on the values of mass and lifetime or
the fact that η evolves with time in the ALP scenario, which we have checked is sufficient. We find
σNUCL = 4.5 × 10−7 using AlterBBN which takes the αi to be principal components in the nuclear
reaction rate parameter space [Fiorentini et al., 1998]. Numerically evaluating the D/H derivative
and taking posterior likelihoods from Planck +WP+highL, we find σETA = 6.9×10−7 . When added
in quadrature these lead to an effective deuterium constraint of
(4.20)
D
= (2.53 ± 0.091) × 105
H
63
which is meant to be compared to a theoretical prediction calculated for the particular values of η
and αi given in Eqn. 4.18.
The effects of these D/H constraints on the (mφ , τφγ ) plane appear in Fig. 4.5. We see that the
effect of an ALP is always to decrease D/H due to the ALP’s effective increase of ηBBN winning
BBN . Moreover, we see that the high precision of the D/H measurements
out over the increase in Neff
leads to a tight constraint on the ALP space in all regions where the decays occur after neutrino
decoupling. Indeed, D/H is now a very powerful probe of ALPs.
The primordial 4 He abundance is inferred astronomically from observations of emission spectra
of highly ionized gas in primitive nearby dwarf galaxies, i.e., in low-metallicity extragalactic HII
regions. The primordial abundance is traditionally inferred by extrapolation to zero metallicity.
To derive helium and metal abundances from the observed spectra requires characterization of
the thermodynamic properties of the emitting gas (i.e., temperature, density). The analysis of
[Aver et al., 2013] derives these quantities simultaneously in a self-consistent manner, and finds a
primordial abundance
(4.21)
Yp = 0.2465 ± 0.0097
where the uncertainty is quantified with an MCMC analysis. We adopt this as our fiducial primordial 4 He constraint. Using a similar data set, Izotov and Thuan [2010] give a helium constraint
of, Yp = 0.254 ± 0.003 where the errors are derived in a less conservative manner. In Fig. 4.5 we
see that, as expected, the effect of an ALP is to increase Yp for almost all of the parameter space
where the decays occur after neutrinos decouple. However, the constraints are not as strong as
those of D/H. There is an island of parameter space around mφ > 100 MeV and τφγ ∼ 1 sec where
Yp decreases. This region corresponds to decays happening between neutrino decoupling and the
start of BBN. Here we have ηBBN = ηCMB and Neff < 3, the latter of which serves to decrease Yp .
It is interesting to note this low helium region does not extend along the entire constant decay-time
contour, cutting off once we enter the in-equilibrium decay side. This occurs because in-equilibrium
decays reduce the ALP energy density more slowly than do out-of-equilibrium ones, and thus the
entire decay cannot fit in the short time between neutrino decoupling and BBN.
Finally, the primordial 7 Li abundance is inferred from observations of the atmospheres of lowmetallicity (extreme Population II) stars in the Galactic stellar halo. Down to a metallicity of
(Fe/H) ∼ 10−2.8 (Fe/H) , these stars have lithium abundances that are the same to within a small
64
scatter consistent with observational errors. The independence of Li with Fe in this “Spite plateau”
indicates that lithium is primordial [Spite and Spite, 1982], and implies a primordial abundance
(4.22)
Li
= (1.6 ± 0.3) × 10−10
H
[Sbordone et al., 2010]. At lower metallicity, however, the Li/H abundance scatter increases dramatically, but always below the Spite plateau value. This suggests that in these very metal-poor
stars some lithium destruction has occurred; the reason for this remains unclear.
The astronomically-inferred lithium abundance in Eqn. 4.22 is inconsistent with the primordial
value expected from standard BBN theory combined with CMB determinations of η. The observed
Li/H value is low at the ∼ 5σ level. This is the “lithium problem” [reviewed in, e.g., Fields, 2011].
Stellar astrophysics uncertainties may be the origin of the problem, but solutions to date require
fine tuning and do not explain the observed Li/H “meltdown” at very low metallicities. A more
radical and intriguing solution is the presence of new physics during or after BBN. The challenge for
such scenarios is to reduce lithium substantially without drawing other light elements–particularly
deuterium–from their concordant primordial abundances. The ALP scenario tends to aggravate
the problem by increasing lithium slightly, as seen in Fig. 4.5, and as expected due to the ALP
effect on η. Thus, awaiting a resolution to the lithium problem, we do not consider lithium bounds.
