PRECISION PREDICTIONS OF PRIMORDIAL POWER SPECTRA
By DANIEL BROOKER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2018 © 2018 Daniel Brooker I dedicate this to my beautiful wife Elizabeth without whom I would be hopelessly lost. ACKNOWLEDGMENTS This work would not have been possible without my wonderful advisor Richard Woodard as well as our incredible collaborators Nick Tsamis and Sergei Odintsov. In the long course of research that went into this thesis and the works which comprise it we profited at various times from conversations with: P. K. S. Dunsby, L. Patino, M. Romania, S. Shandera, M. Sloth, J. Garcia-Bellido, and M. Sasaki. This work was supported by NSF grant PHY1506513, by a travel grant from the UF International Center, by the UF’s McLaughlin fellowship, and by the UF’s Institute for Fundamental Theory.
4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... 4 LIST OF TABLES ...... 7 LIST OF FIGURES ...... 8 ABSTRACT ...... 10
CHAPTER 1 INTRODUCTION, MOTIVATION, AND BACKGROUND ...... 11 1.1 Introduction ...... 11 1.2 Cosmic Preliminaries ...... 11 1.2.1 Basic Facts and Definitions ...... 11 1.2.2 The Need for Inflation ...... 13 1.2.2.1 The horizon problem ...... 13 1.2.2.2 The flatness and monopole problems ...... 13 1.3 An Easy Way To Produce Inflation ...... 14 1.4 Generic Predictions Of Inflation ...... 17 1.4.1 Spontaneous Particle Production During Inflation ...... 17 1.4.2 Quantum Fields and Mode Equations ...... 18 1.4.3 The Constant Epsilon Solution ...... 21 1.4.4 How Valid is Constant ϵ(t) as an Approximation? ...... 23 2 MODE FUNCTIONS AND SPECTRA ...... 27 2.1 Our Evolution Equation ...... 27 2.1.1 An Evolution Equation for the Norms Squared of u(t, k) and v(t, k) .. 27 2.1.2 Factoring Out the Constant ϵ Part ...... 29 2.1.3 Simplifications ...... 30 2.2 Simple Analytic Approximations ...... 36 2.3 Improvements to Our Approximation for the Scalar Power Spectrum ...... 40 2.4 Concluding Remarks on the Power Spectra ...... 42 3 RECONSTRUCTION ...... 44 3.1 Developing The Reconstruction Proceedure ...... 44 3.2 Using The Reconstruction Procedure on Novel Spectra ...... 51 4 IMPROVING THE SINGLE SCALAR CONSISTENCY RELATION ...... 54
2 2 4.1 Constructing ∆h(k) from ∆R(k) ...... 55 4.2 Comparison Using Simulated Data ...... 60 4.3 Discussion ...... 63
5 5 PRECISION PREDICTIONS FOR PRIMORDIAL POWER SPECTRA FROM F (R) MODELS OF INFLATION ...... 65 5.1 Introduction ...... 65 5.2 Numerical Equality but Form Dependence ...... 67 5.2.1 The Model in the Jordan Frame ...... 67 5.2.2 The Model in the Einstein Frame ...... 70 5.2.3 Relating Backgrounds and Perturbation Fields ...... 72 5.2.4 Starobinsky Inflation ...... 74 5.3 Constructing Models from Power Spectra ...... 76 5.3.1 Reconstructing a Scalar Potential Model ...... 76 5.3.2 Reconstructing an f(R) Model ...... 78 5.4 Comparing Analytic and Numerical Results ...... 80 2 2 5.4.1 How We Compute ∆R(k) and ∆h(k) ...... 80 5.4.2 The Two Models ...... 83 5.4.3 Power Spectra of the Two Models ...... 84 5.4.4 Comparison with Analytic Results ...... 85 5.5 Discussion ...... 87 6 OUTLOOK AND CONCLUSIONS ...... 89 REFERENCES ...... 95 BIOGRAPHICAL SKETCH ...... 103
6 LIST OF TABLES Table page 1-1 Simple solutions to the Friedmann equation ...... 12 4-1 Simulated Data ...... 61 4-2 Predicted Results ...... 62
7 LIST OF FIGURES Figure page
2ϵ 1 2 2 1 ϵ − 1−ϵ 1-1 Graph of π (1+ϵ) Γ ( 2 + 1−ϵ )[2(1 ϵ)] as a function of ϵ...... 23 1-2 The PLANCK 2015 strength of temperature variations against their angular sizes. . 24 2-1 Graphs of the local slow roll correction factor C(ϵ) versus a better approximation. . 35 2-2 The ϵ = 0 Green’s function and the coefficient of ε′′(n) in the small ϵ form for Sh(n, k)...... 37
′ 2 ′ 2-3 The coefficients of [ε (n)] and ε (n) in the small ϵ form for Sh(n, k)...... 37 2-4 The Hubble parameter and the first slow roll parameter for a model with features. . 39 2-5 The scalar power spectrum compared with the local slow roll approximation and the scalar power spectrum compared with our analytic approximation...... 40 2-6 Our approximation for the scalar power spectrum of the step model with the non-linear and extra source terms included...... 42 2-7 The tensor power spectrum for the step model compared with the local slow roll approximation...... 43 3-1 Numerical values of exponents 1, 2 and 4 for the step model...... 45
3-2 Numerical values of exp3(n) and exp5(n) for the step model...... 46 3-3 Various choices for the left hand side of the first pass reconstruction equation for the step model...... 46 3-4 Numerical reconstructions of ln[ϵ(n)] for the power spectrum of Figure 2-5...... 48 3-5 Numerical evaluation of the integral I(s) and various approximations...... 49
2 3-6 The spectrum ∆R whose geometry we will reconstruct...... 51 3-7 The values of ln [ϵ(n)] which we have reconstructed...... 52 3-8 The spectrum from the reconstructed geometry of Figure 3-7 compared with the mock spectrum we started with...... 52 4-1 The first slow roll parameter for the step model, the resulting scalar power spectrum and the result of our analytic approximation (4–14)...... 57 5-1 Comparison of first slow roll parameter in the two frames for Starobinsky inflation. . 76 5-2 Comparison of the potentials V (φ) for Starobinsky inflation and the exponential model...... 82
8 5-3 Comparison of the scalar power spectrum for Starobinsky inflation and the exponential model...... 84 5-4 Comparison of the tensor power spectrum and the tensor-to-scalar ratio r(k) for Starobinsky inflation and the exponential model...... 85 5-5 The various spectra for Starobinsky inflation in the Jordan frame and the Einstein frame...... 85 5-6 The various spectra for Starobinsky inflation in the Jordan frame and the Einstein frame...... 85 5-7 The various spectra for the exponential model in the Jordan frame and the Einstein frame...... 86 5-8 The various spectra for the exponential model in the Jordan frame and the Einstein frame...... 86 5-9 Comparison of the exact results with the leading slow roll approximation for Starobinsky inflation...... 86 5-10 Fractional error of our linearized approximation to the scalar and tensor power spectra for Starobinsky inflation...... 87
9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRECISION PREDICTIONS OF PRIMORDIAL POWER SPECTRA By Daniel Brooker August 2018 Chair: Richard Woodard Major: Physics In this work we develop a powerful new formalism for computing the mode functions of scalars and tensors both numerically and analytically in an arbitrary inflating geometry. The formalism is used to derive simple, accurate, and analytic expressions for the primordial power spectra from a given model of inflation as well as for reconstructing an inflationary model from primordial data. Our formalism is used to derive an improved version of the single scalar consistency relation allowing us to test inflationary scenarios with sparse tensor data. Finally, our work is extended to include f(R) modified gravity models of inflation.
10 CHAPTER 1 INTRODUCTION, MOTIVATION, AND BACKGROUND 1.1 Introduction
The theory of inflation, simply put, predicts an era of rapid growth in the expansion ofthe early universe. This theory was first proposed as a means of solving three problems plaguing big-bang models of cosmology at the time: the flatness problem, the horizon problem, and the lack of monopoles. Inflation has since been supported experimentally and has become apart of the ‘standard model’ of cosmology [1–15]. In spite of being now over 35 years old however, there remain problems with the lack of more detailed predictions and the vast multiplicity of models which fit the current data for inflationary cosmology. In this work we will attempt to resolve some of these issues by developing new analytic schemes for better understanding the predictions of a generic inflationary model. This chapter is arranged as follows. Westart by defining the relevant cosmological quantities which will be necessary for our discussion of inflation. Next we will examine the observational reasons why a theory of inflation isdesirable for our understanding of cosmology. We will then examine the background geometry for a general scalar potential dominated cosmology and see how such a model can produce inflation. Finally we will close the introduction by enumerating the predictions of single scalar inflation and we will point the way toward the original work which is contained in the rest of this dissertation. 1.2 Cosmic Preliminaries
1.2.1 Basic Facts and Definitions
We will not have time to study the whole of homogeneous cosmology here, a topic which is the subject of many great textbooks [16, 17]. We will instead simply recall the basic information which will be necessary for the rest of this work. Current data suggests that on the largest scale the universe is spatially flat, homogeneous, and isotropic18 [ –26]. The metric which has all of these properties is called the spatially flat Friedmann Robertson Walker (FRW)
11 metric. It has the following simple form:
2 µ ν 2 2 2 2 2 ds = gµνdx dx = −dt + a (t)(dx + dy + dz ) (1–1)
It is useful to define the following combinations of a(t) and its first two derivatives:
a˙(t) H(t) ≡ (1–2) a(t) H˙ (t) ϵ(t) ≡ − . (1–3) H(t)2
The two nontrivial Einstein equations for the spatially flat FRW metric are:
3H2(t) = 8πGρ (1–4)
−2H˙ (t) − 3H2 = 8πGp (1–5)
where ρ and p are the density and pressure of the fluids which comprise the universe. Ifwe take these fluids to be perfect fluids with equation ofstate p = w × ρ then one finds that we can get any simple behavior for a(t) that we wish by choosing the equation of state parameter (w). A few commonly referenced solutions are given below in Table (1-1).
Table 1-1. Simple solutions to the Friedmann equation
Name a(t)/ai w
2 Matter Domination t 3 0
1 1 2 Radiation Domination t 3 − De Sitter e Hit −1
Inflation in this context simply corresponds to any model where a(t) is accelerating or in other words (˙a, a¨ > 0). This is equivalent to the statement that (H > 0, ϵ < 1). Making use of Equations (1–2, 1–4, 1–5) we see that yet another equivalent requirement for producing − ρ inflation is that p < 3 . Now that we have the necessary definitions to talk about inflation, let us discuss why a theory of inflation is useful for understanding cosmology.
12 1.2.2 The Need for Inflation
1.2.2.1 The horizon problem
The extreme degree of isotropy in the cosmic microwave background (CMB) once posed a theoretical problem for cosmologists.To understand why this is a problem consider the propagation of light in the spatially flat FRW geometry. Since light moves along null geodesics we will have: dt ds2 = 0 −→ = dr (1–6) a(t)
We can integrate (1–6) to find the size of the light cones at atime t2 for light which was
emitted at a time t1. We have: ∫ [ ] t2 1 1 t2 r (t , t ) = dt = − + O (˙ϵ) (1–7) H 1 2 a(t) Ha (1 − ϵ) t1 t1
The horizon problem results from the following two assumptions: (1) the universe has a finite age, (2) the universe was either matter or radiation dominated in its early history.In that case one finds that the integral1–7 in( ) will be dominated by its upper limit hence
r(tCMB, t0) ≫ r(0, tCMB). This would imply that we are seeing many patches on the sky which are not in causal contact and yet the homogeneity of the CMB implies that the whole sky is very nearly in thermal equilibrium. This is a massive fine tuning problem which is avoided by adding a phase of inflation in the early universe which allows the integral1–7 ( ) to be
dominated by its lower bound so that one can have r(tCMB, t0) < r(0, tCMB). In general one finds the the scale factor must grow by a factor of atleast e60 for the whole sky to be in causal contact today [16]. 1.2.2.2 The flatness and monopole problems
A phase of inflation in the early universe explains two other puzzles right at theoutset. These are known as the flatness problem and the monopole problem. In both cases thereis a relevant parameter which will be driven to zero dynamically during inflation. The flatness puzzle asks why the universe is spatially flat. In general the universe might be either positively or negatively curved and the flatness that we observe poses a fine tuning problem without
13 inflation. It is simple to include the effects of spatial curvature into our cosmological model. We modify the metric and Einstein equations as: ( ) dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (1–8) 1 − kr2 3k2 3H2(t) + = 8πGρ (1–9) a2(t) 3k2 −2H˙ (t) − 3H2 − = 8πGp (1–10) a2(t)
with k = {−1, 0, 1}. Since the spatial curvature always enters the Einstein equations with a factor of a−2 one typically finds that a sufficient amount of inflation to solve thehorizon problem will also cause any spatial curvature to be unobservable at the present. The monopole problem asks why magnetic monopoles which are ubiquitous in grand unified theories have not been observed in nature. Much the same as the flatness problem, as long as all ofthe monopoles are produced before inflation, the density (which scales as a−3) will fall below presently observable values during inflation. 1.3 An Easy Way To Produce Inflation
We will now show one simple and extremely popular way to produce inflation. We will discuss other models which produce inflation when we come to modified f(R) gravity in Chapter 5. For now, inflation is accomplished by having a scalar potential dominated universe in which a homogeneous scalar field rolls slowly down its potential6 [ , 7]. The Lagrangian for a general scalar potential model is,
1 √ 1 √ √ L = R −g − ∂ φ∂ φgµν −g − V (φ) −g . (1–11) 16πG 2 µ ν
We assume a homogeneous, isotropic and spatially flat background characterized by φ0(t). Note that the assumption of homogeneity removes any dependence on the spatial coordinates from the background scalar field. Recalling the stress energy tensor for a scalar field and
14 inserting it into (1–4,1–5) we find
ρ =φ ˙ 2(t) + V (φ(t)) , p =φ ˙ 2(t) − V (φ(t)) , (1–12) [ ] 2 1 2 3H = 8πG φ˙ 0 + V (φ0) , (1–13) [2 ] 1 −2H˙ − 3H2 = 8πG φ˙ 2 − V (φ ) . (1–14) 2 0 0
From (1–13,1–14) one sees immediately that if the potential energy of the field dominates over the kinetic energy then one will have an equation of state which produces inflation. If H(t) is known, or if a particular form is desired, there is a simple way of using Equations (1–13-1–14) to construct a potential V (φ) which supports any function H(t) that obeys H˙ (t) < 0 [27–31]. One first adds (1–13) and (1–14) to obtain an equation for the scalar background, √ ∫ t ′ ′ −H˙ (t ) φ0(t) = φi dt . (1–15) ti 16πG
By graphing this relation and then rotating the graph by 90◦ one can easily invert (1–15) to solve for time t(φ). The final step is to subtract (1–14) from (1–13) to find the potential, [ ] 1 V (φ) = H˙ (t)+3H2(t) . (1–16) 8πG t=t(φ)
Rather than specifying the expansion history a(t) and using relations (1–15-1–16) to reconstruct the potential, it is more usual to specify the potential and then solve for the expansion history a(t). This is greatly facilitated by making the slow roll approximation, √ [ ] 1 1 V ′(φ) 2 H ≈ 8πGV (φ) , ϵ ≈ .1 (1–17) 3 16πG V (φ)
1 Some authors will distinguish between the Hubble parameterization definition of ϵ (1–3) ′ which we have used in the above equation, and a potential formalism ϵV ≡ V (φ)/V (φ). In either case ϵ is colloquially referred to as the ’first slow roll parameter’ and to avoid confusion we will always use the geometric (or Hubble) parameterization. In the context of slow roll the two are equivalent up to higher slow roll corrections [32].
15 N It is desirable to express the scale factor a = aie in terms of the number of e-foldings N from the beginning of inflation. Then one can use the slow roll approximation (1–17) to solve for
the scalar’s evolution from initial value φi by inverting the relation, ∫ φ − V (ψ) N = 8πG dψ ′ . (1–18) φi V (ψ)
An important special case is power law potentials,
[ ] α α ϵi 4ϵi 4 V (φ) = Aφ =⇒ ϵ = ,H = Hi 1− N . (1–19) − 4ϵi 1 α N α More generally, we will also consider models which violate the slow roll for brief periods of time.To accomplish this we must solve the full Klein-Gordon equation for the homogeneous scalar field numerically. In evolving these models it is best to convert to dimensionless fields and potentials, √ ψ(n) ≡ 8πG φ(t) ,U(ψ) ≡ (8πG)2V (φ) . (1–20)
When we also convert from co-moving time t to the number of e-foldings n ≡ ln[a(t)/ai], the exact scalar equation becomes,2 ( ) ( ) 1 1 U ′(ψ) ψ′′ + 3− ψ′2 ψ′ + 3− ψ′2 = 0 . (1–21) 2 2 U(ψ)
The Hubble parameter and first slow roll parameter follow from ψ(n) through the exact relations, U(ψ) 1 8πGH2 = , ϵ = ψ′2 . (1–22) − 1 ′2 2 3 2 ψ Higher derivatives of ϵ (which we will need later) are best evaluated by taking the derivatives of (1–22) and using the equation of motion (1–21) to remove things like (ψ′′, ψ′′′...). Using this trick we can reliably compute the geometry for any scalar potential model of inflation.
2 One really only needs ψ(0) to solve (1–21). The other initial condition can be taken as ψ′(0) = U ′(ψ(0))/U(ψ(0)) using the slow roll approximation.
