\section{Introduction}\label{intro}

Cosmological perturbations from primordial inflation
\cite{Starobinsky:1979ty,Mukhanov:1981xt,Starobinsky:1982ee} have a
crucial importance for fundamental theory because they are first
quantum gravitational effect ever detected \cite{Mukhanov:2007zz,
Woodard:2009ns}, because they provide strong evidence for primordial
inflation \cite{Mukhanov:1990me,Liddle:1993fq,Lidsey:1995np}, and
because they set the initial conditions for structure formation in
cosmology \cite{Dodelson:2003ft,Mukhanov:2005sc,Weinberg:2008zzc}.
Two sorts of perturbations are produced during single-scalar
inflation: a scalar perturbation characterized by the field
$\zeta(t,\vec{x})$, and a tensor perturbation characterized by the
transverse-traceless field $h_{ij}(t,\vec{x})$. The scalar signal
has been imaged in the anisotropies of the cosmic microwave
background \cite{Komatsu:2010fb,Keisler:2011aw}, and by measuring
the matter power spectrum with large scale structure surveys
\cite{Reid:2009xm}. The tensor signal has not been imaged so far
\cite{Komatsu:2010fb,Keisler:2011aw} but strenuous efforts are
underway to detect it through the polarization of the cosmic
microwave background \cite{PLANCK,EBEX,SPIDER,PIPER}.

The two perturbation fields are defined on the homogeneous,
isotropic and spatially flat geometry characterized by scale factor
$a(t)$, with Hubble parameter $H(t)$ and slow roll parameters
$\epsilon(t)$ and $\eta(t)$,
\begin{equation}
ds^2 = -dt^2 + a^2(t) d\vec{x} \!\cdot\! d\vec{x} \; , \; H(t)
\equiv \frac{\dot{a}}{a} \; , \; \epsilon(t) \equiv
-\frac{\dot{H}}{H^2} \; , \; \eta(t) \equiv \epsilon -
\frac{\dot{\epsilon}}{2 H \epsilon} \; . \label{geometry}
\end{equation}
Only the quadratic parts of their Lagrangians are relevant to
current measurements (see \cite{Kahya:2010xh} for an explanation of
the full formalism),
\begin{eqnarray}
\mathcal{L}^{(2)}_{\zeta} & = & \frac{\epsilon a^3}{8 \pi G}
\Biggl\{ \dot{\zeta}^2 \!-\! \frac{\partial_k \zeta \partial_k
\zeta}{a^2} \Biggr\} \; , \label{Lzeta} \\
\mathcal{L}^{(2)}_{h} & = & \frac{a^3}{64 \pi G} \Biggl\{
\dot{h}_{ij} \dot{h}_{ij} \!-\! \frac{\partial_k h_{ij} \partial_k
h_{ij}}{a^2} \Biggr\} \; . \label{Lh}
\end{eqnarray}
The reported results
\cite{Komatsu:2010fb,Keisler:2011aw,Reid:2009xm} for the scalar and
tensor power spectra are consistent with evaluating the following
2-point correlators long after the first horizon crossing time $t_k$
such that $k = H(t_k) \, a(t_k)$,
\begin{eqnarray}
\Delta^2_{\mathcal{R}}(t,k) & \equiv & \frac{k^3}{2 \pi^2} \int \!\!
d^3x \, e^{-i \vec{k} \cdot \vec{x}} \Bigl\langle \Omega \Bigl\vert
\zeta(t,\vec{x}) \zeta(t,\vec{0}) \Bigr\vert \Omega \Bigr\rangle \;
, \label{DeltaR} \\
\Delta^2_{h}(t,k) & \equiv & \frac{k^3}{2 \pi^2} \int \!\! d^3x \,
e^{-i \vec{k} \cdot \vec{x}} \Bigl\langle \Omega \Bigl\vert
h_{ij}(t,\vec{x}) h_{ij}(t,\vec{0}) \Bigr\vert \Omega \Bigr\rangle
\; . \label{Deltah}
\end{eqnarray}