In closing we note that our calculations have neglected the effects of photoerosion of the light
elements. This occurs 1) when the ALP mass exceeds light-element binding energies mφ >
∼ B ∼ 10
MeV, and 2) for decay time scales long enough so that the decay photons interact with light
elements before thermalization. This leads to some deuterium destruction via γd → np, but a net
production due to e.g., γ 4 He → dd. Thus constraints arise from D/H, Yp , and 3 He/D [Ellis et al.,
1984, Kawasaki and Moroi, 1995, Cyburt et al., 2003]. These were recently computed for purely
electromagnetic decays by Cyburt et al. [2009], assuming that the decay photons provide a negligible
contribution to the energy density and thus expansion rate. In this case, the constraints are only
7
4
important for τX >
∼ 10 sec and mX >
∼ 10 eV, with X the decaying particle. This regime shows
a Yp drop due to photoerosion and a corresponding D/H increase. These trends could potentially
bring Yp and D/H predictions back into agreement with observations, but would require a more
detailed calculation. Since the regions of parameter where this can happen are already ruled out
by CMB observations, we ignore the effects of photo-erosion.
65
4.2.3. Laboratory. Laboratory bounds on ALPs come from a variety of different experimental setups. At lower masses, roughly m eV, some examples include photo-regeneration experiments (“shining light through walls”), microwave cavities, and helioscopes [for a review, see Hewett
et al., 2012, Essig et al., 2013, Olive and Group, 2014]. For the larger masses considered here, the
best constraints come from electron-positron colliders and beam dumps.
The presence of the ALP-photon interaction allows for the possibility of single-photon final
states at electron-positron colliders. Early interpretation of searches for these events in terms of
constraints on the ALP coupling gφγ was done by Massó and Toldrà [1995]. Both Kleban and
Rabadan [2005] and Mimasu and Sanz [2014] have further shown the ability of current and future
colliders to improve these bounds. Here we reproduce the constraint from LEP given by Mimasu
and Sanz [2014] of,
(4.23)
gφγ < 4.5 × 10−4 GeV−1 ,
valid in the entire lifetime range considered here. The excluded region is labeled “Collider” and
shown in magenta in Fig. 4.3.
Additional constraints come from beam dump experiments, where ALPs would be produced in
the beam dump, penetrate through shielding, then decay to photons which can be detected by a
downstream detector. We use the constraints from Bjorken et al. [1988] which find that at 95%
confidence,
(4.24)
mφ τφγ > 1.4 keV sec.
This is labeled “Beam Dump” and shown in red in Fig. 4.3.
4.2.4. Globular Clusters and SN1987A. ALPs offer a new means for energy loss from stars
if they can both be produced in stellar interiors and have sufficiently weak interaction strengths
to subsequently escape. In the case of SN1987A, the energy loss can affect the duration of the
neutrino pulse from the handful of neutrinos which were detected, placing constraints on the ALP
interaction strength [Massó and Toldrà, 1995, 1997]. These bounds are reproduced in Fig. 4.3.
We note that they assume the ALP has only a two-photon coupling, although constraints based
on other couplings exist. Energy loss can also affect the duration of the red giant phase and of
the horizontal branch, leading to a different observed ratio of such stars in globular clusters. We
66
use the bounds from Cadamuro and Redondo [2012] based on arguments of Raffelt and Dearborn
[1988] and Raffelt [1996]. These are also reproduced in Fig. 4.3.
4.3. Discussion
4.3.1. The MeV-ALP Window. An interesting feature of the exclusion regions prior to this
work is the allowed window bounded on all sides near m ∼ 1 MeV and τ ∼ 100 ms corresponding
to an ALP decay during BBN. We will henceforth call this the MeV-ALP window. It can be seen
in the left panel of Fig. 4.4 as well as in Hewett et al. [2012], Mimasu and Sanz [2014]. Further
interest is driven by the fact that a particle in this window could actually be a DFSZ axion, in which
case its symmetry breaking scale is close to the electroweak scale. One of the main conclusions of
this work is to show that this region is now, in fact, ruled out by the combination of CMB+D/H
measurements.
