%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % LATEX FILE OF THE LETTER: % Massless (pseudo-)scalar seeds of CMB anisotropy % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt,titlepage]{article} %\input epsf \def\baselinestretch{1.1} \setlength{\oddsidemargin}{0.0cm} \setlength{\textwidth}{16.5cm} \setlength{\topmargin}{-.9cm} \setlength{\textheight}{22.5cm}% %\renewcommand{\thesection}{\arabic{section}} %\renewcommand{\theequation}{\thesection.\arabic{equation}} \font\small=cmr8 scaled \magstep0 \font\grande=cmr10 scaled \magstep4 \font\medio=cmr10 scaled \magstep2 \outer\def\beginsection#1\par{\medbreak\bigskip \message{#1}\leftline{\bf#1}\nobreak\medskip \vskip-\parskip \noindent} \def\obdot{\hskip-8pt \vbox to 11pt{\hbox{..}\vfill}} \def\obbdot{\hskip-8pt \vbox to 14pt{\hbox{..}\vfill}} \def\odot{\hskip-6pt \vbox to 6pt{\hbox{..}\vfill}} %minore o circa uguale \def\laq{\raise 0.4ex\hbox{$<$}\kern -0.8em\lower 0.62 ex\hbox{$\sim$}} %maggiore o circa uguale \def\gaq{\raise 0.4ex\hbox{$>$}\kern -0.7em\lower 0.62 ex\hbox{$\sim$}} % ungefahr gleich %\def\simeq{\raise 0.4ex\hbox{$\sim$}\kern -0.7em\lower 0.62 %ex\hbox{$=$}} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\bean{\begin{eqnarray*}} \def\eean{\end{eqnarray*}} \def \bk {{\bf k}} \def \pa {\partial} \def \ra {\rightarrow} \def \fb {\overline \phi} \def \fbp {\dot{\fb}} \def \bp {\dot{\beta}} \def \rb {\overline \rho} \def \pb {\overline p} \def \pr {\prime} \def \se {\prime \prime} \def \H {{a^\prime \over a}} \def \fp {{\phi^\prime}} \def \ti {\tilde} \def \la {\lambda} \def \ls {\lambda_s} \def \La {\Lambda} \def \Da {\Delta} \def \b {\beta} \def \a {\alpha} \def \ap {\alpha^{\prime}} \def \ka {\kappa} \def \Ga {\Gamma} \def \ga {\gamma} \def \sg {\sigma} \def \da {\delta} \def \ep {\epsilon} \def \r {\rho} \def \om {\omega} \def \Om {\Omega} \def \noi {\noindent} \def \pfb {\Pi_{\fb}} \def \pM {\Pi_{M}} \def \pbe {\Pi_{\b}} \def \pa {\partial} \def \dd {\partial} \def \ra {\rightarrow} \def \fb {\overline \phi} \def \fbp {\dot{\fb}} \def \bp {\dot{\beta}} \def \rb {\overline \rho} \def \pb {\overline p} \def \pr {\prime} \def \se {\prime \prime} \def \H {{a^\prime \over a}} \def \fp {{\phi^\prime}} \def \ti {\tilde} \def \al {\alpha} \def \la {\lambda} \def \ls {\lambda_s} \def \La {\Lambda} \def \Da {\Delta} \def \De {\Delta} \def \de {\delta} \def \b {\beta} \def \a {\alpha} \def \ap {\alpha^{\prime}} \def \ka {\kappa} \def \Ga {\Gamma} \def \ga {\gamma} \def \sg {\sigma} \def \Sg {\Sigma} \def \si {\sigma} \def \da {\delta} \def \ep {\epsilon} \def \r {\rho} \def \om {\omega} \def \Om {\Omega} \def \noi {\noindent} \def \pfb {\Pi_{\fb}} \def \pM {\Pi_{M}} \def \pbe {\Pi_{\b}} \def \lap {\triangle} \begin{document} \bibliographystyle {unsrt} \titlepage \begin{flushright} DFTT-27/98 \\ CERN-TH/98-169 \\ astro-ph/9806015 \\ \end{flushright} \vspace{15mm} \begin{center} {\grande Massless (pseudo-)scalar seeds of CMB anisotropy}\\ \vspace{15mm} R. Durrer${}^{(1)}$, M. Gasperini${}^{(2)}$, M. Sakellariadou${}^{(1),(3)},$ G. Veneziano${}^{(4)}$ \\ \vspace{6mm} ${}^{(1)}$ {\sl D\'epartement de Physique Th\'eorique, Universit\'e de Gen\eve, \\ 24 quai E. Ansermet, CH-1211 Geneva, Switzerland }\\ % ${}^{(2)}$ {\sl Dipartimento di Fisica Teorica, Universit\a di Torino, \\ Via P. Giuria 1, 10125 Turin, Italy }\\ % ${}^{(3)}$ {\sl D\'epartement d'Astrophysique Relativiste et de Cosmologie, \\ UPR 176 du Centre National de la Recherche Scientifique, \\ Observatoire de Paris, 92195 Meudon, France }\\ % ${}^{(4)}$ {\sl Theory Division, CERN, CH-1211 Geneva 23, Switzerland} \\ % \end{center} \vskip 2cm \centerline{\medio Abstract} \noindent A primordial stochastic background of very weakly coupled massless (pseudo-)scalars can seed CMB anisotropy, when large-scale fluctuations of their stress-tensor re-enter the horizon during the matter-dominated era. A general relation between multipole coefficients of the CMB anisotropy and the seed's energy spectrum is derived. Magnitude and tilt of the observed anisotropies can be reproduced for the nearly scale-invariant axion spectra that are predicted in a particularly symmetric class of string cosmology backgrounds. \vspace{5mm} \vfill \begin{flushleft} CERN-TH/98-169\\ May 1998\\ \end{flushleft} \newpage In this letter we point out a possible new mechanism for generating large-scale CMB anisotropies. We will show that a cosmologically amplified stochastic background of massless (pseudo-)scalar perturbations, if primordially produced with a properly normalized nearly scale-invariant spectrum, can seed CMB temperature anisotropies in a way consistent with observations \cite{COBE}. Axionic perturbations with the needed characteristics can be produced \cite {Copeland}, \cite{Buon}, \cite{Hadad}, for instance, in the so-called pre-big bang scenario \cite{PBB} of string cosmology. For the sake of generality we define a massless (pseudo-)scalar seed'' field $\si$ through the way it enters the low-energy effective action: \beq S_{eff} = - {1\over 2} \int d^4 x ~ \sqrt{-g}~ A~ (\partial_\mu \si)^2 , \label{action} \eeq and by the two additional conditions: \beq \langle\si\rangle = 0~~~~~ , ~~~~~~~~ ~~~~~~\Omega_{\si} \ll 1. \label{def} \eeq In Eq.~(\ref{action}), $A$ is, in general, a $\si$-independent scalar combination of fields providing, together with the metric, the cosmological background. In Eq.~(\ref{def}), the brackets denote spatial average (or expectation value if perturbations are quantized), and $\Omega_{\si}$ is the fraction of critical energy density carried by the seed field. Such a fraction being small, seeds do not influence the background itself. The above conditions, together with the restriction to massless seeds, make it essentially mandatory for such seeds, if they exist, to consist of very weakly coupled pseudo-scalar (rather than scalar) particles \cite{arion}. An example would be the gravitationally coupled universal axion'' of string theory, on which we shall come back at the end of this letter. We have in mind, typically, a situation in which, in the absence of $\si$, large-scale CMB anisotropies directly induced by quantum fluctuations of the metric are too small; we want to investigate under which conditions seeds may provide the dominant source for them. Seed vacuum fluctuations, after being amplified outside the horizon during inflation, re-enter during the standard Friedman-Robertson-Walker (FRW) era as stochastic Gaussian fields, and give rise to non-trivial -- and not necessarily Gaussian -- fluctuations of the energy-momentum tensor. Within this general context, we derive a simple relation between the usual coefficients $C_l$ of the multipole expansion of the CMB temperature fluctuations, \beq \left\langle{\delta T\over T}({\bf n}){\delta T\over T}({\bf n}') \right\rangle_{{~}_{\!