The Pennsylvania State University The Graduate School The Eberly College of Science

NON-GAUSSIAN STATISTICS AS A PROBE OF THE EARLY

UNIVERSE

A Dissertation in Physics by Elliot Luke Nelson

© 2015 Elliot Luke Nelson

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

August 2015 The dissertation of Elliot Luke Nelson was reviewed and approved∗ by the following:

Sarah Shandera Assistant Professor of Physics Dissertation Advisor, Chair of Committee

Martin Bojowald Professor of Physics

Donghui Jeong Assistant Professor of Astronomy & Astrophysics

Eugenio Bianchi Assistant Professor of Physics

Nitin Samarth Professor of Physics Head of the Department of Physics

∗Signatures are on file in the Graduate School.

ii Abstract

We study effects from mode coupling or non-Gaussianity in the primordial curvature perturbations. For local-type non-Gaussianity, which couples fluctuations on very different scales, we study the influence of long-wavelength background modes on the statistical properties of short-scale modes in a finite volume such as the observable universe. We show that background modes can introduce a shift or bias to observed parameters, 2 and quantify this influence for the primordial power spectrum ∆ζ , non-Gaussianity parameter fNL, spectral index ns, and dark matter halo power spectrum. We show that a non-Gaussian field with sufficiently strong coupling of modes on long and short scales will appear nearly Gaussian in typical subvolumes that are sufficiently small, indicating the naturalness of the weakly non-Gaussian local ansatz. In light of these results, we discuss the implications of observing the universe in a finite volume for our understanding of inflation. In addition to studying the statistical properties of curvature perturbations on a fixed spatial slice at the end of inflation, we also consider an example of mode coupling arising from nonlinear inflationary dynamics. A non-vacuum initial state at the onset of inflation can allow scalar curvature perturbations to couple strongly to long-wavelength tensor perturbations, resulting in an anisotropic contribution to the power spectrum of density perturbations.

iii Table of Contents

List of Figures vii

List of Symbols ix

Acknowledgments xii

Chapter 1 Introduction1 1.1 Overview of Cosmic History...... 1 1.2 Primordial Perturbations...... 3 1.3 Cosmological Collider Physics...... 8 1.4 Cosmic Variance ...... 8 1.5 Observing a Finite Volume...... 9 1.6 Super Cosmic Variance...... 10 1.7 Non-Gaussianity ...... 12 1.8 Statistical Formalism for Primordial Perturbations...... 13 1.8.1 Gaussian Statistics...... 14 1.8.2 Non-Gaussian Statistics ...... 16 1.8.3 Local-type non-Gaussianity ...... 17 1.9 Long Wavelength Modes in Cosmology...... 19 1.9.1 When Are Infrared Modes Relevant? ...... 19 1.9.2 Long Wavelength Backgrounds in Inflationary Models . . . . 20 1.9.3 Long-Wavelength Modes in Large Scale Structure ...... 22 1.9.4 Lyth’s Study of ζ(x) in a Subvolume...... 23 1.9.5 Recent Work on Super Cosmic Variance ...... 24 1.9.6 Primordial Gravitational Waves and Non-Vacuum Initial States 25 1.10 Outline of Chapters2-4...... 25

Chapter 2 Super Cosmic Variance from Local Non-Gaussianity 29

iv 2.1 Introduction...... 29 2.2 Statistics of ζ in a Subvolume with Local Non-Gaussianity . . . . . 32 2.2.1 Long and Short Wavelength Split...... 33 2.2.2 The Long Wavelength Background ...... 36 2.2.3 Spatial Curvature from Long-Wavelength Modes ...... 38 2.3 Example I: Weakly Non-Gaussian Initial Conditions...... 39 2.3.1 Biasing of Parameters with Local Non-Gaussianity . . . . . 39 2.3.2 Implications for Model Builder...... 43 2.4 Example II: Strongly Non-Gaussian Initial Conditions ...... 44 2.5 Example III: Two-field Initial Conditions...... 49 2.5.1 Case 1: σ Is Weakly Non-Gaussian ...... 50 2.5.2 Case 2: σ Is Strongly Non-Gaussian...... 52 2.6 General Local-type Non-Gaussianity ...... 55 2.6.1 Running of Non-Gaussian Parameters...... 56 2.7 Behavior of N-Point Function Shapes...... 57 2.8 Conclusions...... 60

Chapter 3 Super Cosmic Variance with Scale-Dependence 64 3.1 Introduction...... 64 3.2 Subsampling the local ansatz with scale-dependence in single- and multi-source scenarios...... 66 3.2.1 The power spectrum ...... 66 3.2.2 The bispectrum and the level of non-Gaussianity ...... 69 3.3 Observational consequences ...... 72 3.3.1 The shift to the power spectrum...... 73 3.3.2 The shift to the spectral index, ∆ns ...... 74 3.3.3 The shift to the scale dependence of the bispectrum . . . . . 88 3.3.4 Generalized local ansatz and single source vs. multi source effects ...... 89 3.4 Mode coupling effects from a non-local factorizable bispectrum . . . 90 3.5 Tensor mode running as a test of inflation?...... 93 3.6 Discussion and conclusions...... 95

Chapter 4 Fossilized Gravitational Wave Relic from Squeezed Limit Coupling 98 4.1 Introduction...... 98 4.2 A generalized initial state and the gravitational fossil ...... 100 4.2.1 Scalar-scalar-tensor bispectrum ...... 101

v 4.2.2 Enhanced fossil signature in locally anisotropic scalar fluctu- ations ...... 105 4.3 General condition for scalar-scalar-fossil correlation ...... 112 4.4 Quadrupolar anisotropy and superhorizon modes ...... 113 4.5 Conclusion...... 115

Chapter 5 Discussion of Results 118 5.1 Summary of Thesis Work...... 118 5.2 Super Cosmic Variance and Inflation ...... 119 5.2.1 Inflationary Trajectories for Finite Volumes ...... 120 5.2.2 Implications for Inflation...... 121 5.2.3 Statistical Naturalness for Inflationary Trajectories . . . . . 123 5.2.4 Statistical and Inflationary Landscapes ...... 124 5.3 Conclusion and Outlook ...... 125

Appendix A Diagrammatic Representations of N-point Functions 127 A.1 Feynman-like Rules for Momentum Space Diagrams ...... 131

Appendix B Behavior of Local-type N-point Functions 134

Appendix C Mapping Between Statistics in Vl and Statistics in Vs 141

Bibliography 144

vi List of Figures

1.1 A gaussian random field...... 6 1.2 The causal past of the observable universe...... 10 1.3 Correlation of long and short scales for a non-Gaussian random field. 11

2 2.1 The variance hζGli of the long-wavelength background...... 37 2.2 Probability distribution for the global power spectrum, conditioned on an observed power spectrum...... 41 2.3 Probability distribution for the global fNL, conditioned on an ob- obs served fNL ...... 42 2.4 Variation between the global and observed levels of non-Gaussianity, fNL∆ζ ...... 45 obs 2.5 Dependence of observed parameter fNL on superhorizon scales and global spectral index...... 46 obs 2.6 Probability distribution for the global power spectrum and fNL , conditioned on observed parameters, with two sources for curvature perturbations and weak non-Gaussianity...... 52 obs 2.7 Probability distribution for the global power spectrum and fNL , conditioned on observed parameters, with two sources for curvature perturbations and strong non-Gaussianity ...... 54

2 3.1 Parameter space (fNL, hζGli) of scale-invariant local non-Gaussianity with linear and quadratic terms...... 78 2 3.2 Parameter space (fNL(kpiv), hζGli) for local non-Gaussianity with scale-dependent power and fNL(k)...... 81 3.3 Probability distribution for the shift or bias to the locally observed spectral index, in terms of the amplitude of the long-wavelength

background, global fNL(kpiv) on observable scales, and running of fNL(k)...... 84 2 2 obs 3.4 Global power spectrum ∆ζ (k) and locally observed spectrum (∆ζ (k)) , with a shifted spectral index on observable scales...... 85

vii 2 3.5 Parameter space (fNL(kpiv), hσGli) for scale-dependent local non- Gaussianity with two sources φG and σNG...... 87 4.1 Contour plot for survey size needed to detect primordial gravitational waves, in terms of tensor power spectrum and parameter β for non- vacuum initial state...... 109 4.2 Contour plot for survey size needed to detect primordial gravitational waves from scalar-scalar-tensor bispectrum with stronger-than-local squeezed limit, in terms of tensor power spectrum and bispectrum amplitude...... 114

B.1 Correction factor for the power spectrum Pζ (k) and other polyspec- tra in local non-Gaussianity, from a nonzero fNL...... 139

viii List of Symbols

Used primarily in Chapters1–3:

ζ(x) The (non-Gaussian) curvature perturbation. ζ(k), ζk A Fourier mode of the curvature perturbation. Pζ (k) The power spectrum of ζ. 2 ∆ζ (k) The dimensionless power spectrum or variance of fluctu- 2 3 2 ations for ζ, ∆ζ (k) ≡ k Pζ (k)/(2π ). ζG A Gaussian random field used to generate ζ. PG(k) The power spectrum of ζG. 2 ∆G(k) The dimensionless power spectrum for ζG. 2 ns The scalar spectral index ns − 1 = d ln ∆ /d ln k. Bζ (k1, k2, k3) Bispectrum for curvature perturbations. Ωk, H0 The spatial curvature and Hubble scale today. Vl The volume over which ζNG is defined, e.g. the entire post-inflationary patch Vs, Ws A subvolume of Vl and the corresponding window func- R 3 −ix·k tion, Ws(k) = Vs d x e . For most of this thesis we take Vs to be our Hubble volume. ? k , kmax,L Wavenumbers corresponding to the subvolume scale, ? 1/3 k = 2πVs , shortest scale, and size of the global volume, 1/3 L = Vl . kL, kS Wavenumbers referring to widely separated scales kL  kS. ζGs, ζGl The short and long wavelength components of ζG. See Eq. (2.20), Eq. (2.21) 1 N, Nsub The number of super-horizon e-folds, 3 ln(Vl/Vs), and subhorizon e-folds.

ix obs X The value of the quantity X measured in Vs. fNL, gNL, τNL, hNL Non-Gaussian parameters given in Eq. (2.29). σG, φG, ξ Gaussian random fields we use to define ζ in the two- field example in section 2.5, and the ratio of their power 2 spectra, ξ ≡ Pφ/Pσ. n Nn Coefficients of higher order nonlinear terms ζG in the local ansatz. hζn(x)i Mn Dimensionless cumulants Mn ≡ hζ2(x)in/2 of the non- Gaussian distribution in real space. ρ[ζ(x)], ρG[ζ(x)] Non-Gaussian and Gaussian probability distributions for configurations ζ(x).

Used primarily in Chapter3:

nf Parameter describing power-law scale-dependence of fNL(k). nζ Spectral index for Gaussian field ζG in two-field model. nσ Spectral index for Gaussian field σG in two-field model. (m) nf Parameter describing power-law scale-dependence of ξ(k). αs(k) Running of the spectral index, αs ≡ d ln ns(k)/d ln k. nsq.(k) Parameter describing the scale-dependence of the bispec- trum in the squeezed-limit. Pσ(k), Pσ,NG(k) Power spectra for the Gaussian field σG and for its non- linear contribution to the curvature perturbation. CMB fNL Observed value fNL defined in terms of CMB scales. χ A proxy field which couples to tensor perturbations and shifts the tensor power spectrum.

x Used primarily in Chapter4:

γij(x) Tensor metric perturbation. ij Antisymmetric spatial tensor. γs(x) Tensor metric perturbation with polarization s. s s Pγ (k), Aγ Power spectrum for tensor perturbation γ (k), and di- 3 s mensionless amplitude Aγ = k Pγ (k). η Conformal time during inflation. (s),(t) uk (η) Mode functions for scalar (s) or tensor (t) perturbations during inflation. (s),(t) (s),(t) αk , βk Complex paramaters describing non-vacuum initial state for scalar or tensor perturbations. (s),(t) Θk Relative phases between αk and βk parameters, defined iΘk by βk/αk = e |βk/αk|, for scalar or tensor modes. n Pp (k) Noise power spectrum, or variance of estimator for tensor mode γp(k) from scalar curvature perturbations. 2 σγ Variance for optimal estimator of Aγ, amplitude of fluc- tuations for tensor modes. fζζγ Dimensionless fNL-like parameter for amplitude of scalar- scalar-tensor three-point correlation. Λ Maximum comoving scale at which scalar perturbations are excited with βk 6= 0.

xi Acknowledgments

I thank Sarah Shandera for her consistent guidance over the past few years, in particular for pushing me to become more independent in my research, giving me freedom to explore topics of my own choice, and taking the time to answer my many questions about cosmology. I thank Suddhasattwa Brahma for stimulating conversations about many different topics in theoretical physics. The chapters included in this thesis are based on published papers with collabo- rators. Chapter2 is based on reference [1], co-authored with Sarah Shandera, and reference [2], co-authored with Marilena LoVerde and Sarah Shandera. Chapter3 is based on reference [3], co-authored with Joseph Bramante, Jason Kumar, and Sarah Shandera. Chapter4 is based on reference [4], co-authored with Suddhasattwa Brahma and Sarah Shandera. I thank my collaborators on these projects for their contributions to the work presented here. I am also grateful to Niayesh Afshordi, Latham Boyle, and Chris Byrnes for valuable input on the work presented in this thesis. Lastly, I am grateful to the members of my thesis committee for their guidance and their review of this dissertation. I especially thank my parents, Peter and Cheryl Nelson, for their encouragement for many years. My research has been supported by the Eberly Research Funds of The Pennsyl- vania State University. The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University. In addition, the work of chapter4 was supported by the New Frontiers in Astronomy and Cosmology program at the John Templeton Foundation.

xii Chapter 1 | Introduction

1.1 Overview of Cosmic History

Arguably the most important fact about the universe is its expansion, originally observed by Hubble in the 1920’s in the form of increasing recession velocities for more distant galaxies [5]. An immediate consequence of an expanding universe is its cooling. In the same way that a container of gas cools down as the walls of the container are expanded to give it more volume, all particles in the universe lose energy due to its expansion. As a consequence, the history of the universe is a history of different energy scales, with earlier times probing physics at higher energies. The observed expansion of the universe forces us to ask what were the highest energies reached in the history of the universe, and at what time. Over the past fifty years, observations have confirmed the picture of an incredibly high-energy early universe, highly contracted in size compared to today. Perhaps the strongest piece of evidence for a very hot and dense early universe is the Cosmic Microwave Background (CMB), a background of thermal radiation filling the universe. The distribution of energies of photons comprising the CMB was found to fit a blackbody distribution to very high accuracy, indicating the thermal nature of the radiation [6,7]. Since the density of CMB photons is very low today, as is the density of any other particles through which CMB photons might exchange energy through scattering and come to thermal equilibrium, we are forced to conclude that thermal equilibrium occurred at a much earlier point in cosmic history, when the universe was hot and dense enough for photons to thermalize. The picture of a universe expanding and cooling from a hot and dense primordial

1 state is also confirmed by Einstein’s theory of gravity, general relativity [8–10], which has been experimentally confirmed with a wide variety of tests [11]. Although it wasn’t understood at first, general relativity predicted either an expanding or contracting universe. The spatial size of the universe is described with a scale factor a(t), which is proportional to the distance between any two points in space. A static universe of constant size, neither growing nor shrinking, would be unstable, like a carefully balanced upside-down pendulum. According to the field equations of general relativity applied to a spatially uniform spacetime1, the amount of matter in the universe would have to be delicately balanced with both a cosmological constant2 and spatial curvature3 in order to prevent either contraction from the attractive gravitational force of matter (da/dt < 0), or accelerated expansion from the pressure of the cosmological constant (da/dt > 0). Without the pressure of a cosmological constant to stabilize the acceleration of the universe at zero, even a universe with the density of matter and radiation carefully set to fix the expansion rate of the universe to zero (da/dt = 0) would being to contract due to the tendency of matter to slow down expansion (d2a/dt2 < 0). And without spatial curvature, which observations constrain to be close to zero, a universe with matter must expand or contract (da/dt 6= 0). In light of the instability of a static universe, we can say that general relativity predicted an expanding or contracting universe. The expanding solution to Einstein’s gravity, along with the soon-to-be-measured Hubble parameter H ≡ (da/dt)/a or expansion rate, was first pointed out by Georges Lemaîtrue, a physicist and Catholic priest, in 1927 [12]. General relativity predicts a relationship between the expansion history of the universe and the matter and energy filling it. For any combination of non-relativistic matter (particles or collections of particles moving at speeds much less than the speed of light) and relativistic matter or radiation (anything moving at speeds close to the speed of light), the expansion history of the universe is determined in a simple way. If we know the amount of matter and radiation in the universe, general relativity allows us to rewind the clock back to moments when the universe

1See section 1.2 for more on the uniformity of the universe on large scales. 2The cosmological constant is a source of constant energy per unit volume of space, which leads to an accelerated expansion of the universe. 3Spatial curvature describes the amount of curvature in space at a fixed moment in time. Just as the curvature of the earth can be related to its radius, the spatial curvature of the universe can be related to a radius of curvature. Observations of the CMB fix the spatial curvature of our universe to be close to zero, so this “static solution” of Einstein’s equations does not apply.

2 was extremely small and hot, with particles scattering at extraordinary energies. The average energy of particles becomes limitlessly large at a finite time in the past. General relativity allows us to infer this time - the age of the universe - in terms of measured amounts of matter and radiation.4 It turns out to be roughly 13.8 billion years [13]. We cannot reliably extrapolate the history of the universe to an initial time of infinite density and zero size, since generally relativity does not apply for arbitrarily large energies. However, we can reliably map out the history of the universe through the “hot Big Bang,” that is, back to a tiny fraction of a second after the would-be singularity, through times when the average energy of particles falls within the range of energies understood in the Standard Model of particle physics. At even earlier epochs, the energy density of the universe was likely far beyond the range accessible to current particle accelerators such as the Large Hadron Collider (LHC) [14], currently the world’s most powerful collider. At this point, we don’t know exactly what to expect, since we have no access to such energy scales here on earth - new physics likely becomes important. Any high energy physics at play when the universe was hotter and more contracted can only leave an indirect signal for us today. All radiation and matter has been cooled down by the expansion of the universe, so we can’t observe high- energy processes directly. But it is true that high energy physics in the early universe can affect the subsequent expansion history of the universe as it cools down, or the process of structure formation. Radiation observed today, although cooled down by billions of years of expansion, is still seen with its primordial distribution, largely untainted. The universe then becomes a laboratory for studying physics at high energies. In fact, since we cannot accelerate and scatter particles with energies beyond several TeV today, cosmology may be our only observational window to these energy scales.

1.2 Primordial Perturbations

Observations of the universe can be categorized in terms of the spatial scales that they probe. On shorter scales, smaller than roughly 10 Mpc, the universe is

4Most of the matter in the universe is dark matter - it does not interact with particles in the Standard Model, and can thus only be measured through its gravitational effects on other matter and radiation.

3 very inhomogeneous, characterized by complex structures such as galaxy clusters, superclusters, filaments, and voids. However, if we smooth the mass distribution on larger scales, we find a remarkably uniform universe. This uniformity is most clearly seen in the Cosmic Microwave Background, which has a temperature that is uniform in all directions across the sky, up to small variations or “anisotropies” of roughly one part in a hundred thousand.5 Until a few hundred thousand years after the earliest moment, the density of matter and radiation was high enough to maintain thermal equilibrium, but after this time, the expansion of the universe diluted space enough so that CMB photons have since travelled largely uninterrupted through space. Consequently, the small spatial variations of temperature that we measure give us a snapshot of a time when the universe was less than a ten thousandth of its present age. Their very near homogeneity shows us a very smooth and uniform early universe, but also non-uniformities present from the beginning. One of the most important features of the CMB anisotropies is their primordial origin. Because these anisotropies were already present at a very early epoch, and add up in a coherent way to produce correlations over large distances, they could not have been created since the hot Big Bang. No causal signal could traverse the long distances over which these correlations are spread in such a short time [15]. The leading paradigm for understanding both the near uniformity of the universe and the origin of its small variations in temperature and density on large scales is known as inflation [16–19], a period of accelerated expansion before the hot Big Bang, occurring on a timescale of less than 10−30 seconds, and at a very high energy scale of around 1015 GeV. This accelerated expansion allows for correlations over enormous distances to share a common causal past, and finds a quantum mechanical origin for primordial perturbations [19–21]. In order to keep the expansion rate H(t) constant, a matter source is needed with equation of state p ≈ −ρ, where p and ρ are the pressure and energy density, respectively. Equivalently, the energy density must remain constant, rather than diluting away due to the expansion, in order to sustain inflation. The matter source yielding this energy is usually described as a scalar field with a potential V (φ) that is nonzero and changing slowly as the background field evolves according to its

5If the earth was spherical to this precision, altitudes around the globe would vary by only a few hundred feet!

4 equation of motion during inflation. (Like the Higgs field, it would have a nonzero expectation value; unlike the Higgs, though, its potential would also be nonzero.) This “inflaton” field may be a fundamental field, or a low-energy effective field arising from additional degrees of freedom at higher energies. Although inflation involves a near-exponential expansion of space, it is a dynam- ical process that is approximately time translation invariant, since an exponentially expanding spacetime can be obtained as a foliation of de Sitter space – a maximally symmetric spacetime for which all points in time are geometrically equivalent – into spatial slices [22]. As a consequence of this time symmetry, as well as the fact that fluctuations of a given wavelength λ = 2π/k are generated quantum mechanically at a corresponding time tk during inflation when the physical wavelength becomes larger than the curvature or Hubble scale (a(tk)/k ' 1/H(tk)), the large-scale per- turbations generated by inflation inherit a symmetry of scale: the root-mean-square amplitude of perturbations is the same on all spatial scales (this will be quantified below in section 1.8.1). However, inflation must end at some point in time, so in general the time symmetry is not exact. The breaking of time translation invariance is typically quantified with a “slow-roll” parameter   1 quantifying the variation in the potential V (φ). As a consequence, the resulting spectrum of perturbations is not exactly scale-invariant. This near but not exact scale-invariance (see section 1.8.1 below) has been measured observationally in the dependence of temperature fluctuations in the CMB on angular scale [13,23], yielding perhaps the strongest piece of evidence for inflation. The underlying high-energy particle physics at play during inflation, however, is unknown, and much work has been invested in understanding potential signatures of inflationary physics in the statistics of primordial perturbations. The temperature anisotropies in the CMB, as well as other inhomogeneities which led to the formation of structure in the universe, can be traced back to primordial perturbations in the spacetime metric and energy density of matter. At the end of inflation, the metric may be written in the form6

2 2 2 h i i j ds = −(1 + 2Φ(x))dt + a (t) δij(1 − 2Ψ(x)) + γij(x) dx dx . (1.1)

6Different coordinate choices or gauges lead to perturbations in different components of the metric, although the degrees of freedom are the same. Our choice here is the conformal Newtonian gauge.

5 Figure 1.1: A two-dimensional slice of a random realization of the primordial potential Φ(x), with color indicating variation in Φ: dark purple coloration indicates an over density (Φ > 0) and white indicates an under density (Φ < 0). CMB measurements reveal random fluctuations – the density fluctuations and gravitational potential vary randomly in space, with values at any point drawn from a Gaussian distribution – as well as scale invariance (see section 1.8.1), which we assume here.

ij ij The tensor fluctuations γij are transverse (∂iγ = 0) and traceless (δijγ = 0). Primordial tensor perturbations have not been observed and play only a small role in the history of the universe, in contrast with the scalar perturbation Φ, which creates potential wells where gravity acts to form structures. For the most part, we will ignore tensor perturbations. We also omit primordial vector perturbations, which decay exponentially in an expanding universe. Lastly, we assume there is no anisotropic stress, in which case Einstein’s equations for cosmological perturbations allow us to identify Ψ = −Φ. Eq. (1.1) is valid on distance scales k−1 and at times t for which fluctuations in Φ have not become large due to the formation of gravitationally bound objects. We will be interested in the metric at very early times, after inflation but before any later evolution of perturbations takes place. Along with perturbations to the spacetime geometry, we have small variations in the energy density of matter and radiation in the early universe, which are simply related to the metric perturbation Φ: If inflation only involves a single dynamical degree of freedom, perturbations of different quantities, such as the gravitational potential or energy densities of different species, are adiabatic. That is, they all share the same relative amplitude because they all originate in the

6 same quantity during inflation. Multi-field models of inflation may introduce relative differences between perturbations, but such “isocurvature” perturbations are observationally constrained to be very small. Statistical properties of different species of perturbations are then directly related to each other. In particular, because the temperature variations in the CMB are on large scales, and are well preserved from early times, before nonlinear processes such as the formation of gravitationally bound structures, they can be related to the potential Φ (or density perturbations δρ) through a complicated but nearly linear mapping. Primordial perturbations also act as the seeds from which more complicated structures formed: Through the influence of gravity, matter was drawn into potential wells – regions with Φ(x) < 0 – leading to the formation of gravitationally bound objects such as galaxy clusters and their surrounding “halos” of dark matter. The quantity most commonly used to describe primordial perturbations is the curvature perturbation ζ, which is defined as a combination of the gravitational potential Φ and energy density perturbations δρ which is gauge invariant at first order in the perturbations [24,25],

H ζ ≡ Φ − δρ. (1.2) ρ˙

Here, ρ˙ = dρ/dt is the time derivative of the background energy density ρ, which comes from the inflaton during inflation, but Standard Model radiation or cold dark matter in the late universe. The curvature perturbation ζ does not depend on the coordinate system chosen, and in coordinates where the density is spatially uniform (δρ = 0) it is simply the gravitational potential Φ. In this thesis we will work almost entirely with the curvature perturbation. Due to the adiabaticity of primordial perturbations, the curvature perturbation acts as the sole initial condition for the dynamics of cosmological perturbations: its statistical properties determine, along with late-universe dynamics, the statistical properties of cosmological structures. The curvature perturbation is therefore of great interest as a bridge between observable quantities and physics of the early universe. For additional reading on the dynamics of primordial perturbations during and after inflation, a thorough description can be found in [26]. The curvature perturbation ζ is presented clearly in [27]. An introduction to inflation and inflationary fluctuations can be found in [28,28].

7 1.3 Cosmological Collider Physics

In particle physics, the interactions of particles are measured through scattering experiments. Physics at a certain energy scale is studied by accelerating particles to that energy and colliding them. The statistical features of the scattering process contain information about the underlying interactions. In cosmology, the statistical features of primordial perturbations also contain information. In the context of inflation, these perturbations are typically generated from the quantum fluctuations of the inflaton or other fields in their ground state.

Each Fourier mode φk of the inflaton is essentially a quantum harmonic oscillator, and thus has a finite spread around the expectation value hφki = 0, even in the lowest energy state. This quantum uncertainty is converted into the amplitude for classical perturbations. Consequently, the interactions of the quantum field theory describing inflation determine the statistical properties of the resulting cosmological perturbations. More specifically, the degrees of freedom and parameters of that field theory determine a probability distribution on the space of possible configurations for primordial perturbations. In this sense, we can think of primordial perturbations as being the outcome of a cosmological scattering experiment. As in particle physics, the probabilistic predictions are a consequence of the indeterministic, stochastic nature of quantum mechanics, which is propagated to cosmological scales by the exponential expansion of space. Since primordial perturbations act as initial conditions for the subsequent history of the universe, the distributions of observable quantities such as radiation intensity at various wavelengths, bound structures like galaxies, and intergalactic matter act as tracers of their initial conditions. The probability distributions governing these quantities are affected by the probability distribution governing primordial perturbations. Information about early universe physics, then, is mapped into observable quantities in the late universe.

1.4 Cosmic Variance

However, in contrast to the case of particle physics, where a given experiment can be repeated any number of times, we only have one realization of cosmological perturbations - that is, one draw from a probability distribution. This introduces

8 an irreducible sample variance, known as cosmic variance. Statistical properties of a finite sample generally exhibit a sample variance around the statistics of the global population or ensemble distribution, with smaller samples exhibiting larger variance. The average galaxy cluster mass from a sample of ten clusters , for example, will vary significantly around the population mean. More precisely, if we are able to sample a probability distribution P (X) for a variable X a total of N times, then the sample mean X¯ will vary around the ensemble mean hXi ≡ R XP (X)dX, with a variance inversely proportional to N,

σ2 h(X¯ − hXi)2i = X , (1.3) N

2 R 2 where σX ≡ X P (X)dX is the variance of X. In the case of cosmological perturbations, we have a joint probability distribution

P ({ζk}) for a set of Fourier modes of ζ. If we consider the case of a random field

ζ(x) in a finite box of length L, then we have Fourier modes ζk with discrete + wavenumbers ki = 2πni/L, for ni ∈ Z . The number of Fourier modes with q P 2 2 3 k = ki within a small range ∆ ln k scales as k ∆k = k ∆ ln k, so in any finite volume there are many more modes ζk to be sampled on small scales. On the largest scales, only a small number of modes N are available, leading to a large sample variance in Eq. (1.3). Since only one realization of ζ is available, the large sample variance or cosmic variance on large scales places a fundamental limit on our access to the ensemble distribution P [ζ(x)].

1.5 Observing a Finite Volume

In practice, we are also limited in our ability to observe the post-inflationary universe: we only have access to a finite volume of space. The finite age of the universe, combined with the fixed nature of the speed of light, places an upper bound on the distance that any causal signal could have travelled since the hot Big Bang7. This is illustrated in Figure 1.2. The size of the observable universe can also be thought of in terms of its expansion rate. Nearly a hundred years ago, Edwin Hubble observed that more

7Extending back to inflation, a much larger volume falls within our causal past. However, our only observational access to inflation is through primordial perturbations, which are set at the end of inflation at a finite time in our past.

9 Figure 1.2: Only light from within a finite volume can reach us today since the time of the initial conditions. distant galaxies, on average, were moving away from us at speeds proportional to their distance, v ≈ Hd, confirming the prediction of an expanding universe that had been recently demonstrated to be a generic consequence of general relativity. The constant of proportionality H in Hubble’s distance-velocity relation, measured to be roughly 70 km/s/Mpc [13, 29], sets the timescale for the expansion of the universe. With the speed of light fixed, this sets a distance scale of order c/H ≈ 5 Gpc. Points in space separated by more than this distance cannot affect each other causally, since the distance between them is growing faster than light can cross it. Our observation of the universe, therefore, is intrinsically limited by its expansion. Light from regions farther from us than a Hubble distance cannot have reached us since the Big Bang. In the context of inflation, a longer period of inflation corresponds to a larger post-inflationary volume. The length of inflation is described in terms of the number of e-folds, that is, the number of times the scale factor a(t) grows by a factor of e. Inflation longer than a minimum of roughly 60 e-folds corresponds to a −1 post-inflationary volume larger than the observable universe, with the ratio L/H0 between the size of the global and observable universe (as given by the Hubble −1 constant H0) related to the number of additional e-folds N by N ≈ ln(L/H0 ).

1.6 Super Cosmic Variance

We saw in section 1.4 that our limitation to a single realization of primordial perturbations introduced an irreducible uncertainty in our knowledge of the under- lying probability distribution. We also saw in section 1.5 that we are only able to

10 Figure 1.3: Large scale variation in a highly non-Gaussian field (left panel) correlates with the amplitude of short-scale fluctuations of the same field, show in the right panel with large-scale variation removed. Short-scale amplitude in the red box, for instance, would not be representative of the global average, but the correlated large scale variation is uniform and undetectable in the red box. subsample primordial perturbations within a finite volume, and have no access to perturbations on larger scales. This forces us to ask if perturbations in our observed volume are a representative sample of the underlying distribution, or are a biased sample due to the fact that we are sampling in a particular location. If variations in ζ(x) on scales larger than the observable universe are correlated with variations on smaller, observable scales – that is, if the probability distribution

P [ζ] correlates modes ζk with very different k – then a cosmic variance of statistics can occur, in which the probability distributions of small-scale perturbations are influenced by and vary along with fluctuations on larger scales, and are therefore systematically biased when sampled in any one spatial region. We will refer to this as superhorizon or super cosmic variance, since it arises from perturbations −1 on scales larger than the Hubble scale H0 , and illustrate an example in Figure 1.3.A mode coupling of different scales can occur when the probability distribution for the initial conditions is non-Gaussian, a feature which we will discuss in the following section. In the following chapters, we will focus on the effect of the average value of ζ smoothed over a finite subvolume, which sums all background modes to a single number. The cosmic variance associated with any effects from this background will be large, since we only have a N = 1 sample for the average background value.

11 1.7 Non-Gaussianity

For a Gaussian random field, each Fourier mode is a Gaussian random variable, uncorrelated with other modes. All of the information about the underlying probability distribution is included in the variance of this distribution as a function of scale k, that is, the average amplitude of fluctuations or power spectrum. If the underlying distribution is non-Gaussian, however, Fourier modes of the field are no longer uncorrelated, and more statistics are needed to describe the distribution. As we will see in detail in section 1.8, this information is captured in higher order correlation functions (in much the same way as scattering amplitudes in particle physics are determined by correlation functions of quantum fields). Non-Gaussianity or mode coupling can be generated in the initial conditions by interactions that occur during inflation, and can thus carry crucial information about the physics of inflation. For this reason, a detection of primordial non- Gaussianity is highly sought after. At present, no such signal has been observed. The strongest current constraints come from the Planck satellite, which has measured the temperature fluctuations of the CMB to a remarkable precision [30–32], constraining primordial8 non-Gaussianity in the CMB anisotropies to no more than one part in 104. Even in the simplest single-field models for slow roll inflation, in which the inflaton potential is very flat, a small amount of non-Gaussianity is inevitable due to the gravitational self-interactions of the curvature perturbation. However, in this case the amplitude of non-Gaussianity is very small [33], suppressed by the slow roll parameter  ≡ −H/H˙ 2, which quantifies the change of the Hubble parameter during inflation. Models beyond single-field slow roll inflation can generate non- Gaussianity at a detectable amplitude (for reviews, see [34,35]). The Gaussianity, scale-invariance, and adiabatic nature of the observed primordial perturbations are consistent with the simplest models of inflation with a single scalar degree of freedom among the fluctuations during inflation, but do not yet rule out other possibilities such as multiple dynamical fields during inflation. Future measurements of Large Scale Structure are expected to access shorter

8Secondary non-Gaussianities do arise in the CMB due to nonlinear projection effects such as gravitational lensing, as CMB photons travel through the late universe. Non-Gaussianity arising from the primordial perturbations themselves, however, is consistent with zero.

12 scales, thus gaining access to shorter Fourier modes ζk which can be sampled in greater abundance (see section 1.4), and allowing for tighter constraints or more precise measurements of non-Gaussianity. Because Large Scale Structure observa- tions access a three-dimensional volume rather than a two-dimensional projection, more modes of the underlying density perturbations are being measured than in CMB experiments. Furthermore, measurements of 21-cm radiation from emission or absorption of photons from neutral hydrogen (which traces the underlying field of density perturbations) will access a larger spatial volume than that accessible via Large Scale Structure, potentially allowing for even more precise measurements. Lastly, distortions to the intensity spectrum of the cosmic microwave background are imprinted at earlier times when curvature perturbations on much smaller spatial scales were still larger than the Hubble scale. Spectral distortions are sensitive to the short-scale curvature perturbation, and their cross-correlations with probes of the curvature on larger scales have the potential to greatly improve sensitivity to non-Gaussianity, due to the large sample of short-scale modes (see section 1.4). However, as noted above in section 1.6, non-Gaussian mode coupling also introduces a new and significant uncertainty in matching observations to theory, calling into question whether our observable volume provides a representative sample of primordial perturbations. In the following two chapters we will ask the following question: Supposing that inflation lasted sufficiently long that our Hubble volume is small compared to the entire volume generated during inflation, are the statistics of the curvature perturbation observed in our Hubble volume necessarily representative of the statistics of the curvature perturbation in the rest of the universe?

