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PHYSICAL REVIEW D, VOLUME 61, 083518
Nonvacuum initial states for cosmological perturbations of quantum-mechanical origin
Jerome Martin* and Alain Riazuelo†
DARC, Observatoire de Paris, UPR 176 CNRS, 92195 Meudon Cedex, France
Mairi Sakellariadou‡
Theory Division, CERN, CH-1211 Geneva 23, Switzerlandand Centre for Theoretical Physics, University of Sussex, Brighton, Falmer BN1 9QH, United Kingdom
~Received 8 April 1999; published 28 March 2000!
In the context of inflation, nonvacuum initial states for cosmological perturbations that possess a built inscale are studied. It is demonstrated that this assumption leads to a falsifiable class of models. The question ofwhether they lead to conflicts with the available observations is addressed. For this purpose, the powerspectrum of the Bardeen potential operator is calculated and compared with the CMBR anisotropies measure-ments and the redshift surveys of galaxies and clusters of galaxies. Generic predictions of the model are a highfirst acoustic peak, the presence of a bump in the matter power spectrum and non-Gaussian statistics. Thedetails are controlled by the number of quanta in the nonvacuum initial state. Comparisons with observationsshow that there exists a window for the free parameters such that good agreement between the data andtheoretical predictions is possible. However, in the case where the initial state is a state with a fixed number ofquanta, it is shown that this number cannot be greater than a few. On the other hand, if the initial state is aquantum superposition, then a larger class of initial states could account for the observations, even though thestate cannot be too different from the vacuum. Planned missions such as the MAP and Planck satellites and theSloan Survey will demonstrate whether the new class of models proposed here represents a viable alternativeto the standard theory.
PACS number~s!: 98.80.Cq, 98.70.Vc
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I. INTRODUCTION
The observed large-scale structure in the Universebeen currently addressed, within the framework of gravtional instability, by two families of models: initial densitperturbations can either be due to ‘‘freezing in’’ of quantufluctuations of a scalar field~inflaton! during an inflationaryera @1# or they may be seeded by a class of topologicalfects, naturally formed during a symmetry-breaking phatransition in the early Universe@2#. The recent bulk of ob-servational and experimental data and, in particular, themic microwave background anisotropy measurements,the redshift surveys of the distribution of galaxies and clters of galaxies, impose severe constraints on the two falies of models, as well as on the variety of possible scenaintroduced within each family.
The simplest topological defect models of structure fmation show conflicts with observational data. As fishown in Ref.@3#, global topological defects models predistrongly suppressed acoustic peaks. While on large angscales the predicted cosmic microwave background radia~CMBR! spectrum is in good agreement with the CosmBackground Explorer~COBE! measurements, on smaller agular scales the topological defect models cannot reprodthe data of the Saskatoon experiment. One can manufacmodels@4# with structure formation being induced by scalinseeds, which lead to an angular power spectrum with
*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]
0556-2821/2000/61~8!/083518~15!/$15.00 61 0835
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same characteristics~position and amplitude of acoustipeaks!, as the one predicted by standard inflationary modThe open question is, though, whether such models areoutcome of a realistic theory. However, the most sevproblem for topological defects models of structure formtion is their predicted@5,6# lack of large-scale power in thematter power spectrum, once normalized to COBE. Choing scales of 100h21Mpc, which are most probably unaffected by nonlinear gravitational evolution, standard toplogical defect models, once normalized to COBE, requirbias factor (b100) on scales of 100h21Mpc of b100'5, toreconcile the predictions for the density field fluctuatiowith the observed galaxy distribution. However, the lattheoretical and experimental studies favor a current valueb100 close to unity.
In what follows, we shall place ourselves within thframework of cosmological perturbations of quantummechanical origin in the context of inflationary models. Tinflationary paradigm was proposed in order to explainshortcomings of the standard~big bang! cosmological model.In addition, it offers a scenario for the generation of tprimordial density perturbations, which can lead to the fmation of the observed large-scale structure.
The theory of cosmological perturbations of quantumechanical origin rests on two well-established theories.the one hand,~linearized! general relativity allows a calculation of the evolution and the amplification of perturbatiothroughout the cosmic evolution; the mechanism at working parametric amplification of the fluctuations due to tinteraction of the perturbations with the background@7#. Onthe other hand, quantum field theory permits us to undstand the origin of these perturbations. If the quantum fie
©2000 The American Physical Society18-1
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MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
are initially, i.e., at the beginning of inflation, placed in thvacuum state, then because of the Heisenberg principle,tuations are unavoidable. Moreover, the amplitude of thfluctuations is completely fixed.
Inflation, employing the theory of cosmological perturbtions of quantum-mechanical origin, leads to definite predtions for the anisotropies of the CMBR, as well as for tpower spectrum, which can be tested against experimeand observational data. In particular, simple models preda scale-invariant spectrum, with, provided the quantum fieare initially placed in the vacuum, Gaussian fluctuations.
Let us briefly discuss the observational data, namely,CMBR anisotropies measurements and the redshift survof the distribution of galaxies. The CMBR, last scatteredthe epoch of decoupling, has to a high accuracy a black-bdistribution @8#, with a temperatureT052.72860.002 K,which is almost independent of direction. The DifferentMicrowave Radiometer~DMR! experiment on the COBEsatellite measured a tiny variation in intensity of the CMBat fixed frequency. This is equivalently expressed as a vation dT in the temperature, which was measured todT/T0'1025 @9#. The four-year COBE data are fitted byscale-free spectrum; the spectral index was found to benS51.260.3 and the quadrupole anisotropyQrms-PS
515.322.813.8 mK @9#. The CMBR anisotropies spectrum
usually parametrized in terms of the multipole momentsCl ,defined as the coefficients in the expansion of the tempture autocorrelation function
K dT
T~e1!
dT
T~e2!L U
(e1•e25cosq)
51
4p (l
~2l 11!Cl Pl~cosq!, ~1!
which compares points in the sky separated by an angleq.The value ofCl is determined by fluctuations on angulscales of orderp/ l . The angular power spectrum of anisotrpies observed today is usually given by the power per lorithmic interval in l, plotting l ( l 11)Cl versusl.
On large angular scales, the main physical mechanwhich contributes to the redshift of photons propagating iperturbed Friedmann geometry, originates from fluctuatiin the gravitational potential on the last-scattering surfaThe COBE-DMR experiment, which measured CMBanisotropies on such large angular scales (l &20), confirmedthe predicted scale-invariant spectrum and yields mainlnormalization for the different models of large-scale struture formation.