\!({\bf n\cdot n}'=\cos\vartheta)}} ~~=~~~~ {1\over 4\pi}\sum_\ell(2\ell+1)C_\ell P_\ell(\cos\vartheta)~, \label{cor} \eeq and the fraction of critical density in the seeds. The relation reads: \beq C_\ell = K ~~ \int_0^{\infty} d \log k ~~ |j_\ell (k \eta_0)|^2 ~~ \Omega_{\sg}^2 (k, \eta_{re})~, \label{final} \eeq where $K$ is a numerical fudge factor $O(1)$, $\Omega_{\sg}(k,\eta) \equiv \rho_{cr}^{-1} d \rho_{\sg} / d \log k$ is the seed fraction of critical energy density per logarithmic interval of frequency, evaluated at the conformal time $\eta$, $\eta_0$ is the present (conformal) time and $\eta_{re}(k)$ indicates, for each comoving mode $k$, its time of re-entry, $\eta_{re} \sim k^{-1}$. For the relevant (large) scales, re-entry occurs during the matter-dominated era. A crucial aspect of (\ref{final}) is the appearance of $\Omega_\sg$ at a $k$-dependent time (i.e. at re-entry), rather than at a common (e.g. at recombination) time. This is because, in the interesting cases, the so-called integrated" Sachs-Wolfe (SW) contribution \cite{SW} turns out to dominate over the ordinary" SW term. In order to prove Eq.~(\ref{final}) we will proceed as follows. We start from a general formula expressing the spectrum of primordial seed fluctuations in terms of the early, inflationary evolution of the background. We then compute the inhomogeneities induced by this stochastic field in the energy-momentum tensor as well as the Bardeen potentials. Finally, we estimate the large-scale temperature anisotropy using the (total) SW effect. The result (\ref{final}) will thus implicitly connect the observed CMB anisotropy to the very early (possibly to the pre-big bang) history of the universe. This note is intended to give the essential points in the argument and their main consequences. For more details on the calculation in a specific case we refer the reader to our longer recent paper \cite{1}, where other kinds of seeds, as well as the case of massive seeds, are also discussed. Further generalizations of the massive case will be discussed in \cite{massive}. Our computation of the fluctuations of $\si$ follows closely the general approach of \cite{BGV}. In a conformally flat metric the effective action (\ref{action}) becomes: \beq S_{eff} = {1 \over 2} \int d^3 x d \eta ~ S~\left[( \si')^2 - (\nabla \si)^2\right]~, \label{action1} \eeq where a prime stands for derivative with respect to conformal time $\eta$, and the so-called pump field $S$ is simply $S\equiv a^2 A$, where $a$ is the scale factor of the homogeneous, isotropic, spatially-flat metric resulting from a long inflationary phase\footnote{It is important to note that the results that we will obtain here are not valid for conformally coupled fields, e.g for the electromagnetic field, since these do not couple to the scale factor.