1.8 Statistical Formalism for Primordial Perturbations

We first define a Gaussian random field ζG(x) as a field whose Fourier modes each have a Gaussian distribution. (See also [22] for a clear explanation.) We will consider a fixed box size L, with volume V = L3. The Fourier transform is defined so that 1 X ik·x ζ(x) = ζke . (1.4) V k

We will use ζk interchangeably with ζ(k) throughout the following chapters. Since ∗ ζ(x) is real, the Fourier modes satisfy the constraint ζ−k = ζk.

13 1.8.1 Gaussian Statistics

The joint probability distribution for a configuration ζ(x) or set of modes {ζk} is

Y ρG[ζ] = ρG,k(ζk), (1.5) k+ where k+ indicates that we only include half the modes, because the reality condition ∗ ζ−k = ζk fixes the other half. The single-mode distribution is

" 2 # 1 |ζk| ρG,k(ζk) = 2 exp − 2 . (1.6) 2πσk 2σk

The normalization factor ensures that

Z 2 d ζkρG,k(ζk) = 1, (1.7)

2 iθk where d ζk = |ζk|d|ζk|dθk, with the phase θk defined by ζk ≡ |ζk|e . The absence of statistical correlation between different Fourier modes k is clear from the fact that ρ[ζG] factorizes into a mode-by-mode product of distributions. The two-point correlation for two Fourier modes is

Z  Y 2  2 hζk1 ζk2 i = d ζk ζk1 ζk2 ρG[ζ] = 2σk1 δk1,−k2 . (1.8) k+

Writing the product in Eq. (1.5) as a sum in the exponent, we take the continuous 9 P R d3k limit of ρG by letting k → V (2π)3 ,

" Z 3 # 1 d k 2 V ρG[ζ] = N exp − 3 |ζk| 2 , (1.9) 2 (2π) 2σk where N fixes the normalization. For a continuous range of k we take V → ∞ and replace 2σ2 k → (2π)3P (k), (1.10) V ζ where Pζ (k) is the power spectrum, so that

3 3 hζk1 ζk2 i ≡ (2π) δ (k1 + k2)Pζ (k). (1.11)

9 R 3 P P 3 3 −1 P In other words, discretizing we have d k → k ∆k1∆k2∆k3 = k kmin = (2π) V k.

14 −1 3 Here we have replaced the Kronecker delta function δk1,−k2 with V δ (k1 + k2) in the V → ∞ limit, where δ3(k) is the Dirac delta function. Assuming that the statistics of ζ(x) are invariant under translations and ro- tations, the correlation between any two points in space will only depend on the distance between them,

hζ(x1)ζ(x2)i = ξ(|x1 − x2|), (1.12)

for some function ξ. Expressing the two-point function hζk1 ζk2 i in terms of the real-space correlator, Eq. (1.12), we see that statistical homogeneity – fixing

ξ = ξ(x1 − x2) – leads to the Dirac delta function and thus independence of different Fourier modes, and statistical isotropy – further restricting to dependence only on |x1 − x2| – restricts Pζ to depend only on the magnitude k, as we have implicitly assumed above. In terms of the power spectrum, the amplitude of perturbations in real space is

Z d3k hζ2(x)i = P (k) (1.13) (2π)3 ζ Z dk = ∆2(k), (1.14) k ζ where we have defined the dimensionless power spectrum,

k3 ∆2(k) = P (k). (1.15) ζ 2π2 ζ

2 2 If ∆ζ is constant, then we see that hζ i receives equal contributions from all wavenumbers, spaced logarithmically. In this case, the power spectrum is scale- invariant. The scale-dependence of the power spectrum is parametrized with a spectral index ns(k), defined by

k !ns−4 Pζ (k) = Pζ (kpiv) , (1.16) kpiv where kpiv is an arbitrary pivot scale. Scale-invariance corresponds to ns = const. = 1. 2 −9 Observationally, the Planck satellite has measured ∆ζ (kpiv) ≈ 2.20×10 , where kpiv ≡ 0.05/Mpc, and (assuming a constant spectral index) ns = 0.968 ± 0.006

15 at 68% confidence [13]. The slight “red tilt” of the power spectrum, that is, its enhancement at infrared scales due to ns < 1, is an important piece of information for differentiating models of the primordial universe. (A power spectrum with ns > 1 is correspondingly described as having a blue tilt.)

1.8.2 Non-Gaussian Statistics

The addition of non-Gaussianity adds a statistical correlation between different Fourier modes.10 In general, we can write

" ∞ # X Z (N) ρ[ζ] = ρG[ζ] exp ζk1 ζk2 × ...ζkN Kζ (k1, k2, ..., kN ) , (1.17) N=3 N where we have defined

Z Z N 3 Y d ki 3 X  ≡ 3 δ ki . (1.18) N i=1 (2π)

(N) For a perturbative expansion around a Gaussian distribution, the Kζ kernels (N) N−2 (3) will be controlled by a small parameter  so that Kζ ∝  , and Kζ (k1, k2, k3) will typically describe the leading non-Gaussian contribution. Again, translation invariance leads to the momentum-conserving delta function in Fourier-space correlation functions as well as in Eq. (1.17). Statistical isotropy restricts the dependence of the K(N) kernels to variables describing the size and shape of the (3) momentum-space polygons formed by the N momenta. For example, Kζ (k1, k2, k3) will only depend on the magnitudes of the three momenta. The non-Gaussian probability distribution ρ[ζ] can be completely characterized by the set of N-point correlation functions hζ(x1)ζ(x2) × ...ζ(xN )i, or in Fourier space, 3 3 X  (N) hζk1 ζk2 × ...ζkN ic ≡ (2π) δ ki Pζ (k1, k2, ..., kN ). (1.19)

(N) The polyspectra Pζ describe the momentum dependence of the correlation func- (N) tions, and are directly related to the Kζ kernels in Eq. (1.17). The c subscript indicates the connected part of the N-point function, that is, the (non-Gaussian)

10 We might imagine a non-Gaussian modification to Pk(ζG,k) that still preserved the form of Eq. (1.5). In the cases we study, this is not the case, nor do we expect non-Gaussianity generated by nonlinear physical processes to keep modes uncorrelated.

16 part which is not captured by applying Wick’s theorem to express Gaussian part as a product of two-point functions. In general, the leading-order non-Gaussian (3) statistic is the bispectrum Bζ ≡ Pζ , defined by

3 3 hζk1 ζk2 ζk3 i ≡ (2π) δ (k1 + k2 + k3)Bζ (k1, k2, k3), (1.20) which can be thought of as a function of triangles in momentum space formed by the three wavenumbers. A dimensionless bispectrum, analogous to the dimensionless power spectrum, may be defined by rescaling,

2 Bζ (k1, k2, k3) ≡ (k1k2k3) Bζ (k1, k2, k3). (1.21)

In this case, a scale-invariant bispectrum satisfies

Bζ (λk1, λk2, λk3) = Bζ (k1, k2, k3). (1.22)

The configuration(s) of wavenumbers for which the dimensionless bispectrum is largest is described as the “shape” of the bispectrum.

1.8.3 Local-type non-Gaussianity

In practice, we can generate a non-Gaussian field through nonlinear operations on a Gaussian field. A common implementation of this is the local ansatz, which obtains a non-Gaussian field as a nonlinear function of a Gaussian field locally, at each point in space,

ζNG(x) = f(ζG(x)). (1.23)

For a weakly non-Gaussian distribution, f will be approximately linear, and we can expand around a Gaussian field, writing

3 h i 9 ζ (x) = ζ (x) + f ζ2 (x) − hζ2 i + g ζ3 (x) + O(ζ4 ). (1.24) NG G 5 NL G G 25 NL G G

0 00 0 2 6 000 0 3 We have absorbed f (0) into ζG, defined f (0)/(f (0)) ≡ 5 fNL, f (0)/(f (0)) = 36 25 gNL, and subtracted a term to ensure that hζNGi = 0. The bispectrum may be

17 computed from Eq. (1.24) and is given (up to O(fNL)) by

6 B(local)(k , k , k ) = f [P (k )P (k ) + 2 perms.] , (1.25) ζ 1 2 3 5 NL G 1 G 2 where PG(k) is the power spectrum for the Gaussian field ζG. Here we have used Wick’s theorem, which says that the expectation value of an even number of

Gaussian field ζG may be expressed as a sum of product of two-point functions. The local-type bispectrum is largest for “squeezed” configurations where one wavenumber is very small, k1 ≡ kl  k2, k3. Defining ks ≡ (k2 − k3)/2, the squeezed bispectrum is 1 1 12 B(local)(k , |k − k |, |k + k |) = f P (k )P (k ). (1.26) ζ l s 2 l s 2 l 5 NL G l G s Equivalently, the dimensionless bispectrum is

(local) 1 1 12 2 2 2 ks Bζ (kl, |ks − kl|, |ks + kl|) = fNL(2π ∆ζ ) , (1.27) 2 2 5 kl

2 2 showing clearly the squeezed-limit enhancement. (We have approximated ∆ζ ≈ ∆G here.) The influence of long-wavelength background modes on local statistics, which we will discuss in the following chapters, is due to this squeezed-limit coupling. Observationally, the Planck satellite has placed tight constraints on local non-

Gaussianity: fNL = 0.8 ± 5.0 and gNL = −9.0 ± 7.7 at 68% confidence [31]. Models of inflation where curvature perturbations are sourced by an additional degree of freedom other than the inflaton generically produce local non-Gaussianity. This arises from nonlinear processes on superhorizon scales during or after inflation.

The resulting amplitude of non-Gaussianity is typically characterized by fNL & 1. Constraining fNL at this level would rule out the bulk of the parameter space for multi-field models, up to fine tunings which cancel different O(1) contributions to fNL. As we will see, constraints at the level ∆fNL . 1 will also be an important benchmark for super cosmic variance. There are, of course, a number of theoretically-motivated forms of primordial non-Gaussianity that are not captured by the local ansatz (see, for instance [36] and references therein). We restrict our attention primarily to local non-Gaussianity, although we will comment on more general non-Gaussianity in section 3.4. The mode coupling discussed in chapter4 is even stronger than the local-type, since background modes with longer wavelengths have an increasingly strong effect on

18 measured quantities.

1.9 Long Wavelength Modes in Cosmology

With the cosmological background of previous sections, and the statistical setup of section 1.8, we motivate the work in the following chapters by reviewing related studies in the field. Much work has already been carried out on the subject of long wavelength modes in cosmology and their influence on shorter scales, during inflation or in the late universe. We will focus on work that is most closely related to this thesis.

1.9.1 When Are Infrared Modes Relevant?

It is important to ask under what conditions the mode couplings we will study can arise through a physical mechanism during or after inflation. Different degrees of sensitivity to background modes, or equivalently different behavior of correlation functions in squeezed limits, arise from different inflationary models. In single-clock11 inflation, long-wavelength modes have no effect on statistics in a local region, but merely rescale the local scale factor. As originally shown in [33], and explored further and generalized in [37–41] and related references, restriction to single-clock inflation leads to a consistency relation determining the squeezed limit of the three point function in terms of the two point function, so 5 that fNL = − 12 (ns − 1), reflecting the fact that the coupling to a background mode is fixed in terms of the scale-dependence of fluctuations, and is small in the case of near scale invariance. Furthermore, in [42–44] it was observed that a long-wavelength mode in single-clock inflation will, although apparently influencing short modes when computed in comoving coordinates, have no effect on observable quantities on short scales as measured with physical distances (see also [45]). That is, the observed squeezed limit vanishes in physical coordinates – fNL = 0 – so that

11In single-clock inflation, all quantities at each point in space are controlled by a single quantity or “clock” which evolves during inflation. Specifically, fluctuations in the scale factor, Hubble parameter, energy density, pressure, etc. are all determined by the inflaton field φ(x). Additional freedom may be introduced by additional fields, modifications to the quantum state of the inflaton and metric fluctuations at the onset of inflation, or a non-attractor phase where φ˙ enters as an additional free parameter.

19 the effect of long wavelength modes on local statistics vanishes in the limit of large wavelength. When there are multiple fluctuating degrees of freedom during inflation, on the other hand, statistics in a local region can depend on the long-wavelength background of ζ. Fluctuations in an additional field generated during inflation will lead to different energy densities and hence different expansion rates in different regions, leading to fluctuations in the amount of expansion ζ(x). The relationship is nonlinear because the energy density depends nonlinearly on fluctuations in the fields. (In the single-clock case, the nonlinearities are constrained to be small due to the flatness of the potential, leading to the consistency relation noted above.) An additional field can introduce a new stochastic long-wavelength background which has an influence on local statistics that is not constrained by single-clock conditions. Because the nonlinear evolution takes place after fluctuations have been stretched to wavelengths much longer than the Hubble scale (the scale of causal contact), ζ(x) will only depend on any additional fields at the same point in space, yielding non-Gaussianity of the local type, Eq. (1.23). In the following chapters we consider mode couplings of this type, for which arbitrarily long wavelength modes are correlated to statistics in small volumes. Our statistical results find a dynamical origin in inflationary or post-inflationary dynamics with additional fields, other than the inflaton.

1.9.2 Long Wavelength Backgrounds in Inflationary Models

Several studies have considered the influence of superhorizon modes on locally observed statistics for specific models, illustrating how the statistical features which we will discuss can have a physical, dynamical origin. As noted in section 1.9.1, long-wavelength modes have no effect on physical observables in single-clock inflation. The scenarios here involve local-type non-Gaussianity, which does couple short scales to the local background, due to the presence of additional degrees of freedom. In the curvaton model [46,47], an additional scalar field sources the curvature perturbation after inflation has ended. Fluctuations in the curvaton contribute fluctuations to the energy density, which source fluctuations in the curvature. Linde and Mukhanov [48] pointed out that the density (or curvature) perturbations

20 produced by the curvaton will appear Gaussian on very short scales, but non- 2 2 Gaussian on very large scales. This scenario is an example of the case ζ ≈ ζG − hζGi considered on section 2.4. It was furthermore shown in [49] that the distribution of

locally observed values of fNL in the curvaton model is not spread symmetrically

about the mean value hfNLi, but skewed due to the fact that regions with small 2 average value of the curvaton have a large fNL ∝ 1/σ . (That is, the distribution

of fNL in subvolumes is that of the inverse square of a Gaussian random variable.) Again, the discussion of section 2.4 shows how this comes about statistically. The role of super cosmic variance in the curvaton model was discussed in [45]. The fluctuations δσ of the curvaton field in a global volume may be large compared

to the average value σ? of the curvaton at the end of inflation, which leads to a nonlinear dependence of the curvature perturbation on δσ, and thus to strongly non- Gaussian statistics. However, for sufficiently small subvolumes, the local average

value hσiVol of the curvaton field will typically be much larger than its observable (small scale) fluctuations, leading to nearly Gaussian statistics. This scenario is an example of a physical mechanism which generates the statistical effects from long-wavelength modes which we will study in detail. The influence of a long-wavelength background has also been studied in the context of gauge fields coupled to the inflaton. Bartolo et. al. [50] (see also [51,52]) considered the case of a coupling between the inflaton and an electromagnetic- like sector of multiple gauge fields, showing clearly the effect of a background of superhorizon modes on the observed power spectrum resulting from the coupling. For only a few additional e-folds, an accumulation of vector gauge modes longer than the observable universe leads to a landscape where the observed power spectrum varies spatially with the long-wavelength background. In this case, not only is statistical isotropy broken, leading to an anisotropic quadrupolar contribution to the observed power spectrum, but the axial symmetry of the quadrupolar modulation is broken due to the presence of more than one background vector modes. This model serves as an example of super cosmic variance, with the additional feature of a unique directional dependence of the power spectrum due to the vector background. The breaking of statistical isotropy in finite volumes introduced in these models is similar to the effect of superhorizon gravitational waves or tensor modes, studied in section 4.4, in the case of an enhanced squeezed-limit coupling. In this case, the tensor nature of the coupling leads to angular dependence in the squeezed-limit

21 bispectrum. In the studies described above, local statistics are influenced by a constant long- wavelength background. A number of studies have also investigated the effect of a background gradient ∇φ(x) across the observable universe from an additional field, which may be the inflaton or curvaton, for example. A coupling of short scale modes to this background can lead to statistical anisotropies such as a hemispherical power asymmetry in the CMB, and perhaps other large scale anomalies [53–56]. However, these models must satisfy constraints from the absence of power asymmetry at very large multipoles [57].

1.9.3 Long-Wavelength Modes in Large Scale Structure

Although we will mainly consider the case where observable scales are coupled to scales too large to observe, the same discussion can be applied to couplings of different scales that can be observed. A growing literature focuses on the influence of observationally accessible (k . H0) long-wavelength modes in Large Scale Structure. Closely related to the present study is the influence of local-type non-Gaussianity on the dark matter halo power spectrum. The number density of dark matter halos depends on the the underlying primordial density perturbations, but also independently on the curvature power spectrum Pζ . With local non-Gaussianity

Pζ will – as we will see in section 2.3 – vary spatially, with the contribution from nonzero fNL scaling as

∆Pζ (k; x) ∝ fNLζl(x), (1.28) where ζl(x) varies over much larger scales than the argument k. Since the curvature perturbations grow on large scales relative to density perturbations due to Poisson’s 2 equation, δk ≡ δρk/ρ0 ∝ k ζk, a nonzero fNL adds a contribution to the dark matter halo number density that grows on large scales relative to the underlying matter distribution – that is, a scale-dependent halo bias. A positive fNL increases the power Pζ – and hence Pδ – in regions where ζl > 0, or equivalently δl > 0, so that density fluctuations in overdense regions are amplified, pushing the value of δ over the threshold for gravitational collapse in a greater fraction of volume, increasing the amount of structure. (A negative fNL similarly decreases the abundance of clusters.) This effect was originally demonstrated analytically and in cosmological

22 N-body simulations in [58], used to constrain primordial non-Gaussianity with available quasar data [59], explore further in [60] and a number of subsequent works,

and developed in [61,62] for local non-Gaussianity with nonzero gNL. More generally, local non-Gaussianity leads to a spatial variation of correlation functions [63]. A long-wavelength mode can lead to a spatial modulation of an N-point function if the (N + 1)-point function has a strong squeezed limit. Even more generally, an N- and M-point function may be spatially correlated on large scales if the N + M-point function is large enough in the limit where a certain sum of wavenumbers adds up to the long wavelength mode. Recently, the position dependence of the density power spectrum was developed as a probe of primordial non-Gaussianity and studied in N-body simulations in [64]. Our study in the following chapters is also a study of statistical inhomogeneity or “spontaneously broken translation invariance,” but now in the case where statistical averages vary on scales larger than the observable universe.

1.9.4 Lyth’s Study of ζ(x) in a Subvolume

The scenario of a subvolume within a much larger volume, where the statistics of ζ(x) follow local-type non-Gaussianity, was considered by Lyth [65] (see also [66–68]), who raised the question of how statistics averaged in a subvolume relate to those averaged throughout the global volume. Lyth discussed the tree and one-loop12

contributions to the global power spectrum Pζ , defined in terms of a subvolume scale k? (in the notation of [65], k? = 1/M). The global power spectrum up to 2 O(fNL) may be written as a sum of tree and one-loop contributions

loop Pζ (k) = PG(k) + P (k), (1.29)

2 where the O(fNL) loop contribution, which integrates over pairs of modes p and k − p which add to k, comes from the two-point function of the quadratic term in Eq. (1.24). For a given scale k?  k fixing the subvolume size, this may also be written as tree loop ? Pζ (k) = hPζ (k)|si + P (k; k ). (1.30)

12A loop contribution to a correlation function involves one or more integrals over momenta. While these contributions can be represented diagrammatically with loops, they describe classical nonlinearities, rather than quantum interactions.

23 tree Here, Pζ (k)|s is the tree-level power spectrum averaged in a given subvolume, which is then averaged over subvolumes. This piece includes the part of the original loop integral with modes p < k?. The remaining term P loop(k; k?) is the loop contribution as defined for the subvolume: it only integrates over pairs of modes |p|, |k − p| > k? accessible within the subvolume. (The original loop contribution is thus P loop(k) = P loop(k; L−1).) It was shown in [65] that although the tree and loop contributions depend on the subvolume scale k?, the total power spectrum does not. This is of course expected, since averages in any volume should not depend on the arbitrary choice of a smaller scale. The k?-dependent quantities in [65] were tree and loop contributions to the global power spectrum, averaging over the global volume, rather than quantities for a given subvolume with fixed background. In the following chapters, we will follow Lyth’s use of the local ansatz for a subvolume in a larger volume, but focus on statistics in a fixed subvolume, rather than averaging over subvolumes. (Of course, the increasing contribution from the tree-level power spectrum in smaller volumes, as we will see, is due to an increasing amplitude for the linear term in Eq. (1.24), and the dominance of the tree-level contribution for small subvolumes is due to a running towards nearly Gaussian statistics. We also note that Lyth and other authors [68] have compared the running of non-Gaussian statistics with the subvolume scale k? to running of parameters in a quantum field theory under renormalization group equations, an analogy which we will note in a similar context in section 2.6.

1.9.5 Recent Work on Super Cosmic Variance

The work most closely related to this thesis is that of Nurmi, Byrnes, and Tasinato [69], who studied the influence of background modes for local-type non-Gaussian parameters in a fixed subvolume, emphasizing the consistency of a non-detection of obs fNL with a large global fNL. Their work shares some overlap with the discussion obs of chapter2. In [70], the relation between fNL and the global parameter gNL was studied. It was emphasized that constraints on the observed three-point function restrict the amplitude of the global four-point function, since the latter would induce an observed bispectrum in typical subvolumes.

24 1.9.6 Primordial Gravitational Waves and Non-Vacuum Initial States

In chapter4 we will move away from the purely statistical approach of chapters 2-3, and consider squeezed-limit mode coupling generated by inflationary dynamics. We focus consider a three-point correlation between two modes of the curvature perturbation and an additional field. The additional field affects the statistics of ζ, so the amplitude of its fluctuations can be estimated from observations of ζ. This method was originally described in [71] in the context of primordial gravitational waves as the additional field, and was subsequently applied to cosmic microwave background statistics in [72]. We follow [71], but consider a different three-point correlation, with stronger coupling to long wavelength tensor modes. The correlation we consider is the result of a non-vacuum initial state at the onset of inflation, and our study therefore builds on previous studies of the impact on observable statistics from excited pre-inflationary states. In particular, the three-point function for scalar curvature perturbations resulting from non-vacuum initial states with minimal gravitational couplings can become very large in the squeezed limit [73–76]; our computation for the cross-bispectrum of two scalar modes and one tensor mode is closely related.

1.10 Outline of Chapters2-4

In comparison to these studies, the present thesis is unique in the following ways: (1) We work in a purely statistical context, asking how a long-wavelength background affects observables of a non-Gaussian random field, given its statistical properties; (2) we consider observables of ζ(x) in a fixed subvolume, rather than averaging over subvolumes; (3) we focus on local-type non-Gaussianity, for which observable quantities are affected equally by all background fluctuations, as opposed to only those with wavelengths near the scale of the subvolume. Although a number of works may share one or more of these points, the approach taken here is unique in that it is characterized by all of them. The benefits of this approach are that the statistical effects of subsampling are clear, the relation between locally observed and globally averaged quantities is clear, and we are able to accommodate a wide range of inflationary models into the same statistical framework.

25 Chapters2-4 are outlined as follows: Super Cosmic Variance from Local Non-Gaussianity. For local-type non- Gaussianity, we compute N-point functions of the curvature perturbation as seen by local and global observers. We show that long-wavelength background modes ζl can systematically bias locally measured statistics away from their global averages,

hOiobserved = hOi|ζl 6= hOiglobal, (1.31) leading to a cosmic variance due to spatial variation of statistics from subvolume to subvolume. For three different scenarios of non-Gaussianity in the global volume, obs we compute the observed power spectrum Pζ and amplitude of non-Gaussianity obs fNL , and show probability distributions for the global values given measurements of the local quantities. In particular, for large global non-Gaussianity, the measured non-Gaussianity is controlled by the local background,

obs 1 fNL ∼ . (1.32) ζl

We describe the dependence of the local background ζl, and consequently the degree of bias, on the number of additional e-folds of inflation and the power spectrum on superhorizon scales. We describe the running of non-Gaussian parameters with the bias from the long-wavelength background, and study the effect of the background on the momentum dependence of correlation functions. The running towards Gaussian statistics in small volumes indicates a “naturalness” for the weakly non-Gaussian local ansatz. Super Cosmic Variance with Scale-Dependence. We extend the study of chapter

2 to weakly scale-dependent non-Gaussian parameters, fNL(k), gNL(k), etc. and demonstrate that mode coupling can generate cosmic variance uncertainty in the observed spectral index at a cosmologically significant level, |∆ns| ∼ O(0.04), where the sign of the observed spectral index may differ from the global spectral index, over a finite range of scales. This results from two or more contributions to Pζ (k) on observable scales with different runnings, which are shifted in their relative amplitude by the local background. We show that this is possible for a blue-tilted fNL(k), increasing on short scales. This shift is a contribution to the observed power spectrum from the scale-dependence in the global bispectrum. This scale-dependence can also shift the observed bispectrum, leading to a signature of

26 multiple sources of density perturbations (eg. inflaton and curvaton) in the dark matter halo power spectrum being mimicked by perturbations from a single source, in a subvolume. If tensor modes are coupled to long wavelength modes of a second field, the locally observed tensor power and spectral index can also vary. All of these effects, which can be introduced in models where the observed non-Gaussianity is consistent with bounds from the Planck satellite, loosen the constraints that observations place on the parameters of theories of inflation with mode coupling. We suggest observational constraints for which future measurements could aim in order to close this window of cosmic variance uncertainty. Fossilized Gravitational Wave Relic from Squeezed-Limit Coupling. If long wavelength primordial tensor modes are coupled to short wavelength scalar modes, the scalar curvature two-point function will have an off-diagonal component, that is hζk1 ζk2 i does not vanish for k1 + k2 6= 0. This ‘fossil’ remnant is a signature of a mode coupling that cannot be achieved in single clock inflation. We use the example of a non-Bunch Davies initial state for scalar (and possibly tensor) 2 modes, which leads to a squeezed-limit coupling Bζζγ(kS, kS, kL) ∼ (kS/kL) of the scalar-scalar-tensor cross-bispectrum that is even stronger than the local-type coupling of Eq. (1.27), to demonstrate that physically reasonable fossils, consistent with current data, can be observable in the near future. This illustrates how the fossil off-diagonal power spectrum is a complementary probe to the squeezed limit bispectra of the scalar and tensor sectors individually. We also quantify the relation between the observable signal and the squeezed limit bispectrum for a general scalar-scalar-fossil coupling, and note the effect of superhorizon tensor modes on statistical anisotropy in the curvature power spectrum. We will assume throughout that we have perfect access to primordial per- turbations, and will not address the challenges of understanding the nonlinear physical processes and projection effects in the late universe which convert the initial conditions to quantities we observe directly. Furthermore, we assume that all long-wavelength modes are confined to a finite volume, fixing a minimum wavenumber kmin, so we do not deal with the question of infrared divergences in an infinite volume. (For a review of this subject, see [77].) Technically, our discussion would remain valid in the case of an infinite volume, as long as global perturbations were still small, ζ  1, which holds as long as the power spectrum Pζ (k) is sufficiently convergent in the infrared: ns − 1 > 0. However, for

27 2 a scale-invariant or red spectrum, the amplitude of fluctuations hζG(x)i diverges for an infinite volume, so our discussion would only apply to a finite volume with small perturbations (within which one could consider smaller subvolumes). Naively, if one were to take the infinite volume limit L → ∞, locally observed non-Gaussianity obs would vanish, fNL ∼ 1/ ln L → 0, but as we will discuss, our framework does not apply in this limit.

28 Chapter 2 | Super Cosmic Variance from Lo- cal Non-Gaussianity

2.1 Introduction

In single-field models of inflation the statistics of ζ are inherited from quantum fluctuations in the inflaton itself [16,19–21,24]. If multiple light fields are present during inflation, such as in the curvaton scenario [46, 47, 78–80] or modulated reheating [81,82], fields other than the inflaton may generate ζ and the relationship between the post-inflationary curvature and quantum fluctuations generated during inflation can be non-linear (see e.g. [35]). In these scenarios, the inflaton and curva- ton field may obey Gaussian statistics while the observed curvature perturbation is non-linearly related to the Gaussian fluctuations, allowing for the phenomenological parameterization:

3   9   ζ(x) = ζ (x) + f ζ2 (x) − hζ2 i + g ζ3 (x) − 3hζ2 iζ (x) + ..., (2.1) G 5 NL G G 25 NL G G G known as the local ansatz. Here, ζG is a Gaussian random field and fNL, gNL are constants specified by the particular inflationary model [83–87]. The field ζ in Eq. (2.1) obeys non-Gaussian statistics and the level of non-Gaussianity can be q 2 2 characterized by the products fNL hζGi, gNLhζGi... In this chapter, we will how the statistics of the curvature perturbation measured in our Hubble volume relate to statistics averaged in a much larger volume, in light of the mode coupling introduced by local-type non-Gaussianity. In the context of inflation, we suppose that inflation lasted sufficiently long that our Hubble

29 volume is small compared to the entire post-inflationary volume, and ask how observed statistics ought to be compared to those predicted by inflationary models. However, we do not ask what dynamics generated the curvature perturbations. Taking them as initial conditions on a fixed time slice in the early universe, we work out the observational consequences of mode coupling in the post-inflation curvature perturbations without asking which dynamics generated the fluctua- tions. We suppose only that the universe is considerably larger than what we see (which is the natural outcome in many inflation models) and that there exists a homogeneous and isotropic spectrum of primordial fluctuations in the gravitational field on all scales in the entire volume. We do not consider a spatial slice that is significantly inhomogeneous on large scales, so in scenarios that do enter the eternal inflation regime, one may consider the large volume to be the largest finite volume encompassing our Hubble patch for which large-scale perturbations are still small. The cosmological principle might seem to suggest that statistics should be the same in all spatial regions, but (as has been noted by a number of authors [48, 53, 63, 66, 69, 88–91]) even if the universe is statistically homogeneous and isotropic, mode-coupling introduced by non-linear terms like those in Eq. (2.1) correlates the statistics of ζ on very different scales. The locally observed, smaller- than-Hubble-scale statistics depend on the unobservable, super-horizon modes of

ζG (i.e. long-wavelength modes of the Gaussian field that are nearly constant across our Hubble volume). Changes to the local statistics depend on a small

parameter, O(ζsuperhorizon)  1, but can nevertheless be important. In particular, the background mode in our Hubble volume is roughly a sum over all modes

with wavelength π/k larger than c/H0, so the variance of the total background fluctuations is larger than the locally observed variance in a single k mode by an amount dependent on the number of superhorizon e-folds. This situation, in which long-wavelength modes of ζ bias the local statistics, is in stark contrast to the case where ζ is Gaussian and different Fourier modes are strictly uncorrelated. For a Gaussian field, the local power spectrum may be randomly different from the globally averaged one, but is not systematically biased. We will consider various examples of local-type non-Gaussianity, including the general case of a polynomial function of a Gaussian field, as well as a nonlinear function of two Gaussian input fields, and find generically that one may generate statistics in a Hubble-size patch that are weakly Gaussian and consistent with

30 observations despite the fact that the statistics in the larger, post-inflationary patch look very different, and may be strongly non-Gaussian. An easy way to see that Gaussian statistics are recovered on small scales is to 2 consider the simple case ζ = ζG. Breaking ζG into pieces comprised of its long and short-wavelength modes (see section 2.2 below), we have

2 2 ζ(x) = ζGl + 2ζGlζGs(x) + ζGs(x). (x ∈ subvolume) (2.2) = constant + linear + quadratic.

We have removed the spatial argument from ζGl here since it is approximately constant within a small volume. The constant piece will just shift the local background. If the number of background modes is much greater than the number 2 2 of short-wavelength modes, so that hζGli  hζGsi, then in typical subvolumes the linear term will dominate, leading to nearly Gaussian statistics. The relevance of this effect increases as the subvolume size decreases because there are more long-wavelength modes. In smaller subvolumes, local statistics are typically more biased due to a larger background ζGl, and vary more from region to region. The bias depends on the local background value of ζ, which includes contributions from all modes with wavelength k <∼ H0 and is therefore enhanced if the entire post- inflationary patch is large compared with our Hubble volume (see Eqs. (2.25)-(2.26) below). Our results also lead to a statistical notion of naturalness, which may be compared to particle physics notions of naturalness as applied to quantum field theory for the inflaton [92] or its perturbations on a near de Sitter background [36]. We will find a notion of statistical naturalness for typical small volumes where a family of well-behaved correlation functions is generated from a parent volume with arbitrarily fine-tuned statistics in the same family. The “running” of parameters towards those of a weakly non-Gaussian field as the subvolume scale is decreased suggests that the weakly non-Gaussian local ansatz, Eq. (2.1), is statistically natural. In section 2.2 we introduce our notation and formulate the calculation of local statistics of ζ in terms of a short-long wavelength split of the Gaussian field ζG. The long and short wavelength split has been used to show the effect of local non-Gaussianity on the dark matter halo bias [58,61], and applied to the study of

31 infrared superhorizon modes in cosmology [89,93–95]. In section 2.3 we study the mapping between the global and local values of non-Gaussian parameters in the case when the statistics in the larger universe are weakly non-Gaussian as in Eq. (2.1) (see also [69]). In section 2.4 we present full calculations of a somewhat counterintuitive example in which the local statistics of ζ can appear to be nearly Gaussian while the global ones are strongly non-Gaussian. In section 2.5 we study the relation between globally and locally determined statistics for two-field initial conditions. In section 2.6 we consider a general polynomial of a Gaussian field, and see that the running of parameters for nonlinear terms can be expressed in differential form in a manner similar to renormalization group equations. In section 2.7 we point out that although the amplitude of local non-Gaussianity may vary among subvolumes, the shape or momentum dependence of correlation functions is essentially unchanged. Finally, in section 2.8 we summarize our results and discuss some possible implications for interpreting potential measurements of primordial non-Gaussianity. AppendixA contains a diagrammatic formulation of non-Gaussian statistics for a general local model, which is useful both computationally and conceptually for understanding the mapping between local and global statistics. Calculations of global and local statistics for generic local-type non-Gaussian initial conditions are given in AppendicesB andC.

