Article
Reference
CMB anisotropies from pre-big bang cosmology
VERNIZZI, Filippo, MELCHIORRI, Alessandro, DURRER, Ruth
Abstract
We present an alternative scenario for cosmic structure formation where initial fluctuations are due to Kalb-Ramond axions produced during a pre-big bang phase of inflation. We investigate whether this scenario, where the fluctuations are induced by seeds and therefore are of isocurvature nature, can be brought in agreement with present observations by a suitable choice of cosmological parameters. We also discuss several observational signatures which can distinguish axion seeds from standard inflationary models. We finally discuss the gravitational wave background induced in this model and we show that it may be well within the range of future observations.
VERNIZZI, Filippo, MELCHIORRI, Alessandro, DURRER, Ruth. CMB anisotropies from pre-big bang cosmology. Physical Review. D, 2001, vol. 63, no. 063501
DOI : 10.1103/PhysRevD.63.063501 arxiv : astro-ph/0008232
Available at:
http://archive-ouverte.unige.ch/unige:968
Disclaimer: layout of this document may differ from the published version.
CMB anisotropies from pre-big bang cosmology
F. Vernizzi1, A. Melchiorri2;1 and R. Durrer1
1Departement de Physique Theorique, Universite de Geneve, 24 quai Ernest Ansermet, CH-1211 Geneve 4, Switzerland
2 Universita Tor Vergata, Roma, Italy
We present an alternative scenario for cosmic structure formation where initial uctua- tions are due to Kalb-Ramond axions produced during a pre-big bang phase of ination. We investigate whether this scenario, where the uctuations are induced by seeds and therefore are of isocurvature nature, can be brought in agreement with present observations by a suitable choice of cosmological parameters. We also discuss several observational signatures which can distinguish axion seeds from standard inationary models. We nally discuss the gravitational wave background induced in this model and we show that it may be well within the range of future observations.
PACS Numbers : 98.80.Cq 98.80.E
I. INTRODUCTION
It is commonly assumed that an inationary phase is necessary in order to construct a consistent cosmolog- ical model. The familiar adiabatic inationary scenario deserves its popularity to the fact that it solves the horizon and atness problem and at the same time provides a consistent model for the origin of cosmological perturbations. In particular, it naturally leads to a at (Harrison-Zel'dovich) spectrum of perturbations on large scale and to coherent acoustic oscillations on intermediate scales which manifest themselves as \peaks"
in the Cosmic Microwave Background (CMB) anisotropies.
After the recent measurements of the intermediate scale CMB anisotropy power spectrum [1{4], at adiabatic models seem to be favored [6{8]. Nevertheless none of the many inationary scenarios which have been developed during the last 19 years has been constructed consistently on the bases of a serious theory of high energy physics; ination has always been seen as an eective model pointing to a greater more fundamental theory which has not been claried so far. We believe that superstrings are presently the most promising candidate for such a theory but on the other hand it is well known that it is not possible to derive an inationary model from a string theory eective action on a generic background, the reason being that the non-minimal coupling between the dilaton and the metric slows down the expansion of the universe spoiling the solution of the problems for which the ination has been invoked.
The pre-big bang idea [9] represents in this context one of the rst and most interesting attempts to develop a new cosmological scenario which solves the horizon and atness problems, based on string theory. In this radically new picture, the underlying duality symmetry [10] present in the low energy sector of string theory naturally selects perturbative initial conditions and automatically leads to an inationary phase prior to the big bang during which curvature and the dilaton are growing [9,11]. Unless its many appealing features, this scenario is known to face several problems such as the lack of a complete and consistent description of the high coupling and high curvature regime where the transition between the pre-big bang and the post-big bang phase and the stabilization of the dilaton should take place [12]. Furthermore, opinions vary as to whether the initial conditions in the pre-big bang need a large amount of ne tuning [11,13]. On a more phenomenological side, it is nevertheless important to study whether this scenario can provide the features that we observe in the universe today.