}. The corresponding effective Hamiltonian reads \beq H_{eff} = {1 \over 2} \int d^3 x ~\left[ S^{-1} \pi^2 + S (\nabla \si)^2\right] ~~~,\label{ham} \eeq where $\pi = S \si'$ is the canonical variable conjugate to $\si$. The Fourier modes of $\sigma$, when correctly normalized to the vacuum before they exit" the horizon at the time $|k\eta_{ex}| \sim 1$, are given by \beq \si (k, \eta) = \frac{1}{\sqrt{k S}} ~~ e^{-i k\eta + i\varphi_{{k}} }~~,~~~~~~ \pi (k, \eta) = {\sqrt{kS}} e^{-i k\eta +i\varphi'_{{k}} } ~~, ~~~~\eta < \eta_{ex} \sim - k^{-1} , \label{42} \eeq ($\varphi_{{k}}, \varphi'_{{k}}$ are random phases, originating from the random initial conditions). Furthermore, as far as the computation of energy spectra goes, fluctuations on superhorizon scales can be consistently truncated to their frozen modes \cite{BGV} through \beq \si (k, \eta) = \frac{1}{\sqrt{k S_{ex}}} ~~ e^{i\varphi_{{k}} } ~~,~~~~~ \pi (k, \eta) = {\sqrt{kS_{ex}}} e^{i\varphi'_{{k}} } ~~, ~~~~~\eta_{ex} < \eta <\eta_{re} \sim k^{-1} . \label{43} \eeq The matching at re-entry finally gives, for $k \eta>1$, \begin{eqnarray} \si (k, \eta) &=& \frac{1}{\sqrt{k S}} \left[\left(S_{re}\over S_{ex}\right)^{1/2} \cos (k\eta)~ e^{i\varphi_{\vec{k}} } + \left(S_{ex}\over S_{re}\right)^{1/2} \sin(k\eta)~ e^{i\varphi'_{\vec{k}} } \right]\, , \nonumber\\ \pi (k, \eta) &=& {\sqrt{kS}} \left[ \left(S_{ex}\over S_{re}\right)^{1/2} \cos (k\eta)~ e^{i\varphi'_{\vec{k}} } - \left(S_{re}\over S_{ex}\right)^{1/2} \sin(k\eta)~ e^{i\varphi_{\vec{k}} }\right]~~. \label{44} \end{eqnarray} A nice feature of these results is their generality. They hold for any kind of background and irrespectively of whether a perturbation re-enters during the matter- or the radiation-dominated epoch. Furthermore, these equations respect an invariance \cite {BGV} of cosmological perturbations under the duality transformation $S \rightarrow S^{-1}, \nabla \si \leftrightarrow \pi$. For the sake of simplicity, we shall consider here the case of a growing pump field, keeping only the leading terms (those proportional to $S_{re}/ S_{ex} \gg1$) in the fluctuations. This will result in simpler formulae at the price of losing manifest duality. The basic information to be extracted from the preceding formulae is the stochastic (spatial) average of $\si$: \beq \langle \si(\bk) \si^\ast(\bk')\rangle= {(2\pi)^3} \da^3(k-k')\Sg(\bk, \eta) \label{sstoch} \eeq where, according to Eqs. (\ref{43}) and (\ref{44}), \bea \Sg(\bk, \eta) = (k S_{ex})^{-1} ~~~~,~~~~ k \eta <1 ~~, \nonumber \\ \Sg(\bk, \eta) = (k S_{ex})^{-1} {S_{re}\over S(\eta)} ~~~~,~~~~ k \eta >1 ~~. \label{stoch} \eea Equations (\ref{sstoch}) and (\ref{stoch}) allow us to compute the correlation functions of the seed energy-momentum tensor: \beq T_{\mu\nu}^{(\sigma)} \equiv T_{\mu\nu} = {S\over a^2}\left [\partial_{\mu} \si \partial_{\nu} \si - {1 \over 2} ~~g_{\mu\nu} (\partial_a \si)^2 \right]. \label{tmunu} \eeq Let us start with the average energy distribution $d \rho_{\sg}(k) / d \log k= (k^3/a^4)\langle H \rangle$ which, after re-entry, can be computed from the Hamiltonian (\ref{ham}) as \cite{BGV}: \beq {d \rho_{\si}(k) \over d \log k} \simeq \left(k\over a \right)^4 {S_{re} \over S_{ex}}(k) \theta(k_1 -k). \eeq The end-point of the spectrum $k_1$ is the maximal amplified frequency (the frequency that re-entered just after exiting), for which just one quantum is produced per unit phase space. Above $k_1$ the spectrum is exponentially depressed and we thus neglect it. Below $k_1$, we can express $\rho_{\si}$ in units of critical energy density, $\r_c=3H^2/(8\pi G)$ as \beq \Omega_{\sg}(k,\eta) \simeq G\left({k^4\eta^2\over a^2}\right) \left({a_{re}\over a_{rad}}\right)^2\left( S_{rad} \over