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Probing inflation with the cosmic microwave background

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THE UNIVERSITY OF CHICAGO
PROBING INFLATION WITH THE COSMIC MICROWAVE BACKGROUND
A DISSERTATION SUBMITTED TO
THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES
IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
DEPARTMENT OF ASTRONOMY AND ASTROPHYSICS
BY
VINICIUS MIRANDA BRAGANCA
CHICAGO, ILLINOIS
AUGUST 2015
ProQuest Number: 3724448
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TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Non-Canonical Kinetic Inflation . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Generalized Slow Roll . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
2 STEPS IN DBI INFLATION . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . .
2.2 Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Analytical Solution for Small and Sharp Warp Steps
2.2.2 Analytical Solution for Large and Sharp Steps . . . .
2.3 Constraining Sharp Steps . . . . . . . . . . . . . . . . . . . .
2.3.1 Power Spectrum Accuracy . . . . . . . . . . . . . . .
2.3.2 Parameter Settings . . . . . . . . . . . . . . . . . . .
2.3.3 Minimization . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Steps in the Canonical Limit . . . . . . . . . . . . . .
2.3.5 Steps in the Low Sound Speed Limit . . . . . . . . .
2.3.6 Future Tests . . . . . . . . . . . . . . . . . . . . . . .
2.4 Constraining Broad Steps . . . . . . . . . . . . . . . . . . .
3 POLARIZATION PREDICTIONS OF
FIELD INFLATION . . . . . . . . . .
3.1 Inflationary Reconstruction . . .
3.2 Results . . . . . . . . . . . . . . .
3.2.1 Curvature Only . . . . . .
3.2.2 Curvature and Tensors . .
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BROAD BAND FEATURES IN SINGLE. . . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
9
9
11
15
17
24
35
36
43
45
48
52
54
57
63
64
68
69
72
82
LIST OF FIGURES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
GSR source function for a warp step and its approximate form based on Tφφ .
GSR vs exact solution for the power spectrum . . . . . . . . . . . . . . . . . .
Analytic vs. GSR0 solution for a small amplitude sharp step . . . . . . . . . .
Evolution of cs and H across a warp step. . . . . . . . . . . . . . . . . . . . .
Maximum oscillation amplitude C2 for a warp step as a function of csa . . . .
Evolution of cs (upper) and H (lower) across a potential step . . . . . . . . .
GSR approximation vs the exact solution for the curvature power spectrum . .
GSR approximation vs the exact solution for the temperature power spectrum
Minimum χ2 relative to the best fit smooth (no step) model . . . . . . . . . .
Best fit models for a potential step and a warp step . . . . . . . . . . . . . . .
χ2 improvement as a function of csa for a warp and potential steps . . . . . .
Polarization and temperature polarization cross power spectra . . . . . . . . .
Temperature power spectrum derivatives for the best fit model . . . . . . . . .
Temperature power spectra, including tensors . . . . . . . . . . . . . . . . . .
Step in tensor-scalar ratio parameter H cs relative to no step . . . . . . . . . .
Polarization EE power spectrum . . . . . . . . . . . . . . . . . . . . . . . . .
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17
18
22
29
32
33
39
40
50
51
53
55
56
59
61
62
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Curvature power spectrum constraints . . . . . . . . . . . . . . . . . . . . .
Temperature power spectrum constraints . . . . . . . . . . . . . . . . . . . .
Polarization predictions for the various model-data combinations . . . . . . .
Principal component amplitude constraints for the curvature source function
Posterior probability distributions of the first 3 PC parameters . . . . . . . .
First 3 PC eigenvectors for the different model-dataset combinations . . . . .
3 PC filtered curvature source G03PC . . . . . . . . . . . . . . . . . . . . . .
Posterior distribution of the tensor-to-scalar ratio . . . . . . . . . . . . . . .
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69
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78
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iii
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LIST OF TABLES
2.1
2.2
2.3
2.4
3.1
3.2
Best fit flat ΛCDM cosmological model without a step . . . . . . .
Foreground model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Best fit potential step model with cs = 1 . . . . . . . . . . . . . . .
Likelihood for models with tensors and steps with non-inflationary
fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
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parameters
. . . . . . .
43
44
47
Models and datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameter constraints (68% CL) for various model-dataset combinations . . . .
67
81
iv
60
ACKNOWLEDGMENTS
I would like to express my deep appreciation and gratitude to my advisor Prof. Wayne
Hu for all his support and guidance. He was indeed a wonderful advisor, with a brilliant
intuition on physics! I will never forget our infinite chains of e-mails written at all times,
both at day and night, about all the tiny details of the Mukhanov-Sasaki equation - a single
ordinary differential equation that encodes almost an infinite amount of interesting physics!
I would also like to thank Prof. Hu for sharing with me his vision for detail, the 0.1%
detail. I have to assume that this shocked me at first, but with time I started to understand
that it is indeed the 0.1% detail that distinguish inspiring research against trivial results!
Another amazing characteristic of Prof. Hu is his devotion to his students. His door was
always open to discuss the many problems I faced with my research over these past years.
Finally, I will never forget the times I came worried to his office with dozens of plots
summarizing strange results that came out from projects that took me thousands of C++
lines of code to develop and, almost inexplicably, Prof. Hu was able of find exactly where
I was wrong using just a very basic calculator - his legendary ten dollars Casio calculator!
Thanks for believing in me Prof. Wayne Hu!
I would also like to express my gratitude to the committee members Prof. Scott Dodelson,
Lian-Tao Wang and Steve Meyer for reading carefully my thesis and for providing me very
good suggestions. I would also like to express my deep appreciation for Prof. Peter Adshead
and Prof. Cora Dvorkin for collaborating with me!
Finally, I would like to thank my friends Sean Johnson, Ke Fang, Alessandro Manzotti,
Alan Zablocki, Pierre Gratia, Yin Li and Monica Mocanu among others! You all helped me
a lot!
v
ABSTRACT
The existence of a quasi-deSitter expansion in the early universe, known as inflation, generates the seeds of large-scale structures and is one of the foundations of the standard cosmological model. The main observational predictions from inflation include the existence of a
nearly scale-invariant primordial power spectrum that is imprinted on the cosmic microwave
background (CMB), which has been corroborated with remarkable precision in recent years.
In single-field slow-roll inflation, a field called the inflaton dominates the energy density of
the universe and slowly rolls in an almost perfectly flat potential. In addition, the motion of
the inflaton field is friction dominated, with its velocity being completely specified by its position in the field space. This basic scenario is known as the slow-roll approximation and its
validity is controlled by the magnitude of the so-called slow-roll parameters. Generalizations
of single-field slow-roll inflation provide a wealth of observational signatures in the CMB
temperature power spectrum, CMB polarization spectrum, primordial non-Guassianity and
in lensing reconstruction. This thesis provides a series of consistency checks between these
observables that can distinguish slow-roll violations from alternative explanations.
vi
CHAPTER 1
INTRODUCTION
1.1
Non-Canonical Kinetic Inflation
In single-field inflation there is a field, the inflaton, that dominates the energy density of
the background metric and that generates the seeds of large-scale structure through its
perturbation on flat slicing. The dynamical evolution of this field in non-canonical models
is dictated by the action (Armendariz-Picon et al. 1999; Hu 2011)
Z
Sφ =
√
d4 x −gp(X, φ)
(1.1)
where X is the kinetic term of the inflaton field
1
X = − ∇µ φ∇µ φ .
2
(1.2)
Examples of such models include the string motivated Dirac-Born-Infeld (DBI) inflation
(Silverstein & Tong 2004; Alishahiha et al. 2004) and K-Inflation (Armendariz-Picon et al.
1999; Garriga & Mukhanov 1999). These models can be viewed as generalizations of the
canonical action, p(X, φ) = X − V (φ) where fluctuations of the inflaton field in the rest
gauge travel with unity effective speed of sound (Garriga & Mukhanov 1999; Gordon & Hu
2004).
The energy-momentum tensor associated with the non-canonical scalar field is given by
(Armendariz-Picon et al. 1999; Hu 2011)
µ
T ν=
∂p(X, φ) µ
µ
∇ φ∇ν φ + p(X, φ)δ ν .
∂X
(1.3)
In the Friedmann-Robertson-Walker metric, the scalar field behaves as a perfect fluid with
1
pressure p(X, φ) and energy density
ρ(X, φ) = 2X
∂p(X, φ)
− p(X, φ) .
∂X
(1.4)
In addition, the sound speed of perturbations is given by (Gordon & Hu 2004)
c2s =
∂p(X, φ)/∂X
.
(∂p(X, φ)/∂X) + 2X(∂ 2 p(X, φ)/∂X 2 )
(1.5)
Note that this equation reduces to c2s = 1 for canonical actions.
While for canonical fields the slow-roll parameters can be defined as a function of the
potential, the relation between slow-roll parameters and the function form of the action needs
to be generalized for non-canonical fields. Instead of depending explicitly on the canonical
potential, H , ηH and δ2 are now defined as
d ln H
,
d ln a
1 d ln H
,
ηH ≡ H −
2 d ln a
2 − dηH .
δ2 ≡ H ηH + ηH
d ln a
H ≡ −
(1.6)
(1.7)
(1.8)
Similar to the canonical case, sufficient inflation requires H to remain small for many e-folds
and this requires |ηH | 1 to leading order in the slow-roll approximation. In the standard
slow-roll approximation, ηH is also a slowly varying function and then δ2 ηH . In addition
to these parameters, time variations of the sound-speed have a direct impact in the scale
dependence of the curvature power spectrum. Then, the standard slow-roll approximation
also requires that the parameters
d ln cs
,
d ln a
dσ1
σ2 ≡
.
d ln a
σ1 ≡
2
(1.9)
(1.10)
are small and constant to leading order. As a concrete example, DBI inflation will be
extensively study throughout this thesis.
1.2
The Generalized Slow Roll
The Mukhanov-Sasaki equation can be written in the form (Garriga & Mukhanov 1999;
Gong & Stewart 2001; Stewart 2002)
d2 y
2
g(ln s)
2
+
k
−
y
=
y,
ds2
s2
s2
(1.11)
where s is the sound horizon of the perturbations, and the source on the RHS is given by
g=
f 00 − 3f 0
,
f
(1.12)
with
cs aHs 2
f 2 = 8π 2 H 2
.
cs
H
(1.13)
This equation specifies the evolution of each k-mode of curvature fluctuations on comoving
slicing
R=
ksy
,
f
(1.14)
for both canonical and non-canonical fields. The main advantage of using curvature perturbations on comoving slicing is that they freeze-out in the limit when the k-mode is outside
the horizon, which corresponds to x ≡ ks 1. In this limit, the curvature power spectrum
becomes
∆2R ≡
xy 2
2
2
k 3 PR
2 [Re(y)] + [Im(y)] .
=
lim
=
lim
x
x1
x1 f
2π 2
f2
(1.15)
The mode constancy outside the horizon guarantees that the power spectrum is independent
on the details of the re-heating physics that happened after inflation ended.
The explicit integration of the Mukhanov-Sasaki equation has the advantage of provid3
ing the curvature power spectrum without any assumptions about the magnitude and the
time variation of the slow-roll parameters. However, the exact equation of motion can only
handle models on a case-by-case basis, which is not ideal for providing model-independent
constrains. In addition, numerical methods for solving the Mukhanov-Sasaki equation on
models where the inflaton field transverse rapid transitions that break the slow-roll approximation temporarily are too slow for practical usage in Monte Carlo Markov Chains (MCMC).
In this case, approximate schemes for constructing accurate analytical solutions greatly enhances the efficiency of the search for signatures predicted by these models in the Cosmic
Microwave Background (CMB).
Following Dvorkin & Hu (2010b), Hu (2011) and Stewart (2002) very closely, I start
the derivation of the Generalized-Slow-Roll approximation - which will be extensively used
throughout this thesis to overcome some of the limitations of the direct integration of the
Mukanov-Sasaki equation - with the construction of an exact, but formal, solution of equation
(1.11) via the Green function technique
Z ∞
du
g(ln u)y(u)Im[y0∗ (u)y0 (x)] .
y(x) = y0 (x) −
2
u
x
(1.16)
Here y0 (x) is the solution for the homogeneous LHS of the Mukhanov-Sasaki equation
i ix
e ,
y0 (x) = 1 +
x
(1.17)
which generates a scale invariant power spectrum predicted by perfect slow roll where H = 0,
ηH = 0 and δ2 = 0. The formal solution then shows that the source g(ln s) carries information
about deviations from this perfect slow roll approximation. Given Eq. (1.16), the formal
solution for curvature perturbation on comoving slicing can be written, in the limit x 1,
4
as
Z
Z
1 ∞ du
x3 ∞ du
lim (xy) =
X(u)FR (u)g(ln u) +
W (u)FR (u)g(ln u)
(1.18)
3 x u
9 x u
x1
Z ∞
Z ∞
i
du x3
du
ih
+ 3−
F
(u)g(ln
u)
+
W
(u)F
(u)g(ln
u)
+ O(x2 ) ,
I
I
3
3
u
u
u
x
x
where
FR (u) ≡
Re[y(u)]
Re[y0 (u)]
and FI (u) ≡
Im[y(u)]
Im[y0 (u)]
(1.19)
correspond to a more explicit parametrization of the unknown source y(u) on the RHS of
the formal solution (1.16). Furthermore, the windows W (u) and X(u) are given by