On intermediate angular scales, 0.1°&q&2°, the maincontribution to the CMBR anisotropies comes from thetrinsic inhomogeneities on the surface of the last scatterdue to acoustic oscillations in the coupled baryon-radiatfluid prior to decoupling. On the same angular scales, thera Doppler contribution to the CMBR anisotropies, due torelative motions of emitter and observer. The sum of thtwo contributions is denoted by the term acoustic peaks.analysis of recent CMBR flat-band measurements on in
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mediate angular scales gives@10# in the best-fit power spectrum a peak@ l ( l 11)Cl /2p#1/2T0576 mK with l 5260.
Among the various experiments measuring CMBanisotropies, the Saskatoon experiment@11# is of particularimportance since it relates@12# CMBR anisotropies to thepower spectrum of matter density perturbations estimathrough clustering properties of galaxies and clusters of gaxies. More precisely, the Saskatoon experiment meastemperature anisotropies for multipoles in the rangel'802400, which corresponds to the range of wavelengthswhich we have data on galaxy clusters.
Analyzing a large number of available data on redshiftsindividual galaxies and Abell galaxy clusters, one obta@12# the power spectrum for clusters of galaxies, overwave-number interval fromk'0.03 h Mpc21 to k'0.3h Mpc21. On very large scales (k,0.03 h Mpc21), thelarge error bars are due to incomplete data. However, nthe turn over, error bars are small, thus both the relaposition and amplitude of the turn over are determined acrately. As discussed in, e.g. Ref.@12#, the power spectrumreveals the existence of a nontrivial feature at a wave numk050.05260.005 h Mpc21. Assuming this peak exists~further studies are necessary to confirm it!, the amplitude ofthe observed power spectrum is larger near the peak bfactor 1.4 @12# with respect to the power spectrum of thstandard cold dark matter model. The existence of this pis not related@13# to acoustic oscillations in the tight couplebaryon-photon plasma. As stated in Ref.@13#, the currentCMBR experimental data combined with observational clter data, favor theoretical models that have built-in a charteristic scale in their initial spectrum.
Recently, the COBE data have also been used to tesGaussianity of the CMBR anisotropies. Three groups@14–16# have now reported results showing that the fluctuatiowould not be Gaussian. The three groups work with differmethods. In Ref.@14#, the estimation of the bispectrumBl isused as a criterion to test Gaussianity. The dominant nGaussian contribution has been found nearl 516. It is clearthat these results should be taken cautiously since, forample, the issue of foreground contamination could chathe conclusions. However, the possibility of non-Gaussstatistics in the CMBR anisotropies should be taken intocount seriously.
The Saskatoon measurements could be explained by ping with the values of the cosmological parameters. In pticular, the value of the cosmological constant,VL'0.6, re-cently inferred from the SNIa measurements could accofor the high position of the first acoustic peak. But the othfeatures~the presence of a peak in the power spectrum; nGaussian fluctuations in the CMBR!, if confirmed, clearly gobeyond the paradigm of cold dark matter~CDM! and slowroll inflation. In order to explain them, different mechanismhave been advocated. For instance, double-inflation@17# ormultiple-inflation@18# models have been used to explain tpresence of the peak in the power spectrum. Another snario can be offered within models where the inflaton fieevolves through a kink in its potential@19#. To explain thenon-Gaussianity, different mechanisms have been propoOf course, this appears naturally when the perturbations
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NONVACUUM INITIAL STATES FOR COSMOLOGICAL . . . PHYSICAL REVIEW D 61 083518
induced by topological defects. However, even in the conof inflation, non-Gaussianity can be present, as, for examin the case of stochastic inflation@20,21#.
All these solutions are in fact different possible modifictions of the power spectrum of the primordial fluctuations.this article, our aim is to discuss the choice of the initquantum state in which the quantum fields are placed. Tchoice is of course crucial for the determination of the pmordial power spectrum, and different quantum stateslead to different power spectra. In the literature, it is~almost,see Ref.@22#! always assumed that the state of the perturtions is the vacuum~In a curved spacetime the definition othe vacuum state is not unique. A more precise definitionthe vacuum used in this paper is given in what follows acoincides with the one in Ref.@23#!:
u0&[ ^
ku0k&. ~2!
Let us examine how this choice can be justified. Sincequestion is a problem of boundary conditions, it mustaddressed by means of a theory of the initial conditionsthe early Universe. Such a theory should rely on full quatum gravity, which is unknown at present. The only candate at our disposal is quantum cosmology. Generally, it pdicts that the initial state is indeed the vacuum. For examthe no-boundary choice for the wave function of the cosmlogical perturbations implies that the Bardeen operatoplaced in the vacuum state, see Ref.@24#. This result doesnot come as a surprise since the Hartle-Hawking proposa generalization of a method that gives the ground-state wfunction of a system in ordinary quantum mechanics.
However, although fascinating, quantum cosmology isyet a well-developed branch of physics and many importquestions remain unsolved to this day. To our knowledthere exists no proof that quantum considerations automcally lead to a vacuum initial state for the perturbations. Sua proof, if it exists, should rely on full quantum gravity.
On the other hand, the choice of the vacuum is also baon the hypothesis that the initial state of the Universe shobe a ‘‘maximally symmetric state’’@25#. Concretely, thismeans that no scale should be privileged. This seems tthe simplest starting point. However, since the choice ofinitial state is supposed to appear naturally in the contexquantum gravity, it could also be argued that such a prleged scale does exist and is equal to the Planck scall Pl5(\G/c3)1/2'10233 cm. This becomes even more intriguing if one recalls that in order to solve the usual problemsthe standard model of cosmology, one needs 60e folds dur-ing inflation. This means that the Planck scale has now bstretched to a scale of at least 60 pc. Accordingly, allwavelengths below 60 pc were sub-Planckian at the momof their generation. Of course, the structure of space-tbelow the Planck scale is unknown and it may be very dferent from the one we are used to. Probably, such notionsub-Planckian wavelengths or even scale factor are meanless in a regime where the gravitational quantum effectsimportant.