A realistic cosmological model has to generate large-scale matter perturbations and to reproduce the slope and the amplitude of CMB anisotropy spectrum. On this side the pre-big bang scenario was thought for some time to be unable to provide a scale-invariant spectrum of perturbations. First-order tensor and scalar perturbations in the metric, as well as perturbations of the moduli elds, were found to be characterized by extremely \blue" spectra [14]. This large tilt, together with a natural normalization imposed by the string cuto at the shortest amplied scales, would have made their contribution to large-scale structure completely negligible.
However, it was later realized that the spectral tilt of the axion, a universal eld in string theory, can assume a whole range of values depending on the behavior of the internal and external dimensions and in particular it can naturally provide a scale-invariant spectrum of perturbations [15{17]. This result reopened the possibility that pre-big bang cosmology may contain a natural mechanism for generating large-scale CMB anisotropies via the \seed" mechanism [18].
This possibility was analyzed in Refs. [19,20] for massless axions and in Ref. [21] for very light axions but these analytical treatments are restricted to large angular scales. We then have extended the study to smaller scales with the help of numerical calculations. First results of this work have been reported in a letter [22], where a strong correlation between the axion spectrum, n, and the height of the peak was noticed. A range of values aroundn = 1:4 (slightly blue spectra) appeared to be favored by a simultaneous t to the normalization on large angular scales observed by COBE [23] and the data on the rst acoustic peak available at that time.
In this companion paper we present a full explanation of the details of these calculations for the CMB angular power spectrum and for the dark matter power spectrum and we study the problem of the \de- coherence" of axion perturbations which has been ignored in the previous work. Furthermore, we expand on the results on the observational signatures presented in the Letter [22] and we discuss them in the light of the new CMB anisotropy data presently available by investigating the cosmological parameter-space of the model. We also discuss CMB polarization for our model and the contribution of the gravitational wave background induced by axion perturbations.
The paper is organized as follows: in the next section we study axion production in the pre-big bang and explain the details of the computation of the axion energy-momentum tensor which plays the role of the \seed" in our model. In Section III we determine the CMB anisotropy and dark matter spectra. We study the problem of decoherence and show that the coherent approximation is very good in this model.
In Section IV we compare our result with CMB and Supernova data and present a cosmological parameter estimation for this scenario. We also examine and discuss the normalization and the kink in the axion spectrum which is required to t observations. Section V is devoted to a novel prediction of axion seeds:
the tensor component of their energy-momentum tensor induces a gravity wave background which might be observable. In Section VI we summarize our conclusions.
II. AXION SEEDS FROM STRING COSMOLOGY A. Extra dimensions in string cosmology
The minimal low energy eective action of the NS-NS sector of string theory in the string frame is given by [24]
S10=
Z
d10xpjg10je?10
R10+ (r10)2? 1 12H102
; (2.1)
where we have included the 10-dimensional antisymmetric tensorH=@[B], but no gauge or fermion elds.
We assume that the 10-dimensional metric can be factorized into a \large" 4-dimensional part and a
\small" 6-dimensional metric,
ds2=gdxdx+e2IJdXIdXJ (2.2)
(; = 0;:::;3 and I;J = 1;:::;6), where depends only on time =(t). If the six dimensional piece is compactied to a very small radius, the lowest energy Kaluza-Klein modes yield the 4-dimensional action [15],
S=
Z
d4xpjgje?R+ (r)2?3(r)2?1
2e2(r)2
: (2.3)
Here we have introduced the 4-dimensional axion eld dened by
H=e?r: (2.4)
The action (2.3) and the denition (2.4) include the 4-dimensional dilaton eld, , the pseudo-scalar axion eld, , which represents the degrees of freedom of the antisymmetric three tensor eldH, and a modulus eld, , which parameterizes the radius, or the \breathing mode", of the 6-dimensional internal space. The axion eld (not to be confused with the Peccei-Quinn axion) is universal in string theory.
Let us assume a homogeneous dilaton background, =(t), and an external 4-dimensional space-time adequately described by a standard, spatially at FLRW metric with scale factora(t),
g = diag[?1;a2(t);a2(t);a2(t)]: (2.5) In the following we shall also make use of the metric
g =a2()diag[?1;1;1;1]; (2.6)
where we have introduced the conformal time given by d = dt=a (we shall use a point to indicate a derivative with respect to conformal time, _ @=@). With this choice of the external metric, the 4- dimensional dilaton is related to the 10-dimensional one by
=10?6: (2.7)
When the axion eld is trivial, _ = 0, or its contribution to the global dynamic of the universe is negligible, the equations derived from the action (2.3) are invariant under duality transformations,
a(t)!1=a(?t); (t)!(?t)?6ln(a(?t)): (2.8) This invariance (scale factor duality) represents one of the motivations behind the pre-big bang scenario [9].