We first briefly review two relevant generic axion models, referred to as the KSVZ and DFSZ
models. Both models introduce a new global U (1) symmetry which is approximately broken at
some energy scale fφ giving rise to an axion with mass mφ . The symmetry breaking scale is related
to the axion mass by non-perturbative effects and given by
√
mφ =
(4.25)
z mπ f π
1 + z fφ
where mπ is the pion mass, fπ its decay constant, z = mu /md the ratio of up to down quark
masses. Axion models differ in what other new fields are introduced to implement the symmetry
breaking and how these, as well as standard model fields, transform under the new U (1). The
KSVZ model [Kim, 1979, Shifman et al., 1980] has the standard model fermions neutral, whereas
in the DFSZ model [Zhitnitskij, 1980, Dine et al., 1981] they can carry U (1) charge. These model
dependent choices in turn affect the axion’s effective photon coupling which arises from fermion
loops, ultimately leading to a consistency relation between axion mass and photon lifetime which
can be written as,
(4.26)
τφγ =
2(4 + z)
E
−
N
3(1 + z)
−1 √
z 16π 3/2
fπ mπ
1+z α
2
m−5
φ
with the model dependence captured by the E/N factor. In evaluating this relation, we will adopt
fixed values of mπ = 135 MeV, fπ = 92 MeV, and z = 0.56, ignoring small uncertainties that lead
to roughly a 10% uncertainty in the axion mass [Cadamuro et al., 2011, Beringer et al., 2012].
67
The KSVZ model has E/N = 0, so that we have,
(4.27)
τ
φγ
sec
= 6.20 × 104
m −5
φ
eV
The DFSZ model we consider here has E/N = 2, hence we refer to it as the “DFSZ-EN2” model,
with consistency relation,
(4.28)
τ
φγ
sec
= 2.66 × 105
m −5
φ
eV
These two consistency relations are shown as the dashed lines in Fig. 4.3. Other values for E/N
are possible, but the DFSZ-EN2 has the distinction of having a particularly weak coupling because
E/N ≈ 2(4 + z)/3(1 + z) and so these terms nearly cancel in Eqn. 4.26 [Kaplan, 1985, Cheng et al.,
1995]. This weaker coupling means that the DFSZ-EN2 is consistent with the collider bounds over
a larger range of masses as compared to the KSVZ. Ultimately it is that the consistency relation
passes through the MeV-ALP window which motivates our interest in this model. The lower mass
limit for the DFSZ-EN2 in the MeV-ALP window is around mφ ∼ 200 keV, corresponding to
fφ ∼ 30 GeV, less than an order of magnitude from the electroweak scale vweak ∼ 246 GeV where
the axion was initially thought to lie.
While it is interesting that this mass range for the DFSZ-EN2 was previously allowed, this part
of parameter space for the DFSZ-EN2, and more generally the entire MeV-ALP window, is now
ruled out by the combination of CMB+D/H data. This region corresponds to in-equilibrium decays
CMB = 2.44. The decay happens essentially in the middle of BBN, increasing N BBN
hence it gives Neff
eff
CMB
and ηBBN , which, as discussed previously, increases Yp and decreases D/H. The decrease in Neff
and increase in Yp moves along the degeneracy direction for CMB measurements, and is allowed
even by our updated CMB constraints coming from Planck (see Fig. 4.6). It is in combination
with the D/H constraints that the MeV-ALP window is closed, with the best fitting model within
the window ruled out at about 3.5σ. If we replace the CMB constraint from Planck with previous
measurements from the combination of WMAP, ACT, and SPT, the window is ruled out at a
similar significance. This is despite the 20% tighter η constraint from Planck because the central
value also shifts lower, increasing D/H back towards the measured value. Conversely, replacing the
D/H measurement with previous bounds does open the MeV-ALP window again, as seen in the left
panel of Fig. 4.4. It is thus the new bounds from Cooke et al. [2014] that are the key improvement.