2.2 Statistics of ζ in a Subvolume with Local Non- Gaussianity

Let us consider a large but finite spatial volume of size L. In the physical context of our universe, we take this to be the largest spatial volume in our post-inflationary region of spacetime for which density and gravitational field perturbations are indeed small compared to the FRW background. (All of our discussion will apply equally for the case of L as a smaller volume, or as a volume that results from primordial dynamics other than inflation.) We also specify a small subvolume of ? ? 3 size 2π/k and volume Vs = (2π/k ) , along with a maximum wavenumber kmax from the smallest scale we smooth over in defining the fluctuations. The curvature perturbation is defined as the fractional fluctuation in the scale factor a,

a(x) = hai(1 + ζ(x)), (2.3)

32 where 1 Z hai = d3xa(x) (2.4) L3 is the average value of a(x) throughout the global volume. We assume throughout that |ζ| < 1.

2.2.1 Long and Short Wavelength Split

We begin by dividing ζ into long and short-wavelength parts compared to the scale ? k , that is ζ(x) ≡ ζl(x) + ζs(x), where the long-wavelength field is

Z d3k ζ (x) = eik·xW (k)ζ(k), (2.5) l (2π)3 s and Z d3k ζ (x) = ζ(x) − eik·xW (k)ζ(k). (2.6) s (2π)3 s The smoothing function is

Z 1 3 −ik·y Ws(k) ≡ d y e , (2.7) Vs Vs so that Eq. (2.5) may be written

Z 1 3 ζl(x) = d yζ(x − y). (2.8) Vs Vs

In Fourier space,

ζl(k) = Ws(k)ζ(k), ζs(k) = ζ(k)(1 − Ws(k)). (2.9)

? ? For k  k , Ws(k) → 0, while for k  k , Ws(k) → 1. For simplicity we make the ? approximation of a top-hat function in Fourier space, Ws(k) ≈ Θ(k − k), so that we can define

Z 3 Z 3 d k ik·x d k ik·x ζs(x) = ζ(k)e and ζl(x) = ζ(k)e . (2.10) |k|≥k? (2π)3 |k|

? 1/3 where k ∼ 2π/Vs . With these definitions, the scale factor, Eq. (2.3), may be written as follows

33 within the subvolume Vs,

obs a(x) = hais(1 + ζ (x)) , x ∈ Vs, (2.11) where

hais ≡ hai(1 + ζl), (2.12) with the mean value of ζ approximated as ζl as defined in Eq. (2.8), using the

Fourier-space top-hat window function Ws, as discussed above. ζl is taken to be constant within the subvolume, and we emphasize that it takes a particular value, which will vary stochastically between subvolumes. We will denote with an “obs” superscript quantities as defined within a subsample volume such as the observable universe, which do not correspond (except perhaps by a coincidence of values) to quantities in the larger volume. (We will also denote averages over the global volume with h i, and averages over the small volume with h is.) The curvature perturbations in the global and subsample volumes are therefore related by

obs 1 + ζ(x) = (1 + ζl)(1 + ζ (x)) , x ∈ Vs. (2.13)

Since ζ = ζl + ζs, obs ζ = ζs/(1 + ζl) , x ∈ Vs . (2.14)

We will consider homogeneous and isotropic correlations, so the (global) power spectrum and bispectrum are defined as

3 3 hζk1 ζk2 i ≡ (2π) δ (k1 + k2)Pζ (k) . (2.15) 3 3 hζk1 ζk2 ζk3 i ≡ (2π) δ (k1 + k2 + k3)Bζ (k1, k2, k3) . (2.16)

From Eq. (2.14), we see that the power spectra in the two volumes are related by

obs 2 Pζ (k) = Pζ (k)/(1 + ζl) . (2.17)

The amplitude of linearized fluctuations is thus rescaled by a factor of 1 + ζl due to the shift in the local background by the same factor: fluctuations appear smaller in overdense regions and larger in underdense regions. However, this shift does not affect the level of non-Gaussianity in the small volume, as quantified for example

34 n 2 n/2 by the dimensionless connected moments Mn ≡ hζ(x) ic/hζ(x) i , nor does it affect the shapes of the n-point functions, which will be our focus, but only reflects obs the rescaling of ζ . In what follows, we will therefore drop factors of 1 + ζl in expressions for subvolume quantities. Suppose the curvature perturbation in the large volume is given by the local ansatz, Eq. (2.1). We define the power spectrum of the Gaussian field as

3 3 hζG,k1 ζG,k2 i ≡ (2π) δ (k1 + k2)PG(k1) . (2.18)

Splitting the Gaussian field into long- and short-wavelength modes in the manner described above,

ζG(x) = ζGs(x) + ζGl(x), (2.19) we have the real-space auto-correlations

Z 3 0 d k ik·(x−x0) hζGs(x)ζGs(x )i = 3 PG(k)e , (2.20) |k|≥k∗ (2π)

Z 3 0 d k ik·(x−x0) hζGl(x)ζGl(x )i = 3 PG(k)e , (2.21) |k|

0 hζGl(x)ζGs(x )i = 0 . (2.22) between the long- and short-wavelength Gaussian fields. In what follows we assume,

2 2 n n hζGsis ≈ hζGsi while hζGlis ≈ ζGl. (2.23)

That is, we assume that the locally measured, small-scale Gaussian power spectrum is representative of the globally defined one and that the variation in long-wavelength modes (k < k∗) across the volume Vs is negligible. In this limit an observer in the small volume Vs is unable to distinguish between ζGl and the background. However,

1 1/3 The more realistic assumption of a top-hat window function in real space with radius Vs gen- 2 1/3 1/3 erates ∼ ∆G(k ∼ 2.5/Vs )/4 corrections to Eq. (2.22), where kVs ∼ 2.5 is the peak of Ws(k)(1− 0 0 Ws(k)) for a top-hat in real space. For ns = 1 this gives hζGl(x)ζGs(x )i/hζGl(x)ζGl(x )i ∼ 1/(4N), 0 0 0 0 1/3 < and hζGl(x)ζGs(x )i/hζGs(x)ζGs(x )i ∼ −1/(4 ln(|x − x |H0)) for |x − x |/Vs ∼ 10.

35 the nonlinear coupling of ζGl to short wavelength modes ζGs will cause the local, small-scale non-Gaussian statistics of ζ to differ from the global ones in a way that depends on the local value of ζGl.

2.2.2 The Long Wavelength Background

The variance of long-wavelength fluctuations, which will determine the degree of bias for observed quantities, is

Z k∗ 2 dk 2 hζGli = ∆G(k) (2.24) Λ k

2 3 2 where ∆G(k) ≡ k PG(k)/2π is the dimensionless power spectrum for ζG, and 1/3 Λ ∼ 2π/Vl , the infrared cutoff corresponding to the larger volume where the perturbations are set up. In the example calculations and plots we take Vs to be −3 our Hubble volume ∼ H0 , although other scenarios can also be considered. For example, the general expressions may be relevant for making comparisons between theory and particular observables measured in a volume smaller than our Hubble volume. ? The variance of ζGl, Eq. (2.24), is sensitive to both the range of scales k < k and the behavior of the power spectrum PG(k) for these scales. Letting the power 2 spectrum for ζG be a power law, d ln ∆G/d ln k ≡ ns − 1 with ns − 1 = const. gives closed-form expressions for the variance of long-wavelength modes. Defining N ≡ ln(k?L) as the number of super-horizon e-folds from the start of inflation to ? the time when the comoving scale k ∼ H0 of the observable universe crossed into the horizon, we have

2 2 hζGli = ∆GN, (2.25) ns=1  −(n −1)N  1 − e s 2 2 ∗ hζGli = ∆G(k ) . (2.26) ns6=1 ns − 1

2 1/2 In Figure 2.1 we plot ±hζGli , the typical amplitude of the unobservable back- ground mode, as a function of the number of super-horizon e-folds N. The power spectrum of ζG is of course unknown for k <∼ H0, but as a starting point we consider constant ns as in Eq. (2.25)-Eq. (2.26). As can be seen from Figure 2.1, even the modest red-tilt that is currently favored (ns = 0.9608) dramatically increases the

36 Figure 2.1: Quantities in our Hubble volume are different from those in the larger universe by terms proportional to ζGl – the long wavelength fluctuations that appear 2 1/2 2 to be constant within our Hubble volume. Plotted is hζGli assuming that ∆G = −9 ns−1 2.464 × 10 (k/kpiv) for two constant values of ns, along with two examples that 2 1/2 −5 include running. Note that hζGli is larger than the 10 amplitude in an individual 2 2 Fourier mode because hζGli is roughly a sum over hζG(k)i for k < H0. Throughout this chapter we assume constant ns = 0.9608 [96] or ns = 1, but keep in mind that the results > 2 for N ∼ 20 are very sensitive to the (unknown) infrared behavior of ∆G(k).

typical amplitude of super-horizon fluctuations relative to that for a flat spectrum −1 ns = 1. This difference becomes significant for N & −(ns − 1) ' 25 e-folds – 2 precisely when O((ns − 1) ) contributions to the running are expected to change 2 1/2 ns − 1 by order unity [97]. The specific shapes of hζGli plotted in Figure 2.1 should therefore be interpreted with caution, particularly for N  1/(ns − 1). For 2 reference, we also plot examples of hζGli with running spectral indices given by 2 −1 ns(k) = ns(kpiv) ± (ns(kpiv) − 1) ln(k/kpiv), with kpiv = 0.002 Mpc .

37 2.2.3 Spatial Curvature from Long-Wavelength Modes

Long wavelength modes of ζ will also contribute to the mean spatial curvature measured within our Hubble volume2

Z 3 obs −2 d k 2 Ωk = 2 3 k ζ(k)Ws(k) . (2.27) 3H0 (2π)

In a given subvolume Vs, knowing the value of the background mode ζGl, is obs insufficient to specify Ωk . Nevertheless, we can estimate the typical amplitude of

Ωk|s in the scenarios we consider

Z 4∆2(H ) 4∆2(H ) obs 2 4 dk 4 2 2 ζ 0  −(ns+3)N  ζ 0 h(Ωk ) i = 4 k ∆ζ (k)|Ws(k)| = 1 − e ≈ 9H0 k 9(ns + 3) 9(ns + 3) (2.28) where in the final ≈ we have assumed that the global power spectrum PG is not obs too red, ns + 3 > 1). So, the dominant contributions to Ωk come from modes with k ∼ H0 and, in contrast to Eq. (2.25)-Eq. (2.26), there is no enhancement from N  1. We therefore ignore constraints on ζGl coming from constraints in Ωk because only the first few modes outside the horizon lead to spatial curvature. This contribution is small compared to the N-enhanced shifts to the observed power spectrum and other parameters which we will find, justifying our working in the limit where ζ(x) is independent of x within our Hubble volume (e.g. Eq. (2.23)). However, it would be interesting to revisit these constraints and their implication for the bias of local statistics, particularly if local-type primordial non-Gaussianity is detected. 2Here, we are using the scalar curvature on spatial hypersurfaces R(3) = −4∇2ζ(x), and obs 1 R 3 (3) taking Ωk = − H0 −3 d x R (x), however see [55,98–102] for more detailed discussions of 6 H0 constraints on ζ(k) contributions to Ωk as measured in our Hubble volume.

38 2.3 Example I: Weakly Non-Gaussian Initial Condi- tions

2.3.1 Biasing of Parameters with Local Non-Gaussianity

In this section we imagine that the statistics in the larger volume Vl can be described by the usual local ansatz3

3   9   ζ(x) = ζ (x) + f ζ2 (x) − hζ2 i + g ζ3 (x) − 3hζ2 iζ (x) G 5 NL G G 25 NL G G G 27   + h ζ4 − 6hζ2 iζ2 (x) + 3hζ2 i2 , (2.29) 125 NL G G G G where the non-zero coefficients satisfy

2 1/2 2 2 3/2 1  fNLhζGi  gNLhζGi  hNLhζGi . (2.30)

The equation above is the definition of weak non-Gaussianity for this model. Single-source non-Gaussian models with coefficients with this scaling will generate n 2 n−1 non-Gaussian polyspectra that scale as hζ ic ∼ (∆ζ ) ; for further discussion see AppendixB. We can then apply the condition in Eq. (2.30) to require that the power spectrum of ζ on CMB scales agrees with the power spectrum of ζG to 2 some accuracy, that is we could require that the O(fNL) terms are not important. Note that depending on the shape of the power spectrum, this requirement may q 2 2 be much stronger condition than requiring that fNL ∆WMAP(k), gNL∆WMAP(k), 3/2 hNL∆WMAP(k)  1 on CMB scales. We have checked that the examples plotted in

Figures 2.2 and 2.3 satisfy Eq. (2.30) for the assumed ns = const. power spectra.

In the larger volume Vl, the field ζ(x) given in Eq. (2.29) has power spectrum

 2  Pζ (k) = PG(k) 1 + O(hζGi) , (2.31) and the bispectrum and trispectrum are characterized by the coefficients fNL, gNL

3 2 Here we subtract the 3gNLhζGiζG(x) so that the power spectrum is unaffected at linear order 2 2 in gNL and subtract 6hNLhζGiζG(x) so that the bispectrum is unchanged at linear order in hNL. This helps to isolate how each coefficient changes the statistics of ζ, particularly in the case where a lower-order coefficient is vanishing (e.g. fNL = 0, but hNL 6= 0). See AppendixB for general expressions relating the coefficients in Eq. 2.29 and series coefficients in a general local map between ζG(x) and ζ(x).

39 6 2 2 and τNL = ( 5 fNL) up to corrections O(hζGi). When we carry out the long-short wavelength split of section 2.2, the long- wavelength pieces of higher order terms can be recollected in the coefficients of lower order terms, since ζl is constant within the subvolume. An observer in a finite region Vs with background field value ζGl will see local statistics described by

3   ζobs(x) = ζobs(x) + f obs (ζobs)2(x) − h(ζobs)2i G 5 NL G G 9   + gobs (ζobs)3(x) − 3h(ζobs)2iζobs(x) + ... (2.32) 25 NL G G G where we’ve defined

 6  ζobs(x) = 1 + f ζ + O(ζ2 ) ζ (2.33) G 5 NL Gl Gl Gs which we require to give the locally observed power spectrum

obs obs 0 3 3 0 obs hζG (k)ζG (k )i = (2π) δ (k + k )PG (k), (2.34) where 2π2 P obs(k) = ∆2 (k), (2.35) G k3 WMAP  obs 2 obs up to corrections at O (fNL ) . The local power spectrum Pζ is related to the globally defined one Pζ through

 12  P obs(k) = 1 + f ζ + O(ζ2 ) P (k), (2.36) ζ 5 NL Gl Gl G and the locally observed non-Gaussian parameters are

9 12 f obs = f + g ζ − f 2 ζ + O(ζ2 ), (2.37) NL NL 5 NL Gl 5 NL Gl Gl 12 18 gobs = g + h ζ − f g ζ + O(ζ2 ) . (2.38) NL NL 5 NL Gl 5 NL NL Gl Gl

Eq. (2.36)-(2.38) show that the connected n + 1-point functions of ζ adjust the obs n-point functions of ζ by terms O(ζGl) and cause the locally observed statistics to differ from the global ones. For a strictly fNL model (i.e. gNL, hNL,··· = 0) with fNL > 0, a positive background fluctuation ζGl boosts the local power relative to the local bispectrum, the net effect is to make the local statistics appear more Gaussian

40 Figure 2.2: The observed amplitude of scalar fluctuations in our Hubble volume systematically differs from the average value in the entire universe by a fractional amount 12/5fNLζGl where ζGl is the (unobservable) background mode in our Hubble patch. Plotted is an estimate of the probability distribution for the true value of ∆2, given the 2 −9 obs locally observed value ∆WMAP = 2.464 × 10 for two values of fNL and two values of N, the number of super-horizon e-folds. An observer in our Hubble volume cannot measure N, and therefore is unable to determine which probability distribution correctly describes our universe. The vertical dashed lines show the 68% confidence interval (±0.072 × 10−9) 2 on ∆WMAP from the eCMB+BAO+H0 dataset [96].

q √ obs obs 2 obs 2 than they are in the larger volume Vl (i.e. fNL < fNL and fNL (∆ ) < fNL ∆ ).

Negative fNL or background fluctuations will, of course, have the opposite effect. On 2 the other hand if gNL ∼ fNL, then leakage from the trispectrum into the bispectrum can compensate and the local fNL value can be representative of the globally defined 9 12 2 one. The cancellation between 25 gNL and 5 fNL in Eq. (2.37) is precisely what happens in the curvaton model when the curvaton dominates the energy density of the universe at the time of decay [103]. However, the level of non-Gaussianity as q obs 2 obs quantified by fNL (∆ ) is still adjusted.

If fNL =6 0, the measured value of the scalar power spectrum in our Hubble 2 volume ∆WMAP differs from the average value in the larger universe Vl by an 12 unknown amount 5 fNLζGl – unknown because we don’t know the values of ζGl or 2 fNL. In Figure 2.2 we plot an estimate of the probability distribution for ∆ζ in 2 Vl for fixed values of fNL, assuming the observed value is ∆WMAP, and that ζGl is

41 obs Figure 2.3: The observed amplitude of fNL in our Hubble volume, fNL , differs from the average value in entire universe depending on ζGl, fNL and gNL. Plotted is an estimate of 2 −9 the probability distribution for the true value of fNL given ∆WMAP = 2.464 × 10 and 3 5 obs fixed gNL = 10 (left panel) or gNL = 10 (right panel). For fNL = 30 and gNL = 1000 2 there is a partial cancellation between the gNL and fNL terms in Eq. (2.37) causing the distribution to be narrower than in the fNL = 10, 50 cases. The width of the distributions depends on the RMS value of ζGl. We assume ns = const. = 0.9608, which in comparison 2 1/2 to ns = 1, gives a difference in hζGli of ∼ 10% at N = 1 and a factor of ∼ 3.5 by N = 100.

drawn from a power law spectrum as in Eq. (2.26). Similarly, the local fNL and

gNL values in Vl are related to the observed ones by amounts dependent on ζGl.

In Figure 2.3 we plot estimates for the distribution of fNL values in Vl assuming obs obs the locally observed power spectrum, and several possible values of fNL , gNL . The probability distributions plotted in Figure 2.2 and Figure 2.3 are estimates of the probability distributions in that: (i) we don’t allow all the observed parameters 2 obs obs obs 2 (∆G) , fNL , gNL to vary simultaneously and (ii) we neglect terms O(ζGl) in relating

values of parameters fNL, gNL measured in our Hubble volume to those in the larger

universe Vl. A more realistic, but more involved calculation would be to calculate 2 the posterior probability distribution of (∆G, fNL, gNL) given the observed values 2 obs obs obs (∆G) , fNL , gNL along with their observational uncertainties, and the fact that

ζGl is Gaussian distributed. We are merely interested in illustrating the range of possibilities and leave a thorough exploration of parameters for another study. We further emphasize that observationally, we don’t have observational access to N – a parameter we have held fixed in Figure 2.2 and Figure 2.3. The variation in the probability distributions for different N values should therefore be interpreted as an additional observational uncertainty.

42 2.3.2 Implications for Model Builder

When constructing a model of inflation, one typically specifies some set of fields relevant for inflation and the primordial fluctuations, as well as any interactions the fields may have. This guarantees the existence of an inflating solution and fluctuations and determines the possible shapes of the correlation functions. Ad- justable parameters then allow the model builder to match the observed amplitude of fluctuations and to tune any non-Gaussianity to an amplitude consistent with observational constraints. The length of slow-roll inflation may or may not be an independently tunable microphysical parameter. How should the model builder decide if a given set of microphysical parameters gives rise to a significant number of Hubble volumes consistent with the one we see? In non-Gaussian models the neces- sarily statistical nature of making predictions from inflation for our Hubble volume becomes much more important, even for relatively short durations of inflation. To illustrate this point, consider a very simple (if unrealistic) model with only quadratic non-Gaussianity, fNL, and a constant spectral index. Expressing the observed parameters in terms of the amount of non-Gaussianity in the large volume, q 2 fNL hζGi, as well as a bias parameter B, gives a sense of how the local statistics can differ from the global statistics:

 12 q  f obs = f 1 − f hζ2 i B , (2.39) NL NL 5 NL G  6 q  f obs∆obs = f ∆ 1 − f hζ2 i B . (2.40) NL ζ NL ζ 5 NL G

2 4 where hζGi is the correlation function at zero separation . The bias

ζGl B ≡ 2 1/2 , (2.41) hζGi like ζGl, is larger for more rare fluctuations and increases as N increases (or as the size of the subvolume considered is decreased within a fixed global volume),

4 2 One might worry that we are scaling quantities by a loop factor hζGi, which is dependent on the power spectrum over the entire range of scales (and, without a cutoff is formally divergent for 2 a scale invariant power spectrum). However, hζGi is merely a placeholder and the actual value cancels when calculating observed quantities – our results do not depend on the unknown UV behavior of the power spectrum PG.

43 allowing more modes to contribute in the background. It’s variance is

? 2 ln(k L) hB i = . (ns = 1) (2.42) ln(kmaxL)

As the range of subhorizon scales becomes small compared to superhorizon scales, so ? 2 that ln(k L) ' ln(kmaxL), the variance hB i → 1. The effect of the long-wavelength background is therefore controlled by the level of non-Gaussianity in the large volume, as well as the size and rarity of the subvolume as captured in the bias B. Notice that for non-Gaussian inflation models, matching parameters in the theory to agree exactly with our local observations makes sense only if the number of e-folds in the model is not too large. One way of visualizing this criteria is plotted in Figure 2.4. If the number of e-folds in the theory is larger, the parameters should not be matched identically to what we observe on CMB scales. In that case sub-volumes that have statistics identical to the parent will be rare, and so our observed universe will not be the typical outcome of those models. Finally, we note that because both the amplitude of fluctuations and the value of fNL are changing in typical subsamples as we look on different scales, it is useful to plot the quantity that shows how non-Gaussian the subsamples are on average. The relative amplitude of non-Gaussianity in the subvolume to that in the large volume is shown in Figure 2.4. Note that for positive fNL, an overdensity ζGl > 0 causes the non-Gaussianity to be smaller in the small volume. Similarly, an underdensity

ζGl < 0 causes the non-Gaussianity to be larger.

2.4 Example II: Strongly Non-Gaussian Initial Condi- tions

Suppose the non-Gaussian curvature perturbation in the larger volume Vl is given by p p ζ(x) = ζG(x) − hζGi (2.43) where p is a positive integer > 1. This field has statistics that are not accurately characterized by an expansion of the form Eq. (2.1), in particular the polyspectra have a different shape and scale dependence from the local shapes given in Eq. (B.11), Eq. (B.13), and Eq. (B.12). Nevertheless, in the squeezed limits that observationally

44 Figure 2.4: Left: The degree of√ variation in non-Gaussianity in subvolumes as quantified (f ∆2 )obs 2 2 2 2 NL √ G
6
2 2 6
2 by the variance ΣNG ≡ 2 − 1 = 5 fNL hζGihB i = 5 fNL hζGli. The fNL ∆G horizontal, dotted line indicates ΣNG = 0.15, roughly when uncertainty in fNL due to super-horizon correlations becomes comparable to the expected error on fNL from

Planck (we assume ±5√), if fNL = 40 in Vl. Right: The fractional change in the level of 2 obs (fNL √∆ ) non-Gaussianity, 2 , vs. number of super-horizon e-folds N. The upper and fNL ∆ 1/2 1/2 lower curves correspond to ζGl = hζGli and ζGl = −hζGli , respectively. In these figures we’ve assumed gNL, hNL... = 0.

100

NG fNL =40, ns =0.9608 Σ

fNL =10, ns =0.9608

fNL =40, ns =1

fNL =10, ns =1

10-1

10-2

-3 variance in the "level" of non-Gaussianity non-Gaussianity of "level" the in variance 10 0 50 100 150 200 number of super-horizon e-folds N

define fNL, gNL, and τNL one finds

1 1 1 fNL ∼ 2 p/2 , gNL ∼ 2 p , τNL ∼ 2 p for p even (2.44) hζGi hζGi hζGi

1 1 fNL = 0 , gNL ∼ 2 p , τNL ∼ 2 p for p odd . (2.45) hζGi hζGi In contrast to the weakly non-Gaussian case in section 2.3, this field has 1 ∼ q 2 2 2 2 2p fNL ∆ζ ∼ gNL∆ζ ∼ τNL∆ζ , where ∆ζ ∼ hζG i is the observed variance. In general, the fNL, gNL and τNL will also be scale-dependent functions of ks, kl, the long and short-wavelengths used to take the squeezed limits in Eq. (B.11), Eq. (B.13), and Eq. (B.12). For a more thorough discussion of weak and strong local non- Gaussianity, see AppendixB.

Consider the local statistics of ζ in a subvolume of size Vs. The local non- Gaussian curvature can be written in terms of short and long wavelength modes of

45 Figure 2.5: Here we suppose that the curvature perturbation in Vl is given by ζ(x) = p p ζG(x) − hζGi, and ask whether the curvature perturbations in our Hubble patch could appear to be described by the weakly non-Gaussian series (e.g. Eq. (2.48)). We require q 2 a large local background fluctuation ζGl  hζGsi and consistency with the observed 2 power spectrum ∆WMAP. Plotted are the corresponding values of fNL (left panels) and q 2 gNL (right panels) for different values of the power law index p and ζGl: ζGl = hζGli q 2 q 2 (solid lines), ζGl = 3 hζGli (dashed lines) and ζGl = 5 hζGli (dotted lines).The upper 2 row uses ns = 0.9608 for ∆G, the lower row uses ns = 1. The bend in the plots for ns = 0.9608 occurs at N(ns − 1)/p ∼ 1.

n =0.9608 n =0.9608 s 105 s p =2 p =3 p =3 p =4 102 p =4 104

NL NL g f 103

101 102 local value of of value local local value of of value local

101

100 100 102 103 102 103 number of super horizon e-folds N number of super horizon e-folds N (a)(b)

n =1 n =1 103 s 105 s p =2 p =3 p =3 p =4 p =4

102 104 NL NL g f

101 103 local value of of value local local value of of value local

100 102 102 103 102 103 number of super horizon e-folds N number of super horizon e-folds N (c)(d)

ζG as in Eq. (2.20), Eq. (2.21) as

p!   p! ζobs(x) = pζp−1ζ (x) + ζp−2 ζ2 (x) − hζ2 i + ζp−3ζ3 + ... Gl Gs 2!(p − 2)! Gl Gs Gs 3!(p − 3)! Gl Gs (2.46) where we have suggestively ordered the series with the term linear in ζGs first. Now,

46 if we happen to be considering a small volume Vs with a background fluctuation satisfying q 2 ζGl  hζGsi , (2.47) then to a local observer ζobs given in Eq. (2.46) appears to be a field described by a weakly non-Gaussian expansion of the form5

3   9   ζobs(x) = ζobs(x)+ f obs (ζobs)2(x)−h(ζobs)2i + gobs (ζobs)3(x)−3ζobs(x)h(ζobs)2i +..., G 5 NL G G 25 NL G G G (2.48) where 2 ! obs p−1 hζGsi ζG (x) = pζGl ζGs(x) + O 2 , (2.49) ζGl 2 2 obs obs obs 2 (the O (hζGsi/ζGl) is because we have subtracted 3gNL ζG h(ζG ) i from the linear term in Eq. (2.48)) and

3 p − 1 hζ2 i! 9 (p − 1)(p − 2) hζ2 i! f obs = + O Gs and gobs = + O Gs . NL p 2 NL 2 2p 2 5 2pζGl ζGl 25 3!p ζGl ζGl (2.50)

Now, ζGl  1 so the field in Eq. (2.48) should have large local non-Gaussianity. However, it is possible for the local statistics to appear only weakly non-Gaussian, i.e. q obs obs 2 obs obs 2 fNL h(ζG ) i, gNL h(ζG ) i, ...  1 (2.51)

on top of sufficiently large background fluctuations. Taking Vs to be our Hubble

volume and assuming that ζG has a power-law spectrum with constant spectral index as in Eq. (2.26), the criterion given in Eq. (2.47) for observing weak non-Gaussianity can be written,

v u ? s ζGl uln(kmax/k ) Ns q  t = for ns = 1 (2.52) 2 ln(k?L) N hζGli s (n −1)N ζGl e s s − 1 q  for ns = const. 6= 1 (2.53) 2 1 − e−(ns−1)N hζGli

5 < p! Here we’re assuming p ∼ 10, say, or small enough that the binomial coefficients k!(p−k)! p! p 2 p 2 don’t spoil the smallness of the quantity k!(p−k)! hζGsi/ζGl when hζGsi/ζGl  1 as given in Eq. (2.46).

47 where, as before N is the number of super-horizon e-folds and we have introduced Ns, q e(ns−1)Ns −1 the number of sub-horizon e-folds. For reference, Ns ∼ 60 gives 1−e−(ns−1)N ∼ 1 q e(ns−1)Ns −1 for N ∼ 15 and 1−e−(ns−1)N ∼ 0.05 by N ∼ 150 when ns = .9608. 2 1/2 For typical subvolumes, |ζGl| ∼ hζGli . Consequently, the condition on the ? ? −1 ratios of scales kmax/k and k /L in order for typical subvolumes to be weakly non-Gaussian is v u ? uln(kmax/k ) t  1. (2.54) ln(k?L) That is, the range of superhorizon scales must significantly exceed the range of observationally accessible scales. In terms of the bias parameter introduced in section 2.3.2, this condition is hB2i ≈ 1. (2.55)

In other words, weak non-Gaussianity will arise in subvolumes with an order one bias. Equations (2.52), (2.53) show that even for strongly non-Gaussian statistics of

ζ in Vl, the statistics in subvolumes appear weakly Gaussian on top of very rare background fluctuations. But for sufficiently small subvolumes, statistics in Vs appear weakly Gaussian even for typical values of ζGl. In regions where Eq. (2.47) obs is satisfied, the possible values of fNL in subvolumes depends qualitatively on p: for obs p even p, fNL > 0 in all subvolumes, whereas for odd p the sign of ζGl is significant obs obs and fNL can be negative. On the other hand, gNL > 0 for all values of p. 2 Now we ask what the restrictions on p, ∆G, and ζGl are in order to generate a curvature perturbation as in Eq. (2.48) that satisfies the observational constraints on the power spectrum, fNL and gNL in our Hubble volume. If we fix the ratio q 2 ζGl/ hζGli (which is a measure of the rarity of our Hubble patch), the index p, and 2 2 the observed level of fluctuations ∆χ = ∆WMAP, Eq. (2.49) and Eq. (2.50) allows us to solve for the variance of fluctuations in the (unobservable) background field

ζG along with the observed values of fNL and gNL as a function of N. The results are plotted in Figure 2.5. We see that current constraints on the observed level of non-Gaussianity are indeed compatible with a scenario in which our Hubble patch is a biased subsample of a larger universe with strongly non-Gaussian initial p p curvature perturbations ζ(x) = ζG(x) − hζGi. The trend towards Gaussian statistics may also be viewed in terms of the

48 real-space dimensionless cumulants,

n 2 n/2 Mn ≡ hζ(x) ic/hζ(x) i , (2.56) which are measures of the non-Gaussianity that may be straightforwardly computed 6 from the real-space field ζ(x). Averaged throughout the large volume, the Mn from Eq. (2.43) are all O(1). In the subvolume, we find from Eq. (2.46) that

1 !n−2 Mn ∝ p , 2 < n ≤ p. (2.57) 2ζGl

? ? ? q We see that for k L  kmax/k or k  kmax/L, in typical subvolumes the dimen- sionless cumulants follow a hierarchical scaling, falling off with the characteristic 2 1/2 scale hζGsi /ζGl. This hierarchical scaling is of course the same scaling for the

Mn that results from the original weakly non-Gaussian local ansatz, Eq. (2.1).

2.5 Example III: Two-field Initial Conditions

In this section we consider initial conditions inspired by a version of the curvaton model [46,47,78 –80,103] in which perturbations from both the inflaton φ and the curvaton σ are responsible for generating ζ (see e.g. [104–107]). In this “inflaton- curvaton” scenario, the curvature perturbation in the larger volume is given by

3 ζ(x) = φ (x) + σ (x) + f˜ (σ2 (x) − hσ2 i) (2.58) G G 5 NL G G

We make the simplifying assumption that φG(x) and σG(x) are statistically inde- pendent (i.e. hφ(k)σ(k0)i = 0), Gaussian random fields with proportional power spectra   2 Pφ 2 18 ˜2 ξ ≡ so that Pζ (k) = Pσ(k) 1 + ξ + fNLIσ(k) (2.59) Pσ 25 2 where Iσ(k) ∼ ∆σ is defined in Eq. (B.5) and for simplicity we assume ξ is a constant (however, see e.g. [106,108]).

6 In contrast, parameters such as fNL which assume a particular form of non-Gaussianity can only be estimated from the data, rather than directly measured, by looking at its overlap with a particular non-Gaussian template such as the local ansatz.

49 2.5.1 Case 1: σ Is Weakly Non-Gaussian

First, we make the usual assumption that the curvaton contributions to the curvature q ˜ 2 perturbation are only weakly non-Gaussian. That is, we assume that fNL ∆σ  1. The non-Gaussian parameters that characterize the bispectrum and trispectrum of ζ are 2  6 ˜  f˜ fNL f = NL , τ = 5 (2.60) NL (1 + ξ2)2 NL (1 + ξ2)3 and gNL = 0. In a subvolume Vs, a local observer will see statistics described by

12 f˜ ! P obs = P 1 + NL σ (2.61) ζ ζ 5 1 + ξ2 Gl f˜ 12 ξ2 − 1 ! f obs = NL 1 + f˜ σ (2.62) NL (1 + ξ2)2 5 ξ2 + 1 NL Gl obs gNL = 0 (2.63) 2  6 ˜  ! fNL 12 ξ2 − 2 τ obs = 5 1 + f˜ σ . (2.64) NL (1 + ξ2)3 5 ξ2 + 1 NL Gl

In contrast to the case in section 2.3, the local statistics are now modulated by long-wavelength modes of σG only, as opposed to fluctuations in the total curvature

fluctuation ζGl = φGl + σGl. To compare with section 2.3, we rewrite Eq. (2.61)-

Eq. (2.64) in terms of fNL, τNL in Vl,

 12    P obs = P 1 + f 1 + ξ2 σ (2.65) ζ ζ 5 NL Gl   2    5  2 12 τNL − 2fNL   obs   6  2  fNL = fNL 1 +   1 + ξ σGl (2.66) 5 fNL

Now, the amount by which the power spectrum and fNL vary from place to place 2 is the same as in Eq. (2.36) and Eq. (2.37) with ζGl → (1 + ξ )σGl. The typical q q 2 2 2 2 2 size of the modulation of statistics in Vs is h(1 + ξ ) σGli = (1 + ξ )hζGli. So for fixed fNL, the typical modulation in the power spectrum is larger relative to the case where a single field, ζG, generates density perturbations. That is, a one-sigma fluctuation in σ generates a larger change than a one-sigma fluctuation √Gl 2 in ζGl, larger by a factor 1 + ξ . The σ field itself must be more non-Gaussian to maintain a fixed fNL in the curvature as the power from σ decreases (ξ increases).