16 Now that we know one way to produce inflation we must consider what will happen during inflation beyond the background level so that we can make further predictions fromthe theory. 1.4 Generic Predictions Of Inflation
1.4.1 Spontaneous Particle Production During Inflation
An important prediction of inflation is that the genesis of perturbations in cosmology can be attributed to the amplification of quantum fluctuations as the universe expands during inflation. This is important since these perturbations provide the seeds for galaxy formation and large scale structure which are observed at the present. Let us see how these perturbations are formed then we will consider how to describe their statistics. We start by considering how the expansion of space will modify the energy-time uncertainty relation. In flat space we have:
1 1 ∆E∆t ≥ −→ ∆t ∼ √ (1–23) 2 m2 + k2
In an expanding universe however the coordinate distance between two events and hence the wavelength and energy of a particle which emerges from the vacuum become time dependent. Thus we have in an expanding space time: ∫ ∫ √ t+∆t t+∆t k2 1 ′ ′ ′ 2 ≥ dt E(t , k) = dt m + 2 . (1–24) t t a 2
We can simplify this expression by taking the limit of massless particles as well as the De Sitter limit without altering the conclusion. In this case we find the following:
k E(t, k) = (1–25) a(t) ∫ ∫ t+∆t t+∆t k k [ ] ′ ′ ′ −Hi∆t dt E(t , k) = dt ′ = 1 − e ∼ 1 (1–26) Hit t t aie Hia
From (1–26) we can infer that if Hia > k then the persistence time of a pair of particles which emerged from the vacuum spontaneously can tend towards infinity. Thus we find that an inflating geometry will spontaneously emit particles. In this way we can start inflation in a vacuum and by the end of inflation the universe will be full of matter. Modes which
17 have extremely high occupation numbers will become effectively classical providing the initial inhomogeneities which evolve to into all of the structure observed in the universe today. It is historically interesting to note that the existence of the primordial perturbation spectra were actually predicted before the theory of inflation was even developed33 [ , 34]. From the stand point of quantum field theory we will be able to understand the cosmological implications of this particle creation effect by looking at the N-point correlators of fields which experience the inflationary particle production. There are two such fields of interest in single scalar inflation. They are the minimally coupled scalar and the graviton. It is the study of these perturbations which will comprise the main body of this work. We will close this introductory chapter by looking at the generation of perturbations for scalars and tensors as it was understood previously within the context of a particular approximation for the inflationary geometry called the constant ϵ approximation. Examining the limitations of this approximation will point the way for the work which is to come. 1.4.2 Quantum Fields and Mode Equations
In this section we discuss the formulation of quantum gravity for single scalar inflation and define the correlators which correspond to the scalar and tensor power spectra. Aswe have shown, there is a correspondence between scalar potentials and FRW geometries thus specifying either the potential or the geometry will specify the model completely. We will work exclusively with geometric variables. Since General relativity is a gauge theory we will need to pick a gauge to work in. A convenient choice will be to fix the scalar field to follow its background trajectory and to fix the tensor perturbations to be transverse35 [ –37]. The conditions are expressed mathematically as:
e e φ(t, ⃗x) = φ0(t) (1–27) e e ∂ihij(t, ⃗x) = 0 . (1–28)
18 After enforcing these gauge conditions the the g00 and g0i components of the metric are constrained fields and the perturbed spatial metric may be written as: [ ] 2 2ζ(t,k) h(t,k) gij = a(t) e e ij (1–29)
where ζ carries the inflaton degree of freedom and hij carries the two graviton degrees of freedom. Since we wish to study quantum effects we must promote these perturbations to quantum fields. The quadratic Lagrangians which we quantize are[38]:
2 ϵa 2 ∂kζ∂kζ L 2 = (ζ˙ − ) (1–30) ζ 8πG a2
2 a ∂khij∂khij L 2 = (h˙ h˙ − ) (1–31) h 64πG ij ij a2 Next we expand into and subsequently quantize the normal modes of these fields by writing: ∫ √ d3k ζ(t, ⃗x) = 4πG [v(t, k)ei⃗k·xα(⃗k) + v∗(t, k)e−i⃗k·xα†(⃗k)] (1–32) (2π)3 ∫ √ d3k ∑ h (t, ⃗x) = 32πG [u(t, k)ei⃗k·⃗xϵ (⃗k, λ)β(⃗k, λ) + c.c.] (1–33) ij (2π)3 ij λ= In the above expression we have used α and β as the creation/annihilation operators and
ϵij is the polarization tensor for flat space gravitons. The only thing left to specify isthe mode functions u(t,⃗k) and v(t,⃗k). They will be given as solutions of the following differential equations, subject to Wronskian normalization:
k2 i u¨ − 3Hu˙ + u = 0 ,W = uu˙ ∗ − uu˙ ∗ = (1–34) a2 u a3 ϵ˙ k2 i v¨ + (3H + )v ˙ + v = 0 ,W = vv˙ ∗ − vv˙ ∗ = . (1–35) ϵ a2 v ϵa3 It is important to note that the scalar mode function v is related to its tensor cousin u by a simple transformation given by [39, 40]:
√ ∂ 1 ∂ a(t) −→ ϵ(t)a(t) , −→ √ (1–36) ∂t ϵ(t) ∂t
19 The primordial power spectra will correspond to the two point correlators of these fields which are given by: ∫ k3 2 ≡ 3 −i⃗k·⃗x | | ∆h(t, k) 2 d xe < Ω hij(t, ⃗x)hij(t, 0) Ω > (1–37) 2π ∫ k3 ∆2 (t, k) ≡ d3xe−i⃗k·⃗x < Ω|ζ(t, ⃗x)ζ(t, 0)|Ω > (1–38) R 2π2
In order to compute the primordial spectra at tree order we simply substituting in our field definition1–33 ( ) to find:
k3 ∆2 (t, k) = × 32πG × 2 × |u(t, k)|2 (1–39) h 2π2 k3 ∆2 (t, k) = × 2πG × 2 × |v(t, k)|2 (1–40) R 2π2
At early times the universe is very small (aH ≪ k) so the frequency term will dominate the friction term in Equations (1–34, 1–35) and we will use the WKB approximation to set the initial functional forms for u and v as: [ ∫ ] 1 t dt′ v(t, k) −−−−→ √ exp −ik , (1–41) k ≫ Ha 2 ′ 2kϵ(t)a (t) tia(t ) [ ∫ ] 1 t dt′ u(t, k) −−−−→ √ exp −ik . (1–42) k ≫ Ha 2 ′ 2ka (t) tia(t )
When we evolve the early time forms to late times (aH ≫ k) the frequency term tends to zero and the solution approaches a constant. This process is called ’freeze in’ and it is the reason why the spectra are still observable today. Note that since freeze in is determined by the quantity k/aH the time at which it occurs will be different for each mode. It is commonly assumed that the modes which contribute to CMB anisotropies experienced Hubble crossing (k = Ha) 50-60 e-foldings before the end of inflation41 [ ]. Many authors will work exclusively with the time independent spectra which simply result from evaluating (1–39,1–40) at very late times in inflation when all the modes of interest are frozen in. To make contact with observations the time independent spectra are convolved with a transfer function which encodes the effects of post-inflationary cosmology. Observing these
20 spectra today then allows cosmologists to not only fit a model of late time (post-inflation) cosmology but also to deconvolve the late time effects and fit the primordial power spectra from CMB temperature and polarization data [42–44]. 1.4.3 The Constant Epsilon Solution
The task of predicting the spectra of scalar and tensor perturbations generated during inflation has now been turned into the problem of solving the mode Equations1–34 ( , 1–35). In general these mode equations do not have analytic solutions since ϵ can be any function during inflation as long as it stays between zero and one. An important special case inwhich the mode equations can be found analytically is when ϵ(t) is constant, for which the Hubble parameter and scale factor are,
[ ] 1 Hi ϵi ϵ(t) = ϵi =⇒ H(t) = , a(t) = 1+ϵiHi∆t , (1–43) 1+ϵiHi∆t
ϵ where ∆t ≡ t − ti. Note that the combination H(t)[a(t)] is constant. The appropriate tensor mode function for constant ϵ(t) is, √ 1 πz (1) k 3 ϵ u0(t, k) = √ × H (z) , z(t, k) ≡ , ν ≡ + . (1–44) a(t) 2k 2 ν (1−ϵ)Ha 2 1−ϵ
From the small argument expansion of the Hankel function we can infer the constant late time limit of (1–44), √ ( ) πz iΓ(ν) 2 ν u (t, k) −−−−→ ×− , (1–45) 0 k ≪ Ha 4a2k π z 1 ϵ [ ] ϵ [ ϵ ] 1 i(1+ϵ)Γ( + ) 1−ϵ H(t)a (t) 1−ϵ = − √ 2 1−ϵ 2(1−ϵ) . (1–46) 2πk3 kϵ
It is usual to evaluate the constant factor of H(t)aϵ(t) at horizon crossing,
1 ϵ [ ] ϵ −−−→ H(tk) i(1+ϵ)Γ( 2 + 1−ϵ ) 1−ϵ u0(t, k) k ≪ Ha √ ×− √ 2(1−ϵ) . (1–47) 2k3 π
The transformation rule (1–36) tells us that once we know u0 in this case we can find the
corresponding v0. Throughout this work we will see that this is a running theme that the two spectra are really two sides of the same coin in the case of single scalar inflation. Thus the
21 constant ϵ scalar modes are given by: √ 1 πz (1) k 3 ϵ v0(t, k) = √ × H (z) , z(t, k) ≡ , ν ≡ + , (1–48) a(t) 2kϵ 2 ν (1−ϵ)Ha 2 1−ϵ 1 ϵ [ ] ϵ −−−→ H(tk) i(1+ϵ)Γ( 2 + 1−ϵ ) 1−ϵ v0(t, k) k ≪ Ha √ ×− √ 2(1−ϵ) . (1–49) 2k3ϵ π
Substituting (1–47,1–49) in expressions (1–39-1–40) gives the famous constant ϵ predictions for the power spectra,
( ~ ) 2 2 2 1 ϵ [ ] 2ϵ 2 GH (tk) (1+ϵ) Γ ( 2 + 1−ϵ ) 1−ϵ ∆R(k) = × × 2(1−ϵ) , (1–50) ϵ˙=0 c5 πϵ π ( ) 2 2 1 ϵ [ ] 2ϵ ~ 16 (1+ϵ) Γ ( + ) 1−ϵ 2 × 2 × 2 1−ϵ − ∆h(k) = GH (tk) 2(1 ϵ) . (1–51) ϵ˙=0 c5 π π
One consequence of the relation between the scalar and tensor modes (1–36) is that the two spectra will be related in the same way that the mode functions are. Inspecting Equations (1–50,1–51) we see that the following relation holds:
∆2 (k) h ≡ 2 r(k) = 16ϵ . (1–52) ∆R(k)
This can be rewritten in a form which is independent of the geometric quantities by defining:
∂ log [∆2 (k)] n ≡ h −→ r = −8n (1–53) T ∂ log (k) T
This form is excellent because it provides a relationship between two measurable quantities, and it must hold for all models of inflation whose perturbations obey1–34 ( ,1–35). This will provide a consistency check for models of inflation and we will explore this in depth in Chapter 4. The final factor in expressions (1–50,1–51) contains an ϵ-dependent correction which is not usually quoted because it is so near unity for small ϵ,
2 2 1 ϵ [ ] 2ϵ (1+ϵ) Γ ( + ) − C(ϵ) ≡ 2 1−ϵ 2(1−ϵ) 1 ϵ . (1–54) π
Figure 1-1 shows the dependence of C(ϵ) versus ϵ for the full inflationary range of 0 ≤ ϵ < 1. Note that C(ϵ) is a monotonically decreasing function of ϵ. In particular, it goes to zero for
22 2ϵ 1 2 2 1 ϵ − 1−ϵ Figure 1-1. Graph of π (1+ϵ) Γ ( 2 + 1−ϵ )[2(1 ϵ)] as a function of ϵ.
ϵ → 1−. If we assume the single-scalar relation of r = 16ϵ then the current upper bound of r < 0.09 implies ϵ < 0.0056. At this upper bound the constant ϵ correction factor is about 0.997. It would be even closer to unity for smaller vales of ϵ. 1.4.4 How Valid is Constant ϵ(t) as an Approximation?
Having now explored the constant ϵ solutions to the mode equations and the resulting form of the tensor and scalar spectra, we need to now consider the extent to which this solution can be used to approximate actual models of inflation. The monomial potentials whose geometry are given in (1–19) can be seen to be very slowly varrying. For a ϕ2 potential with ϵ = 0.005 at a time 60 e-foldings before the end of inflation, the fractional change in ϵ over the next ten e-foldings will be only 11%. We can do even better than this however by approximating for each mode a constant ϵ geometry with a value of ϵ corresponding
to ϵ = ϵ(tk). This approximation goes by many names including the constant epsilon approximation and the Bessel approximation or the horizon crossing approximation and it has accuracy on the order of a few percent [45–47]. More generally one can see from (1–19) that
23 the following relations hold:
32 16 4 α ⇒ ′′ ≃ 3 ′2 ≃ 4 ′ ≃ 2 V (φ) = Aφ = ϵ 2 ϵ , ϵ 2 ϵ , ϵ ϵ , (1–55) ( ) α ( α) α ϵ′ ′ 16 ϵ′ 2 16 ϵ′ 4 =⇒ ≃ ϵ2 , ≃ ϵ2 , ≃ ϵ . (1–56) ϵ α2 ϵ α2 ϵ α
In fact, for any model which obeys slow roll it will be the case that derivatives of ϵ are much smaller than ϵ itself and we can safely approximate the solutions to the mode equations and hence the power spectra according to (1–50,1–51). The question then is, does the data support slow roll as a model of inflationary dynamics? The answer is not so simple as we might hope. There are two anomalous features currently observed in the scalar spectrum which can be observed in the temperature-temperature (TT) correlations of the CMB at large angular scales.They can be seen in Figure 1-2 at l = 22 and l = 40.
Figure 1-2. The PLANCK 2015 strength of temperature variations against their angular sizes. The PLANCK 2015 [25] strength of temperature variations (vertical) against their angular sizes (horizontal). The line is the standard cosmological model, the dots are the data.
It has been known for some time that features in the power spectrum can be induced by nontrivial geometries during inflation48 [ –50] however such models will necessarily violate slow
24 roll. It has been confirmed independently from both WMAP and Planck that these features are present and there are even models of how they are produced [50–55]. Before this work there were two main approaches to dealing with models which contain slow roll violations. The oldest and most straightforward approach to computing the primordial power spectra beyond slow roll is of course to solve the mode equations in a purely numerical fashion [45, 56–59]. This has been done in a number of different ways both to characterize the validity of slow roll and to study slow roll violating models. Our formalism will actually yield an entirely new way to solve the mode equations numerically which is explicitly real and nonlinear as opposed to complex but linear. Another formalism which is commonly used for understanding slow roll violating geometries is the ’Generalized Slow Roll’ (GSR) formalism which develops a perturbation theory for the scalar and tensor mode functions themselves and it has been carried out to second order [60–63]. Further, GSR has been extended to include all inflation models which involve only a single scalar degree of freedom [64–66]. For applications GSR has been used to study the bispectra (three point correlator), reconstruction of the inflationary geometry, and recently models of inflation which produce primordial black holes67 [ –70]. Our formalism is also capable of making all of these studies, however we feel that our approach is advantageous for a number of reasons. In particular the GSR approach is rather ad-hoc and relies motivating the results after the fact by comparing with numerical results whereas in our formalism everything will be derived in a straightforward fashion. What has been lacking until now was a good analytic understanding of how models of inflation lead to features in the spectra as well as a quick and simple way to compute the spectra for such models. Chapter 2 will be devoted to deriving a new formalism for computing the primordial power spectra which addresses these needs. In chapter 3 we will address the question of how to take the primordial spectra and determine which geometry produced them. Finally Chapters 4 and 5 will focus on applications of our formalism namely improving the single scalar consistency relation (1–53) and on extending our formalism to other models of inflation. We will postpone doing a detailed
25 comparison between our work and previous works until the conclusion in Chapter 6 so that our formalism may be fully developed first.
26 CHAPTER 2 MODE FUNCTIONS AND SPECTRA
2.1 Our Evolution Equation
In this chapter we give full analytic and numerical treatment for the scalar and tensor mode functions in an arbitrary inflating geometry. We begin by reviewing the derivation of an evolution equation for the norms squared M(t, k) ≡ |u(t, k)|2 and N(t, k) ≡ |v(t, k)|2 [40, 73]. We will factorize the two functions M(t, k), N(t, k) to separate the local slow roll corrections which we covered in the introduction from the non-local corrections which we wish to understand. By linearizing the resulting equations we derive excellent approximations for the power spectra in a general inflationary expansion history. Finally we will close by presenting numerical results for several models of inflation some of which have non-negligible corrections, and by giving an error analysis for our analytic results. 2.1.1 An Evolution Equation for the Norms Squared of u(t, k) and v(t, k)
The power spectra (1–39, 1–40) depend upon the norm-squared of the tensor mode functions u(t, k) and v(t, k). It is numerically wasteful to follow the irrelevant phases using the evolution Equations (1–34, 1–35), especially during the early time regime of k ≫ H(t)a(t) when oscillations are rapid. The better strategy is to use (1–34) to derive an equation for M(t, k) ≡ |u(t, k)|2 directly. Due to the relation (1–36) all of what follows can be repeated with the exact same steps for N(t, k) ≡ |v(t, k)|2 and the results will be very similar. We will do the calculation for the tensor modes and then present the scalar results at the end and highlight the differences between the two then. We begin by computing the first twotime
Compiled and edited with permission from the following works [46, 47, 71, 72].