The eld equations for a; and are solved [9] by the following power laws, known as dilaton-vacuum solutions in the pre-big bang for <?1:
a() =
? 1
1?; e()=
? 1
1? ; e()=
? 1
3 ?1
1? ; (2.9)
whereand satisfy the Kasner constraint,
32+ 62= 1: (2.10)
Here?1is the (conformal) time at which curvature and dilaton become so large that loop corrections from string theory have to be taken into account. It is hoped that these corrections then lead to a radiation dominated Friedman universe with \frozen" dilaton at > 1.
From these solutions one can see that, during the pre-big bang phase, i.e. for negative conformal time, a negativeand a positiveare required to make the external 3-dimensional space expand and the internal 6-dimensional space contract. Thereforehas to lie in the interval?1=p3 <0, which leads always to a growing dilaton and growing 4-curvature,R(_a=a2)2/1=(a)2/(?)1??2.
B. Amplication of axion quantum uctuations
In this subsection we briey review the mechanism for the generation of a primordial quasi-scale-invariant spectrum from the pre-big bang phase and we discuss the dependence of the spectral index on the evolution of the internal and external dimensions of the pre-big bang universe. Secondly, using as initial conditions the axion eld obtained during the pre-big bang phase, we analyze its evolution after the big bang in a critical FLRW universe with and without cosmological constant, paying particular attention to the frequency modes that enter into the calculation of the CMB anisotropy power spectrum.
As in previous works [19,21,22] we suppose that the contribution of the axion eld to the equations of motion for,aand is negligible and that the evolution of the dilaton, the moduli, and the scale factor are governed by the dilaton-vacuum solutions (2.9). Nevertheless, quantum uctuations of all the elds are of course present and we will show that quantum uctuations of the axion eld can seed density perturbations and CMB anisotropies in the post-big bang era. To this goal we have to study the axion evolution equation and the spectrum of axions produced during the pre-big bang phase due to their coupling to the background gravitational eld and the dilaton.
Varying the action (2.3) with respect to the eld in the string frame yields the equation of motion
r(e?2r) = 0: (2.11)
The study of this equation is conveniently performed by use of the canonical variable given by
aAae=2; (2.12)
which \diagonalizes" the perturbed action expanded up to second order. The factoraAis the so called pump eld of the axion. The Fourier modes k() satisfy a canonical linear second-order equation, completely decoupled from the other elds,
k+
k2?aA
aA
k= 0: (2.13)
This is the evolution equation for the axion eld.
Eq. (2.13) is equivalent to the equation for a classical harmonic oscillator with parametric evolution driven by the time dependent eective potential aA=aA. When the time evolution of the velocity of the pump eld, a_A, is suciently slow such that, for a given mode k, aA=aA k2, we are in the adiabatic regime with the result that no particles are created. When the acceleration in the pump eld is high enough to violate the adiabatic regime, quantum particle production starts. The evolution of the axion eld and the resulting spectrum of particles are fully determined by the time behavior of the pump eld in the dierent phases of the universe. In particular, a strong dierence in this behavior exists between the pre-big bang phase and the standard radiation and matter dominated eras in the post-big bang universe.
The pre-big bang phase is characterized by an accelerated evolution of the pump eld, aA/(?); = 5?1
2(1?); (2.14)
where < 0 is the power which characterizes the evolution of the external dimensions, Eq. (2.9). Using Eq. (2.13), the evolution equation of the axion can be written as
k+k21?(?1) x2
k= 0; (2.15)
wherexk. This equation is solved analytically in terms of the Hankel functions1=2H(1) and1=2H(2)
with=j?1=2j.
