aH_{eq}> k.
\label{413a}
\eeq
In this case we may neglect the effects of an additional axion
production in the matter-dominated era, since
$a^2m^2>\ddot{a}/a$ at $\eta\geq \eta_{eq}$. The axion fluctuations
are amplified by the inflation $\ra$ radiation transition, but are to
be evaluated in
the non-relativistic regime ($\eta>\eta_{eq}$), where the mass
contribution is already important.
For non-relativistic, super-horizon modes, the Fourier components of
the axion field become (see the non-trivial calculation
reported in Appendix~C):
\beq
\sg (\bk, \eta) = {c (\bk)\over a\sqrt {ma}}
\left(k\over k_1\right)^{1/2}\left(H_1\over m\right)^{1/4} \sin
\left(m\over H\right) ,~~~~~~k k$.
This is the reason why we always obtain a white noise spectrum.
On the other hand, the behaviour in $\eta$ depends on which region
of $p$ dominates. Specifically we find:
1) For $3/4 \leq \mu \leq 3/2$ the leading contribution to $I$
comes from $p\sim \eta^{-1}$, and gives the single term appearing
in eq. (\ref{347}).
2) For $\mu < 3/4$ the leading contribution comes either from
$p\sim k_1$ (giving the first term in the square brackets of
(\ref{347})), or (for $\mu$ very close to $3/4$)
from $p\sim \eta^{-1}$ (giving the second term
in the same brackets).
\subsection{Massive Axions}
For massive axions, the energy density spectrum is determined by Eq.
(\ref{418}), with $3/4<\mu\leq 3/2$. This integral is dominated by the
region $p\sim k$, and its rough behaviour can be easily obtained this
way. For a more precise evaluation we proceed as
follows: the angular integration gives
\beq
k^3 |f_\rho|^2 \left(M\over a\right)^4
= {mH_1\over 16\pi^2 z(\mu-1)}\left(k_1\over a\right)^6\left(k\over
k_1\right)^3 \int_0^1 dy~
y^{1-2\mu} \left[(z-y)^{2-2\mu}-(z+y)^{2-2\mu}\right].
\eeq
Defining $t=y/z$ we obtain
\beq
k^3 |f_\rho|^2 \left(M\over a\right)^4
= {mH_1\over 16\pi^2 (\mu-1)}\left(k_1\over a\right)^6\left(k\over
k_1\right)^3 z^{3-4\mu}
\left(A-B\right)
\eeq
where, after some manipulation \cite{GrRy},
\bea
A&=&\int_0^\infty dt\,\, t^{1-2\mu}
\left[(1-t)^{2-2\mu}-(1+t)^{2-2\mu}\right] =\nonumber\\
&=&
{2^{4\mu-4}\over \sqrt \pi}\Ga (2-2\mu)\Ga(2\mu-3/4)
\left[\cos 2\pi (\mu-1) -1\right]
\eea
and
\beq
B=\int_{1/z}^\infty dt\,\, t^{1-2\mu}
\left[(1-t)^{2-2\mu}-(1+t)^{2-2\mu}\right] .
\eeq
By evaluating this second integral in the limit $z \ra 0$, we
obtain
\beq
B\sim z^{4\mu-3} \ll A.
\eeq
so that
\beq
k^3 |f_\rho|^2 \left(M\over a\right)^4
= {mH_1 A\over 16\pi^2 (\mu-1)}\left(k_1\over a\right)^6\left(k\over
k_1\right)^{6-4\mu},
~~~~~~~~~~~~~~~~ 3/4<\mu <3/2 ,
\eeq
as reported in eq. (\ref{419}). Note that there is no singularity for
$\mu=1$, as
\beq
\lim_{\mu \ra 1}{\Ga (2-2\mu)\over (\mu-1)} \left[\cos 2\pi (\mu-1)
-1\right] = {4\pi^2\over (\mu-1)^2}(\mu-1)^2={\rm const}
\eeq
\section{Non-relativistic corrections to the axion spectrum}
\label{C}
For a massive-axion perturbation $\sg$, the string frame action
\beq
S={1\over 2}\int d^4x \sqrt{-g} e^\phi \left[(\pa_\mu \sg)^2
-m^2\sg^2\right],
\eeq
in a conformally flat background, can be written in terms of the
canonical variable
\beq
\psi =z \sg, ~~~~~~~~~~~~~~~~ z= a e^{\phi/2},
\eeq
as
\beq
S={1\over 2}\int d^3x d\eta \left[\dot {\psi}^2 -(\pa_i\psi)^2 +
{\ddot z\over z}\psi^2 -m^2a^2 \psi^2 \right]
\eeq
(the dot denotes differentiation with respect to the conformal time
$\eta$). The Fourier modes $\psi_k$ satisfy the perturbation
equation
\beq
\ddot \psi_k +\left(k^2 -{\ddot z\over z}+m^2a^2\right)\psi_k=0.
\label{c4}
\eeq
We shall consider the background transition at $\eta=\eta_1$ from
an initial pre-big bang phase in which the axion is massless, to a
final radiation-dominated phase in which the dilaton freezes to its
present value, and the axion acquires a small (in string units) mass.
For $\eta>\eta_1$ the solution of Eq. (\ref{c4}) depends on the
kinematics of the pump field $z$ and, after normalization to an
initial vacuum spectrum, it can be written in terms of the
second-kindHankel functions \cite{11} as:
\beq
\psi_k(\eta)=\eta^{1/2}H_\mu^{(2)} (k\eta)
\label{c5} .