3
3 sin(2u) 3 cos(2u) 3 sin(2u)
W (u) ≡ − Im[y0 (u)]Re[y0 (u)] =
,
(1.20)
−
−
u
2u
2u3
u2
3
3 cos(2u) 3 sin(2u) 3 cos(2u)
3
X(u) ≡
−
+
Re[y0 (u)]Re[y0 (u)] = −
+ 3 (1 + u2 ) .
3
2
u
2u
2u
u
2u
A close inspection of Eq. 1.19 shows that the real part of the mode function contribute
to the curvature power spectrum quadratically in powers of the source g(ln s). As a first
approximation, Dvorkin & Hu (2010b) and Stewart (2002) disregard this term under the
assumption that the replacement y(u) → y0 (u) in the RHS of the green function recovers a
good solution. With the auxiliary replacement
Z
x3 ∞ du −3
1f0 lim
u g(ln u) = −
,
u
3 f
x1 3 x
(1.21)
the curvature perturbation becomes
Z
1
1 f 0 1 ∞ ds
lim |R| =
1+
+
W (ks)g(ln s) .
f
3f
3 smin s
x1
(1.22)
Equation (1.22) can be viewed as a first attempt to develop the GSR approximation.
5
However, there are problems in this expression that need to be addressed, the first one being
the fact that Eq. (1.22) allows the curvature perturbation to cross zero for large deviations
from slow-roll scale invariance. This problem can be ameliorated with the assumption that
it corresponds to the linearized expansion of the exponential function
ln ∆2R (k)
Z
2 ∞ ds
= G(ln smin ) +
W (ks)g(ln s) ,
3 smin s
(1.23)
where
G(ln s) ≡ ln
1 2 f0
.
+
3f
f2
(1.24)
Equation (1.23) still possesses the fundamental issue of not matching the fully non-linear
result for super-horizon modes that guarantees the constancy of curvature perturbations on
comoving slicing (Dvorkin & Hu 2010b). The relation (1.14) between curvature perturbations and mode functions mandates the solution y to deviate substantially from the slow-roll
solution y0 whenever f evolves noticeably outside the horizon and this breaks the approximation that the replacement y(u) → y0 (u) on RHS of the green function recovers a good
solution.
The enforcement of the conservation of curvature perturbations on comoving slicing outside the horizon requires a new form for the equation (1.23) that does not depend on the
lower integration limit as long as ksmin 1. This can be achieved with the replacement
g → G0 if the second-order term (f 0 /f )2 in the slow-roll approximation is neglected given
that the exact relation between the sources g(ln s) and G0 (ln s) is given by
2g
2 f 0 2
0
=G +
.
3
3 f
(1.25)
Even for models that break the slow-roll approximate, the replacement g → G0 remains
valid as long as the inequality |f 00 /f | (f 0 /f )2 is not violated. In terms of the slow roll
parameters, this translate into a good approximation if η ∼ O(1) only where |δ2 | 1.
6
Then, the final expression for the leading order GSR approximation is given by (Dvorkin
& Hu 2010b; Hu 2011)
ln ∆2R (k)
Z ∞
Z ∞
ds 0
ds
0
W (ks)G (ln s) = −
W (ks)G(ln s) ,
= G(ln smin ) +
smin s
smin s
(1.26)
where
2 f 00
f 0 f 02 −3 − 2
(1.27)
3 f
f
f
2 2 aHs
2 aHs
2
(2H − 2ηH − σ1 ) +
−1 +
− 1 (4 + 2H − 2ηH − σ1 )
=
3
3 cs
3 cs
2 h
i
1 aHs
2
2
2
2δ2 + 2H − 2ηH − 2ηH − 3σ1 + 2ηH σ1 + σ1 − H (4ηH + σ1 ) − σ2 .
+
3
cs
G0 =
Now, the curvature spectrum does not depend on the evolution of the source f outside the
R
horizon since − 0∞ d ln xW 0 (x) = 1 and limx→0 W 0 (x) → 0.
The leading order GSR approximation shown on Eq. 1.26 is exact to linear order in
the source g(ln s) and its accuracy is sufficient for searches of large slow-roll violations in
the WMAP data (Dvorkin & Hu 2011, 2010a; Miranda et al. 2012; Adshead et al. 2012).
However, the remarkable accuracy of the new Planck data (Ade et al. 2013b,c) mandates an
approximation that is exact to second order in the source g(ln s). Following Dvorkin & Hu
(2010b) and Hu (2011) the next to leading order GSR approximation is given by
ln ∆2R (k)
Z ∞
1 2
1
1 2
ds
0
2
= G(ln smin ) +
W (ks)G (ln s) + ln [1 + I1 (k) + I2 (k)] + I1 (k) ,
4
2
2
smin s
(1.28)
where
Z ∞
1
ds
I1 (k) = √
X(ks)G0 (ln s) ,
2 0 s
Z ∞
ds
1
f0
I2 (k) = −4
[X(ks) + X 0 (ks)] F (s) ,
s
3
f
0
7
(1.29)
with
Z ∞
dv f 0
F (s) =
.
2
s v f
.
8
(1.30)
CHAPTER 2
STEPS IN DBI INFLATION
2.1
Introduction
In this chapter, I develop the theoretical tools necessary to probe rapid oscillations in
the curvature power spectrum generated by steps in DBI inflation. My original motivation
to study this class of models came from hints in the WMAP7 temperature data, which
favored canonical models with steps in the potential at the level of ∆χ2 ∼ 10 − 20 in
comparison to power law inflation (Flauger et al. 2010; Adshead et al. 2012; Peiris et al.
2013; Meerburg et al. 2013). Models of inflation that generate high-frequency oscillations
in the curvature power spectrum also predict oscillatory signals in the E-mode polarization
spectrum and Non-Gaussianity (Miranda & Hu 2014; Adshead et al. 2012, 2013). These
predictions provide additional ways to test the presence of such oscillations in the primordial
spectrum against the null hypothesis that the fluctuations seen in the temperature power
spectrum are caused by instrument noise or cosmic variance.
The 2013 Planck collaboration analysis also favored the presence of such oscillations at
the level of ∆χ2 ∼ 10−20 in comparison to the power law slow-roll inflation (Ade et al. 2013c;
Easther & Flauger 2013; Benetti 2013; Meerburg & Spergel 2013). This signal seemed to
have faded away in the 2015 Planck release, and this new analysis carried out by the Planck
was based on theoretical developments that are presented in this thesis (Ade et al. 2015b).
Rapidly oscillating power spectra can be generated during inflation if the inflaton field
rolls over rapid transitions in a time period much less than an e-fold. In canonical inflation,
for example, this transition can be induced by a step in the potential (Adams et al. 2001).
Dirac-Born-Infeld (DBI) inflation, a single-field model with a non-canonical Lagrangian (Silverstein & Tong 2004; Alishahiha et al. 2004), may also generate high-frequency oscillations
through transient but rapid changes in the sound speed (Hailu & Tye 2007; Bean et al.
2008a; Miranda et al. 2012). DBI models with a step in the sound speed will be discussed
9
in greater detail in the sections that follow.
The presence of oscillations in the inflaton potential is another interesting possibility that
can generate rapidly oscillating power spectra. Such models generate high-frequency oscillations in the curvature power spectrum that scale logarithmic in k-space, while step models
produce linear oscillations in k-space, (Chen et al. 2007; Silverstein & Tong 2004). These
models are all good examples where the generalized slow-roll provides a good framework for
the construction of analytical solutions for the inflationary power spectrum, which greatly
enhances the efficiency of the analysis of these models against the data (Adshead et al. 2012;
Miranda & Hu 2014; Motohashi & Hu 2015).
This chapter is structure as follows. Section 2.1.1 provides a basic introduction on DBI
inflation, including expressions for the slow-roll parameter required when evaluating the
GSR power spectrum sources. Section 2.2 introduces steps in the potential and in the warp
brane, and discusses the accuracy of the generalized slow-roll in for steps that generate order
unity changes in the curvature power spectrum. Sections 2.2.1 and 2.2.2 construct analytical
solutions that extend previous analyses (Adshead et al. 2012) by analytically treating large
amplitude sharp steps in both the potential and warp at arbitrary sound speeds including
new second order corrections that are required by the enhanced precision of the Planck data.
Section 2.3.1 provides a jointly fit for step and cosmological parameters, and argue that
a joint variation is crucial for interpreting constraints on step parameters. Although other
analyses in the literature have also jointly varied parameters, not including 2013 Planck
collaboration analysis (Ade et al. 2013c), they did so in a different context where the oscillations persist out to arbitrarily high multipoles (Easther & Flauger 2013; Meerburg &
Spergel 2013). In particular, I show that joint variation is important for finite width steps
and misleading constraints arise when cosmological parameters are fixed. The considerations presented in this section were indeed recognized and followed by the new 2015 Planck
collaboration analysis (Ade et al. 2015b).
Finally, section 2.4 analyses how broad steps, generated when the inflation rolls over
10
a transition in a time period around one e-fold, could alleviate one tension in the 2013
Planck temperature data given by an excess of power in the temperature spectrum at low
multipoles. Such problem is aggravated in the presence of tensors at any level, and I quantify
its significant for a model with no tensors and for two extreme values of the scalar-tensor
ration. I also show predictions for the E-mode polarization and lensing reconstruction that
can be tested with the 2015 Planck release.
2.1.1
Basic Definitions
I consider DBI inflation to be a phenomenological model with the Lagrangian density
i
h
p
L = 1 − 1 − 2X/T (φ) T (φ) − V (φ),
(2.1)
where the kinetic term 2X = −∇µ φ∇µ φ. In braneworld theories that motivate the DBI
Lagrangian, φ determines the position of the brane, T (φ) gives the warped brane tension,
and V (φ) is the interaction potential.
As a consequence of the non-canonical kinetic structure, field perturbations propagate at
the sound speed
cs (φ, X) =
p
1 − 2X/T (φ).
(2.2)
The inflaton energy density and pressure can be expressed in terms of the sound speed as
ρ(φ, X) =
1
− 1 T (φ) + V (φ),
cs
p(φ, X) = (1 − cs )T (φ) − V (φ).
(2.3)
Note that for X/T 1, cs = 1 and the Lagrangian, ρ and p take on their canonical forms.
For the background equations of motion, the acceleration equation
φ2N
HN
=−
≡ −H ,
H
2cs
11
(2.4)
is taken where X = H 2 φ2N /2, the Hubble parameter satisfies the Friedmann equation H 2 =
ρ/3, and, finally, the field equation
φN N = −
Vφ
Tφ
1
HN
2
+ 3cs φN − c3s 2 + (1 − cs )2 (1 + 2cs ) 2 .
H
2
H
H
(2.5)
Here and throughout this chapter the subscript N denotes d/d ln a, the subscript φ likewise
denotes d/dφ. If warp and potential features are absent near the initial conditions, initial
values for {φ, φN , H} can be set on the slow-roll attractor
φN ≈ −
cs Vφ
,
3 H2
H2 ≈
V
,
3
(2.6)
where I assume that the Vφ term dominates over Tφ . Given that I choose to solve Eq. (2.4)
and (2.5), I must ensure that the Friedmann equation is exactly satisfied on the initial
condition (Mortonson et al. 2009). This can be achieved by first choosing the initial φ(Ni ),
then taking
r
HφN Ni
=−
V Vφ
cs ,
3 V
cs N =
i
2
V Vφ
1+
3T V 2
!−1/2
,
(2.7)
and calculating ρ, H exactly through Eq. (2.3). Since φN = HφN /H, now there is a selfconsistent set of initial conditions {φ, φN , H} at Ni . This technique remains valid for all cs
in the slow-roll approximation. On the other hand, for cs 1 the slow-roll approximation
can remain valid even for steep potentials in Eq. (2.6).
These equations are evolved until the field reaches φ = φend , assumed to be the end of
inflation, and define N = 0 to be this epoch
Z ln aend
N=
d ln a,
(2.8)
ln a
such that N < 0 during inflation. For the purposes of calculating the power spectrum, it
is useful to express the efolding number, N , in terms of the sound horizon, the comoving
12
distance sound can travel from N to the end of inflation
Z 0
Z 0
cs
1
cs
s(N ) =
dÑ
=
dÑ
.
aH
aend N
N
eÑ H
(2.9)
The unknown aend gives the number of e-folds from the end of inflation to today. By defining
the effective reheat temperature as Treheat ≡ T0 /aend , where the present CMB temperature
is T0 = 2.726K = 3.673 × 1025 Mpc−1 , the sound horizon can be expressed as
Z 0
cs
s(N )
−65.08
=e
Treheat
dÑ
.
500Mpc
N
eÑ H
(2.10)
The next step is to obtain the slow-roll parameters from the solution to the background
equations of motion (2.4) and (2.5). Here I start by using the field equation (2.5),
1 (1 − cs )(1 + 2cs )
φN N
σ̃1 ,
= H − c2s η̃H +
φN
2
1 + cs
(2.11)
where
Vφ cs
η̃H ≡
3+
,
φN H 2
Tφ
φ .
σ̃1 ≡
T N
(2.12)
These auxiliary parameters η̃H and σ̃1 quantify slow-roll deviations generated by steps in
the potential Vφ and features in the warp Tφ respectively.
In terms of the auxiliary parameters, the slow roll parameters themselves become
ηH =
1 + c2s
cs 1 − cs
η̃H −
σ̃ ,
2
2 1 + cs 1
σ1 = (1 − cs )σ̃1 + (1 − c2s )η̃H .
(2.13)
Note that for ηH , the term involving σ̃1 is suppressed both as cs → 0 and cs → 1. Further13
more η̃H is slow roll suppressed on the attractor of Eq. (2.6) and for cs = 1, ηH = η̃H . The
remaining slow roll parameters σ2 and δ2 can be constructed by taking the derivatives of σ1
and ηH
dη̃
dσ̃1
− cs σ1 σ̃1 + (1 − c2s ) H − 2c2s σ1 η̃H ,
dN
dN
2
2
1 + cs dη̃H
2 − c2 σ η̃ + cs 1 − cs dσ̃1 + cs 1 − 2cs − cs σ σ̃(2.14)
+ H ηH + ηH
δ2 = −
,
s 1 H
2 dN
2 1 + cs dN
2 (1 + cs )2 1 1
σ2 = (1 − cs )
where
cs V φ
Vφφ
φN N
dη̃H
σ1 −
= cs 2 +
+ 2H
dN
φN
H
φN H 2
Vφφ
1 − cs σ̃1
= cs 2 + (η̃H − 3) H + η̃H +
,
1 + cs 2
H
2
Tφφ 2
Tφ
Tφ
dσ̃1
=
φN −
φN
+
φ
dN
T
T
T NN
Tφφ 2
(1 + cs + 2c2s ) 2
=
φN −
σ̃1 + (H − c2s η̃H )σ̃1 .
T
2(1 + cs )
(2.15)
With the slow-roll parameters defined, the GSR sources f 0 /f and G0 can be calculated.
The GSR technique applies to steps in the warped brane tension T (φ) and V (φ) at arbitrary
sound speed and this will be studied in great detail in the remaining chapters. Here I will
limit myself to some considerations about the GSR sources in the slow roll limit, where one
can iteratively substitute the attractor solution Eq. (2.6) into the field equation to obtain
G0
≈ 4H − 2ηH + σ1 ≈
cs (2 + c2s )
Vφφ
Tφ Vφ
Vφ 2
− 2c3s
− cs (1 − c2s )
,
V
V
T V
(2.16)
where cs (φ) is given by the attractor solution Eq. (2.7). The absence of a Tφφ term in Eq.
(2.16) can be attributed to the fact that the attractor solution is determined by Vφ .
Furthermore, in the slow roll limit, evolution in G0 is second order in slow roll parameters
