\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). In conclusion, irrespectively of its possible model-dependent origin, we believe that a cosmic background of massless pseudo-scalar fluctuations may provide a consistent and interesting explanation of the anisotropies observed in the CMB temperature, at large angular scales. It is unclear, at present, whether such an axion-induced anisotropy may lead to significant differences in the acoustic peak structure of the CMB anisotropy spectrum at smaller angular scales. If it does, this (plus possibly some non-Gaussianity of the fluctuations) should allow tests of our axionic-seed mechanism through the high-precision measurements planned for the near future \cite{35}. The discussion of this possibility is postponed to further work. \vspace{2cm} \begin{thebibliography}{99} \newcommand{\plb}{{\em Phys. Lett. B}\ } \newcommand{\prl}{{\em Phys. Rev. Lett.}\ } \newcommand{\prd}{{\em Phys. Rev. D}\ } \newcommand{\npb}{{\em Nucl. Phys. B}\ } \newcommand{\bb}{\bibitem} \bb{COBE}G. F. Smoot et al., {\em Ap. J.} {\bf 396}, L1 (1992); C. L. Bennett et al., {\em Ap. J.} {\bf 430}, 423 (1994). \bb{Copeland}E. J. Copeland, R. Easther and D. Wands, \prd {\bf 56}, 874 (1997); E. J. Copeland, J. E. Lidsey and D. Wands, \npb {\bf 506}, 407 (1997). \bb{Buon} A. Buonanno, K. A. Meissner, C. Ungarelli and G. Veneziano, {\em JHEP01}, 004 (1998). \bb{Hadad} R. Brustein and M. Hadad, \prd {\bf 57}, 725 (1998). \bibitem{PBB}G. Veneziano, {\em Phys. Lett. B} {\bf 265}, 287 (1991); M. Gasperini and G. Veneziano, {\em Astropart. Phys.} {\bf 1}, 317 (1993); {\em Mod. Phys. Lett. A} {\bf 8}, 3701 (1993); {\em Phys. Rev. D} {\bf 50}, 2519 (1994). An updated collection of papers on the pre-big bang scenario is available at {\tt http://www.to.infn.it/\~{}gasperin/}. \bb{arion}A. A. Anselm and N. G. Uraltsev, {\em Phys. Lett. B} {\bf 114}, 39 (1982). \bb{SW} R. K. Sachs and A. M. Wolfe. {\em Ap. J.} {\bf 147}, 73 (1967). \bibitem{1} R. Durrer, M. Gasperini, M. Sakellariadou and G. Veneziano, {\sl Seeds of large-scale anisotropy in string cosmology}, CERN-TH/98-69, gr-qc/9804076. \bb{massive} M. Gasperini and G. Veneziano, {\sl Constraints on pre-big bang models for seeding large-scale anisotropy by massive Kalb-Ramond axions}, CERN-TH/98-180. \bb{BGV} R. Brustein, M. Gasperini and G. Veneziano, {\sl Duality in Cosmological Perturbation Theory}, Phys. Lett. B (1998), in press (hep-th/9803018). \bb{6a} R. Durrer and N. Straumann, {\em Helv. Phys. Acta} {\bf 61}, 1027 (1988). \bb{Durrer} R. Durrer, {\em Phys. Rev. D} {\bf 42}, 2533 (1990); {\em Fund. of Cosmic Physics} {\bf 15}, 209 (1994). \bb{7a} H. R. Harrison, {\em Phys. Rev. D} {\bf 1}, 2726 (1970); Y. B. Zel'dovich, {\em Mon. Not. Roy. Ast. Soc.} {\bf160}, 1 (1972). \bb{JamesBond} R. Bond and G. Efstathiou, {\em MNRAS} {\bf 226}, 655 (1987). \bb{smootscott} G.F. Smoot and D. Scott, in L. Montanet {\sl et al.}, {\em Phys. Rev. D} {\bf 50}, 1173 (1994) (1996 update). \bb{5b} M. Gasperini and M. Giovannini, \prd {\bf 47}, 1519 (1993);\\ R. Brustein, M. Gasperini, M. Giovannini, V. F. Mukhanov and G. Veneziano, \prd {\bf 51}, 6744 (1995). \bb{33} A. J. Banday et al., {\em Astrophys. J.} {\bf 475}, 393 (1997). \bb{35} M. Bersanelli et al., {\em COBRAS/SAMBA, Report on the phase A study}, ESA Document D/SCI(96) 3 (1996). \end{thebibliography} \end{document}