27 derivatives,
M˙ (t, k) = u(t, k)×u˙ ∗(t, k) +u ˙(t, k)×u∗(t, k) , (2–1)
M¨ (t, k) = u(t, k)×u¨∗(t, k) + 2u ˙(t, k)×u˙ ∗(t, k) +u ¨(t, k)×u∗(t, k) . (2–2)
Now use (1–34) to eliminate u¨ and u¨∗ in (2–2),
2k2 M¨ = −3HM˙ − M + 2u ˙u˙ ∗ . (2–3) a2
Squaring (2–1) and subtracting the square of the Wronskian (1–34) gives u˙u˙ ∗,
M˙ 2 = +u2u˙ ∗2 + 2Mu˙u˙ ∗ +u ˙ 2u∗2 , (2–4) 1 = −u2u˙ ∗2 + 2Mu˙u˙ ∗ − u˙ 2u∗2 . (2–5) a6
Hence the desired evolution equation for M(t, k) is [40, 73], [ ] 2k2 1 1 M¨ + 3HM˙ + M = M˙ 2 + . (2–6) a2 2M a6
As already noted, the transformation (1–36) converts (2–6) into an equation for the norm-squared of the scalar mode function N(t, k) ≡ |v(t, k)|2, so both power spectra follow from M(t, k). The equation for N(t, k) is: ( ) [ ] ϵ˙ 2k2 1 1 N¨ + 3H + N˙ + N = N˙ 2 + . (2–7) ϵ a2 2N ϵ2a6
Note that the only differences are the presence of an extra 1/ϵ2 in the final term and an additional friction term on the left hand side. One indication of how much more efficient it is to evolve2–6 ( ) than (1–34) comes from comparing the asymptotic expansions of u(t, k) and M(t, k) in the early time regime of k ≫ H(t)a(t). The expansion for u(t, k) is in powers of 1/k and is not even local at first
28 order, { } ∫ ( ) − t dt′ iα(t) β(t) 1 exp[ ik ′ ] u(t, k) = 1 + + + O × √ ti a(t ) , (2–8) k k2 k3 2ka2(t) ∫ [ ] 1 t α(t) = dt′ 2−ϵ(t′) H2(t′)a(t′) , (2–9) 2 ti [ ] 1 1 β(t) = − α2(t) + 2−ϵ(t) H2(t)a2(t) . (2–10) 2 4
In contrast, M(t, k) gives a series in 1/k2 which is local to all orders, { } ( ) α(t) β(t) 1 1 M(t, k) = 1 + + + O × , (2–11) k2 k4 k6 2ka2(t) ( ) 1 α(t) = 1− ϵ H2a2 , (2–12) [ 2 ] ( )( ) 9 2 1 9˙ϵ 3ϵϵ˙ ϵ¨ β(t) = ϵ 1− ϵ 1− ϵ + − + H4a4 . (2–13) 4 3 2 8H 4H 8H2
2.1.2 Factoring Out the Constant ϵ Part
A further improvement comes by factoring out an (at this stage) arbitrary approximate solution, M0(t, k), to derive a damped, driven oscillator equation (with small nonlinearities) for the residual exponent. We begin by writing,
M(t, k) ≡ M0(t, k) × ∆M(t, k) . (2–14)
Differentiating (2–14) results in the relations,
˙ ˙ ˙ M = M0 × ∆M + M0 × ∆M, (2–15) ¨ ¨ ˙ ˙ ¨ M = M0 × ∆M + 2M0 × ∆M + M0∆M, (2–16) M˙ 2 M˙ 2 ∆M˙ 2 = 0 × ∆M + M˙ × ∆M˙ + M × , (2–17) 2M 2M 0 0 2∆M 0 [ ] 1 1 × 1 × − 1 6 = 6 ∆M + 6 ∆M + . (2–18) 2a M 2a M0 2a M0 ∆M
29 Substituting relations (2–15-2–18) into (2–6) and dividing by M(t, k) gives, [ ] ( ) [ ] ¨ ˙ ˙ ˙ 2 ∆M M0 ∆M − 1 ∆M 1 − 1 + 3H + + 6 2 1 2 ∆M M0 ∆M 2 ∆M 2a M0 ∆M ( ) ¨ ˙ 2 ˙ 2 −M0 − M0 − 2k 1 M0 1 ≡ = 3H 2 + + 6 2 Sh(t, k) . (2–19) M0 M0 a 2 M0 2a M0
This is an evolution equation for ∆M(t, k), which is driven by a source Sh(t, k). 2.1.3 Simplifications
Relation (2–19) can be improved by changing to the dimensionless time parameter
n = ln[a(t)/ai] and by expressing ∆M(t, k) in terms of a new dependent variable h(t, k) as ≡ − 1 ∆M(t, k) exp[ 2 h(t, k)].This leaves us with:
′ [ ] − 1 h ′′ ω ′ 2 1 ′2 2 h M = M ×e 2 =⇒ h − h + ω h = S + h − ω e −1−h . (2–20) 0 ω h 4
Here the frequency ω(n, k) and the tensor source Sh(n, k) are, ( ) ( ) 1 ω′ ′ ω′ 2 4k2 ≡ ⇒ − ′ − − 2 − 2 ω 3 = Sh = 2 + + 2ϵ (3 ϵ) + 2 2 ω . (2–21) Ha M0 ω ω H a
The source of course is the same as the source in (2–19) but for simplicity we have redefined many of the terms. Evaluating h(n, k) numerically is rather straight forward. The Equation (2–20) is ordinary, second order, and quasi-linear so there a vast number of ways to treat it on the computer. It is more desirable however at this point to develop a perturbative solution as this will yield simple analytic expressions for the power spectra. It is easy to develop a Green’s function solution to the linearized version of (2–20) so we will start our perturbation theory by doing just that. Note that the homogeneous equation takes the form,
ω′ χ′′ − χ′ + ω2χ = 0 , χ′ ≡ ∂ χ(n, k) , ω′ ≡ ∂ ω(n, k) . (2–22) ω n n
The two linearly independent solutions of (2–22) can be expressed in terms of the integral of
ω(n, nk), ∫ [ n ] ⇒ ′ − ′ χ(n, nk) = exp i dm ω(m, nk) = χ+χ− χ+χ− = 2iω . (2–23) 0
30 Hence the retarded Green’s function we seek is, ∫ ′ [ n ] ′ θ(n−n ) G(n; n ) = ′ sin dm ω(m, nk) . (2–24) ω(n , nk) n′
And the Green’s function solution to the linearized (2–20) is, ∫ ∫ N [ N ] ′ ′ Sh(n, Nk) h(t, k) = dn sin dn ω(n ,Nk) . (2–25) 0 n ω(n, Nk)
It is an amazing fact that an exact Green’s function exists for the left hand side of Equation
(2–20), valid for any choice of the approximate solution M0 [46] and we have now constructed
it. This permits us to solve (2–20) perturbatively h = h1 + h2 + ... by expanding in the nonlinear terms, ∫ n h1(n, k) = dm Gh(n; m)Sh(m, k) , (2–26) 0 { } ∫ [ ] [ ] n 2 2 1 ′ − 1 h2(n, k) = dm Gh(n; m) h1(m, k) ω(m, k)h1(m, k) . (2–27) 0 4 2
Now we quickly present the equivalent results for scalars. The steps are the exact same so we will gloss over the details for the sake of expediency. Factoring out by an arbitrary
approximate solution N0(t, k) produces another damped, driven oscillator equation for the residual exponent,
′ [ ] − 1 g ′′ Ω ′ 2 1 ′2 2 g N = N ×e 2 =⇒ g − g + Ω g = S + g − Ω e −1−g . (2–28) 0 Ω g 4
Here the frequency Ω(n, k) and the scalar source Sg(n, k) are,
1 Ω ≡ , (2–29) ϵHa3N ( )0 ( ) ( ) ( ) Ω′ ′ Ω′ 2 ϵ′ 2 ϵ′ ′ 4k2 S = −2 + + 2ϵ′ − 3−ϵ+ − 2 + − Ω2 . (2–30) g Ω Ω ϵ ϵ H2a2
Making the replacement ω → Ω in (2–25) gives an exact Green’s function which is valid for N any choice of 0, [∫ ] θ(n−m) n Gg(n; m) = sin dℓ Ω(ℓ, k) . (2–31) Ω(m, k) 0
31 And we can of course develop a perturbative solution to (2–28) g = g1 + g2 + ... , ∫ n g1(n, k) = dm Gg(n; m)Sg(m, k) , (2–32) 0 { } ∫ [ ] [ ] n 2 2 1 ′ − 1 g2(n, k) = dm Gg(n; m) g1(m, k) Ω(m, k)g1(m, k) . (2–33) 0 4 2
sectionChoosing M0(t, k) and N0(t, k) Effectively The formalism of the previous section is valid for all choices of the approximate solutions
M0(t, k) and N0(t, k). Of course the correction exponents h(n, k) and g(n, k) will be smaller if the zeroth order solutions are more carefully chosen. In previous work we used the instantaneously constant ϵ solutions [46, 47], ( ) ( ) z(t, k)H ν(t), z(t, k) z(t, k)H ν(t), z(t, k) M (t, k) ≡ , N (t, k) ≡ , (2–34) inst 2ka2(t) inst 2kϵ(t)a2(t) where we define,
π 2 1 1 k H(ν, z) ≡ H(1)(z) , ν(t) ≡ + , z(t, k) ≡ . (2–35) 2 ν 2 1−ϵ(t) [1−ϵ(t)]H(t)a(t)
However, the choice (2–34) has the undesirable effect of complicating the late time limits. The physical quantities M(t, k) and N (t, k) freeze in to constant values soon after first horizon crossing, but continued evolution in ϵ(t) prevents M0(t, k) and N0(t, k) from approaching constants. Hence the residual exponents h(n, k) and g(n, k) must evolve so as to cancel this effect. We can make the late time limits simpler by adopting a piecewise continuous choice for the approximate solutions,
M0(t, k) = θ(tk −t)Minst(t, k) + θ(t−tk)M inst(t, k) , (2–36)
N0(t, k) = θ(tk −t)Ninst(t, k) + θ(t−tk)N inst(t, k) . (2–37)
32 By M inst(t, k) and N inst(t, k) we mean the solutions which would pertain for the ersatz geometry,
∆n −ϵk∆n a(n) = a(n) = ake , H(n) = Hke , ϵ(n) = ϵk . (2–38)
Here and henceforth ∆n ≡ n − nk stands for the number of e-foldings from horizon crossing. To be explicit about the over-lined quantities,
− H H (1 ϵk)∆n ≡ z (νk, z) N ≡ z (νk, z) ≡ e M inst 2 , inst 2 , z . (2–39) 2ka 2kϵka 1−ϵk
With the choice (2–36, 2–37) the approximate solutions rapidly freeze in to constants,
H2 H2 lim M (t, k) = k ×C(ϵ ) , lim N (t, k) = k ×C(ϵ ) . (2–40) ≫ 0 3 k ≫ 0 3 k t tk 2k t tk 2ϵkk
Recalling the relationship between M(t, k) and N(t, k) and their respective spectra (1–39, 1–40) we arrive at the following forms for the power spectra and fixes the non-local correction exponents to,
2 2 2 16GH (tk) τ[ϵ](k) 2 GH (tk) σ[ϵ](k) ∆h(k) = C (ϵ (tk)) e , ∆R(k) = C (ϵ( tk )) e (2–41) π πϵ(tk) 1 1 τ[ϵ](k) = − lim g(t, k) , σ[ϵ](k) = − lim h(t, k) . (2–42) 2 t≫tk 2 t≫tk
It remains to specialize the sources to (2–36, 2–37). First note the simple relation between the scalar and tensor frequencies,
ϵ Ω(n, k) = θ(n −n)ω(n, k) + θ(n−n )ω(n, k)× k . (2–43) n k ϵ(n)
This means the scalar source (2–30) consists of the tensor source (2–21) minus a handful of terms mostly involving ϵ(n), [( ) ( ) ] ϵ′ ′ 1 ϵ′ 2 ϵ′ Sg(n, k) = Sh(n, k) − 2θ(nk −n) + + (3−ϵ) ϵ [(2 ϵ ) ϵ ( )] ϵ′ ω′ ϵ′ ϵ2 +2δ(n−n ) − 2θ(n−n ) 3−ϵ+ + ω2 k −1 . (2–44) k ϵ k ω ϵ ϵ2
33 To obtain an explicit formula for the tensor source we first note that the tensor frequency is,
2(1−ϵ) 2(1−ϵk) H ω(n, k) = θ(nk −n) + θ(n−nk) × . (2–45) H(ν, z) H(νk, z) H Hence the n derivative of its logarithm is, [ ] [ ′ ] ω′ ϵ′ H′ H = θ(n −n) − − + θ(n−n ) ∆ϵ − , (2–46) ω k 1−ϵ H k H
≡ − H ≡ H 1 1 where ∆ϵ ϵ(n) ϵk and (νk, z). Before horizon crossing ν = 2 + 1−ϵ is time dependent and z = k/[(1 − ϵ)Ha] so we have, [ ] ϵ′ ϵ′ ν′ = , z′ = − 1−ϵ− z (1−ϵ)2 1−ϵ [ ] H′ ϵ′ ϵ′ =⇒ − = − A + 1−ϵ− B , (2–47) H (1−ϵ)2 1−ϵ
where A and B involve derivatives of H(ν, z) with respect to ν and ζ = ln(z), [ ] [ ] ζ ζ A ≡ ∂ν ln H(ν, e ) , B ≡ ∂ζ ln H(ν, e ) . (2–48)
The analogous result after horizon crossing is much simpler,
H′ − = (1−ϵ )B , (2–49) H k
ζ where B means B with ν specialized to νk and e specialized to z. Taking the derivative of ω′/ω before horizon crossing, ( ) [ ] [ ] ω′ ′ ϵ′′ ϵ′2 ϵ′′ 2ϵ′2 ϵ′′ ϵ′2 = − − − + A− ϵ′ + + B ω 1−ϵ (1−ϵ)2 (1−ϵ)2 (1−ϵ)3 1−ϵ (1−ϵ)2 [ ] [ ] ϵ′2 2ϵ′ ϵ′ ϵ′ − C + 1−ϵ− D − 1−ϵ− E , (2–50) (1−ϵ)4 (1−ϵ)2 1−ϵ 1−ϵ
requires three second derivatives of ln[H], [ ] [ ] [ ] C ≡ 2 H ζ D ≡ H ζ E ≡ 2 H ζ ∂ν ln (ν, e ) , ∂ζ ∂ν ln (ν, e ) , ∂ζ ln (ν, e ) . (2–51)
34 (1) ζ Bessel’s equation and the Wronskian of Hν (e ) imply, ( ) 2(1−ϵ) 2 − 2E − 2B2 − − 2 − 2 2ζ − 2(1 ϵ) + (1 ϵ) (3 ϵ) + 4(1 ϵ) e H = 0 . (2–52)
Substituting relations (2–46), (2–47), (2–50) and (2–52) in the definition of the tensor source (2–21) gives,
′′ [ A ] [ AB D ] 2ϵ ′ 2 2 t < tk =⇒ Sbefore = 1+ +B + 2ϵ 1− −B − − 2E { 1−ϵ 1−ϵ 1−ϵ 1−}ϵ [ ] 2ϵ′2 1 1 A 2 A C 2D + − + 2+ +B + + + +E . (2–53) (1−ϵ)2 2 2 1−ϵ 1−ϵ (1−ϵ)2 1−ϵ
The analogous result after horizon crossing is, [ ]
t > tk =⇒ Safter = 2∆ϵ 3−ϵk +(1−ϵk)B [ ( ) ][ 2 ] 2 − 2 k − 1 ϵk H − +4 2 1 . (2–54) H a2 H H2
There is also a jump at horizon crossing so that the complete result is, [ ] 2ϵ′ A S = θ(n −n) S − δ(n−n ) 1+ +B + θ(n−n ) S . (2–55) h k before k 1−ϵ 1−ϵ k after
ℭ(ϵ) C(ϵ)
1.0 1.000● ● 0.998 0.8 ● ● 0.996 ● 0.6 ● 0.994 ● 0.4 ● 0.992 ● 0.2 0.990 ● ● ϵ ϵ 0.2 0.4 0.6 0.8 1.0 0.005 0.010 0.015 0.020
Figure 2-1. Graphs of the local slow roll correction factor C(ϵ) versus a better approximation. The left hand graph shows the local slow roll correction factor C(ϵ) (solid blue), which was defined inexpression (1–54). Also shown is its global approximation of 1 − ϵ (dashed yellow) over the full inflationary range of 0 ≤ ϵ < 1. The right hand graph shows C(ϵ) (solid blue) versus the better approximation of 1 − 0.55ϵ (large dots) relevant to the range 0 ≤ ϵ < 0.02 favored by current data.
35 2.2 Simple Analytic Approximations
The exact analytic results of the previous section are valid for all single-scalar models of inflation. However, they can be wonderfully simplified by exploiting the factthat the first slow roll parameter is very small. The 95% confidence bound on the tensor-to-scalar ration of r < 0.12 [24, 25] implies ϵ < 0.0075. This suggests a number of approximations. First, the local slow roll correction factor C(ϵk), defined in (1–54), may as well be set to unity. From
Figure 2-1 we see that the bound of ϵ < 0.0075 implies 1.0000 < C(ϵk) < 0.9959. This is not currently resolvable. Another excellent approximation is taking ϵ = 0 in the tensor and scalar Green’s functions of expressions (2–25) and (2–31),
lim Gh(n; m) = lim Gg(n; m) ≡ G0(n; m) ϵ=0 ϵ=0 [ ] [ ] { ( )} θ(n−m) n = e∆m +e3∆m sin −2 e−∆ℓ −arctan e−∆ℓ , (2–56) 2 m where ∆m ≡ m − nk and ∆ℓ ≡ ℓ − nk. Note that this expression is valid before and after horizon crossing. An important special case of (2–56) is when n becomes large, which gives the function G(e∆m) we define as, ( ) [ ( )] 1 2 1 G(x) ≡ x+x3 sin −2 arctan . (2–57) 2 x x
From the graph in Figure 2-2 we see that G(e∆n) suppresses contributions more than a few e-foldings before horizon crossing. We can also take ϵ = 0 in H and the derivatives of it in expressions (2–48) and (2–51). This leads to exact results for H, B and E in terms of the parameter x ≡ e∆n,
3 lim H ≡ H0(x) = x + x , (2–58) ϵ=0 −1−3x2 lim B ≡ B0(x) = , (2–59) ϵ=0 1+x2 4x2 lim E ≡ E0(x) = . (2–60) ϵ=0 (1+x2)2
36 Δn Δn Gⅇ ℰ1ⅇ
0.4
Δn -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.3
-0.05 0.2
0.1 -0.10
Δn -5 -4 -3 -2 -1 -0.15
-0.1 -0.20
Figure 2-2. The ϵ = 0 Green’s function and the coefficient of ε′′(n) in the small ϵ form for Sh(n, k). The left hand graph shows the ϵ = 0 Green’s function G(e∆n) given in expression (2–57). The ′′ right hand graph shows the coefficient of ε (n) in the small ϵ form (2–65) for Sh(n, k). This function E1(x) is defined by expressions (2–59), (2–61) and (2–66). The solid blue curve gives the exact numerical result while the large dots give the approximation resulting from the series expansion on the right hand side of expression (2–61).
Δn Δn ℰ2ⅇ ℰ3ⅇ
0.05
0.2 Δn -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 Δn -0.05 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5
-0.2 -0.10
-0.4 -0.15
-0.6 -0.20
-0.8 -0.25
′ 2 ′ Figure 2-3. The coefficients of [ε (n)] and ε (n) in the small ϵ form for Sh(n, k). ′ 2 ′ The coefficients of [ε (n)] (left) and ε (n) (right) in the small ϵ form (2–65) for Sh(n, k) . In each case the solid blue curve gives the exact numerical result, while the large dots give the result of using the series approximations on the far right of (2–61-2–63) in expressions (2–67) and (2–68).