At tree order the correlators (\ref{DeltaR}-\ref{Deltah}) can be
expressed in terms of the scalar mode function $v(t,k)$ and its
tensor cousin $u(t,k)$,
\begin{eqnarray}
\Delta^2_{\mathcal{R}}(t,k) & = & \frac{k^3}{2 \pi^2} \times 4\pi G
\times \vert v(t,k) \vert^2 + O(G^2) \; , \label{treezeta} \\
\Delta^2_{h}(t,k) & = & \frac{k^3}{2 \pi^2} \times 32 \pi G \times 2
\times \vert u(t,k) \vert^2 + O(G^2) \; . \label{treeh}
\end{eqnarray}
The relevant mode equations and normalization conditions are,
\begin{eqnarray}
\ddot{v} + \Bigl(3 H + \frac{\dot{\epsilon}}{\epsilon}\Bigr) \dot{v}
+ \frac{k^2}{a^2} v = 0 \quad & , & \quad W_v \equiv v \dot{v}^* -
\dot{v} v^* = \frac{i}{\epsilon a^3} \; , \label{veqns} \\
\ddot{u} + 3 H \dot{u} + \frac{k^2}{a^2} u = 0 \quad & , & \quad W_u
\equiv u \dot{u}^* - \dot{u} u^* = \frac{i}{a^3} \; . \label{ueqns}
\end{eqnarray}
The initial conditions for the mode functions (which correspond to
the choice of Bunch-Davies vacuum) derive from the WKB solutions in
the far ultraviolet (that is, for $k \gg H(t) a(t)$),
\begin{eqnarray}
v(t,k) & \longrightarrow & \frac{\exp[-ik \int^t dt'/a(t')]}{\sqrt{2
k a^2(t) \epsilon(t)}} \Biggl\{ 1 + O\Bigl(\frac{H a}{k}\Bigr) \Biggr\}
\; , \label{initialzeta} \\
u(t,k) & \longrightarrow & \frac{\exp[-ik \int^t dt'/a(t')]}{\sqrt{2
k a^2(t)}} \Biggl\{1 + O\Bigl(\frac{Ha}{k}\Bigr) \Biggr\} \; .
\label{initialh}
\end{eqnarray}
As is evident from these asymptotic forms, both mode functions
oscillate and fall off in the far ultraviolet. They become
approximately constant near the time of first horizon crossing
$t_k$. (One can infer the existence of constant solutions quite
generally when the $\frac{k^2}{a^2(t)}$ terms in equations
(\ref{veqns}-\ref{ueqns}) become irrelevant.) It is these constant
amplitudes which determine the crucial theoretical predictions for
the power spectra through relations (\ref{treezeta}-\ref{treeh}).

One of the frustrating things about this formalism is the need to
employ approximation techniques, even for evaluating tree order
formulae such as (\ref{treezeta}-\ref{treeh}), because equations
(\ref{veqns}-\ref{ueqns}) for the mode functions cannot be solved
analytically for general scale factor $a(t)$
\cite{Tsamis:2002qk,Tsamis:2003zs}. Examples of such approximation
techniques include \cite{Stewart:1993bc,Wang:1997cw,Martin:1999wa}:
\begin{itemize}
\item{Assuming the slow roll parameter $\epsilon(t)$ is constant;}
\item{Matching the leading ultraviolet and infrared forms at $t = t_k$; and}
\item{Employing the full WKB solutions.}
\end{itemize}
When greater accuracy is needed, one must resort to numerical
evolution of (\ref{veqns}-\ref{ueqns}) from the known initial
conditions (\ref{initialzeta}-\ref{initialh}) until well past the
time of first horizon crossing \cite{Wang:1997cw}. Excellent
numerical solution techniques exist
\cite{Easther:2010qz,Mortonson:2010er,Easther:2011yq}, but a large
fraction of their power is wasted on reproducing the oscillations of
the mode functions, which contribute nothing to the power spectra.
This can be especially time-consuming for models in which there is
an extended phase of inflation, or when scanning properties of
classes of models.