At very early time, a perturbation of given wave numberkis well inside the horizon,jxj=jkj1, and the solutions of Eq. (2.15) are harmonic oscillations which can be consistently normalized to the vacuum uctuation spectrum for!?1. This initial condition implies that theH(1) mode is absent and
k() = (?)1=2H(2)(k); = 12?= 1?3
1? ; for <?1: (2.16) Here ?1 ! 1 is the transition time scale between the pre-big bang phase and the standard radiation dominated era.
After the singularity, during the standard radiation and matter dominated eras, the dilaton is frozen, = const, and the pump eld is proportional to the standard scale factor, aA /a. The scale factor, a, and its second derivative, a, are given by Friedman's equations. For a critical universe, which we consider throughout our calculations and which is certainly a good approximation until redshiftsz 5, we have
a
a = 4G
3 a2(?3p) + 2a2
3 ; (2.17)
a_2
a2 = 8G
3 a2+a2
3 : (2.18)
Energy conservation for radiation (r) and matter (m) yieldsr /1=a4 andm /1=a3, with =r+m
andp=r=3;ris the radiation energy-density, m is the matter energy-density, andpthe pressure of the radiation uid. At early times, when is negligible, these equations have a simple analytical solution,
a=aeq
=+ 14(=)2
;
3
4Geq
1=2
= eq
2(p2?1) '1:2eq; (2.19) whereeq is the transition time between the radiation and the matter dominated era, r(eq) =m(eq) = eq=2. The eective potential during the post-big bang becomes
aA aA = a
a = 21+122: (2.20)
When is non vanishing, the solution for the eective potential can be found numerically but since the contribution of a small cosmological constant to the scale factor becomes important only at late time, the solutions to (2.15) are almost unaected; this has been checked by numerical tests. In the radiation dominated era, < eq, the eective potential can be approximated by aA=aA'1=(2).
We now study the axion evolution in the post-big bang era. Let us write the term in parenthesis on the left hand side of the axion equation of motion, Eq. (2.13), as
k2?a a
=k21?(a=a)2 x2
=k21?=(2+=2) x2
: (2.21)
In order to study the solution of Eq. (2.13) we have to study the ratio of the dimensionless eective potential (a=a)2 andx2 to be compared with unity. As long as we are well in the radiation dominated era, , the dimensionless eective potential is small and particle creation induced by the pump eld is negligible at early times. Eq. (2.13) then is a harmonic equation solved by free plane waves,
k() = 1pk[c+(k)e?ik+c?(k)eik]: (2.22) By matching the two solutions (2.16) and (2.22) at the transition time 1 we obtain, for jk1j 1 and eq > 1,
c(k) =c(k); with hjc(k)j2i'
(
kk1?2?1 k < k1
0 k > k1 ; (2.23) so that
k= c(k)
pk sin(k(?1)); for 1< eq : (2.24) Here k1 = 1=1 represents the maximal amplied frequency of the pre-big bang phase. We suppose that modes with frequencies much lower thank1are unaected by the unknown details of the transition from the pre- to the post-big bang phase.
The energy-density distribution of the produced axions is then d(k)
dlogk '12
k a
4
hjc(k)j2i'
k1 a
4 k
k1
3?2
/kn?1: (2.25)
The axion spectral indexnis related to the power which characterizes the evolution of the external dimen- sions by
n= 4?2= 3 + 2= 2
1 + 1?
; (2.26)
which follows from Eq. (2.16). In order not to over-produce infrared axions we have to require 3=2, orn1, which implies ?1=3. As already pointed out in [19], the limiting value= 3=2 corresponds precisely to a Harrison-Zel'dovich spectrum of CMB anisotropies on large scale. In terms of the evolution of the scale factor, this corresponds to an isotropic expansion and contraction respectively of the external and internal dimensions,
a/e1 /(?)?1=3: (2.27)
Notice that only for a 10-dimensional space-time, symmetrical expansion and contraction corresponds to a at axion spectrum which induces a Harrison-Zel'dovich spectrum of CMB uctuations [20,19]!
Nevertheless, as will be discussed in Section III, at very large scales and very early (negative) times, we will need a slightly blue axion spectrum to t CMB data. This requires a somewhat larger value of, i.e. a slower expansion of the external dimensions and, correspondingly, a somewhat faster contraction of internal dimensions at early time.