50 2 In Figure 2.6 we plot estimates for the distributions of ∆ζ and fNL in the total volume, given several values of ξ2. As in section 2.3 these are estimates of the 2 probability distributions in that we (i) neglect terms O(ζGl) in relating values of

parameters measured in our Hubble volume to those in the larger universe Vl and obs 2 obs (ii) we don’t allow the observed parameters ((∆ζ ) , fNL ) to all vary simultaneously. We have again fixed the number of e-folds for illustrative purposes even though this is also an unobservable quantity. Note, that while the observed amplitude of the three-point and four-point q 2 2 functions are characterized by fNL ∆ζ , τNL∆ζ , when one considers the entire series of correlation functions neither product alone quantifies the level of non-Gaussianity ˜ fNL∆ζ in the field ζ. The single quantity that controls the level of non-Gaussianity is 1+ξ2 ˜ fNL∆ζ (rather than fNL∆ζ = (1+ξ2)2 ). When this quantity is small, the series of cumulants is ordered and the amplitude of each consecutive cumulant is smaller by this factor. Each cumulant also has an extra factor of 1/(1 + ξ2) which does not affect their relative importance. In terms of the observed non-Gaussian parameters given in ˜ fNL∆ζ Eq. (2.60), this criterion for weak non-Gaussianity, 1+ξ2  1, is equivalent to 2 q 2 requiring the kurtosis to be much smaller than the skewness: τNL∆ζ  fNL ∆ζ . As in the single field case, we can ask how the total amplitude of non-Gaussianity differs in biased subvolumes:

q q f˜ ∆2 f˜ ∆2 " ˜ 2 1/2 # NL ζ NL ζ 6 fNLhζGi = 1 − B (2.67) 1 + ξ2 1 + ξ2 5 1 + ξ2 obs

where the bias here is defined as

σGl B = 2 1/2 . (2.68) hζGi

The relationship between the amplitude of non-Gaussianity in Vs and Vl has the same structure as in the single field case, but the bias will generally be smaller q 2 (assuming the same total amplitude of fluctuations, ∆ζ ) since the fluctuating field contributes only part of the total power. Notice that when ξ = 0 this reduces to the single field expression, Eq.(2.40).

51 Figure 2.6: In the two-field “weak non-Gaussianity” case the amplitude of scalar 2 obs obs perturbations, (∆ ) and of fNL in our Hubble volume, fNL , differ from the average value due to the background value of σGl, while the total curvature is set by σGl + φGl. p 2 For fixed fNL, changes to the local statistics are typically larger by a factor of 1 + ξ relative to the case in §2.3. Plotted are estimates of the probability distributions for 2 ∆ (left panel) and fNL (right panel) in Vl, given the observed values in Vs for different 2 2 obs 2 values of ξ ≡ Pφ/Pσ. Note that for ξ = 1, fNL = fNL + O(σGl).

2.5.2 Case 2: σ Is Strongly Non-Gaussian

Now we assume the perturbations coming from the curvaton are strongly non- q ˜ 2 Gaussian fNL ∆σ ∼ 1, but a subdominant contribution to the total curvature 2 ˜2 2 (ξ  fNL∆σ ∼ 1). To understand the dependencies, it’s helpful to define the ˜2 O(fNL) fractional change in the globally defined power spectrum in Vl

2  3 ˜  fNL Iσ(k) ˜(k) ≡ 2 5 (2.69) ξ2

2 2 By assumption ˜ ∼ 1/ξ and again Iσ ∼ ∆σ as defined in Eq. (B.5). In the larger box Vl the power spectrum is

2 Pζ (k) = ξ Pσ(k) (1 + ˜(k)) . (2.70)

52 Taking the squeezed and squashed limits in Eq. (B.11), Eq. (B.13), Eq. (B.12) gives scale-dependent non-Gaussian parameters:

˜ 2 fNL 6  1 fNL(kl) = 2 ˜(kl) , gNL = 0 , τNL(kl) = fNL(kl) (2.71) ξ 5 ˜(kl) where kl is the magnitude of the long-wavelength mode used to calculate the squeezed and squashed limits. The scale dependence of fNL, τNL is given by the 2 function Iσ(k) in Eq. (B.5). For k  Λ, where Λ is the infrared cutoff in ∆ (k), 2 the scale dependence of Iσ(k) is generally weak: for ns = 1, Iσ(k) ∼ 2∆σ ln(k/Λ). In this example, the field ζ is weakly Gaussian with hierarchical cumulants in q 2 2 that 1  fNL hζGi  τNLhζGi. However, in contrast to the weakly non-Gaussian, n+1 n 2 single-source case in §2.3 where the cumulants scale as hζ ic/hζ ic ∼ hζ i, the hierarchy of cumulants in this example scales as

hζn+1i q hζ2i1/2 hζ2i1/2 c ˜ 2 n ∼ (fNL hσGi) ∼ . (2.72) hζ ic ξ ξ

We have assumed that ξ2  1, but depending on the relative magnitudes of ξ and hζ2i, the higher-order cumulants may be more important relative to the lower order ones than in the examples considered in section 2.5.1 and §2.3.

In a subvolume Vs, an observer will see a local power spectrum

2 !! obs 2 2σGl Pζ (k) = ξ Pσ(k) 1 + ˜s(k) 1 + (2.73) Iσs (k) and non-Gaussian parameters

 2 2 ! 6 obs f˜ 2σ fNL f obs(k ) = NL ˜ 1 + Gl , gobs = 0 , τ obs(k ) = 5 (2.74) NL l 2 s NL NL l  2  ξ Iσ (kl) 2σGl s ˜s 1 + Iσs (kl)

2 where Iσs is Eq. (B.5) with P (k) → Pσ(k)|1 − Ws(k)| and ˜s is Eq. (2.69) with

Iσ → Iσs . In this case the difference between the local and global statistics is more complicated: since Iσ 6= Iσs the local statistics in Vs differ from those in Vl even if

53 2 Figure 2.7: In the two-field “strong non-Gaussianity” case σGl modulates the local statistics leading to skewed probability distributions for the observed non-Gaussian obs parameters. The solid curves are the distributions for the observed values of fNL (left obs panel) and τNL (right panel) for different values of N – the number of super-horizon 3 ˜ 2 2 e-folds (in this plot we set ns = 1). In each case we’ve fixed ( 5 fNL) ∆σ = 1 and chosen ˜ the values of fNL and ξ (see Eq. (2.58) - Eq. (2.59)) to produce fNL(k = 0.002/Mpc) = 10 when averaged throughout Vl (however, the corresponding τNL values are different). The values of fNL and τNL when averaged over Vl are indicated by the vertical dotted lines.

2 7 σGl = 0. Averaging over σGl will recover the parameters in Vl. To see this, note 2 2 that PσIσ ∼ Iσs Pσs + 2Pσs hσL i + IσL PσL so that Iσ(k) → Iσs (k) + 2hσL i for scales > 1/3 2 k ∼ Vs , so that h˜s(1 + 2σGl/Iσs )i → ˜. To study the statistics in subvolumes Vs, consider the following example: fix 3 ˜ 2 2 ( 5 fNL) ∆σ = 1, then we find expressions for the locally measured non-Gaussian 2 parameters in terms of ξ and the observed power spectrum ∆WMAP,

 2 2 2 2 Iσ(k) 10 1 Iσ(k) 6 ξ ∆σ ˜ → 2 2 , fNL → ± q 3 2 , τNL → fNL . ξ ∆ 2 ξ ∆ 5 2 Iσ(k) σ 3 ∆WMAP σ (2.75)

For simplicity we’ll assume that ns = 1 so that we can use the analytic expression

7The average of the small-volume polyspectra over the long wavelength modes must recover the large-volume polyspectra. However, since the parameters fNL, gNL etc. are ratios of quantities dependent on the random variable ζGl, when terms non-linear in ζGl are important the relationship obs between fNL and fNL, say, is generally more complicated. In this example the non-Gaussian polyspectra are dependent on σ, but power spectrum is dominated by the Gaussian field φ, and therefore averaging over σGl doesn’t change the denominator in the ratios used to define the obs non-Gaussian parameters, and the expressions for fNL and τNL are easily recovered from fNL obs 2 and τNL by averaging over σGl.

54 for Iσ(k) given in Eq. (B.6). Eq. (2.75) allows us to choose values of the ratio of inflaton to curvaton power that give particular values of fNL, τNL. We now choose

ξ such that fNL(kpiv) = 10 when averaged over the entire volume Vl. However, 2 since the local value of fNL depends on σGl, observers in a finite volume can easily measure fNL(kpiv) 6= 10. In Figure 2.7, we plot the probability distribution of obs obs possible observed values of fNL and τNL . We see that unlike the cases considered in section 2.3, section 2.4, and §2.5.2 the probability distributions are extremely skewed and there is a large offset between the median and modes of the distribution obs obs of values of fNL and τNL .

2.6 General Local-type Non-Gaussianity

In this section we consider arbitrary local-type non-Gaussianity with a single source, given by a general polynomial in ζG,

1 1 ζ = N ζ + N ζ2 + N ζ3 + ..., (2.76) 1 G 2! 2 G 3! 3 G where we implicitly shift the field so the mean hζi throughout the large volume is zero, and we take the Ni to be constants. Under the long- and short-wavelength split, the observed curvature perturbation in a subvolume is

1 1 ζobs = ζobs + N obs(ζobs)2 + N obs(ζobs)3 + ..., (2.77) G 2! 2 G 3! 3 G where we have set N1 ≡ 1 and defined

1 2 obs Nn + Nn+1ζGl + 2! Nn+2ζGl + ... N = n , (2.78) n h 1 2 i N1 + N2ζGl + 2! N3ζGl + ... and  1  ζobs ≡ N + N ζ + N ζ2 + ... ζ . (2.79) G 1 2 Gl 2! 3 Gl Gs Eq. (2.77) can be checked by plugging in Eqs. (2.78)-(2.79) to recover Eq. (2.76). The level of non-Gaussianity in ζ introduced by any one of the nonlinear terms 2 (n−1)/2 in Eq. (2.76) can be quantified by NnhζGi , which is roughly the amplitude for the nth term in comparison to the Gaussian term. Using Eqs. (2.79) and (2.78),

55 it is easy to show that the corresponding quantity for the small volume is given by

1 2 obs obs 2 (n−1)/2 Nn + Nn+1ζGl + 2! Nn+2ζGl + ... 2 (n−1)/2 λn ≡ Nn h(ζG ) i = 1 2 hζGsi . (2.80) N1 + N2ζGl + 2! N3ζGl + ...

As before, for unbiased subsamples where the long wavelength modes average obs to zero ζGl = 0, we have Nn = Nn, so non-Gaussian statistics are unchanged. Generically, if the series in the large volume was a good Taylor expansion with

|Nn+1ζG| < |Nn|, the observed coefficients will not be too different from the original, global parameters. Note that if we truncate the series at two terms, where N1 = 1 6 and N2 = 5 fNL, we recover Eqs. (2.36), (2.37).

2.6.1 Running of Non-Gaussian Parameters

We can think of the parameters of the local ansatz as running with the long- wavelength background ζGl, which accounts for running with the scale of the subvolume, as well as dependence on the location of the subvolume,

? λn = λn(ζGl) = λn(k , xVol), (2.81)

? where xVol is the location of the subvolume, so that ζGl = ζGl(k , xVol) ≈ const. The running of the parameters of the series can be expressed in differential form, analogous to renormalization group equations. Defining

1 N obs ≡ N + N ζ + N ζ2 + ..., (2.82) n n n+1 Gl 2! n+2 Gl we have obs dNn obs = Nn+1, (2.83) dζGl and from Eq. (2.76) smoothed on the subvolume scale,

obs dζl dN0 obs ≡ = N1 . (2.84) dζGl dζGl

2 1/2 Changing variables to the bias B ≡ ζGl/hζGi , it is straightforward to show from Eq. (2.80) that d ln λn λn+1 = − λ2. (2.85) dB λn

56 This equation is valid for any set of initial conditions λn(0), that is, for any set of coefficients Nn, although one must take care when B = 0 in cases where there is no linear term in the large volume (N1 = 0), because in the small volume, Eq. (2.77), we have normalized the linear term to have a coefficient 1. Eq. (2.85) describes n n+1 a gain to the ζG term from the ζG term in the expansion as B increases, and a loss from the quadratic term, which directly boosts the amplitude of the linear term, decreasing non-Gaussianity. Writing a similar differential equation for the n 2 n/2 dimensionless (connected) moments Mn ≡ hζ(x) ic/hζ(x) i would avoid that problem and be more complete, but it is also more notationally cumbersome so we do not write it here.

2.7 Behavior of N-Point Function Shapes.

We have seen that the amplitude of local non-Gaussianity may vary significantly with volume location and size. In this section we point out that the shape of non-Gaussianity is unchanged, with correlation functions characterized by the same squeezed-limit scalings, independently of the sample volume. In specifying ζ, we determine shapes for N-point correlation functions on all scales. In a subvolume, these shapes are still present, but can be dominated by soft limits from higher N-point functions induced by the background, resulting in an influence from the background on observed parameters. For example, the shift to the observed power spectrum – Eq. (2.36) – comes from configurations of the global squeezed bispectrum where the long mode cannot be resolved in a subvolume. One might expect the shapes of lower N-point functions to be modified by soft limits of higher N-point functions. For the local ansatz one might think that arbitrarily non-linear terms in ζ could give arbitrary k-dependence to N-point functions. Then the usual local-shape N-point functions could be recovered in sufficiently biased small subsamples from very different shapes in the large volume. In particular, for the strongly non-Gaussian case of section 2.4, N-point functions will involve many loops (momentum space integrals8), whereas in subvolumes for which a dominant linear term is regenerated, the dominant shape will come from tree diagrams. We

8Although these momentum integrals may be represented diagrammatically with loops – see sectionA – they should not be confused with quantum mechanical loop corrections to correlators. In particular, the latter also involve time integrals. The loops we encounter are not from dynamics, but merely add up the effects of all the modes coupled to a mode of interest at a particular time.

57 will see, however, that for local non-Gaussianity loop integrals introduce at most very mild additional momentum dependence. The spatial locality of the nonlinear transformation on the Gaussian field ensures that all squeezed limits depend the same way on long wavelength modes, so shapes will remain unchanged. p+1 Let us consider the p-loop two-point function for the case ζ = ζG . The power spectrum is defined by

p+1 p+1 3 3 p−loop h(ζG )k1 (ζG )k2 i = (2π) δ (k1 + k2)Pζ , (2.86) and scales as   Z p ! p−1 p-loop Y 3 1 Y 1 1 Pζ ∝ d pi 3  3  3 . (2.87) i=1 |k − pp| i=1 |pi+1 − pi| p1

After evaluating m such integrals starting from the right, with a momentum cutoff −1 −1 L for all factors in denominators and taking the limit L  pi  kmax, an m p-loop −3 p additional factor of ln (pm+1L) appears, giving Pζ ∝ k ln (kL). Additional terms are also introduced, but either have weaker momentum dependence or can be discarded in the limit pi/kmax  1. The appearance of the scale L in these expressions should not be interpreted as measurability of L, since its value is completely degenerate with the amplitude of the power spectrum, the spectral index, and analogous quantities for higher order correlation functions (see, eg, [109]). This analysis can be generalized to the three-point and higher N-point functions; an N-loop contribution to the bispectrum will involve terms of the form [110]

1 m1 m2 m3 3 3 ln (k1L) ln (k2L) ln (min(k1, k2)L) + perms., (2.88) k1k2

P where mi = n and m1,2,3 are the number of loops coming from contractions between different pairs among three terms in the series contributing to the bis- pectrum. In the squeezed limit, k1 → 0 and k2 ' k3, only terms with m1 = 0 −3 will contribute, so the squeezed limit will still be characterized by the usual k1 dependence. We conclude that a local ansatz with arbitrarily fine-tuned coefficients

Nn can contribute additional logarithmic k-dependence to N-point functions, but the behavior in the squeezed limit remains unchanged. The question of shape is also more complex for higher N-point functions in that

58 there are more tree level shapes. For the local model, there are two trispectrum shapes typically discussed (see Eqs. (B.12)-(B.13)):

54 T = g P (k )P (k )P (k ) + 3perms. g 25 NL G 1 G 2 G 3 1 T = τ P (k )P (k )P (|k + k |) + 23perms., τ 2 NL G 1 G 2 G 1 3

6 2 In our case τNL = ( 5 fNL) . One might worry that coupling to a long-wavelength background can change the tree-level shape contributing most strongly to the trispectrum or a higher correlation function. To see that this is not the case, consider a fine-tuned local ansatz,

1 ζ(x) = ζ (x) + N (ζp (x) − hζp i) , (2.89) G p! p G G with a dominant Gaussian term, and nonlinear terms with powers smaller than p set to zero. For this ansatz the global four-point function is9

1 hζ ζ ζ (N ζp ) i + 3 perms. (odd p), p! G,k1 G,k2 G,k3 p G k4 which has the same momentum dependence as the Tg shape, up to the logarithmic loop corrections discussed above. In typical subvolumes, cubic and quadratic terms will be regenerated with

obs 2 (fNL ) h 2 (p−1)/2ip−1 obs ∼ NphζGi  1, (2.90) gNL so that again the Tg shape will dominate the Tτ shape. Consequently, the tree-level shape that dominates the trispectrum is not changed by the influence of background modes. p p The same is true in the strongly non-Gaussian case ζ ≈ ζG −hζGi. In subsamples obs  obs 2 the quadratic and cubic terms will satisfy gNL = O (fNL ) (assuming p > 2), so the two shapes contribute equally. In the large volume this is also true because the loop integrals in the trispectrum can be contracted diagrammatically in different

9 p+1 If p is even, the leading contribution to the trispectrum comes from the ζG nonlinear term instead.

59 ways, contributing terms that approximate both tree level shapes [110].10 For higher N-point functions, the same argument holds: nonlinear terms will regenerate lower order terms, allowing additional shapes to contribute in subvolumes, but these contributions will not exceed the original (global) shape(s) in amplitude.

2.8 Conclusions

Local type non-Gaussianity couples the small-scale statistics measured by an observer restricted to a small volume Vs to the unobservable, long wavelength modes ζGl that are nearly constant across Vs. In this paper we have systematically calculated the relationship between local and global statistical quantities (the power spectrum, bispectrum, and trispectrum) in models with local-type primordial non- Gaussianity. We demonstrate through explicit calculation, that for homogeneous and isotropic curvature perturbations in a large volume characterized by an arbitrary set of local terms, one recovers a weakly non-Gaussian series in typical subvolumes on sufficiently small scales. Broad classes of statistical distributions for the curvature perturbation in the larger universe Vl are thus consistent with nearly Gaussian statistics observed in our Hubble volume. This many-to-one nature of the mapping between statistics in Vl and Vs is potentially a challenge for using statistics measured in our Hubble volume to infer the statistics in the entire universe. We have studied three examples in detail: the usual local ansatz in section 2.3, an example with strongly non-Gaussian initial conditions coming from a single field in section 2.4, and finally a two-field example in section 2.5. The framework outlined in section 2.2, however, is general, and we applied it to generic local non-Gaussianity in section 2.6 (see also and AppendixB). For the weakly non-Gaussian statistics for ζ in section 2.3 we find, in agreement with [69], that values of the non-Gaussian parameters fNL, gNL consistent with current constraints can cause the statistics measured within our Hubble volume to differ from those in the larger universe, even for a modest number of super horizon e-folds (N ∼ O(10), say). This is illustrated in Figures 2.2 and 2.3. These results are dependent on the unknown behavior of the curvature power spectrum on super-horizon scales (examples for different IR extrapolations of the power spectrum

10 As an exception, for p = 2 the τNL shape dominates in both volumes because the quadratic term is abnormally large compared to the cubic term.

60 are plotted in Figure 2.1). Figures 2.2 and 2.3 assume that the power law spectrum −N on super-horizon scales remains unchanged out to k/H0 ∼ e , which may be 2 false – the true behavior of ∆ (k) for k < H0 could increase or decrease the typical amplitude of ζGl. Typical changes to the level of non-Gaussianity in Hubble-size subvolumes are plotted in Figure 2.4 for both ns = 0.9608 and ns = 1. In a universe with local non-Gaussianity, constraints on global statistics (and therefore inflationary parameters) from observations in our Hubble patch are necessarily probabilistic because the locally observed power spectrum, bispectrum, trispectrum are dependent upon the unknown value of the random variable ζGl. While this has been known for a long time in the context of slow-roll inflation [65,111–113], we have shown that for inflationary models with local non-Gaussianity – either strong non-Gaussianity or merely observable levels of non-Gaussianity – the probabilistic relationship between theory and observations is important. p In section 2.4 we consider strongly non-Gaussian statistics, ζ(x) ∼ ζG in the larger universe Vl that can appear only weakly non-Gaussian on sufficiently large background fluctuations ζ . We determine the restrictions on p, ∆2 and ζ to Gl ζG Gl produce statistics consistent with observations in our Hubble volume. The main results are illustrated in Figure 2.5. We see that for sufficiently large N, typical q 2 subvolumes (e.g. ζGl/ hζGli ∼ 1, corresponding to ∼ 30% of Hubble-sized patches in the universe) will have statistics consistent with constraints on parameters in the weakly non-Gaussian ansatz Eq. (2.1), even if the curvature perturbation in the rest of the universe is strongly non-Gaussian. At very large N, this can be true of the vast majority of subsamples (not only 1σ and higher fluctuations) depending on the infrared behavior of the power spectrum. In this sense weakly non-Gaussian statistics may be considered ‘natural’. Furthermore, the dimensionless non-Gaussian cumulants measured in biased subvolumes will follow a hierarchical n−2 scaling Mn+1 ∼ M3 , indicating once again the statistical naturalness of the weakly non-Gaussian local ansatz, which generates the same scaling. While section p 2.4 focuses on initial conditions that are a single power law ζG the qualitative results should hold for more general forms of strongly non-Gaussian initial conditions and we provide a framework for these calculations in AppendicesA andB. We qualify this result by noting that under the assumption of a perturbative long-wavelength obs background, |ζ|  1, Eq. (2.50) implies that |fNL |  1 for all subvolumes, so a residual non-Gaussianity will remain at an observationally accessible level. We will

61 comment more on this point in chapters3. In section 2.5 we consider an example in which the initial curvature perturbation is given by a sum of two uncorrelated fields, one Gaussian φ and one non-Gaussian ˜ 2 2 with a quadratic coupling, σ = σG(x) + fNL(σG(x) − hσGi). Initial conditions ˜ of this type are consistent with observations for a range of values of fNL and/or 2 ξ = Pφ/Pσ. The qualitative difference between this scenario and those in section

2.3 and section 2.4 is that the field that modulates the local statistics, σGl, is only partially correlated with the total curvature perturbation ζ = φ + σ. There is therefore greater freedom in finding statistics in Vl that map to weakly Gaussian statistics in Vs. For σ only weakly non-Gaussian, the results are similar to those in 2 section 2.3 but, for fixed fNL and ∆ζ , the typical size of the modulation in local q statistics is enhanced by a factor 1 + Pφ/Pσ (see Figure 2.6). On the other hand, ˜ 2 1/2 if fNLhσGi ∼ 1, the results are qualitatively different: the probability distribution for observed values of fNL, τNL is highly skewed (see Figure 2.7). In section 2.6, we studied local-type non-Gaussianity in its most general form, where ζ is a polynomial in ζG. We saw that non-Gaussian parameters “run” with the background ζGl or equivalently the bias B in a way that can be described with a simple set of differential equations. Lastly, although the amplitude of non-Gaussianity is volume-dependent, we saw in section 2.7 that the form of the local ansatz is protected against changes of scale: although the finiteness of the observable universe means a one-to-one map between observations and theory parameters may not be possible, subsampling does not lead to correlation functions with arbitrary shape in momentum space. In particular, the characteristic squeezed-limit behavior of the local bispectrum cannot be erased by fine-tuning the coefficients, and is therefore a reliable observational signal of local non-Gaussianity even if the universe is larger than what we observe. We have not attempted to understand how differences between local and global statistics alter inferences about particular inflationary scenarios. The parametric forms of initial conditions we have considered in section 2.3, section 2.4, and section 2.5 are simplified examples of initial conditions that can arise in the curvaton, or 2 modulated reheating scenarios, but we have assumed that the parameters (∆G and the coefficients of the non-linear terms, for instance) can be freely adjusted to tune the statistics in Vs, which is not necessarily the case. In chapter5 we will comment on inflationary models as they relate to our results, and discuss the relevance of our

62 results for relating observations to parameters describing the physics of inflation. Throughout this paper we have made the simplifying assumption that the background mode ζGl is precisely constant across our Hubble volume (see Eq. (2.23)).

In reality slight variations in ζGl from modes with wavelengths not too much larger than c/H0, and correlations between these variations and the small-scale statistics of ζ may be detectable [55, 98–102, 114]. While the assumptions we have made in Eq. (2.23) should be sufficiently precise for sub-horizon scales k  H0, and contributions to ζGl from k  H0, it would be interesting to explore the potentially observable corrections for k ∼ H0.

63 Chapter 3 | Super Cosmic Variance with Scale- Dependence

3.1 Introduction

In Chapter2 we considered the influence of superhorizon modes on the ob- served power spectrum and amplitude of non-Gaussianity in the context of lo- cal non-Gaussianity. We saw that even with globally small fluctuations, weak non-Gaussianity, and of order 10-100 extra e-folds of inflation, the amplitude of

fluctuations and the amplitude of non-Gaussianity (the observed fNL) can vary significantly throughout the entire post-inflationary volume. Also important for understanding the early universe are observables describing scale-dependence. In this chapter we consider mode coupling to a long-wavelength background with weakly scale-dependent parameters in the local ansatz. We focus on a generalization of the local ansatz above, allowing fNL, gNL, etc. to be scale- dependent. In that case, the curvature fluctuations measured in subvolumes do not all have the same spectral index and bispectral indices as the parent theory. In other words, the possibility of mode-coupling, even at a level consistent with Planck bounds on non-Gaussianity, relaxes the restrictions that the precisely measured red tilt places on the theory of the primordial fluctuations. We will see that the observed spectral index can vary around the global average at a significant level, |∆ns| ' 0.04. This can occur when ζ receives contributions with different running, and when a long-wavelength background can adjust the relative amplitude of these contributions in a subvolume, modifying the spectral

64 index over a finite range of scales. We work with curvature perturbations that are assumed to be output by some inflationary model, and the mode-coupling effects we discuss are only significant if modes of sufficiently different wavelengths are physically coupled. Purely single field models of inflation do not generate such a coupling [33, 37, 43]. We will demonstrate how to see directly from the shape of the bispectrum that single field type bispectra do not lead to cosmic variance from subsampling. So, although the curvature perturbations with local-type non-Gaussianity may have a single source, the inflationary scenario they come from must be multi-field. In section 3.2 we introduce a two-source ansatz with scale-dependence that changes the momentum dependence of the correlation functions. We compute features of the power spectrum and bispectrum observed in subvolumes and show obs that the locally observed spectral index of primordial scalar perturbations (ns ) can be shifted by scale dependent coupling to modes that are observationally inaccessible. In section 3.3 we illustrate how cosmic variance from mode-coupling affects the relationship between observation and theory for the spectral index and the amplitude of the power spectrum and bispectrum. There we illustrate, for example, that a spectrum which is scale-invariant on observable scales may look locally red or blue, and a red spectrum may look locally redder, scale-invariant, or blue when scale-dependent non-Gaussianity is present. It is unlikely to find subvolumes with an observed red tilt inside of a large volume with nearly Gaussian fluctuations with a blue power spectrum. However, a large volume with a blue power spectrum on observable scales due to a significant non-Gaussian contribution may have subvolumes with power spectra that are nearly Gaussian and red. We similarly show that the scale dependence of the bispectrum and higher order spectra in our Hubble volume can be shifted by non-Gaussian correlations with modes that are observationally inaccessible. Section 3.4 calculates the effect of a generic factorizable bispectrum on the amplitude and scale dependence of the power spectrum in subvolumes and verifies that not all bispectra lead to a variation in the locally observed statistics. The effects of mode coupling on the power spectrum and spectral index of tensor modes is considered in section 3.5. We summarize our results in section 3.6 and suggest future observational limits that could rule out the need to consider these statistical uncertainties in using observations to constrain (the slow-roll part of) inflation

65 theory.

3.2 Subsampling the local ansatz with scale-dependence in single- and multi-source scenarios

3.2.1 The power spectrum

We introduce a generalized local ansatz for the curvature perturbation, similar to the two-source ansatz of section 2.5, but with scale-dependent coefficients,1

3 9 ζ(x)=φ (x) + σ (x) + f ? [σ (x)2 − hσ (x)2i] + g ? σ (x)3 + ... (3.1) G G 5 NL G G 25 NL G where the dots also contain terms that ensure hζ(x)i = 0. We have defined the fields

φG and σG to absorb any coefficient of the linear terms, which typically appear, for example, in relating the inflaton fluctuations to the curvature. The Fourier space field, which we will use to do the long and short wavelength split, is

3 Z d3p ζ(k) = φ (k) + σ (k) + f (k) σ (p)σ (k − p) G G 5 NL (2π)3 G G 9 Z d3p d3p + g (k) 1 2 σ (p )σ (p )σ (k − p − p ) + ... (3.2) 25 NL (2π)3 (2π)3 G 1 G 2 G 1 2

For most of the paper, we will take fNL, gNL, etc. to be weakly scale-dependent functions:

k !nf fNL(k) = fNL(kp) (3.3) kp k !ng gNL(k) = gNL(kp) . (3.4) kp

To determine the mapping between statistics in subsamples to those in the large volume, we follow the same procedure as in section 2.2, splitting the Gaussian part of the curvature perturbation into long and short wavelength parts: φG ≡ φGl +φGs, ? σG ≡ σGl + σGs. The division happens at the intermediate scale k , which we

1 ˜ Note that the parameter fNL as defined here corresponds to the parameter denoted as fNL in the scale-invariant two-field model considered in chapter2.

66 take to be roughly the largest sub-horizon scale today. Splitting into short and long wavelength parts results in a splitting of the convolution integrals. The non-

Gaussian curvature perturbation is also split into ζl and ζs. As described in section

2.2, the local background ζl shifts the amplitude of the fluctuations as defined in obs the small volume, ζ = ζs/(1 + ζl), but for our purposes it is safe to neglect this obs 2 shift, so ζ ' ζs. Carrying out the long- and short-wavelength split, we have

 6 27  ζobs = φ + 1 + f (k)σ + g (k)σ2 σ k Gs,k 5 NL Gl 25 NL Gl Gs,k 3 27  9 + f (k) + g (k)σ (σ2 ) + g (k)(σ3 ) + ..., (3.5) 5 NL 25 NL Gl Gs k 25 NL Gs k where we have neglected corrections from quartic and higher terms. We see that the scale-dependent coefficients are corrected by long-wavelength pieces from higher terms and may scale differently in the small volume.

With the assumptions that the two fields are not correlated, hφk1 σk2 i = 0, and neglecting gNL and higher order terms, the total power spectrum in the large volume is

Z kmax 3 18 2 d p Pζ (k) = Pφ(k) + Pσ(k) + fNL(k) Pσ(p)Pσ(|k − p|) 25 L−1 (2π)3  36    ' P (k) + 1 + f 2 (k) hσ2 i + hσ2 (k)i P (k), (3.6) φ 25 NL Gl Gs σ where Pφ and Pσ are the power spectra for the Gaussian fields φG and σG, and 2 R k d3p 2 R k∗ d3p we identify hσGs(k)i = k∗ (2π)3 Pσ(p), hσGli = L−1 (2π)3 Pσ(p), and for future use we 2 d ln ∆2 2 2 2 d ln ∆σ φ define hσG(k)i = hσGli + hσGs(k)i. We also define nσ ≡ d ln k and nφ ≡ d ln k for 2 k3 2 k3 future reference, where ∆σ(k) ≡ 2π2 Pσ(k) and ∆φ(k) ≡ 2π2 Pφ(k). In the second line of (3.6) we have split the integral of the first line at the scale k∗ after using the approximation [115–117],

3 3 Z pmax d p Z k d p Pσ(p)Pσ(|k − p|) ' 2Pσ(k) Pσ(p). (3.7) L−1 (2π)3 L−1 (2π)3

2 R d3p Note that splitting the momentum space convolution integrals yields factors of (2π)3 σGl(p), 3 R d p ip·x0 which can be set equal to σGl(x0) = (2π)3 σGle in the long-wavelength limit p · x0  1, up to O(p) corrections, which we neglect. Here, x0 is the location of the subvolume.

67 The fractional contribution of the non-Gaussian source to the total power is3

Pσ,NG(k) ξm(k) ≡ , (3.8) Pζ (k)

3 18 2 R kmax d p where Pσ,NG(k) ≡ Pσ(k) + 25 fNL(k) L−1 (2π)3 Pσ(p)Pσ(|k − p|) includes all contri- butions from the σ sector of the perturbations. In the weakly non-Gaussian regime,

ξm(k) ≈ Pσ(k)/Pζ,G(k) ≡ Pσ(k)/(Pσ(k) + Pφ(k)). This ratio is a weakly scale- dependent function if the power spectra are not too different, so we parametrize it as n(m)(k) k ! f ξm(k) = ξm(kp) . (3.9) kp The superscript (m) indicates a running from multiple sources, and we use a subscript f because, like the running of fNL, this running will affect the bispectrum scale-dependence. Splitting off the long-wavelength background in Eq. (3.5), the curvature observed in a subvolume is

 12 36  P obs(k) = P (k) + 1 + f (k)σ + f 2 (k)σ2 P (k) ζ φ 5 NL Gl 25 NL Gl σ Z kmax 3 18 2 d p + fNL(k) Pσ(p)Pσ(|k − p|) 25 k∗ (2π)3  12 36    ' P (k) + 1 + f (k)σ + f 2 (k) σ2 + hσ2 (k)i P (k) (3.10) φ 5 NL Gl 25 NL Gl Gs σ

Note that we recover Eq. (3.6) by averaging over the background σGl. This expression shows that the local power on the scale k in typical subvolumes may be nearly Gaussian even if the global power on that scale is not. In other words, 36 2 2 2 2 consider Eq. (3.6) in the case that 25 fNL(k)hσGli > 1 and hσGli  hσGs(k)i. The field σ on the scale k is strongly non-Gaussian. However, in subvolumes with 2 2 σGl ' hσGli the contribution to the local power spectrum quadratic in σGl (the last term in the first line of Eq. (3.10)) will give the dominant contribution to the 2 Gaussian power while the local fNL term (the term in the second line of Eq. (3.10)) can be dropped. The locally observed σ field on scale k is weakly non-Gaussian. When the locally observed field is nearly Gaussian (although the global field

3 p Note that in chapter2 the parameter ξ was defined differently, as ξ = Pφ/Pσ.

68 need not be) the observed relative power of the two sources will vary in small volumes and is given by

  6 2 36 2 2 obs  (1 + 5 fNL(k)σGl) + 25 fNL(k)hσGs(k)i  ξm (k) = ξm(k)    .  36 2 2 12 3 2 2  1 + 25 fNL(k)hσG(k)i + 5 ξm(k)fNL(k) σGl + 5 fNL(k)(σGl − hσGli) (3.11) obs obs Notice that hξm (k)i = ξm(k), and that ξm(k) = 1 implies ξm (k) = 1.

