1 Introduction

UFIFT-QG-16-08 , CCTP-2016-10

CCQCN-2016-147 , ITCP-IPP 2016/08

Final Thoughts on the Power Spectra of Scalar Potential Models

D. J. Brooker, N. C. Tsamis and R. P. Woodard

Department of Physics, University of Florida,

Gainesville, FL 32611, UNITED STATES

Institute of Theoretical Physics & Computational Physics,

Department of Physics, University of Crete,

GR-710 03 Heraklion, HELLAS


We give final shape to a recent formalism for deriving the functional forms of the primordial power spectra of single-scalar potential models and theories which are related to them by conformal transformation. An excellent analytic approximation is derived for the nonlocal correction factors which are crucial to capture the “ringing” that can result from features in the potential. We also present the full algorithm for using our representation, including the nonlocal factors, to reconstruct the inflationary geometry from the power spectra.

PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g

e-mail: djbrooker@ufl.edu

e-mail: tsamis@physics.uoc.gr

e-mail: woodard@phys.ufl.edu

1 Introduction

The simplest models of primordial inflation are based on general relativity (for a spacelike metric ) plus a single, minimally coupled scalar ,


A key prediction is the generation of tensor [1] and scalar [2] perturbations. These are the first observable quantum gravitational phenomena ever recognized as such [3, 4, 5]. They are also our chief means of testing the viability of scalar potential models [6, 7, 8], and of reconstructing [9, 10, 11].

Reconstruction is simplest in terms of the Hubble representation [12] using the Hubble parameter and first slow roll parameter of the homogeneous, isotropic and spatially flat background geometry of inflation,111The connection to the potential representation is [13, 14, 15, 16, 17],


Let stand for the time of first horizon crossing, when modes of wave number obey . The tensor and scalar power spectra take the form of leading slow roll results at , multiplied by local slow roll corrections also at , times nonlocal factors involving times near [18, 19],


The local slow roll correction is,


The nonlocal correction exponents, and , vanish for and effectively depend on the geometry only a few e-foldings before and after [18, 19].

The purpose of this paper is to rationalize and simplify our formalism for evolving the norms of the mode functions, rather than the mode functions [20], and then to derive an excellent analytic approximation for the nonlocal correction exponents and . We also demonstrate how this approximation can be used to reconstruct the inflationary geometry from the power spectra, even for models which possess features. These topics represent sections 2-3, 4 and 5, respectively. In section 6 we discuss some of the many applications [21, 22] this formalism facilitates.

Figure 1: The left hand graph shows one model’s scalar power spectrum as a function of , the number of e-foldings from the beginning of inflation to first horizon crossing. The right hand graph shows the same power spectrum versus , the number of e-foldings until the end of inflation. Early times correspond to small and large , whereas late times correspond to large and small .

We shall often employ the alternate time parameter provided by , the number of e-foldings since inflation’s onset. This is superior to the co-moving time by virtue of being dimensionless and relating evolution to the size of the universe. We shall abuse the notation slightly by writing and , instead of the correct but cumbersome expressions and . Which time parameter pertains should be clear from context, and from our exclusive use of , and to stand for e-foldings. Over-dots represent time derivatives and primes stand for derivatives,


We caution readers against confusing with the common parameter , the number of e-foldings until the end of inflation (at ). Figure 1 illustrates the difference.

2 Our Formalism in General

The tree order tensor power spectrum is obtained by evolving the graviton mode function past the time of first horizon crossing [9, 10, 11],


We do not possess exact solutions for for realistic geometries , but we do know the evolution equation, the Wronskian and the form at asymptotically early times [9, 10, 11, 23],


Because the power spectrum depends upon the norm-squared, rather than the rapidly-varying phase, it is better to convert (8) into a nonlinear evolution equation for [20],


If necessary, the mode function can be easily recovered [19],


Relation (9) can be improved by changing to the dimensionless time parameter ,


A further improvement comes by factoring out an (at this stage) arbitrary approximate solution, , to derive a damped, driven oscillator equation (with small nonlinearities) for the residual exponent [18],


Here the frequency and the tensor source are,


It is an amazing fact that an exact Green’s function exists for the left hand side of equation (12), valid for any choice of the approximate solution [18],


This permits us to solve (12) perturbatively by expanding in the nonlinear terms,


The tree order scalar power spectrum is obtained by evolving the mode function past the time of first horizon crossing [9, 10, 11],


Just as for its tensor cousin, we lack exact solutions for for realistic geometries , but we do know the evolution equation, the Wronskian and the form at asymptotically early times [9, 10, 11, 23],


Converting to the norm-squared gives [19],


The scalar mode function mode can be recovered from [19],


Converting from co-moving time to gives,


Factoring out by an arbitrary approximate solution produces another damped, driven oscillator equation for the residual exponent,