S_{ex} \right)~ \simeq G \left(k\over a_{re}\right)^2 \left( a_{re} \over a_{rad} \right)^2\left( S_{rad} \over S_{ex} \right)~ \left( a_{re} \over a \right)~. \label{omega} \eeq We have denoted by $rad$ the beginning of the radiation era, and we have assumed the background field $A$ to be constant for $\eta > \eta_{rad}$. Also, we have limited our attention to scales relevant to the COBE DMR data, which re-enter during the matter-dominated era. The suppression factor $(a_{eq} / a)$, naively expected for massless particles, is actually replaced by $(a_{re} / a )$. This is due to the additional amplification of modes which are still outside the horizon during (part of) the matter-dominated phase. In particular, just at re-entry, we find: \beq \Omega_{\sg}(k,\eta_{re}) \simeq G \omega^2 \left( a_{re} \over a_{rad} \right)^2 \left( S_{rad} \over S_{ex} \right)~. \label{omegare} \eeq Clearly, some condition has to be imposed on the behaviour of $S$ during inflation to ensure that $\Omega_\sg \ll 1$ at all times. Let us consider next the fluctuations of the various components of the energy-momentum tensor and, in particular, their power spectra $P_{\mu}^{\nu}$ defined by (no sum over $\mu, \nu$ being implied): \beq \langle T_\mu^\nu (x) T_\mu^\nu (x') \rangle- \langle T_\mu^\nu (x)\rangle \langle T_\mu^\nu (x') \rangle = \int {d^3 k \over (2\pi k)^3}e^{i{\bf k}\cdot ({\bf x-x'})} P_{\mu}^{\nu}(k)~. \eeq One easily finds that all the relevant components of $P_{\mu}^{\nu}$ behave similarly, and are controlled by a convolution of the form: \beq P_{\mu}^{\nu}(k,\eta) \sim \left( S \over a^2 \right)^2~ \left(k^3\over a^4 \right) \int d^3 p~ p^2 |k-p|^2 \Sigma(p) \Sigma(k-p) . \eeq Using Eqs. (\ref{stoch}), it is not hard to analyse the various integration regions in $p$ in the above integral while always keeping $k \eta \le 1$. In the region $0< p < \eta^{-1}$ the integrand is proportional to $dp~ p^4~ S_{ex}^{-2}(p)$. Imposing that seeds never be dominant makes this integrand peaked at its {\em upper} end. On the other hand, in the region $\eta^{-1} \eta_{rad}$, we obtain \bea k^{3/2} |\Psi - \Phi|(k, \eta) &\simeq & G (k \eta)^{-5/2}\left(k\over a\right)^2{S(\eta) \over S_{rad}}~ {S_{rad} \over S(-\eta)}~ \simeq ~ G (k \eta)^{-5/2}\left(k\over a_{rad}\right)^2 {S_{rad} \over S(-\eta)} \nonumber \\ &\simeq & (H_1/M_P)^2~(k \eta)^{-5/2} (k/k_1)^2 \left[S_{rad}/S(-\eta)\right]~, \label{Bardeen} \eea where $H_1 \equiv k_1/a_{rad}$ is the Hubble parameter at the beginning of the radiation era. Equations (\ref{Bardeen}) and (\ref{omega}) together provide the interesting relation: \beq k^{3/2} |\Psi - \Phi|(k, \eta) \simeq (k \eta)^{-5/2} \left[S_{ex}(k)/S(-\eta)\right] \Omega_\sigma(k, \eta_{re}) \label{relation} \eeq and, in particular: \beq k^{3/2} |\Psi - \Phi|(k, \eta_{re}) \simeq \Omega_\sigma(k, \eta_{re}) \simeq (H_1/M_P)^2~(k/k_1)^2~\left[S_{rad}/S_{ex}(k)\right]. \label{relpart} \eeq At this point we insert the above result in the formula of the SW effect, which is known to dominate the temperature anisotropies at large angular scales, $\ell \ll 100$. Combining the so-called ordinary'' and integrated'' SW contributions, a standard analysis \cite{1,Durrer} yields: \beq C_\ell^{SW} = {2\over\pi}\int {dk\over k} \left\langle\left[\int_{k\eta_{dec}}^{k\eta_0} k^{3/2} (\Psi -\Phi)({\bf