3.2.2 The bispectrum and the level of non-Gaussianity

From the generalized local ansatz in Eq. (3.1), the large volume bispectrum is

6 B(k , k , k ) = f (k )ξ (k )ξ (k )P (k )P (k ) + 2 perms. + O(f 3 ) + ... 1 2 3 5 NL 3 m 1 m 2 G,ζ 1 G,ζ 2 NL (3.12) where the total Gaussian power, PG,ζ , comes from ζG ≡ φG + σG. The terms proportional to three or more powers of fNL (evaluated at various scales) come both from the contribution from three copies of the quadratic σG term from Eq. (3.1) and from the conversion between PG,σ and PG,ζ . Those terms may dominate the bispectrum if the model is sufficiently non-Gaussian over a wide enough range of scales. The same quantity as observed in a weakly non-Gaussian local subvolume is

  obs obs obs 6 9 ξm (k1)PG,ζ (k1) Bζ (k1, k2, k3) = fNL(k3) + gNL(k3)σGl 6 27 2 5 5 1 + 5 fNL(k1)σGl + 25 gNL(k1)σGl obs obs ξm (k2)PG,ζ (k2) × 6 27 2 + 2 perms. + ... (3.13) 1 + 5 fNL(k2)σGl + 25 gNL(k2)σGl where the ... again denote terms proportional to more power of fNL. Comparing this expression to the previous equation in the weakly non-Gaussian regime, the observed bispectrum is again a product of functions of k1, k2, and k3. However, those functions are no longer equivalent to the Gaussian power and ratio of power in the two fields that would be measured from the two-point correlation. In other words, the coupling to the background not only shifts the amplitude of non-Gaussianity, but can also introduce new k-dependence which alters the shape of the small-volume bispectrum. Although a full analysis of a generic local type non-Gaussianity would

69 be very useful, for the rest of this section we set gNL and all higher terms to zero for simplicity. These bispectra now have a more complicated shape than in the standard local ansatz, but for weak scale-dependence they are not still not too different. In practice one defines an fNL-like quantity from the squeezed limit of the bispectrum:

obs 3 eff 1 Bζ (kl, ks, −kl − ks) fNL(ks, kl) ≡ lim , (3.14) k →0 obs obs 5 4 l Pζ (kl)Pζ (ks)

obs obs obs eff where Pζ and Bζ are defined in terms of ζ . The definition of fNL in Eq. (3.14) is imperfect in any finite volume, since we cannot take the exact limit kl → 0.

Instead, we must choose the long and short wavelength modes (kl  ks) from within some range of observable scales. Since the best observational constraints over the widest range of scales currently come from the CMB, we will fix kl and ks in terms of the range of angular scales probed by Planck and define

CMB eff fNL (ks, kl) ≡ fNL(ks, kl)|ks=kCMB max, kl=kCMB min . (3.15)

The observed non-Gaussianity in a subvolume can then be expressed in terms of the large volume quantities as

   6 fNL(ks)ξm(ks)ξm(kl) 1 + 5 fNL(ks)σGl ks → kl CMB fNL =    , 36 2 2 12 3 2 2 1 + 25 fNL(ks)hσG(ks)i + 5 ξm(ks)fNL(ks) σGl + 5 fNL(ks)(σGl − hσGli) ks → kl (3.16) where (ks → kl) indicates the same term as the preceding, except with ks replaced by kl, and this expression is evaluated with kl, ks equal to the limiting wavemodes observed in the CMB. As in the discussion below Eq. (3.10), this expression is valid obs even for fNL(k)σGl & 1 as long as we can neglect the 1-loop contribution to Pζ ; this must always be the case for our observed universe with very nearly Gaussian CMB statistics. Keep in mind that fNL is not a small volume version of the parameter fNL(k) defined in Eqs. (3.1), (3.2), which is a function of a single scale. Rather, CMB fNL corresponds to the observed amplitude of nearly local type non-Gaussianity over CMB scales for given values of fNL, ξm, ks, kl, σGl. In the discussions above, we have considered the case when modes of a particular

70 scale k may be strongly coupled (the term quadratic in fNL(k) dominates in P (k)). However, it is also useful to have a measure of total non-Gaussianity that integrates the non-Gaussian contributions on all scales. For this we use the dimensionless skewness hζ3(x)i M ≡  1. (3.17) 3 hζ2(x)i3/2

There are two important things to notice about this quantity compared to fNL(k) CMB in the local ansatz itself and fNL (ks, kl) as defined in Eq. (3.16). 2 1/2 First, fNL(k)hσGli & 1 does not necessarily imply M3 > 1, even in the single source case. The behavior of the power spectrum and bispectrum over the entire hζ3(x)i span of superhorizon and subhorizon e-folds enter M3. Evaluating M3 ≡ hζ2(x)i3/2 in the single-source case (σG → ζG, nσ → nζ , as defined in section 2.2) for a scale-dependent scalar power spectrum, nζ 6= 1, yields

1/2 Nsub(ns−1) ! −N(nf +2nζ −2) 36 2 1/2 e − 1 (nζ − 1)e M3 ' fNL(kl)hζ i 1 + Gl −N(nζ −1)  2 5 1 − e 1 − e−(N+Nsub)(nζ −1) " # e(N+Nsub)(nf +2nζ −2) − 1 e(N+Nsub)(nf +nζ −1) − 1 × − , (3.18) (nf + 2nζ − 2) (nf + nζ − 1) where we have used the approximation shown in Eq. (3.7), and neglected 1-loop and higher contributions to hζ3i and hζ2i. Splitting the total e-folds on a scale appropriate for our cosmology, the number of subhorizon e-folds is Nsub = 60. For a scale-invariant spectrum, nζ = 1, the expression above becomes

36  N 1/2 1 M 2 1/2 sub 3 ' fNL(kl)hζGli 1 + 2 2 (3.19) 5 N nf (N + Nsub) h i nf Nsub −nf N × e (−1 + nf (N + Nsub)) + e .

For the scale-independent case, nf = 0, this reduces to

18 M = f ∆ (N + N )1/2 . (3.20) 3 5 NL ζ sub

Notice that for nζ = 1 and nf < 0 (the case of increasing non-Gaussianity in the

IR), M3 grows rapidly with N. On the other hand, if the power spectrum has a red tilt, nζ < 1, M3 will stay small for a wider range of nf values.

71 The second thing to keep in mind about the Mn is that the series gives a more CMB accurate characterization of the total level of non-Gaussianity than fNL or M3

alone would. The level of non-Gaussianity as determined by Mn+1/Mn is also what controls the size of the shift small volume quantities can have due to mode coupling. For example, in the two-field case the quantity controlling the level of

non-Gaussianity of ζ is ξmfNL∆ζ , where ξm(k) is the fraction of power coming from

σG in the weakly non-Gaussian case. This quantity determines the scaling of the dimensionless non-Gaussian cumulants,

hζ(x)ni M c n−2 n ≡ 2 n/2 ∝ ξm [ξmfNL∆ζ ] . (3.21) hζ(x) i kp

(We specify the scale-dependent functions at some pivot scale as the cumulants involve integrals over these functions at all scales.) Recall from chapter2 that the

quantity ξmfNL∆ζ as defined in a subvolume differs from the large-volume quantity due to coupling to background modes:

 6  ξ f ∆ | = ξ f ∆ 1 − ξ f hζ2 iB , (3.22) m NL ζ obs m NL ζ 5 m NL G

where we have suppressed the scale-dependence, and the bias is now defined as

2 1/2 B ≡ σGl/hζGi , (3.23)

so that it is larger when σ, which biases the subsamples, is a larger fraction of the curvature perturbation.

3.3 Observational consequences

In this section we illustrate the range of large-volume statistics that can give rise to locally observed fluctuations consistent with our observations. In considering the relationship between Planck CMB data and inflation theory, we set the scale of ? the subvolume to be k ≈ H0.

72 3.3.1 The shift to the power spectrum

Expressed in terms of the large volume power spectrum Eq. (3.6), the small volume power spectrum Eq. (3.10) is

" 3 2 2 2 # obs 12 fNL(k)σGl + 5 fNL(k)(σGl − hσGli) Pζ (k) = Pζ (k) 1 + ξm(k) 36 2 2 , (3.24) 5 1 + 25 fNL(k)hσG(k)i

In the single field, scale-independent, weakly non-Gaussian limit, ξm = 1 and

fNL = const., and Eq. (3.24) reduces to Eq. (2.36). The shift to the local power 2 1/2 spectrum is proportional to the level of non-Gaussianity ξm(k)fNL(k)hζGi coupling subhorizon modes to long-wavelength modes. We will see in section 3.3.2 below 2 1/2 that if mode coupling is weaker on superhorizon scales, ξm(k)fNL(k)hζGi & 1 can be consistent with weak global non-Gaussianity. 6 Depending on the value of ξm(k) and on the biasing quantity 5 fNL(k)σGl on the scale k, this shift is approximately

 obs  1 + 12 ξ (k)f (k)σ , 6 f (k)σ  1 Pζ (k)  5 m NL Gl 5 NL Gl ≈  2 2  (3.25) σGl−hσGli 6 Pζ (k)  1 + ξm(k) 2 , fNL(k)σGl  1  hσG(k)i 5

6 In the 5 fNL(k)σGl  1 limit, the shift to the observed power comes from the O(fNLσGl) term, which increases or decreases the power from the field σ. In addition, the spectral index can change if the non-Gaussianity is scale-dependent (note the

additional k-dependence from the fNL(k)σGl term in Eq. (3.10) as compared to Eq. (3.6)). New scale dependence can also be introduced if there are two sources 6 contributing to ζ and one is non-Gaussian. In the 5 fNL(k)σGl  1 limit, where the global power Pσ(k) on subhorizon scales is dominated by the 1-loop contribution, 2 2 the O(fNLσGl) term dominates. If the size of the background fluctuation is larger (smaller) than 1σ, the power from the field σ will be increased (decreased) relative to the global average,4 but with the same scale-dependence. Consequently, a shift

in ns comes from the difference in running between the two fields: the observed obs running ns will be shifted by the running of the fields φG or σ, depending on whether the power from the field σ is increased or decreased (see Eq. (3.27) below).

4Note that there is a decrease in power in the majority of subvolumes, which is balanced by a strong increase in power in more rare subvolumes, so that the average power over all subvolumes obs recovers the large-volume power, hPζ isubvolumes = Pζ .

73 Alternatively, if fNL(k)σGl = O(1) on or near observable scales, ns can be shifted due to the relative change in power of the linear and quadratic pieces of σ; this scenario is shown below in Figure 3.4.

3.3.2 The shift to the spectral index, ∆ns Eq. (3.24) shows that the presence of a superhorizon mode background causes the 2 d ln ∆ζ 5 spectral index d ln k ≡ ns − 1 to vary between subvolumes. Taking the logarithmic derivative of Eq. (3.24) with respect to k, we find

obs ∆ns(k) ≡ ns − ns 12  (m) 3 2 2 (m)  5 ξmfNL σGl(nf + X1nf ) + 5 fNL(σGl − hσGli)(nf + X2nf ) =   , (3.26) 36 2 2 12 3 2 2 1 + 25 fNLhσG(k)i + 5 ξmfNL σGl + 5 fNL(σGl − hσGli)

d ln fNL (m) d ln ξm where from Eq. (3.3) and Eq. (3.9), nf ≡ d ln k , nf ≡ d ln k ,

36 2 2 1 − 25 fNLhσG(k)i 2 X1 ≡ 36 2 2 ,X2 ≡ 36 2 2 , 1 + 25 fNLhσG(k)i 1 + 25 fNLhσG(k)i and we have mostly suppressed the k-dependence. From either (3.25) or (3.26) we see that depending on the value of ξm(k) and on the level of non-Gaussianity 6 5 fNL(k)σGl on the scale k, this shift is approximately

 12 (m) 6  5 ξmfNLσGl(nf + nf ), 5 fNL(k)σGl  1 ∆ns ≈  2 2  (3.27) (m) σGl−hσGli 6  nf −1 2 2 2 , 5 fNL(k)σGl  1 ξm hσG(k)i+σGl−hσGli where these expressions are approximate, and in particular the single-source limit 6 (m) 5 fNL(k)σGl  1 cannot be taken simply as the nf → 0, ξm → 1 limit of Eq. (3.27). That limit requires the full expression, (3.26), from which we find that 6 2 2 in the single source case when 5 fNL(k)σGl  1 and σGl  hσGs(k)i, the correction 3 to the power spectrum vanishes, ∆ns ' −nf /( 5 fNL(k)σGl) → 0. This equation indicates that these scenarios will also in general have a non-constant spectral index. Although we have not done a complete analysis, Eq.(3.26) shows that

5 Recall that ns is the running of the total field, in contrast with nσ,φ (nζ ) for the Gaussian fields σG, φG (ζG) in the multi-source (single-source) case.

74 obs obs αs (k) ≡ dns /d ln k =6 αs(k) should generically be of order slow-roll parameters squared, which is consistent with Planck results [118]. The shift to the spectral index is thus determined by runnings in the large- volume bispectrum, the level of non-Gaussianity on scale k (the strength of mode coupling between this scale and larger scales), and the amount of bias for the subvolume, which will depend on the number of superhorizon e-folds along with the size and running of the power spectrum outside the horizon. We stress that this shift depends on the non-Gaussianity and non-Gaussian running of the statistics at the scale being measured, and does not depend directly on the superhorizon behavior of the bispectrum parameters fNL(k), ξm(k). We will see below that even if fNL(k) or ξm(k) fall swiftly to zero outside the observable volume, the shift ∆ns will be −1 significant if subhorizon modes k > H0 are strongly coupled to superhorizon modes. Note also that the bias from a given background mode does not depend on the 2 scale of the mode (except through the scale-dependence of ∆σ) as σGl simply adds up all the background modes equally. We will see in section 3.4 that this is not true for nonlocal mode coupling: infrared modes of different wavelength can be weighted differently. 6 For the purpose of model building, it should be pointed out that when 5 fNLσGl < 0, equations (3.11) and (3.26) can diverge. For instance for a single source model 2 with a hundred superhorizon e-folds (hζGsi ∼ 0), equation (3.26) is inversely 6 2 proportional to factors of (1 + 5 fNLσGl) . This would be cause for concern – naively 6 it implies extremely large corrections to the spectral index when 5 fNLσGl ∼ −1. However because of the same proportionality, Eq. (3.16) will also diverge, indicating that the subsamples in this phase space would observe extremely non-Gaussian obs statistics (fNL  10). Hence the Planck satellite’s bound on non-Gaussianity has already excluded the worst-behaved phase space for a negative combination of 6 6 2 1/2 parameters and background fluctuation, 5 fNLσGl = 5 fNLhσGli B < 0. We saw in chapter2 that strong non-Gaussianity in a large volume can be consistent with weak non-Gaussianity measured in typical subvolumes. Furthermore, for scale-dependent non-Gaussianity, large fNL(k) on a given scale can be consistent with weak total non-Gaussianity (adding over all scales). In light of this, we would like to better understand for what values of the parameters, and in particular the global spectral index and bispectral indices, it is possible for a shift |∆ns| ∼ 0.04

75 to be typical in Hubble-sized subvolumes, while satisfying the following theoretical and observational conditions:

1. ζ is a small perturbation. We will impose this by requiring that the amplitude of fluctuations is small for each term in the local ansatz.

2 obs −9 −1 2. The observed power spectrum (∆ζ ) (kp) = 2.2×10 , where kp = 0.05 Mpc [119], should be typical for subvolumes. We will enforce this condition by 2 obs setting (∆ζ ) (kp) as given in Eq. (3.24) equal to the power in a subvol- ume with a typical background fluctuation. This determines the number of 2 superhorizon e-folds, N, in terms of nζ , fNL(kp), and hσGli in the case of

single-source perturbations, while for two sources a choice of ξm(kp) is also needed to fix N.

3. The observed level of non-Gaussianity in typical Hubble-sized subvolumes is CMB consistent with Planck satellite bounds. Using Eq. (3.15) we require |fNL | ≤ 10 for a typical background fluctuation, although a more precise analysis

could be done. The maximum and minimum multipoles (lmax, lmin) = (2500, 1) CMB used to estimate fNL in [120] translate into 3-dimensional wavenumbers −1 −4 −1 kmax = 0.2 Mpc , kmin = 10 Mpc [118], in Eq. (3.16).

4. The total non-Gaussianity is weak, M3  1. Our formulae are strictly correct for scenarios where the large volume is weakly non-Gaussian on all scales, and when some scales in the large volume are strongly coupled, but in typical subvolumes weakly non-Gaussian statistics are observed. To give some sense of the regime in which our expressions are not exact, our plots will indicate the parameter ranges where the total non-Gaussianity summed

over all scales is not small, M3 ≥ 1. If smaller modes are more strongly

coupled, nf > 0, this constraint is generally weaker than the requirement of matching the observational constraints on non-Gaussianity. However, if the

long wavelength modes are strongly coupled, nf < 0, this restriction can be quite important.

A further possible criteria might be to require |∆ns| . 0.1; for larger values the obs observed near scale-invariance ns ' 1 might be an unlikely accident given the large variation in scale-dependence among subvolumes. However, Eq. (3.27) shows that

76 2 1/2 this condition is satisfied even for large fNL(k)hσGli as long as the non-Gaussian (m) runnings nf , nf are not too large, which is also necessary to preserve conditions 1 and 4 above.

Example I: Single source perturbations with constant fNL. To understand how the conditions above affect the parameter space, consider

first the simple case of single-source, scale-invariant non-Gaussianity with only fNL non-zero: 3 ζ = ζ + f (ζ2 − hζ2 i), (3.28) G 5 NL G G 2 where fNL is constant. In Figure 3.1, we show the parameter space (hζGli, fNL) consistent with ζ  1, the observed power spectrum and observed bounds on non-Gaussianity. The dashed black line divides the parameter space where the entire volume is on average weakly or strongly non-Gaussian by setting 18 2 1/2 M3 ' 5 fNLhζGi = 1. This dashed line levels off in parameter space with 2 2 very small superhorizon contributions to M3, hζGli  hζGsi (meaning subhorizon fluctuations dominate the cumulative skewness), which for a nearly scale-invariant

power spectrum implies N  Nsub. (Here and in the rest of this section, we set the number of subhorizon e-folds Nsub = 60.) When M3 & O(1), the dominant contribution to bispectrum in the large volume is given by the higher order terms not explicitly written in Eq. (3.12). The shaded region to the right of the thin gray solid lines shows where ζ is no longer a small perturbation, either due to a large linear or quadratic term. The CMB shaded region on the left shows where fNL in typical subvolumes is inconsistent with constraints from Planck. We see that in the weakly non-Gaussian regime,

consistency with Planck reduces to fNL < 10, whereas in the strongly non-Gaussian 2 1/2 2 regime the amplitude of fluctuations must be large enough, hζl i ∼ fNLhζGli & 1 10 , to sufficiently bias Hubble-sized subvolumes so that weak non-Gaussianity is typical. In this regime there is only a small window where 1σ fluctuations give CMB subvolumes consistent with observation, and requiring fNL to be a factor of CMB 10 smaller would essentially remove this small window. This is because fNL ∼ 2 CMB 1/fNLζGl ∼ 1/ζl, so if fNL is constrained to O(1), ζ is forced to be nonperturbative. Thus, for strongly non-Gaussian, scale-invariant superhorizon perturbations on a homogeneous background geometry to be consistent with observation, the degree of non-Gaussianity in our subvolume would have to exceed the observed degree of

77 Figure 3.1: Parameter space for single-source, scale-invariant non-Gaussianity. All lines and regions assume a flat spectral index, nζ = 1, except as indicated. The shaded region 2 3 2 2 2 on the right is marked off by lines where hζGli = 0.1 and h[ 5 fNL(ζGl −hζGli)] i = 0.1. The shaded region on the left shows the constraint on non-Gaussianity from Planck; outside CMB 6 2 this region, fNL = fNL/(1 + 5 fNLζGl) < 10 in subvolumes with a +1σ background 2 1/2 fluctuation ζGl = hζGli (for a −1σ fluctuation, the constraint is similar but stronger). 18 2 1/2 The dashed black line denotes M3 ' 5 fNLhζGi = 1, dividing the weakly and strongly non-Gaussian regions (here we take nζ = 1). The dotted lines, from left to right, denote curves of constant N = 350 for nζ = 1.04, 1, and 0.96.

105

104 Nonperturbative

103

102

NL CMB f fNL too large Strongly 101 NG Weakly 0 10 NG

10-1

10-2 10-8 10-6 10-4 10-2 100 2 <ΖGl > inhomogeneity, 1 part in 105. The remaining lines denote curves of constant N = 350 for different values of nζ (which we take to be constant), fixed by the requirement that the ob- 2 obs served amplitude of fluctuations be typical of subvolumes: (∆ζ ) (kp) = (1 + 6 2 2 nζ −1 −9 fNLζGl) hζ i −N(n −1) = 2.2 × 10 for a typical +1σ background fluctuation 5 Gl 1−e ζ 2 1/2 (ζGl = hζGli ). The entire unshaded parameter space is consistent with the ob- served amplitude of fluctuations, once we impose this relationship between N and   2 1/2 2 obs 2 2 nζ −1 the parameters plotted. For fNLhζ i  1, (∆ ) ≈ ∆ = hζ i −N(n −1) Gl ζ G Gl 1−e ζ is fixed by the observed power spectrum, so curves of constant N approach a fixed

78 2 2 1/2 2 obs 2 2 2 value of hζGli. On the other hand, for fNLhζGli  1, (∆ζ ) ∝ fNLζGl∆G and 2 curves of constant N approach lines of constant fNLhζGli. The variation with nζ shows that for a red (blue) tilt, a given number of superhorizon e-folds corresponds 2 to a much larger (smaller) amplitude of fluctuations hζGli. For a red tilt or flat 2 spectrum, there is a maximum number of e-folds consistent with hζGli < 1, whereas 2 obs −9 2 for a blue tilt as small as nζ − 1 ∼ (∆ζ ) ∼ 10 , hζGli will remain perturbatively

small for an arbitrarily large number of e-folds. For instance, for nζ = 1.04, having 2 more than 50 superhorizon e-folds does not appreciably change the value of hζGli 2 1/2 in the region where fNLhζGli  1 (the vertical part of the blue dashed line in Figure 3.1 will not shift right with the addition of more superhorizon e-folds).

Note that, in the case where fNL is scale-invariant, M3 is a function of super-

horizon e-folds N (Eq. (3.20)). In order to calculate the dashed line for fixed M3

in Figure 3.1 we assume nζ = 1, which along with Eq. (2.26) fixes the number of 2 superhorizon e-folds in terms of fNL and hζGli. In the strongly non-Gaussian regime, moving along the allowed window in parameter space (along curves of constant N) 2 2 does not change the amplitude of fluctuations hζ i or statistics ζ ∝ ζG, but only 2 gives a relative rescaling to fNL and hζGli. That is, requiring weak non-Gaussianity of a given size in typical subvolumes from a strongly non-Gaussian large volume singles out (in the scale-invariant case) a particular amplitude of fluctuations in the large volume, and as described above, this amplitude becomes nonperturbative obs when fNL ∼ 1 is typical. In the following section we will see how this condition can be removed in the case of scale-dependent non-Gaussianity: a blue running of

fNL implies the level of non-Gaussianity attenuates at large scales.

Example II: Single source with running fNL(k). Next, consider a single source local ansatz with scale-dependent non-Gaussianity

d ln fNL parameterized by nf ≡ d ln k . The parameter spaces for large volume statistics 2 with nf = ±0.1 and a red or scale-invariant power spectrum ∆G are shown in Figure 3.2. All plots here assume an overdense subsample with a +0.5σ background fluctuation. Remarkably, the upper left panel shows that the super-Hubble universe

could have a flat spectral index nζ = 1, while still being consistent with Planck’s observations at the Hubble scale. Conversely, the right panels demonstrate that

models with running non-Gaussianity which predict nζ = 0.96 over a super-Hubble obs volume will typically yield a range of values for ns on observable scales in Hubble-

79 sized subsamples. (The spectral index ns(kp) on observable scales is only well 6 2 1/2 approximated by nζ if 5 fNL(kp)hζGli is sufficiently small; we will see that this is still consistent with a sizeable shift |∆ns|.) CMB In these plots we require fNL < 10 for a typical background fluctuation 2 1/2 CMB ζGl = 0.5hζGli . Due to the dependence of fNL on nf this condition is slightly stronger for positive nf , which can be seen by comparing the upper and lower diagrams in Figure 3.2. On the other hand, for larger background fluctuations, 2 1/2 CMB |ζGl| > 0.5hζGli , the condition fNL < 10 excludes less parameter space.

In Figure 3.2 we compare only two types of spectral indices, nζ < 1 and nζ = 1.

While the spectral index nζ does not directly affect the parameter space constrained CMB by fNL and ζG < 1, it does have the following two effects:

1. A red tilt in the power spectrum gives superhorizon modes more power, and

biases the subvolumes more strongly (for fixed fNL(k)). Thus, a given value 2 of hζGli corresponds to a smaller (larger) number of e-folds in the case of a red tilt (blue tilt), as shown in Figure 3.1, so it is easier to realize a large shift to a global red tilt than to a global blue tilt. In fact, as previously noted in 2 obs the discussion of Figure 3.1, imposing the requirement that (∆ζ ) be typical 2 of subvolumes for scenarios with a blue tilt nζ − 1 causes hζGli to converge to a particular value as N is increased.

2. A red tilt in the power spectrum can relax the constraint from requiring weak global non-Gaussianity, as seen by comparing the right panels in Figure 3.2 to

the left panels. For example, when nf > 0, a red tilt in the power spectrum

gives more relative weight in M3 to the more weakly coupled superhorizon modes and damps the power of strongly coupled subhorizon modes. Note that the bottom two panels in Figure 3.2 permit about the same number of super-horizon e-folds of weakly non-Gaussian parameter space. In the right

panel the power removed from subhorizon e-folds by nζ < 1 is balanced by 2 power added to superhorizon e-folds leading to a larger background hζGli per e-fold as compared to the bottom left panel.

For these reasons single-source scenarios with a red power tilt in the large volume have the most significant range of cosmic variance due to subsampling.

The solid black lines in Figure 3.2 show ∆ns(kp) = −0.04 in subvolumes with 2 1/2 a +0.5σ background fluctuation (ζGl = +0.5hζGli ), and thus show part of the

80 Figure 3.2: Parameter space for single source non-Gaussian models with nf = 0.1 in the upper panels and nf = −0.1 in the lower panels. Left and right panels show parameter space for globally flat and red spectral indices, nζ = 1, 0.96. The solid black lines show ∆ns = −0.04 for +0.5σ background fluctuations and positive fNL (or −0.5σ background fluctuations and negative fNL). The dotted-dashed lines indicate 2 1/2 where fNL(kp)hζGli = 10, above which ∆ns will approach zero and ns(kp) ' nζ + 2nf . 2 The far right region, hζGli & 0.1, is nonperturbative, along with the nonperturbative 3 2 2 2 region h[ 5 fNL ? (ζGl − hζGli)] i & 0.1, which excludes parameter space for a red tilt of CMB fNL(k) (nf < 0). The upper left regions show the observational constraint fNL < 10 from Planck. The dashed curves show M3 ' 1, and thus divide weakly and strongly non-Gaussian parametrizations. The dotted lines indicate how many superhorizon e-folds 2 are implied by the choice of nf , nζ , hζGli, and fNL(kp). As discussed after Eq. (3.26) and 6 2 1/2 indicated in the upper right of the top panels, 5 fNL(kp)hζGli  1 implies ∆ns → 0. The black squares mark phase space for ∆ns probabilities plotted in Figure 3.3.

nζ = 1, nf = 0.1 nζ = 0.96, nf = 0.1 105 105 Strongly Dns > 0 Dns > 0

4 4 e

10 ns > 1.2 e 10 NG ns > 1.16

v v

i i

t t

a a

b b r

3 r 3

u u t

10 t 10

r r

e e

p p

n n o 2 o 2 L L

p p N 10 N 10 D D k Strongly n k n H s > H s > CMB -0.04 CMB -0.04 NL NG f > 10 D NL fNL > 10 f 1 NL n f 1 Dn 10 s > 10 s Weakly -0.04 Weakly >-0.04 NG NG 100 100

10-1 10-1

N = 10, 100, 103, 105, 107 N = 10, 50, 100, 200, 350 10-2 10-2 10-8 10-6 10-4 10-2 100 10-8 10-6 10-4 10-2 100 2 2 <ΖGl > <ΖGl >

nζ = 1, nf = −0.1 nζ = 0.96, nf = −0.1 105 105

104 104

103 103

2 2 L L

p 10 p 10 k k H H CMB CMB fNL > 10 f > 10 NL NL NL

f 1 f 1 10 10 Strongly Strongly Weakly Weakly NG 0 NG 0 NG 10 NG 10 Nonperturbative Nonperturbative 10-1 10-1

N = 10, 100, N = 10, 50, 100, 10-2 10-2 10-8 10-6 10-4 10-2 100 10-8 10-6 10-4 10-2 100 2 2 <ΖGl > <ΖGl > 81 parameter space where |∆ns(kp)| can be observationally significant. Here we have 2 2 neglected the subhorizon one-loop correction hζGs(kp)i  ζGl; this breaks down for small N but is valid outside of the region of parameter space excluded by the CMB requirement fNL < 10. Rewriting Eq. (3.26) for a single source scenario (ξm = 1),

  12  36 2 2  72 2 2 2 nf 5 fNLζGl 1 − 25 fNLhζGli + 25 fNL(ζGl − hζGli) single source ∆ns ' 2 . (3.29)  6   36 2 2  1 + 5 fNLζGl 1 + 25 fNLhζGli

2 1/2 Assuming nf = 0.1 and ζGl = 0.5hζGli , we can solve this equation to show that single source 2 1/2 ∆ns = −0.04 when fNL(kp)hζGli = 0.94 or 5.9, which are the equations of the two black lines plotted in Figure 3.2. These lines assume positive fNL in the large volume, fNL > 0, but they remain the same for fNL < 0 and a −0.5σ background fluctuation. For values of |nf | larger or smaller than 0.1, the distance between these lines grows or shrinks in parameter space. Of course, for the full expression of ∆ns and a different set of parameter choices, there can be more than two solutions of |∆ns| = 0.04. For positive fNL(kp)ζGl (see below), the typical 6 size of ∆ns is largest in the region between these lines ( 5 fNL(kp)ζGl ∼ 1) and falls towards zero on either side. 6 2 1/2 The upper dotted-dashed lines mark where 5 fNL(kp)hζGli is large (O(10)). −nf When that quantity is large, ∆ns ' 3 2 1/2 and thus approaches zero as 5 fNLhζGli indicated in Figure 3.2. Note that in this region the observed spectral index is obs ns ' ns ' nζ + 2nf , so for the parameter choices in Figure 3.2 the Planck satellite excludes the region above the dotted-dashed lines. All lines and contours in Figure 6 3.2 assume that 5 fNL(kp)ζGl > 0 (eg, overdense fluctuations with positive fNL). If 6 this figure assumed 5 fNL(kp)ζGl < 0 (eg, overdense fluctuations with negative fNL), 6 2 1/2 the area in parameter space near the line 5 fNL(kp)hζGli = 1 would be excluded. 6 For further discussion of parameter space with 5 fNLζGl < 0, see the discussion after Eq. (3.26). Figure 3.2 shows that, under the conditions we have imposed and the spectral indices considered, only scenarios where the bispectral tilt is not very red have typical subvolumes where the observed spectral index varies by an amount that is cosmologically interesting for us, |∆ns| & 0.01. A blue bispectral index may avoid the current observational constraints, which do not probe particularly small scales, and easily remain globally perturbative and weakly non-Gaussian (see paragraph

82 below). In contrast, the bottom panels of Figure 3.2 illustrate that for either spectral index, a scenario with nf < 0 will be nonperturbative in the interesting part of parameter space where |∆ns| ∼ 0.04. (In addition, there is only a small window with strongly non-Gaussian but perturbative global statistics.) If both the power spectrum and non-Gaussianity increase in the IR, as in the lower right panel of Figure 3.2, the statistics will be strongly non-Gaussian across parameter space for a small number of superhorizon e-folds. The upper panels of Figure 3.2 illustrate a feature discussed in section 3.2.2: 6 2 1/2 5 fNL(kp)hζGli & 1 does not necessarily imply a large cumulative skewness, M3 & 1. The dashed curves fix M3 = 1 as a function of superhorizon e-folds, which are determined at each point in parameter space by the observed level of 2 the power spectrum along with nf , fNL and hζGli. In regions where M3 < 1 2 1/2 but fNL(kp)hζGli & 1, there are a sufficient number of superhorizon modes with weaker coupling (nf > 0) damp the total non-Gaussianity. To elaborate, 2 1/2 in the limit nf (N + Nsub)  1, Eq. (3.19) gives M3 ∝ [hζGli/N(N + Nsub)] . 2 1/2 2 2 obs For fNL(kp)hζGli  1, N = hζGli/(∆ζ ) and so M3 becomes independent 2 2 1/2 of hζGli in the limit N  Nsub. For fNL(kp)hζGli  1, on the other hand, 2 3/2 2 M3 ∝ 1/fNL(kp)hζGli , so large fNL(kp) and sufficiently large hζGli are needed to 2 obs −9 keep the total non-Gaussianity small, and (∆ζ ) ∼ 2 × 10 typical in subvolumes, as seen in the upper left panel of Figure 3.2. Note that throughout this analysis, we have assumed nf is constant for all Nsub = 60 subhorizon e-folds, so that for blue nf non-Gaussianity continues to grow on subhorizon scales where nonlinear evolution has taken over. If this condition is relaxed, the conditions from weak non-Gaussianity are less restrictive.

Figure 3.3 shows the probability distribution for the shift ∆ns for the parameters in part of the range of interest for the blue bispectral index shown in the top panels of Figure 3.2. Both panels show examples that (for appropriate choices of large CMB volume parameters) give local power spectra amplitude and fNL consistent with our observations. Notice that the distribution on the right is substantially less Gaussian than the distribution on the left. This trend continues if one considers 2 larger hζGli while keeping all other parameters fixed. To conclude this section, Figure 3.4 illustrates a single-source scenario in which a power spectrum which appears blue-tilted in the large volume on short scales can appear red on the same scales in a subvolume. On scales where

83 2 4 2 3 fNL(kp)=5, ζ = 10− fNL(kp)=5, ζ = 10− Gl Gl 2 ￿3 ￿ ￿ 2 ￿4 P ￿ns for fnl￿5, nf￿0.05￿ ￿and 0.1, ￿ΖGl￿￿10 P (∆Pns￿)ns for fnl￿5, nf￿0.05 and 0.1, ￿ΖGl￿￿10 P (∆ns)

100 nf =0.05 nf =0.05 10 ￿ ￿ ￿ ￿nf =0.1 nf =0.1 10 1 1 0.1 0.1

0.01 0.01

0.001 0.001 ￿0.10 ￿0.05 0.00 0.05 ￿0.4 ￿0.3 ￿0.2 ￿0.1 0.0 0.1 ∆ns ∆ns

Figure 3.3: The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with 4 nζ = 0.96 or 4 × 10 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with nζ = 0.96 or 5 × 105 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of nf = 0.05 while the dotted blue lines show nf = 0.1. In the right panel about 24% (6%) of subvolumes in the nf = 0.1 (nf = 0.05) have ∆ns ≥ 0.02 and 17% (5%) have ∆ns ≤ −0.04. The points in parameter space that correspond to the dotted lines (nf = 0.1) are shown with black squares in Figure 3.2.