68
4.3.2. A Loophole in the Presence of Extra Radiation. While the MeV-ALP region is
now excluded by the CMB+D/H measurements, this result depends on the assumption of having
no extra radiation besides neutrinos and the ALP. In some scenarios, for example as predicted
by the string axiverse [Arvanitaki et al., 2009], it is natural to have many ALPs, some of which
could also contribute to Neff but be light enough to remain otherwise invisible. Motivated by this
possibility, we explore constraints when in addition to the ALP mass and lifetime, we also allow an
extra arbitrary addition to Neff .
The MeV-ALP region is ruled out largely because it predicts too low an abundance of primorBBN increases D/H and can bring it back into agreement with
dial deuterium. An addition to Neff
measurements. The penalty is a further increase in Yp , but because the helium constraints are not
as tight as D/H, an allowed window now opens up again.
We explore this window with Markov Chain Monte Carlo (MCMC) 2. We run two MCMC
chains, one for the ALP case where both mass and lifetime are free parameters, and another for
the DFSZ-EN2 case with only the mass free and the lifetime given by Eqn. 4.28. In both cases
we also leave free the quantity we call ΔNeff which controls any extra relativistic energy density
preBBN
= 3 + 4/7 + ΔNeff . The ALP contributes
at some early time before BBN, meaning that Neff
4/7 because it is one bosonic degree of freedom and is fully thermalized in all regions of parameter
space explored by the chain. The likelihood includes all of the bounds in Fig. 4.3. In either the
ALP of DFSZ-EN2 cases, we find that the MeV-ALP window is again allowed and the best-fitting
model consistent with all of the data at 1σ.
The mass posterior distributions for the ALP and DFSZ-EN2 chains are shown in Fig. 4.7.
In both cases masses below 200 keV are excluded. Masses above 1 MeV in DFSZ-EN2 case are
excluded by the collider bound, but are allowed in the ALP case because models can evade this
constraint by having a smaller photon coupling. We also show a forecast for a next generation
electron-positron collider SUPER-KEKB after two years of integration (discussed in Sec. 4.3.4)
which can probe down to almost exactly the 200 keV minimum.
The corresponding likelihoods for Neff are given in the right panel. These show that the data
preBBN
∼ 4.7 and diluting this down to
accommodate the ALP scenario by initially having Neff
CMB ∼ 3.4 via the ALP decay. Constraints on the extra radiation are ΔN
Neff
eff = 1.13 ± 0.30. Thus,
one or (marginally) two extra neutrino-like particles allow for an ALP in the MeV-ALP window.
2https://github.com/marius311/cosmoslik
70
Table 4.1 Best-fit parameters
preBBN
CMB 105 D/H
m [keV] τ [ms] Neff
Neff
Yp
1010 Li7
ΛCDM
3
3.046
2.56
0.247
4.58
1936
4.6
4.73
3.52
2.46
0.255
5.02
ΛCDM+ΔNeff +ALP
ΛCDM+ΔNeff +DFSZ-EN2
734
6.2
4.61
3.30
2.45
0.258
5.15
Table 4.2 Best-fit χ2
ΛCDM
ΛCDM+ΔNeff +ALP
ΛCDM+ΔNeff +DFSZ-EN2
Planck(2) Cooke(1) Aver(1) Izotov(1) Planck+Cooke+Aver(4) Planck+Cooke+Izotov(4)
0.96
0.10
0.00
5.62
1.06
6.68
0.22
0.57
0.74
0.08
1.53
0.87
0.02
0.91
1.43
1.86
2.35
2.79
Alternatively, the decay of such an ALP can hide the existence of one or two additional neutrino-like
particles from the tight CMB constraints which would otherwise rule them out. A similar loophole
allowing for extra radiation has been proposed by Ho and Scherrer [2013].