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The arguments presented in the previous discussion sthat it is worth studying nonvacuum initial states for cosmlogical perturbations. Rather than relying on theoreticalguments, our goal will be to allow for the possibility of nonvacuum initial states and to establish the consequencesthe observables described at the beginning of the Introdtion. We study whether choices other than the vacuum amatically lead to inconsistencies or conflicts with the avaable observations or if, on the contrary, there existswindow for the free parameters of the model, which fits tobservational data.
Our choice of a nonvacuum initial state is guided byvery simple idea: the initial state could have a built-in chacteristic scale since this seems to be the simplest wageneralize the vacuum state. The question now arises athe physical origin of this scale. A possible answer is thatnatural scale is the Planck length stretched by the cosmolcal expansion. It is clear that, so as not to be in conflict wobservations, we would like this fundamental scale tonow translated to the characteristic scalel C'200 Mpc56.231026 cm ~here, as in the rest of this article we takh50.5). Since the ratiol C/ l Pl is given by l C/ l Pl'1027eN,whereN is the number ofe-folds during inflation, this meansthatN'75. Interestingly enough, we note that this leads tnumber of e folds greater than the minimum number rquired, i.e., 60. In the context of Linde’s chaotic inflation,is assumed that, initially, the inflaton potentialV(w) is suchthat V(w i)'mPl
4 wheremPl is the Planck mass. If the potential is given by, e.g.,V(w)5(l/4!)w4, this leads to an initialvalue of the scalar field greater than 4.4mPl , which is neededto get the usual 60e folds. Consequently, this leads to a hunumber ofe folds, N'108. It is clear that, with such a number, the Planck length cannot be stretched to 200 Mpc pently. Let us note that these models~with a large number ofe folds! suffer from the ‘‘super-Planck scale problem’’@27#:all the scales of cosmological interest now were suPlanckian at the beginning of inflation. Since quantum fietheory is expected to break down in this regime, the predtions of these models could be questionable. On the ohand, in the spirit of chaotic inflation itself, there exist rgions of space in which the initial value of the field wasw i'4.9mPl . This value leads to a number ofe folds equal to75. Therefore, the model presented in this article is certamore relevant in the case where inflation does not last folong period. In chaotic inflation, the probability of havinglong period of inflation is greater than the probability to ga small number ofe folds. Thus, our model does not fit verwell within the chaotic inflation approach.
It should also be mentioned that it has been shown in R@28# that a large class of initial states approaches the BunDavis vacuum in the de Sitter spacetime. However, this cof initial states has a Gaussian wave functional and therethe argument that the choice of a nonvacuum initial stwould involve exponential fine tuning does not apply to tcase considered here.
Recently, a model with a small number ofe folds hasbeen constructed in Ref.@29#. This kind of models naturallyarises in the context of supersymetric~SUSY! and supergrav-ity ~SUGRA! inflation. They are particulary well suited t
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MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
the model put forward in this article. They consist in multipbursts of inflation which in total last for'75 expansiontimes. In addition, the last stage of inflation is precededother inflationary epochs. The ‘‘initial state’’ of this last epoch is the result of the evolution of the ‘‘true initial statethrough the multiple preceding bursts of inflation. Clearthere is no reason for assuming this ‘‘effective initial statto be the vacuum. Let us emphasize that this argument hfor every model with many stages of inflation since, in thcase, the origin of the characteristic scale could no longethe Planck length stretched tol C but could correspond to thtime where one of the fields starts rolling down.
Our model possesses a privileged scale and thereforelongs to the class of models already envisaged in Ref.@26#.However, we would like to emphasize that the origin of thscale is physically completely different and we will point othat there exist observables, which, in principle, allow usdistinguish between the different models. A last commentthe fine tuning issue is in order here. It is true that the potion of the characteristic scale must be chosen carefully. Oerwise the model would simply be in contradiction with tavailable data. We would like to emphasize that this is nofeature of our model only but in fact of all the BSI~brokenscale invariant! models @26#. In this respect our model issimilar to the other BSI models.
This paper is organized as follows. In Sec. II we discunonvacuum initial state for the cosmological perturbatioWe first briefly describe the theory of perturbationsquantum-mechanical origin. We then describe the noncuum initial states considered in this article. We finally cculate the power spectra of the Bardeen potential for thstates and show that it possesses either a step or a bumSec. III we examine the observational consequences ofcalculated power spectra; we compare our theoretical pretions with current experimental and observational dawhich will fix the parameters of our model. We end with thconclusions given in Sec. IV.
II. NONVACUUM INITIAL STATEFOR THE PERTURBATIONS
A. Perturbations of quantum-mechanical origin
The background model is described by a FriedmaLemaıtre-Robertson-Walker~FLRW! metric whose spacelike sections are flat (c51): ds25a2(h)(2dh21dx2).We assume that inflation is driven by a single scalar fiew0(h). It is convenient to define the background quantitH(h) andg(h) by:
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NONVACUUM INITIAL STATES FOR COSMOLOGICAL . . . PHYSICAL REVIEW D 61 083518
w~h,x!51
a~h!
1
~2p!3/2E dkx~h,k!eik•x, ~9!
where we have renormalized the time-dependent amplitwith the scale factor. The Fourier component satisfiesx(h,2k)5x* (h,k), because the field is real. The action, givby
S5E d4xL ~10!
51
2E d4xa2F @w8~h,x!#22(i
@] iw~h,x!#2G ,~11!
can also be written in terms of the Fourier componx(h,k). The result reads
S5E dhER31
dkL ~12!
5E dhER31
dkH ux8~h,k!u21S a82
a22k2D ux~h,k!u2
2a8
a@x8~h,k!x* ~h,k!1x8* ~h,k!x~h,k!#J . ~13!
The variation of the action leads to the equation of motfor the equation componentx91x@k22a9/a#50. Again, wefind a parametric oscillator-type equation. Of course, if this no expansion, or if the Universe is in the radiatiodominated era, it reduces to the usual equation of motioan harmonic oscillator.
We now turn to the Hamiltonian formalism. The first steis the calculation of the momentum conjugate tow(h,x) de-fined by
p~h,x![]L
]@w8~h,x!#5a2w8~h,x!. ~14!
p(h,x) can be expressed in terms of the momentum congate tox(h,k),
p~h,k![]L
]@x8* ~h,k!#5x8~h,k!2
a8
ax~h,k!, ~15!
through the relation
p~h,x!5a~h!
~2p!3/2E dkp~h,k!eik•x. ~16!