k}, \eta)j_{\ell}'\left(k\eta_0-k\eta\right) d(k \eta)\right]^2\right\rangle ~, \label{Cell} \eeq where $j_l$ are the usual spherical Bessel functions and $\eta_0, \eta_{dec}$ are, respectively, the present time and the time of decoupling between matter and radiation (a prime stands here for the derivative of the Bessel function with respect to its argument). We exploit the previously determined $\eta$-dependence of the Bardeen potentials, assuming that, after re-entry, these potentials are dominated by a cold dark matter (CDM) component and are therefore constant. We find \cite{1} that the $\eta$ integral in Eq.~(\ref{Cell}) is dominated by the region $k \eta \sim 1$, leading to: \beq C_\ell^{SW} \sim \int d~(\log k) \left \langle \left[k^{3/2} (\Psi -\Phi)({\bf k}, \eta_{re})j_{\ell}\left(k\eta_0\right)\right]^2 \right\rangle . \label{Cell2} \eeq Inserting (\ref{relpart}) we immediately recover the desired result (\ref{final}). Note that temperature fluctuations are controlled, for each scale $k$, by the value of the Bardeen potentials at the time it re-enters the horizon. Roughly: \beq (\Delta T/ T) (k) \sim (\Phi - \Psi) (\eta)|_{\eta \sim k^{-1}} \sim \Omega_\sg (k, \eta_{re})~. \eeq In this way, the $\eta$ dependence of $(\Phi - \Psi)$ gets translated into a $k$ (or $l$) dependence of the temperature fluctuation spectrum. Thus a scale-invariant $\Omega_\sg$ leads to scale-invariant Harrison-Zeldovich \cite{7a} spectrum of CMB fluctuations. For a simple power-law behaviour of the pump field, $S_{rad}/S_{ex}(k) = (k/k_1)^{\alpha-2}$, Eq.~(\ref{Cell2}) can be integrated analytically with the result: \beq C_{\ell}^{SW} \approx K (k_1\eta_0)^{-2\al}\left(H_1\over M_p\right)^4{\Ga(2-2\al)\over 4^{(1-\al)}\Ga(3/2-\al)} {\Ga(\ell+\al)\over\Ga(\ell + 2-\al)}. ~~~~~~~ \label{simple} \eeq Comparing (\ref{simple}) with the standard inflationary result for CDM \cite{JamesBond}, where the spectral index $n$ is defined by \cite{JamesBond} \beq C_\ell^{SW} \propto {\Ga(\ell -1/2+ n/2)\over\Ga(\ell+5/2-n/2)} ~, \label{inflat} \eeq leads to the identification $(n-1)= 2\al$. More generally, we can relate an effective (i.e. $k$-dependent) spectral index $n_{eff}$ to the behaviour of the pump field during inflation via the relation: \beq (n_{eff}-1)/2 = \al_{eff} \equiv 2 - \left[d \log S_{ex}(k)/d \log k\right]. \label{indexrel} \eeq The nearly scale-invariant spectrum, measured by the DMR experiment aboard the COBE satellite \cite{smootscott}, requires \beq 0.8\le n_{eff} \le 1.4 \label{259a} \eeq and thus, allowing for generous error bars, COBE's observations imply \beq -0.1\le \al_{eff} \le 0.2 \label{alpourga0} \eeq in the very small $k$ region. Thus, through the definition of $\al_{eff}$ in Eq. (\ref{indexrel}), one is able to relate COBE's data to the early-time evolution of the pump field. In this paper we have concentrated our attention on scalar perturbations. However, since the seeds are of second order in the scalar field, we also expect the presence of vector and tensor perturbations with roughly similar amplitudes. Turning to the absolute normalization, we see from Eq.