Pζ (k) ' PG(k), ns(k) ' nζ , whereas on scales where the 1-loop contribution 2 1-loop 36 2 2 2 dominates (∆ζ ) (k) ' 25 fNL(k)hζGli∆G(k) and the spectral index will be ns(k) ' nζ + 2nf . If the transition of power takes place on a scale near the 2 1/2 observable range of scales (fNL(kp)hζGli = O(1)), the observed spectral index 2 2 2 2 can be shifted. For example, if ζGl < hζGli, the blue-tilted fNLhζGli contribution

loses power in the subvolume, and if fNL(kp)ζGl > 0, the red-tilted piece gains power (compare Eqs. (3.6), (3.10)). This scenario is shown in Figure 3.4. Note

that as long as fNL(kp) is not extremely large (which would violate the constraint CMB 2 1/2 2 1/2 on fNL for the value of fNL(kp)hζGli chosen here), ζGl  hζGs(k)i and the 2 obs 2 2 1-loop contribution to (∆ζ ) is very small, suppressed by a factor of hζGs(k)i/ζGl.

Example III: Multiple sources with running ξm(k). In the single-source case, a large shift to the observed spectral index could only occur if the 1-loop contribution to the power spectrum dominated on small

scales. With two sources, a significant shift to ns can be consistent with weak 1/2 non-Gaussianity ξm(k)fNL(k)hσGli < 1 on all scales. If the running of the 1-loop

84 2 Figure 3.4: Top panel: The contributions to the power spectrum ∆G(k) and 2 1-loop 36 2 2 2 (∆ζ ) (k) ' 25 fNL(k)hζGli∆G(k) are shown, for the following parameter choices: 2 1/2 nζ = 0.95, nf = 0.05, fNL(kp)hζGli = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0.1σ background fluctuation is shown with a thick black line. The vertical scale can 2 obs be fixed so (∆ζ ) matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with nζ = 0.95 and nf = 0.05 is shown. The dotted-dashed 2 1/2 line indicates fNL(kp)hζGli = 10, both black lines indicate ∆ns = −0.065 for a +0.1σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0.1σ bias. The exclusion regions are marked the same as those in Figure 3.2, but these assume a +0.1σ bias.

ln P

PΖ

obs PΖ 1- loop PΣ PΣ

H0 k p ln k

nζ = 0.95 nf = 0.05 104

3 e v

10 i

t

a

b

r

u

t

r e

L p

p n

k o

H 2

10 N NL f D n s CMB > fNL > 10 - 0.065 101

N=10, 50, 100, 200, 350 100 10-8 10-6 10-4 10-2 100 2 <ΣGl >

85 2 d ln ∆2 (k) d ln ∆σ(k) φ contribution lies between the runnings nσ ≡ d ln(k) and nφ ≡ d ln(k) of the Gaussian contributions to the total power, then it will be subdominant on large and small scales.

The transition of power between σG and φG takes place over a finite range of

scales, over which ns changes from nσ to nφ. If the power spectrum of φG is blue and dominates on small scales (ξm(k & H0)  1), and the Gaussian contribution from σ is red and dominates on large scales (ξm(k << H0) ' 1), then the background

ζl ' σl for any subvolume couples to and biases the local statistics. For example, a obs globally flat or blue spectral index ns(k > H0) > 1 can again appear red, ns < 1,

in a subvolume. The shift to ns can come only from the modulation of power in

σ relative to φG, and need not rely on running non-Gaussianity nf 6= 0. That is, a large running of the difference in power of the fields can be achieved without a large level of running non-Gaussianity. This becomes apparent upon inspecting the

running of ξm,

 36 2 2  (m) d ln ξm(k) 2nf 25 fNL(k)hσG(k)i nf (k) ≡ = (1 − ξm(k)) nσ − nφ + 36 2 2 . (3.30) d ln k 1 + 25 fNL(k)hσG(k)i

If φG is more red-tilted than σG, the background is uncorrelated with short-

wavelength modes because φG dominates on large scales, ζl ' φGl, so local statistics

are not biased. Thus, both nσ ≤ 1 and nφ > nσ are needed for a significant bias. In Figure 3.5 we show the parameter space for the two-source scenario described above,

with nσ(kp) = 0.93, nφ(kp) = 1.005, and ξm(kp) = 0.1. We also fix nf = 0.001 so that mode coupling is weaker on superhorizon scales. As before, the upper left obs region shows where fNL & 10 in typical subvolumes. We see that adding the second 2 1/2 source relaxes the constraint on fNL in the fNLhσGli  1 regime. This makes 2 it possible to achieve a large shift ∆ns for smaller values of hσGli and thus fewer superhorizon e-folds.

The condition ξm(kp) = 0.1 makes the field φG dominant on Planck scales, so from the perspective of the large volume, the power spectrum has a blue tilt

ns(kp) ' nφ = 1.005 on scale kp. However, for significant biasing (3σ) and a small

(or zero) non-Gaussian running of the coupled field nf = 0.001, the black lines

in Figure 3.5 denote where ∆ns = −0.03, which would be consistent with Planck

observations. Here the shift in ∆ns is coming not from nf but from the difference (m) in running of Pσ,NG and Pφ, nf , as the red-tilted Pσ,NG is amplified due to the

86 Figure 3.5: Multifield parameter space for ξm(kp) = 0.1, nσ = 0.93, nφ = 1.005, nf = 0.001. The black lines show ∆ns ' −0.03 for a +3σ background fluctuation. The 2 1/2 dotted-dashed line shows fNL(kp)hσGli = 10. The upper left region shows the Planck CMB constraint on fNL for a +3σ background.

105

e

v

i

t a

4 b r

10 u

t

r e

ó p n n s o CMB > 3 fNL > 10 - N 10 0.03

L ó n p s

k > H - 0.03 NL f 102

101

100 10-8 10-6 10-4 10-2 100 2 <ΣGl >

strong background overdensity. Summary. In summary, a significant shift to the observed spectral index from correlations with long-wavelength background modes is possible under the following conditions:

1. A red tilt for the field with mode coupling, nσ ≤ 1 (nζ ≤ 1 in the single-source 2 case), is necessary for the cumulative power hσGli on superhorizon scales to be large enough to significantly bias local statistics.

2. A blue bispectral index nf ≥ 0 for fNL(k) (assuming constant nf ) is needed to remove the power from the non-Gaussian term on large scales so that strong coupling of short-scale modes to background modes is consistent with weak global non-Gaussianity and ζ being perturbative, while having enough background modes to give a large bias.

3. In a two-source scenario, the ratio of power in the non-Gaussian field to total (m) power should have a red spectrum (nf (kp) ≤ 0) so that the non-Gaussian

field σG grows relative to φG on large scales, causing the background ζl to be

sufficiently correlated with local statistics. If φG contributes on observable

87 scales (ξm(kp) < 1), larger values of fNL(kp) are consistent with observational

constraints on non-Gaussianity, so a smaller background σGl is needed to give obs the same shift to ns .

Introducing scale-dependence into the spectral indices would relax the conditions for large |∆ns|. Although the scenario becomes more complicated in this case, the qualitative features remain valid: scale-dependence of power spectra and non- Gaussian parameters must allow for sufficient cumulative superhorizon power that a large background σGl from the source with mode coupling is typical. We note that for given large-volume statistics, the observed red tilt may not be equally consistent with a local overdensity or underdensity in σG. In the single- source case with nf > 0, for example, an overdensity (underdensity) corresponds to an increase (decrease) of power on small scales. Thus, for a scale-invariant power obs spectrum in the large volume, the observed red tilt ns ' 0.96 could be accounted for in terms of a blue-tilted global bispectrum and local underdensity. However, without information about the global power spectrum, it would be difficult to infer whether we sit on a local underdensity or overdensity.

3.3.3 The shift to the scale dependence of the bispectrum

The bispectrum may also be shifted by mode coupling coming from the soft limits of the large-volume trispectrum and from any non-Gaussian shifts to power spectrum. We can define a spectral index for the squeezed limit of the bispectrum within any particular volume as

d ln Bζ (kL, kS, kS) nsq. ≡ − (ns − 1) (3.31) d ln kL where kL and kS are long wavelength and short wavelength modes, respectively. The obs small volume quantity, nsq. , should be calculated using the observed bispectrum and the observed spectral index. For a single source, scale-invariant local ansatz, nsq. = −3. For the single source, weakly non-Gaussian, scale-dependent scenario with gNL absent, the shift in this bispectral index between the large volume and what is observed in the small volume is

obs LargeVol. Single Source : ∆nsq.(k) ≡ nsq. (k) − nsq. (k) (3.32)

88 6 5 fNL(kL)σGl nf ≈ − 6 . 1 + 5 fNL(kL)σGl

6 6 2 1/2 6 2 1/2 If 5 fNL(kL)σGl = 5 fNL(kL)hζGi B  1, then ∆nsq.(k) ≈ − 5 fNL(kL)hζGi B nf . This shift is less than one in magnitude, but still relevant for interpreting bispectral indices of order slow-roll parameters. In the two source case, there can be additional scale dependence coming from the ratio of power of the two fields. Considering only the weak coupling case, 6 5 fNL(k)σGl  1 (and again setting gNL = 0 for simplicity),

12 6 5 fNL(k)σGl 5 fNL(k)σGl Two Source : ∆nsq.(k) = 12 nf − 6 nf (3.33) 1 + 5 fNL(k)σGl 1 + 5 fNL(k)σGl 12 5 ξm(k)fNL(k)σGl (m) − 12 (nf + nf ) 1 + 5 ξm(k)fNL(k)σGl 6 12 ≈ f (k)σ n − ξ (k)f (k)σ (n + n(m)) . 5 NL Gl f 5 m NL Gl f f

Reintroducing gNL and higher terms would lead to additional terms, introducing scale-dependence even if fNL in the large volume is a constant.

3.3.4 Generalized local ansatz and single source vs. multi source effects

The two source, weakly scale dependent local ansatz in Eq. (3.1) is representative of the properties of inflation models that generate local type non-Gaussianity.

For example, the scale-dependence fNL(k) can come from curvaton models with self-interactions [121,122]. The function ξm(k) comes from the difference in power spectrum of two fields (eg, the inflaton and the curvaton) contributing to the (m) curvature fluctuations. In typical multi-field models, the bispectral indices nf , nf are of order slow-roll parameters (like the scale dependence of the power spectrum), and are often not constant. Generic expressions for the squeezed limit behavior of a multi-field bispectrum are given in [123]. The scale-dependent functions fNL(k) and

ξm(k) are observationally relevant for tests for primordial non-Gaussianity using the bias of dark matter halos and their luminous tracers (eg. quasars or luminous red galaxies). The power law dependence of the squeezed limit on the long wavelength, small momentum mode (nsq. from Eq. (3.31)) generates the scale-dependence of the

89 non-Gaussian term in the bias. The dependence on the short wavelength modes generates a dependence of the non-Gaussian bias on the mass of the tracer (which is absent in the usual local ansatz). In principle, if local non-Gaussianity is ever detected, it may be within the power of future large scale structure surveys to detect some amplitude of running [106]. However, as demonstrated above, the same shape of bispectrum can be generated locally by a single source for the curvature perturbations, so the presence of the non-trivial function ξm in the observed bispectrum does not necessarily indicate that two fundamental fields contributed to the primordial curvature perturbations. On the other hand, the presence of one Gaussian source and one non-Gaussian source for the local curvature perturbations is in principle detectable by comparing power spectra that are sensitive in different ways to the total curvature field and to just the non-Gaussian part [105]. Eq. (3.13) shows that in a single source scenario the local background σGl can act as a second field to generate the full, multi-source shaped bispectrum, but σGl is constant within a single volume. This ‘second field’ does not have fluctuations on all scales, but its variations are relevant for considering a collection of subvolumes of a particular size.

3.4 Mode coupling effects from a non-local factoriz- able bispectrum

We have considered the effect of superhorizon modes only for the case of nearly local non-Gaussianity, but inflationary theory has generated an expanding space of models exhibiting different types of mode coupling. Intuitively, any scenario that does not couple modes of sufficiently different wavelengths should not lead to correlation functions whose amplitudes or shapes change under subsampling. As a first step towards considering the observational consequences of subsampling general non-Gaussian scenarios, it is straighforward to find corrections from the background to small-volume quantities in the case of a factorizable quadratic kernel in Fourier space with power-law dependence.

90 Consider a curvature perturbation in the large volume given by

Z 3 3 d p1 d p2 3 3 ζk = φG,k + σG,k + (2π) δ (p1 + p2 − k)F (p1, p2, k)σG,p1 σG,p2 + ..., L−1 (2π)3 (2π)3 (3.34) where !m1,j !m2,j !m3,j X k1 k2 k3 F (k1, k2, k3) = aNL,j(kp) (3.35) j kp kp kp is a sum of factorizable terms with power law dependence on the momenta. On the P right hand side the aj are amplitudes defined at a pivot scale kp. When i mi,j ' 0 for every term j, the bispectrum is approximately scale-invariant. The kernel

F (k1, k2, k3) can be chosen to generate a desired bispectrum with well behaved one-loop corrections to the power spectrum [124]. Splitting the modes into long and short, the locally defined short wavelength modes with shifts induced from coupling to long wavelength modes from one term in the series above are

 !m1+m3 !m2+m3  k (m2) k (m1) ζk = φG,k + σG,k + σG,kaNL(kp)  σGl + σGl  kp kp Z d3p k !m3 |k − p|!m2 p !m1 + aNL(kp) 3 σG(p)σG(|k − p|) (3.36) k∗ (2π) kp kp kp where Z k∗ 3 !mL (mL) d p p σGl ≡ 3 σp . (3.37) L−1 (2π) kp When the local field is weakly non-Gaussian, the second line is small and we can rewrite the first line as

ζks ≈ φG,ks + σG,ks [1 + ∆σ(k)] (3.38)  !m1+m3 !m2+m3  k (m2) k (m1) ∆σ(k) = aNL(kp)  σGl + σGl  . kp kp

obs The leading shift to the power spectrum Pζ in a subvolume from unobservable infrared modes in one term of the series above (and assuming weak non-Gaussianity) is:

obs n h 2io Pζ (k) = Pζ (k) 1 + ξm(k) 2∆σ(k) + ∆σ(k) ; (3.39)

91 where ξm(k) is still the ratio of power in the non-Gaussian source to the total power, defined in Eq.(3.8). In the two-field case with weak non-Gaussianity on all scales, the observed ratio of power in the two fields is related to the same ratio in the large volume by

2 obs [1 + ∆σ(k)] ξm (k) = ξm(k) 2 . (3.40) 1 + ξm(k)[2∆σ(k) + ∆σ(k) ]

The induced shift to the spectral index has two terms, but assuming that, say, the

first term in the square brackets in ∆σ is dominant and defining mS = m1 + m3,

mS m2 = mL, and aNL(k) = aNL(kp)(k/kp) it is

(mL) (m) ∆ns(k) ≈ 2aξm(k)aNL(k)σGl (nf + mS) . (3.41)

The bispectrum in the large volume is

!m3 k3 Bζ (k1, k2, k3) = aNL(kp) Pζ (k1)ξm(k1)Pζ (k2)ξm(k2) (3.42) kp " k !m1 k !m2 k !m2 k !m1 # × 1 2 + 1 2 + 2 perm. kp kp kp kp while the observed bispectrum is

!m3 " obs obs #" obs obs # obs k3 Pζ (k1)ξm (k1) Pζ (k2)ξm (k2) Bζ (k1, k2, k3) = aNL(kp) kp 1 + ∆σ(k1) 1 + ∆σ(k2) " k !m1 k !m2 k !m2 k !m1 # × 1 2 + 1 2 + 2 perm. (3.43) kp kp kp kp

Consider k1 = kL  k2 ≈ k3. If m2 < m1 (so the second term in the second line of the equation above dominates), and mS ≡ m1 + m3, then in the squeezed limit the large volume bispectrum has

(m) nsq.(k) = −3 + nf + m2 . (3.44)

The shift to the observed running of the squeezed-limit bispectrum is

∆σ(k) ∆n (k) = ∆n(m)(k) − m ≈ ∆σ m − 2ξ ∆σ(n(m) + m ) . (3.45) sq. f 1 + ∆σ(k) S S m f S

92 In the case of the generalized, two source local ansatz considered in sections 3.2 3 and 3.3.3, aNL(k) = 5 fNL(k), m3 = nf , and m1 = m2 = 0 so mS = nf , and both terms in the square brackets of ∆σ, Eq.(3.38) contribute equally, so we recover the weakly non-Gaussian limits of Eqs. (3.24), (3.26), and Eq. (3.33). As a second example, consider single field inflation (with a Bunch-Davies vacuum and inflation proceeding along the attractor solution). In this case, the squeezed limit of the bispectrum diverges with the long wavelength mode no more strongly than [33,37,42,43],

2 kL  Bζ (kL, kS, kS) ∝ O Pζ (kL)Pζ (kS). (3.46) kS

A bispectrum with this squeezed limit can be obtained by using the equilateral 2 template [125] to generate a kernel F (p1, p2, k) ∝ −3 − 2p1p2/k + 2(p1 + p2)/k + 2 2 2 (p1 + p2)/k [124]. This yields a squeezed-limit bispectrum with nsq. = −1 and 2 mL = 2 in Eq.(3.37). That is, this bispectrum generates a bias B ∝ ∇ ζGl, so there is no sensitivity of locally measured quantities to long wavelength, nearly constant modes. In single field inflation, there is a direct map between local observables and the parameters of the inflationary Lagrangian. Finally, suppose modes are coupled through a bispectrum with a very strong

squeezed-limit (eg, nsq. = −4 and mL = −1). Then the biasing of local statistics may come predominantly from background modes farthest in the infrared, which are shared by many neighboring subvolumes. In other words, the dependence of the global bispectrum on the long wavelength mode is related to the average spatial gradient of the bias the large volume.

3.5 Tensor mode running as a test of inflation?

2 d ln∆t If the scale dependence of the tensor power spectrum, nt ≡ d lnk , can someday be measured, a red tilt would be (nearly) definitive evidence for inflation and against a contracting or ekpyrotic scenario (an interesting special case is ‘solid

inflation’ [126]). Would it be possible to induce a blue tilt nt > 0 in a subvolume the size of the observable universe when the larger volume exhibits a more typical

red tilt? If so, a measurement of nt > 0 would not necessarily rule out standard

scalar field models of inflation. Conversely, if a red tilt nt < 0 can be induced

93 in a large fraction of subvolumes from non-Gaussianity in a contracting universe

scenario, a measurement of nt < 0 may not be a smoking gun for inflation . Consider a three-point interaction

X hχ γs1 γs2 i ≡ (2π)3δ3( k )B(k , k , k )δ (3.47) k1 k2 k3 i 1 2 3 s1s2

between two tensor modes γki with polarizations si and one mode from a field χ (here, a scalar field for example). In the squeezed limit, this three-point function will induce a dependence of the local tensor power spectrum on superhorizon χ modes. Any choice of the Fourier space kernel that gives the correct squeezed limit of the bispectrum should show the correct shift to the local power spectrum. So, with a simple choice we find that the tensor power spectrum is shifted by the correlation with long wavelength modes p as

Z 3 3 si si d p1 d p2 3 3 si si γk = γG,k+ (2π) δ (p1+p2−k)F (p1, p2, k)(γG,p χG,p2 +γG,p χG,p1 )+..., L−1 (2π)3 (2π)3 1 2 (3.48) where we take

!m1,j !m2,j !m3,j eff X k1 k2 k3 F (k1, k2, k3) = fγγχ(kp) aj . (3.49) j kp kp kp

For long wavelength modes of the χ field, the tensor power spectrum is shifted by

obs 2 Pγ = Pγ [1 + ∆χ(k)] (3.50)  !m1+m3 !m2+m3  eff k (m2) k (m1) ∆χ(k) = a fγγχ(kp)  χGl + χGl  kp kp Z k∗ 3 !mL (mL) d p p χGl ≡ 3 χp . L−1 (2π) kp

With this parameterization, long wavelength modes of the χ field can shift the locally observed tilt of the tensor power spectrum. In the case that the first term

in ∆χ dominates, we can again define mS = m1 + m3, mL = m2 and then the shift is approximately

∆nt(k) ≈ 2∆χ(k) mS . (3.51)

The quantity mS is zero for an exactly scale-invariant, local type model and more

94 generally cannot be too large if we want to require weak non-Gaussianity for all fields. Depending on the coupling of χ to the scalar curvature, this physics may also introduce a shift in the locally observed scalar power spectrum, the tensor- to-scalar ratio, and a ‘fossil’ signature in the off-diagonal part of the scalar power spectrum [71], which would be an interesting complementary observable. From these expressions, it looks possible to find scenarios where the locally observed tensor power spectrum would be shifted from red to blue and vice-versa, but a full analysis along the lines of Section 3.3 should be performed to check consistency with all observables.

3.6 Discussion and conclusions

The scaling of the squeezed limit of the bispectrum can also be shifted, which is relevant for constraints on non-Gaussianity from galaxy bias. These results show that in spite of the excellent precision of the measurements from the Planck satellite obs (especially ns = 0.9603 ± 0.0073 [118] and constraints on non-Gaussianity), the door is open for a significant cosmic variance uncertainty in comparing our observed patch of the universe to any particular inflation theory - even leaving aside issues with eternal inflation. Moreover, rather than just presenting a new source of uncertainty from the super-Hubble background, the correlation between bispectral running in a super-Hubble volume and subvolume power spectrum measurements reopens the door for inflationary models with flat or bluer super-Hubble spectral

indices, ns 6= 0.96, provided they also have scale-dependent local non-Gaussianity. This may be particularly useful for hybrid inflation. The numbers measured by the Planck satellite are consistent with a range of levels of non-Gaussianity in a post-inflationary volume, given a model for the statistics in that volume. For example, we recover the observed power spectrum CMB and spectral index, and satisfy current constraints on fNL for a post-inflationary volume with

• No local type non-Gaussianity, an arbitrary number of extra e-folds, and any behavior of the power spectrum on superhorizon scales.

local local • Constant fNL = 5. We observe fNL = 7 if, for example, the spectral index

is a constant ns − 1 = 0.96 over about 200 extra e-folds of inflation and our

95 Hubble patch sits on top of a 2-sigma under density.

local local • Local non-Gaussianity with constant fNL = 15. We observe fNL = 10 if, for

example, the spectral index is a constant ns − 1 = 0.96 over about 150 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma over-density.

• Scale-dependent non-Gaussianity with fNL(kp) = −2 and nf = 0.04 and

ns = 0.94. We would observe fNL(kp) = −1 and ns − 1 = 0.965 if our Hubble patch sits on top of a 2-sigma under density in a volume with about 190 extra e-folds.

• Scale-dependent non-Gaussianity with fNL(kp) = 5 and nf = 0.05 and ns = 1.

We would observe fNL(kp) = 2.5 and ns − 1 = 0.98 if our Hubble patch sits on top of a 0.5-sigma over density in a volume with about 260 extra e-folds.

In contrast, we could design an inflation model to have parameters consistent with

Planck data, say fNL(kp) = 5, nf = 0.1, and ns − 1 = 0.983. However, if the model allows about 400 extra e-folds of inflation, and our Hubble patch were to sit on a

2-sigma over density, we would observe fNL(kp) = 4 and ns − 1 = 1.02. These results demonstrate that predictions for our observations in any scenarios with local type non-Gaussianity must be given statistically. To turn the picture around, they also suggest a new route to understanding whether observations can give us any hints about the size of the universe beyond what is directly observable. Previous ideas focused on topologically finite universes (also significantly constrained by Planck [127]) or on evidence for or against a nonperturbatively connected multiverse from bubble collisions [128–130] or curvature [101,102]. We have found that on statistical grounds, a strongly non-Gaussian global universe will leave obs a residual non-Gaussianity fNL & 1 in any finite subvolume, so constraints or obs measuring fNL at this level could reveal whether super cosmic variance in a much larger universe is possible. Even if |fNL| > 1 is observed, tests for the running of the spectral index, any scale-dependence of |fNL|, and any evidence for extra fields through isocurvature modes or ‘fossil’ relics hiding in the off-diagonal power spectrum could still limit the size of any subsampling uncertainty. For example, if a blue tilt to fNL is ruled out, biasing of the spectral index is unlikely for single- source models with nζ and nf constant on all scales. Of course, making these observations statistically well-defined depends on comparing particular competing

96 models. It would be particularly interesting if those models had other cosmological implications related to the size of the universe [131]. It would also be worthwhile to investigate the generic behavior of the local ansatz with higher nonlinear terms and scale-dependent coefficients, along the lines of the analysis in section 2.6 It may be that there are statistically natural values for the spectral index in typical small subvolumes. Then, stronger conclusions about generic cosmic variance of the spectral index might be possible. However, it is already clear that if improved limits on the amplitude and scale-dependence of non-Gaussianity can be reached, we could close the window of observational access to a perturbatively connected larger universe.

97 Chapter 4 | Fossilized Gravitational Wave Relic from Squeezed Limit Coupling

4.1 Introduction

Our study of mode coupling effects thus far has focused on statistical features of non- Gaussian fields at a fixed time. In this chapter we discuss an example of coupling of modes on very different scales that can be generated by inflation. Thus far, we have considered mainly the curvature perturbation ζ(x), which are produced by inflaton and scalar metric perturbations. Although tensor metric perturbations have not been observed and do not couple to density perturbations in the post-inflationary universe at linear order, they are generically produced during inflation, and can be coupled to density perturbations during or before inflation. Consequently, although tensor modes subsequently decay upon re-entering the horizon in the late universe (when a(η)H(η) becomes smaller than k for a given mode), a primordial coupling to scalar modes can leave an observable “fossil” signature in the statistics of density perturbations, which persists in the distribution of Large Scale Structure. Here we will study a scenario in which the short-scale scalar modes are coupled to long- wavelength tensor modes even more strongly than in local-type non-Gaussianity, potentially allowing for the detection of a long-wavelength tensor background. The observation of primordial gravitational waves would be an extremely im- portant verification of the inflationary scenario and would give us a key piece of information - the energy scale of inflation. A measurement of a red tilt in the tensor mode power would be especially strong evidence for inflation itself. Further-

98 more, the characteristics of inflationary gravitational waves and the range of their observable consequences extend beyond the power spectrum of B-modes in the CMB. Primordial tensor fluctuations can also affect observed Large Scale Structure statistics [132,133] and 21-cm radiation [134,135] through intrinsic distortion of the geometry as well as lensing effects. Tensor fluctuations sourced during inflation have self-interactions as well as interactions with the scalar fluctuations, as computed in [33, 136]. Although measuring the non-Gaussianity of just the tensor modes may be out of reach for the foreseeable future [137] (or maybe not [138]), it has been suggested [71, 72, 89, 134] that in some cases the three-point correlation of two scalar modes and one tensor mode hζ ζ γij i (more generally, any primordial k1 k2 k3 “fossil” field coupled to scalar modes) may lead to an anisotropic contribution to the scalar power spectrum, which could be observed in Large Scale Structure or the CMB. In single-clock inflation the squeezed limit of this tensor-scalar-scalar bispectrum, in which short-wavelength scalar modes couple to long-wavelength tensor modes, is fixed in terms of the power spectra by a consistency relation [33, 37]. The observed bispectrum vanishes in the exact squeezed limit [43], so long wavelength modes merely rescale the local background and have only a small, infrared-suppressed effect on local statistics [139]. For the scalar perturbations alone, there are several ways to relax the single- clock conditions necessary for the consistency relation and to generate a physical coupling between long and short wavelength modes. All of them work by allowing additional dynamical freedom in the scalar fluctuations that is not associated purely to the evolution of the nearly de Sitter background. For example, the curvature mode can evolve outside the horizon if there are scalar fields other than the inflaton (isocurvature modes) or if a non-attractor phase of inflation preceded the usual slow-roll [140–144]. Allowing a non-Bunch Davies initial state for the fluctuations can also lead to a non-trivial squeezed limit for the scalar bispectrum [73, 145]. Depending on the range of initial states considered, the full bispectrum can also be enhanced in other configurations compared to single-field inflation where modes begin in the Bunch-Davies state [75,76,146–153]. There has so far not been as much theoretical effort put into trying to break the standard slow-roll relationships between the properties of the tensor fluctuations and the inflationary background, although a few proposals exist for generating a

99 blue tensor index from inflation (solid inflation [126] and ‘generalized G-inflation’ that gives the tensors an evolving sound horizon [154]). Here, to be illustrative, we will alter the typical inflationary signatures of correlations beyond the power spectrum by considering a general class of non-Bunch-Davies initial states for the tensor and scalar modes. This will allow us to explore the qualitative observational features of allowing up to three separate clocks: one for the background, one for the scalar modes, and one for the tensors. We are interested in the case with a fossil signature, which is generically accompanied by a non-negligible squeezed limit bispectrum of scalars and/or tensors. Depending on how many separate clocks are physically realizable, a “fossil” might be accompanied by related observational signals in the scalar sector (for example, the halo bias [155, 156]) and/or in the gravitational sector. The plan of this chapter is as follows: In section 4.2 we compute the fossil gravitational wave signature following the method of [71]. We begin with the three- point function with generalized initial states for the scalar and tensor modes and compute the signal-to-noise for the local anisotropy in the scalar power spectrum. We study its dependence on the occupation numbers for scalar and tensor excitations and on the range of excited modes. In section 4.3 we consider a general scalar-scalar- fossil correlation and show the dependence of the signature on the fossil power spectrum amplitude and bispectrum amplitude and squeezed-limit, considering in particular the case of primordial tensor modes. In section 4.4 we comment on the effect of superhorizon tensor modes on local anisotropy in the scalar power spectrum. We conclude in section 4.5.

4.2 A generalized initial state and the gravitational fossil

Since it is unlikely that inflation lasted forever, and since any theory of inflation is anyway likely to be only an effective description valid below some energy scale, it is quite reasonable to allow a non-Bunch-Davies initial state for primordial scalar or tensor modes. While an application of the cosmological principle suggests that it is a fine-tuning to insist that the deviation from Bunch-Davies is significant on the scales corresponding to those we observe in the CMB, it is not at all clear how

100 to put a measure on models of inflation and how long inflation might have lasted. The possibility of deviations from Bunch-Davies is an important conceptual point about the inflationary paradigm and understanding the observational possibilities, which can be ruled out, is useful in deciding which aspects of inflation theory can be robustly tested. In the scalar sector, deviations from a Bunch-Davies state at the onset of inflation can result, for example, from a previous non-attractor phase [152], a previous phase with anisotropic expansion [157,158], tunneling from a false vacuum followed by inflation within the bubble [151,153, 159], or ultraviolet completions of inflation such as loop quantum gravity [160]. An effective field theory treatment of the scalar fluctuations that parametrizes non-Gaussian effects in terms of an √ energy scale M (with Mp ≤ M > H), should also incorporate the possibility of a generic initial state [150] for modes near the scale M. Less work has been put into examples of modifications of the initial state for tensor modes. Carney et. al. [161] found that pre slow-roll dynamics for the inflaton will not affect the quadratic action for tensor modes, which will remain in the vacuum state. However, more generically it seems reasonable to allow the initial state of both tensor and scalars to be modified in independent ways. In the next section we work out the bispectrum for one tensor mode and two scalars from the standard inflationary action but allowing non-Bunch Davies states.