14
and
f0
1
1
≈ −2H + ηH − σ1 ≈ − G0
f
2
2
(2.17)
and so
π
√ (4H − 2ηH + σ1 ),
2 2
0 2
f
≈ − (4H − 2ηH + σ1 )2 .
I2 ≈ −4
f
I1 ≈
(2.18)
Thus, the total second order correction involves a near cancellation of the I1 and I2 terms
2
1 2
1 2
1
π
2
2
ln [1 + I1 (k) + I2 (k)] + I1 (k) ≈ I1 + I2 ≈
− 1 (4H − 2ηH + σ1 )2 . (2.19)
4
2
2
8
2.2
Steps
I will consider steps that appear in either the warp or the potential
φ4
[1 + bT F (φ)],
λB
1 2
V (φ) = V0 1 − βφ [1 + bV F (φ)].
6
T (φ) =
(2.20)
with
F (φ) = tanh
φ − φs
d
− 1.
(2.21)
but for simplicity simultaneous steps in both the warp and the potential will not be allowed.
A convention that after the feature, T (φ) or V (φ) goes back to its bT,V = 0 value is chosen
since physical scales are matched to the end of inflation through Eq. (2.9). Consequently, this
choice simplifies the comparison to the smooth featureless case. For simplicity, Treheat =
1/4
V0
is taken following (Bean et al. 2008b). Thus the DBI step model is specified by 4
parameters {λB , V0 , β, φend } controlling the underlying smooth spectrum and 3 parameters
15
describing the step feature {φs ,bT,V ,d}.
Since the GSR approximation reproduces the exact second order expansion in slow roll
parameters by construction when they are all small, the technique will be extensively tested
throughout this chapter for the nontrivial case of steps that generate order unity changes
in the curvature power spectrum. To provide an illustrative example, I show on Fig. 2.1
the GSR source function for warp steps with bT = −0.4 (top) and bT = −0.005 (bottom).
Both cases appear like the second derivative of the step in T (φ) with a width determined by
the number of e-folds it takes for the inflaton to cross the step δ ln s ≈ δN ≈ φd . Indeed,
N
Fig. 2.1 also shows that the approximation
Vφφ
2
1
1 − cs Tφφ 2
G0 ≈ δ2 − σ2 ≈ −2cs
−
φ .
3
3
V
3(1 + cs ) T N
(2.22)
fairly reproduces the numerical result.
Fig. 2.2 shows the corresponding power spectrum for the more extreme bT = −0.4 case.
In the top panel, I compare the power spectrum from the full GSR approximation, denoted
“GSR2” here and throughout the thesis, of Eq. (1.28) to the exact solution. In the middle
panel, I show that the approximation is accurate at the 1-2% level for the order unity feature.
Moreover, the second order corrections remain small as shown in the bottom panel where
here and throughout the thesis “GSR1” denotes setting I2 = 0 and “GSR0” denotes setting
both I1 = I2 = 0 in Eq. (1.28). Here the maximum value that |I1 | attains is 0.37. As in
√
the canonical case, max|I1 | < 1/ 2 ensures accuracy in the power spectrum of the GSR
approximation, typically to a few percent in observables such as the CMB power spectrum
(Dvorkin & Hu 2011).
Note I2 provides a negligible absolute correction for order unity and smaller features.
For small features I1 and I2 corrections do become comparable but in that case both are
negligible (see Eq. 2.18). Since both leading order and I1 terms depend only on a single
source function G0 (ln s), observational constraints from the power spectrum may be directly
16
Figure 2.1: GSR source function G0 for a warp step with bT = −0.4 (top) and bT = −0.005
(bottom) with d = 2.81 × 10−11 . Also shown is the approximate form based on Tφφ in the
sharp, small amplitude limit from Eq. (2.22) which is an excellent approximation for the
small bT case and remains in good qualitative agreement for the high bT case.
mapped onto constraints on this GSR source function (Dvorkin & Hu 2010b; Miranda et al.
2015), and I will explore that in the next chapter.
2.2.1
Analytical Solution for Small and Sharp Warp Steps
Steps in DBI inflation can generate high frequency oscillations in the curvature power
spectrum that span several decades in k-space, covering the whole observable region. In this
case, the computational demands to solve the exact Mukhanov-Sasaki equation or the GSR
integrals is beyond the point of applicability in Markov Chain Monte Carlo methods. Consequently, analytical approximations for the curvature power spectrum enhance substantially
the search for step signatures in the data.
17
Figure 2.2: GSR vs exact solution for the power spectrum (top panel), the fractional difference between the two (middle panel), and the impact of second order corrections (bottom
panel) corresponding to the bT = −0.4 model in Fig. 2.1 (top). While the GSR2 solution
captures effects at the 1 − 2% level for bT = −0.4, the leading order GSR0 is accurate at the
10% level.
In this section, I will develop an analytical approximation for the curvature power spectrum valid for small and sharp warp steps. The analytical form of this solution will be the
foundation to the more general treatment that accurately models both potential and warp
steps at arbitrary sound speed that produce order unity changes in the power spectrum. In
the small step limit, there is no need for second order I1 and I2 corrections and, consequently,
the power spectrum can be written in the following form
ln ∆2R
= ln As
k ns −1
+ I0 (k),
k0
where I0 (k) is the GSR0 correction to the leading order slow-roll power-law.
18
(2.23)
Following Adshead et al. (2012), I begin by expanding the field as
φ = φ0 + φ1 ,
(2.24)
where φ = φ0 when bT = 0, and calculate to zeroth order in the unperturbed slow-roll
parameters. Furthermore, the expansion rate is also unaffected by the warp feature since
H 1 throughout. The field equation for φ1 then becomes
T
3
1 (1 − cs0 )2 (1 + 2cs0 ) 2 Tφ1
φN N 1 = −3φN 1 + (1 − c2s0 )φN 0 1 +
φN 0
,
2
T0 2
T0
1 − c2s0
(2.25)
where again 0 and 1 denote unperturbed bT = 0 and finite bT perturbations respectively.
Here I have used the fact that
(c2 − 1)
cs1
= s0 2
cs0
2cs0
T
φ
2 N1 − 1
φN 0 T0
.
(2.26)
The time variable can be further transformed from e-folds N to background field value
φ0 by taking φN 0 ≈ const.
d
dφ0
3φ
3φ0 h
0 dφ
3
1 T1 1 (1 − cs0 )2 (1 + 2cs0 ) T1φ i
1
φN 0
e
= e φN 0 (1 − c2s0 )
+
.
dφ0
2
φN 0 T0 2
T0
1 − c2s0
(2.27)
The first term on the RHS can be integrated by parts to make the whole source proportional
to T1φ . For sharp features, T1φ is very concentrated around the feature and, consequently,
the background quantities can be approximated by their values at φs . Combined with the
boundary condition that the field is on the attractor before the step
dφ1
= −(1 − c2s0 )bT ,
φφs dφ0
lim
(2.28)
I obtain
1 − c2s0
dφ1
=
bT F (φ0 ) −
dφ0
2
3(φs −φ0 )
c2s0 1 − cs0
b [F (φ0 ) + 2]e φN 0 .
2 1 + cs0 T
19
(2.29)
Using this result in Eq. (2.26) and replacing
φs − φ0 = φN 0 (Ns − N ),
(2.30)
the perturbation on the sound speed is given by
1 − c2s0
cs1
(1 − cs0 )2
=
bT F (φ0 ) +
bT [F (φ0 ) + 2]e3(Ns −N ) .
cs0
2
2
(2.31)
From H = φ2N /2cs I obtain
1 − c2s0
1 + c2s0 (1 − cs0 )
H1
=
bT F (φ0 ) −
b [F (φ0 ) + 2]e3(Ns −N ) .
H0
2
2 (1 + cs0 ) T
(2.32)
Then, the sound speed and H before (”b”; csb ; Hb ), immediately after (”i”; csi , Hi ) and
well after (”a”; csa ; Ha ) the step are given by
csa = cs0 ,
csb /csa = 1 − (1 − c2s0 )bT ,
csi /csa = 1 + (1 − cs0 )2 bT ,
(2.33)
and
Ha = H0 ,
Hb /Ha = 1 − (1 − c2s0 )bT ,
Hi /sa = 1 −
(1 − csa )(1 + c2sa )bT
.
1 + csa
(2.34)
From these quantities, G0 can be calculated by taking H → 0, bT → 0, d → 0
2
5
8 aHs
1
− 1).
G0 ≈ − σ2 + δ2 − σ1 − 2ηH + (
3
3
3
3 cs
20
(2.35)
In this approximation,
5 − 2cs0 − 3c2s0
aHs
3(1 − cs0 )2
−1≈−
bT F (φ0 )eN −Ns −
bT [F (φ0 ) + 2]e3(Ns −N ) . (2.36)
cs
8
8
After several integrations by parts I obtain that the change in ln ∆2R from Eq. (1.28) due to
the feature, with I1 = I2 = 0, is given by
I0 (k) = C1 W (kss ) + C2 W 0 (kss ) + C3 Y (kss )
(2.37)
Here,
8
Y (x) = − x
3
Z
d ln x̃
W 0 (x̃)
6x cos(2x) + (4x2 − 3) sin(2x)
=
,
x̃
x3
(2.38)
and
C1 = 2(1 − c2s0 )bT ,
2 1 − cs0
b ,
3 1 + cs0 T
5 − 2cs0 − 3c2s0
C3 =
bT .
4
C2 = −
(2.39)
Note that in this derivation the parameter G(ln smin ) was replaced in Eq. (1.28) with the
power spectrum normalization As at k0 .
Given that
lim W (x) = 1,
x→0
lim W 0 (x) = 0,
x→0
lim Y (x) = 0,
x→0
lim W (x) = 0,
x→∞
lim W 0 (x) = −3 cos(2x),
x→∞
lim Y (x) = 0,
x→∞
(2.40)
I can further interpret the meaning of the Ci coefficients. C1 represents a step in the power
spectrum and its amplitude is determined by the fact that the inflaton is on the attractor
21
solution before and well after the step. C2 provides a constant amplitude oscillation whose
value is determined by the sharpest part of the feature. This is exactly the same form as
oscillations produced by a step in the potential for a canonical kinetic term (see Adshead
et al. (2012) Eq. 32). Finally C3 modifies the shape of the first few oscillations due to the
aHs/cs − 1 source. Since even for bT 1 a small error in the location of the feature ss ,
which controls the frequency of the oscillation, causes a noticeable change in the phase of
the oscillation over many cycles, it is defined such that
δG0 (ln ss ) = 0
(2.41)
for the change from the smooth b = 0 model. This definition differs slightly from the sound
horizon at φs for large b as shown in Fig. 2.1.
Figure 2.3: Analytic vs. GSR0 solution for a small amplitude sharp step bT = −0.005,
d = 0.005, φN 0 = 2.44 × 10−12 , with cs0 = 0.0507 (left) and 0.50 (right). Top panel:
difference in ln PR between this model and the same bT = 0 model. Bottom panel: difference
between the curves in the top panel divided by the smooth envelope of the oscillations (see
text).
For finite step width d in field space, the inflaton traverses the step in ∆s/ss ≈ |d ln s/dφ|d ≈
22
d/φN 0 . The window functions W and W 0 oscillate on a time scale ∆s = 1/k. Thus the
integral over G0 is damped for kss > φN /d. For the tanh step, this causes a damping such
that the Ci coefficients are replaced by (Adshead et al. 2012)
Ci → Ci D
kss
xd
where
φ
dφ 1
≈ N0 =
xd =
d ln s πd
πd
,
√
2H0 cs0
.
πd
(2.42)
(2.43)
To obtain the full power spectrum, ln ∆2R1 is added to a calculation of the bT = 0 model. This
can be an exact numerical solution, a slow-roll approximation, or the GSR approximation.
In Fig. 2.3, I test the analytic approximation for a small amplitude sharp step bT =
−0.005 and two values of the sound speed. In the lower panel I divide the difference between
the analytic and GSR0 solutions by the envelope function
3C2 D
kss
xd
.
(2.44)
Agreement is at the 1% level except on scales much larger than the step kss 1 and those
affected by damping kss & xd . In the former case differences from the change in φN from
φN 0 due to the different slow-roll attractor change the mapping between φ and ln s. Near
the damping scale, small changes in xd are amplified in the fractional difference due to the
exponential nature of the damping even though the absolute prediction remains accurate.
Next, I will generalize the analytical solution derived in section for order unity steps. This
generalization open the possibility of probing very sharp steps given that projection effects
damp the amplitude of primordial oscillations in the temperature power spectrum (Adshead
et al. 2012). At very high frequencies, percent level oscillations in the C` are generated by
order unity amplitude oscillations in the curvature power spectrum as I will show further in
this chapter.
23
2.2.2
Analytical Solution for Large and Sharp Steps
In this section I will generalize the analytical solution previous derived to order unity
step on both warp and potential steps at arbitrary sound speed. The basic idea behind the
technique presented here is to use global arguments such as the energy conservation and the
slow-roll attractor to set the functional form of the sound speed and H parameters. With
the addition of I1 corrections, I will show that this procedure works remarkably well for
order unity feature as long as the GSR approximation itself does not break down.
I start with some general considerations dictated by energy conservation and the slow-roll
attractor. If the inflaton crosses a step in δN 1 then the energy loss to the expansion can
be ignored and the total energy ρ in Eq. (2.3) can be set to be equal before and immediately
after the crossing (Bean et al. 2008a). Kinetic energy in excess (or deficit) of the attractor
after the step will then dilute away on the δN ∼ 1 timescale. Denoting with ∆ the change
in quantities going through the step, I have immediately after the step
1 − cs
∆T
∆cs
=
.
cs
1 + cs ∆T /T T
(2.45)
Note that for a decrease in T , energy conservation restricts an amplitude of |∆T /T | = |2b| <
1/cs . For a small amplitude warp feature, I can linearize
∆cs
∆T
≈ (1 − cs )
.
cs
T
(2.46)
Thus for the case of a small, sharp step in T , the sound speed takes a fractional step of
comparable amplitude. Furthermore the slow-roll parameters σ1 and σ2 follow by taking
derivatives of ∆T /T during the interval around the step. Similarly
3ρ+p
3
≈
H =
2 ρ
2
24
1
− cs
cs
T
,
V
(2.47)
and so
∆H
1 − cs ∆T
cs ∆cs
cs
=
.
=
H
1 + cs ∆T /T 1 + cs T
1 + cs cs
(2.48)
Note that at low sound speed, the relative effect of the step on H is suppressed vs cs by
cs /(1 + cs ), as are ηH compared with σ1 , and δ2 compared with σ2 , in agreement with
Eq. (2.13).
After crossing the step, the inflaton hits the attractor solution (2.6) as the kinetic energy
from the step decays after several e-folds. For a small amplitude step
∆T
∆cs
1
,
= (1 − c2s )
cs
2
T
(2.49)
∆cs
∆H
=
.
H
cs
(2.50)
and
Given that the change in H is determined by the change in the sound speed I seek to
quantify the full evolution of cs from the step through to the attractor regime.
To keep the treatment general to either steps in the warp T (φ) and in the potential V (φ)
of the DBI Lagrangian (2.1), first the evolution of these quantities is parameterized relying
on energy conservation and the attractor solution to define their functional form. Following
that, the correspondence of this parameterization to specific step parameters is given.