Let us therefore investigate what happens if the universe expands with some expansion law described by ? at early times, < b <?1 and then switches to an expansion law given by + after b. Suciently short wavelength modes which are inside the horizon during the entire epoch < b, which satisfyjkbj<1, are not inuenced by this change in the expansion law. The term aA=aA is indeed sub-dominant in the equation of motion for kduring this epoch and hence the Bogoliubov coecientjc(k)j2of Eq. (2.23) is not inuenced by the transition; we just obtain the result (2.23) with=+.
The situation is dierent if a mode exits the horizon before b. Then the \incoming" solution ( <
b) = (?)1=2H(2)?(k) diers from the vacuum solution and matching it to the general \outgoing" solu- tion, ( > b) = b1(?)1=2H(1)+(k) +b2(?)1=2H(2)+(k), yields b2?b1 = (?(?)=?(+))jkb=2j+??. Correspondingly, the coecientjc(k)j2 is changed by a factor jb2?b1j2. In a model where the expansion law changes at a well dened timeb?1=kb, we therefore get the following Bogoliubov coecients in the post-big bang radiation era (see Fig. 1):
hjc(k)j2i'
k k1
?1?2+(k=kb)2+?2? for kkb
1 for kkb: (2.28)
1 10 100 1000
Log[kb]
1
Log[Ω σ(k)]
g12
Log[k]
FIG. 1. Evolution of the axion spectral index during the pre-big bang. The valueg1is the string coupling constant given by the string scale divided by the Planck scale,g1= (k1=a1)=mPlanck(see Section III).
We do not want to specify the event which may have triggered such a transition from n(k < kb) = 4?2? = 1 + to n(k > kb) = 1, but there are certainly dierent possibilities. For example, it is interesting to note that isotropic expansion and contraction,a/1=b, in a 26-dimensional space-time gives ? = 1=5, or n = 1:33, which corresponds to = 1=3, just about the \tilt" needed to t the observed CMB anisotropies (see Section III). Therefore, if we start out the pre-big bang phase with a 26-dimensional bosonic string vacuum (which we know to be unstable due to the presence of tachyons) which then \decays"
to a supersymmetric and 10-dimensional string vacuum at some time b, which corresponds to a comoving energy scalekb, this could induce the required tilt.
We now study the modication in the axion spectrum during the post-big bang era, whereaA=a. As we have seen above, during the radiation era, < , the dimensionless eective potential is small. Furthermore, once a mode enters the horizon,k >1, thek2-term always dominates over the eective potential and there is no more particle creation. Therefore modes which enter the horizon before equality, k > 1, are not amplied any further in the post-big bang phase. The spectrum of axion perturbations for these modes remains unaected. However, the low frequency tail of the spectrum is further modied as soon as we enter in the matter dominated era, where the dimensionless eective potential becomes of order unity. The modes which enter the horizon after the equality, k < 1, are amplied. This amplication of low frequency modes has important consequences on the angular spectrum of the CMB as will shall discuss in detail in Subsection IIIB.
10−2 10−1 100 101 102
η/ηeq
2
k=0.1k
k=10k
eq
eq
FIG. 2. The dimensionless potential (a=a)2 (thick line) and two modes that enter the horizon before and after equality. The mode that enters the horizon before equality, '(k = 10keq;), is unaected by the pump eld and begins to oscillate without being amplied. The mode that enters the horizon after equality, '(k = 0:1keq;), is amplied and begins to oscillate later.
The behavior of the dimensionless eective potential (a=a)2 together with two modes that enter the horizon before and after equality have been plotted in Fig. 2. As one can see, only modes that enter the horizon after equality are amplied. Deep in the matter eraeq, the dimensionless eective potential is constant and Eq. (2.13) becomes
k+
k2?22
k= 0: (2.29)
This equation can again be solved in terms of Hankel functions,
k() =1=2[AH3(2)=2(k) +BH3(1)=2(k)]; for eq; (2.30) whereAandBare constants to be determined by matching conditions (see [20]). The post-big bang solutions (2.24) and (2.30) are only correct far from matter-radiation equality eq and in order to compute CMB
anisotropies we need to require better precision on these solutions also for eq. We therefore solve the axion equation of motion Eq. (2.13) numerically, from the early radiation era through the radiation-matter transition.