The three derivatives with respect to ν do not lead to simple expressions even for ϵ → 0, but they can be well approximated over the range we require by short series expansions in powers
37 of x2,
1.5x2 +1.8x4 −1.5x6 +.63x8 lim A ≡ A0(x) ≃ , (2–61) ϵ=0 1+x2 x2 +6.1x4 −3.7x6 +1.6x8 lim C ≡ C0(x) ≃ , (2–62) ϵ=0 (1+x2)2 −3x2 −6.8x4 +5.5x6 −2.6x8 lim D ≡ D0(x) ≃ . (2–63) ϵ=0 (1+x2)2
We can express the ratio of H/H in terms of the deviation ∆ϵ(n) ≡ ϵ(n) − ϵk, ∫ ∫ 2 [ n ] n H − − ≃ 2 1 = exp 2 dm ∆ϵ(m) 1 2 dm ∆ϵ(m) . (2–64) H nk nk
All of this gives an approximation for the tensor source (2–55), [ ] ′′ ∆n ′2 ∆n ′ ∆n ′ Sh(n, k) ≃ −2θ(−∆n) ϵ E1(e )+ϵ E2(e )+ϵ E3(e ) + 2δ(∆n)ϵ E1(1) { } ( )∫ 4+2e2∆n n 2 +2θ(∆n) ∆ϵ(n)+ 2∆n dm ∆ϵ(m) 2∆n , (2–65) 1+e nk 1+e
where the three coefficient functions are,
E1(x) = −1 − A0(x) − B0(x) , (2–66) [ ] 1 1 2 E (x) = −A (x)−C (x)−2D (x)−E (x)− 2+A (x)+B (x) , (2–67) 2 2 0 0 0 0 2 0 0 E − A B B2 D E 3(x) = 1+ 0(x) 0(x)+ 0(x)+2 0(x)+2 0(x) . (2–68)
Figures 2-2 and 2-3 show the various coefficient functions. The smallness of ϵ means that the factors of 1/ϵ which occur in the scalar source (2–44)
are hugely important. By comparison we can ignore the Sh(n, k) terms and simply write, [( ) ( ) ] ϵ′ ′ 1 ϵ′ 2 ϵ′ ϵ′ ϵ′ 2 S (n, k) ≃ −2θ(−∆n) + +3 + 2δ(∆n) − 2θ(∆n) . (2–69) g ϵ 2 ϵ ϵ ϵ ϵ 1+e2∆n
Because ϵ < 0.0075 we expect Sg to be more than 100 times as strong as Sh. The approximations (2–56), (2–65) and (2–69) are valid so long as ϵ is small. If we additionally ignore nonlinear terms in the equations for h(n, k) and g(n, k), the correction
38 H(n) ϵ(n) H(0) 0.5 0.014
0.012 0.4 0.010
0.3 0.008
0.006 0.2
0.004 0.1 0.002
0.0 n 0.000 n 168 170 172 174 176 178 168 170 172 174 176 178
Figure 2-4. The Hubble parameter and the first slow roll parameter for a model with features. The left hand figure shows the Hubble parameter and the right shows the first slow roll parameter for a model with features. This model which was proposed [50, 51] to explain the observed features in the scalar power spectrum at ℓ ≈ 22 and ℓ ≈ 40 which are visible in the data reported from both WMAP [52, 53] and PLANCK [54, 55]. Note that the feature has little impact on H(n) but it does lead to a distinct bump in ϵ(n). exponents of expression (2–41) become, [ ] ∫ [ ] nk 2 ′′ ∆n ′ ∆n ′ ∆n ∆n τ[ϵ](k) ≃ dn ϵ (n)E1(e )+ ϵ (n) E2(e )+ϵ (n)E3(e ) G(e ) 0 { } ∫ ( )∫ ∞ 4+2e2∆n n 2G(e∆n) −ϵ′(n )E (1)G(1) − dn ∆ϵ(n)+ dm ∆ϵ(m) , (2–70) k 1 1+e2∆n 1+e2∆n [ nk nk ] ∫ ( ) nk 1 2 ≃ 2 ∆n σ[ϵ](k) dn ∂n ln[ϵ(n)]+ ∂n ln[ϵ(n)] +3∂n ln[ϵ(n)] G(e ) 0 2 ∫ ∞ 2G(e∆n) −∂ ln[ϵ(n )] G(1) + dn ∂ ln[ϵ(n)] . (2–71) nk k n 2∆n nk 1+e
∆n Recall that ∆n ≡ n − nk, ∆ϵ(n) ≡ ϵ(n) − ϵk, the Green’s function G(e ) was defined in
∆n ∆n ∆n (2–57), and the coefficient functions E1(e ), E2(e ) and E3(e ) were given in expressions (2–66-2–68). How large τ[ϵ](k) and ϵ[ϵ](k) are depends on what the inflationary model predicts for derivatives of ϵ(n). For example, the slow roll approximation of monomial inflation gives,
ϵ V (φ) = Aφα =⇒ ϵ(n) ≃ i . (2–72) − 4 1 α ϵin
39 For these models the various tensor and scalar contributions are small,
32 16 4 α ⇒ ′′ ≃ 3 ′2 ≃ 4 ′ ≃ 2 V (φ) = Aφ = ϵ 2 ϵ , ϵ 2 ϵ , ϵ ϵ , (2–73) ( ) α ( α) α ϵ′ ′ 16 ϵ′ 2 16 ϵ′ 4 =⇒ ≃ ϵ2 , ≃ ϵ2 , ≃ ϵ . (2–74) ϵ α2 ϵ α2 ϵ α
2 2 π Δℛ π Δℛ
2 2 GHi GHi
35 35
30 30
25 25
20 20
15 15 n n 170 172 174 176 178 170 172 174 176 Figure 2-5. The scalar power spectrum compared with the local slow roll approximation and the scalar power spectrum compared with our analytic approximation. These graphs show the scalar power spectrum for the model of Figure 2-4. The left hand 2 figure compares the exact result (solid blue) with the local slow roll approximation ∆R(k) ≈ 2 × GHk /πϵk C(ϵk) (yellow dashed). The right hand figure compares the exact result (solid blue) with the much better approximation (yellow dashed) obtained from multiplying by exp[σ[ϵ](k)], using our analytic approximation (2–71) for σ[ϵ](k)].
The data disfavors monomial inflation24 [ –26], but τ[ϵ](k) and σ[ϵ](k) will be small for any model which has only slow evolution of ϵ(n). Much larger effects occur for models with “features”, which are transient fluctuations above or below the usual smooth52 fits[ ]. Features imply short-lived changes in ϵ(n), which do not have much effect on H(n) but can lead to large values of ϵ′(n) and ϵ′′(n). Figure 2-4 shows H(n) and ϵ(n) for a model that was proposed [50, 51] to explain a deficit at ℓ ≈ 22, and an excess at ℓ ≈ 40, in the data reported by both WMAP [52, 53] and PLANCK [54, 55]. In the range 171 < n < 172.5 the scalar experiences a step in its potential which has little effect on H(n) but leads to a noticeable bump in ϵ(n). 2.3 Improvements to Our Approximation for the Scalar Power Spectrum
As can be seen from Figure 2-5 our approximation is excellent however there is room for improvement near n = 172. Looking back through our formalism we identify two possibilities
40 for why our approximation is lacking here. Firstly it could be that the non-linear corrections become important in this regime. The process for including the first nonlinear correction is straightforward although somewhat tedious. We first compute in the ϵ −→ 0 limit the two
{ 2 ′ 2} nonlinear terms g1(n, k), (g1(n, k)) . They are found using the first order result (2–32) which is given before horizon crossing by: ∫ 1 n ( ) g (n < n , k) = dm S (m)e∆m 1 + e2∆m 1 k 2 g [ { ( 0 ) ( )}] sin 2 e−∆m − tan−1 e−∆m − e−∆n + tan−1 e−∆n (2–75) ∫ n 1 + e2∆m g′ (n < n , k) = dm S (m)e∆m 1 k g 1 + e2∆n [ { ( )0 ( )}] cos 2 e−∆m − arctan e−∆m − e−∆n + arctan e−∆n (2–76)
where as before ∆n = n − nk. Taking the square of these two terms and inserting them into Equation (2–33) yields the first nonlinear correction terms for g(n, k). These correction terms are then to be viewed as source terms for σ[ϵ](k) and can be included in the integrand on the first line of Equation (2–71). The other possibility is that one of the terms which we discarded earlier from the scalar source Sg is not as small as we had hoped. Comparing Figure 2-4 with Figure 2-5 we convince ourselves that the source terms we would be interested in are the ones which occur after horizon crossing and referring back to Equation (2–44) we see that the only term which has been left out is the last one. Taking the ϵ = 0 limit for this term gives us a new source term for σ[ϵ](k) which can be included by adding the following term: ∫ ( ) ∞ 2 ∆n ϵ(nk) − 4G(e ) dn 2 1 ∆n 3∆n 2 . (2–77) nk ϵ(n) (e + e )
It turns out that both of these corrections are necessary to alleviate the slight problems our approximation has as can be seen in the figure below. Further improvements would require either including more source terms, higher nonlinear terms, or making fewer approximations for the scalar and tensor Green’s functions. The result for the spectrum once these improvements are added in are shown in Figure 2-6.
41 Figure 2-6. Our approximation for the scalar power spectrum of the step model with the non-linear and extra source terms included. Our approximation for the scalar power spectrum of the step model with the non-linear and extra source terms included. Note in particular the improvement near nk = 172 over figure 2-5.
2.4 Concluding Remarks on the Power Spectra
Figure 2-5 shows shows the scalar power spectrum for the model of Figure 2-4. The left hand graph compares the exact result to the local slow roll approximation, without including the non-local corrections from σ[ϵ](k). Not even the main feature is correct, and the secondary oscillations are completely absent. There is also a small systematic offset before and after the features. The right hand graph shows the effect of adding σ[ϵ](k) with our approximation (2–71). The agreement is almost perfect, with the small remaining deviations attributable to nonlinear effects and additional source terms as we have discussed. The small offset oftheleft hand graph (before and after the features) is due to the local slow roll approximation missing the steady growth which ϵ(n) needs to reach the threshold of ϵ = 1 at which inflation ends. We conclude:
1. The nonlocal correction σ[ϵ](k) fixes the systematic under-prediction of the local slow roll approximation when ϵ(n) is growing steadily;
2. The nonlocal correction σ[ϵ](k) makes large and essential contributions when features are present; and
3. The nonlocal correction σ[ϵ](k) is well approximated by (2–71). Figure 2-7 shows the tensor power spectrum for the model of Figure 2-4. The left hand graph compares the exact result with the local slow roll approximation. The prominent features
42 2 πΔh (n) 2 16 GHi 0.0020
0.242 0.0015
0.240 0.0010
0.238 0.0005
n 0.236 170 172 174 176
-0.0005 0.234
-0.0010 n 0.232 171.0 171.5 172.0 172.5 Figure 2-7. The tensor power spectrum for the step model compared with the local slow roll approximation. These graphs show the tensor power spectrum for the model of Figure 2-4. The left hand 2 ≈ figure compares the exact result (solid blue) with the local slow roll approximation ∆h(k) 16 2 π GHk C(ϵk) (yellow dashed). The solid blue line on the right hand graph shows the logarithm of 2 the ratio of ∆h(k) to its local slow roll approximation. The yellow dashed line gives the nonlocal corrections of expression (2–70).
of the scalar power spectrum which can be seen in Figure 2-5 are several hundred times smaller, inverted and phase shifted, but they can just be made out. The right hand graph compares our approximation (2–70) for τ[ϵ](k) with the exact result. The agreement is again almost perfect, with the small deviations actually attributable to numerical roughness in the interpolation of the exact computation, rather than to any problem with our approximation (2– 70). Correlating tensor features with their much stronger scalar counterparts might be possible in the far future and would represent an impressive confirmation of single-scalar inflation74 [ ]. Now that we have worked out the problem of taking an arbitrary geometry (or potential) and computing the power spectra, we will turn our attention to the inverse problem. We will begin with by covering how reconstruct the geometry when slow roll is valid. Then we will consider the general case when slow roll may be violated. We will test our reconstruction on the step model we have considered in this chapter then close by applying it to a new spectrum.
43 CHAPTER 3 RECONSTRUCTION
3.1 Developing The Reconstruction Proceedure
We have so far considered the problem of using the inflationary geometry to predict the
2 2 power spectra. Here we wish to consider the inverse problem of using ∆R(k) and ∆h(k) to reconstruct H(n) and ϵ(n). It is well to begin by setting down a few general principles:
2 1. Although ∆R(k) is measured to 3-digit accuracy, the tensor power spectrum has yet to 2 be resolved. When ∆h(k) is finally detected it will take a number of years before much 2 2 precision is attained. Therefore, reconstruction should be based on ∆R(k), with ∆h(k) used only to fix the integration constant which gives the scale of inflation.
2. The first slow roll parameter is so small that there is no point in using anexact 2 evaluation (2–41) for ∆R(k). Figure 2-1 shows that we can ignore the local slow roll correction factor C(ϵk). Although the nonlocal correction exponent σ[ϵ](k) must be included, Figure 2-5 shows that the approximation (2–71) almost perfect.
3. The fact that ϵ(n) is small and smooth, with small transients, motivates a hierarchy between H, ϵ and ϵ′/ϵ based on calculus, [ ∫ ] [∫ ] n n ϵ′(m) H(n) = Hi exp − dm ϵ(m) , ϵ(n) = ϵi exp dm . (3–1) 0 0 ϵ(m) Hence H(n) is insensitive to small errors in ϵ(n), and ϵ(n) is insensitive to small errors in ∂n ln[ϵ(n)].
We begin by converting from wave number k to nk, the number of e-foldings since the beginning of inflation that k experienced first horizon crossing. It is also desirable to factor out the scale of inflation Hi ≡ H(0),
2 ≡ H(n) ≡ π∆R(k) h(n) , δ(nk) 2 . (3–2) Hi GHi
Compiled and edited with permission from the following works [71, 72].
44 exp1,2,4 exp1+2+4 0.6
0.4 0.4
0.2 0.2
n n 170 172 174 176 178 170 172 174 176 178
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
Figure 3-1. Numerical values of exponents 1, 2 and 4 for the step model. Numerical values of exponents 1, 2 and 4 for the model of Figure 2-4. The left hand graph gives separate results for expression (3–4) in dashed blue, expression (3–5) in dot-dashed yellow, and expression (3–7) in solid green. The right hand graph shows the sum of all three exponents.
(Hi is the single number which would come from the tensor power spectrum.) Based on the three principles we base reconstruction on the formula, [ ] h2(n) ∑5 δ(n) ≃ × exp exp (n) , (3–3) ϵ(n) i i=1 where the five exponents follow from our approximation2–71 ( ) for σ[ϵ](k),
exp (n) = −∂ ln[ϵ(n)]×G(1) , (3–4) 1 ∫ n n 2 × m−n exp2(n) = dm ∂m ln[ϵ(m)] G(e ) , (3–5) 0∫ [ ] 1 n 2 × m−n exp3(n) = dm ∂m ln[ϵ(m)] G(e ) , (3–6) 2∫ 0 n × m−n exp4(n) = 3 dm ∂m ln[ϵ(m)] G(e ) , (3–7) ∫0 ∞ m−n × G(e ) exp5(n) = 2 dm ∂m ln[ϵ(m)] 2(m−n) . (3–8) n 1+e
To just reconstruct the Hubble parameter there is no need to include the correction exponents (3–4-3–8). Using only the leading slow roll terms gives,
h2(n) 1 δ(n) ≃ =⇒ h2(n) ≃ ∫ . (3–9) ϵ(n) n 2dm 1+ 0 δ(m)
45 exp3,5 exp3+5 0.10
0.06 0.05
0.04
n 170 172 174 176 178 0.02
-0.05 n 170 172 174 176 178
-0.10
Figure 3-2. Numerical values of exp3(n) and exp5(n) for the step model. Numerical values of exp3(n) and exp5(n) for the model of Figure 2-4. The left hand graph gives separate results for expression (3–6) in dashed blue, and expression (3–8) in solid yellow. Note
that exp5(n) is responsible for correcting the small, systematic under-prediction of the slow roll approximation before and after the feature. The right hand graph shows the sum.
Even for the power spectrum of Figure 2-5 the reconstruction of h(n) given by expression (3–9) is barely distinguishable from the left hand graph of Figure 2-4.
h(n)2 h(n)2 log log δ(n) δ(n)
-4.0 -4.0
-4.2 -4.2
-4.4 -4.4
-4.6 -4.6
-4.8 -4.8
-5.0 -5.0 n n 170 172 174 176 178 170 172 174 176 178 Figure 3-3. Various choices for the left hand side of the first pass reconstruction equation for the step model. Various choices for the left hand side of the first pass reconstruction equation for the model of Figure 2-4. The left hand graph shows the first pass source − ln[δ(n)] + 2 ln[h(n)] in solid blue with ln[ε(n)] overlaid in dashed yellow. The poor agreement between the two curves is why using just ln[ε(n)] as the left hand side of the first pass reconstruction fails to converge when features are present. The right hand graph shows the much better agreement between the same source − − − (solid blue) and ln[ε(n)] exp1(n) exp2(n) exp4(n) (dashed yellow).
Not all the exponents (3–4-3–8) are equally important. Figures 3-1 and 3-2 show that
the set of exp1(n), exp2(n) and exp4(n) are about ten times larger than exp3(n) and exp5(n) for the model of Figure 2-4. That reconstructing features indeed requires the three large
46 exponents is apparent from Figure 3-3. Taking the logarithm of (3–3) and moving the three large exponents to the left gives, ∫ [ ] n [ ] − 2 × m−n 1+G(1)∂n ln[ϵ(n)] dm ∂m +3∂m ln[ϵ(m)] G(e ) 0 ≃ − ln[δ(n)] + 2 ln[h(n)] + exp3(n) + exp5(n) . (3–10)
This becomes a linear, nonlocal equation for ln[ϵ(n)] if we drop exp3(n) and exp5(n) and use expression (3–9) for the Hubble parameter, ∫ [ ] n [ ] − 2 × m−n 1+G(1)∂n ln[ϵ(n)] dm ∂m +3∂m ln[ϵ(m)] G(e ) 0 [ ∫ ] n 2dm ≃ − ln[δ(n)] − ln 1+ . (3–11) 0 δ(m)
The linearity of Equation (3–11) means that it can be solved by a Green’s function, in spite of being nonlocal. The required Green’s function becomes a symmetric function of its
− arguments if we note from Figure 2-2 and expression (2–57) that G(en nk ) is essentially zero more than about N ∼ 4 e-foldings before horizon crossing. The Green’s function equation is, ∫ [ ] n G − 2 G × m−n 1+G(1)∂n (n) dm (∂m +3∂m) (m) G(e ) = δ(n) . (3–12) −N
We can solve (3–11) by integrating against the source on the right hand side,
∫ ∞ ln[ϵ(n)] = dm G(n−m)×Source(m) . (3–13) 0
This might be regarded as the first pass of an iterative solution3–10 to( ). After the first pass solution of (3–11) one would use the resulting ln[ϵ(n)] to construct h(n) and to evaluate
exp3(n) and exp5(n) on the right hand side of (3–10). Then the same Green’s function solution (3–13) could be used with this more accurate source to find a more accurate ln[ϵ(n)], which would lead to a more accurate source, and so on. We are not able to solve (3–12) exactly owing to the factor of G(em−n) inside the integral. Consideration of Figure 2-2 suggests that this troublesome factor might be
47 ¡ Log(¡ [n]) Log( [n])
-4.3 -4.3
-4.4 -4.4
-4.5 -4.5
-4.6 -4.6
-4.7 -4.7
n n 170 172 174 176 170 172 174 176
Figure 3-4. Numerical reconstructions of ln[ϵ(n)] for the power spectrum of Figure 2-5. These graphs show numerical reconstructions of ln[ϵ(n)] for the power spectrum of Figure 2-5. The solid blue line of the left hand graph shows the exact result while the yellow dashed line gives the result of integrating G0(n−m) — using the first six terms of the sum over ℓ in expression (3–16) — against the first pass source on the right hand side of(3–11). The right hand graph shows the result of adding the first order improvement G1(n−m) — computed using the first four terms of the sum over m in expression (3–25).
approximated as a square wave of width ∆ = 0.8,
G(em−n) ≈ G(1)θ(n−m−∆) . (3–14)
Making the approximation (3–14) leads to a still-nonlocal equation,
δ(n) 1 (∂ +3)G (n−∆) − αG (n) = , α ≡ 3 − . (3–15) n 0 0 G(1) G(1)
The “retarded” solution to (3–15) which avoids exponentially growing terms is,
∞ [ ( )] ( ) e3(n+∆) ∑ 1 ℓ G (n) = αe−3∆ n+(ℓ+1)∆ θ n+(ℓ+1)∆ . (3–16) 0 G(1) ℓ! ℓ=0
Figure 3-4 shows the result of using just G0(n) to reconstruct ln[ϵ(n)] with the source taken as the right hand side of (3–11). Further improvement requires a better approximation for the Green’s function G(n). It is instructive to take the Laplace transform, restoring the second argument of the Green’s function, ∫ ∞ Gb(s; m) ≡ dn e−snG(n−m) . (3–17) 0
48 ℐ(s) (s) 0.35 0.0005 0.30
0.25 s 5 10 15 20 0.20
-0.0005 0.15
0.10 -0.0010 0.05
0.00 s -0.0015 0 2 4 6 8 10
Figure 3-5. Numerical evaluation of the integral I(s) and various approximations. The solid blue line of the left hand graph shows a numerical evaluation of the integral I(s) of expression (3–19). The 0th order approximation I0(s) of expression (3–20) is overlaid in large dots. The solid blue line of the right hand graph shows the deviation ∆I(s) ≡ I(s) − I0(s). Our fit I1(s) of expression (3–21) is overlaid in large dots.