Here the frequency and the scalar source are,


Making the replacement in (14) gives an exact Green’s function which is valid for any choice of ,


And we can of course develop a perturbative solution to (22) ,


3 Choosing and Effectively

The formalism of the previous section is valid for all choices of the approximate solutions and . Of course the correction exponents and will be smaller if the zeroth order solutions are more carefully chosen. In previous work we used the instantaneously constant solutions [18, 19],


where we define,


However, the choice (28) has the undesirable effect of complicating the late time limits. The physical quantities and freeze in to constant values soon after first horizon crossing, but continued evolution in prevents and from approaching constants. Hence the residual exponents and must evolve so as to cancel this effect.

We can make the late time limits simpler by adopting a piecewise choice for the approximate solutions,


By and we mean the solutions which would pertain for the ersatz geometry,


Here and henceforth stands for the number of e-foldings from horizon crossing. To be explicit about the over-lined quantities,


With the choice (30-31) the approximate solutions rapidly freeze in to constants,


This establishes the forms (3-4) for the power spectra and fixes the nonlocal correction exponents to,


It remains to specialize the sources to (30-31). First note the simple relation between the scalar and tensor frequencies,


This means the scalar source (24) consists of the tensor source (13) minus a handful of terms mostly involving ,


To obtain an explicit formula for the tensor source we first note that the tensor frequency is,


Hence the derivative of its logarithm is,


where and . Before horizon crossing is time dependent and so we have,


where and involve derivatives of with respect to and ,


The analogous result after horizon crossing is much simpler,


where means with specialized to and specialized to .

Taking the derivative of before horizon crossing,


requires three second derivatives of ,


Bessel’s equation and the Wronskian of imply,


Substituting relations (39), (40), (43) and (45) in the definition of the tensor source (13) gives,


The analogous result after horizon crossing is,


There is also a jump at horizon crossing so that the complete result is,

Figure 2: The left hand graph shows the local slow roll correction factor (solid blue), which was defined inexpression (5). Also shown is its global approximation of (dashed yellow) over the full inflationary range of . The right hand graph shows (solid blue) versus the better approximation of (large dots) relevant to the range favored by current data.

4 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 fact that the first slow roll parameter is very small. The confidence bound on the tensor-to-scalar ration of [24, 25] implies . This suggests a number of approximations. First, the local slow roll correction factor , defined in (5), may as well be set to unity. From Figure 2 we see that the bound of implies . This is not currently resolvable.

Another excellent approximation is taking in the tensor and scalar Green’s functions of expressions (14) and (25),


where and . Note that this expression is valid before and after horizon crossing. An important special case of (49) is when becomes large, which gives the function we define as,


From the graph in Figure 3 we see that suppresses contributions more than a few e-foldings before horizon crossing.

Figure 3: The left hand graph shows the Green’s function given in expression (50). The right hand graph shows the coefficient of in the small form (58) for . This function is defined by expressions (52), (54) and (59). 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 (54).
Figure 4: The coefficients of (left) and (right) in the small form (58) for . 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 (54-56) in expressions (60) and (61).

We can also take in and the derivatives of it in expressions (41) and (44). This leads to exact results for , and in terms of the parameter ,


The three derivatives with respect to do not lead to simple expressions even for , but they can be well approximated over the range we require by short series expansions in powers of ,


We can express the ratio of in terms of the deviation ,


All of this gives an approximation for the tensor source (48),


where the three coefficient functions are,


Figures 3 and 4 show the various coefficient functions.

Figure 5: 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 [28, 29] to explain the observed features in the scalar power spectrum at and which are visible in the data reported from both WMAP [27, 30] and PLANCK [31, 32]. Note that the feature has little impact on but it does lead to a distinct bump in .

The smallness of means that the factors of which occur in the scalar source (37) are hugely important. By comparison we can ignore the terms and simply write,


Because we expect to be more than 100 times as strong as .

The approximations (49), (58) and (62) are valid so long as is small. If we additionally ignore nonlinear terms in the equations for and , the correction exponents of expressions (3-4) become,


Recall that , , the Green’s function was defined in (50), and the coefficient functions , and were given in expressions (59-61).

How large and are depends on what the inflationary model predicts for derivatives of . For example, the slow roll approximation of monomial inflation gives,


For these models the various tensor and scalar contributions are small,

Figure 6: These graphs show the scalar power spectrum for the model of Figure 5. The left hand figure compares the exact result (solid blue) with the local slow roll approximation (yellow dashed). The right hand figure compares the exact result (solid blue) with the much better approximation (yellow dashed) obtained from multiplying by , using our analytic approximation (