The energy density of the inflaton (2.3) is conserved as long as the inflaton rolls across
the step in much less than an efold. This conservation then gives the relationship between
the sound speed before and immediately after the step. The acceleration equation (2.48)
then gives the corresponding change in H . After the step, the rolling of the inflaton
φ2N
≡
dφ 2
= 2cs H
dN
(2.51)
differs from the friction dominated attractor solution in Eq. 2.6, to which it must decay on
the expansion time scale or well after the inflaton has crossed the step. These relations hold
25
for arbitrarily large steps so long as H 1.
Together, they imply that the functional form of cs and H previously derived for small
amplitude steps can be generalized as follows
cs
1 − cb
c −1
= 1+
F+ i
(F + 2)e3(Ns −N ) ,
csa
2
2
H
e −1
1 − eb
F+ i
(F + 2)e3(Ns −N ) ,
= 1+
Ha
2
2
(2.52)
where the quantities are scaled to their values on the attractor after the step for convenience
csb
,
csa
eb = Hb ,
Ha
cb =
c
ci = si ,
csa
ei = Hi .
Ha
(2.53)
Here F represents a step of infinitesimal width at N = Ns normalized to −2 before the step
and 0 after. The impact of the finite width is discussed below.
Last section the source function G0 was derived in the approximation that changes to cs
and H are small by taking their derivatives and integrals to form the quantities in Eq. (2.35).
Note that this limit does not necessarily require the steps in the warp itself to be small. In
the limit of a large warp factor φ2N /T 1, the sound speed approaches unity regardless of
the form of T and hence the change in the sound speed are small even for a large fractional
change in T .
Integrals over G0 are then simply evaluated by recalling that dF/d ln s is a delta function
of amplitude 2. The result of integrating the source by parts for the leading order GSR
contribution in Eq. (1.28) has the exact same form of equation Eq.(2.23) after I replace the
parameter G(ln smin ) in Eq. (1.28) with the power spectrum normalization As at k0 . The
26
coefficients Ci are, however, rescaled to
C1 = −(cb − 1) − (eb − 1),
1
1
C2 = − (ci − cb ) + (ei − eb ),
3
3
1
C3 = (1 − cb ) + (ci − 1),
4
(2.54)
Likewise the first order corrections are given by
√
π
2I1 (k) = (1 − ns ) + C1 X(kss ) + C2 X 0 (kss ) + C3 Z(kss ),
2
(2.55)
and
8
Z(x) = − x
3
Z
d ln x̃
X 0 (x̃)
3 + 2x2 − (3 − 4x2 ) cos(2x) − 6x sin(2x)
.
=−
x̃
x3
(2.56)
Then, the analytic model with the I1 correction is given by
ln ∆2R
= ln As
k ns −1
+ I0 (k) + ln[1 + I12 (k)],
k0
(2.57)
√
and again I will call this solution GSR1. Using the limit I1 2 < 1, the GSR expansion itself
is expected to be under control for kss 1 so long as
|C2 | < 1/3.
(2.58)
At kss ∼ 1, the exact requirement is a model dependent restriction on a combination of C1 ,
C2 , C3 but in the warp and potential step examples this gives roughly the same criteria for
the step height.
In principle this domain of validity includes fractional deviations in cs and H that
approach unity, including the region of interest for Planck. However although the GSR
expansion itself remains under control, Eq. (2.54) is derived by assuming small fractional
27
deviations and requires correction. Just as the validity of the step approximation to nonlinearities in the step amplitude was extended above, a weak nonlinearity in the slow roll
parameters can be approximately corrected by rescaling the Ci coefficients.
The C1 amplitude gives the step in power and hence the slow roll attractor ∆2R ∝
(cs H )−1 determines it as
C1 = − ln cb eb .
(2.59)
The changes in cs and H at the step are already determined nonlinearly and so the only
further correction to C2 comes from the conversion to slow-roll parameters, e.g. σ1 =
c−1
s dcs /dN . Following Ref. (Adshead et al. 2012), cs and H are evaluated at the midpoint
of the step and hence
C2 = −
2 ci − cb 2 ei − eb
+
.
3 ci + cb 3 ei + eb
(2.60)
Finally for C3 , while there is no direct nonlinear constraint to determine its amplitude, by also
renormalizing to the midpoint of the step the relative relationship between the coefficients
that determines the shape of their combined contributions is approximately preserved. This
is especially important for warp steps where cancellations between C1 and C3 occur around
the first oscillation. Thus, I take
C3 = 2
(1 − cb ) + (ci − 1)/4
.
ci + cb
(2.61)
Finally, the finite width of the step can be taken into account with the same approach
described on section 2.2.1. With a replacement of the step function with a tanh function, the
integrals over G0 will not contribute if the windows oscillate many times over the width of
the step k ss d/φN . This damping in the oscillations can be modeled with the replacement
(Adshead et al. 2012)
ks s
Ci → Ci D
,
xd
28
(2.62)
with the damping function
D(y) =
y
,
sinh(y)
(2.63)
and with the damping scale
xd =
1 dφ
.
πd d ln s
(2.64)
Note that the xd linearization shown on equation (2.43) for the small step case hasn’t been
applied here. The derivation of the analytic solution for arbitrary steps is thus complete. I
now turn to specific forms for warp and potential steps.
Figure 2.4: Evolution of cs (upper) and H (lower) across a warp step. The step in both
parameters and transient behaviour right after the step is modeled by Eq. (2.52) and (2.71)
to excellent approximation. Model parameter choices are given in section 2.3.1.
I start analyzing steps in the warp T . In this case, warp and potential are before and
29
after the step
Tb = Ta (1 − 2bT ),
Vb = Va ,
(2.65)
and the energy conservation and the attractor solution can be used to give the relevant c
and e parameters of the model in Eq. (2.53). Our convention is to quote these in terms of
bT and the sound speed on the attractor after the step csa . The attractor solution tells us
that
s
cb =
1 − 2bT
,
1 − 2bT c2sa
(2.66)
and using Eq. (2.48) for H ,
eb = cb ,
(2.67)
is obtained.
Now let us consider the sharp changes immediately after the step. Energy conservation
tells us
1
−1=
csi
1
− 1 (1 − 2bT ),
csb
(2.68)
or
ci =
cb
,
1 − 2bT (1 − csa cb )
(2.69)
and with Eq. (2.48)
1 − c2sa c2i
ei =
.
ci (1 − c2sa )
(2.70)
I show an example of the evolution of cs and H in Fig. 2.4. Since the analytic model only
captures the evolution of the parameters around the step and not the evolution on the slow
roll attractor, I plot
cat
s (N − Ns )
cs (N − Ns ) = can
(N
−
N
)
,
s
s
±
cat
s (0 )
30
(2.71)
at
where can
s is the analytic model of Eq. (2.52), cs is the attractor on either side of the step and
±
at +
cat
s (0 ) is evaluated approaching the step from either side with csa = cs (0 ) approached
from the side after the step. I likewise account for the slow roll evolution of H . In practice,
rather than iterating the attractor solution of Eq. (2.6) in the equations of motion to the
required accuracy I numerically solve the equivalent smooth model before and after the step
to determine cat
s .
The general description of Eqs. (2.59), (2.60), (2.61) can now be used to give the Ci
coefficients of the analytic power spectrum form. Note that in the small step limit,
lim cb = 1 − (1 − c2sa )bT ,
bT →0
lim ci = 1 + (1 − csa )2 bT ,
bT →0
lim ei = 1 −
bT →0
(1 − csa )(1 + c2sa )bT
,
1 + csa
(2.72)
and so
lim C1 = 2(1 − c2sa )bT ,
bT →0
lim C2 = −
bT →0
lim C3 =
bT →0
2 1 − csa
b ,
3 1 + csa T
1
(5 − 2csa − 3c2sa )bT ,
4
(2.73)
in agreement with the results I derived for small and sharp steps. This generalized expression
allow the exploration of the bT → −∞ limit
1
,
csa
bT →−∞
2csa
lim ci =
,
1 + c2sa
bT →−∞
lim
cb =
lim
ei =
bT →−∞
1 1 + 3c2sa
.
2csa 1 + c2sa
31
(2.74)
Note that for finite csa these limits are all finite and so the maximal Ci amplitudes are also
bounded
lim
C1 = 2 ln csa ,
lim
C2 = 4
lim
C3 = −
bT →−∞
bT →−∞
bT →−∞
1 − c4sa
,
9 + 42c2sa + 45c4sa
1 4 − 3csa + 2c2sa − 3c3sa
.
2
1 + 3c2sa
(2.75)
Thus for a fixed observed oscillation amplitude C2 > 0 there is always a maximum cs for
which a warp step cannot explain the data
c2sa max
=−
21C2
+
4 + 45C2
q
2 4 + 36C2 + 9C22
4 + 45C2
.
(2.76)
For example, if C2 = 1/15, c2sa |max ≈ 0.724.
Figure 2.5: Maximum oscillation amplitude C2 for a warp step as a function of csa , the
sound speed after the step. The blue dashed line corresponds to C2 = 1/3 where the GSR
approximation breaks down in the oscillatory regime. Note that for cs & 1/4 the oscillation
amplitude is limited by physicality rather than the GSR approximation.
32
Figure 2.6: Evolution of cs (upper) and H (lower) across a potential step. The transient
change in both parameters is modeled by Eq. (2.52) and (2.71) to excellent approximation.
Model parameter choices are given in section 2.3.1.
I now start the investigation of potential steps. In this case,
Vb = Va (1 − 2bV ),
Tb = Ta .
(2.77)
The attractor solution says that to leading order in H , there is no net change in cs or H
only a transient deviation at the step. Thus Hb = Ha and csa = csb (cb = eb = 1) or
C1 = 0.
33
(2.78)
The attractor can also be used to eliminate T /Va using
3T
Ha =
2 Va
1
− csa .
csa
(2.79)
Energy conservation then gives the transient change as
3bV (1 − c2sa )
,
3bV (1 − c2sa ) − Ha
(2.80)
3bV [−3bV (1 − c2sa ) + (1 + c2sa )Ha ]
.
Ha [−3bV (1 − c2sa ) + Ha ]
(2.81)
ci = 1 −
and using Eq. (2.48)
ei = 1 −
The general description of Eqs. (2.59), (2.60), (2.61) then gives the Ci coefficients of the
analytic power spectrum form. I show an example of the evolution of cs and H in Fig. 2.6.
Again, to capture the slow roll evolution of the smooth model, we plot the analytic model
corrected as in Eq. (2.71).
In the limit of a small potential step
lim ci = 1 + 3
bV →0
1 − c2sa
b ,
Ha V
1 + c2sa
lim ei = 1 − 3
b ,
Ha V
bV →0
(2.82)
and so
2
b ,
lim C2 = −
Ha V
bV →0
3 1 − c2sa
lim C3 =
b ,
4 Ha V
bV →0
(2.83)
which generalizes the results of Ref. (Adshead et al. 2012) to arbitrary sound speed. The
sound speed experiences a transient dip for downward steps bV < 0.
34
Note that in the opposite limit
1 Ha
,
3bV 1 − c2sa
bV →−∞
3bV
lim ei =
,
Ha
bV →−∞
lim
ci =
(2.84)
and so
4
,
3
bV →−∞
1
lim C3 = − .
2
bV →−∞
lim
C2 =
(2.85)
Since these amplitudes are beyond the limits of the GSR approximation itself according to
Eq. (2.58), there is effectively no relevant bound on the oscillation amplitude set by energy
conservation and the attractor solution unlike the warp step case. Likewise, for a given
0 < C2 4/3, there is no bound on the required sound speed.
2.3
Constraining Sharp Steps
In this section, I analyze the Planck data for the presence of sharp inflationary steps which
create high frequency oscillations in the power spectrum. First, I test the accuracy of the
leading order GSR0 approximation, which was used in previous analyses Adshead et al.
(2012), and the first order GSR1 corrections against an exact computation of the power
spectrum from the DBI Lagrangian of Eq. (2.1). Then, I discuss details of the analysis that
enhance the efficiency of the model search. Finally I present results for potential steps in
canonical sound speed models, and then for arbitrary sound speed models where both warp
and potential steps can produce the oscillatory phenomenology favored by the Planck data.
35
2.3.1
Power Spectrum Accuracy
In this section, I test the accuracy of the leading order GSR0 approximation used in previous
analyses Adshead et al. (2012) and the first order GSR1 corrections against an exact computation of the power spectrum from the DBI Lagrangian of Eq. (2.1). Although GSR0 was
previously demonstrated to be sufficiently accurate for WMAP data Adshead et al. (2012),
I show here that the increase in precision to the 10−3 level in Planck requires second order
corrections.
The exact computation of the power spectrum follows from solving Eq. (1.11) for a DBI
step model that is parameterized by {V0 , β, λb , φend }, defining the broadband amplitude
and slope of the power spectrum, and the step parameters {φs , bT , bV , d} defining the step
position, height parameters, and width. For testing purposes, I choose
V0 = 7.1038 × 10−26 ,
β = 5.5895 × 10−2 ,
λb = 2.1771 × 1014 ,
φend = 8.2506 × 10−8 ,
(2.86)
and
φs = 3.8311 × 10−8 ,
d = 9.3835 × 10−13 .
(2.87)
For the warp step, I choose
bT = −3.364,
bV
= 0,
36
(warp)
(2.88)
and for the potential step
bT = 0,
bV
= −6.543 × 10−21 ,
(potential).
(2.89)
These parameters are in fact chosen to be close to the Planck maximum likelihood solution
for the amplitude and frequency of warp step oscillations by inverting the steps in this test.
Notice that in that case, |bT | the fractional change in the warp T exceeds unity. The width
d is set so that damping occurs in the ` ∼ 103 region that Planck is most sensitive to so as
to yield the most stringent test of accuracy. The cosmological parameters for the test are
given in Tab. 2.1 and coincide with the best fit model without a step.
The analytic models are specified by the conversion of the fundamental parameters into
the amplitude parameters {C1 , C2 , C3 }, the sound horizon at the step ss , the effective number
of oscillations before damping xd , as well as the broadband amplitude and tilt parameters
As and ns .
Given a solution to the background equations without the step, I set the parameters
csa = 0.67,
Ha = 1.70 × 10−19 ,
(2.90)
according to their values at N = Ns . The Ci amplitude parameters are then determined by
Eqs. (2.59-2.61) such that
C1 = −0.65,
C2 = 0.071,
C3 = −0.34,
37
(warp)
(2.91)
and
C1 = 0,
C2 = 0.071,
C3 = −0.015,
(potential).
(2.92)
Next, the physical scale associated with the step has to be set very precisely in order not
to have a phase error after many oscillations. I follow Eq. (2.41) in defining it numerically
to be the sound horizon at which the deviation in the GSR source function due to the step
is appropriately centered to a small fraction of the step width. Using this definition I obtain
ss =



