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
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  and scalar  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  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 purpose of this paper is to rationalize and simplify our formalism for evolving the norms of the mode functions, rather than the mode functions , 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.
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
If necessary, the mode function can be easily recovered ,
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 ,
Here the frequency and the tensor source are,
This permits us to solve (12) perturbatively by expanding in the nonlinear terms,
Converting to the norm-squared gives ,
The scalar mode function mode can be recovered from ,
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,
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,
The analogous result after horizon crossing is,
There is also a jump at horizon crossing so that the complete result is,
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.
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.
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,
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 .
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,