4.2.1 Scalar-scalar-tensor bispectrum

Consider an initial state for the scalar (ζ) and tensor (γ) fluctuations that is a general Bogoliubov transformation of the Bunch-Davies state:

(s) (s)∗ † ζk(η) =u ˜k (η)ak +u ˜k (η)a−k, p (t) p (t)∗ p† γk(η) =u ˜k (η)ak +u ˜k (η)a−k, (4.1)

† p p,† where ak, ak and ak, ak are canonical creation and annihilation operators for scalar and tensor modes respectively, p labels the graviton polarization, and u˜(s),(t) includes the Bogoliubov transformation on the scalar and tensor mode functions,

(s) (s) (s) (s) ∗(s) u˜k (η) = αk uk (η) + βk uk (η), (t) (t) (t) (t) ∗(t) u˜k (η) = αk uk (η) + βk uk (η). (4.2)

101 (s) 2 (t) Here, u (η) = H √1 (1 + ikη)e−ikη, and u (η) = H √1 (1 + ikη)e−ikη. The k φ˙ 2k3 k Mp k3 normalization conditions are

(s) 2 (s) 2 |αk | − |βk | = 1, (t) 2 (t) 2 |αk | − |βk | = 1. (4.3)

The power spectra are defined in terms of the two-point functions,

3 3 hζk1 ζk2 i = (2π) δ (k1 + k2)Pζ (k1), 0 hγp γp i = (2π)3δ3(k + k )δpp0 P p(k ). (4.4) k1 k2 1 2 γ 1

We will define η0 as the earliest (conformal) time where we expect the inflationary description to be valid. In specific scenarios with a pre-inflationary era, η0 would be the beginning of standard slow-roll inflation. Alternatively, η0 might be the earliest time when all modes of observational relevance had physical scale below some maximum energy scale M we understand, k/a(η0) . M. The scale M should be well above the Hubble scale for inflation, M  H, in order for the classical inflationary background to be valid. For modes that are well within the horizon during inflation, |kη0|  1, the scalar and tensor power spectra are, respectively,

2 H 1 (s) (s) 2 Pζ (k) = 2 3 |αk + βk | (4.5) 2Mp 2k

φ˙2 where  ≡ 2 2 , and 2H Mp

2 p 4H 1 (t) (t) 2 Pγ (k) = 2 3 |αk + βk | . (4.6) Mp 2k

(s),(t) In both cases, the β (k) → 0 sufficiently rapidly for k > a(η0)M to avoid any back reaction that would spoil the inflationary background. The tensor-to-scalar ratio is P P p |α(t) + β(t)|2 r ≡ p γ = 16 k k . (4.7) P (s) (s) 2 ζ |αk + βk | The extra factors must be consistent with the observed near scale-invariance of the scalar power spectrum but can alter the standard slow-roll consistency relations. In particular, some functions α(s),(t), β(s),(t) could suppress the tensor-to-scalar ratio

102 enough to allow high energy scale or large field models of inflation to be consistent with Planck satellite constraints [162]. Given the normalization condition Eq. (4.3) we can write

q (s) (s) 2 (s) 2 (s) 2 (s) 2 (s) |αk + βk | = 1 + 2|βk | + 2 |βk | (|βk | + 1) cos Θ , (4.8) q (t) (t) 2 (t) 2 (t) 2 (t) 2 (t) |αk + βk | = 1 + 2|βk | + 2 |βk | (|βk | + 1) cos Θ , (4.9)

(s) (s) (s) (t) where Θ is the relative phase between αk and βk , and Θ is defined similarly. During inflation the tensor (γp with two polarizations p) and scalar (ζ) fluctua- tions are coupled gravitationally, giving rise to a three point correlation

p 3 3 X hγ (k1)ζ(k2)ζ(k3)i ≡ (2π) δ ( ki)Bp(k1, k2, k3). (4.10)

Following Maldacena’s calculation [33] using the in-in formalism [163] for the Bunch- Davies case, and using the modified mode functions in Eq. (4.2), we find that for modified initial states

H6 1 Z 0 dη B (k , k , k ) = 4 p ki kj −i p 1 2 3 2 ˙2 Q 3 ij 2 3 2 Mp φ 2ki η0 η

(t) (t) (t)∗ ik1η (t)∗ −ik1η ×(αk1 + βk1 )(αk1 (1 − ik1η)e + βk1 (1 + ik1η)e )

(s) (s) (s)∗ ik2η (s)∗ −ik2η ×(αk2 + βk2 )(αk2 (1 − ik2η)e + βk2 (1 + ik2η)e )

(s) (s) (s)∗ ik3η (s)∗ −ik3η ×(αk3 + βk3 )(αk3 (1 − ik3η)e + βk2 (1 + ik3η)e ) +c.c. H6 1 = 4 p ki kj(α(t) + β(t))(α(s) + β(s))(α(s) + β(s)) 2 ˙2 Q 3 ij 2 3 k1 k1 k2 k2 k3 k3 Mp φ 2ki " (t)∗ (s)∗ (s)∗ (t)∗ (s)∗ (s)∗ × αk1 αk2 αk3 I(k1, k2, k3; η0) + βk1 αk2 αk3 I(−k1, k2, k3; η0)

(t)∗ (s)∗ (s)∗ (t)∗ (s)∗ (s)∗ +αk1 βk2 αk3 I(k1, −k2, k3; η0) + αk1 αk2 βk3 I(k1, k2, −k3; η0) (t)∗ (s)∗ (s)∗ (t)∗ (s)∗ (s)∗ +βk1 βk2 αk3 I(−k1, −k2, k3; η0) + βk1 αk2 βk3 I(−k1, k2, −k3; η0) # (t)∗ (s)∗ (s)∗ (t)∗ (s)∗ (s)∗ +αk1 βk2 βk3 I(k1, −k2, −k3; η0) + βk1 βk2 βk3 I(−k1, −k2, −k3; η0) +c.c., (4.11)

103 where

P i>j pipj p1p2p3 I(p1, p2, p3) = −(p1 + p2 + p3) + + 2 p1 + p2 + p3 (p1 + p2 + p3)  P p p p p p (1 − i(p + p + p )η ) i  i(p1+p2+p3)η0 i>j i j 1 2 3 1 2 3 0 −e + 2 + , p1 + p2 + p3 (p1 + p2 + p3) η0 (4.12)

and pt = p1 + p2 + p3. For the permutations in Eq. (4.11), pt will be either kt or combinations such as k1 − k2 + k3. To evaluate the time integrals in the first expression of Eq.(4.11), we have used, for example, that the integral for the ααα permutation term is

0 (P ) 0 Z dη kikj k1k2k3 i Y ikiη iktη i

i iktη = I(k1, k2, k3) + lim e . η→0 η

i 1−iktη The divergent η piece comes from the η2 part of the integrand and is purely imaginary and independent of the ki in the η → 0 limit. We can therefore factor this piece outside of the permutations, with a change of sign in each of the conjugate pieces to see that the Bogoliubov coefficients of this imaginary, divergent, piece multiply to a real quantity,

(t) (t) (s) (s) (s) (s) (t) (t) (s) (s) (s) (s) ∗ (αk1 + βk1 )(αk2 + βk2 )(αk3 + βk3 ) × [(αk1 + βk1 )(αk2 + βk2 )(αk3 + βk3 )] , which is canceled by its complex conjugate in the first expression of Eq.(4.11), leading to the second expression. (s) (t) (s) (t) For Bunch-Davies initial states, αk = αk = 1 and βk = βk = 0, Eq. (4.11) reduces to Eq. (4.10) of [33] if we take η0 → −∞ with the contour prescription used there.1 It follows from Eq. (4.11) that an excited initial state can lead to a large three-point correlation in the squeezed limit in the coordinate frame appropriate for late-time observers. It was shown in [43] that moving to this frame, described with

1We have an extra factor of two; this comes from the tensor power spectrum Eq. (4.6), which is consistent with the consistency relation r = 16 in the Bunch-Davies case and differs from that in [33] by a factor of two.

104 conformal Fermi Normal Coordinates, introduces another term in the three-point

function in the squeezed limit kL ≡ k1  k2 ' k3 ≡ kS,

obs 1 p p ˆi ˆj ∂ ln Pζ (kS) Bp (kL, k2, k3) = Bp(kL, k2, k3) + Pγ (kL)Pζ (kS)ijkSkS . (4.13) 2 ∂ ln kS

When the consistency relation of single-clock inflation is satisfied the additional term in Eq. (4.13) cancels the first term in the exact squeezed limit (to order 2 (kL/kS) ). In that case there is no physical correlation with long wavelength modes and the apparently nonvanishing squeezed limit is a gauge artifact (see [42] for an earlier version of this argument). With a general initial state, the consistency relation need not hold: for a finite range of modes, a small physical coupling between long and short wavelength modes is consistent with inflationary expansion of the background even though the correlation is not generated by the background2. The permutations in Eq. (4.11) proportional to β(i) can dominate the squeezed limit on some scales, yielding a physical coupling between a long wavelength tensor mode with the short wavelength fluctuations that can be seen in the observer’s reference frame. The details of the shape, however, depend on the k−dependence (s) (t) of the α and β coefficients. In the βk = βk = 0 case these terms (and hence the observed squeezed limit) vanish [33,43].

4.2.2 Enhanced fossil signature in locally anisotropic scalar fluc- tuations

The scalar-scalar-tensor correlation can be used to estimate the magnitude of an unseen tensor mode γp(K) through observations of local anisotropy in scalar modes. In general, if scalar curvature perturbations ζ are coupled to an unobservable fossil field with an isotropic scalar-scalar-fossil three-point function, the local power spectrum evaluated in the presence of a tensor field realization is [71,139]

3 3 hζ(k1)ζ(k2)iγ = (2π) δ (k1 + k2)Pζ (k)

2 In other words, the consistency relation in the exact squeezed limit kL/kS → 0 is not violated. We are only computing correlation functions for modes that are subhorizon at the onset of the inflationary era, |kLη0| & 1, and longer wavelength modes are treated as part of the background. (s) Taking η0 farther into the past tightens the back-reaction bound on β [75], so that taking (s) η0 → −∞ returns us to the Bunch-Davies state, β = 0. Sufficiently short wavelength, observable modes will also have β(s) = 0.

105 Z 3 X d K 3 3 ∗ Bp(K, k1, k2) + 3 (2π) δ (k1 + k2 + K)γp (K) p . (4.14) p (2π) Pγ (K)

Each pair of scalar modes whose momenta add to K can be used to give an estimator for γp(K). The minimum variance estimator obtained from all such pairs n has some uncertainty, which is quantified by the noise power spectrum Pp (K), as defined in Eq. (5) of [71]:

2 n −1 X Bp (K, k, K − k) [Pp (K)] = p 2 tot tot , (4.15) k 2VPγ (K) Pζ (k)Pζ (|K − k|)

tot where Pζ (k) is the measured scalar power spectrum, including signal and noise, 3 and V ≡ (2π/kmin) is the volume of the survey. Going to the continuous limit P R 3 3 k → V d k/(2π) and making use of Eqs. (4.5), (4.6) and (4.11), we find

Z kmax 3 4 4 2 n −1 (t) (t) −4 d k (s) (s) −2 (s) (s) −2 k sin θ cos 2φ [Pp (K)] = |αK + βK | 3 |αk + βk | |α|K−k| + β|K−k|| 3 3 kmin (2π) 8k |K − k| h (t) (t) (s) (s) (s) (s) ˜ 2 × (αK + βK )(αk + βk )(α|K−k| + β|K−k|)I +c.c.] , (4.16)

where I˜ denotes the quantity in brackets in Eq. (4.11). Here we have followed [71] by tot tot approximating Pζ (k)/Pζ (k) ' 1 for k < kmax and Pζ (k)/Pζ (k) ' 0 for k > kmax,

where kmax is the smallest scale at which the power spectrum can be measured. p We have also taken K to be in the zˆ direction, so that the polarization tensor ij + + × × takes the form xx = −yy = 1 for the + polarization, and xy = yx = 1 for the × polarization, with all other components zero. This yields the factor cos2 2φ for the + polarization, as shown above; the × polarization yields a factor of sin2 2φ instead, but the integral is identical. (s),(t) (s),(t) We will take αk and βk to be constant for the observable range kmin < 3 k < kmax, and consider possible k-dependence later in this section. We will also

take the time η0 at which we specify the initial state for inflation to be early enough

so that all modes of interest are well inside the horizon, |kiη0|  1. Furthermore, due to the limited resolution of modes in a finite volume, only configurations for

which k1 − k2 + k3 & kmin (and similarly for other permutations) will contribute

3 (s),(t) In the limit of large k, βk must approach zero sufficiently quickly or vanish above some scale so that back-reaction constraints on the energy density are met [75, 164].

106 to the observable signal. Since we assume |kminη0|  1, the terms in the second line of Eq. (4.12) oscillate rapidly and will average to zero when we integrate over 4 k, so we discard them from now on. The k → kmax limit dominates the integral , 5 k and the second and third terms of Eq. (4.12) are enhanced by a factor of K in the middle four permutations in Eq. (4.11), so the bispectrum scales as k−2K−4 in

the squeezed limit K  k. As long as βk is not too small (which we will quantify below), these terms dominate the integral. (The coordinate transformation term from Eq. (4.13) scales as k−3K−3, as do the subdominant permutations in Eq. (4.11), and can thus be ignored.)

The angular dependence in the denominator of these terms, eg. k1 − k2 + k3 = K(1 + cos θ), contributes a divergence in the collinear or flattened limit. Dropping factors of (constant) α and β for now and cutting off the angular integral at

θmin ' kmin/K and θmax ' π − kmin/K, we find

5 Z θmax  2 n −1 π kmax 5 4 [Pp (K)] ∝ 3 2 dθ sin θ 4 40(2π) K θmin sin θ 5  2 16π kmax K ∝ 3 2 (4.17) 40(2π) K kmin

Restoring factors of α and β and using the normalization conditions, Eq. (4.3), we have 3 !2 n −1 kmax kmax −1 (s) (t) (s) (t) [Pp (K)] = 2 F (β , β , Θ , Θ ), (4.18) 20π kmin where

F −1(β(s), β(t), Θ(s), Θ(t)) = β(s)2(1 + β(s)2)  q  × 1 + 2β(t)2 + 2 β(t)2(β(t)2 + 1) cos Θ(t)

 q −2 × 1 + 2β(s)2 + 2 β(s)2(β(s)2 + 1) cos Θ(s)

"q 2q  × 1 + β(s)2eiΘ(s) + β(s) 1 + β(t)2eiΘ(t) + β(t)

4There is also a divergence in the collinear limit for K < k, but upon cutting off the integral with kmin, this gives a subdominant contribution, which is completely negligible in the K → kmin limit that dominates the final result, Eq. (4.20). 5In fact, only the third term will matter, but we will include both for completeness.

107 2 q  # × 1 + β(t)2e−iΘ(t) − β(t) e−iΘ(s) + c.c. , (4.19)

(s) (s) (t) (t) (s) (t) where β ≡ |βk |, β ≡ |βk |, and Θ and Θ are the relative phases between (s) (s) (t) (t) 2 α and β , and α and β , respectively. Note that the factor of (kmax/kmin) in Eq. (4.18) comes from the k−2K−4 scaling of the non-Bunch-Davies bispectrum as described above. The estimates for each mode γp(K) can be combined to give a minimum variance 3 p H2 (t) (t) 2 estimator for the power Aγ ≡ K P (K) = 2 2 |α + β | in tensor fluctuations γ Mp k k (we assume a nearly scale invariant tensor power spectrum). The overall variance is obtained by summing over the inverse variances from each K [71]:

−2 1 X h 3 n i−2 σγ ≡ K Pp (K) . (4.20) 2 K,p

n We see from Eq. (4.18) that Pp (K) is independent of K in the regime we are considering, and the sum over polarizations simply gives a factor of two, so the 2 3 6 n 2 variance of the estimator for Aγ is given by σγ = 4π kmin(Pp ) . If we require a 3σ detection of Aγ, the minimum detectable amplitude for the tensor power spectrum is √ !−5 kmax (s) (t) (s) (t) 3σγ = 30π 3π F (β , β , Θ , Θ ). (4.21) kmin

The minimum value of kmax/kmin needed for a survey capable of detecting primordial tensor modes of a given amplitude is found by setting the detection threshold of

Eq. (4.21) equal to the tensor amplitude Aγ. In Figure 4.1 we show the dependence of the survey size required to detect tensor modes on the initial state parameters β(s) and β(t), for two values of the tensor-to-scalar ratio r and two pairs of values for the angles Θ(s), Θ(t). For a 3 given survey volume V = (2π/kmin) , the contour lines determine values of kmax required to make a detection. (The effect of the overall bispectrum and tensor power spectrum amplitudes on the signal is somewhat obscured by their complicated dependence on the Bogoliubov transformation parameters, and is shown more simply in Figure 4.2 below.) As expected, the three-point correlation is stronger for larger β(s), resulting in a smaller minimum survey size. Although one might expect the increase in the tensor

108 Figure 4.1: Contour plot of the minimum survey size kmax/kmin needed to detect the effect of primordial tensor fluctuations through off-diagonal contributions to the scalar power spectrum, in terms of scalar and tensor Bogoliubov parameters β(s), β(t), for different values of the tensor-to-scalar ratio r and non-Bunch-Davies phases Θ(s), Θ(t). The scales kmin and kmax are the longest and shortest observable scales (at which scalar modes are excited). The dashed line at β(s) = 0.1 indicates the back-reaction bound (s) (s) β . 0.1 (this does not apply in the Θ = π case [164]). The dark shaded region is NBD2 (s),(t) ruled out by the Planck constraint fNL = 0.2 ± 0.4. For Θ = π the power spectra decrease rather than increase with β(s),(t), leading to different behavior.

(t) power spectrum Pγ for larger β to universally decrease the minimum survey size, (t) a larger β also affects the variance of the estimator for Pγ and can, depending on the phases, increase or decrease the minimum survey size. In order to have inflation we require  < 1, which constrains the (β(s), β(t)) parameter space if we also fix r in Eq. (4.7). This is shown in Figure 4.1 in the form of curves of fixed . Furthermore, it was shown in [164] that requiring (i) that the energy density from the excited modes not spoil the slow roll evolution,

109 and (ii) that the observed power spectrum be nearly scale-invariant, rules out large (s) (s) occupation numbers β  1 but allows for β . 0.1. This bound is indicated in Figure 4.1 as a dashed line, and is relaxed in the Θ(s) = π case [164]. Planck constraints on non-Gaussianity [165] impose an observational bound on (s) NBD2 NBD2 β by constraining fNL = 0.2 ± 0.4, where fNL parameterizes the amplitude of the bispectrum.6 This parameter is related to the squeezed limit of the bispectrum,7

1 Bζ (kL, |kS − kL/2|, | − kS − kL/2|) fNL ≡ lim , (4.22) kLkS 4 Pζ (kL)Pζ (kS)

NBD2 kL 3 by a factor of f = O(1) fNL. In Eq. (4.22), Bζ is the hζ i bispectrum [73,156] NL kS for the initial state in Eq. (4.2). Taking the squeezed limit we find8

 (s) kS (s) (s)  2|β | cos Θ , |β |  1  kL f = 1  kS f(Θ(s)), |β(s)|  1, Θ(s) 6= π (4.23) NL 2 kL   −8|β(s)|4 kS , |β(s)|  1, Θ(s) = π, kL where f(Θ(s)) = (3 + 2 cos Θ(s) − cos 2Θ(s))/(1 + cos Θ(s))2. From Eq. (4.7) we NBD2 see that for a given r the Planck constraints on fNL impose a bound on the Bogoliubov parameters. This bound is also shown in Figure 4.1 (which includes the full squeezed-limit dependence on β(s), although we have only shown the limiting cases in Eq. (4.23)). Note that depending on the phases, the dependence on β(s) and β(t) can be quite different [164]. For Θ(s) = 0 the theoretical constraint from requiring  to be small is stronger than the observational constraint from Planck bounds on non-Gaussianity, consistent with the conclusions of [164, 166] that observable non-Gaussianity is not expected in the CMB from Bogoliubov initial states. On the other hand, for Θ(s) = π the observational constraints play a significant role. 3 In the case of a smaller speed of sound cs < 1, the hζ i bispectrum receives a new 2 1−cs kS contribution that is zeroth order in slow roll [150], with amplitude f ∝ 2 . NL cs kL (s) (s) This results in a stronger constraint on β for smaller cs. For |β |  1 the

6 NBD2 NBD1 Note that in Eq. (14) of [165], fNL is written as fNL . 7 NBD2 In [165], fNL is the amplitude for a slightly different bispectrum template than that obtained from a Bogoliubov initial state, which has additional dependence on β(s) and Θ(s) that mildly affects the momentum dependence, although the behavior in the squeezed and flattened limits is the same. The constraints shown in 4.1 are therefore approximate. 8We omit O(1) factors from the angular dependence in the bispectrum.

110 (s) amplitude is proportional to |β |, so for cs = 0.02, saturating the constraint from (s) −2 Planck on equilateral and orthogonal non-Gaussianity [165], we have |β | . 10 . Returning to the present calculation, note that for very small β(s), all terms with β(s) in the bispectrum, Eq. (4.11), become negligible, and the threshold amplitude √ −3 appears to be 3σγ = 30π 3π(kmax/kmin) [71]. In this limit the coordinate transformation Eq. (4.13) to the observer’s frame of reference becomes significant and the squeezed-limit signal vanishes. In order for the non-Bunch-Davies terms to dominate in Eq. (4.16) we need F −1 ∝ β(s)2 for β(s)  1, so

k !2 (β(s))2 max  1. (4.24) kmin

If we were to include the other terms in Eq. (4.16) in the above calculation, we

1 kmin would have found fractional corrections to the result Eq. (4.21) of order (s) . β kmax In the case of a sharp cutoff Λ < kmax within the observable range, above which (s) scalar modes are not excited, all terms with βk vanish for k > Λ, so we cut off the −5 integral in Eq. (4.16) at Λ. The final result is then modified to σγ ∝ (Λ/kmin) , so the plots in Figure 4.1 indicate a lower bound on the required kmax, which is (s) saturated for βk nonzero over the entire range of observed modes. Alternatively, the contour lines can be thought of as showing Λ/kmin, giving the scale Λ up to which scalar modes in the initial state would have to be excited to leave a detectable signature,9 assuming it is possible to probe this scale observationally. The signal from the non-Bunch-Davies modes is dominant as long as the terms considered n above in Pp (K) still dominate the integral. This is true if

5 (s) 2 Λ (β ) 3 2  1. (4.25) kmaxkmin

5 In comparison to Eq. (4.24), an extra factor of (Λ/kmax) comes from the change in the upper limit. If Λ is too small this condition will no longer be satisfied: there will be too few excited modes in the observable range to produce a detectable signal. 2 2 −δ We can also consider a power law parameterization, |βk| = β (k/k∗) , where 2 k∗ functions like the cutoff Λ. If we consider the limit of small β, so O(β ) terms in −δ the bispectrum can be dropped, then we simply pick up an extra factor of (k/k∗)

9We assume tensor modes are also excited up to this scale; for tensor modes excited over fewer scales, the dependence on the occupation number β(t) would be weakened.

111 −δ in the integral in Eq. (4.16), which leads to an extra factor of (k∗/kmax) in Eq. −δ (4.21), and a factor (kmax/k∗) in Eq. (4.24). The step function therefore mimics the power law behavior for δ = 5.

4.3 General condition for scalar-scalar-fossil correla- tion

Although we have computed the very specific effect of a modified initial state with Bogoliubov coefficients, we can easily generalize the result to see the effect of long-wavelength fossil modes on local scalar fluctuations more generally. We will consider the squeezed limit of a factorizable three-point function with power law

dependence, and compute σf (in the following we will change ‘γ’ subscripts to ‘f’

for a generic fossil field). The squeezed limit is parameterized by an amplitude fζζf , with additional power law dependence for the long and short-wavelength modes,

!mS !mL kS kL B(kL, k2, k3) = fζζf Pf (kL)Pζ (kS) , (4.26) kp kp

where kL  k2 ' k3 ≡ kS, and we ignore any angular dependence. For a nearly

scale-invariant power spectrum and bispectrum, nf ' 0 and mS + mL ' 0. We n −3 Af  k  f parameterize the power spectrum as P (k) ≡ 3 . If we require a detection f k kp of significance of α ≡ Af /σf standard deviations, then we arrive at an inequality relating the survey size, strength of the fossil correlation, amplitude of curvature and fossil fluctuations, and runnings of the tensor power spectrum and squeezed-limit three-point function, !2mS +3 1 2 kmax fζζf Af > α (4.27) C kmin where 3/2 1/2 C ≡ 2π (2mS + 3)(3 − 4mL − 2nf ) , (4.28)

and we have chosen kp = kmin. We have assumed that the integral over k for the noise power spectrum is dominated by the upper limit, and the integral over K for −2 σf is dominated by the lower limit, so that the signal comes from the squeezed

limit K  k; this requires that mS > −3/2 and 4mL + 2nf − 3 < 0. Eq. (4.27) shows that a detectable fossil signal would require either a nonzero

112 positive value of mS (and thus for scale invariant bispectra a strong divergence in the K → 0 limit) as in the case above, a large bispectrum amplitude, and a sufficiently large amplitude of fossil field fluctuations. If the bispectrum is not proportional to Af , that is, if fζζf as defined above depends on Af , the effect of the fossil amplitude on the signal strength can be different and may even be reversed, counterintuitively. In the case considered above of a primordial gravitational wave fossil field and non-Bunch-Davies initial state with constant Bogoliubov coefficients, mL = −1 and mS = 1, leading to a stronger signal. Letting f → γ in Eq. (4.27) and setting

α = 3 we can find the required kmax/kmin for a 3σ detection. In 4.2 we show the 10 dependence of this number on fζζγ and the tensor-to-scalar ratio. As expected, increasing either the bispectrum or tensor power spectrum increases the level of detectability of the tensor modes. For a sufficiently large three-point correlation with K−4 behavior in the squeezed limit, and large enough tensor fluctuations, the signal would be within the observable range. q −1 (s) (t) (s) (t) For the Bogoliubov initial state, we have fζζγ = O(1)× F (β , β , Θ , Θ ). Thus, each point in Figure (4.2) at r = 0.1 or r = 0.01 corresponds to a one- parameter family of values of β(s),(t) in Figure 4.1, if we fix Θ(s),(t).

4.4 Quadrupolar anisotropy and superhorizon modes

We have computed the fossil signature in off-diagonal correlations of the scalar power spectrum, which is insensitive to superhorizon tensor modes, since we cannot resolve scalar modes k2 and k3 separated by |k2 + k3| = K < a0H0. However, the tensor nature of the gravitational fossil can also give rise to a quadrupolar modulation of the scalar power spectrum [139], introducing both statistical inhomogeneity and anisotropy. This signature is sensitive to superhorizon fossil modes if the squeezed-limit bispectrum is strong enough. Consider a general scalar-scalar-tensor bispectrum with squeezed limit

!mL !mS p kL kS p ˆi ˆj lim Bp(kL, kS, −kL−kS) = fζζγPζ (kS)Pγ (kL) ijkSkS, (4.29) kLkS kP kp

10 p ˆiˆj We include the angular dependent factor ijk k to the bispectrum with a spin-2 polarization 4 tensor as described above in section 4.2.2, which leads to an extra factor of 15 in Eq. (4.27).

113 Figure 4.2: Contour plot of kmax/kmin for a 3σ detection of the gravitational fossil in off-diagonal contributions to the scalar power spectrum, in terms of the tensor-to-scalar ratio r and scalar-scalar-tensor bispectrum amplitude fζζγ, assuming a stronger-than-local 2 squeezed limit (mS = 1, mL = −1 in Eq. (4.27)). The signal is proportional to fζζγAγ. −9 We set r = Aγ/(2.2 × 10 ) and take nt ' 0.

where kp and kP are pivot scales, which we allow to be different as we are interested in kL and kS in different ranges, ie. superhorizon kL coupling to subhorizon kS. This bispectrum contributes to the scalar two-point function on a fixed tensor background, Eq. (4.14):

" k !mS # (mL) ˆiˆj Pζ (k, x)|γ = Pζ (k) 1 + fζζγγij (x)k k , (4.30) kp where Z 3  mL (mL) X d K iK·x p p K γij (x) = e γ (K)ij . (4.31) −1 3 p L (2π) kP

114 The quadrupolar anisotropy varies spatially with the tensor background, and depending on the squeezed limit behavior, it is more or less sensitive to longer- wavelength, superhorizon tensor modes. The infrared cutoff scale L−1 is introduced because we do not expect Eq. (4.29) to be valid for arbitrarily small kL (for example, bispectra computed from an effective field theory will be limited by the range of scales described by the theory). Note that the additional k-dependence from mS will affect the observed spectral index, as we saw in chapter3. (s) In the non-Bunch-Davies case with excited scalar modes βk 6= 0 overlapping with observable scales, mL = −1 and the local quadrupole anisotropy is modulated by the longest wavelength (∼ L) tensor modes and thus varies less spatially. Note that whereas the off-diagonal fossil signature computed in section 4.2 probes the correlation of observable scalar modes with longer but sub-Hubble tensor modes, the quadrupole signature probes their correlation with superhorizon tensor modes. Both effects, however, originate in the excitation of the scalar modes in the initial state.

4.5 Conclusion

We have computed the gravitational fossil signature in off-diagonal correlations in the scalar two-point function, from a primordial scalar-scalar-tensor three-point p function hγ (k1)ζ(k2)ζ(k3)i for excited Bogoliubov initial states for both the scalar and tensor modes. The bispectrum is largest in the squeezed and flattened limit k2 ≈ k3  k1, k2 + k1 ≈ k3 (or k2 − k1 ≈ k3), characterized by an unusually −4 (s),(t) (s),(t) strong k1 dependence, under the assumption that α and β are constant. The fossil signature is obtained by summing over all measurable off-diagonal pairs of scalar modes. Depending on the amplitudes and phases of the Bogoliubov 2 coefficients, the survey size needed for a detection can be as small as kmax/kmin ∼ 10 , within the reach of current surveys. We have also given, in section 4.3 a phenomenological parameterization for the off-diagonal fossil signature in terms of the amplitude of fossil fluctuations, and amplitude and squeezed-limit scaling d ln B of a scale-invariant scalar-scalar-fossil d ln kL bispectrum. This illustrates the necessary criteria for any primordial correlation to satisfy in order to leave an observable imprint. The mode coupling introduced by the initial state can also lead to a gravitational fossil signature in the form of

115 a quadrupole anisotropy in the scalar power spectrum, with greater sensitivity to longer wavelength tensor modes, as described in section 4.4. Here we have studied the effect on the scalar and tensor perturbations from a non- Bunch-Davies initial state imprinted by unknown pre-inflationary dynamics. In the Bunch-Davies case the clock of the inflationary background dynamics determines the statistics of the fluctuations. The attractor behavior forbids correlations forming with modes that have crossed the horizon, and correlations forming with very subhorizon modes are exponentially suppressed [164]. Consequently, correlations between modes of very different wavelength are disallowed, and the squeezed-limit bispectrum vanishes. Pre-inflationary dynamics, on the other hand, can result in excited modes at the onset of inflation so that the amplitude of fluctuations √ on scale k is not purely determined by the background evolution (H2/  when the mode crosses out of the inflationary horizon). If subhorizon scalar modes are excited, they can now correlate to longer tensor modes when the tensor modes cross the horizon, before they become part of the classical background, resulting (s) in a large squeezed-limit bispectrum. From Eq. (4.11) we see that βk =6 0 is the necessary and sufficient condition for the B ∝ k−2K−4 squeezed limit scaling (β(t) 6= 0 allows the tensor modes to be influenced by an additional clock, and to correlate with other modes before crossing the horizon, but they could not couple to shorter scalar modes unless they too were excited). In the same way, in order for short tensor modes to couple to long scalar modes through the ζγγ interaction, the tensor modes would have to be excited when deep inside the horizon, requiring (t) that βk 6= 0. (Classically, a realization of short-wavelength modes of one field XS can be adjusted to be correlated with a fixed realization of long-wavelength modes

of a field XL, but not vice versa. So it is the XS modes that must be influenced by dynamics other than those of the inflationary background if they are to couple to

XL.) A new clock influencing the fluctuations can also come from additional dy- namics during inflation from a non-inflaton sector. For example, although tensor perturbations always arise from the vacuum, they can also be generated via particle production and decay of other fields during inflation. Depending on whether this production dominates over the vacuum fluctuations and on the relation between the inflaton dynamics and dynamics governing particle production, the scalar and/or tensor perturbations may be governed by a clock other than the inflationary back-

116 ground. The usual intuition that non-Gaussianity in the gravitational sector is small compared to that in the scalar sector doesn’t have to hold, as was recently shown by [138]. In the typical particle production scenario the non-Gaussianity is not of the local type, but it would be very interesting to explore the range of physical mechanisms, and conditions on additional dynamics before or during inflation that would be necessary to generate squeezed-limit mode coupling of scalar and/or tensor modes. For example, one might ask if governing both scalar and tensor perturbations with the same dynamics, but different from the inevitable fluctuations from inflation, could remove any coupling between scalar and tensor modes. If this is possible, the tensor modes may not be as clear a diagnostic of the inflationary background (and classical evolution of the inflaton) as we currently hope they are. The fossil signature studied here comes from the minimal gravitational coupling of the inflaton. Fossil signatures may also come from coupling of the inflaton to other sectors or from nonminimal coupling to gravity. It could be interesting to investigate conditions on such couplings that allow for an observable fossil signature, subject to observational constraints, for example on isocurvature modes. Modulation of local statistics by a fossil field may also be worth investigating in light of anisotropy features on large scales in the CMB [167]. Superhorizon scalar curvature modes coupled to scalar modes on observable scales with an anisotropic hζζζi bispectrum can lead to an observable power asymmetry [90]. It would be interesting to see under what conditions the squeezed-limit coupling to superhorizon modes of another field can be strong enough to give rise to such effects. A non- Bunch-Davies initial state or additional dynamics before or during inflation may introduce greater sensitivity of ζ to background modes of other fields, as seen here for tensor modes.

117 Chapter 5 | Discussion of Results

In this chapter we will step back and consider the implications of the preceding chapters for our understanding of the early universe. We begin with a brief overview of the results presented in the preceding chapters.

5.1 Summary of Thesis Work

We saw in chapters2-3 that weakly non-Gaussian statistics observed in a finite volume may be generated from perturbations in a strongly non-Gaussian global obs obs volume, with the small parameter fNL ∆ζ  1 characterizing the observed level of non-Gaussianity arising from a hierarchy of scales in the large volume,

v 2 1/2 u ? hζ i uln(kmax/k ) obs obs Gs t fNL ∆ζ ∼ ∼ ?  1, (5.1) ζGl ln(k L)

2 2 for typical subvolumes with ζGl ∼ hζGli. (The second relation here assumes scale- invariance.) However, non-Gaussianity cannot be completely concealed by observing only a small volume. For fluctuations in a larger universe that are still perturbative,

|ζGl|  1, we have obs 1 fNL ∼  1. (5.2) ζGl obs The smallest fNL can be is therefore O(1), in the nonperturbative limit ζl = O(1), in which our approximation of a homogeneous background breaks down. If observational constraints on local-type non-Gaussianity can reach this level, the scenario of a strongly non-Gaussian larger universe would be under pressure from

118 data, perhaps only surviving as a viable description of the larger universe in the 2 context of eternal inflation, for which ∆ζ = O(1), and for which primordial physics before the final 60 e-folds of inflation would be very uncertain, regardless of mode coupling considerations.

In chapter3 we extended this study to include scale-dependence in fNL(k) and other parameters, and studied the influence of background modes on the observed spectral index, dark matter distribution, and tensor power spectrum. In particular, scale-dependence in a global bispectrum can reverse the sign of the observed spectral obs index ns as compared to the globally averaged spectral index ns; this occurs over a range of scales for which different contributions to the total power are shifted in their relative amplitude due to the long-wavelength background. This and other degeneracies with background mode effects limit our ability to infer parameters of the global volume. However, we found that cosmic variance of the spectral index occurs more readily for a blue-tilted fNL(k), so constraining local non-Gaussianity on shorter scales than those accessible in the CMB has the potential to rule out this type of statistical cosmic variance. In chapter4 we explored an example of the dynamical origins of mode coupling during inflation. We studied an alternative method for detecting primordial gravita- tional waves via their coupling in the early universe to scalar curvature fluctuations, hζ ζ hij i, which generates anisotropy in the observed scalar power spectrum k1 k2 k3 hζk1 ζk2 i. We found that a non-vacuum initial state for curvature perturbations at the onset of inflation could strengthen the squeezed-limit coupling between short scalar modes and infrared tensor modes, yielding a potentially observable fossil signal of primordial gravitational waves, which persists after their decay. The statistical anisotropy from the modulation by background tensor modes could also provide evidence for tensor modes on superhorizon scales.

5.2 Super Cosmic Variance and Inflation

Our discussion of the influence of background fluctuations through mode coupling has focused on the question of how statistics observed in a subvolume relate to statistics averaged throughout a larger global volume. In order to understand how these results translate into a relationship between observables and parameters of in- flationary models, we must ask how long-wavelength modes affect finite subvolumes

119 during inflation. We will briefly discuss this question in the next subsection for the case of multi-field inflation, and comment on the implications of super cosmic variance for inflation in section 5.2.2. Note that although we will focus on the case of multiple fields contributing to the background inflationary dynamics, similar conclusions apply in the case of a second field sourcing curvature perturbations after inflation, as in the case of the curvaton model. In that case, superhorizon curvaton modes shift the accessible region of the curvaton potential for any finite volume.