The Laplace transform of the Green’s function Equation (3–12) is, [ ] 1+G(1)s−(s+3)s×I(s) Gb(s; m) = e−ms , (3–18) where we define, ∫ ∞ I(s) ≡ dℓ e−sℓ ×G(e−ℓ) . (3–19) 0 The problem of approximating G(n − m) is therefore related to the one of approximating (3–19), and of recognizing the resulting inverse Laplace transform of Gb(s; m). Making the approximation (3–14) in (3–19) gives, [ ] G(1) I (s) = 1 − e−0.8s . (3–20) 0 s
Figure 3-5 reveals that this is indeed a good approximation. Figure 3-5 also shows that the small residual is well fit by the function, [ ] ( ) 0.154 I (s) = sin 1.76 1−e−0.262(s−3.78) . (3–21) 1 (s + 8.97)2
49 b To obtain the first correction to G0(n − m) we begin by expanding G(s; m) in powers of
I1(s),
−ms Gb ≃ e (s; m) −∆s , (3–22) G(1)[(s+3)e −α]−s(s+3)I1(s) e−ms s(s+3)I (s)e−ms = + 1 + ..., (3–23) G(1)[(s+3)e−∆s −α] G2(1)[(s+3)e−∆s −α]2 b b ≡ G0(s; m) + G1(s; m) + ... (3–24)
We can recognize the inverse Laplace transform by expanding I1(s), { [ ] ∞ [ ] a a ∑ (−1)m 2m sin b−be−c(s+d) = sin(b) be−c(s+d) (s+e)2 (s+e)2 (2m)! m=0 } ∞ [ ] ∑ (−1)m 2m+1 − cos(b) be−c(s+d) . (3–25) (2m+1)! m=0
Figure 3-4 shows the effect of using G0(n − m) + G1(n − m) to solve Equation (3–11) approximately for ln[ϵ(n)]. Figure 3-4 shows that additional improvements are needed before our technique gives
good results for ∂n ln[ϵ(n)] when features are present. However, our results for ϵ(n) are already reasonable, and those for h(n) are staggeringly accurate. For the model of Figure 2-4 the largest percentage error on in reconstructing ϵ(n) is 2.2%, and the percentage error for h(n) never exceeds 0.04%. It is significant that our Green’s function G(n − m) depends only on the difference of its arguments, and we just need it over a range of about ten e-foldings. Further, its Laplace transform is defined by relations3–18 ( -3–19). Figure 3-5 shows that there is only a single, simple pole on the real axis, somewhat below s = 3. If nothing else worked we could therefore evaluate I(s0 + iω) numerically for some s0 > 3 and then numerically compute the inverse
Laplace transform, ∫ 1 ∞ (s0+iω)n b G(n − m) = dω e G(s0 +iω; m) . (3–26) 2πi −∞ No matter how time-consuming the computation proved, it would only need to be done once.
50 3.2 Using The Reconstruction Procedure on Novel Spectra
We will now demonstrate the true advantage of our reconstruction algorithm by applying it to a new toy model where the scalar spectrum is made to mimic the present data by having two large features but being otherwise flat.The functional form of the scalar spectrum which we consider is:
2 2 −9 −11 −9 −7(172.296−nk) ∆R(Nk) = 19.08 × 10 − 9.65 × 10 nk − 1.21 × 10 e
− − − 2 +1.18 × 10 9 e 26(172.85 nk) . (3–27)
where nk is the number of e-foldings from the start of inflation, and inflation ends at ne = 225.626. A graph of this spectrum is shown in Figure 3-6. We will imagine that the tensor
2 × −11 × amplitude is ∆h(nk = 165.626) = 3.1 10 so that Hi has the nominal value 2.8 √ 10−5/ 8πG at a time 60 e-foldings before the end of inflation. We stress that the exact time
(or wave number) at which we fix Hi is inconsequential. In the event of a positive detection of primordial B modes we will use whichever wave number has the most well determined value for the tensor amplitude.
2 Δ R 3.5 × 10-9
3. × 10-9
2.5 × 10-9
2. × 10-9
1.5 × 10-9
n 168 170 172 174 176
2 Figure 3-6. The spectrum ∆R whose geometry we will reconstruct.
Since we have already discussed the reconstruction proceedure at length we will simply present the results then interpret them. The results for reconstructing the geometry of our mock spectrum are show in Figure 3-7.
51 ¡(n) H(n) 0.0000280
-5.3 0.0000278
-5.4 0.0000276
-5.5 0.0000274
0.0000272 -5.6
0.0000270
n n 168 170 172 174 176 168 170 172 174 176
Figure 3-7. The values of ln [ϵ(n)] which we have reconstructed. The left hand graph shows the values of ln [ϵ(n)] which we have reconstructed from (3–27). On the right hand side are the two different interpolations of the Hubble parameter H(n). The orange curve is the result of integrating ϵ(n) which is shown in the left hand graph while the blue dashed curve comes from the leading slow roll terms in (3–9). Note that the curves agree on the edges of the figure but disagree near the feature.
What we find is that in order to produce a spectrum with exactly two features one would require two features in the geometry as opposed to the usual case where only one is considered. The first peak induces the ringing in the system just like we see inthestep model however in this case the slight dip after the peak has the effect of canceling out the
secondary peaks. We can perform a check of our reconstruction by integrating ϵ0(n) to obtain a new value of H(n) and then inserting both into our approximation for the scalar spectrum (2–41, 2–71) and comparing with our original model. This can be seen in Figure 3-8 and the results speak for themselves. We must stress that what we have done here is truely novel.
2 Δ R 3.5 × 10-9
3. × 10-9
2.5 × 10-9
2. × 10-9
1.5 × 10-9
n 170 172 174 176 Figure 3-8. The spectrum from the reconstructed geometry of Figure 3-7 compared with the mock spectrum we started with.
52 We have not proposed any analytic model for either the potential or the geometry. We have explicitly constructed an excellent approximation to the geometry which would produce a model spectrum which mimics current data. At this point we have done two remarkable things. We have shown how to take any scalar potential model which supports inflation and compute its power spectra. We have also seen how to take any spectra which contain only transient features and reconstruct the geometry which produced those spectra. Given the power of this formalism, the natural next steps will be to examine how we might test the theory of inflation as a whole using the insights we have gleaned thus far and to see how to extend this formalism to other models of inflation.
53 CHAPTER 4 IMPROVING THE SINGLE SCALAR CONSISTENCY RELATION The theory of primordial inflation1 [ –8] has had a profound effect on cosmology and fundamental theory. Particularly striking is the prediction that primordial tensor [33] and scalar [34] perturbations derive from quantum gravitational fluctuations which fossilized near the end of inflation. This not not only affords us access to quantum gravity at an intoxicating energy scale [75–77], it also provides information about the mechanism that powered inflation. This
2 information can be accessed by comparing observations of the two power spectra, ∆R(k) and
2 ∆h(k), to predictions from the many models [12–14]. For example, the simplest models of inflation are driven by the potential of a single, minimally coupled scalar. These modelsall obey the single-scalar consistency relation [78–80],
r ≈ −8nt , (4–1)
where r is the tensor-to-scalar ratio and nt is the tensor spectral index,
∆2 (k) ∂ ln(∆2 (k)) ≡ h ≡ h r(k) 2 , nt(k) . (4–2) ∆R(k) ∂ ln(k)
A statistically significant violation of4–1 ( ) would falsify the entire class of single-scalar models, as well as all models which are related to them by conformal transformation, such as f(R) inflation [81]. Although the single-scalar consistency relation was a brilliant theoretical insight, the progress of observation has rendered it somewhat inconvenient. The scalar power spectrum was first resolved in 1992 [18], and is now quite well measured [19–22]. The tensor power spectrum has not yet been resolved [24, 82], but polarization measurements are now providing the strongest limits on it [25]. It is not known if the current generation of polarization experiments
Reprinted with permission from: D.J. Brooker, N.C. Tsamis, & R.P. Woodard, ’Improving the Single Scalar Consistency Relation’, Phys. Lett. B, 773, 225-230 [74]
54 [83–87] can resolve the tensor power spectrum at all, and it is very unlikely that they will measure it well enough to constrain the tensor spectral index with much accuracy. In view of the observational situation, it makes sense to develop a test of single-scalar
2 inflation that is based primarily on the abundant datafor ∆R(k), and does not require taking
2 derivatives of the sparse data for ∆h(k) likely to result from the first positive detections. There is no reason not to do this because the close relation between the tensor and scalar mode functions of single-scalar inflation implies that either power spectrum determines the other, up to some integration constants. That is the purpose of this paper. In the next section we fix notation, recall the relation between the two power spectra, and infer the tensor power spectrum from the scalar one. Section 3 gives a comparison between the single scalar consistency relation and the scatter test we propose, using simulated data based on a hypothetical detection of r = 0.01 with the same number of data points and the same fractional error as was in fact reported by the recent spurious BICEP2 detection [23]. The final section mentions applications.
2 2 4.1 Constructing ∆h(k) from ∆R(k) We work in spatially flat, co-moving coordinates with scale factor a(t), Hubble parameter H(t) and first slow roll parameter ϵ(t),
a˙ H˙ ds2 = −dt2 + a2(t)d⃗x·d⃗x =⇒ H(t) ≡ , ϵ(t) ≡ − . (4–3) a H2
We assume a(t) is known, with the scalar background and potential determined to enforce the background Einstein equations [27–31], ∫ √ t ϵ(t′) φ (t) = φ (t ) dt′H(t′) ⇐⇒ t(φ) , (4–4) 0 0 i 4πG ti
[3−ϵ(t)]H2(t) V (φ) = . (4–5) 8πG t=t(φ)
We fix the gauge so that the full scalar agrees with its background value and the graviton field
hij is transverse, with g00 and g0i regarded as constraints. The two dynamical fields are hij and
55 2 2ζ h ζ, which reside in the 3-metric gij = a e [e ]ij. At quadratic order their Lagrangian is [38], [ ] [ ] a3 h h ϵa3 ζ ζ L = h˙ h˙ − ij,k ij,k + ζ˙2 − ,k ,k . (4–6) 2 64πG ij ij a2 8πG a2
The spatial plane wave mode functions of the graviton are u(t, k), with exactly the same polarization tensors as in flat space. From (4–6) we see that the evolution equation, Wronskian and asymptotically early form of the tensor mode functions u(t, k) are, ∫ t ′ 2 − dt k i exp[ ik ′ ] u¨ + 3Hu˙ + u = 0 , uu˙ ∗ −uu˙ ∗ = , u(t, k) −→ √ ti a(t ) . (4–7) a2 a3 2ka2(t)
The scalar perturbation ζ has spatial plane wave mode functions v(t, k). From (4–6) we see that their evolution equation, Wronskian and asymptotically early form are, ∫ t ′ ( ) 2 − dt ϵ˙ k i exp[ ik ′ ] v¨ + 3H + v˙ + v = 0 , vv˙ ∗ − vv˙ ∗ = , v(t, k) −→ √ ti a(t ) . (4–8) ϵ a2 ϵa3 2kϵ(t)a2(t)
The two power spectra are determined (at tree order) by evolving their respective mode functions from their early forms through the time tk of first horizon crossingk ( ≡ H(tk)a(tk)), after which they approach constants,
3 2 2 2 k GH (tk) ∆R(k) = × 4πG × v(t, k) ≈ , (4–9) 2π2 t≫t πϵ(t ) k k k3 2 16GH2(t ) 2 × × × ≈ k ∆h(k) = 2 32πG 2 u(t, k) . (4–10) 2π t≫tk π
The relations (4–7) which define u(t, k) are carried into the relations (4–8) which define v(t, k) by making simultaneous changes in the scale factor and the co-moving time [39, 40],
√ ∂ 1 ∂ a(t) −→ ϵ(t) a(t) , −→ √ . (4–11) ∂t ϵ(t) ∂t
To understand what this means for the power spectra we must consider them as nonlocal functionals of the expansion history a(t), which will involve integrals and derivatives with respect to time. We denote this functional dependence with square brackets, so relation (4–11)
56 implies, [ ] [ ] 1 √ √ ∆2 a, dt (k) = ∆2 ϵa, ϵdt (k) . (4–12) R 16 h Relation (4–12) is easy to check at leading slow roll order by comparing the slow roll
2 approximation for ∆R(k) on the right hand side of (4–9) with the effect of making transformation (4–11) on the Hubble parameter in the right hand side of expression (4–10), [ ] ∂ 1 ∂ √ H + ϵ˙ H(t) ≡ ln[a(t)] −→ √ ln ϵ a = √ 2ϵ . (4–13) ∂t ϵ ∂t ϵ
However, we stress that relation (4–12) is exact, not just valid at leading slow roll order,
2 2 provided one employs the exact expressions for ∆h(k) and ∆R(k).
(N) for the Step Model 2 2
R / i 0.014 GH Exact Result vs Approximation
0.013 10
0.012 2 i
) 8 GH N / ( 0.011
2 ¡
R ¡
0.010 6 Exact
0.009 Approximation
4
50 52 54 56 58 60 50 52 54 56 58 60 E-Folds Until End of Inflation E-Folds Until End of Inflation Figure 4-1. The first slow roll parameter for the step model, the resulting scalar power spectrum and the result of our analytic approximation (4–14). The left hand figure shows the first slow roll parameter for a model which was50 proposed[ , 51] to explain the observed features in the scalar power spectrum at ℓ ≈ 22 and ℓ ≈ 40 which are visible in the data reported from both WMAP [52, 53] and PLANCK [54, 55]. The right hand figure shows the resulting scalar power spectrum (in blue), with the result of our analytic approximation (4–14) (in yellow). The slow roll approximation (4–9) does not give a very accurate fit even to the main feature in the range 54.5 < N < 53 e-foldings before the end of inflation, and it completely misses the secondary oscillations visible in the range 53.5 < N < 51.5. The nonlocal contributions (4–17) are essential for correctly reproducing these features.
We should also point out that very accurate functional expressions are now available for
the power spectra of single scalar inflation, valid to all orders in the slow roll parameter ϵ(tk), and even including nonlocal effects from times before and after first horizon crossing46 [ , 47].
57 These expressions take the form [71], ( ) [ ] GH2(t ) ∆2 (k) ≃ k ×C ϵ(t ) ×exp σ[ϵ](k) , (4–14) R πϵ(t ) k k ( ) [ ] 16GH2(t ) ∆2 (k) ≃ k ×C ϵ(t ) ×exp τ[ϵ](k) , (4–15) h π k
where the local slow roll correction factor is,
( )[ ] 2 1 1 1 1−ϵ C(ϵ) ≡ Γ2 + 2(1−ϵ) ≈ 1 − ϵ . (4–16) π 2 1−ϵ
For the nonlocal corrections σ[ϵ](k) and τ[ϵ](k) it is best to abuse the notation by writing the first slow parameter ϵ(n) ≡ ϵ(t(n)) as a function of n ≡ ln[a(t)/ai], the number of e-foldings since the start of inflation, [ ] ∫ ( ) ( ) nk 2 2 1 ∆n σ[ϵ](k) = dn ∂n ln[ϵ(n)]+ ∂n ln[ϵ(n)] +3∂n ln[ϵ(n)] G e 0 2 ∫ ∞ 2G(e∆n) −∂ ln[ϵ(n )] G(1) + dn ∂ ln[ϵ(n)] , (4–17) nk k n 1+e2∆n [ nk ] ∫ ( ) ( )( ) ( ) ( ) nk 2 ∆n ′′ ∆n ′ ∆n ′ ∆n τ[ϵ](k) = dn E1 e ϵ (n)+E2 e ϵ (n) +E3 e ϵ (n) G e 0 ∫ { ∫ } ∞ ( 2∆n ) n ∆n − ′ E − 4+2e 2G(e ) ϵ (nk) 1(1)G(1) dn ∆ϵ(n)+ 2∆n dm ∆ϵ(m) 2∆n . (4–18) nk 1+e nk 1+e
∆n Here ∆n ≡ n − nk, ∆ϵ(m) ≡ ϵ(m) − ϵk, and the functions of x ≡ e are, ( ) [ ( )] 1 2 1 G(x) = x+x3 sin −2arctan , (4–19) 2 x x 1 x2 −1.8x4 +1.5x6 −0.63x8 E (x) ≃ = 2 , (4–20) 1 1+x2 2.8x4 −7x6 +3x8 +1.8x10 −2.3x12 +0.95x14 −0.20x16 E (x) ≃ , (4–21) 2 (1+x2)2 9 x2 −11.9x4 +7.1x6 −1.3x8 −1.9x10 E (x) ≃ 2 . (4–22) 3 (1+x2)2
The 95% confidence bound on the tensor-to-scalar ratio of r < 0.12 [24, 25] implies ϵ < 0.0075, so τ[ϵ](k) is about a hundred times smaller than σ[ϵ](k). Models with smooth potentials typically have ϵ′ ∼ ϵ2 and ϵ′′ ∼ ϵ3, so the leading contributions in σ[ϵ](k) come
58 from the 3rd and 5th terms of expression (4–17). In particular the 5th (final) term is needed to correct for a systematic under-prediction of the local slow roll approximation [71]. For models with features the leading contributions to σ[ϵ](k) come from the 1st, 3rd and 4th terms of expression (4–17)[71]. These corrections can be very important for realistic models such as the one depicted in Figure 4-1.
2 2 To keep the analysis simple, we illustrate the procedure for predicting ∆h(k) from ∆R(k) using only the leading slow roll terms in expressions (4–14-4–15), without either of the nonlocal corrections or even the slow roll factor C(ϵ). The conversion from wave number to time is, dk k = H(t )a(t ) =⇒ = (1−ϵ)Hdt ≈ Hdt . (4–23) k k k The leading slow roll approximation (4–14) for the scalar power spectrum can be recognized as a differential equation for the Hubble parameter,
GH2(t ) GH4(t ) ∆2 (k) ≃ k = − k . (4–24) R ˙ πϵ(tk) πH(tk)
We can integrate this equation from some arbitrary time t∗ to tk, ∫ ( ) ln(k/k∗) ′ 1 ≃ 2G d ln(k) ⇒ 1 − 1 ≃ 2G d ln(k ) d 2 2 = 2 2 2 ′ . (4–25) H π∆R(k) H (tk) H (t∗) π 0 ∆R(k )
Substituting the reconstructed Hubble parameter (4–25) into the leading slow roll approximation (4–10) for the tensor power spectrum gives, [ ] ∫ −1 2 ln(k/k∗) 2 16GH (t ) r(k∗) ∆ (k∗) 2 ≃ k ≃ 2 ′ R ∆h(k) ∆h(k∗) 1 + d ln(k ) 2 ′ . (4–26) π 8 0 ∆R(k )
Equation (4–26) is in some sense an integrated form of the single-scalar consistency relation (4–1) which can be applied more reliably. Both relations are valid to leading slow roll order,
2 but whereas (4–1) compares a single value of the high quality data in ∆R(k) with a derivative
2 of the poor data on ∆h(k), our relation (4–26) combines a single measurement of the tensor
2 power spectrum at k = k∗ with the high quality scalar data to predict what ∆h(k) should be
59 2 for other wave numbers. This seems to be a better way of exploiting the sparse data on ∆h(k) which is likely to persist for some years after a first positive detection. 4.2 Comparison Using Simulated Data
It is illuminating to compare the single scalar consistency relation with the method we propose using simulated data. Let us suppose that the actual tensor power spectrum 1 − 1 corresponds to single scalar inflation with r = 100 , and which implies nt = 800 . We further suppose the simplest possible k dependence, ( ) [ ] [ ] [ ] k nt k 2 ⇒ 2 × ∆h(k) = rAS = ln ∆h(k) = ln rAS + nt ln , (4–27) k0 k0
−9 where the scalar amplitude (at the tensor pivot k0) is AS = 2.5 × 10 . Let us assume that the first positive detection of this tensor power spectrum consists of results for fivebinned wave numbers, the same as was in fact reported for the spurious BICEP2 detection [23]. To simplify matters we assume a linear relation for logarithms of the observed wave numbers,
1 2 ln[ki+1/ki] = 3 , and that each measurement of ln[∆h] has the same 1-sigma uncertainty of 1 σ = 4 . These numbers are roughly consistent with what BICEP2 actually reported [23]. Hence the detection consists of five observations yi obeying the relation, [ ] { } i y = ln 2.5× 10−11 − + e , i ∈ 1, 2, 3, 4, 5 , (4–28) i 2400 i where the ei are independent Gaussian random variables with mean zero and standard
1 deviation σ = 4 . Table 4-1 simulates the five data points using a random number generator to find the ei. Because the relation (4–27) is linear we can use least squares to determine the parameters. The least squares fit for N data points obeying the relation yi = α + βxi
60 Table 4-1. Simulated Data × −11 − i i ln(2.5 10 ) 2400 ei yi 1 −24.412145 −0.000417 +0.226742 −24.185820 2 −24.412145 −0.000833 −0.176041 −24.589020 3 −24.412145 −0.001250 −0.091555 −24.504950 4 −24.412145 −0.001667 −0.164330 −24.578142 5 −24.412145 −0.002083 +0.331640 −24.082589 Simulated data from relation (4–28), representing a hypothetical first detection of the tensor 1 − 1 power spectrum with r = 100 and nt = 800 . The random errors ei follow a normal distribution 1 with mean zero and standard deviation σ = 4 .