A more effective technique was developed recently for the tensor
power spectrum \cite{Romania:2011ez} in order to work out the
gravitational wave signal from a novel model of inflation in which
the Hubble parameter oscillates for a brief time
\cite{Tsamis:2009ja,Romania:2010zq}. The key to the new technique is
to convert relations (\ref{ueqns}) into an equation for the
norm-squared of the mode function, $M(t,k) \equiv \vert u(t,k)
\vert^2$. This is the quantity that actually enters the tensor power
spectrum (\ref{treeh}), and it has a much more sedate evolution than
the mode function. Evolving $M(t,k)$ avoids the wasted effort of
numerically simulating the oscillations of $u(t,k)$. Further, an
excellent asymptotic expansion can be derived for $M(t,k)$ which
must be valid until only a few e-foldings before first horizon
crossing, no matter how long inflation persists
\cite{Romania:2011ez}. It is therefore only necessary to numerically
evolve $M(t,k)$ for a few e-foldings, from just before $t_k$ until
it becomes constant.

The purpose of this paper is to extend this more effective technique
to the scalar mode functions using a trick \cite{Tsamis:2003zs} for
converting solutions to $u(t,k)$ into solutions for $v(t,k)$. To
further simplify the formalism, we express our results in terms of
the tree order power spectra (\ref{treezeta}-\ref{treeh}) directly,
without ever mentioning the mode functions. Section \ref{eqn}
derives the key equations for $\Delta^2_{\mathcal{R}}(t,k)$ and
$\Delta^2_{h}(t,k)$, and section \ref{expand} gives their asymptotic
expansions. Our discussion comprises section \ref{discuss}.

\section{Differential Equations for $\Delta^2_{\mathcal{R}}$ and
$\Delta^2_{h}$}\label{eqn}

We begin by reviewing how one derives an equation for $M(t,k) \equiv
\vert u(t,k) \vert^2$. The first step is to write out the first and
second time derivatives,
\begin{eqnarray}
\dot{M}(t,k) & = & \dot{u} u^* + u \dot{u}^* \; , \label{Mdot} \\
\ddot{M}(t,k) & = & \ddot{u} u^* + 2 \dot{u} \dot{u}^* + u
\ddot{u}^* \; . \label{Mddot}
\end{eqnarray}
One next eliminates $\ddot{u}$ and $\ddot{u}^*$ in (\ref{Mddot}) by
using (\ref{ueqns}), and then recognizing factors of $M$ and
$\dot{M}$,
\begin{equation}
\ddot{M}(t,k) = -3 H \dot{M} - \frac{2 k^2}{a^2} M + 2 \dot{u}
\dot{u} \; . \label{Mstep2}
\end{equation}
The final factor involving $\dot{u} \dot{u}^*$ can be expressed in
terms of $M$ by subtracting the square of the Wronskian $W_u$ in
expression (\ref{ueqns}) from the square of $\dot{M}$ in expression
(\ref{Mdot}),
\begin{equation}
\dot{M}^2 - W^2_u = 2 M \times 2 \dot{u} \dot{u}^* = \dot{M}^2 +
\frac1{a^6} \qquad \Longrightarrow \qquad 2 \dot{u} \dot{u}^* =
\frac{\dot{M}^2}{2 M} + \frac1{2 M a^6} \; . \label{dotuu*}
\end{equation}
Substituting (\ref{dotuu*}) in relation (\ref{Mstep2}) gives the
desired equation for $M(t,k)$,
\begin{equation}
\ddot{M} + 3 H \dot{M} + \frac{2 k^2}{a^2} M = \frac{ \dot{M}^2}{2
M} + \frac1{2 M a^6} \; . \label{Meqn}
\end{equation}