\eeq
In the radiation era, $\eta>\eta_1$, the ``effective potential"
${\ddot z/z}$ is vanishing, and the perturbation equation
reduces to
\beq
\ddot \psi_k +\left(k^2 +\a^2\eta^2\right)\psi_k=0,
\label{c6}
\eeq
where we have put
\beq
m^2a^2 =\a^2\eta^2, ~~~~~~~~~~~~~~~~
\a= mH_1a_1^2,
\eeq
using the time behaviour of the scale factor, $a \sim \eta$.
Assuming that the mass term is negligible at the transition scale, $
m \ll k/a$, we can match the solution (\ref{c5}) to the plane-wave
solution
\beq
\psi_k= {1\over \sqrt k}\left[c_+(k) e^{-ik\eta}+
c_-(k) e^{ik\eta}\right] ,
\label{c8}
\eeq
and obtain:
\beq
c_\pm=\pm c(k) e^{\pm ik\eta}, ~~~~~~~~~
|c(k)|\sim (k/k_1)^{-\mu-1/2} .
\label{c9}
\eeq
(We are neglecting, for simplicity, numerical factors of order 1,
which are not very significant in view of the many approximations
performed. Their contribution will be included into an overall
numericalcoefficient in front of the final spectrum.) In the
relativistic regime, the amplified axion perturbation then takes the
form:
\beq
\sg({\bf k}, \eta) = {c({\bf k})\over a \sqrt k}\sin (k\eta),
\label{c10}
\eeq
used in Section \ref{III3} for the massless-axion case.
In the radiation era the proper momentum is red-shifted
with respect to the rest mass, and all axion modes tend to become
non-relativistic. When the mass term is no longer negligible, the
general solution of Eq. (\ref{c6}) can be written in terms of
parabolic cylinder functions \cite{11}. For an approximate estimate
of the axion field in the non-relativistic regime, however, it is
convenient to distinguish two cases, depending on whether a mode
$k$ becomes non-relativistic inside or outside the horizon. Defining
as $k_m$ the limiting comoving frequency of a mode that becomes
non-relativistic ($k_m=ma_m$) at the time it re-enters the horizon
($k_m=H_ma_m$), we find, in the radiation era,
\beq
k_m= k_1 \left(m\over H_1\right)^{1/2}.
\label{c11}
\eeq
We will thus consider the two cases $k \gg k_m$ and $k\ll k_m$.
In the first case, we rewrite the perturbation equation (\ref{c6}) as\beq
{d^2\psi_k\over dx^2}+\left({x^2\over 4} -b\right)\psi_k=0,
~~~~~ x=\eta (2 \a)^{1/2}, ~~~~ -b= k^2/2\a , \label{c12}
\eeq
and we give the general solution in the form
\beq
\psi=A W(b,x)+B W(b,-x)~,\label{c13}
\eeq
where $W(b,x)$ are the Weber parabolic cylinder functions
(see \cite{11}, chap.~19). In order to fix the integration constants
$A$ and $B$ we shall match the solutions (\ref{c13}) and (\ref{c10})
in the relativistic limit
\beq
{k^2 \over m^2 a^2}=
{k^2 \over \a^2 \eta^2}= {-4b\over x^2} \gg1.
\label{c14}
\eeq
In this limit, as we are considering modes that become
non-relativistic when they are already inside the horizon,
\beq
\left(k\over k_m\right)^2 \sim {k^2\over \a} \sim (-b) \gg1,
\label{c15}
\eeq
we can expand the $W$ functions for $b$ large with $x$ moderate
\cite{11}. Matching to the plane-wave solution (\ref{c10}), we obtain
$A=0$, and
\beq
\psi_k \simeq {c({\bf k})\over \a^{1/4}} W(b,-x) .
\label{c16}
\eeq
In the opposite, non-relativistic limit $x^2 \gg |4b|$, the expansion
of the Weber functions gives \cite{11}
\beq
\psi_k \simeq {c({\bf k})\over (\a \eta)^{1/2}} \sin\left(m\over
H\right)
\label{c17}
\eeq
(we have used $x^2/4=ma\eta/2\sim m/H$). The corresponding
axion field is (inside the horizon)
\beq
\sg({\bf k}, \eta) = {c({\bf k})\over a \sqrt{ma}}\sin \left(m\over
H\right), ~~~~~~~~~~~~~k>k_m .
\label{c18}
\eeq
Consider now the case of a mode that becomes non-relativistic when it
is still
outside the horizon, $k\ll k_m$. In this case, we cannot use the
large $|b|$ expansion as $|b| <1$, and it is convenient to express the
general solution of Eq. (\ref{c12}) as
\beq
\psi=A y_1(b,x)+B y_2(b,x)~,\label{c19}
\eeq
where $y_1$ and $y_2$ are the even and odd parts of the parabolic
cylinder functions \cite{11}. Matching to (\ref{c10}), in the
relativistic limit $x \ra 0$, gives $A=0$ and
\beq
\psi_k \simeq c({\bf k})\left(k\over 2\a \right)^{1/2} y_2(b,x) .
\label{c20}
\eeq
In the non-relativistic limit $x^2 \gg |b|$ we use the relation
\cite{11}
\beq
y_2 \sim \left[W(b,x)-W(b,-x)\right] \sim {1\over \sqrt x} \sin
{x^2\over 4} ,
\label{c21}
\eeq
which leads to
\beq
\psi_k \simeq {c({\bf k})\over (\a \eta)^{1/2}} \left(k^2\over \a
\right)^{1/4} \sin\left(m\over H\right) .
\label{c22}
\eeq
Using Eqs.~(\ref{c15}) and (\ref{c11}) for $k^2/\a$, we finally arrive
at the non-relativistic axion field presented in Eq. (\ref{414}):
\beq
\sg({\bf k}, \eta) = {c({\bf k})\over a \sqrt{ma}}
\left(k\over k_1\right)^{1/2}\left(H_1\over m \right)^{1/4}
\sin \left(m\over H\right), ~~~~~~~~~~~~~k