As a preparation to the quantization, the normal variaa(h,k) is introduced. Its definition is given by
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MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
This equation can be solved by a Bogoliubov transformatiexpressed as
ck~h!5uk~h!ck~h0!1vk~h!c2k† ~h0!, ~24!
ck†~h!5uk* ~h!ck
†~h0!1vk* ~h!c2k~h0!,~25!
where the functionsuk(h) and vk(h) only depend on thenorm of the vectork. These functions are such thatuuku2
2uvku251, so that the commutation relation between tcreation and annihilation operators is preserved in time.time h0 must be thought of as the time where the initconditions are fixed. Whatever these last ones are, we huk(h0)51 and vk(h0)50. The differential equations thaallow the determination ofuk(h) andvk(h) are
iuk85kuk1 ia8
avk* , ivk85kvk1 i
a8
auk* . ~26!
If we introduce the Bogoliubov transformation given by Eq~24! and~25! in the expression of the field operator, Eq.~21!,we obtain
w~h,x!5A\
a~h!
1
~2p!3/2E dk
A2k
3@ck~h0!~uk1vk* !~h!eik•x
1ck†~h0!~uk* 1vk!~h!e2 ik•x#. ~27!
From Eq. ~26!, it is easy to see that the function (uk
1vk* )(h) satisfies the same equation asx(h,k). In the high-frequency regime, the terma9/a becomes negligible and whave limk→1`(uk1vk* )(h)5e2 ik(h2h0). This means that, in
this regime, the operatorx(h,k) is given by
limk→1`
x~h,k!5A\Fck~h0!e2 ik(h2h0)
A2k1c2k
† ~h0!eik(h2h0)
A2kG .
~28!
We can now come back to our initial problem, which cosists in finding the correct normalization of the perturbscalar field and of the scalar perturbations. We can identhe Fourier component operator of the perturbed fiw1(h,k), with x(h,k)/a(h), both operators being considered in the high-frequency regime. Let us emphasize agthat this identification is valid only in this regime. Otherwisthe field w1(h,k) does not behave as the free fieldw(h,x)and the time dependence of the modes is no longer givethe function (uk1vk* )(h). The normalization of the perturbed scalar field fixes automatically the normalizationthe scalar perturbations of the metric since they are linthrough Einstein’s equations. We only need this link in thigh-frequency regime. It can be expressed as~see Refs.@30,31#!
limk→1`
m~h,k!52A2ka~h! limk→1`
w1~h,k!, ~29!
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wherek[8pG. From the last expression and Eq.~28!, weimmediately deduce that
limk→1`
m~h,k!
524Ap l PlFck~h0!e2 ik(h2h0)
A2k1c2k
† ~h0!eik(h2h0)
A2kG ,
~30!
where l Pl5(G\)1/2 is the Planck length. As announced, thnormalization of the scalar perturbations is now completdetermined. In general, the operatorm(h,k) will be given by
m~h,k!524Ap l Pl@ck~h0!jk~h!1c2k† ~h0!jk* ~h!#.
~31!
The function jk(h) is the solution of Eq.~8! such thatlimk→1`jk5e2 ik(h2h0)/A2k. If we introduce the functionf k(h) defined by
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NONVACUUM INITIAL STATES FOR COSMOLOGICAL . . . PHYSICAL REVIEW D 61 083518
where Hb11/2(2) is the second-kind Hankel function of orde
b11/2. It is straightforward to deduce the correspondequation forf k(h):
f k~h!52pA2HAg
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~36!
The previous expression forf k(h) guarantees that the fiel
F(h,x) possesses the correct behavior in the high-frequeregime. Roughly speaking,
limk→1`
f k~h!;e2 ikh/A2k,
with the correct amplitude. This should be the case of afield, regardless of the initial conditions one may choose
B. Quantum states
The formulation of quantum field theory used in the pvious subsection was written in the Heisenberg picture.far, we have only calculated the time dependence ofBardeen operator. In order to describe completely the stem, one must in addition specify in which quantum statefield F(h,x) is placed. As we already mentioned, it is usally found in the literature that the initial state of the pertubations is taken to be the vacuum. Here we address thepothesis that the perturbations are initially in a nonvacustate. Our choice of nonvacuum states is guided by thethat one must introduce a scale in the theory. We denotecorresponding wave number byk0. We examine three differ-ent nonvacuum states.
Let D be a domain in the momentum space characteriby the numbersk0 ands, such thatk0 is the privileged scaleands the dispersion around it. Concretely, we takeD as thespace between the spheres of radiusk02s andk01s, i.e., ashell of width 2s in k space. The domainD is invariantunder rotations and therefore is compatible with the assution of isotropy of the Universe. Our first state is given b
uC1~k0 ,s,n!&[ )kPD(k0 ,s)
~ck†!n
An!u0k& ^
p¹D(k0 ,s)u0p&,
~37!
5 ^
kPD(k0 ,s)unk& ^
p¹D(k0 ,s)u0p&. ~38!
The stateunk& is an n-particle state satisfying, ath5h0 :ckunk&5Anu(n21)k& andck
†unk&5An11u(n11)k&.More complicated states can be constructed by consi
ing quantum superpositions ofuC1(k0 ,s,n)&. We will con-sider the following state:
uC2~k0 ,n!&[E dsg~s!uC1~k0 ,s,n!&, ~39!
where,a priori, g(s) is an arbitrary function ofs. It is clearfrom the definition of the stateuC1& that the transition be-
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tween the empty and the filled modes is sharp. Physicathis is probably not very realistic. The functiong(s) will bechosen in order to ‘‘smooth out’’ the quantum stateuC1&.Also it should be clear that the writing ofuC2& in Eq. ~39! issymbolic. An accurate definition of this state requires to tainto account subtleties, which will be considered whencalculate the spectrum in the next section.
Finally a third state can be defined according to
uC3~k0!&[ (n50
`
h~n!uC2~k0 ,n!&. ~40!
The functionh(n) is arbitrary. As demonstrated below, thstate will allow us to work with an effective number oquanta, which will no longer be an integer. This state seeto be the most natural rotational-invariant, smooth, quantstate that privileges a scale. Typically, the quantum sgiven in Eq.~40! depends onk0 and on the free parametercharacterizing the functionsg(s) and h(n). Their valueswill be determined later by confronting our theoretical prdictions versus experimental and observational data. Ouris now to calculate the power spectra of the Bardeen potial operator in the three statesuC i&, i 51,2,3.