The axion eld is then given by
(k;) = 1a() k() = c(k)
apk'(k;); (2.31)
where the variable'is the solution of equation '+
k2?a a
'= 0 (2.32)
with initial condition (obtained from the pre-big bang solution Eq. (2.24))
'(k;) = sin(k); : (2.33)
We have solved Eq. (2.32) numerically in this work using the eective potential (2.20). The pre-factorc(k) is a stochastic Gaussian eld with power spectrum
hjc(k)j2i= (k=k1)n?5; (2.34) where n is again the primordial spectral index (2.26), our free parameter which depends on the higher- dimensional pre-big bang phase.
C. Axion quantum uctuations as seeds
We are now ready to consider the axion eld as a source of the linear cosmological perturbation equations.
As in previous works [19,20,22] we suppose that the contribution of the axions to the cosmic uid can be neglected and that they interact with it only gravitationally. They then play the role of seeds which, by their gravitational eld, induce uctuations in the cosmic uid [18]. The back-reaction of the metric perturbations on the evolution of seeds is second order and can be neglected in rst order perturbation theory. The evolution of axions can be computed by using the solutions of the axion eld equation in the unperturbed background geometry, Eq. (2.13).
The axion eldis a Gaussian stochastic variable. Its contribution to the perturbation equations is given in terms of its energy-momentum tensor,
T()=@@?12(@)2; (2.35)
which is quadratic in and therefore not Gaussian. Moreover, although the axion eld evolves according to a linear equation, it will enter into the perturbation equations throughT()which evolves non-linearly.
The perturbations in the dark matter and radiation components are set to zero in the initial conditions and are subsequently induced by the gravitational eld of the axion. Hence, axion seed perturbations belong to the class of isocurvature perturbations. However, they dier from topological defects by being \acausal", i.e. they have non-vanishing correlations on super-Hubble scales, since they are due to eld excitations induced during an inationary era.
As we have seen above, the axion power spectrum obeys a simple power law with cuto and is in general not analytic at k = 0. Furthermore, axion perturbations do not, in general, display the scaling behavior expected from topological defects. In the pre-big bang we have an additional scale, the string scalek1, which breaks scale-invariance. The axion spectrum on large scales is therefore not determined by dimensional arguments since there are dimensionless factors of the form (k=k1) which may alter the spectrum1. The signicance of these points will become clearer later in the paper.
1Actually the radiation { matter transition scalerepresents a scale which is also present in models with topological defects, but deep in the radiation or matter era this scale has no signicance, whereas as we shall see the above factors multiply the entire power spectrum of uctuations.
As in [22], we rst consider a critical universe (total density parameter = 1) consisting of cold dark matter, baryons, photons, and three types of massless neutrino, with or without a cosmological constant. We choose the baryonic density parameter b = 0:05 and the value of the Hubble parameter H0= 100hkms?1Mpc?1 with h= 0:65.
The linear perturbation equations for this universe in Fourier space are of the form
DX=S; (2.36)
whereX is a long vector containing all the uid perturbation variables which depends on the wave number
k and conformal time , S is a source vector which consists of certain combinations of the seed energy momentum tensor and D is a linear ordinary dierential operator. More details on the linear system of dierential equations (2.36) can be found in [25] and references therein.
For a given initial condition, this equation can in general be solved by means of a Green's function,G(;0), in the form
X(k;0) =
Z 0
inG(k;0;)S(k;)d: (2.37)
We want to determine power spectra or, more generally, quadratic expectation values of the form
hXi(k;0)Xj(k;0)i; (2.38)
which, according to Eq. (2.37), are given by
hXi(k;0)Xj(k;0)i=
Z 0 in
Z 0
inGil(0;)Gjm(0;0)hSl()Sm(0)idd0: (2.39) (Sums over double indices are understood.)
We therefore have to compute the unequal time correlators,hSl()Sm(0)i, of the seed energy-momentum tensor. This problem can, in general, be solved by an eigenvector expansion method [25,26], as it will be done in Subsection IIIB. However, if the source evolution is linear, the problem becomes especially simple.