We next test the extent to which the data prefer these extended models. We perform a simple
test using best-fit χ2 values given in Tab. 4.2. If the χ2 for the extended model decreases significantly
as compared to ΛCDM, then roughly that model is preferred. Although all bounds from Fig. 4.3 are
included in the fit, we only give χ2 for those which are not hard cutoffs. When using the combination
of Planck +D/H+Yp , we find the baseline data choice slightly disfavors both the DFSZ-EN2 and
ALP models. The only case where there is a preference for the extend model is in the ALP case
when using the helium constraint from Izotov and Thuan [2010]. Here we find an improvement in
χ2 of 5.81 when we have added 3 new free parameters, something we expect to happen by chance
only 12% of the time. If the high helium value inferred by Izotov and Thuan [2010] is confirmed,
then this scenario is a natural explanation as it can increase helium compared to the standard value
CMB roughly unchanged.
while keeping the deuterium abundance and Neff
4.3.3. A Simple Expression for Exclusion Bounds. Given the improved constraints from
CMB+D/H measurements, we suggest a simple expression for ALP bounds which can be adopted
by those who prefer a simpler picture than the many probes shown in Fig. 4.3. The CMB+D/H
data alone now essentially rule out any energy injection after neutrino decoupling, giving allowed
parameters of
(4.29)
τφγ
mφ
> 107 and
< 10−2 .
eV
sec
71
This assumes no extra radiation besides ALPs, and is valid roughly until masses become small
enough or lifetimes long enough that decays happen after CMB last scattering. These late decays
are analyzed in more detail by Cadamuro and Redondo [2012], who find approximately
(4.30)
τφγ
mφ
< 101 or
> 1024 ,
eV
sec
are once again allowed.
4.3.4. Forecasts. Measurements relevant for placing bounds on ALP parameters have been
recently improving and will continue to do so in the near future. It is expected that several probes
will soon have the sensitivity to further test the MeV-ALP window. In this section we compute
forecasts for some of them.
Currently CMB anisotropies alone are not enough to rule out the MeV-ALP window where
there is a maximum of Neff = 2.44 and an increase in Yp , but which lie along the CMB degeneracy
direction and are thus allowed. Abazajian et al. [2013] show that a Stage-IV CMB experiment could
CMB to within 0.02 at 1-σ. Given such tight constraints, the arguments of Sec. 4.2.1.2
measure Neff
may need to be revisited; while it is true that the ALP decays before any modes relevant for the
CMB enter the horizon, the difference is only an order of magnitude in scale factor. Assuming any
eff
such corrections do not provide loopholes, if a Stage-IV CMB measurement found a value of Neff
consistent with the standard value of 3.046, the MeV-ALP window would be strongly ruled out 3.
For the CMB constraints, however, there will always be the possibility of extra radiation exactly
canceling the dilution due to the ALP decay (as in Sec. 4.3.2).
Helium and deuterium measurements will continue to improve as more systems are discovered
and systematic errors are better understood. Additionally, D/H measurements can be significantly
improved by better measuring nuclear reaction rates in the laboratory. Fig. 4.8 shows that in the
D/H-Yp plane, the ΛCDM+ΔNeff +ALP scenario is not continuous with the standard model. This
allows for sufficiently tight Yp and D/H constraints around the standard values to rule out the
presence of ALPs, independent of assumptions about extra radiation. We find that the minimum
requirement for all points in the ΛCDM+ΔNeff +ALP chain to be ruled out at > 3σ by the combination of Yp and D/H measurements is a factor of two improvement in the D/H error and a
3Their forecast assumes standard BBN, however we believe it unlikely that freeing Y could degrade it so much as to
p
allow 2.44
72
factor of three improvement in the Yp error bar. We note the former is possible by eliminating the
uncertainty due to nuclear reaction rates alone.
On the laboratory side, in Sec. 4.2.3 we used the constraint from LEP which limited gφγ 4.5 × 10−4 GeV−1 . If a search for single-photon events were performed using the entire 1000 fb−1
of currently existing KEKB data and the standard model background was found, forecasts from
Kleban and Rabadan [2005] show that the constraints could improve to gφγ 10−6 GeV−1 . Similar
improvement could come from reinterpreting the constraints on dark photons from 500 fb−1 of
BABAR data given in Collaboration [2014] in terms of ALPs. Attempting either of these is outside
of the scope of this paper, but could make significant improvements in the mass bounds shown
in Fig. 4.7. SUPER-KEKB, an ongoing upgrade to KEKB, plans to improve on the integrated
luminosity of KEKB by a factor of ten with two years of integration, and by a factor of fifty with
ten years [Abe et al., 2010]. Taking the constraint on gφγ to scale with the square root of the
integrated luminosity, we find that within the first two years, SUPER-KEKB can rule out the last
remaining part of the MeV-ALP window through which the DFSZ-EN2 passes (the mass limit
forecast is shown in Fig. 4.7). The full ten year forecast is exactly enough to close the MeV-ALP
window entirely, bringing the collider bound up to gφγ 10−6 GeV−1 where the SN1987a constraint
begins. This simple forecast is in broad agreement with a more sophisticated calculation given by
Mimasu and Sanz [2014].