C. Power spectra
From now on, for convenience, we consider that the stem is in a box whose edges have lengthL. As a conse-quence, the wave vector possesses discrete compongiven by ki5@(2p)/L#mi , where mi is an integer. TheBardeen operator can be written as
F~h,x!5l Pl
L3/2 (k
@ck~h0! f k~h!eik•x
1ck†~h0! f k* ~h!e2 ik•x#. ~41!
We pass from the discrete representation to the continuone by sendingL to infinity and by applying the rule1/(2p)3*dk→1/L3(k . It is clear that the final result doenot depend onL.
The power spectrum ofF(h,x) in the stateuC&, denotedby PF(k;uC&), is defined through the calculation of the twopoint correlation functionK2(r ;uC&). In the continuouslimit,
K2~r ;uC&)[^CuF~h,x!F~h,x1r !uC&
^CuC&
5E0
`dk
k
sinkr
krk3PF~k;uC&). ~42!
In this definition, we have taken into account the fact thatstateuC& is not automatically normalized to 1. The powspectrum is a time-dependent function but in the lonwavelength limit this dependence disappears. In order toform the computation of the correlation function for the stauC1&, one needs the following quantities:
8-7
f
thi-n
ve
onm
tolo
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.
ins
at
of
lt
,
er,
MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
^C1~k0 ,s,n!ucpcquC1~k0 ,s,n!&
5^C1~k0 ,s,n!ucp†cq
†uC1~k0 ,s,n!&50, ~43!
^C1~k0 ,s,n!ucpcq†uC1~k0 ,s,n!&5nd~qPD!dpq1dpq ,
~44!
^C1~k0 ,s,n!ucp†cquC1~k0 ,s,n!&5nd~qPD!dpq .
~45!
In these formulas,d(qPD) is a function that is equal to 1 iqPD and 0 otherwise.
As a warm up, we calculate the power spectrum forstateuC1& with n50, i.e., for the vacuum. Using the prevous equations in the definition of the correlation functioEq. ~42!, one finds
K2~r ;u0&)5l Pl2
L3 (k
u f ku2e2 ik•r, ~46!
which in the continuous limitL→1` goes to
l Pl2
~2p!3E d3kW u f ku2e2 ik•r. ~47!
After having performed the angular integrations, we recothe power spectrumPF(k;u0&):
k3PF~k;u0&)5l Pl2
2p2k3u f ku2. ~48!
Let us turn to the calculation ofK2(r ;uC1&. It can be ex-pressed as
K2~r ;uC1&)5l Pl2
L3 (k
~ u f ku2e2 ik•r@112nd~kPD!# !.
~49!
From this equation and from the definition of the functid(kPD), we deduce the expression of the power spectru
k3PF~k;uC1&)5l Pl2
2p2k3u f ku2$112n@H~k2k01s
2H~k2k02s!#%, ~50!
whereH is a Heaviside function. We see that, in additionthe usual vacuum spectrum, there is a new contributioncated around the wave numberk0. This new contributionvanishes ifn50, as expected.
This spectrum is not continuous. As already mentionthis is not physically very realistic. It has for origin the vecrude definition of the stateuC1&. We thus turn to the caswhere the quantum state is given byuC2&. This refinementwill allow us to obtain a smooth and physical spectrum.
Since the system is placed in a box, the stateuC2& can bedefined by a discrete sum according to
08351
e
,
r
-
,
uC2~k0 ,n!&[(i 50
N
gi uC1~k0 ,s i ,n!&, ~51!
wheregi and s i , i 50, . . . ,N, are just series of numbersWe choose thes i ’s such that
^C1~k0 ,s i ,n!uC1~k0 ,s j ,n!&5d i j . ~52!
This is satisfied if the number of modes in the domaD(k0 ,s i) and D(k0 ,s j ), N(Di) and N(Dj ), respectively,are such thatN(Di)2N(Dj )>1. This condition boils downto s i2s j>p2/@L3(k0
21s i2)#, where we have assumed th
s i2s j!1. Therefore, we can always find a value ofL suchthat the condition be fulfilled. Then, the calculation
^C2uF(h,x)F(h,x1r )uC2& can be performed. The resureads
^C2uF~h,x!F~h,x1r !uC2&
5l Pl2
L3 (k
u f ku2e2 ik•rH F(i 50
N
ugi u2G12nF(
i 50
N
ugi u2d„kPD~k0 ,s i !…G J . ~53!
Our aim is to calculateG(k)[( i 50N ugi u2d(kPDi) @for con-
venienceD(k0 ,s i) is denoted byDi ]. By symmetry,G(k02k8)5G(k01k8), so that we will consider the casek[k01k8, k8>0. In this sumk8 is fixed. As a consequencethere exists an integeri 0 such that ifi , i 0 , d(kPDi)50 andif i> i 0, then d(kPDi)51, or, equivalenty, s i 0
,k8
<s i 011. This means that the sum( i 50N ugi u2d(kPDi) is in
fact equal to( i 5 i 0Nugi u2. We choose thes i ’s and the coef-
ficientsgi according to
s i[ iXmax
N, ugi u2[2
Xmax
N3
dF
dxUx5s i
, ~54!
whereF is any decreasing function such thatF(Xmax)50.Then we have
(i 50
N
ugi u2d~kPDi !52Xmax
N (i 5 i 0
N
F8S iXmax
N D . ~55!
The last step is to sendN to infinity. This means that weconsider a continuous series of intervalsDi . We obtain
limN→1`
(i 50
N
ugi u2d~kPDi !52Ek8
XmaxF8~x!dx5F~k8!,
~56!
by definition of the Riemann integral. In the same mann( i 50
N ugi u25F(0). In what follows, we take
G~k!5F~k8
tl
i
ibthte
i
to
s
ein
ca
ibletoethe
lled
r is
r-
ing
enle
on
theh
tion
he
iscil-orre-es.on-nd,m
ela-e-e
tly
autthels
NONVACUUM INITIAL STATES FOR COSMOLOGICAL . . . PHYSICAL REVIEW D 61 083518
whereS is a free parameter. This function does not exacsatisfy the assumptions made previously, but it is easyshow that the final result is free of these limitations and isfact valid for any functionF. It is clear that other functionsare possible, but only the approximate shape of the distrtion is important and the Gaussian is the prototype offunction we have in mind. The spectrum is obtained afhaving introduced the previous result in Eq.~53! and havingtaken the limitL→1`. We obtain
k3PF~k;uC2&)5l Pl2
2p2k3u f ku2~112ne2(k2k0)2/S2
!.