In this \coherent" case, we have
Sj() =Fji(;in)Si(in); (2.40) with a deterministic transfer functionFji. In this situation we can, by a simple change of variables, diago- nalize the hermitian, positive initial equal time correlation matrix,
hSl(in)Sm(in)i=llm: Inserting this in Eq. (2.39) yields
hXi()Xj(0)i=
Z 0
inGil(0;)til(;in)pldZ 0
inGjm(0;0)tjm(0;in)pmd0lm: (2.41) We therefore obtain exactly the same result as the one obtained by replacing the stochastic variableS by the deterministic source termS(jdet) given by
S (det)
j ()S(idet)() =Fjl(;in)Fil(;in)l= exp(iji)qhjSj()j2ihjSi()j2i; (2.42) wherejiis a, in principle unknown, phase which has to be determined case by case. Clearlyjj = 0. When the stochastic variableS is real (as in our case) exp(iji) =1. This linear or coherent approximation will be fully used in this paper. We shall test its validity in Subsection IIIB.
It is useful to split the energy-momentum tensor of the axion seeds (2.35) into a scalar, vector, and tensor part since the perturbations generated by each of these components are evolved independently. Due to statistical isotropy these three modes are uncorrelated. This also corresponds to a decomposition of the source
termS into a scalar, vector, and tensor contributions,S(S),S(V), andS(T). A suitable parameterization of the decomposition of the Fourier components ofT()is [18]
T00()=f;
Tj(0)=?ikjfv+vj; (2.43)
Tij()=ijfp?
kikj?k2 3ij
f+ 12(wikj+wjki) +ij;
wheref,fv,fp, andf are random function ofk;wandvare transverse vectors,wk=vk= 0, andij
is a symmetric, traceless, transverse tensor,ii=ijkj = 0. The variables (f), (v,w) and (ij) represent the scalar, vector, and tensor degrees of freedom ofT() respectively. They are the source of the perturbation equations.
The goal of the next three subsections is to express the correlators of the source componentsS(S), S(V), andS(T) in terms of these variables. These expressions, inserted in the perturbation equation (2.36), then allow us to compute the CMB anisotropy and dark matter power spectra numerically.
D. Axion seeds { Scalar component
We rst consider the scalar contribution given by the four variablesf of Eq. (2.44). Only two of these functions are independent, the other two are related by energy and momentum conservation. We shall use two linear combinations of the three scalar seed-functionsf,fv, and f,
f(k;) =a2()=T00()(k;); (2.44)
fv(k;) = ikjT0(j)(k;)
k2 ; (2.45)
f(k;) = 32k4
?Tij()(k;)kikj+ 13k2klTkl()(k;)
: (2.46)
In the presence of seeds and in the linear perturbation approximation, the scalar component of the total geometric perturbations can be separated into a part induced by the seeds, s and s, given by
k2s= 4G[f+ 3(_a=a)fv)]; s+ s=?8Gf; (2.47) and a part induced by the perturbations of the cosmic uid, mand m. The total geometric perturbation are given by the sums,
= s+ m; = s+ m: (2.48)
The variables and are the (Fourier components of the) Bardeen potentials. They are gauge invariant and fully describe scalar perturbations of the Friedman geometry (for details look in [27,28]).