~(\ref{simple}) that it is controlled to a large extent by the crucial parameter $(H_1/M_P)^4$. The appearance of the fourth power of $H_1/M_P$ rather than the (more usual) second power is precisely the consequence of using seeds --- rather than first-order fluctuations of the scalar field --- for generating anisotropies. Thus $\Delta T/T$ goes like the square of the original fluctuations. For the same reason, although the fluctuations of $\si$ are expected to be Gaussian, some non-Gaussianity is expected in the fluctuations of $\Delta T/T$, since they are sourced by the seed energy-momentum tensor, which is quadratic in the Gaussian variable $\si$. Thus, we rather expect $\Delta T/T$ to obey a $\chi^2$ statistics (note that this is one of the few non-Gaussian examples where we really have a handle on the statistics). Let us finally turn our attention to a specific example of our new mechanism, that of the universal axion in pre-big bang (PBB) cosmology. The universal axion of superstring theory is just the (pseudo-scalar) partner of the dilaton in the string effective action, and is massless in perturbation theory because of a Peccei-Quinn (PQ) symmetry. While the dilaton is expected to acquire a mass as soon as supersymmetry is broken, the axion could remain massless, or almost massless, because of its Nambu-Goldstone origin. Although the PQ symmetry is broken by instantonic effects, in the presence of various axions coupled to the same topological current, a linear combination, mainly lying along the invisible axion's axis, is expected to remain very light or massless (for the purpose of this work a mass of order $H_0$ can be considered to be zero). Such a light particle, being a gravitationally coupled pseudo-scalar, should not lead to phenomenological difficulties. The field $A$ of Eq.~(\ref{action}) turns out to be, in this case, $e^{\phi}$ (where $\phi$ denotes the dilaton) and is related to the effective Newtonian constant, in the conventions used in PBB cosmology \cite{PBB}, by $G_N^{eff} \sim e^{\phi}$. The pump field $S$ is thus $a^2 e^{\phi}$, and grows very fast during PBB inflation since both $a$ and $e^{\phi}$ have accelerated behaviour (on the contrary, the pump field for dilaton and gravity-wave perturbations is $a^2 e^{-\phi}$, and the two factors tend to cancel out, giving too little power at large scales \cite{5b}). In the axion case the exponent $\al$ of Eq.~(\ref{simple}) can be evaluated \cite{Copeland} from the known background solutions \cite{PBB}. As noticed in \cite{Buon}, the desired value $\al=0$ is reached, in particular, for a highly symmetric, ten-dimensional PBB background in which the six extra dimensions evolve like the three ordinary ones (up to an irrelevant $T$-duality transformation). Precisely in this case, a scale-invariant spectrum for $\Delta T/T$ will result. Concerning the overall normalization, controlled by $(H_1/M_P)^2$, we note that in the PBB scenario the inflationary scale $H_1$ is typically of order of the string scale $M_s$, usually taken to be around $5\times 10^{17}$ GeV. Typically, $(H_1/M_P)^2$ thus varies between $10^{-2}$ and $10^{-4}$. We have to add the fudge factor $K$, which is hard to evaluate precisely, but is expected to contain factors like $(16 \pi ^2)^{-1}$. Thus, amusingly enough, the right order of magnitude \cite{33} for $C_2$ ($C_2 \sim 10^{-10}$) may come out naturally from $K (M_s/M_P)^4$ (taking also into account the possibility that the spectrum be slightly tilted, see \cite{1} for a quantitative discussion). 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