3699 Mpc (warp)
.
(2.93)


3708 Mpc (potential)
Note that although the step is at the same position in field space in both cases, the sound
horizon differs slightly due to the change in cs .
For the damping parameter, I likewise convert the field width d to a physical width ss /xd
with the numerical solution for φ(ln s) through
dφ 1 d ln s πd = xd .
(2.94)
s=ss
For the test cases, I obtain
xd =



170.0 (warp)
.


169.9 (potential)
Finally there are the broadband power parameters ns and As . For the tilt parameter,
which is slowly varying and essentially independent of the step, I take the slope at k = k0 =
38
Figure 2.7: Comparison between the GSR approximation and the exact solution for the
curvature power spectrum (top panel) and the fractional error of the GSR0 and GSR1
analytical solutions (bottom panel). The models are warp step (left) and potential step
(right).
0.08 Mpc−1 of the model with bT,V = 0. I have chosen parameters in Eq. (2.86) so that ns
coincides with the value given in Tab. 2.1. On the other hand, the effective amplitude As
depends on the presence of the step as well as the order of the GSR approximation used. In
Eq. (2.23), the broadband power gains a contribution from the average of the oscillations
0
heC1 W +C2 W +C3 Y i
kss
3
kss 2
≈ I0 3C2 D
≈ 1 + C2 D
+ O(C24 ), (2.95)
xd
2
xd
where I0 here is the modified Bessel function, not to be confused with the GSR integral
I0 . This non-zero average is the fundamental reason why cosmological parameters must be
varied jointly with the step parameters when analyzing the Planck data. In the first order
correction Eq. (2.55), there is the analogous averaging effect
hI12 i
π2
3
kss 2
2
≈
(1 − ns ) + C2 D
,
8
2
xd
39
(2.96)
Figure 2.8: Comparison between GSR approximation and the exact solution for the temperature power spectrum (top panel) and the fractional error of the GSR0 and GSR1 analytical
solutions (bottom panel). The models are warp step (left) and potential step (right).
which is also O(C22 ) despite being higher order in the GSR approximation. Moreover, around
the damping scale set by xd the broadband average of the oscillation changes with k in
Eq. (2.95)-(2.96) and is not purely an amplitude shift. Note that the error induced by this
average term scales as δCl /Cl ∝ C22 and so rapidly increases with the amplitude of the
oscillations.
Since the best choice for As depends on both the method and the data set considered, I
choose As as the amplitude which gives the best agreement between the exact computation
and the given GSR computation for the Planck dataset. I therefore use the Planck likelihood
itself to define As for each method. In order to remove the ambiguity caused by the exact
model not possessing the maximum likelihood normalization, I in practice maximize both
the Planck likelihood over As to obtain A0s and a rescaling of the amplitude of the exact
model by R to obtain its best normalization. I then set As = A0s /R to remove the rescaling.
In Fig. 2.8 I show the residual errors in the GSR0 and GSR1 after the normalization has
40
been set in this way. Note that the residuals δCl /Cl cross zero at ` ∼ 103 , reflecting the
pivot or best constrained portion of the Planck spectrum. For scales much smaller than or
much larger than the damping scale of the oscillation, the difference between GSR0, GSR1,
and exact is nearly constant and can be absorbed into the normalization. However in my
test case, which I have chosen to be the worst case scenario, the damping falls exactly at the
pivot. The result is that even with the best fit normalization, GSR0 produces ∼ 1% errors
that pivot around ` ∼ 103 . While the error in GSR0 can mainly be absorbed by adjusting
cosmological parameters such as the tilt, they are large enough to bias such parameters nonnegligibly. These residuals are reduced to the ∼ 0.1% level with the GSR1 approximation.
More quantitatively, for these specific test cases the residuals produce a change in the
Planck likelihood versus exact of
∆χ2 =



−8.6
GSR0
(warp),
(2.97)
(potential).
(2.98)


−0.97 GSR1
and
∆χ2 =



−7.4
GSR0


−0.33 GSR1
Thus the GSR1 approximation is sufficiently accurate for the Planck analysis. In fact, the
χ2 errors would be even smaller at its global minimum.
The error in these approximations also depends on the step parameter model. For reference if xd → ∞ the error in the GSR0 approximation becomes ∆χ2 = 0.5 for the warp step
and ∆χ2 = 0.7 for the potential step. Here the error is significantly lower since it takes the
form of a constant amplitude rescaling which can be absorbed into As .
41
In the examples I present here the result of the normalization procedure is to set,
As =



2.1432 × 10−9
GSR0


2.1295 × 10−9
GSR1
(warp),
(2.99)
(potential),
(2.100)
for the warp step and
As =



2.1425 × 10−9
GSR0


2.1288 × 10−9
GSR1
for the potential step. In the absence of a step (bT = bV = 0), the same procedure would
yield As = 2.1554 × 10−9 which is 1% different from the GSR1 value. These changes reflect
the broadband power introduced by the oscillations in each case. Given the 0.1% precision
of the Planck data these differences are significant and As cannot be held fixed when fitting
to step models.
For the minimization procedure, which I will discuss in further detail later, it is nonetheless useful to have an approximate prescription for the renormalization of As in the presence
of oscillations. Given the average of the oscillatory pieces in Eq. (2.95-2.96), the normalization parameter that the data should hold approximately fixed is
Ãs = As e−2τ (1 + Ō),
(2.101)
where Ō contains the average of the oscillatory pieces in each approximation
Ō =



 9 C 2 D2 k0 ss
4 2
GSR0
xd
.
(2.102)

2

 π (1 − ns )2 + 9 C 2 D2 k0 ss
GSR1
xd
8
2 2
Here the e−2τ factor accounts for the change in the heights of the acoustic peaks due to
42
109 Ãs
1.8027
ns
0.9607
100θA
1.04144
10Ωc h2
1.1995
100Ωb h2
2.2039
100τ
8.952
H0
67.22
109 As
2.1562
D∗ (Mpc)
13893.1
χ2
9802.8
Table 2.1: Best fit flat ΛCDM cosmological model without a step with 6 varied parameters
(top) and derived parameters (bottom). This model provides the baseline χ2 for the smooth
model but its parameters require adjustment in the presence of a step. Ãs is an effective
normalization parameter defined in Eq. (2.101).
anisotropy suppression by scattering during reionization. In my test cases
Ãs =