(with xi = i/3) is, ∑ ∑ ∑ ∑ ∑ ∑ N 2 N N N N N x yj − xi xjyj xi(xi −xj)yj α = i=1 i ∑j=1 ∑i=1 j=1 = ∑i=1 ∑j=1 , (4–29) N 2 N 2 N i−1 2 N x −( xi) (xi −xj) ∑ i=1 ∑i i∑=1 ∑ ∑ i=2 j=1 N N N N i−1 N xiyi − xi yj (xi −xj)(yi −yj) β = i=1∑ i∑=1 j=1 = i=2∑ j=1∑ . (4–30) N 2 − N 2 N i−1 − 2 N i=1 xi ( i=1 xi) i=2 j=1(xi xj)
Even in this general form it is obvious that expression (4–29) for α represents a sort of average whereas expression (4–30) is a kind of numerical derivative. So we expect the fractional error on β to be larger than that on α. That becomes even more apparent when specializing to
N = 5 and xi = i/3,
(8y +5y +2y −y −4y ) α −→ 1 2 3 4 5 ≃ −24.453306 0.262202 , (4–31) 10 (−6y −3y +3y +6y ) β −→ 1 2 4 5 ≃ +0.065202 0.237171 . (4–32) 10
Hence the simulated data of Table 4-1 implies a reasonably accurate reconstruction of the tensor-to-scalar ratio, [ ( )] r = exp α − ln 2.5×10−9 = 0.0096 0.0027 , (4–33)
but a miserably inaccurate value for the tensor spectral index,
nt = β = 0.065 0.237 . (4–34)
61 Table 4-2. Predicted Results − r × − i α 24 i zi yi zi 1 −24.453306 −0.000400 −24.453706 +0.267886 2 −24.453306 −0.000800 −24.454106 −0.134914 3 −24.453306 −0.001200 −24.454506 −0.050445 4 −24.453306 −0.001599 −24.454906 −0.123236 5 −24.453306 −0.001999 −24.455306 +0.372717 Predicted results according to relation (4–36), with the parameters α and r taken from expressions (4–31) and (4–33), respectively.
The resulting check of the single scalar consistency relation is not very sensitive,
0.010 0.003 = −0.522 1.897 . (4–35)
Because of the large (but statistically allowed) positive fluctuation e5 the measured tensor spectral index (4–34) does not even have the right sign! We propose to instead use the much better measured scalar spectral index to predict the tensor spectral index, up to an integration constant, and then to compare the fluctuation of the observed data around this prediction. For the model in question this might amount to assuming predictions of the form, r z = α − ×i , (4–36) i 24 where α is (4–31) and r is (4–33). Table 4-2 reports these predictions, along with the
difference between each simulated observation yi and the associated prediction zi. Of course the parameter r comes from the parameter α through relation (4–33), so the final column of Table 4-2 represents four statistically independent measurements. The resulting estimate for the scatter between measurement and prediction is, v u u ∑5 t1 (y −z )2 ≃ 0.246614 . (4–37) 4 i i i=1
1 This is quite consistent with our assumed 1-sigma fluctuation of σ = 4 for each observation.
62 4.3 Discussion
2 Resolving the tensor power spectrum ∆h(k) is crucial for the progress of inflation because it constrains the causative mechanism. This is already evident from the angst [88–91] elicited by the increasingly tight bounds on the tensor-to-scalar ratio r [26]. A positive detection at several different wave lengths has the potential to falsify entire classes of models. For example, any model in which inflation is driven by the potential of a minimally coupled scalar mustobey
relation (4–1) between r and the tensor spectral index nt [78–80]. Unfortunately, relation (4–
2 1) requires taking a derivative of ∆h(k), and the first generation of detections will probably be too sparse to provide a good bound because numerical differentiation makes bad data worse.
2 It makes more sense to integrate the high quality data we already possess for ∆R(k). If the leading slow roll expressions (4–9-4–10) are assumed then the prediction (4–26) from
2 2 ∆R(k) requires only a single integration constant from ∆h(k). (The same thing would be true even if the more accurate approximations (4–14-4–15) were employed [71].) Fixing this
2 constant uses up one combination of whatever data we have for ∆h(k), leaving the scatter of the remaining data about the prediction as a legitimate test of single scalar inflation. Hence relation (4–26) is a sort of integrated form of the single-scalar consistency relation (4–1) which can be applied more reliably. Section 3 compares this sort of scatter test with checking
r = −8nt for simulated data based on a hypothetical detection of r = 0.01 at five wave lengths with fractional errors similar to those reported in the spurious BICEP2 detection [23]. Of course no massaging of poorly resolved data is going to extract a precision bound, but the scatter test seems clearly better. Note that it is simple to adapt the scatter test to data fits. For example, the usual parameterization of the scalar data [19–22] implies, [ ] −1 ( ) − [( ) − ] k ns 1 r(k∗) k 1 ns 2 ≃ ⇒ 2 ≃ 2 − ∆R(k) As = ∆h(k) ∆h(k∗) 1 + 1 . (4–38) k0 8(1−ns) k∗
Here As is the scalar amplitude, ns is the scalar spectral index, and k0 is a fiducial wave number.
63 Finally, we can look forward to the day, in the far future, when the tensor power spectrum is well resolved. Then the sort of scatter test we propose could be employed to search for correlations between features in the two power spectra. For example, Figure 4-1 depicts the bump in the first slow roll parameter from a model50 [ , 51] introduced to explain the scalar power spectrum’s dip at ℓ ≈ 22 and peak at ℓ ≈ 40 [52–55]. These features are caused by the way the scalar nonlocal corrections (4–17) depend upon derivatives of ϵ(n). The tensor nonlocal corrections (4–18) involve the same derivatives — although lacking the large factors of 1/ϵ — so it is obvious there will be corresponding features [71]. Resolving this sort of correlation probes the functional relation between the two power spectra far more deeply than the single scalar consistency relation.
64 CHAPTER 5 PRECISION PREDICTIONS FOR PRIMORDIAL POWER SPECTRA FROM F (R) MODELS OF INFLATION
5.1 Introduction
The proposal that the evolution of the universe is caused mainly by gravitation attracts more and more attention. However, it has been realized that gravity is not as simple as we thought and could be modified from standard General Relativity in several ways. Modified gravity theories are especially attractive to explain the current phase of cosmic acceleration. The first complete model of primordial inflation was the 1980 proposal by Starobinsky to modify the gravitational Lagrangian by the addition of a term quadratic in the Ricci scalar [2]. Although this model was for decades eclipsed by scalar potential models, the increasingly tight bounds on the tensor-to-scalar ratio [25], and the consequent elimination of the simplest potentials [26], have combined to produce a resurgence of interest in it [92]. It has been realized lately that more general modifications of the Hilbert Lagrangian, from R to f(R), may provide a consistent description of late time acceleration [93, 94], or even provide a unified description of primordial inflation and dark energy95 [ ]. A number of modified gravities which may consistently describe such a unified evolution of the universe are known [96, 97]. f(R) gravity has attracted the main interest because it is ghost-free and reasonably simple. It is quite remarkable that f(R) gravity appears as a two-faced Janus: in the Jordan frame it is a modified gravity theory, whereas it is a kind of scalar-tensor theory after conformal transformation to the Einstein frame. The equivalence of the two frames has
Reprinted with permission from: D.J. Brooker, S.D. Odintsov, & R.P. Woodard, ’Precision Predictions for Primordial Power Spectra from f(R) Models of Inflation’, Nucl. Phys. B, 911, 318-337 [81]
65 been demonstrated for some important observables [98–101], however, that may not be the whole story for a number of reasons:
• Singularities (typical for super-acceleration) can lead to a breakdown of the mathematical equivalence between the two frames [102–104];
• The non-gravitational sector of the theory knows the difference because matter is minimally coupled in the Jordan frame whereas the coupling is highly non-minimal in the Einstein frame [105, 106]; and
• It can happen that the universe accelerates in one frame while decelerating in the other [107]. Nevertheless, it is expected that, for regular geometries, and in the absence of matter, the two frames are indeed equivalent. Studies of f(R) inflation have been made in the Jordan frame [108–111], but the normal, and much easier approach, is to work in the Einstein frame. Although we shall have to discuss the issue of frame dependence somewhat, the purpose of this paper is to extend to f(R) inflation a new formalism for computing the scalar and tensor power spectra. The formalism is based on first replacing the usual linear evolution equations for the mode functions with nonlinear evolution equations for the norm-squared mode functions which go into the power spectra [40]. This avoids the wasted effort of keeping track of the irrelevant phase. We then factor out the exact solutions which exist for constant first slow roll parameter, and derive a Green’s function solution for the residual46 factor[ , 47] which can be written for an arbitrary inflationary geometry. The power inherent is this analytic functional representation has been recently exploited to derive an improved version [74] of the famous single scalar consistency relation [78–80]. In section 2 we show how primordial perturbations appear in the Jordan and Einstein frames. Section 3 is devoted to the issue of using the power spectra (when the tensor power spectrum is eventually resolved) to reconstruct either a scalar potential model or an f(R) model which would generate them. In section 4 we apply the new technique to two models of f(R) inflation. Our conclusions comprise section 5.
66 5.2 Numerical Equality but Form Dependence
The purpose of this section is to show that the scalar and tensor perturbation fields of the Jordan and Einstein frames agree, but their power spectra nonetheless take highly different forms when expressed in terms of the geometrical quantities of each frame. Webegin with a careful definition of the two frames, their backgrounds, their natural gauges andtheir perturbation fields. We then give the relation between the backgrounds and perturbations of each frame. The Starobinsky model provides a nice illustration of frame dependence because the standard slow roll approximations for the power spectra are valid in the Einstein frame but completely incorrect in the Jordan frame. 5.2.1 The Model in the Jordan Frame
The (spacelike) metric of the Jordan frame is gµν, which couples minimally to matter and gives physical distances and times. The Lagrangian of this frame is, √ f(R) −g L = . (5–1) 16πG
Its equation of motion is, [ ] 1 f ′(R)R − f(R)g + g − D D f ′(R) = 0 , (5–2) µν 2 µν µν µ ν
µν where Dµ represents the covariant derivative operator and ≡ g DµDν is the covariant d’Alembertian. The background geometry of the Jordan frame takes the form, [ ] ds2 = −dt2 + a2(t)d⃗x·d⃗x = a2(t) −dη2 + d⃗x·d⃗x . (5–3)
One can see from (5–2) one can see that this background obeys the equations, ( ) ( ) ( ) ˙ 2 ′ 1 ′ 0 = −3(H + H )f R0(t) + f R0(t) + 3H∂tf R0(t) , (5–4) ( ) (2 ) ( ) 1 0 = (H˙ +3H2)f ′ R (t) − f R (t) − (∂2 +2H∂ )f ′ R (t) . (5–5) 0 2 0 t t 0
67 Here and henceforth H(t) ≡ a/a˙ is the Hubble parameter of the Jordan frame and R0(t) ≡ 6H˙ (t) + 12H2(t) is the background value of the Ricci scalar. Adding (5–4) to (5–5) gives a relation we shall exploit later, [ ] ′′ ˙ ˙ ′ ′′ ˙ ∂t f (R0)R0 = −2Hf (R0) + Hf (R0)R0 . (5–6)
The natural temporal gauge condition for the Jordan frame is R(t, ⃗x) = R0(t) [112].
In this gauge the g00 and g0i components of the metric are constrained fields. The gij components take the form, [ ] 2 2ζ(t,⃗x) h(t,⃗x) gij(t, ⃗x) = a (t) × e × e , hii(t, ⃗x) = 0 . (5–7) ij
Note that requiring hii = 0 is not a gauge condition but rather how one defines the breakup
between ζ and hij. The spatial gauge condition is,
∂ihij(t, ⃗x) = 0 . (5–8)
The homogeneity and isotropy of the Jordan frame background implies that the perturbation fields have the following free field expansions, ∫ { } √ d3k ∑ h (t, ⃗x) = 32πG u(t, k)ei⃗k·⃗xϵ (⃗k, λ)α(⃗k, λ) + c.c. , (5–9) ij (2π)3 ij λ= ∫ { } √ d3k ζ(t, ⃗x) = 4πG v(t, k)ei⃗k·⃗xβ(⃗k) + c.c. . (5–10) (2π)3
⃗ The polarization tensor ϵij(k, λ) obeys the same relations as in flat space, and is identical to the flat space result,
⃗ ∗ ⃗ kiϵij = 0 = ϵii , ϵij(k, κ)ϵij(k, λ) = δκλ . (5–11)
The creation and annihilation operators also obey the flat space relations, [ ] [ ] ⃗ † 3 3 ⃗ ⃗ † 3 3 ⃗ α(k, κ), α (⃗p,λ) = δκλ(2π) δ (k−⃗p) , β(k), β (⃗p) = (2π) δ (k−⃗p) . (5–12)
68 It is best to define the (tree order) power spectra as the asymptotic late time formsof equal-time correlators, ∫ ⟨ ⟩ k3 ∆2 (t, k) ≡ d3x e−i⃗k·⃗x Ω h (t, ⃗x)h (t,⃗0) Ω h 2π2 ij ij k3 × × ×| |2 = 2 32πG 2 u(t, k) , (5–13) ∫ ⟨ 2 π ⟩ k3 k3 ∆2 (t, k) ≡ d3x e−i⃗k·⃗x Ω ζ(t, ⃗x)ζ(t,⃗0) Ω = ×4πG×|v(t, k)|2 , (5–14) R 2π2 2π2
The equations obeyed by the tensor mode function u(t, k) are fairly easy to read off by linearizing (5–2) and applying canonical quantization,
( ′′ ˙ ) 2 f (R0)R0 k ∗ − ∗ i u¨ + 3H + ′ u˙ + 2 u = 0 , uu˙ uu˙ = ′ 3 . (5–15) f (R0) a f (R0)a
Obtaining the scalar mode equations is much more difficult because one must first solve the constraints. A long calculation reveals that v(t, k) obeys,
( ′′ ˙ ˙ ) 2 f (R0)R0 E k ∗ − ∗ i v¨ + 3H + ′ + v˙ + 2 v = 0 , vv˙ vv˙ = ′ 3 , (5–16) f (R0) E a Ef (R0)a where the function E(t) is, ′′ ˙ f (R0)R0 2 3( ′ ) 2f (R0)H E = ′′ ˙ . (5–17) f (R0)R0 2 (1+ ′ ) 2f (R0)H Differential equations such5–15 as( -5–16) define the mode functions up to initial conditions. The usual (Bunch-Davies-like) initial conditions are that the WKB forms apply in the distant past, [ ∫ ] 1 t dt′ u(t, k) −→ √ exp −ik , (5–18) ′ 2 ′ 2kf (R0(t))a (t) ti a(t ) [ ∫ ] 1 t dt′ v(t, k) −→ √ exp −ik . (5–19) ′ 2 ′ 2kE(t)f (R0(t))a (t) ti a(t )
One can see from (5–15-5–16) that the mode functions must approach constants when the term k2/a2(t) becomes insignificant. Those constants can be found by using(5–15-5–16) to evolve u(t, k) and v(t, k) from their initial forms (5–18-5–19). Substituting those constants
69 into the time-dependent power spectra (5–13-5–14) gives the model’s predictions for the primordial power spectra. 5.2.2 The Model in the Einstein Frame
The transformation from the Jordan frame to the Einstein frame is effected by first introducing an auxiliary scalar ϕ which obeys the equation,
ϕ = f ′(R) ⇐⇒ R = R(ϕ) . (5–20)
We then construct a potential U(ϕ) by Legendre transforming, ( ) U(ϕ) ≡ ϕR(ϕ) − f R(ϕ) ⇐⇒ U ′(ϕ) = R(ϕ) . (5–21)
The Einstein frame Lagrangian is, [ ] 1 √ Le = ϕR − U(ϕ) −g . (5–22) 16πG
The two field equations associated with5–22 ( ) are,
0 = R − U ′(ϕ) , (5–23) [ ] [ ] 1 0 = ϕR − ϕR−U(ϕ) + g −D D ϕ . (5–24) µν 2 µν µ ν
Of course (5–23) reproduces (5–20), whereupon we recognize (5–24) as the Jordan frame Equation (5–2). We reach the final form of the Einstein frame by making a field redefinition whichisthe conformal transformation, [ √ ] 16πG ge ≡ ϕg ⇐⇒ g = exp − φ ge , (5–25) µν µν µν 3 µν √ [√ ] 3 16πG φ ≡ ln(ϕ) ⇐⇒ ϕ = exp φ . (5–26) 16πG 3
Substituting (5–25-5–26) in (5–22) gives the classic form of a minimally coupled scalar, √ Re −ge 1 √ √ Le = − ∂ φ∂ φgeµν −ge − V (φ) −ge . (5–27) 16πG 2 µ ν
70 where the scalar potential is, [ √ ] ( [√ ]) 1 16πG 16πG V (φ) ≡ exp −2 φ U exp φ . (5–28) 16πG 3 3
The background geometry of the Einstein frame takes the form, [ ] dse2 = −det2 + ea2(et)d⃗x·d⃗x = ea2(et) −dη2 + d⃗x·d⃗x . (5–29)
e It relates to the background scalar field φ0(t) through the Einstein equations, [ ( )] e 2 e 1 2 e e 3H (t) = 8πG φ˙ 0(t) + V φ0(t) , (5–30) [2 ( )] ˙ 1 −2He(et) − 3He 2(et) = 8πG φ˙ 2(et) − V φ (et) . (5–31) 2 0 0 e e The natural temporal gauge condition in the Einstein frame is φ(t, ⃗x) = φ0(t) [35]. In
this gauge the ge00 and ge0i components of the metric are constrained fields and the spatial components take the form, [ ] e 2 e 2ζe(et,⃗x) eh(et,⃗x) e geij(t, ⃗x) ≡ ea (t) × e × e , hii = 0 . (5–32) ij e e Note that requiring hij(t, ⃗x) to be traceless is not a gauge condition but rather part of the e e e e definition of ζ(t, ⃗x). The true spatial gauge condition is the transversality of hij(t, ⃗x),