Finally, PIXIE is a proposed mission which would greatly improve constraints on CMB spectral
distortions [Kogut et al., 2011]. Expected bounds on the μ parameter are,
(4.31)
|μ| < 5 × 10−8 .
Bounds due to spectral distortion constitute constant decay-time boundaries (Sec. 4.1), and we find
that the PIXIE forecast moves up the exclusion region by only a factor of five in decay time as
compared to FIRAS. This is much less than needed to reach the MeV-ALP window. Evidently it
is very difficult to constrain decays happening very much into the T era using spectral distortions.
4.4. Conclusion
We have shown how cosmological, astrophysical, and laboratory bounds can provide complementary constraints in the mass-lifetime parameter space of axions and ALPs. We have updated
73
the work of Cadamuro and Redondo [2012] with constraints from the Planck satellite and the latest inferences of primordial D/H and helium abundances, and provided a more detailed calculation
of spectral distortions. The most important change is that CMB+D/H constraints now rule out
the entire region corresponding to decays happening after neutrino decoupling but before CMB
last scattering. This includes closing the MeV-ALP region of parameter space which we have also
shown can correspond to a type of DFSZ axion. The presence of additional radiation can relax
the exclusion regions and once again allow the MeV-ALP window. Although it is allowed in this
case, including such a particle slightly degrades the overall fit to the CMB+BBN data if our most
robust data combination is used. Alternatively, this model can provide a natural explanation for
a high value of helium such as found by Izotov and Thuan [2010]. Forecasts for future primordial
abundance measurements and for SUPER-KEKB are promising; both have the ability to test the
MeV-ALP window even in the presence of extra radiation. A detection by either would be very
exciting. Even the null result, however, would signify a new level of precision in our understanding
of the contents of the primordial plasma.
74
CHAPTER 5
Conclusion
In this thesis we have examined the challenges involved in analyzing precision data from the
CMB damping tail, such as that from Planck or SPT. One such challenge involves the removal of
extra-galactic foregrounds, and we have developed a realistic model for extra-galactic foregrounds,
and shown that such a model is necessary to extract unbiased cosmological information from the
CMB. Additionally, we have given a new method to put priors on this model with auxiliary data in
the form of source count data from SPT, and shown that the results from Planck generally agree
very well with these priors. The exception is when including highL data, which we leave for further
examination.
We have also used Planck data, whose analysis involved some of the methods discussed in this
thesis, to constrain a model of axions and axion-like particles. We found that the interesting region
in parameter space which we termed the MeV-ALP window is newly ruled out by the combination
of Planck and D/H data. If additional radiation is present aside from the axion or ALP, then the
bounds relax. Depending on the position one takes, this can be interpreted as either a way to
allow one or two extra thermalized neutrino-like species, or a way to allow an MeV axion, both
interesting possibilities.
The future of CMB science continues to be exciting. Forecasts for next generation experiments
promise significant improvements in constraints and even guaranteed detection of new physics
[Abazajian et al., 2013]. The precision era has forced us to improve and scrutinize our analyses,
some examples of which we have demonstrated here. We expect this will be a continued necessity
and look forward to it eagerly.
75
APPENDIX A
Principal Component Analysis for Power Spectra
In Ch. 2 we use a principal component analysis (PCA) to reduce the dimensionality of the
tSZ astrophysical parameter space (Sec. 2.1.2) and of the entire foreground contribution to the
CMB linear combination (Sec. 2.5.2). Here we present in more detail the procedure used in those
sections.