Deriving an equation for $N(t,k) \equiv \vert v(t,k)\vert^2$ entails
the same first step,
\begin{eqnarray}
\dot{N}(t,k) & = & \dot{v} v^* + v \dot{v}^* \; , \label{Ndot} \\
\ddot{N}(t,k) & = & \ddot{v} v^* + 2 \dot{v} \dot{v}^* + v
\ddot{v}^* \; . \label{Nddot}
\end{eqnarray}
We next employ expression (\ref{veqns}) to eliminate $\ddot{v}$ and
$\ddot{v}^*$ in favor of $N$ and $\dot{N}$,
\begin{equation}
\ddot{N} = -\Bigl(3 H + \frac{\dot{\epsilon}}{\epsilon}\Bigr)
\dot{N} - \frac{2 k^2}{a^2} N + 2 \dot{v} \dot{v}^* \; .
\label{Nstep2}
\end{equation}
Subtracting the squares of the $\dot{N}$ from (\ref{Ndot}) and the
Wronskian $W_v$ in equation (\ref{veqns}) gives $\dot{v} \dot{v}^*$,
\begin{equation}
\dot{N}^2 - W^2_v = 2 N \times 2 \dot{v} \dot{v}^* = \dot{N}^2 +
\frac1{\epsilon^2 a^6} \qquad \Longrightarrow \qquad 2 \dot{v}
\dot{v}^* = \frac{\dot{N}^2}{2 N} + \frac1{2 N \epsilon^2 a^6} \; .
\label{dotvv*}
\end{equation}
And the final equation for $N(t,k)$ comes from substituting
(\ref{dotvv*}) in (\ref{Nstep2}),
\begin{equation}
\ddot{N} + \Bigl(3 H + \frac{\dot{\epsilon}}{\epsilon}\Bigr) \dot{N}
+ \frac{2 k^2}{a^2} N = \frac{\dot{N}^2}{2 N} + \frac1{2 N
\epsilon^2 a^6} \; . \label{Neqn}
\end{equation}

Because equations (\ref{Meqn}) and (\ref{Neqn}) are nonlinear, they
incorporate the Wronskian normalization as well as the evolution
equations for the mode functions. Converting them to equations for
$\Delta^2_{\mathcal{R}}(t,k)$ and $\Delta^2_{h}(t,k)$ requires only
changing the coefficients of the final terms,
\begin{eqnarray}
\ddot{\Delta}^2_{\mathcal{R}} + \Bigl(3 H +
\frac{\dot{\epsilon}}{\epsilon} \Bigr) \dot{\Delta}^2_{\mathcal{R}}
+ \frac{2 k^2}{a^2} \Delta^2_{\mathcal{R}} =
\frac{(\dot{\Delta}^2_{\mathcal{R}})^2}{\Delta^2_{\mathcal{R}}} +
\frac{2 G^2 k^6}{\pi^2 \epsilon^2 a^6}
\frac1{\Delta^2_{\mathcal{R}}} \; , \label{scalareqn} \\
\ddot{\Delta}^2_{h} + 3 H \dot{\Delta}^2_{h} + \frac{2 k^2}{a^2}
\Delta^2_{h} = \frac{(\dot{\Delta}^2_{h})^2}{\Delta^2_{h}} +
\frac{2^9 G^2 k^6}{\pi^2 a^6} \frac1{\Delta^2_{h}} \; .
\label{tensoreqn}
\end{eqnarray}

\section{Asymptotic Expansions for $\Delta^2_{\mathcal{R}}$ and
$\Delta^2_{h}$} \label{expand}