e e ∂ihij(t, ⃗x) = 0 . (5–33)
Homogeneity and isotropy are also symmetries in the Einstein frame so we can expand the perturbation fields the same way as in the Jordan frame, only with different mode functions, ∫ { } √ d3k ∑ eh (et, ⃗x) = 32πG ue(et, k)ei⃗k·⃗xϵ (⃗k, λ)α(⃗k, λ) + c.c. , (5–34) ij (2π)3 ij λ= ∫ { } √ d3k ζe(et, ⃗x) = 4πG ve(et, k)ei⃗k·⃗xβ(⃗k) + c.c. . (5–35) (2π)3
Note that the polarization tensor of the Einstein frame is identical to that of the Jordan frame, as are the creation and annihilation operators. The time dependent power spectra are defined
71 in the same way as for the Jordan frame to give,
k3 ∆e 2 (et, k) ≡ ×32πG×2×|ue(et, k)|2 , (5–36) h 2π2 k3 ∆e 2 (et, k) ≡ ×4πG×|ve(et, k)|2 . (5–37) R 2π2
By solving the constraint equations and employing canonical quantization one finds that the mode functions obey the following equations and Wronskian normalization conditions, [ ] ∂2 ∂ k2 ∂ue∗ ∂ue i + 3He + ue = 0 , ue − ue∗ = , (5–38) e2 e ea2 e e ea3 [ ( ∂t )∂t ] ∂t ∂t ∂2 1 deϵ ∂ k2 ∂ve∗ ∂ve i + 3He + + ve = 0 , ve − ve∗ = . (5–39) ∂et2 eϵ det ∂et ea2 ∂et ∂et eϵea3
The assumption of Bunch-Davies-like vacuum corresponds to the following asymptotic early time forms, ∫ [ et ′ ] e e −→ √ 1 − dt u(t, k) exp ik ′ , (5–40) e ea(t ) 2kea2(et) ti ∫ [ et ′ ] e e −→ √ 1 − dt v(t, k) exp ik ′ . (5–41) e ea(t ) 2keϵ(et)ea2(et) ti
The model’s predictions for the primordial power spectra are obtained by using (5–38-5– 39) to evolve ue(et, k) and ve(et, k) from their initial forms (5–40-5–41) to find their late time constant values, and then substituting these constants into the time dependent power spectra (5–36-5–37). 5.2.3 Relating Backgrounds and Perturbation Fields
Comparison of expression (5–7) with (5–32), and relations (5–25-5–26), implies that the perturbation fields agree between the two frames101 [ ],
ζ(t, ⃗x) = ζe(et, ⃗x) , (5–42) e e hij(t, ⃗x) = hij(t, ⃗x) . (5–43)
72 This means that the scalar and tensor power spectra also agree numerically between the two frames. However, expressions for those power spectra are quite frame dependent because the expansion histories and co-moving times of the two frames do not agree, [ √ ] √ ( ) 4πG a(t) = exp − φ(et) ×ea(et) ⇐⇒ ea(et) = f ′ R (t) ×a(t) , (5–44) 3 0 [ √ ] √ ( ) 4πG dt = exp − φ(et) ×det ⇐⇒ det = f ′ R (t) ×dt . (5–45) 3 0
It follows that the Hubble parameter of the Einstein frame is, [ ] [√ ] e e d e 1 d ′ H(t) ≡ ln ea(t) = √ ln f (R0(t)) a(t) , (5–46) det ′ dt [ f (R0(t))] H f ′′(R )R˙ = √ 1 + 0 0 . (5–47) ′ ′ f (R0) 2f (R0)H
Using relation (5–6) the first slow roll parameter is,
[ √ ] ′′ ˙ ′ f (R0)R0 2 d 1 1 d f (R ) 3( ′ ) e e ≡ √ 0 2f (R0)H ϵ(t) = ′′ ˙ = ′′ ˙ . (5–48) e e e ′ dt f (R0)R0 f (R0)R0 2 dt H(t) f (R0) H + ′ [1 + ′ ] 2f (R0) 2f (R0)H Both parameters depend critically on the function X,
′′ ˙ ′′ [ ] ≡ f (R0)R0 −f (R0)R0 ϵ˙ X ′ = ′ ϵ + , (5–49) 2f (R0)H f (R0) 2(2−ϵ)H
2 where the final form on the right follows from R0 = 6(2 − ϵ)H and hence,
˙ 3 2 R0 = −12ϵ(2−ϵ)H − 6˙ϵH = −2H(ϵR0 +3˙ϵH) . (5–50)
Combining relations (5–47-5–48) with the usual slow roll results for the power spectra in the Einstein frame (and hence also in the Jordan frame) gives,
GHe 2 GH2(1+X)4 2 ≃ ∆R(k) = ′ 2 , (5–51) πeϵ 3πf (R0)X 16 16GH2(1+X)2 2 ≃ e 2 ∆h(k) GH = ′ . (5–52) π πf (R0)
73 Therefore, the tensor-to-scalar ratio is,
∆2 (k) 48X2 ≡ h ≈ e r(k) 2 16ϵ = 2 . (5–53) ∆R(k) (1+X)
′′ ′ Successful models of f(R) inflation typically have f (R0)R0/f (R0) ∼ 1, so relation (5–49) implies X ∼ −ϵ. Substituting into relation (5–53) means that slow roll inflation in the Einstein frame, with r ≈ 16eϵ, typically implies r ≈ 48ϵ2 when expressed using the Jordan frame geometry. 5.2.4 Starobinsky Inflation
Starobinsky inflation corresponds to,
8πGR2 16πGR 16πG f(R) = R + =⇒ f ′(R) = 1 + =⇒ f ′′(R) = . (5–54) 6M 2 6M 2 6M 2
Substituting (5–54) into the background Equations (5–4-5–5) reveals a good approximate solution with, M 2 H˙ (t) ≃ − ≡ −ϵ H2 , (5–55) 48πG i i
where Hi and ϵi are the initial values of the Hubble and first slow roll parameters. Hence the various geometrical parameters are,
≃ ϵi ϵ(t) 2 , (5–56) [1−ϵiHi∆t]
H(t) ≃ Hi[1−ϵiHi∆t] , (5–57) [ ] 1 a(t) ≃ a exp H ∆t− ϵ (H ∆t)2 . (5–58) i i 2 i i
Expressing these parameters in terms of the number of e-foldings n from the start of inflation gives, √ ϵi n ϵ = ,H = Hi 1−2ϵin , a = aie . (5–59) 1−2ϵin
Under the usual assumption that 0 < ϵi ≪ 1 we have, ( ) ( ) 2 f ′′ R (t) R (t) ≃ ≃ f ′ R (t) . (5–60) 0 0 3ϵ(t) 0
74 Substituting into relation(5–49) implies,
X(t) ≃ −ϵ(t) . (5–61)
Hence the first slow roll parameter of the Einstein5–48 frame( ) is much smaller than the first slow roll parameter of the Jordan frame, as depicted in Fig. 5-1. The power spectra and their ratio are, [ ] 2 2 2 2 GH GHi ∆R(k) ≃ ≃ 1−2ϵink , (5–62) 2πϵ 2πϵi 24 24 ∆2 (k) ≃ GH2ϵ ≃ GH2ϵ , (5–63) h π π i i 48ϵ2 ≃ 2 ≃ i r(k) 48ϵ 2 , (5–64) (1−2ϵink)
where nk ≃ ln(k/aiHi) is the e-folding of first horizon crossing. This model actually obeys the famous single-scalar consistency relation [78–80] but one would need to carry the expansion
2 ≃ 24 2 × − of ∆h(k) π GH ϵ (1 3ϵ + ... ) one more order to give a nonzero result for the tensor spectral index. However, relations (5–62-5–64) deviate extensively from the usual slow roll results when expressed in terms of the Jordan frame geometry. Starobinsky inflation obeys the general rule of f(R) inflation that its power spectra are numerically the same, for fixed wave number k, in both Jordan and Einstein frames. However, what this “k” means geometrically is very different in the two frames. One way to see the difference is by expressing the spectra in terms of the number of e-foldings until theendof inflation. From relation (5–44) we infer,
e ne a(t) ai 1 ea(t) ≡ eaie ≃ √ =⇒ eai ≃ √ , ne ≃ n + ln(1−2ϵin) . (5–65) 3 3 2 2 ϵ(t) 2 ϵi
1 1 1 1 Inflation ends at nend ≃ − , which corresponds to neend ≃ + ln(ϵi). The number of 2ϵi 2 2ϵi 2
Jordan frame e-foldings until the end of inflation is N ≡ nend − n, so the number of Einstein frame e-foldings until the end of inflation is,
1 Ne ≡ ne − ne ≃ N − ln(1+2N) . (5–66) end 2
75 Figure 5-1. Comparison of first slow roll parameter in the two frames for Starobinsky inflation. Comparison of first slow roll parameter in the two frames for Starobinsky inflation. Theblue curve gives Jordan frame result ϵ whereas the yellow curve shows the much smaller Einstein frame result eϵ of expression (5–48).
Therefore, a feature which occurs N = 50 Jordan frame e-foldings before the end of inflation appears at about Ne ≃ 47.7 e-foldings Einstein frame e-foldings before the end of inflation. 5.3 Constructing Models from Power Spectra
Because the perturbation fields of the Einstein and Jordan frames are identical, thepower spectra in each frame are the same functions of the wave number k. Given only these functions
2 2 ∆R(k) and ∆h(k), one cannot tell whether primordial inflation was driven by a scalar potential model or by an f(R) model. The purpose of this section is to explain how to reconstruct either
2 2 sort of model. We begin by using ∆R(k) and ∆h(k) to infer the scalar potential model which would produce them. We then construct the f(R) model that would produce the same results. 5.3.1 Reconstructing a Scalar Potential Model
If the inflationary expansion history a(t) is driven by the potential of a single, minimally coupled scalar then the resulting (tree order) scalar and tensor power spectra can be expressed in terms of the geometry near the time tk of first crossing, k ≡ H(tk)a(tk). The exact formulae take the form of leading slow roll results, times local slow roll corrections, multiplied
76 by nonlocal factors [46, 47], ( ) GH2(t ) ∆2 (k) = k ×C ϵ(t ) ×S(k) , (5–67) R πϵ(t ) k k ( ) 16 ∆2 (k) = GH2(t )×C ϵ(t ) ×C(k) . (5–68) h π k k
The local slow roll correction C(ϵ) is a monotonically deceasing function well approximated by 1 − ϵ (see Figure 2 of [46]),
( )[ ] 2 1 1 1 1−ϵ C(ϵ) ≡ Γ2 + 2(1−ϵ) ≈ 1 − ϵ . (5–69) π 2 1−ϵ
The nonlocal correction factors, S(k) and C(k), are unity for ϵ˙ = 0 and depend in a completely
known way [46, 47] upon conditions only a few e-foldings before and after tk. It would be simple enough to give an successive approximation technique for exactly
2 reconstructing H (tk) from the full expressions (5–67-5–68) but we will here work with just the
2 leading slow roll results. First, express ∆R(k) as a differential equation for H(tk),
GH2(t ) 1 d 1 2G 1 2 ≃ k ⇒ ≃ ∆R(k) = 2 2 . (5–70) πϵ(tk) H(tk) dtk H (tk) π ∆R(k)
2 Now multiply by H(tk)dtk ≃ dk/k, integrate to solve for H (tk), and express the integration
2 constant in terms of the leading slow roll result for ∆h(k),
2 π 2 H (t∗) ∆ (k∗) 2 ≃ ∫ ≃ 16G h H (tk) 2 ′ ∫ ′ 2 . (5–71) 2GH (t∗) k dk 1 1 k dk ∆R(k∗) ′ ′ 1+ ∗ 2 1 + r(k∗) ′ 2 ′ π k k ∆R(k ) 8 k∗ k ∆R(k ) One finds the scale factor by, k a(tk) = . (5–72) H(tk)
The construction is completed by integrating the differential relation H(tk)dtk ≃ dk/k and then inverting to solve for k(t), ∫ k dk′ t = t∗ + ⇐⇒ k = k(t) . (5–73) ′ ′ k∗ k H(tk )
77 Of course these operations would have to be performed numerically, but we stress that, by going beyond the leading slow roll forms, the reconstruction could be accomplished to a
2 2 precision limited only by the quality of the data for ∆R(k) and ∆h(k). Note also that the construction depends much more heavily on the well-measured scalar power spectrum, with its tensor cousin used only to supply integration constants. By comparing this reconstruction
2 with ∆h(k), when it is finally resolved, one can test the consistency of assuming single scalar inflation [74]. Given the expansion history a(t) and its derivatives, we can apply a well known construction [27–31, 114] to find the scalar and its potential from the two nontrivial Einstein equations, [ ( )] 1 3H2(t) = 8πG φ˙ 2(t) + V φ(t) , (5–74) [2 ( )] 1 −2H˙ (t) − 3H2(t) = 8πG φ˙ 2(t) − V φ(t) , (5–75) 2
By adding (5–74) to (5–75) we can reconstruct the scalar, up to its initial value and an arbitrary sign choice, ∫ √ t − ˙ 2 2H(s) − 2H(t) = 8πGφ˙ (t) =⇒ φ(t) = φ(ti) ds . (5–76) ti 8πG
Expression (5–76) makes sense as long as H˙ (t) < 0, which is the usual case. Under the same assumption, the scalar φ(t) is a monotonically growing or falling function of time, and we can invert (5–76) to find t(φ). The final step is substituting this expression into the difference of (5–74) and (5–75) in order to reconstruct the potential,
H˙ (t(φ)) + 3H2(t(φ)) V (φ) = . (5–77) 16πG
5.3.2 Reconstructing an f(R) Model
The previous subsection explained how the power spectra could be used to reconstruct a
2 2 scalar potential model which would produce the observed power spectra ∆R(k) and ∆h(k). Suppose that this has been done this. To find the f(R) model which would produce the very
78 same power spectra, one begins by regarding the reconstructed expansion history (5–72) as the Einstein frame scale factor ea(et) of some f(R) model, expressed as a function of the Einstein frame time et. Similarly, consider the reconstructed scalar (5–76) as the Einstein frame scalar φ(et), also expressed as a function of et. The next step is to reconstruct the geometry of the Jordan frame. This is accomplished by integrating Equation (5–45) and inverting to express the Einstein frame time as a function of the Jordan frame time, ∫ [ √ ] et 4πG e t = ti + ds exp − φ(s) =⇒ t(t) . (5–78) e ti 3
Now substitute into relation (5–44) to find the Jordan frame expansion history, [ ] √ ( ) ( ) 4πG a(t) = exp − φ et(t) ×ea et(t) . (5–79) 3
Of course this gives us the Hubble parameter H(t) and the first slow roll parameter ϵ(t) as well.
The final step is to reconstruct the function f(R). First, invert the relation for R0(t) to express time as a function of the Ricci scalar, [ ] 2 R0(t) = 6 2−ϵ(t) H (t) ⇐⇒ t(R) . (5–80)
Now note that the differential of the Ricci scalar is, { } [ ] 3 dR0(t) = −12ϵ(t) 2−ϵ(t) − 6˙ϵ(t)H (t) dt . (5–81)
One finds f(R) by integrating the relation for f ′(R) and using (5–80), ∫ [√ ] t(R) ( ) ′ 16πG e ′ f(R) = f(Ri) + dR0(t ) exp φ t(t ) . (5–82) ti 3
79 5.4 Comparing Analytic and Numerical Results
The purpose of this section is to compare analytic and numerical results for Starobinsky inflation and another representative f(R) model. We begin by explaining how the analytic results are derived. Then the models are described and numerical results for their power spectra are given. The section closes by comparing with various analytic approximations.
2 2 5.4.1 How We Compute ∆R(k) and ∆h(k) We use the Hubble representation [32] of the Einstein frame, in which one assumes that
e ne e e e ea(t) ≡ eaie , H(t) and eϵ(t) are known, or can be generated numerically. Because the Einstein frame is a scalar potential model we represent the power spectra the same as expressions (5–67-5–68) but using the Einstein frame geometry, ( ) 16 ∆2 (k) = GHe 2(et )×C eϵ(et ) ×Ce(k) , (5–83) h π k k ( ) GHe 2(et ) 2 k × e e ×Se ∆R(k) = e C ϵ(tk) (k) . (5–84) πeϵ(tk)
Here the slow roll correction factor C(ϵ) was defined in (5–69). Of course the terms involving e e e e e H(tk) and eϵ(tk) are clear enough so it is the nonlocal correction factors, C(k) and S(k) which require explanation. Our technique for determining the nonlocal correction factors is based on nonlinear evolution equations [40] for the norm-squared mode functions Mf(et, k) ≡ |ue(et, k)|2 and Ne(et, k) ≡ |ve(et, k)|2 which appear in expressions (5–36) and (5–37) for the power spectra. We then factor out the instantaneously constant eϵ solutions and express the residuals in terms of the number of e-foldings ne since the beginning of inflation [46, 47], [ ] f e [ ] f e f e 1e e e M0(t, k) 1 M(t, k) ≡ M0(t, k) × exp − h(n,e k) , N(t, k) ≡ × exp − ge(n,e k) , (5–85) 2 eϵ(et, k) 2 where the instantaneously constant eϵ solution involves a Hankel function, ( ) 2 ( −e) f e π (1) k 1 3 ϵ M(t, k) ≡ H e , νe ≡ . (5–86) [1−eϵ(et)]He(et)ea3(et) νe(t) 1−eϵ(et)]He(et)ea(et) 2 1−eϵ
80 The nonlocal correction factors come from the late time forms of the residuals eh(n,e k) and ge(n,e k),
e [ ] 2ϵe(t) [ ] 2 −e e −e e [ ] ea(et) 1 ϵ(t) He(et) 1 ϵ(t) C(eϵ(et)) 1 Ce(k) = lim × × × exp − eh(n,e k) , (5–87) e≫e e e e e e e t tk a(tk) H(tk) C(ϵ(tk) 2 e [ ] 2ϵe(t) [ ] 2 e e ea(et) 1−ϵe(t) He(et) 1−ϵe(t) Se(k) = lim × e≫e e e e e t tk a(tk) H(tk) e e e e [ ] ×C(ϵ(t)) × ϵ(tk) × −1e e e e exp g(n, k) . (5–88) C(eϵ(tk) eϵ(t) 2
The residuals are damped, driven oscillators with small nonlinearities [46, 47],
′ ( ) [ ] ωe 1 2 e e′′ − e′ e2e e e′ e2 e− h h e h + ω h = S + h + ω 1+h e , (5–89) ω 4 ( ) [ ] ωe′ 1 2 ge′′ − ge′ + ωe2ge = Se + ∆Se + ge′ + ωe2 1+ge−ege . (5–90) ωe 4
Here and henceforth a prime denotes differentiation with respect to ne. It is remarkable that both the tensor and scalar residual have the same frequency,
1 ωe(n,e k) ≡ . (5–91) e 3 e f e H(ne)ea (t)M0(t, k)
The source for the tensor residual vanishes for constant eϵ [46] and is typically small, [ ] ( ) 4k2 Mf′′ 1 Mf′ 2 Mf′ Se(n,e k) ≡ − ωe2 + 2 0 − 0 + (3−eϵ) 0 . (5–92) e 2 2 f f f H ea M0 2 M0 M0
In contrast, the extra source for the scalar residual can be large if the potential has features
[47], [ ] ( ) eϵ′′ 1 eϵ′ 2 eϵ′ ∆Se(ne) ≡ −2 − + (3−eϵ) . (5–93) eϵ 2 eϵ eϵ Another remarkable fact is that the linear differential operators on the left hand side of (5–89-5–90) possess a Green’s function which is known analytically for an arbitrary inflationary expansion history [46, 47], [∫ ] e− e ne e e e θ(n m) e G(n; m) = e e sin dℓ ω(ℓ, k) . (5–94) ω(m, k) me
81 e e e This means we can express both residuals analytically as series expansions h = h1 + h2 + ... and ge = ge1 + ge2 + ... , whose first two terms are, ∫ ne e e e h1(n,e k) = dme G(ne; me )S(m,e k) , (5–95) ∫0 ne [ ] e e e ge1(n,e k) = dme G(ne; me ) S(m,e k) + ∆S(me ) , (5–96) ∫0 [ ] ne 1 1 e e e e e e e′2 e − e2 e e2 e h2(n, k) = dm G(n; m) h1 (m, k) ω (m, k)h1(m, k) , (5–97) 0 4 2 ∫ [ ] ne 1 1 e e e e e e e′2 e − e2 e e2 e g2(n, k) = dm G(n; m) g1 (m, k) ω (m, k)g1(m, k) . (5–98) 0 4 2
e The higher terms — h2(n,e k), ge2(n,e k) and so on — are only necessary if the residuals or their derivatives become order one or larger.