1.8000 × 10−9
GSR0


1.7999 × 10−9
GSR1
(warp),
(2.103)
(potential),
(2.104)
for the warp step,
Ãs =



1.7993 × 10−9
GSR0


1.7993 × 10−9
GSR1
for the potential step and Ãs = 1.8021 × 10−9 without a step. Note that Ãs absorbs most of
the changes in the As normalization given in Eq. (2.99-2.100) from the presence of the step.
2.3.2
Parameter Settings
In order to compare the analytical GSR approximation with the exact Mukhanov solution,
the DBI fundamental parameters must be set to have a fixed power spectrum amplitude As
43
γ CIB
0.538
APS
217
112.4
PS
r143×217
0.906
ACIB
143
6.18
ACIB
217
27.5
c100
1.000580
AtSZ
143
6.71
c217
0.9963
ξ tSZ−CIB
0.2
β11
0.55
APS
100
152
AkSZ
3
APS
143
50.8
CIB
r143×217
0.365
Table 2.2: Foreground model. These parameters are jointly minimized with those of Tab. 2.1
in the smooth model and held fixed for the step analysis.
and tilt ns − 1 at some wavelength pivot for a smooth model. This condition fixes the
parameters {φend , λB , β} of the model and I set V0 = 7.10 × 10−26 to a constant. The
phenomenological and fundamental parameters are related through the slow-roll attractor in
Eq. 2.6, which determines X at φ
r
√
dφ
V d ln V
= 2X ≈ −
cs .
H
d ln a
3 dφ
(2.105)
Through this relation, the phenomenological parameters p = {As , ns − 1, cs } are defined
given the field position which completes the set of fundamental parameters . Explicitly, I set
"
cs (π) ≈
V
1+
3T
d ln V
dφ
2 #−1/2
,
(2.106)
and use the Friedmann equation to set H
3H 2 (π)
=
1
− 1 T (φ) + V (φ) .
cs
44
(2.107)
The amplitude and tilt are then given as
2
H
As (π) ≈
,
2πdφ/d ln a
d ln As dφ
.
(ns − 1)(π) ≈
dφ d ln a
(2.108)
This gives the phenomenological parameters as a nonlinear function of the fundamental
parameters p(π). To set the sound horizon at φ to equal ss , the appropriate φend is chosen
such that φ is effectively replaced with φend in the fundamental parameter set after the fact.
Of course, I actually want the fundamental parameters as a function of the phenomenological parameters π(p). For cs 1, these relations are easily inverted
4
1
4
,
λB ≈
4π 2 As ns − 1
3 ns − 1
β ≈ −
,
4 cs
√ s
2 3
V0
φ ≈
.
3π
As (ns − 1)2
(2.109)
For larger cs , the expressions are not readily invertible but from the solution at small cs ,
I can approximate small changes by linearizing and inverting the response. Starting from
some parameter set p0 , I move to a new set p by iterating
ln(π/π0 ) =
∂ ln p −1
ln(p/p0 ),
∂ ln π
(2.110)
until convergence. Here the inverse factor is the Jacobian matrix inverse. I repeat this
procedure until I obtain all the desired values of the sound speed cs .
2.3.3
Minimization
In this subsection I provide details of the effective χ2 minimization for the various models
presented in this chapter. In each case, I use the MIGRAD variable metric minimizer from
45
the CERN Minuit2 code (James & Roos 1975).
I begin with the smooth ΛCDM cosmology specified by the cosmological parameters
{Ãs , ns , θA , Ωc h2 , Ωb h2 , τ }, and 14 foregrounds parameters defined in the Planck likelihood
Ade et al. (2013a), where θA is the angular acoustic scale at recombination, Ωc h2 parameterizes the cold dark matter density, Ωb h2 the baryon density, τ the Thomson optical depth
to reionization. I include the Planck low-` spectrum (Commander, l < 50), the high-`
spectrum (CAMspec, 50 < l < 2500) and WMAP9 polarization (lowlike) likelihoods in my
analysis (Ade et al. 2013a; Bennett et al. 2012). Ãs is the effective normalization defined
in Eq. (2.101); in the absence of a step Ãs = As e−2τ . In the standard ΛCDM model, the
effective number and sum of the masses of neutrinos are held fixed to Neff = 3.046 and
P
mν = 0.06eV respectively with the helium fraction YP = 0.2477. The best fit model is
given in Tab. 2.1 and 2.2 and its χ2 is the baseline from which I quantify any improvements
due to the step parameters. The foreground parameters is fixed to these values for the following analysis but I have spot checked that reoptimization over the foreground parameters
does not substantially change my results.
For the step analysis, the starting point is the canonical cs = 1 potential step where
C1 = C3 = 0. As the oscillatory features from C2 dominate the fit to the Planck data, the
other cases are built from this model. In this case the step is described by {C2 , ss , xd }. While
I could directly minimize the χ2 in this joint cosmological and step parameter space, the
efficiency of the search is greatly improved by choosing parameters that are better aligned
to the principle axes of the χ2 surface.
The angular frequency of the oscillation changes with cosmological parameters at fixed
ss . It is thus advantageous to replace ss with
θs =
ss
,
D∗
(2.111)
where D∗ is the distance to recombination. Note that the oscillations in C` are then described
46
GSR1
GSR0
xd
105
2000
105
2000
10θs
2.665
2.667
2.665
2.666
10Ac
1.17
0.663
1.11
0.707
109 Ãs
1.8021
1.8024
1.8020
1.8021
ns
0.9690
0.9608
0.9640
0.9606
100θA
1.04140 1.04145 1.04136 1.04140
10Ωc h2
1.2091
1.1995
1.2035
1.1993
100Ωb h2
2.1974
2.2039
2.2053
2.2039
100τ
9.421
9.117
9.361
9.205
H0
66.82
67.23
67.07
67.22
109 As
2.1669
2.1420
2.1701
2.1565
D∗ (Mpc)
13874.7 13893.2
13882.9 13893.9
ss (Mpc)
3696.9
3704.7
3699.2
3704.5
C2
0.075
0.043
0.071
0.045
∆χ2
−14.0
−11.4
−13.8
−11.3
Table 2.3: Best fit potential step model with cs = 1 showing the 9 parameters jointly varied
(top) and derived parameters (bottom) using the GSR1 approximation of the main paper
versus the less accurate GSR0 approximation. The global minimum is at xd = 105 but
xd = 2000 where there is no damping of oscillations for the Planck data gives a comparable
fit, albeit with lower oscillation amplitude Ac . Ãs is an effective normalization parameter
defined in Eq. (2.101) that determines the broadband observed CMB power in the presence
of τ and a step.
by sinusoids such as sin(2`θs ). Next, due to projection effects, a fixed amplitude C2 produces
an oscillation in C` that decays as C2 (`θs )−1/2 . In Ref. Adshead et al. (2012) this scaling
was approximately accounted for in the curvature power spectrum description by introducing
the amplitude parameter
"r
Ac = 3C2
ss
1Gpc
#−1
;
(2.112)
I adopt this convention rather than the more orthogonal angular approach in order to com47
pare with the previous literature. Note that the original version of the Planck collaboration
analysis erroneously conflated this parameter with C2 Ade et al. (2013c).
Finally, given that the oscillations produce excess broadband power, I use the normalization parameter Ãs as defined in Eq. (2.101). For the best fit models, this parameter rather than As itself is nearly constant.
The optimized parameters are therefore
{Ãs , ns , θA , Ωc h2 , Ωb h2 , τ } for the smooth cosmology and {Ac , θs , xd } for the step. The
minimum χ2 potential step model with cs = 1 is given in Tab. 2.3 (GSR1 column) and represents an improvement of ∆χ2 = −14.0 over the smooth model of Tab. 2.1. For reference
I also show here the best fit model at xd = 2000, where the oscillations are undamped all
the way to the maximum of ` = 2500 for Planck. Note that most of the improvement due
to the step remains. I also repeat the minimization for the GSR0 approximation used in
previous treatments for comparison. After adjusting cosmological parameters, steps in either
approximation fit equally well but the recovery of such parameters would be biased by using
the less accurate GSR0 approximation.
For the arbitrary sound speed warp and potential step models, C1 and C3 are set consistently with the step amplitude {bT , bV } and slow roll parameters after the step {csa , Ha }
through Eqs. (2.59, 2.60, 2.61). These parameters mainly change the power spectrum around
` ∼ 1/θs and hence produce only small changes in the χ2 due to the limitations of cosmic
variance. I therefore keep the other parameters fixed to the values of xd = 105 model listed
in Tab. 2.3 when considering the additional freedom in these models. Given a fixed C2 ,
which fixes the amplitude of the step, this freedom is parameterized by csa , the sound speed
after the step.
2.3.4
Steps in the Canonical Limit
I begin the analysis with the simplest case of cs = 1 models. Here, warp steps have no
effect and potential steps give C1 = C3 = 0. Step models are thus described by three
parameters {ss , C2 , xd }, the oscillation frequency, amplitude and damping scale respectively.
48
I am interested in the question of whether the step parameters significantly improve the fit
to the Planck data rather than marginalized constraints on the parameters themselves. Since
the Monte Carlo Markov chain technique is highly inefficient for these purposes, I instead
directly maximize the likelihood or minimize the effective χ2 = −2 ln L in the step and
cosmological parameter space jointly. For these oscillatory spectra, the likelihood is a rapidly
varying function of frequency ss with many local minima. Fortunately previous works have
shown that ss ≈ 3700 Mpc is the frequency range that contains the global minimum (Ade
et al. 2013c; Meerburg & Spergel 2013). I therefore search only around this global minimum
region. Even so, for efficiency in the minimization it is important to choose combinations
of the parameters that are close to the principal components of the curvature or covariance
matrix, as discussed on section 2.3.3.
Unlike the 2013 released of the Planck collaboration analysis (Ade et al. 2013c), I simultaneously vary the cosmological and step parameters in the minimization. This step is
crucial as discussed in section 2.3.1 since the presence of rapid oscillations also changes both
the amplitude and shape of the broadband power in multipole space. If the cosmological
parameters are not readjusted, the Planck data would falsely suggest that the oscillations
cannot continue into the ` ∼ 103 regime where the data is most constraining. For this reason,
the minimum found in Ref. Ade et al. (2013c) is not the global minimum nor is there strong
evidence for damping of the oscillations at high multipole. The minimum χ2 as a function
of the damping scale xd is shown in Fig. 2.9 and is nearly flat for xd > 102 .
I find that the global minimum is given by
C2 = 0.075,
ss = 3696.9 Mpc,
xd = 105.0,
∆χ2 = −14.0,
49
(2.113)
Figure 2.9: Minimum χ2 relative to the best fit smooth (no step) model as a function of
the oscillation damping scale xd for the cs = 1 potential step model (top panel). The
minimization is performed jointly over cosmological and step parameters with step position
ss and amplitude of oscillations C2 shown in the middle and bottom panels respectively.
where the χ2 improvement is measured against the best fit smooth model. In contrast,
the best fit model of Ref. Ade et al. (2013c) had xd = 87 and a similar amplitude and
frequency with a ∆χ2 = −11.7.1 As shown in Tab. 2.3, this difference is not due to the
inclusion of second order corrections from I1 in Eq. (2.23) though their omission would bias
cosmological parameters such as As and ns . The Planck 2013 data thus favor oscillations at
the few percent level in C` , with a peak-to-peak spacing of ∆` ∼ 12. Note that at kss = xd ,
the damping suppression D = 0.85 and for this model corresponds to `d ≈ 400. Thus the
oscillations persists at least out to the second acoustic peak (see Fig. 2.10).
1. The original version of the Planck collaboration analysis Ade et al. (2013c)
[arXiv:1303.5082v1] erroneously conflated C2 with Ac (see Eq. 2.112) and correcting this
definition also brings the amplitude into agreement with that found for WMAP Adshead
et al. (2012).
50
Figure 2.10: Best fit models for a potential step with csa = 1.0 (red) and a warp step at
csa = 0.7 (blue). Both models have the same few percent oscillations around the best fit
smooth model at the first and second peaks (lower panel). The warp step is a marginally
better fit of ∆χ2 = −15.2, versus the potential step with ∆χ2 = −14.0, due to suppression
of the lowest multipoles but introduces the sound speed as an additional parameter.
The complete list of cosmological parameters for the best fit step model is given in Tab.2.3.
The oscillations add broadband power and generally require a lower normalization. Because
in this model they damp near the well-constrained third peak, this also requires a higher
tilt to keep the total power fixed at this best constrained region. As a result, the model has
slightly smaller broadband power at low multipoles, reaching ∼ −5% at the quadrupole.
Furthermore, the Planck data are also compatible with oscillations that persist out to
the highest multipole measured in the data ` = 2500. Taking xd ≈ 2000, which is indistin-
51
guishable from infinity for Planck,
C2 = 0.043,
ss = 3704.7 Mpc,
∆χ2 = −11.4.
(2.114)
This fit only differs in χ2 by 2.6 from that of the global minimum and is comparable to
the best fit found in Ref. Ade et al. (2013c). Note that even with no damping scale in
curvature fluctuations, oscillations in C` decline with ` due to projection and lensing effects
(see Adshead et al. (2012) and Eq. 2.112). Had I fixed cosmological parameters to the best
fit smooth model then this xd would be falsely penalized by ∆χ2 = 9. The cosmological
parameters for this model are also given in Tab. 2.3. Notably with no damping scale, the tilt
no longer requires significant adjustment. The change in ss mainly reflects slightly different
cosmological parameters that produce correspondingly different distances to recombination
rather than a change in the angular scale.
In summary, the Planck 2013 data favor percent level oscillations in C` produced by
potential step features by ∆χ2 = −11.4 with two parameters that control the oscillation
frequency and amplitude. Minimizing the χ2 for damping of the oscillations confines the
oscillations to roughly the first and second peaks and marginally improves the fit with an
additional ∆χ2 = −2.6 for a total of −14.0 with one additional parameter for a total of
three.
2.3.5
Steps in the Low Sound Speed Limit
Low sound speed DBI models allow for two different classes of steps with two different
phenomenologies that impact low multipoles in the Planck data. Both steps in the potential
V (φ) and warp T (φ) produce the same high multipole oscillations driven by the amplitude
parameter C2 . Given a C2 that minimizes the Planck χ2 at high multipole, the remaining
52
Figure 2.11: χ2 improvement as a function of the sound speed after the step csa for the
warp (blue +) and potential (red ×) steps. Other parameters are fixed to their minimum χ2
values in the cs = 1 potential step model including the oscillation amplitude C2 . Note that
warp steps with csa & 0.7 cannot generate the required C2 (see Eq. 2.76) as marked by the
vertical line.
freedom is in choosing a sound speed after the step csa . In both the potential and warp
scenarios, this uniquely fixes the two remaining step parameters C1 and C3 . Recall that
C1 controls the step in the power spectrum around the first oscillation and C3 controls the
shape of the first few oscillations. Since the fit is driven by the C2 oscillations with only small
impact from C1 and C3 , I fix all the other parameters to the global minimum of Eq. (2.113)
when examining the impact of csa .
For potential steps, C1 = 0 and −3/8 < C3 /C2 < 0. Even for the maximal case of −3/8
and csa → 0, there is very little impact on the CMB power spectrum. Consequently as
shown in Fig. 2.11 the χ2 surface is essentially flat across csa .
For warp steps both C1 /C2 and C3 /C2 can be greater than unity and the sound speed
has a larger fractional effect on C` . However, their impact is still limited to the first few
oscillations and, given the preference for a horizon scale ss , severely cosmic variance limited.
Raising csa mainly enhances the step in the power spectrum relative to the oscillations thus
lowering the first few multipoles. Both C1 and C3 are important in establishing the shape
due to a cancellation in their effects at the first oscillation.
53
Since warp steps do not produce oscillatory features as cs → 1, there is a maximum
csa ∼ 0.7 for which they can explain the oscillations (see Fig. 2.11). The best fit has the
maximal possible sound speed
csa = 0.70,
∆χ2 = −15.2 (warp),
(2.115)
which implies C1 = −0.70, C3 = −0.37 given the fixed parameters in Eq. (2.113). While the
step in the curvature power spectrum is approximately 50% (see Fig. 2.7) in C` , this and
the changes in the cosmological parameters translates into a ∼ 20% suppression of power at
the quadrupole relative to the smooth model (see Fig. 2.10). Note that the drop between
2 ≤ ` ≤ 5 is particularly sharp for warp steps due to a local maximum in the curvature
spectrum oscillations. Nonetheless, with cosmic variance these changes have only a small
impact on the fit. As a consequence, while warp steps have interesting phenomenology that
may ameliorate low multipole anomalies, there is no statistically significant preference for
cs < 1.
2.3.6
Future Tests
While an improvement of ∆χ2 ≈ −11 in the Planck 2013 data for two parameters and
up to −15 in the full step parameter space may sound significant, it has been shown that for
more flexible oscillatory models, where not only the amplitude and frequency but also the
phase of the oscillation is fit, realizations of smooth models with noise often recover this level
of improvement, albeit typically with a smaller oscillation amplitude (Meerburg et al. 2013)
(see also Easther & Flauger (2013)). Furthermore the improvement in the WMAP likelihood
(Adshead et al. 2012; Meerburg et al. 2013) is comparable to that of the Planck likelihood
despite the higher precision of Planck whereas one would have expected the latter to increase
for a true signal. For these reasons, it is important to have more definitive tests for the origin
54
Figure 2.12: Polarization (left) and temperature polarization cross (right) power spectra
for the best fit models of Fig. 2.10. Step oscillations provide falsifiable predictions for the
polarization which would not be mimicked by chance features in the noise.
of these improvements. In this section I discuss predictions of the best fit models identified
above that may be used to verify or falsify the hypothesis of their primordial origin.
As emphasized by Ref. Mortonson et al. (2009), the most incisive consistency test for inflationary features is the E-mode polarization power spectrum and cross spectrum. In Fig. 2.12