Equations (\ref{scalareqn})-\ref{tensoreqn}) require initial values
for the power spectra and their first time derivatives. Of course
these derive from the ultraviolet limits
(\ref{initialzeta}-\ref{initialh}), which imply the zeroth order
terms for an expansion in powers of $(H a/k)^2$,
\begin{eqnarray}
\Delta^2_{\mathcal{R}}(t,k) & \longrightarrow & \frac{G k^2}{\pi
\epsilon a^2} \Biggl\{ 1 + O\Bigl( \frac{H^2 a^2}{k^2}\Bigr)
\Biggr\} \; , \label{zeta0term} \\
\Delta^2_{h}(t,k) & \longrightarrow & \frac{16 G k^2}{\pi a^2}
\Biggl\{ 1 + O\Bigl( \frac{H^2 a^2}{k^2} \Bigr) \Biggr\} \; .
\label{h0term}
\end{eqnarray}
We shall develop the expansion for $\Delta^2_{h}(t,k)$ to second
order, and then employ a trick for converting tensor mode functions
to scalar ones \cite{Tsamis:2003zs} to obtain the corresponding
expansion of $\Delta^2_{\mathcal{R}}(t,k)$. These expansions are so
accurate that there seems little point to numerically evolving for
anything except the last few e-foldings before first horizon
crossing.

From the asymptotic limit (\ref{h0term}) one can see that the
``big'' terms in the evolution equation (\ref{tensoreqn}) are those
with no derivatives. It is best to segregate these terms,
\begin{equation}
\Delta^2_{h} - \Bigl( \frac{16 \pi G k^2}{\pi a^2} \Bigr)^2
\frac1{\Delta^2_{h}} = \frac{a^2}{2 k^2} \Biggl[-\ddot{\Delta}^2_{h}
- 3 H \dot{\Delta}^2_{h} + \frac{ (\dot{\Delta}^2_{h})^2}{2
\Delta^2_{h}} \Biggr] \; . \label{perteqn}
\end{equation}
The desired expansion is now easy to read off \cite{Romania:2011ez},
\begin{eqnarray}
\lefteqn{ \Delta^2_{h}(t,k) = \frac{16 G k^2}{\pi a^2} \Biggl\{ 1 +
\Bigl[1 - \frac12 \epsilon\Bigr] \Bigl( \frac{H a}{k}\Bigr)^2 }
\nonumber \\
& & \hspace{.5cm} + \Biggl[ \frac94 \epsilon \!-\! \frac{21}8
\epsilon^2 \!+\! \frac34 \epsilon^3 \!+\! \Bigl( \frac74 \!-\!
\frac34 \epsilon\Bigr) \frac{\dot{\epsilon}}{H} \!+\! \frac18
\frac{\ddot{\epsilon}}{H^2} \Biggr] \Bigl( \frac{H a}{k}\Bigr)^4 +
O\Bigl( \frac{H^6 a^6}{k^6} \Bigr) \Biggr\} \; . \qquad
\label{tensorexp}
\end{eqnarray}
Note that this expansion is in powers of $(\frac{H a}{k})^2$, rather
than the expansion in powers of $\frac{H a}{k}$ one gets for the
mode function. The better convergence for small $\frac{H a}{k}$ is
one more indication of the superiority of our method.

The trick for converting $\Delta^2_{h}(t,k)$ into
$\Delta^2_{\mathcal{R}}(t,k)$ involves simultaneously changing the
scale factor and the meaning of time \cite{Tsamis:2003zs},
\begin{eqnarray}
a(t) & \longrightarrow & \sqrt{\epsilon(t)} \times a(t) \; ,
\label{newa} \\
\frac{\partial}{\partial t} & \longrightarrow &
\frac1{\sqrt{\epsilon(t)}} \times \frac{\partial}{\partial t} \; .
\label{newdt}
\end{eqnarray}
One can see that these two replacements convert the $u(t,k)$ mode
relations (\ref{ueqns}) into the relations (\ref{veqns}) for
$v(t,k)$. Replacements (\ref{newa}-\ref{newdt}) also convert the
$\Delta^2_{h}(t,k)$ equation (\ref{tensoreqn}) into expression
(\ref{scalareqn}) for $\Delta^2_{\mathcal{R}}(t,k)$. We could have
used them to derive that equation had the straightforward derivation
not been so trivial.