Given nsim realizations of an n -length power spectrum, drawn from a statistically significant
sample of parameter space, we first form the [n × nsim ] matrix Y . In each column of Y we place
the deviation from the mean power spectrum for that realization. This matrix is then subject to a
singular value decomposition,
(A.1)
Y = U SV T ,
where the columns of U contain the orthogonal basis vectors, S is a diagonal matrix of the singular
values and the columns of V are the principle component weights. The i-th realization can be
written as
(A.2)
(i)
y =
Φμ wμ(i) ,
μ
where the singular value-weighted orthogonal basis vectors are
(A.3)
Φμ = √
1
Uμ Sμμ .
nsim
and the wμ are the weights,
(A.4)
wμ(i) =
√
nsim Viμ .
Because the singular values are in decreasing order, we can truncate the sum in Eq. A.2 at
some small value of μ and still accurately describe each realization. Furthermore, the distribution
of weights P (wμ ) sampled over nsim realizations provides a prior on our principal component
amplitudes equivalent to the parameter space which was sampled to produce the Y matrix.
76
APPENDIX B
CMB Linear Combination Generalization to Off-Diagonal
Correlations
The method for constructing a best estimate of the CMB presented in Section 2.5 assumes only
temperature power spectra, and a covariance which is diagonal in . The generalization to include
polarization and mode-mode coupling induced by sky masking is presented here. The math is,
infact, indentical for the two scenarios, so in this appendix we’ll refer to polarization types with
the understanding that we could just as well be talking about different values of .
The added difficulty in dealing with different power spectrum types (for simplicity here just
TT and EE) comes from the fact that we cannot arbitrarily create linear combinations which sum
TT
EE
them. For example, C = C100
GHz − C100 GHz neither preserves CMB normalization, nor can we be
sure it is CMB–free independent of model. To remedy this, we make sure that in our construction,
any linear combination we consider must have the CMB signal cancel out for all but one type. For
TT
EE
EE
example, C = C100
GHz − C100 GHz + C143 GHz is a valid linear combination.
We start by considering the covariance matrix for the TT and EE spectra.
⎡
⎣
(B.1)
ΣT T
···
···
ΣEE
⎤
⎦
By creating the single-type weight matrix (Eq. 2.29) for each of the diagonal blocks, we can cancel
the CMB out of all but two weightings. The new covariance will look like,
(B.2)
⎡
⎣
⎡ ⎛
⎤⎡
WTTT
0
0
T
WEE
⎦⎣
ΣT T
···
···
ΣEE
⎤⎡
⎦⎣
0
0
WEE
77
⎤
⎢ ⎝ σT T 0 ⎠
⎥
···
⎢
⎥
..
⎢
⎥
.
0
⎦=⎢
⎞ ⎥
⎛
⎢
⎥
⎢
⎥
0
σ
⎢
⎠ ⎥
⎝ EE
⎦
⎣
···
..
.
0
⎤
WT T
⎞
Under a permutation to place the two CMB weightings at the front, the covariance becomes,
⎡ ⎛
⎢ ⎝
⎢
⎢
···
⎣
(B.3)
⎞
···
σT T
σEE
···
⎤
⎤
⎡
⎠ ··· ⎥
T
Σ
Σ
cross
⎥ ⎣ cmb
⎦
⎥≡
⎦
Σ
Σ
cross
dif f
..
.
where we’ve labeled Σcmb as the covariance between TT and EE estimates, Σdif f as the covariance
of the CMB–free differenced spectra, and Σcross as the cross-correlation between the two. We
now would like to do one final reweighting in an attempt to zero out the cross-correlation. The
reweighting should leave the differenced spectra unchanged, should not add TT and EE together,
but will add CMB and CMB–free power spectra. Note that this will continue to satisfy our earlier
condition that all but one CMB type canceling out. The reweighting matrix will look like,
⎤
⎡
⎣
(B.4)
I
0
W
I
⎦
The new covariance must satisfy,
⎡
(B.5)
⎣
I
0
W T
I
⎤⎡
⎦⎣
Σcmb
Σcross
ΣTcross
⎤
⎤⎡
⎦⎣
Σdif f
I
0
W
I
Solving for W yeilds,
(B.6)
W = −Σ−1
dif f Σcross
78
⎡
⎦=⎣
⎤
Σcmb
0
0
Σdif f
⎦
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