(left panel), I show the predicted E-mode power spectrum of the models in Fig. 2.10. Consistency with inflationary oscillations demands that oscillations appear at the same frequency
while modulated by the acoustic transfer to have nodes that are out of phase with the temperature. Furthermore due to projection effects, the polarization oscillations are twice as
prominent in polarization. In principle, the low ` polarization can also more than double
the distinguishing power between the warp and potential fits, albeit limited in practice by
galactic foregrounds and uncertainties in the reionization model.
Finally, the temperature-polarization cross spectrum must also exhibit consistent oscillations as shown in Fig. 2.12 (right panel). These predictions should be tested in the next
release of the Planck data. More generally they can be tested in any CMB polarization
55
data set that has sufficient amounts of sky to distinguish modes separated by ∆` ≈ 12 and
oscillations in power of 3-10%.
Figure 2.13: Temperature power spectrum derivatives for the best fit models of Fig. 2.10.
Low amplitude, high frequency oscillations produce new signals for lensing reconstruction
and squeezed bispectra.
The best fit oscillatory models also predict different CMB lensing effects. High frequency
features act in a similar fashion as the acoustic peaks in providing a signal for lensing. The
∆` ≈ 12 fineness of the features compared with the acoustic spacing of ∆` ≈ 300 offsets
the smallness of the amplitude. In Fig. 2.13, I quantify this expectation by showing for the
smooth and best fit step models
d ln `2 C`
,
d ln `
(2.116)
which controls lensing and squeezed bispectrum effects (Lewis et al. 2011). Note that what
was a small effect for the power spectrum is an order unity effect for certain lensing effects.
Features during inflation also produce primordial non-Gaussianity in mainly the equilateral configuration (Chen et al. 2007; Adshead et al. 2011; Adshead & Hu 2012; Achucarro
et al. 2012b). For the best fit step models these should also be observable in Planck (Adshead
et al. 2012, 2013). Extracting these signals though will require using specific templates that
56
include these rapid oscillations (Fergusson et al. 2012). Since the equilateral bispectrum
amplitude scales as x2d , the lack of a strong bound on the damping scale implies that the bispectrum signal could be very large at high multipole, though these models would be beyond
the regime of validity of the effective field theory that underlies their calculation (Baumann
& Green 2011). Thus if the high-frequency oscillatory fits really reflect inflationary features,
there is a battery of consistency tests that the CMB temperature and polarization anisotropy
must satisfy.
2.4
Constraining Broad Steps
My original motivation to study broad steps was the 2014 release of the BICEP2 measurement which restricted the tensor-scalar ratio to r = 0.2+0.07
−0.05 from degree scale B-mode
polarization of the CMB (Ade et al. 2014). This was in “moderately-strong” tension with
slow-roll inflation models that predict scale-free, albeit slightly tilted (1 − ns 1) power-law
power spectra. This tension is due to the implied excess in the temperature spectrum at
low multipoles which is not observed and restricts r0.002 < 0.11 (95% CL) in this context
(Ade et al. 2013c). Even in the absence of tensors, this excess at low multipole persists
in the Planck data, although with smaller significance. This section will also show models
with r = 0.1 which is still allowed by the new 2015 Planck joint analysis with the BICEP2
experiment (Ade et al. 2015a).
These findings can be reconciled in the single-field inflationary paradigm by introducing
a scale into the scalar power spectra to suppress power on these large-angular scales. For
example a large running of tilt, dns /d ln k ∼ −0.02, is possible as a compromise (Ade
et al. 2014). Here the scale introduced is associated with the scalar spectrum transiently
passing through a scale-invariant slope near observed scales. However, such a large running
is uncomfortable in the simplest models of inflation which typically produce running of order
O[(1 − ns )2 ]. Moreover, a large running also requires further additional parameters in order
that inflation does not end too quickly after the observed scales leave the horizon (Easther
57
& Peiris 2006).
The temperature anisotropy excess implied by tensors is also not a smooth function of
scale, but rather cut off at the horizon at recombination. To counter this excess, a transition
in the scalar power spectrum that occurs more sharply, though coincidentally near these
scales, would be preferred. Such a transition can occur without affecting the tensor spectrum
if there is a slow-roll violating step in the tensor-scalar ratio while the Hubble rate is left
nearly fixed (Hu 2014).
In this section I will consider the effects of placing such a feature near scales associated
with the horizon at recombination, thereby suppressing the scalar spectrum on large scales.
I jointly fit the Planck 2013 temperature results, WMAP9 polarization results, and BICEP2
2014 release to models with and without steps in the tensor-scalar ratio parameter H cs .
Similar to the analysis of sharp steps, I use the MIGRAD variable metric algorithm from
the CERN Minuit2 code (James & Roos 1975) and a modified version of CAMB (Lewis
et al. 2000; Howlett et al. 2012) for model comparisons. The Planck likelihood includes the
Planck low-` spectrum (Commander, ` < 50) and the high-` spectrum (CAMspec, 50 < ` <
2500), whereas the BICEP2 likelihood includes both its E and B contributions. Finally, the
GSR power spectrum is numerically evaluated since the analytical approximation previously
derived in this chapter is only suitable for sharp steps.
I begin with the baseline best fit 6 parameter slow-roll flat ΛCDM model with r = 0 given
by Tab. 2.1. When considering alternate models we fix the non-inflationary parameters to
these values while allowing the inflationary parameters, including As and ns to vary. As
shown in Tab. 2.4, this r = 0 model is strongly penalized by the BICEP2 2014 data. Moving
to the r = 0.2 model with the same parameters removes this penalty at the expense of making
the Planck likelihood worse by 2∆ ln L = 9.6 due to the excess in the ` . 100 temperature
power spectrum shown in Fig. 2.14.
Next I fit for a step with parameters C1 , ss , xd controlling the amplitude, location and
width of the step. The best fit model at r = 0.2 more than removes the penalty from the
58
Figure 2.14: Total temperature power spectra showing the unobserved excess produced by
adding tensors of r = 0.2 to the best fit 6 parameter ΛCDM model and its removal by adding
a step in the tensor-scalar parameter H cs . Planck 2013 data in fact favor removing more
power than the tensor excess, preferring a step even if r = 0. Step model parameters are
given in Tab. 2.4.
temperature excess for Planck while fitting the BICEP2 BB results equally well. The net
result is a preference for a step feature at the level of 2∆ ln LP = −14.2 over no feature. The
inclusion of BICEP2 results slightly degrades the fit to 2∆ ln Ltot = −13.7 due to changes
in the EE spectrum (see below). The r = 0.2 model with a step is very close to the global
maximum with further optimization in r allowing only an improvement of 2∆ ln Ltot = −0.1.
With the addition of the step, there remains a small high-` change in the vicinity of
the first acoustic peak in Fig. 2.14 which is interestingly marginally favored by the data.
Note that I have fixed the non-inflationary parameters to their values without the step, for
example τ . Thus the likelihood may in fact increase in a full fit (see Fig. 2.16). Conversely,
I do not consider any compromise solutions where non-inflationary cosmological parameters
59
r
C1 ss (Mpc)
0
0
0 -0.15
337.1
0
0.1
0.1 -0.22
339.2
0
0.2
0.2 -0.31
351.8
xd As × 109
2.1972
1.58
2.2003
2.1961
1.60
2.2000
2.1939
1.47
2.2002
ns
−2 ln LP
0.961
9802.7
0.957
9798.6
0.962
9806.5
0.958
9797.8
0.963
9812.3
0.959
9798.1
−2 ln LB
89.1
89.2
47.9
48.2
39.4
39.9
−2 ln ∆Ltot
40.1
36.1
2.7
-5.7
0
-13.7
Table 2.4: Likelihood for models with tensors and steps with non-inflationary parameters
fixed. LP is the likelihood for the Planck low-` spectrum, high-` spectrum and WMAP9
polarization; LB is that for the BICEP2 E and B likelihood. The change in the total is
quoted relative to the r = 0.2 no feature case.
ameliorate the tension without a step.
The best fit step also predicts changes to the EE polarization. Like the T T spectrum,
the excess power from the tensor contribution is partially compensated by the reduction in
the scalar spectrum for ` & 30. This is a signature of the step model which requires only a
moderate increase in data to test as witnessed by the change in the BICEP2 2014 likelihood
of 2∆ ln LB ∼ 0.5 it induces. Differences at ` . 30, shown here at fixed τ , are largely
degenerate with changes in the ionization history (Mortonson et al. 2009)
I also test models at r = 0.1, which is still allowed be the new Planck 2015 joint analysis
with BICEP2 (Ade et al. 2015a). Even in this case, the Planck portion of the likelihood
improves with the inclusion of a step though the preference is weakened to 2∆ ln LP = −8.6
versus no step. At r = 0, the Planck 2013 data still prefers a step to remove power at a
reduced improvement of 2∆ ln LP = −4.1, a fact that was already evident in the Planck
2013 collaboration analysis of anticorrelated isocurvature perturbations (Ade et al. 2013c).
Such an explanation should also help resolve the tensor-scalar tension albeit outside of the
context of single-field inflation. Interestingly, the addition of tensors at both r = 0.1 and
0.2 in fact further helps step models fit the Planck 2013 data due to the changes shown in
Fig. 2.14 independent of the BICEP2 result.
To conclude, a transient violation of slow-roll which generates a step in the scalar power
spectrum at scales near to the horizon size at recombination can alleviate problems of pre60
Figure 2.15: Step in tensor-scalar ratio parameter H cs relative to no step, from the best fit
r = 0.2 solution centered at the efold Ns at which the inflaton crosses the step. Planck 2013
data favor a step that is traversed in about an efold.
dicted excess power in the temperature spectrum, present already in the best fit ΛCDM
spectrum. Such a step may be generated by a sharp change in the speed of the rolling of the
inflaton H or by a sharp change in the speed of sound cs over a period of around an efolding
which combine to form the tensor-scalar ratio. Preference for a step from the temperature
power spectrum in the 2013 Planck data is at a level of 2∆ ln LP = −14.2 for r = 0.2 and
is still −8.6 at r = 0.1, which is still allowed by the new joint analysis between Planck 2015
data and BICEP2 (Ade et al. 2015a).
Such an explanation makes several concrete predictions. Since slow-roll is transiently
violated in this scenario, there will be an enhancement in the associated three-point correlation function. However, I do not expect this signal to be observable as it impacts only a
small number of modes (Adshead et al. 2011; Adshead & Hu 2012). E-mode fluctuations on
similar scales would be predicted to have a smaller enhancement then with tensors alone.
While I have used a DBI type Lagrangian to illustrate the impact of a change in the
tensor-scalar ratio parameter H cs due to a step in the sound speed, I do not expect that
61
Figure 2.16: EE power spectrum for the models in Fig. 2.14 showing the change from the
best fit r = 0 ΛCDM power spectrum. Excess E-modes from the tensors at r = 0.2 are
partially compensated by the step at ` & 30 while changes at lower ` can be altered by
changing the reionization history. Preference for removing power at substantially smaller r
would predict a deficit of power as the r = 0 model shows.
the results presented here require this form, though precise details of the fit may change.
Transient shifts in the speed of sound have been found to occur in inflationary models where
additional heavy degrees of freedom have been integrated out (Achucarro et al. 2012a).
62
CHAPTER 3
POLARIZATION PREDICTIONS OF BROAD BAND
FEATURES IN SINGLE-FIELD INFLATION
Ever since the first release of WMAP data (Hinshaw et al. 2003), the large-angle temperature power spectrum has shown several anomalous features when compared with the
simplest power law or scale-free inflationary ΛCDM model. In particular, there is a glitch in
the power spectrum at multipoles ` ∼ 20 − 40 (Peiris et al. 2003) and a deficit of large-angle
correlations (Spergel et al. 2003; Copi et al. 2007).
The significance and interpretation of these features change with temperature measurements by the 2013 and 2015 Planck satellite (Ade et al. 2013d) and the 150 GHz measurement
of degree scale B-mode polarization by the BICEP2 experiment (Ade et al. 2014). Relative
to 2013 Planck data at higher multipole moments, the significance of the power deficit at low
multipoles in the ΛCDM model increases. Although the BICEP2 measurement is partially
contaminated by galactic dust based on subsequent Planck measurements at dust dominated
frequencies (Adam et al. 2014; Ade et al. 2015a), any contribution from inflationary gravitational waves near a tensor-to-scalar ratio of r ∼ 0.1 exacerbates the power spectrum deficit
and increases its significance (Ade et al. 2013c).
There is an extensive literature on converting these temperature measurements into
model-independent constraints on the primordial curvature spectrum (e.g. (Hannestad 2001;
Hu & Okamoto 2004; Tegmark & Zaldarriaga 2002; Hannestad 2004; Bridle et al. 2003;
Mukherjee & Wang 2003; Leach 2006; Peiris & Verde 2010; Hlozek et al. 2012; Gauthier &
Bucher 2012; Vazquez et al. 2012; Hunt & Sarkar 2014; Aslanyan et al. 2014; Hazra et al.
2014b)). Features could have an origin in inflation if its near time-translation invariance is
broken at least transiently when these scales left the horizon during inflation (e.g. (Peiris
et al. 2003; Contaldi et al. 2003; Martin & Ringeval 2004; Freivogel et al. 2006; Covi et al.
2006; Joy et al. 2008; Hazra et al. 2010; Achucarro et al. 2014; Contaldi et al. 2014) and
63
(Miranda et al. 2014; Abazajian et al. 2014; Hazra et al. 2014a; Bousso et al. 2014)). Specific
models that fit features also make predictions for E mode polarization by which they can
be verified as I showed last chapter (Miranda et al. 2014; Bousso et al. 2014). While I have
previous focused on DBI models, here I seek to generalize these results and polarization
predictions for any single-field inflation model that fits the temperature features.
Model independent constraints on curvature power spectrum features cannot be directly
applied to inflationary features. In particular, not all possible curvature power spectra are
allowed in single field inflation. This restriction is especially important when considering
sharp features in the temperature power spectrum. As shown on last chapter, temporal
feature that is localized to less than an e-fold during inflation does not produce a feature in
the curvature spectrum localized to a comparable range in wavenumber as implied by the
slow-roll approximation. Instead the features oscillate or ring across an extended range in
wavenumbers.
The generalized slow-roll (GSR) approximation is well suited for model-independent studies of power spectrum reconstruction, because it has the property that the curvature power
spectrum depends on integrals which are linear in a single source function (Dvorkin & Hu
2010a, 2011). In canonical single-field inflation, this source function is related to the shape of
the inflationary potential in the same way the tilt is in the slow-roll approximation. In this
chapter, I adapt this reconstruction technique for the study of large-angle power spectrum
anomalies in the presence of potentially non-negligible tensor contributions from inflation.
3.1
Inflationary Reconstruction
The ordinary slow-roll approximation corresponds to a parameterization of the curvature
source function by a constant G0 (ln s) = 1 − ns , and results in a power-law curvature power
spectrum. I therefore look for parameterized deviations from this constant behavior. In
general, given some set of basis functions Bi (ln s) the source function can be described with
64
a set of coefficients pi as
δG0 (ln s) ≡ G0 (ln s) − (1 − ns ) =
X
pi Bi (ln s).
(3.1)
i
The advantage of the GSR form in Eq. (1.28) is that the integrals are linear in G0 and hence
the impact of the individual components can be precomputed separately
Z ∞
ds
Wi (k) =
W (ks)Bi (ln s),
s∗ s
Z ∞
ds
Xi (k) =
X(ks)Bi (ln s),
s
0
(3.2)
so that the power spectrum becomes a sum over the basis
ln ∆2R (k) = ln As
"
#
1 + I12 (k)
k ns −1 X +
pi Wi (k) − Wi (k0 ) + ln
,
k0
1 + I12 (k0 )
(3.3)
π
1 X
I1 (k) = √ (1 − ns ) + √
pi Xi (k).
2 2
2 i
(3.4)
i
where
Note that the normalization constant G(ln s∗ ) have been absorbed into the amplitude of the
power spectrum at the scale k0
As = ∆2R (k0 ).
(3.5)
In Ref. (Dvorkin & Hu 2010a, 2011), the basis functions Bi were chosen to be the principal
components (PCs) of the Fisher matrix for the full WMAP range of scales. Since the Fisher
matrix is constructed from the expected errors of a given experiment, this technique is blind
to the presence of anomalies in the actual data. The drawback for studying known largeangle anomalies is that the basis does not efficiently encode them. Here I take an alternate
65
approach that is better suited to making polarization predictions for such anomalies rather
than searching for them.
These anomalies appear on scales larger than the acoustic scale at recombination but
smaller than the current horizon scale, and so I choose to restrict the parameterization to
200 <
s
< 20000.
Mpc
(3.6)
Next I follow Ref. Dvorkin & Hu (2011) in defining a band limit for the frequency of deviations
by sampling δG0 (ln sj ) at a rate of 10 per decade or about 4 per e-fold of inflation. This rate
was determined to be sufficient to capture large-scale features in the power spectrum.
The parameterized δG0 function is then the natural spline of these sampling points pi =
δG0 (ln si ). In the Bi language of Eq. (3.1), its basis is constructed by splining the set of
sampling points
Bi (ln sj ) =