~58!
In this equation, it is clear thatn is an integer. We now showthat this condition can be relaxed if the system is placedthe stateuC3&.
To calculate the spectrum for this state, it is sufficientnotice that ^C2(k0 ,n)uC2(k0 ,m)&5dmn . Using this for-mula, straightforward calculations lead to
k3PF~k;uC3&)5l Pl2
2p2k3u f ku2~112neffe
2(k2k0)2/S2!,
~59!
where the effective number of quantaneff is given by
neff5
(n50
`
nuh~n!u2
(n50
`
uh~n!u2. ~60!
An attractive choice for the functionh(n) is obviouslyh(n)[e2bn @this b has of course nothing to do with thebdefined in Eq.~34!#. In this case,neff is given by
neff5e22b
12e22b. ~61!
The spectra of Eqs.~58! and~59! are the main results of thisection. Clearly, they possess a peak around the scalek0. Theposition of the peak is controlled by the value ofk0, its widthby S and its height byn or neff ~in fact byb in the last case!.
We will need the primordial spectrum only for largwavelengths. In the case of power law inflation, everythcan be calculated exactly. In this limit, we have
k3PF~k;u0&)5ASknS21, ~62!
with
AS5l Pl2
l 02
g~11b!2
22b14cos2~bp!G2~b15/2!, nS52b15.
~63!
The above expression is strictly speaking not applicablethe case of a de Sitter universe, since then there are no smetric perturbations and the functiong(h) turns out to be
08351
yton
u-er
n
g
inlar
zero. The generation of density perturbations is only possafter the transition from the exponential inflationary erathe radiation-dominated Universe. If during inflation thUniverse was very close to the de Sitter space-time, thenspectrum of density perturbations today is the so-caHarrison-Zel’dovich spectrum (nS51). All expressions de-rived in this section are still valid forb&22. The initialpower spectrum in the case where the Bardeen operatoplaced in the stateuC2& can be written as
k3PF~k;uC2&)5ASknS21~112ne2(k2k0)2/S2
!. ~64!
If the state isuC3&, we just have to replace the integern withthe real numberneff . Let us note that if, instead of consideing intervals of the form@k02s,k01s#, one considers in-tervals such as ]0,k01s] or @k02s,`@ , which still privilegea scale, it is possible to build steplike spectra, the step belocated at the scalek0.
Recently in the literature, BSI spectra have also bestudied, see Ref.@26#. In these articles, the privileged scaarises as a privileged energy in the inflaton potential~moreprecisely, a discontinuity, or a rapid variation, in the inflatpotential if first derivatives are present!. We would like tostress that, in our case, the different physical origin ofprivileged scale would in principle allow us to distinguisthe different models. Indeed, in Ref.@26#, the fluctuations areGaussian. In the case studied here, the three-point correlafunction still vanishes
K dT
T~e1!
dT
T~e2!
dT
T~e3!L 50, ~65!
but the four-point correlation function no longer satisfies trelation
K S dT
T~e! D 4L 53F K S dT
T~e! D 2L G2
, ~66!
which is typical of Gaussian statistics. The reason for thisclear. The ground-state wave function of an harmonic oslator is a Gaussian and, as a consequence, the CMBR clation functions for the vacuum exhibit Gaussian propertiOn the other hand, the wave function of a state with a nvanishing number of quanta is no longer a Gaussian acorrespondingly, the correlation functions deviate froGaussianity. Therefore, a measure of the four-point corrtion function ~as well as any higher-order even-point corrlation function! would permit to distinguish between thclass of models presented here and the models of Ref.@26#.If it turns out that the type of non-Gaussianity apparendetected recently@14–16# is really present in the CMBRmap, then these two classes of BSI models~as well as stan-dard inflation! are ruled out, because they both predictvanishing three-point correlation function. But if it turns othat some non-Gaussianity is present in the CMBR atlevel of the four-point correlation function then the modepresented here could account for this.
8-9
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e
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otherin
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hestithr-ave
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ts-by
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MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
III. COMPARISON WITH OBSERVATIONS
The aim of this section is to confront the power specgiven by Eq.~64! with observations. We will not use anaccurate statistical methods to find the best values of theparametersk0 , S, andn/neff , because we just want to obtacrude constraints. For this purpose we will use observatiof the CMBR anisotropies and of the matter power spectru
We choose to work with the following cosmological prameters: the Hubble parameter ish50.5, the baryonicmatter-density parameter isVb50.05, the density parameteV0[Vc1Vb1VL is equal to 1 (Vc and VL are, respec-tively, the CDM and theL-density parameters!, there is nosignificant reionization and the spectral index isnS51, whenthere is no contribution from the gravitational waves. Letemphasize again that in this case, since it corresponds toSitter phase, the Eq.~63! giving the normalization of thespectrum is strictly speaking not applicable. However,expressions derived in the previous section can be appfor nS&1. We will later discuss the case of small deviatiofrom a scale-invariant~Harrison-Zel’dovich! spectrum, in-cluding the contribution of gravitational waves in the CMBanisotropies. We will consider two different values for tcosmological constant density parameterVL[L/(3H0
2) andthe sum of baryon-matter density parameter and CDM dsity parameterVm[Vb1Vc , that is VL50, Vc50.95~hereafter denoted SCDM!, and VL50.6, Vc50.35 ~here-after denotedLCDM). We point out that we have not assumed any biasing in the galaxy distribution with respecthe underlying mass fluctuations~the bias parameter is equto 1!.
The spectrum must be normalized, i.e., the value ofASmust be determined. For this purpose, we use the valuQrms-PS5T0(5C2/4p)1/2;18 mK (T052.7 K) measuredby the COBE satellite. We use the valueQrms-PS;18 mKbecause we have assumed that the spectrum is scale inant. In the large-wavelength approximation, we havedT/T;(1/3)F. In addition the transfer function for the Bardeepotential can be taken equal to 1~with an appropriate nor-malization!. As a consequence the multipole can be writtas
Cl54p
9 E0
1`dk
k@ j l@k~h02hLSS!#
2AS~nS!knS21
3~112ne2(k2k0)2/S2!#, ~67!
wherej l is a spherical Bessel function of orderl, andh0 andhLSS denote, respectively, the conformal times now andthe last scattering surface. Let us remark that theAS in thelast expression is not exactly theAS in Eqs. ~62! and ~63!.Since the difference is not important for our purpose,have kept the same notation. The previous expression caevaluated explicitly. For the quadrupole, the result reads
C254p
9AS~nS!~h02hLSS!