To convert the asymptotic expansion (\ref{tensorexp}) into an
expansion for $\Delta^2_{\mathcal{R}}(t,k)$ we first note that the
replacements (\ref{newa}-\ref{newdt}) imply the following
geometrical replacements,
\begin{eqnarray}
H & \longrightarrow & \frac{H}{\sqrt{\epsilon}} \times \Bigl[ 1
\!+\! \epsilon \!-\! \eta \Bigr] \equiv \frac{H}{\sqrt{\epsilon}}
\times D \; , \label{newH} \\
\epsilon & \longrightarrow & \frac{ (1 \!+\! \epsilon \!-\! \eta) (2
\epsilon \!-\! \eta) \!-\! (\frac{\dot{\epsilon} - \dot{\eta}}{H})}{
(1 \!+\! \epsilon \!-\! \eta)^2 } \equiv \frac{N}{D^2} \; , \label{neweps} \\
\frac{\dot{\epsilon}}{H} & \longrightarrow & \frac{D \dot{N} \!-\! 2
N \dot{D}}{H D^4} \; , \label{newdoteps} \\
\frac{\ddot{\epsilon}}{H^2} & \longrightarrow & \frac{D^2 \ddot{N}
\!-\! 4 D \dot{N} \dot{D} \!-\! 2 D N \ddot{D} \!+\! 6 N
\dot{D}^2}{H^2 D^6} - \frac{ (\epsilon \!-\! \eta) (D \dot{N} \!-\!
2 N \dot{D})}{H D^5} \; . \label{newddoteps}
\end{eqnarray}
After many tedious manipulations the result is,
\begin{eqnarray}
\lefteqn{ \Delta^2_{\mathcal{R}}(t,k) = \frac{G k^2}{\pi \epsilon
a^2} \Biggl\{ 1 + \Biggl[ 1 \!-\! \frac12 (1 \!+\! \epsilon \!-\!
\eta) (2 \epsilon \!-\! \eta) - \Bigl( \frac{\dot{\epsilon} \!-\!
\dot{\eta}}{2 H} \Bigr) \Biggr] \Bigl( \frac{H a}{k} \Bigr)^2 }
\nonumber \\
& & \hspace{-.7cm} + \Biggl[ \frac38 (1 \!+\! \epsilon \!-\! \eta)
(3 \!-\! \epsilon \!-\! \eta) (2 \!-\! \eta) (2 \epsilon \!-\! \eta)
\!+\! \Bigl[ \frac54 \!+\! \frac92 \epsilon \!-\! \frac{17}4 \eta
\!-\! \frac78 \epsilon^2 \!-\! \frac32 \epsilon \eta \!+\!
\frac{13}8 \eta^2 \Bigr] \frac{\dot{\epsilon}}{H} \nonumber \\
& & \hspace{-.7cm} + \Bigl[ \frac12 \!-\! \frac{21}8 \epsilon \!+\!
\frac{13}8 \eta \!+\! \epsilon^2 \!+\! \frac12 \epsilon \eta \!-\!
\frac34 \eta^2 \Bigr] \frac{\dot{\eta}}{H} \!+\! \frac74 \Bigl(
\frac{\dot{\epsilon}}{H}\Bigr)^2 \!-\! \frac{25}8
\Bigl(\frac{\dot{\epsilon}}{H} \Bigr) \Bigl(
\frac{\dot{\eta}}{H}\Bigr) \!+\! \frac{11}8 \Bigl(
\frac{\dot{\eta}}{H} \Bigr)^2 \nonumber \\
& & \hspace{-.7cm} - \Bigl[ \frac32 \!+\! \frac38 \epsilon \!-\!
\frac34 \eta \Bigr] \frac{\ddot{\epsilon}}{H^2} \!+\! \Bigl[
\frac{13}8 \!+\! \frac{\epsilon}2 \!-\! \frac78 \eta\Bigr]
\frac{\ddot{\eta}}{H^2} \!-\! \Bigl( \frac{\dddot{\epsilon} \!-\!
\dddot{\eta}}{8 H^3} \Bigr) \! \Biggr] \! \Bigl( \frac{Ha}{k}
\Bigr)^4 \!\!\!+\! O\Bigl( \frac{H^6 a^6}{k^6} \Bigr) \! \Biggr\} .
\label{scalarexp} \quad
\end{eqnarray}
Unless $\epsilon(t)$ or $\eta(t)$ is changing rapidly on the Hubble
scale, the fractional error in (\ref{scalarexp}) should be of order
$(\frac{Ha}{k})^6$. Had we used this formula to estimate
$\Delta^2_{\mathcal{R}}(t,k)$ for $t$ as close as one e-folding
before first horizon crossing, the fractional error would only be
about $2.5 \times 10^{-3}$; for two e-foldings that would fall to
about $6.1 \times 10^{-6}$; and the fractional error would only be
about $1.5 \times 10^{-8}$ at three e-foldings before horizon
crossing. This is comparable to what loop corrections might give
\cite{Miao:2012xc}.