Figure 5-2. Comparison of the potentials V (φ) for Starobinsky inflation and the exponential model. Comparison of the potentials V (φ) for Starobinsky inflation (yellow) and the exponential model (blue).
Although expressions (5–95-5–98) involve integrations over the entire range of e-foldings from the beginning of inflation, the only net contributions come from the few e-foldings around first horizon crossing. The reason nothing happens before is that the frequency term issolarge
82 at early times, [ ] ( ) ( ) 2k 2 He 2ea2 Early Times : ωe2(n,e k) = 1 + O . (5–99) Heea k2
This means that the early time form of the scalar residual is small, the tensor residual is very small, and both are local [46, 47], ( ) ( ) Heea 2 He 4ea4 Early Times : ge(n,e k) = ∆Sb(ne)× + O , (5–100) 2k k4 [ ] ( ) ( ) Heea 4 He 6ea6 Early Times : eh(n,e k) = −4 eϵ′′ + (9−7eϵ)eϵ′ × + O . (5–101) 2k k6
Shortly after first horizon crossing the frequency drops to zero, [ ] ( ) 6−2ϵe 2 ( 2 ) 2k 1−ϵe π k Late Times : ωe2(n,e k) = + O . (5–102) e 4 3 eϵ e Hea −e 1−ϵe 4 H2ea2 [4(1 ϵ)] Γ ( 2 + 1−eϵ )
Although the residuals eh(n,e k) and ge(n,e k) have some small late time dependence due to continued evolution of eϵ(et), the full solutions Mf(et, k) and Ne(et, k) freeze in to constant values less than two e-foldings after horizon crossing. 5.4.2 The Two Models
We studied two models, both of which take the form (5–1). The first was Starobinsky inflation5–54 ( ), with the parameter and initial conditions chosen as,
−5 2 × −9 M = 10 , ϵi = 0.00221 , GHi = 7.55 10 . (5–103)
We also studied a model which has been proposed to describe cosmology from inflation to the current phase of acceleration [113], [ ] [ ( ) ] R 4 R2 f(R) = R − Λ 1−exp − + . (5–104) 2Λ 4Λ
The parameter and initial conditions were chosen as,
−16 2 × −15 GΛ = 10 , ϵi = 0.00501 , GHi = 2.22 10 . (5–105)
83 Figure 5-3. Comparison of the scalar power spectrum for Starobinsky inflation and the exponential model. 2 Comparison of the scalar power spectrum ∆R(k) for Starobinsky inflation (yellow) and the exponential model (blue). Both are displayed as a function of Ne, the number of Einstein frame e-foldings before the end of inflation at which horizon crossing occurs.
Despite the different functions f(R) between (5–54) and (5–104), the two models are quite similar as far as inflation is concerned. This shows up clearly from Figure 5-2 which gives their potentials. Although there are some significant differences for low potential, inflation is governed by the behavior for large potential, which is almost identical. 5.4.3 Power Spectra of the Two Models
We numerically simulated each model exactly. Figure 5-3 shows that the scalar power spectrum of the Starobinsky inflation is slightly larger than for exponential model, although both have roughly the same shape. From Figure 5-4 we see that the tensor power spectrum of Starobinsky inflation slight exceeds that of the exponential model. However, the difference is so slight that the tensor-to-scalar ratio of the exponential model exceeds that of Starobinsky inflation. Figure 5-6 displays the spectra of the Starobinsky model as functions of the number of e-foldings N to the end of inflation in the Jordan frame, and the number of e-foldings Ne to
84 Figure 5-4. Comparison of the tensor power spectrum and the tensor-to-scalar ratio r(k) for Starobinsky inflation and the exponential model. 2 Comparison of the tensor power spectrum ∆h(k) (left) and the tensor-to-scalar ratio r(k) (right) for Starobinsky inflation (yellow) and the exponential model (blue). All results are displayed as a function of Ne, the number of Einstein frame e-foldings before the end of inflation at which horizon crossing occurs.
Figure 5-5. The various spectra for Starobinsky inflation in the Jordan frame and the Einstein frame.
Figure 5-6. The various spectra for Starobinsky inflation in the Jordan frame and the Einstein frame. 2 2 The various spectra — ∆R(k) (left), ∆h(k) (middle) and r(k) (right) — for Starobinsky inflation, as functions of the number of e-foldings from first horizon crossing until the end of inflation. For the yellow plots the x axes give N, the number of e-foldings in the Jordan frame, whereas the x axes of the blue plots give Ne, the number of e-foldings in the Einstein frame. the end of inflation in the Einstein frame. In each case, features at N appear to be displaced e ≃ − 1 to N N 2 ln(1 + 2N), in agreement with Equation(5–66). Figure 5-8 gives the relation between N and Ne for the exponential model. 5.4.4 Comparison with Analytic Results
A major point of this paper has been to develop good analytic approximations for how the power spectra of f(R) models depend functionally upon the geometry. For all the spectra, and
85 Figure 5-7. The various spectra for the exponential model in the Jordan frame and the Einstein frame.
Figure 5-8. The various spectra for the exponential model in the Jordan frame and the Einstein frame. 2 2 The various spectra — ∆R(k) (left), ∆h(k) (middle) and r(k) (right) — for the exponential model, as functions of the number of e-foldings from first horizon crossing until the endof inflation. For the yellow plots the x axes give N, the number of e-foldings in the Jordan frame, whereas the x axes of the blue plots give Ne, the number of e-foldings in the Einstein frame.
Figure 5-9. Comparison of the exact results with the leading slow roll approximation for Starobinsky inflation. Comparison of the exact results (yellow) with the leading slow roll approximation (blue) for Starobinsky inflation. The left graph shows the scalar power spectrum (5–51), the middle graph shows the tensor power spectrum (5–52), and the right graph show the tensor-to-scalar ratio (5–53).
for both models, the leading slow roll approximations are pretty accurate,
GHe 2(et ) GH2(t ) 2 ≃ k ≃ k ∆R(k) e , (5–106) πeϵ(tk) 2πϵ(tk) 16 24 ∆2 (k) ≃ GHe 2(et ) ≃ GH2(t )ϵ(t ) , (5–107) h π k π k k e 2 r(k) ≃ 16eϵ(tk) ≃ 48ϵ (tk) . (5–108)
Figure 5-9 shows this for Starobinsky inflation. Including the slow roll corrections, and just the linearized approximations for S(k) and C, makes the agreement essentially perfect. Figure 5-10 shows that the relative error of the scalar
86 Figure 5-10. Fractional error of our linearized approximation to the scalar and tensor power spectra for Starobinsky inflation. Fractional error of our linearized approximation to the scalar (left) and tensor (right) power spectra for Starobinsky inflation.
power spectrum is less than 0.3% for Starobinsky inflation. The relative error for the tensor power spectrum is actually at the 0.002% accuracy of our numerical simulation. 5.5 Discussion
We have developed a good functional form for the primordial power spectra of f(R) inflation, after discussing (in section 2) the relation between Jordan and Einstein frames. When the Einstein frame potential lacks features, the leading slow roll results (5–106-5–108) are accurate. This is shown for Starobinsky inflation by Figure 5-9. (An f(R) model will agree with Starobinsky inflation if the parameter X(t) of equation (49) obeys X(t) ≃ −ϵ(t).) When features are present (for which there continues to be observational support [55]), one gets essentially perfect agreement by using just the first two terms of the nonlocal correction factors (5–95-5–98) in expressions (5–83-5–84)[47]. One cannot distinguish f(R) models from scalar potential models with just the power spectra. In section 3 we showed how the same data could be used to reconstruct either kind of model. Even for de Sitter-like models this changes if one has information about what the wave number “k” means in terms of other scales. There is a shift of 2-3 e-foldings between the same feature of the scalar potential reconstruction and the f(R) reconstruction, with the scalar potential model feature appearing nearer to the end of inflation. One can see this from Figures 5-6 and 5-8.
87 Finally, we mention that an interesting and very topical application of this formalism is perturbations for Higgs inflation [115, 116]. More generally, scalar models with a nonminimal coupling involve similar conformal transformations between Jordan and Einstein frames.
88 CHAPTER 6 OUTLOOK AND CONCLUSIONS We have seen in the introduction why a theory of inflation is important, and how it fits into our larger understanding of cosmology. In Chapters 2 and 3 the bulk of the theoretical developments were performed as we developed a new and unparalleled formalism for computing the primordial power spectra from an inflationary geometry and for computing the inflationary geometry from an arbitrary non-flat primordial scalar power spectrum. Finally in Chapters 4 and 5 we examined two applications of our formalism. The first application was to improve the single scalar consistency relation and the second was to extend our formalism to any f(R) model of inflation. Before we finish it is important that we do a point by point analysis ofhow our formalism compares with and improves upon previous work in this area. Another formalism which is commonly used for understanding slow roll violating geometries is the ’Generalized Slow Roll’ (GSR) formalism which develops a perturbation theory for the scalar and tensor mode functions themselves and it has been carried out to second order [60–63]. In spite of the overlap we feel that our approach is advantageous for a number of reasons which we will expand upon here. Evolving the amplitudes instead of the mode functions is a much better way to approach the problem of computing the power spectra because the phases are ultimately irrelevant to the spectra. One way to see why it is more efficient to evolve M(t, k) and N(t, k) than the mode functions themselves comes from comparing the asymptotic expansions of the tenor modes u(t, k) and tensor amplitudes M(t, k) in the early time regime of k ≫ H(t)a(t). Due to the fact that the two mode functions are related, the conclusions which we will draw for the tensor modes will extend to the scalar modes as well. Recall from Chapter 2 that the expansions for
89 u(t, k) is in powers of 1/k and is not even local at first order, { } ∫ ( ) − t dt′ iα(t) β(t) 1 exp[ ik ′ ] u(t, k) = 1 + + + O × √ ti a(t ) , (6–1) k k2 k3 2ka2(t) ∫ [ ] 1 t α(t) = dt′ 2−ϵ(t′) H2(t′)a(t′) , (6–2) 2 ti [ ] 1 1 β(t) = − α2(t) + 2−ϵ(t) H2(t)a2(t) . (6–3) 2 4
In contrast, the expansion for M(t, k) is in powers of 1/k2 and is local to all orders, { } ( ) α(t) β(t) 1 1 M(t, k) = 1 + + + O × , (6–4) k2 k4 k6 2ka2(t) ( ) 1 α(t) = 1− ϵ H2a2 , (6–5) [ 2 ] ( )( ) 9 2 1 9˙ϵ 3ϵϵ˙ ϵ¨ β(t) = ϵ 1− ϵ 1− ϵ + − + H4a4 . (6–6) 4 3 2 8H 4H 8H2
Taking the norm-squared of (6–1) demonstrates why our formalism is so much more accurate than the generalized slow roll approximation, { }{ } ( ) ( ) 2 iα β 1 iα β 1 1 u(t, k) = 1+ + +O 1− + +O × , (6–7) k k2 k3 k k2 k3 2ka2 { } ( ) [α2 +2β] 1 1 = 1+ +O × . (6–8) k2 k4 2ka2
Comparing (6–8) with (6–4) shows that we must expand u(t, k) to second order to recover the first order correction to M(t, k). This is important because at late times At late times just as at early times, our expansion for the amplitudes will converge faster than an expansion for the mode functions. We can see this by doing a direct comparison between our formalism and the various formulations established in reference [62]. In both this paper and the paper of Dvorkin and Hu the step model is considered so we can even compare directly their accuracy and our own. In the rest of this paragraph we will refer exclusively to this reference when making comparisons between our formalism and GSR. We can see from Figures 10 and 11 of the Dvorkin and Hu
90 work that an expansion based on computing the mode functions iteratively (GSRS) does not converge very well. More over this formalism is ambiguous in the sense that they get different results when evaluating at different times which physically should not be the case as longas the spectrum is being evaluated well after horizon crossing. This is not however the primary method used in the for approximating the power spectrum for precisely this reason. The other main expansion which they study is called the ’Iterative GSLR expansion’ which has better convergence properties. Still however as the authors point out themselves their first corrections ’come half from the first order calculation of the real part of the field and half from iterating the imaginary part to second order’ [62]. This is in contrast to our first correction which is an entirely first order result. Once the corrections are computed accomplished however, the GSR approximation achieves comparable accuracy with our work. In both cases the error for the leading approximation is on the order of 20% at its highest for the step model. There are two points of contact between our formalism and GSR which are important to mention since it explains why we achieve similar accuracy levels. In the De Sitter limit our Green’s function is the same as their ’window function’ W (x) up to an overall multiplicative factor after a change of variables. Second, our source function ∆S(n) is contained in their source function g(ln u) along with some additional factors which are not present in our formalism. Importantly however we stop evaluating this source at horizon crossing and use a different source afterwords. It is also important to keep in mind that our Green’s function is based on making what we call the ’Instantaneously Constant Epsilon’ approximation for the zeroth order amplitudes, which is also sometimes called the ’Bessel Approximation’ or the ’Horizon Crossing Approximation’ [45]. Since our formalism has now been applied to the question of reconstructing features from primordial power spectra we shall discuss the literature on reconstruction as well. There are several proposals in the recent literature concerning reconstruction in the presence of slow roll violations. The key advantage of our approach is that it is easy to implement and since it is based on Green’s function techniques it is very straight forward to compute. With
91 regards to the step model our formalism only requires us to go to first order to achieve less than 5% accuracy for ϵ(t) and less than 0.04% accuracy for H(n). Other proposals which deal with slow roll violating models have been based on the General Slow Roll approximation [68] and inverse-scattering [117]. In the case of inverse scattering a detailed error analysis is provided and the authors obtain errors in H(n) on the order of a few percent. For the case of Generalized Slow Roll there is no error analysis provided, however since our approximation for the spectra has comparable accuracy to the Generalized Slow Roll approximation for the spectra, our reconstruction formulas should also be comparable. In spite of the similarities between our approach and the GSR approach we still feel that our method is superior because of its incredibly rapid convergence and for a few more reasons which we now emphasize. One key aspect of our formalism is that the amplitudes which we compute carry the exact same amount of information as the mode functions themselves. This is because the phase information that we discard when switching from (u, v) to (M,N) can be recovered as follows. We define the tensor and scalar mode functions as, √ √ u(t, k) ≡ M(t, k)×e−iθ(t,k) , v(t, k) ≡ N (t, k)×e−iϕ(t,k) . (6–9)
Next we simply use the Wronskian normalization conditions which the mode functions obey to express the phases in terms of the magnitudes (M(t, k),N(t, k),
i 1 uu˙ ∗ − uu˙ ∗ = =⇒ θ˙(t, k) = , (6–10) a3 2a3(t)M(t, k) i 1 vv˙ ∗ − vv˙ ∗ = =⇒ ϕ˙(t, k) = . (6–11) ϵa3 2ϵ(t)a3(t)N (t, k)
Equations (6–10-6–11) give us the desired phases and these can be used to compute products of mode functions at different times which enter both the propagator and the tree order non-Gaussianity, [ ∫ ] √ t dt′′ ∗ ′ ′ − u(t, k)u (t , k) = M(t, k)M(t , k) exp i 3 ′′ ′′ , (6–12) ′ 2a (t )M(t , k) [ ∫ t ] √ t dt′′ v(t, k)v∗(t′, k) = N (t, k)N (t′, k) exp −i . (6–13) ′′ 3 ′′ N ′′ t′ 2ϵ(t )a (t ) (t , k)
92 Finally, as we have mentioned before, our approach constructs a Green’s function for the amplitudes out of instantaneously constant ϵ solutions instead of using the De Sitter solutions which GSR is based upon. This means that our formalism will give a very good, rapidly converging representations of the mode functions at finite times. This is critical for studies of loop computations in quantum gravity. There are two big questions which are begging to be answered and our formalism is perfect for studying them. The two questions are:
• The zeta propagator must contain a factor of 1/ϵ(n), but at what value of n should we evaluate it?
• The quantum gravitational loop-counting parameter is GH2(n), but at what value of n should we evaluate it? The first point is especially subtle because it is usually assumed that factors of 1/ϵ(n) are cancelled by powers of ϵ(t) from the vertices [38]. However, it since the propogator depends nonlocally on ϵ(t) it might receive significant contributions from small, early values of ϵ(t). If so, one might expect loop corrections from vertices at late times to be enhanced by large factors of ϵlate/ϵearly. One other study which would be facilitated by our formalism concerns nonlocal modified gravity models of cosmology which may arise due to quantum gravitational effects that became nonperturbatively strong during inflation118 [ –120]. These quantum gravitational effects derive from secular growth in the graviton propagator which is known for de Sitter [121–123], but not for geometries in which ϵ(t) evolves [124]. Our formalism will facilitate better extrapolations of these growth factors to general geometries, which should motivate more realistic models. With that said there are however some key developments which must be left to future endeavors. They are:
1. Extending our formalism to include other modes which may be present during inflation.
2. Extending our formalism to models with non-minimal coupling between the scalar and gravity as mentioned in Chapter 5.
3. Using our better understanding of the mode functions to construct better approximations for propagators during inflation.
93 4. Developing better approximations for 3-point correlators. On this last point there has been work done in the context of models which violate slow roll but the work is limited to either the squeezed limit (when two of the three ⃗k’s are equivalent) or the equilateral case [36, 125, 126]. What is needed in this case, and what should be easily supplied by our formalism are simple analytic expressions for the three point functions in an arbitrary triangular configuration. The dominant term in the presence ofslow roll violations is proportional to: ∫ ∗ ∗ ∗ 3 ′ v (n, k1)v (n, k2)v (n, k3) a (m)H(m)ϵ (m)
′ ′ ×v (m, k1)v (m, k2)v(m, k3)dm + c.c. + permutations . (6–14)
Such an expression would lend itself nicely to study using our formalism.
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102 BIOGRAPHICAL SKETCH Daniel Brooker started his career in physics at the College of William and Mary where he double majored in Physics and French. While there he did research on infrared optics and condensed matter physics. After graduating he came to the University of Florida where in his first year courses he was inspired to make the switch to theoretical cosmology.
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