12nS
3S p
242nS
G@32nS#G@21~nS21!/2#
G2@~42nS!/2#G@42~nS21!/2#12nI D ,
~68!
08351
a
ee
s.
sde
lled
n-
o
of
ari-
n
t
ebe
where
I[E0
1` du
u22ns@ j 2~u!#2e2(u2u0)2/U2
, ~69!
with u0[k0(h02hLSS) andU[S(h02hLSS). In what fol-lows, we takeh02hLSS51. The integral will be evaluatednumerically for different values of the free parameters. Wjust have to specialize the last equation to a scale-invarspectrum to obtain the following value forAS:
AS5108
5T02
Qrms-PS2 1
1124nI5
9.4310210
1124nI. ~70!
In terms of the band powerdTl defined bydTl[T0@ l ( l11)Cl /2p#1/2, we finddT25A12/5Qrms-PS527.9 mK.
We must choose the three parametersk0 , n, andS. Re-cently, it has been emphasized by many authors@12# that thepower spectrum seems to contain large amplitude featurethe scalel C'100h21 Mpc, which corresponds to a wavnumber equal to 0.062h Mpc21. No other value for a privi-leged scale has been detected so far, and therefore anychoice would either lie in an unobservable range, or beconflict with the data available at present. Consequently,choose
k050.062h Mpc2150.031 Mpc21, ~71!
with our value of the Hubble constant. Let us turn to tchoice of the varianceS. We have seen that the simplenonvacuum initial states can lead to a power spectrum weither a bump or a step. In this article, we will restrict ouselves to the study of the bump case. Steplike spectra halready been studied in Ref.@26# and our conclusions wouldbe similar. Therefore we will consider~rather arbitrarily, butthe conclusion does not depend on the exact value ofS, aslong as it is not too large!:
S50.3k050.0186h Mpc21. ~72!
From now on, we will always take these two values fork0and S in any of the plots shown. In Fig. 1 we display thinitial power spectrum for a few values ofneff . The differ-ence betweenneff50 andneffÞ0 is obvious.
In the case considered here, the integralI is equal toI (S50.3k0)'1.331026. It is completely negligible andwill be taken equal to zero. This arises from the fact thatquadrupole is mainly fed by very large wavelengths~of theorder of today’s Hubble radius!, whereas the bump occurs amuch smaller wavelengths~of the order of the Hubble radiuat the time of decoupling!. Thus, the calculation of the quadrupole, and therefore the normalization, is not modifiedthe presence of the bump.
Let us discuss the matter power spectrum. The pospectrum can either be obtained by the Boltzmann codeveloped by one of us~A.R.! or by means of analytical fits. Inthis case, the baryon power spectrum is given by
8-10
bei-
-
-m
eng
um
e
ld
noyInini-
. Asher
ereeming
. Letec-rumig.
d
eenthe
ATn as
the
NONVACUUM INITIAL STATES FOR COSMOLOGICAL . . . PHYSICAL REVIEW D 61 083518
drb
rb[ud~k!u25AT2~k!
g2~V0!
g2~Vm!k@112ne2(k2k0)2/S2
#,
~73!
where the different terms in this equation are explainedlow; T(k) is the transfer function, which can be approxmated by the following numerical fit@33#:
T~k!5ln~112.34q!
2.34q
[email protected]~16.1q!21~5.46q!31~6.71q!4#21/4,
~74!
with q[k/@(hG)Mpc21# whereG is the so-called shape parameter, which can be written as@34#
G[Vmhe2Vb2Vb /Vm. ~75!
The functiong(V) takes into account the modification induced in the power spectrum by the presence of a coslogical constant. Its expression can be written as@35#
g~V
ou
nd
-l
y
tn
itatlscnnathierd
s
ns
be
lt
ith
l-etheibleE
ons,uct
e
en
MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
As a conclusion of this rapid analysis, we stress thatmodel is much more constrained if one imposesneff to be aninteger instead of a real number. Moreover, our model teto favor a moderate value ofneff as well as a low value of thecosmological constant if the data of Einastoet al. are con-firmed, or a low value ofneff and a high value of the cosmological constant~i.e., the currently popular cosmologicamodel, with vacuum initial state! in the other case. It is easto notice from Eq.~60!, that sinceneff is quite small,h(n) ispeaked around small values ofn, and therefore the allowedwindow for the effective number of quanta is constrainedbe around small values. In conclusion, the initial state fouis not too far from the vacuum.
B. Scalar and tensor modes
One should also consider the contribution of the gravtional waves in the CMBR anisotropies. The data currenavailable are in fact the sum of the scalar plus the tencontributions to the CMBR anisotropies. We recall that sinthere are two modes of polarization for the CMBR photoand that one of them is only generated by gravitatiowaves, it is in principle possible to distinguish betweenscalar and tensor contributions to the CMBR anisotropsee Ref.@42#. In what follows, we consider some standainflationary predictions for gravitational waves: we takenS
FIG. 3. Power spectrum for the SCDM model, withneff rangingfrom 0 to 2 with step of 0.5~from the bottom to the top!. Diamondsrepresent the APM data, squares the velocities field measuremand crosses the data by Einastoet al.
FIG. 4. Same as Fig. 2, but for theLCDM model, with VL
50.6.
08351
r
s
od
-yoresl
es,
50.9, nT5nS21'20.1 ~the last equation being rigorouin the case of power-law inflation only!, and the ratio ofscalar to tensor amplitudeC2
T/C2S'27nT . In Fig. 6, we de-
compose CMBR anisotropies, showing the contributiofrom scalar and tensor modes separately.