5.2.1 Inflationary Trajectories for Finite Volumes

Given an inflationary potential V (φI ) for one or more fields φI , the trajectory in field space during inflation is determined by the potential and initial field values at (0) the start of inflation. We will call this the global or background trajectory φI (t). Now let us consider a given finite (comoving) volume of space, say the volume which we can observe. (We will assume that the size 1/k∗ of this volume is smaller than the Hubble radius H(t0) at the onset of inflation, so volumes this size would be in the sub-horizon regime.) As inflation proceeds, fluctuations δφI,k on scales much larger than this volume (k < k∗) will become larger than the Hubble scale, and can then be treated as classical perturbations shifting the local background region values of the fields φI for the finite region of interest. At a given time t, the scales contributing to this local background will be those that have crossed the Hubble scale at that time. Consequently,

Z a(t)H(t) d3k region (0) ik·x0 φI (t) = φI (t) + 3 δφI,ke , (5.3) kmin (2π)

∗ where x0 is the location of the region of interest, and kmin  k is an infrared scale which we can take to be of order the initial comoving Hubble scale a(t0)H(t0). region For any finite region, then, the “local” trajectory in field space is given by φI (t) rather than φ(t). (We are especially interested in the case of multiple fields, for which the trajectory moves through a field space of two or more dimensions, so that the local trajectory may be nontrivially shifted away from the global trajectory.) Consequently, at the time t∗ defined by a(t∗)H(t∗) ≡ k∗, when the volume of interest becomes of order the Hubble scale, the local trajectory has shifted the

120 (0) ∗ region ∗ 1 position in field space from φI (t ) to φI (t ). The potential relevant for the subsequent inflation of the region of interest will then be the potential along the region ∗ (0) ∗ trajectory starting at φI (t ), rather than the trajectory starting at φI (t ). If we consider the comoving region of space occupied by our observable patch of the universe during inflation, then we see that the inflationary potential which we hope to map out through cosmological observations will allow us to explore a limited region of field space, but that we cannot be sure what the potential V (φI ) looks like away from our local trajectory. This discussion relates to the study of the preceding chapters in the following way. If we consider the global inflating volume on the spatial hypersurface at the end of inflation, then different subvolumes within that volume will have followed different trajectories in field space until they stopped inflating. Since the correlation functions of the curvature perturbation ζ(x) are determined by the inflationary potential, they will differ from subvolume to subvolume. This variation, of course, originates in the realization of long-wavelength modes δφI,k. The curvature perturbation ζk is a nonlinear function of these perturbations [168,169], so we expect that a study of the variation of local trajectories in multi-field inflation would reproduce the spatial variation of statistics considered in this thesis.

5.2.2 Implications for Inflation

With a possible dynamical inflationary origin of super cosmic variance in mind, we can ask to what extent it introduces a new complication to inferring the physics of inflation from cosmological observations. First of all, it should be clear that super cosmic variance does not obscure our access to the inflationary potential everywhere in field space. As we saw in section 5.2.1, the influence of background modes is to displace the initial field values for our region prior to the last 60 or so e-folds of inflation, shifting the starting point of our local trajectory in field space in a nontrivial way if there is a multi-dimensional field space. For the region of field space traversed by this trajectory, measurements in our volume provide information that is not obscured by background modes. Because the long-wavelength modes stochastically shift the trajectory followed by

1 region ∗ (0) ∗ We note that since the potential V (φI (t )) differs from the potential V (φI (t )) along the global trajectory, the local Hubble scale will be shifted from H(t∗). We will assume that the potential is sufficiently flat for this difference to be negligible.

121 our volume away from the global trajectory, that local trajectory may not be a representative trajectory in field space. For a given Lagrangian for inflation and initial point in field space, parameters of the theory determine the statistics of ζ throughout the post-inflation volume. Since measurements in a subvolume can only yield probabilistic statements about the global statistics, the initial condition and parameters in the Lagrangian and can only be inferred with some probability, up to the unknown values of the local background modes. Equivalently, the background trajectory can only be known up to this uncertainty, given a segment of a random local trajectory. The salient question, then, is how the limitation to probing a random trajectory in field space could obscure the inflationary potential. Second, super cosmic variance does not introduce a new limitation to our access to the inflationary potential traversed by the inflaton prior to the final 60 e-folds. In single-field inflation, it is still true that inflation may last for many additional e-folds, in which case there would still exist a part of the potential which we could not probe.

Our ignorance of the long-wavelength modes δφI,k could introduce an additional region ∗ difficulty in relating the starting point φI (t ) of inflation in our region to the point in field space higher on the potential at which inflation started, because multiple local trajectories, originating at a range of initial field values, could could region ∗ reach the field-space point φI at the time t when the scale k ≈ H0 crossed the horizon. This ambiguity may or may not complicate the task of understanding how the inflationary potential accessible through observations connects to pre- inflationary physics. The degree of complication would depend on how well the potential could be understood based on the information that observations yield about its behavior along the path in field space traversed by our volume. Furthermore, it is possible that super cosmic variance may hide a feature of the potential V (φi) by shifting the trajectory followed by our subvolume away from it. Indeed, we ought to suspect as much given that sampling only a finite volume can hide non-Gaussianity in a larger volume. We consider this possibility in the following subsection, in light of the naturalness of nearly Gaussian statistics in small volumes.

122 5.2.3 Statistical Naturalness for Inflationary Trajectories

A potential dynamical description of this may follow from the work of Peterson and Tegmark [170], who studied trajectories on two-field potentials. It was argued in [170] that non-Gaussianity will be large for inflationary trajectories along a potential for which nearby trajectories are diverging, for example along the top of a narrow ridge. For trajectories that are converging with nearby trajectories, local however, near Gaussianity fNL ∼ 1 will be the outcome. In order to obtain large non-Gaussianity, a fine-tuning is needed of the trajectory, and consequently of both the potential and initial field values. (0) If the global or background trajectory φI (t) is diverging from nearby trajectories region and produces large non-Gaussianity, it may be that typical local trajectories φI (t) for finite volumes will diverge away into regions where nearby trajectories are converging. For example, if the background trajectory proceeds along a ridge in the region potential, long-wavelength modes may shift the field φI away from the ridge, causing typical trajectories to move off the ridge. It was found in [170] that a small change of direction in the trajectory could dramatically decrease non-Gaussianity. It may be that such a description would underly the fact that subsampling a non-Gaussian field in typical subvolumes results can result in nearly Gaussian statistics. If this is correct, the statistical naturalness of weak non-Gaussianity may be mapped to a notion of geometrical stability or naturalness, where typical trajectories converge towards certain regions in field space, possibly described by the concavity of the potential. It would be interesting to identify whether it is generically true that regions in field space where inflationary potentials produce large non-Gaussianity are unstable, in the sense that trajectories for finite volumes will typically leave such regions within a certain number of e-folds. It may be possible to quantify the degree of geometric fine-tuning of the potential (or initial conditions) needed in order for typical local trajectories to produce large non-Gaussianity. (Since local non-Gaussianity may also result from kinetic couplings due to a non-flat field space metric, as in quasi single-field inflation [171,172] for example, it would also be interesting to understand if there is a characteristic geometry in field-space for typical local trajectories.) Such a study would make use of the probability distribution of trajectories for a given potential, which could be computed from the classical trajectory along

123 with the multi-field Fokker-Planck equation for diffusion away from the classical path. Furthermore, attractor regions of the potential (regions to which trajectories converge) may restrict the range of values that observable parameters could take, perhaps determining “natural” values for the spectral index or tensor to scalar ratio. Of course, not all ways of generating local-type primordial non-Gaussianity rely on features of the inflationary potential. The approach sketched above may however offer some insight into the influence of infrared modes and the naturalness of weak non-Gaussianity in cases where the curvature is generated during inflation, and multiple fields contribute to the background dynamics.

5.2.4 Statistical and Inflationary Landscapes

In section 5.2.1 and 5.2.3 we considered the spatial variation of long-wavelength modes as it relates to the background evolution for finite subvolumes. The in- flationary dynamics of the background (for a local or global region) influences the Lagrangian for the fluctuations, that is, the operators that appear and their coefficients. Consequently, the influence of long-wavelength modes will also enter in the Lagrangian for fluctuations in a finite volume. Because super-Hubble modes vary spatially on large scales, the parameters describing sub-Hubble modes, such as masses and sound speeds, may also vary from region to region. For models with local-type cubic operators, parameters would be sensitive to long wavelengths, 2 leading to a landscape. For example, a cubic coupling of the form L3 ∼ σ(∂iδφ) – appearing for example in quasi- single-field models of inflation [171] – may lead to a shift to the quadratic Lagrangian for δφ in subhorizon regions with a large 2 accumulation σl of superhorizon modes, ∆L2 ∼ ∆σl(∂iδφs) . This can lead to a shift ∆cs in the sound speed for δφ fluctuations, as noted in the context of solid inflation [173] (see also [174]). In general, the statistical landscape of spatially varying correlation functions, if due to super-Hubble modes generated during infla- tion, could originate in a dynamical landscape of subhorizon inflationary physics, modulated by superhorizon perturbations.

124 5.3 Conclusion and Outlook

Because we live in an expanding universe at a specific time and location, our observation of the universe is fundamentally limited, and it is an open question how severely this spatial confinement will limit our understanding of Nature. Mode coupling of primordial perturbations carries information about the physics of inflation, but it also couples observable quantities to fluctuations on scales too large to be observed, leading to a systematic bias due to our local environment in a larger volume. We have seen that in a large enough volume, this bias can affect measured quantities in a significant way, affecting our conclusions about inflation. Our observations can only tell us directly about the last 60 e-folds of inflation. In the context of eternal inflation, an enormously long period of inflation may precede these final e-folds, and observations in a finite volume may carry very little information about earlier inflationary dynamics. Here we have seen that even for non-eternal inflation, our inability to probe earlier e-folds can lead to a large uncertainty in the prior history of inflation. Optimistically, constraints obs at the level ∆fNL . 1 have the potential to rule out the bulk of the parameter obs space in inflationary models where super cosmic variance occurs. If fNL & 1 is detected, there may be more ambiguity in the history of inflation prior to the final 60 e-folds. At the same time, observational access to an additional degree of freedom at inflationary energy scales would be an extraordinary opportunity for high energy physics. In summary, super cosmic variance leads us to ask the following question: How much can we learn about the theory governing inflation from mapping out the region of the inflationary potential that was traversed by the inflaton in our observable volume, during its last 60 e-folds of inflation? Cosmology is in an exciting time period, with unprecedented quality and quantity of data expected to be gathered from observations in the coming years, possibly providing much more detailed information about this region of inflationary field space. On the theory side, string theory or other UV completions may lead to a more detailed understanding of how that patch can be extended, perhaps narrowing down the possibilities. As observers within the universe, we must observe from some particular vantage point. Although we may be limited in the range of observations that we can

125 make, our position gives us a unique view of the universe. At earlier epochs, fewer cosmological scales would have been accessible – the largest scales, which offer the clearest and most direct picture of the primordial universe – would lie beyond the horizon. At later epochs dark energy may accelerate structures away from our galaxy, again hiding the universe from view. The linearity of the early universe allows us to understand physics back to the Big Bang and to access information about high energies. Whether this window is broad enough to see even further back remains to be seen. In any case, observational access to the energy scales and timescales of inflation is an extraordinary opportunity, and one which we are only beginning to explore.

126 Appendix A| Diagrammatic Representations of N-point Functions

1 We want to calculate N-point correlation functions of the non-Gaussian field ζNG defined by a local function of a Gaussian field,

ζNG(x) = f(ζG(x)) − hf(ζG)i . (A.1)

The n-point functions of ζNG can be written entirely in terms of two-point func- tions of ζG and the derivatives of f. However, the expressions quickly get messy so it’s helpful to use connected diagrams to keep track of the terms (see also [91,175]).

Dictionary of Diagrams: A line segment connecting two points 1 and 2 represents the real-space correlation function between the Gaussian fields ζG at two spatial points x1 and x2.

1 2 hζG(x1)ζG(x2)i ≡ ¡ (A.2)

1 We add a NG superscript to ζ to emphasize the non-Gaussian nature of the connected diagrams shown below.

127 while a double line segment indicates the square of the Gaussian correlation function

2 1 2 hζG(x1)ζG(x2)i ≡ ¡ (A.3) and vertices with multiple line segments indicate products of correlation functions connected to different points 3 hζG(x1)ζG(x2)ihζG(x1)ζG(x3)i ≡ 1¡ (A.4) 2

Circles represent ζG(x) contracted with itself, which is independent of x. For instance, we can write

1 2 hζ (x )ζ (x )ihζ2 (x )i = G 1 G 2 G 1 ¡

1 2 = ¡ × ¡

so it doesn’t matter which vertex a loop is connected to.

The Two-point Function:

The two point correlation function of the non-Gaussian field ζNG is given by

    1 2  (1) 2 (1) (3)  hζ (x )ζ (x )i = (f ) + f f + ...  NG 1 NG 2 ¡  ¡   

1 2 1  + ¡ (f (2))2 + ... 2

128 + ...

where the ... in each parenthesis represent terms with higher-order loop contribu- tions, and the ... in the last line indicate terms with additional shapes (but still 6 O(ζG)).

The Three-point Function: 6 The three point function of ζNG at O(ζG) can be grouped into three distinct shapes, a tree-level shape (with loop corrections differing only in amplitude) and two 1-loop shapes:

hζNG(x1)ζNG(x2)ζNG(x3)ic =        (1) 2 (2) 1 (1) 2 (4) (1) (2) (3)  (f ) f + (f ) f + f f f + ...   ¡   2  3  3 3  ×  + + 1   1¡ 1¡ ¡  2 3 2 3 2  3 1  + f (1)f (2)f (3) + ... 1 + 1 + 2 ¡ ¡ 1¡ 2 3 3 2 3 2 + + 1 + 1  1¡ ¡ ¡ 2 3 2 2   + (f (2))3 + ... 1¡ 2 + ...

The Connected Four-point Function:

129 8 Finally, we compute the connected four-point function up to O(ζG). Up to this order there are two tree-level shapes (commonly parametrized with gNL and τNL), and give 1-loop shapes:

hζNG(x1)ζNG(x2)ζNG(x3)ζNG(x4)ic =

       (1) 3 (3) 3 (1) 2 (3) 2 1 (1) 3 (5)  (f ) f + (f ) (f ) + (f ) f   ¡  2 2  2 3 ! × ¡ + 3 perm. 1 4      (1) 2 (2) 2  (1) (2) 2 (3) (1) 2 (2) (4)  + (f ) (f ) + f (f ) f + (f ) f f   ¡   2 3 2 3    ¡ ¡  ×  + + 5 perm. 1 4 1 4 2 3   (2) 4  ¡  + (f )  + 2 perm. 1 4 2 3 2 3   1 (1) (2) 2 (3)  ¡ ¡  + f (f ) f  + + 11 perm. 2 1 4 1 4 2 3 2 3 1 ! + (f (1))2(f (3))2 + + 5 perm. 2 ¡ ¡ 21 34 1 4   (1) (2) 2 (3)  ¡  +f (f ) f  + 11 perm. 1 4 2 3 1 ! + (f (1))2f (2)f (4) + 11 perm. 2 ¡ 1 4

130 A.1 Feynman-like Rules for Momentum Space Dia- grams

The real-space diagrams shown above can also be calculated in Fourier space:

Z n d3k P Y i i ki·xi real-space expression of (x1,... xn) = 3 [k-space expression] e . i (2π) (A.5) Here we will show how momentum-space expressions such as those given above in AppendixB for the bispectrum and trispectrum can be quickly recovered from their corresponding diagrams. P 1 (m) m m For the local ansatz ζNG(x) = m m! f (ζG (x)−hζG (x)i), a particular n-point function is given by

(m1) (m2) (mn) X X X f f f m1 m2 mn hζk1 ζk2 . . . ζkn i = ··· ... h(ζG )k1 (ζG )k2 ... (ζG )kn i, m1 m2 mn m1! m2! mn! (A.6) where the modes are given by the convolution integrals

Z m−1 3  m−1  m Y d pi X (ζG )k = 3 ζG(pi)ζG k − pj . (A.7) i=1 (2π) j=1

A given term in (A.6) is specified (up to permutations in the ki) by a set of m numbers Vm, where Vm is the number of times the ζG term appears. Thus, P Vm = n. Restricting to the connected part hζk1 ζk2 . . . ζkn ic imposes the condition 1 P 2 mVm − n + 1 ≥ 0. This is not a sufficient condition for the contribution to be connected; the contractions must be made so that the corresponding diagram is P connected, as discussed below. Note that mVm is even for nonzero contributions. m There are m − 1 momentum-space convolution integrals for each of the (ζG )ki P for a total of mVm − n integrals. The factors of ζG in (A.6) are contracted 1 P using Wick’s theorem, giving 2 mVm − 1 delta functions, not counting the final 3 P 1 P overall δ ( ki), so in the final expression there are L ≡ 2 mVm − n + 1 integrals

131 remaining. This is the number of loops that will appear in the diagram. If L = 0 the graph is a tree graph. Tree graphs dominate contributions to the n-point functions for a weakly non-Gaussian series. If L < 0 the diagram is disconnected. m The (ζG )ki factors will be represented as m-point vertices, with Vm of each type in the diagram, and a total of n vertices. V1 ≡ E denotes the number times the linear term contributes; these 1-point vertices appear as external lines.

Finally, each contraction between two factors of ζG yields a factor of the power spectrum and is represented by a line connecting two vertices, with a total of 1 P P ≡ 2 mVm = n + L − 1 lines. The rules for diagrams are as follows:

1. Assign a momentum label ki (i = 1, 2...n) to each vertex, including external 1-point vertices. Each m-point vertex is equivalent to a factor of f (m). (The 1 m! is cancelled by the m! ways of contracting into the vertex.) 2. Assign a momentum label to each line, with a direction. Lines contracted with 1-point vertices share their momentum label. L internal lines can be labelled

with integrated momenta pj (j = 1, 2...L); these can be chosen arbitrarily among the lines forming loops. The remaining P −E−L = (n−E)−1 internal P lines can be labelled with momenta kI + qk, where kI is the momentum of one of the vertices contracting with the line (either can be chosen), and

qk (k = 1, 2...m − 1) denote the incoming momenta of the other lines being contracted into that vertex. This imposes momentum conservation at each vertex. These labels can be made by working into the diagram starting from

the external lines. Each line is then equivalent to a factor of PG(q), where q is the momentum for that line.

R d3p 3. Integrate over the loop momenta by adding a factor (2π)3 for each loop. 2 Note that loops at a single vertex contribute a factor hζGi.

4. Divide by the symmetry factor of the diagram. As in standard quantum field theory, the symmetry factor is determined by counting the number of ways of exchanging identical vertices or identical lines, as well as lines contracted at a single vertex.

5. Sum over permutations of the kj (momenta for the vertices). Sum over connected diagrams with n vertices, to desired loop order, or level of ap-

132 3 3 P proximation, and multiply by (2π) δ ( ki) to obtain the n-point function

hζk1 ζk2 . . . ζkn ic.

These diagrams are essentially equivalent to those considered in [175] (see also [116]), where the more general case of multiple fields contributing to the curvature perturbation was considered.

133 Appendix B| Behavior of Local-type N-point Functions

Scaling of the Non-Gaussian Cumulants: Suppose that the statistics of the curvature perturbation can be written as a non-linear transformation of a Gaussian field that is local in real space:

ζNG = f(ζG(x)) − hf(ζG(x))i . (B.1)

2 We can calculate the statistics of ζNG in terms of hζGi  1 and derivatives of f

∂(n)f (n) f ≡ n . (B.2) ∂ζ G ζG=0

For the moment, we’ll ignore the shape dependence of the N-point functions of 2 ζNG and just consider the scaling of the cumulants in terms of hζGi (we consider the shape dependence in sectionB). In terms of f and ζG we have,

 1   hζ2 i = hζ2 i (f (1))2 + (f (2))2 + 2f (1)f (3) hζ2 i NG G 2 G

1    + 5(f (3))2 + 6f (2)f (4) + 3f (1)f (5) hζ2 i2 + ... 12 G

  3   hζ3 i = hζ2 i2 3(f (1))2f (2) + (f (2))3 + 6f (1)f (2)f (3) + (f (1))2f (4) hζ2 i + ... NG G 2 G

4 2 3  (1) 3 (3) (1) 2 (2) 2  (2) 4 (1) (2) 2 (3) hζNGic = hζGi 4(f ) f + 12(f ) (f ) + 3(f ) + 36f (f ) f

134 (1) 2 (3) 2 (1) 2 (2) (4) (1) 3 (5) 2  +12(f ) (f ) + 18(f ) f f + 2(f ) f hζGi + ...

5 2 4  (1) 4 (4) (1) 2 (2) 3 (1) 3 (2) (3)  hζNGic = hζGi 5(f ) f + 60(f ) (f ) + 60(f ) f f + ...

...

n 2 n−1  (1) n−1 (n−1) (1) n−2 (2) (n−2)  hζNGic = hζGi n(f ) f + n(n − 1)(n − 2)(f ) f f + ...

where the subscript c indicates the connected part and ... indicate terms higher- 2 2 order in hζGi. We’ve kept a number of subleading (in hζGi) terms in order to help illustrate the following points:

0 (n) (1) 2 (n−1)/2 n • For f 6= 0 and f /f hζGi  1, cumulants scale as hζNGic ∼ 2 n−1 2 n−1 hζGi ≈ hζNGi . So, higher cumulants are suppressed by powers of the observed variance. We refer to this type of statistics as weakly non-Gaussian.

0 2 • If f = 0 the cumulants still scale with increasing powers of hζGi, but the 2 2 p observed variance is hζNGi ∼ hζGi where p is the order of the first nonzero n 1 derivative of f, so the relative scaling of each hζNGic is different . In particular the suppression of higher cumulants can be weaker than in the f 0 6= 0 case.2 We refer to this type of statistics as strongly non-Gaussian.

(n+1)q 2 (n) (n) Further note that for series coefficients f hζGi  f when f is non-zero, non-Gaussianity will first be evident in either the skewness or kurtosis. That is, 3 n 4 n hζNGi ≥ hζNGic and/or hζNGic ≥ hζNGic for n > 4.

Shape and Scale-dependence of the N-point Functions: We now consider the shape and scale-dependence of the power spectrum, bispec-

1 (n) (n) 2
What we really mean by f = 0 is f ∼ O hζG,si , our expansion parameter, so that the next order terms are comparable. 2To be more precise, the scaling of cumulants with the non-Gaussian variance depends on P 1 (n) n which terms are non-vanishing in f(x) = n n! f x . If the only non-vanishing term has an 2 odd power n, then the cumulants scale with powers of hζNGi, but generically higher terms scale 2 2 0 with ∼ hζGi > hζNGi and are less suppressed than in the f 6= 0 case.

135 trum, and trispectrum of ζNG in terms of the Gaussian field ζG. The statistics of

ζG are completely specified by the two point function:

0 3 3 0 2 3 2 hζG(k)ζG(k )i = (2π) δ (k + k )PG(k) , ∆G(k) = k PG(k)/(2π ). (B.3)

The power spectrum, bispectrum, and trispectrum of ζNG are defined through

3 3 hζNG(k1)ζNG(k2)i ≡ (2π) δ (k1 + k2)PNG(k1) 3 3 hζNG(k1)ζNG(k2)ζNG(k3)i ≡ (2π) δ (k1 + k2 + k3)B(k1, k2, k3) 3 3 hζNG(k1)ζNG(k2)ζNG(k3)ζNG(k4)ic ≡ (2π) δ (k1 + k2 + k3 + k4)T (k1, k2, k3, k4)

Fourier transforming the real-space diagrammatic expressions in AppendixA we find that the non-Gaussian power spectrum is given by

  1   P (k) = P (k) (f (1))2 + f (1)f (3)hζ2 i + ... + I (k) (f (2))2 + ... + ... , NG G G ζG 2 (B.4) 2 where hζGi is the two-point function at zero separation (a constant) and we have defined, Z 3 0 1 d k 0 0 IζG (k) ≡ 3 PG(k )PG(|k + k |) . (B.5) PG(k) (2π)

For a scale-invariant spectrum, ns = 1, this becomes

( Λ2 k2 ) k ! I (k) = −∆2 ln + and hζ2 i = ∆2 ln max (B.6) ζG G k2 − Λ2 k2 − Λ2 G G Λ where k is the UV cutoff of the power spectrum and Λ ∼ 2π/L is the IR cutoff. max√ √ 2 2 For k < ekmaxΛ, hζGi > IζG (k), but for k > ekmaxΛ, hζGi < IζG (k). IζG (k) is plotted in Figure B.1 for ns = 1. The bispectrum is given by

B(k1, k2, k3) = (PG(k1)PG(k2) + 2 perm.)

  1   × (f (1))2f (2) + f (1)f (2)f (3) + (f (1))2f (4) hζ2 i + ... 2 G

1   + (J(k , k )P (k )P (k ) + 2 perm.)) (f (2))3 + ... 3 1 3 G 1 G 3

136 f (1)f (2)f (3) ! + (I(k )P (k )P (k ) + 5 perm.) + ... + ... 2 G 1 G 2 2

2 where J(k1, k2) ∼ O(∆G) is a function that depends on both the magnitudes of k1 and k2, and the angle between them

Z 3 0 1 d k 0 0 0 J(k1, k2) = 3 PG(|k1 − k |)PG(|k2 + k |)PG(k ) . (B.7) PG(k1)PG(k2) (2π)

2 2 Notice that in the squeezed limit J(kl, ks) → I(kl) + O(kl /ks ). In another squeezed 0 0 0 0 0 limit: |ks| = |ks| = |ks + ks|, J(ks, −ks − kl) + J(ks, −ks − kl) + J(ks, ks + ks) →

3βζG (ks) where,

Z 1 2 3 n −4 2 ns −2 2 0 ns −2 β (k ) = ∆ (k ) d x x s (1 + x − 2x · kˆ ) 2 (1 + x + 2x · kˆ ) 2 (B.8) ζG s 4π G s s s

So, in the squeezed limit the angular dependence vanishes and the J functions are just dependent on the magnitudes ks, kl. Finally, the trispectrum is given by

T (k1, k2, k3, k4) = (PG(k1)PG(k2)PG(k3) + 3 perm.)  3 1   (f (1))3f (3) + (f (1))2(f (3))2 + (f (1))3f (5) hζ2 i + ... 2 2 G

+ (PG(k1)PG(k2)(PG(|k1 + k3|) + PG(|k1 + k4|)) + 5 perm.)

 (1) 2 (2) 2  (1) (2) 2 (3) (1) 2 (2) (4) 2  (f ) (f ) + f (f ) f + (f ) f f hζGi + ...

 (2) 4  + (K(k1, k2, k3)PG(k1)PG(|k2 + k3|)PG(k3) + 2 perm.) (f ) + ...

+ ((IζG (k1) + IζG (k2))PG(k1)PG(k2)(PG(|k1 + k3|) + PG(|k1 + k4|)) + 5 perm.)

1  f (1)(f (2))2f (3) + ... 2

137  (1) (2) 2 (3)  + (PG(k1)PG(k2)PG(k4)J(k2, k4) + 11 perm.) f (f ) f + ...

+ (PG(k1)PG(k2)(IζG (|k1 + k3|)PG(|k1 + k3|) + IζG (|k2 + k3|)PG(|k2 + k3|))

1  +5 perm.) (f (1))2(f (3))2 + ... 2

1  + (I (k )P (k )P (k )P (k ) + 11 perm.) (f (1))2f (2)f (4) + ... ζG 1 G 1 G 2 G 3 2

2 where K ∼ O(∆G) and depends on the magnitudes of k1, k2, k3 and the relative angles between them

R d3k0 0 0 0 0 (2π)3 PG(k )PG(|k1 + k |))PG(|k1 + k2 + k |)PG(|k3 − k |) K(k1, k2, k3) = . PG(k1)PG(|k1 + k2|)PG(k3) (B.9)

In the squeezed limit needed to calculate gNL we find

0 K(ks, ks, kl) → I(kl) (B.10)

0 0 where in taking the limit we have fixed |ks| = |ks| = |ks + ks|. It is perhaps more useful to consider the local parameters as defined in terms of squeezed limits:

3 1 B(kl, ks, −kl − ks) fNL ≡ lim (B.11) 5 4 kl→0 P (ks)P (kl) 0 0 1 T (ks, −ks + kl, ks, −ks − kl) τNL ≡ lim 0 (B.12) 4 kl→0 P (ks)P (ks)P (kl) 0 0 9 1 T (ks, ks, kl, −ks − ks − kl) 1 gNL ≡ lim 0 − τNL (B.13) 25 18 kl→0 P (ks)P (ks)P (kl) 3

0 0 where in the expression for gNL we fix |ks| = |ks| = |ks + ks|. Substituting the power spectrum, bispectrum and trispectrum into Eq. (B.11)-(B.13) gives

3 f (2) f (4) f (2)f (3) ! f (2)f (3) 1 f (2)f (3) (f (2))3 ! f = + − hζ2 i+ I (k )+ − I (k )+... 5 NL 2(f (1))2 4(f (1))2 2(f (1))3 G 4(f (1))3 ζG l 4 (f (1))3 (f (1))4 ζG s

138 102

101 2 G

0 /

) 10 k ( 1 I

10-1

10-2 100 101 102 103 104 k/Λ

Figure B.1: Plotted is IζG (k), the higher-order correction to polyspectra of ζNG given 2 in Eq. (B.5), shown here for a scale invariant spectrum ∆G = const.

9 f (3) 1 f (5) 1 (f (3))2 ! (f (3))2 (f (2))2f (3) f (2)f (4) ! g = + − hζ2 i + − + I (k ) 25 NL 6(f (1))3 12 (f (1))3 4 (f (1))4 G 6(f (1))4 3(f (1))5 6(f (1))4 ζG s

1 f (2)f (4) (f (3))2 1 (f (2))2f (3) ! 1 (f (2))2f (3) + − + I (k ) + β(k ) + ... 12 (f (1))4 6(f (1))4 12 (f (1))5 ζG l 6 (f (1))5 s and

(f (2))2 f (2)f (4) (f (2))2f (3) ! 1 (f (3))2 (f (2))2f (3) ! τ = + − 2 hζ2 i + + I (k ) NL (f (1))4 (f (1))4 (f (1))5 G 2 (f (1))4 (f (1))5 ζG l (f (2))2f (3) (f (2))4 ! I (k ) + I (k0 ) + − ζG s ζG s + ... (f (1))5 (f (1))6 2

The observed values of the non-Gaussian parameters fNL, gNL, τNL include the 2 scale-independent loop contributions at O(hζGi), rewriting the series ζNG(x) = P 1 (n) n P (n) 2 n/2 2 1/2 n n! f ζG(x) as ζNG(x) = n h hζGi Hen(ζG(x)/hζGi ), where Hen are the

139 Hermite polynomials (as we have done in Eq. (2.29)) and the h(n) refer to the parameters fNL, gNL, etc., cancels these lowest order loop terms. If instead we have f (1) = 0 and f (2) 6= 0, then

3 1 4 fNL = (2) + . . . τNL = (2) 2 0 + . . . gNL = 0 + ... 5 IζG (ks)f (f ) IζG (ks)IζG (ks) (B.14)

140 Appendix C| Mapping Between Statistics in Vl and Statistics in Vs

obs We would like to calculate the local statistics, that is correlation functions of ζNG. First, we rewrite the locally observed non-Gaussian field as

! f (3)   f (4) ζobs (x) = f (1) + f (2)ζ + ζ2 − hζ2 i + ζ3 + ... ζ NG Gl 2 Gl Gl 3! Gl Gs ! 1 f (4)   f (5)   + f (2) + f (3)ζ + ζ2 − hζ2 i + ζ3 + ... ζ2 − hζ2 i 2 Gl 2! Gl Gl 3! Gl Gs Gs

! 1 f (5)   + f (3) + f (4)ζ + ζ2 − hζ2 i + ... ζ3 3! Gl 2 Gl Gl Gs

! 1 f (6)     + f (4) + f (5)ζ + ζ2 − hζ2 i + ... ζ4 − 3hζ2 i2 4! Gl 2 Gl Gl Gs Gs

+ ...

g(2)   g(3) g(4)   = g(1)ζ + ζ2 − hζ2 i + ζ3 + ζ4 − hζ4 i + ... Gs 2 Gs Gs 3! Gs 4! Gs Gs

(n) (n) where g ≡ f(ζGl). The coefficients g are equal to f up to corrections (n+1) O(f ζGl). So, the local statistics are similar to the global ones as long as m (n) (n+m) the amplitude of the background mode obeys ζGl < m!f /f . Under the approximation that the coefficients g(n) are constant across the volume

141 Vs we can use the expressions from appendixB with f → g and IζG (k) → IζGs (k) :

3 g(2) g(4) g(2)g(3) ! g(2)g(3) f obs = + − hζ2 i + I (k ) 5 NL 2(g(1))2 4(g(1))2 2(g(1))3 Gs 4(g(1))3 ζGs l 1 g(2)g(3) (g(2))3 ! + − I (k ) + ... 4 (g(1))3 (g(1))4 ζGs s

9 g(3) 1 g(5) 1 (g(3))2 ! (g(3))2 (g(2))2g(3) g(2)g(4) ! gobs = + − hζ2 i + − + I (k ) 25 NL 6(g(1))3 12 (g(1))3 4 (g(1))4 Gs 6(g(1))4 3(g(1))5 6(g(1))4 ζGs s

1 g(2)g(4) (g(3))2 1 (g(2))2g(3) ! 1 (g(2))2g(3) + − + I (k ) + β (k ) + ... 12 (g(1))4 6(g(1))4 12 (g(1))5 ζGs l 6 (g(1))5 ζGs s and

(g(2))2 g(2)g(4) (g(2))2g(3) ! 1 (g(3))2 (g(2))2g(3) ! τ obs = + − 2 hζ2 i + + I (k ) NL (g(1))4 (g(1))4 (g(1))5 Gs 2 (g(1))4 (g(1))5 ζGs l (g(2))2g(3) (g(2))4 ! I (k ) + I (k0 ) + − ζGs s ζGs s + ... (g(1))5 (g(1))6 2

1/3 where ks, kl ≥ 2π/Vs are the long and short wavelength modes used to measure fNL, gNL τNL within Vs. So that

obs (3) (2) !  (2) !2 (3)  fNL f f f f 2 = 1 + (2) − 2 (1) ζGl + 3 (1) − 2 (1)  ζGl fNL f f f f ! ! f (4) f (3)   f (3) f (4)   + − ζ2 − hζ2 i + − hζ2 i − hζ2 i 2f (2) f (1) Gl Gl f (1) 2f (2) G Gs

1 f (2) !2 f (3) + (I (k ) − I (k )) − (I (k ) + I (k ) − I (k ) − I (k )) 2 f (1) ζG s ζGs s 2f (1) ζG l ζG s ζGs l ζGs s

142 and when f (1) = 0, f (2) 6= 0,

obs 2 fNL 8ζGl + 2IζGs (kl) = (2) 2 2 + ... (C.1) fNL f (IζGs + 2ζGl)(IζGs + 2ζGl)

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157 Vita Personal Elliot Luke Nelson

Place of Birth: Minneapolis, MN Email: [email protected]

Education

2009–2015 PhD, Physics, The Pennsylvania State University. 2006–2009 B.Sc., Physics, Wheaton College, IL.

Positions

Postdoctoral Researcher, Perimeter Institute for Theoretical Physics, Water- loo, ON, 2015-2018. Visiting Graduate Fellow, Perimeter Institute for Theoretical Physics, May- October 2014. Member, Center for Theoretical and Observational Cosmology, Penn State University, 2013-2015. Member, Institute for Gravitation and the Cosmos, Penn State University, 2012-2015. Research Assistant, Institute for Gravitation and the Cosmos, Penn State University; 2012-2014. Teaching Assistant, Penn State University; 2009-2012, 2015.

Honors

NASA Pennsylvania Space Grant Consortium Graduate Research Fellowship, Penn State University (2013-2015) Honorable Mention, Peter Eklund Lectureship Award, Penn State University (2014) Edward and Rosemary Mebus Graduate Fellowship in Physics, Penn State University (2013-2014, 2014-2015) Edward M. Frymoyer Honors Fellowship, Penn State University (2012-2013) Joseph Spradley Outstanding Physics Student Award, Wheaton College (2008- 09)