1 i = j
,
(3.7)


0 i 6= j
with ln si values in the range specified by Eq. (3.6) and a sampling grid in ln sj that extends
sufficiently further that the basis functions have negligible support thereafter. I choose the
arbitrary ln s∗ epoch to be the large-scale endpoint of the sampling grid and order the points
so that s1 = 200 Mpc is the smallest scale.
In practice, I then precompute Wi (k) and Xi (k) on a fine grid in k-space and use a
modified version of CAMB to evaluate CMB observables. The curvature power spectrum
is then defined by 22 parameters {ln As , ns , p1 , . . . p20 }. The normalization point k0 = 0.08
Mpc−1 is set to be the pivot point for the Planck 2013 dataset, which has the benefit that
it is in the featureless or slow-roll regime by assumption. To these I add the cosmological
parameters of the flat ΛCDM model, and this model, defined by 26 parameters, is called
GΛCDM whereas the ΛCDM model sets pi = 0 and has only 6 free parameters.
For the tensor power spectrum I consider cases where r = 0 or is constrained by the
66
temperature and/or polarization data. Since there is little current information on the slope
of the tensor spectrum, I set nt = −r/8 so as to satisfy the inflationary consistency relation
for cs = 1. I call the model that allows for non-negligible tensors rGΛCDM.
Constraints on these parameters from the datasets are obtained using the Markov Chain
Monte Carlo technique implemented with the CosmoMC code (Lewis & Bridle 2002). The
cosmological parameters are all given non-informative priors except for a global constraint
√
on I1 set by I1 < 1/ 2 beyond which the GSR approach breaks down. I also include the
standard Planck foreground parameters in all analyses.
Model
Dataset
G-T
GΛCDM
T
rG-T
rGΛCDM
T
rG-T B
rGΛCDM T +BICEP2
Table 3.1: Models and datasets. The GΛCDM model includes 20 parameters that sample
curvature source function deviations in addition to the 6 flat power-law ΛCDM parameters, whereas the rGΛCDM includes the tensor-to-scalar ratio r. The T dataset mainly
reflects Planck temperature data, but also includes WMAP9 polarization, Union 2.1 supernovae distance, baryon acoustic oscillation, and H0 measurements to constrain cosmological
parameters. The BICEP2 data set adds polarization constraints that limit r.
Since my choice of parameters oversamples δG0 relative to what the data can constrain,
individual measurements of pi are noisy with any true signal buried in the small covariance
between parameters. For visualization purposes, I therefore also construct the principal
components derived from an eigenvalue decomposition of the MCMC covariance matrix
estimate
Cij = hpi pj i − hpi ihpj i =
X
Sia σa2 Sja ,
(3.8)
a
where Sia is an orthonormal matrix of eigenvectors. Specifically, I define the PC parameters
ma =
X
i
67
Sia pi ,
(3.9)
such that their covariance matrix satisfies
hma mb i − hma ihmb i = δab σa2 .
(3.10)
I then postprocess the MCMC chains to obtain the posterior probability distributions in
these derived parameters. Given a rank ordering of the PC modes from smallest to largest
variance, I can also construct a PC filtered reconstruction of δG0 as Dvorkin & Hu (2011)
δG0bPC (ln si )
=
b
X
ma Sia ,
(3.11)
a=1
where b is chosen to reflect the well-measured eigenmodes.
The differences between this construction and that of Ref. Dvorkin & Hu (2011) are that
the PCs are defined by the covariance matrix inferred from the data itself and change for
different data combinations, their normalization is set by the discrete rather than continuous
orthonormality condition, and that their range is restricted by Eq. (3.6) to be in the region
of known anomalies. Finally, I allow for the possibility of non-negligible tensor contributions
to the observed spectra through the tensor-to-scalar ratio r.
3.2
Results
Here I present results for the curvature source function G0 , which controls deviations from
power-law initial conditions and their polarization predictions. I begin in §3.2.1 with the
case where tensor contributions are assumed to be negligible. In §3.2.2 I study the impact
of tensors, constrained either by the temperature data alone or by the BICEP2 B-mode
measurement, in changing the interpretation of temperature anomalies and their polarization
predictions. These model and dataset choices are summarized in Tab. 3.1.
68
Figure 3.1: Curvature power spectrum constraints derived from those on the curvature
source function in the various model-dataset combinations (68% and 95% CL bands here
and below). With no tensors (top panel) the suppression of power at k . 0.002 Mpc−1
begins at a sharp glitch with slightly larger power on either side. Allowing tensors in the T
dataset (middle panel) absorbs the excess at high k, decreasing the significance of the glitch
but increasing that of the power suppression. Constraining the maximum allowed tensors in
the T B dataset (bottom panel) interpolates between these cases. Lines represent the fiducial
ΛCDM model which I use in the following figures as a baseline for comparison.
.
3.2.1
Curvature Only
I begin with a baseline dataset whose inferences on the source function G0 is mainly driven
by the 2013 Planck temperature power spectrum (Ade et al. 2013b). To these I add the
WMAP9 polarization (Bennett et al. 2012), Union 2.1 supernovae distance , baryon acoustic
oscillation (Anderson et al. 2013; Padmanabhan et al. 2012; Blake et al. 2011), and SHOES
H0 (Riess et al. 2011) datasets in order to constrain other cosmological parameters in the
flat ΛCDM model. I call this combination the “T ” dataset.
69
Figure 3.2: Temperature power spectrum constraints relative to the fiducial ΛCDM model
of Fig. 3.1. In each model-data case, deviations in the curvature source δG0 can model the
∼ 15% glitch feature at ` ∼ 20 − 40 and the suppression of low multipole power but with
different contributions from tensors that lead to different predictions for polarization and
curvature sources.
I first study this T dataset under the assumption that tensors are negligible (r = 0) in the
GΛCDM context and call this the G-T analysis. Tab. 3.2 gives the constraints on parameters.
As expected, the oversampling of the δG0 function relative to what the data can constrain
means that results on individual amplitudes pi = δG0 (ln si ) marginalized over the other
parameters have very low signal-to-noise. Nonetheless, combined with their covariances,
they do favor a suppression of the curvature power spectrum at large scales. To quantify
these features I consider the power spectrum itself ∆2R (k) to be a derived parameter and
show the 68% and 95% CL regions in Fig. 3.1 (top panel). I also show the best fit ΛCDM
model, with parameters given by table 2.1.
Note the coherent suppression of power relative to this fiducial model for k . 0.002 Mpc−1
70
that begins with a fairly sharp, almost oscillatory, dip with a slight preference for larger
power on either side. This position corresponds to the known temperature power spectrum
anomaly at ` ∼ 20 − 40 as shown in Fig. 3.2 (Peiris et al. 2003). Here I similarly consider the
theoretical temperature power spectrum C`T T as a derived parameter. In particular there is
∼ 15% deficit of power at ` . 20 and a slight excess of power at ` ∼ 40.
In this region, the T E cross correlation is small and so the EE power spectrum provides
nearly independent information on this feature. In Fig. 3.3, I show the theoretical C`EE
power spectrum as a derived parameter. Note first the much larger allowed range of fractional
deviations. Since the cosmic variance limit on measuring deviations in T T and EE are the
same fractionally, this indicates the large discovery potential for precision EE measurements
with even 40% measurements across the ` ∼ 20 − 40 band being of interest for verifying or
falsifying the inflationary explanation of temperature features. Because of projection effects,
namely the enhanced sharpness of the transfer of power to polarization Hu & White (1997),
the fractional suppression of polarization power is predicted to begin at a slightly higher
multipole and is allowed to reach lower values at the extrema at around ` ∼ 26. For ` . 20
the predictions are subject to uncertainties in reionization (Mortonson et al. 2009) as well as
possible impact of galactic foregrounds on the WMAP9 polarization used here as a constraint.
They are thus of less immediate relevance for inflationary features.
Given the possibility of confirmation by upcoming polarization measurements, it is interesting to explore in more detail what constraints on inflationary models these features imply.
Since the pi constraints on δG0 are too noisy to visualize the small but statistically significant constraints directly, I transform them to the PC basis ma as described in the previous
section. In Fig. 3.4 I show each of the 20 statistically independent ma measurements. Only
the first 3 PCs show measurements that that deviate from ma = 0 at the 95% CL or more.
In terms of the standard errors in Tab. 3.2, ma = 0 is a 2.2σ deviation in both m1 and m2 .
Fig. 3.5 confirms that ma = 0 indeed lies in the tails of the posterior probability distribution
in both. Given the 20 parameter model, this indicates a preference for a deviation in δG0
71
that is significant but not overwhelmingly so. Given that the corresponding E polarization
features can be twice as large, I can infer that if the ma parameters remain at their central
values, polarization measurements can provide a convincing detection of the deviation.
These first 3 PCs represent coherent deviations in the source function on scales s & 300
Mpc with differences mainly reflecting the location and how sharply the deviations rise
around that scale (see Fig. 3.6). Since the PCs are constructed for each model-dataset
independently, a particular ma does not have a fixed meaning. It is therefore useful to
sum the first three components together to form a 3PC filtered reconstruction of δG0 from
Eq. (3.11) shown in Fig. 3.7. Of course, more rapid deviations or deviations at s 103 Mpc
are allowed by the higher PCs but they are not significantly constrained by the data. In fact
√
these models are mainly limited by the prior I1 < 1/ 2 and the sampling rate. The data
instead favor a relatively sharp suppression of δG0 beginning at s ∼ 300 − 400 Mpc that is
coherent thereafter.
3.2.2
Curvature and Tensors
The preference for features in the curvature source only get more significant if the tensorto-scalar ratio r is allowed to vary as in the rGΛCDM model. I first consider implications
from the temperature-based T dataset and call this the rG-T analysis. In this dataset, the
tensor amplitude is constrained by the shape of the temperature power spectrum due to its
tensor contributions above the horizon at recombination. To achieve the same temperature
power spectrum, the curvature contributions must be further suppressed and hence there is
a near degeneracy between δG0 and r.
Thus instead of the upper limit of r < 0.11 (95% CL) at k = 0.002 Mpc−1 (Ade et al.
2013c), the constraints on r weaken substantially as shown in Fig. 3.8, allowing and even
mildly preferring values of r > 0.2. These large values are mildly preferred because of the
excess of power in the temperature spectrum around ` ∼ 40 (see Fig. 3.2) which can be
explained by a large tensor contribution. Of course, such an explanation would require
72
an even larger suppression of the curvature spectrum on larger scales to produce the same
temperature power spectrum.
These qualitative expectations are borne out in the curvature power spectrum constraints
in Fig. 3.1. The suppression in the curvature power spectrum begins at k & 0.002 Mpc−1
making the feature there appear less like a glitch and more like part of a coherent, but larger
and more significant, suppression of long-wavelength power. Note that the combination of
the curvature and tensor sources leads to the very similar temperature power spectra shown
in Fig. 3.2.
Interestingly, the prediction for E-mode polarization power spectrum differs qualitatively
from the G-T case (see Fig. 3.3). Tensors also contribute E-modes with a larger E to T ratio
than scalars due to projection effects (see e.g. Hu & White (1997), Eq. 25). Thus instead of
a deficit in power there is a preference for an increment in power in the ` ∼ 20 − 70 regime
that is allowed to reach in excess of 20% in contrast to the −40% without tensors. Thus, E
polarization power spectrum can provide a sharp test of models with r > 0.2.
In terms of the principal components, m2 and m3 = 0 are disfavored at 1.7σ and 3.2σ
respectively in Tab. 3.2 and in Fig. 3.4, and lie in the tails of the posterior distributions
of Fig. 3.5. The first component m1 no longer shows a significant deviation. Although the
detailed shape of the PCs vary depending on the model-dataset combination (see Fig. 3.6),
the first component is still associated with a rapid change in deviations around s = 200 Mpc.
With the addition of tensors, the change in the curvature source is more gradual. This can
be seen in the 3 PC filtered reconstruction of Fig. 3.7, where the main difference is a gradual
increase in amplitude to larger s and a broadening of the allowed range.
In fact, the T dataset allows such large values of r that interpreting the BICEP2 B-mode
detection as an upper limit restricts the range of deviations and E-polarization predictions.
In the rG-T B analysis, I assume that there is no dust contamination to the measurement and
hence obtain conservative maximal values for r. Even under this assumption, the addition of
the BICEP2 measurement eliminates models with r & 0.4 at high confidence (see Fig. 3.8).
73
In the curvature power spectrum, this makes the predictions intermediate between the G-T
and rG-T analyses, in particular for k slightly larger than the k ∼ 0.002 Mpc−1 glitch.
Likewise, the E-polarization predictions are intermediate as well. Instead of a deficit or
increment in predicted power, there is little net preference for either. Note however that in
the ` ∼ 20 − 40 regime there still is a shallower relative dip of ∼ 10 − 20% which can still be
used to confirm an inflationary feature with precision measurements (see also Miranda et al.
(2014) for model examples). For the PCs of the rG-T B analysis the first three components
disfavor ma = 0 at the 2.2σ, 3.3σ, 2.3σ levels (see Tab. 3.2 and Figs. 3.4-3.5).
Of course, this larger formal significance of should not be interpreted as enhanced evidence for features given the uncertain level of contamination by dust. Accounting for some
fractional contamination by dust would further interpolate between the G-T and rG-T B
results. In fact, if a polarization dip at ` ∼ 20 − 30 is detected, its depth relative to the
temperature one can be used to constrain r further independently of the B-modes.
74
Figure 3.3: Polarization predictions for the various model-data combinations. With no tensors (top panel) predicted features are twice as large (∼ 30%) as the corresponding temperature ones, implying that comparably precise measurements should conclusively confirm or
falsify their origin as curvature source features. With the tensors allowed by the temperature
based T dataset (middle panel), the relatively larger tensor E contributions fill in the ` < 40
scalar deficit with increments predicted for r & 0.2. Using the BICEP2 measurement to limit
the r bound, the possible increment and a measured decrement would provide independent
constraints on r. Note that ` . 20 predictions are subject to reionization model uncertainties and employ WMAP9 polarization constraints that are subject to galactic foreground
uncertainties.
75
Figure 3.4: Principal component amplitude constraints for the curvature source function
δG0 . Deviations from a featureless ma = 0 spectrum at > 95%CL appear in the first 3 PCs
but are absent in the higher ones. Models with allowed tensor contributions show both larger
and more significant deviations. The 3 PCs are constructed separately in each model-dataset
combination and hence ma does not represent the same parameter between panels.
76
Figure 3.5: Posterior probability distributions of the first 3 PC parameters. In each modeldataset case, the featureless ma = 0 model lies in the tails for two or three components, with
the more extreme deviations for those that allow tensor contributions.
77
Figure 3.6: First 3 PC eigenvectors constructed separately for the different model-dataset
combinations. Although they differ in detail in each combination, the first component mainly
determines how rapidly deviations begin after s = 200 Mpc and the third one carries substantial coherent deviations at s > 400 Mpc. The second component affects the intermediate
regime, and carries different contributions for s > 400 in the different cases.
78
Figure 3.7: 3 PC filtered curvature source G03PC (see Eq. 3.11). Favored deviations correspond to a negative source at s > 400 Mpc whose significance, depth and extent to smaller
scales increases for model cases that allow for tensors.
.
79
Figure 3.8: Posterior probability distribution of the tensor-to-scalar ratio r for the various
model-dataset combinations. In the G-T case it is fixed at r = 0. In the case that tensors are
constrained only by the T dataset, much larger r is allowed in the GΛCDM model compared
with the ΛCDM given the ability to reduce large-scale power in the curvature spectrum. The
BICEP2 data with no dust contamination favors r ≈ 0.2, shown here as the T B dataset,
and accounting for contamination still sets a stronger upper limit on r.
80
h2
Ωb
Ωc h2
θMC
τ
ln(1010 As )
ns
r
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15
p16
p17
p18
p19
p20
m1
m2
m3
G-T
rG-T
rG-T B
0.02218 ± 0.00024 0.02210 ± 0.00025 0.022093 ± 0.00024
0.1183 ± 0.0014
0.1183 ± 0.0014
0.1182 ± 0.0014
1.04150 ± 0.00055 1.04142 ± 0.00054 1.04148 ± 0.00054
0.099 ± 0.016
0.100 ± 0.017
0.103 ± 0.017
3.086 ± 0.033
3.089 ± 0.033
3.094 ± 0.034
0.9612 ± 0.0060
0.9638 ± 0.0063
0.9626 ± 0.0060
0
0.30 ± 0.16
0.229 ± 0.048
−0.05 ± 0.10
−0.17 ± 0.13
−0.11 ± 0.11
−0.13 ± 0.17
−0.20 ± 0.17
−0.16 ± 0.18
0.07 ± 0.24
−0.16 ± 0.27
−0.03 ± 0.24
−0.47 ± 0.35
−0.57 ± 0.35
−0.60 ± 0.34
0.51 ± 0.56
0.37 ± 0.56
0.41 ± 0.53
−0.61 ± 0.92
−0.69 ± 0.94
−0.84 ± 0.90
−0.8 ± 1.5
−0.8 ± 1.5
−0.6 ± 1.4
−0.4 ± 2.5
−0.5 ± 2.4
−0.8 ± 2.4
−0.2 ± 3.3
0.1 ± 3.0
0.4 ± 3.0
2.2 ± 3.5
1.5 ± 3.1
1.3 ± 3.1
−0.4 ± 3.5
−0.1 ± 3.3
0.1 ± 3.3
−1.3 ± 3.3
−1.3 ± 3.3
−1.5 ± 3.2
−0.0 ± 3.2
0.1 ± 3.3
0.2 ± 3.2
1.2 ± 3.3
1.1 ± 3.5
1.0 ± 3.4
−1.4 ± 3.6
−1.2 ± 3.7
−1.1 ± 3.6
−0.1 ± 3.9
−0.1 ± 3.9
−0.3 ± 3.9
0.3 ± 4.1
0.2 ± 4.1
0.3 ± 4.1
−0.1 ± 3.9
−0.1 ± 3.9
−0.2 ± 3.9
0.04 ± 3.7
0.0 ± 3.8
0.1 ± 3.7
0.1 ± 2.9
0.0 ± 2.9
0.0 ± 2.9
−0.155 ± 0.080
−0.018 ± 0.095
−0.188 ± 0.084
−0.26 ± 0.12
−0.27 ± 0.16
−0.41 ± 0.12
−0.22 ± 0.18
−0.65 ± 0.20
−0.40 ± 0.18
Table 3.2: Parameter constraints (68% CL) for the various model-dataset combinations of
Tab. 3.1. pi represent the deviations in the curvature source function from a scale-free power
law and the derived parameters ma represent amplitudes of the principal components of
their covariance matrix, which are not the same between combinations.
81
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