In any model, the gravitational waves contribution canimportant only for multipolesl &100, while it is negligible atsmaller angular scales~roughly speaking, the gravitationawaves contribution is two orders of magnitude smaller al'300 than at the quadrupole!. The effect of gravitationalwaves is therefore to boost power on large angular scales~or,equivalently, to lower the height of the acoustic peaks wrespect to the height of the lowl plateau!. The fact that oneobserves an excess of power on small angular scales~withSaskatoon data!, favors a low contribution from gravitationawaves~which is in agreement with most inflationary models!. In our model, the possibility to have a bump in thinitial power spectrum enables us to boost the height ofacoustic peaks, and therefore to have some non-negligcontribution from gravitational waves: normalizing at COBdata, one imposes the value ofAS1AT instead ofAS. As aresult, the scalar perturbations amplitudeAS is smaller. Sincethe first acoustic peak depends only on scalar perturbatiwe must keep the same value as before for the prodAS(112n exp@2(k02kpeak)
2/S2#), which permits a highervalue ofneff (kpeak is the characteristic wave number of thfirst Doppler peak!.
ts,
FIG. 5. Same as Fig. 3, but for theLCDM model.
FIG. 6. CMBR anisotropies decomposition, showing scalar~dot-ted line! and tensor~dashed line! contributions. The total contribu-tion is given by the solid line.
8-12
thth
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o-
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on
i
onae
n-fullwerth
ba-byter-tes,he
antlcu-esewiththes ofs,eak,t-ink is
i-the
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NONVACUUM INITIAL STATES FOR COSMOLOGICAL . . . PHYSICAL REVIEW D 61 083518
In Figs. 7 and 8 we show the CMBR anisotropies andmatter power spectrum for the SCDM model, including boscalar and tensor contributions. Comparing these figuresFigs. 2 and 3, it can be concluded that if both scalar atensor modes are included in the calculation of the multipmomentsCl ’s, then a higher number of quanta (.4) is re-quired as expected.
Finally, in Figs. 9 and 10 we show the CMBR anisotrpies and the matter power spectrum for theLCDM modelincluding both scalar and tensor contributions. The saconclusions as for the SCDM model hold, but we note agthat, as for the case without gravitational waves, mapower spectrum data favor a higher value ofneff than CMBRanisotropies data. We emphasize that when the gravitatiwaves contribution is not negligible, the standard caseneff50 is excluded, and that extra power in the initial statenecessary.
IV. CONCLUSIONS
In this paper we address the question of whether nvacuum initial states for cosmological perturbations arelowed, or whether they are ruled out on the basis of pres
FIG. 7. CMBR anisotropies for the SCDM model, withneff
ranging from 0 to 4 with a step of 1~from the bottom to the top!.Both scalar and tensor contributions are included. Diamonds resent COBE data, squares the Saskatoon data, and crosses thedata.
FIG. 8. Power spectrum for the SCDM model, withneff rangingfrom 0 to 4 with steps of 1. Diamonds represent the APM dasquares the velocities field measurements, and crosses thegiven by Einastoet al.
08351
e
ithde
einr
al
s
-l-nt
experimental and observational data.The choice of the initial quantum state in which the qua
tum fields are placed should be made on the basis ofquantum gravity. Since this theory is at present unknown,believe, as we discussed in the Introduction, that it is wostudying nonvacuum initial states for cosmological perturtions. Our choice of a nonvacuum initial state is guidedthe idea that the initial state could have a built-in characistic scale. We examined three different nonvacuum stawhich are compatible with the assumption of isotropy of tUniverse. Of particular interest is our choice of stateuC3&,which seems to be the most natural rotational-invarismooth quantum state, which privileges a scale. We calated the power spectra of the Bardeen potential for ththree states and compared their theoretical predictionscurrent experimental and observational data, namelyCMBR anisotropy measurements and the redshift surveythe distribution of galaxies. With our choice of initial statethe power spectra of the Bardeen potential possess a paround the wave number, that corresponds to the builcharacteristic scale of our model. The height of the peacontrolled by the number of quantan of the initial state andits width by another free parameter of our model. If the intial state is a quantum superposition then the height ofpeak is controlled by the numberneff , which does not needto be an integer.
The angular power spectrum of CMBR anisotropies fo
e-AT
,ata
FIG. 9. Same as Fig. 7, but for theLCDM model and withneff
ranging from 0 to 4 with a step of 1~from the bottom to the top!.
FIG. 10. Same as Fig. 8, but for theLCDM model and withneff
ranging from 0 to 4 with a step of 1~from the bottom to the top!.
8-13
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MARTIN, RIAZUELO, AND SAKELLARIADOU PHYSICAL REVIEW D 61 083518
model with vanishing cosmological constant, tells us thatcharacteristics of the first acoustic peak, as revealed bySaskatoon experiment, are compatible with the caseneff51.In the presence of a cosmological constant, CMBR anisropy measurements are in agreement withneff50 or neff51, depending on the value of the cosmological parametThe observational data for the matter power spectra, as gby Einastoet al., favor higher values of the number ofneff ~2or 3!, whatever the value of the cosmological constant.
The most realistic case is the one for which the sumscalar and tensor modes contributions is included. Consiing standard inflationary predictions for gravitational wavwe find that CMBR anisotropies measurements requirhigher value ofneff ~3 or 4! for both types of models, withand without a cosmological constant, than in the case oabsence of tensor modes contribution. This is in agreemwith the matter power spectra. The analysis of the redssurveys by Einastoet al. leads to matter power spectra thfavor higher values ofneff , once tensor contributions are alsincluded. The interpretation of these results for the stauC2& and uC3& leads to the conclusion that sincen andneffcannot be higher than a few, these states must be close tvacuum.
tt
J-’
ys
08351
ehe
t-
s.en
fr-,a
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ift
s
the
In conclusion, if the initial state of the cosmological peturbations is not the vacuum but, instead, has a built-in chacteristic scale, then generic predictions of the model arhigh amplitude of the first acoustic peak, a nontrivial featuin the matter power spectrum, and deviations from Gauanity in the CMBR map. It is too early to say whether thresults of the Saskatoon experiment~see also Ref.@43#!, aswell as the analysis performed recently by Einastoet al., arefirst steps in this direction. More data are needed and fuexperiments will be important in determining whether tclass of models proposed here provides an explanation wallows a better description of the observations than the sdard paradigm of slow roll inflation plus cold dark matter
ACKNOWLEDGMENTS
It is a pleasure to thank Robert Brandenberger for useexchanges of comments. Discussions with Nathalie DerueRuth Durrer, Alejandro Gangui, and David Langlois are aacknowledged. We would also like to thank Volker Mu¨ller,who provided us with the cluster data.
-a-
ys.
ot.
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