Scalar perturbations are seeded by s and s. These are the standard independent variables to use as scalar sources in the perturbation equations. In order to simplify somewhat the computation, we use sand f as our scalar seed degrees of freedom and the scalar source vector becomes
S
(S)(k;) = [s(k;);4Gf(k;)]: (2.49) The energy-momentum tensor of the axion is given by Eq. (2.35), which leads to the following expressions for the seed-functions in terms of the axion eld,
f(k;) = 12Z d3p (2)3
h_(p;)_(jk?pj;)?p(k?p)(p;)(jk?pj;)i; (2.50) fv(k;) =? 1
k2
Z d3p
(2)3k(k?p)_(p;)(jk?pj;); (2.51) f(k;) =? 3
2k
Z d3p (2)
h(kp)[k(k?p)]?1
3k2p(k?p)i(p;)(jk?pj;): (2.52)
The rst two seed-functions,f andfv, together with Eq. (2.47), yield s, s(k;) = 4G
k2
Z d3p (2)3
1
2 _(p;)_(jk?pj;)?1
2p(k?p)(p;)(jk?pj;)?3 _a
ak(kk2?p)_(p;)(jk?pj;)
: (2.53)
The only information about the source random variables which we really need are the unequal time correlators between the Fourier components of the independent variables sand f. These correlators can be written in terms of four real (since the correlatorsh(k;)(k0;0)i are real) scalar source correlation functions,F11,F22, F12, andF21, which completely characterize the scalar component of the source,
hs(k;)s(k0;0)i=(k?k0)F11(k;;0); 4Ghs(k;)f(k0;0)i=(k?k0)F12(k;;0); 4Ghf(k;)s(k0;0)i=(k?k0)F21(k;;0); (4G)2hf(k;)f(k0;0)i=(k?k0)F22(k;;0):
Note that F11(k;;) and F22(k;;) are positive by denition and, since the functions F are real, Fij(k;;0) = Fji(k;0;). In order to compute these functions we make use of Eqs. (2.52) and (2.53) and we exploit the stochastic average conditions of the Gaussian variables and _ (Wick's theorem). We rst introduce three real auxiliary variables 1, 2, and 3, which depend on the power spectrum of the axion eld,hjc(k)j2i, and on the solution'of the evolution equation, Eq. (2.32),
h(k;)(k0;0)i= (2)3(k?k0)1(k;;0);
h_(k;)_(k0;0)i= (2)3(k?k0)2(k;;0);
h(k;)_(k0;0)i= (2)3(k?k0)3(k;;0);
h_(k;)(k0;0)i= (2)3(k?k0)3(k;0;): (2.54) The variables iare given by
1(k;;0) = hjc(k)j2i
ka()a(0)'(k;)'(k;0); (2.55)
2(k;;0) = hjc(k)j2i
ka()a(0)[ _'(k;)?H()'(k;)][ _'(k;0)?H()'(k;0)]; (2.56) 3(k;;0) = hjc(k)j2i
ka()a(0)[ _'(k;)?H()]'(k;0); (2.57) whereHa=a_ . Notice that 1(;) and 2(;) are positive by denition.
Inserting these results in Eqs. (2.52) and (2.53), and making use of Wick's theorem for the \random variable"c(k), we can work out a somewhat lengthy but straight forward expression for the scalar source functions,F11,F22,F12, andF21, in terms of the variables 1, 2, and 3:
F11(k;;0) = (4G)2 k4
Z d3p (2)3
n1
22(p;;0)2(jk?pj;;0)?1
2p(k?p)h3(p;;0)3(jk?pj;;0) + 3(p;0;)3(jk?pj;0;)i?3k(k?p) k2
h
H()2(p;;0)3(jk?pj;;0) +H(0)2(p;;0)3(jk?pj;0;)i+ 12(p(k?p))21(p;;0)1(jk?pj;;0) + 3(pk?p2)(k2?pk) k2
h
H()3(p;0;)1(jk?pj;;0) +H(0)3(p;;0)1(jk?pj;0;)i+ 9H()H(0) k4
h(k(k?p))22(p;;0)1(jk?pj;;0) + (k(k?p))(kp)3(p;0;)3(jk?pj;;0)io; F22(k;;0) = 9(4G)2
2k8
Z d3p (2)3
h(kp)(k(k?p))?1
3k2p(k?p)i21(p;;0)1(jk?pj;;0); F12(k;;0) =?3(4G)2
2k6
Z d3p (2)3
h(kp)(k(k?p))?1
3k2p(k?p)i h3(p;0;)3(jk?pj;0;)?p(k?p)1(p;;1)1(jk?pj;;0)?6H()k(k?p)
k2 3(p;0;)1(jk?pj;;0)i; F21(k;;0) =F12(k;0;):
The scalar source correlators of the perturbation equation (2.36) can be written as a two by two positive and hermitian matrix,
hS (S)
i (k;)S(jS)(k;0)i=
F11(k;;0) F12(k;;0) F21(k;0;) F22(k;;0)
: (2.58)