\section{Discussion}\label{discuss}

We have derived nonlinear, second order differential equations
(\ref{scalareqn}) and (\ref{tensoreqn}) for the correlators
$\Delta^2_{\mathcal{R}}(t,k)$ and $\Delta^2_{h}(t,k)$ whose late
time limits give the scalar and tensor power spectra at tree order.
Numerically evolving these correlators is more economical than
evolving the mode functions (\ref{veqns}-\ref{ueqns}) because one no
longer has to keep track of the phases which drop out of the power
spectra. This economy shows up in the asymptotic expansions
(\ref{scalarexp}) and (\ref{tensorexp}) pertinent to the ultraviolet
regime of $k \gg H(t) \, a(t)$. Our expansions for the correlators
are in powers of the small parameter $(\frac{Ha}{k})^2$, whereas the
analogous expansions for the mode functions are only in powers of
$\frac{Ha}{k}$. In fact our asymptotic expansions
(\ref{scalarexp}-\ref{tensorexp}) should be so accurate that they
could be used to provide the initial value data for
$\Delta^2_{\mathcal{R}}(t,k)$ and $\Delta^2_{h}(t,k)$ only a few
e-foldings before the time $t_k$ of first horizon crossing.

We have employed ``Hubble parametrization'', in which the scale
factor $a(t)$ is assumed known and results are expressed in terms of
background geometrical quantities $H(t)$, $\epsilon(t)$ and
$\eta(t)$ defined in expression (\ref{geometry}). This is especially
useful for the tensor power spectrum because gravitational waves
depend only on the background geometry, no matter what caused it.
Many people prefer ``potential parametrization'', in which the
scalar potential $V(\phi)$ is assumed known (with the field starting
at rest from some fiducial value) and results are expressed as
derivatives of the potential. Approximate conversion formulae are,
\begin{equation}
H^2 \simeq \frac83 \pi G V \quad , \quad \epsilon \simeq \frac1{16
\pi G} \Bigl( \frac{V'}{V} \Bigr)^2 \quad , \quad \eta \simeq
\frac1{16 \pi G} \Biggl[ 2\frac{V''}{V} \!-\! \Bigl( \frac{V'}{V}
\Bigr)^2 \Biggr] \; .
\end{equation}

Finally, we note that the same technology can be applied to derive
an equation for the power spectrum of any field whose plane wave
mode functions obey the Mukhanov equation \cite{Tsamis:2003zs}. Of
course the associated asymptotic expansion is also easy to develop.
Both the equation and its solution in the regime of $ k \gg H(t) \,
a(t)$ can be obtained from the analogous tensor relations,
(\ref{tensoreqn}) and (\ref{tensorexp}), by changing the scale
factor and the time as we did in equations (\ref{newa}-\ref{newdt}).

\vskip .5cm

\centerline{\bf Acknowledgements}

We are grateful to W. Kinney and M. Sasaki for conversations on this
subject. This work was partially supported by European Union program
Thalis ESF/NSRF 2007-2013, by European Union Grant
FP-7-REGPOT-2008-1-CreteHEPCosmo-228644, by NSF grant PHY-1205591,
and by the Institute for Fundamental Theory at the University of
Florida.

