Large-scale anisotropies of the cosmic background radiation in generalized inflationary cosmologies

Volume 166B, n u m b e r 1

PHYSICS LETTERS

2 January 1986

LARGE-SCALE A N I S O T R O P I E S OF T H E C O S M I C BACKGROUND RADIATION IN GENERALIZED INFLATIONARY C O S M O L O G I E S R FABBRI Sezlone dz Fzstca Supertore, Dtpartlmento dt Ftstca, Unwerstta ,:It Fwenze, Vta S Marta 3, 1-50139 Florence, ltal)

F LUCCHIN Dlparttmento dl Ftslca "G Gahlet", Via Marzolo 8 1-35100 Padua, Italy

and S MATARRESE lnternattonal School for Advanced Studtes (ISAS), Strada Costtera 11, 1-34014 Trwste, Italy Received 28 September 1985

We calculate the dipole and quadrupole moments, ( A T / T ) I l = 1, 2, of the cosmic background radmtlon amsotropy due to scalar and tensor waves w~th power-law spectra, produced by q u a n t u m processes during a "generalized" inflationary era of the early universe The observational bounds on ( A T / T ) I are used to give constraints on the parameters of the cosmological models

One of the most Important by-products of the mflattonary models of the early umverse [ 1] xs the prechct~on of a scale-mvarlant, constant curvature spectrum of density perturbattons [2] and grawtaUonal waves [3] According to present ideas, the above (Zel'dovlch [4]) density fluctuation spectrum may prowde, ff with a suitable strength, a quite saUsfactory picture of galaxy formation [5] The requirement that large wavelength density perturbatlons and grawtatlonal waves should not dasturb the observed lsotropy of the cosmxc background rachat.ton [6] allows settmg lmasts on the inflation parameters ,1 However ~t has become clear recently that a successful mflaUon does not need a penod of ("standard") exponenUal expansion of the universe [ 14,15] Simple models of generalized mflaUon have been studied where the cosimc expansxon proceeds faster ("super,1 For density perturbataons see refs [ 7 - 1 1 ], for gravttaUonal waves, refs. [10,12,13]

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mflaUon") or slower ("sub-mflaUon") than exponentially Inflalaonary models of this kind [16], prowded the expansxon occurs as driven by an "effectave" equation of state pip --- w = const (p = pressure, p = energy density, w < - 1 / 3 in order to have acceleration), also prechct the generation of both density perturbations and grawtatlonal waves with power-law (but generally not constant curvature) spectra These spectra are gausslan, bemg generated by quantum fluctuahons of an almost free field It is worth recaUmg that smular spectra have been widely discussed m the framework of the galaxy formataon theory (see, e g, ref [17]), the emphasis on the Zel'dovlch spectrum being due, nowadays, partly to the widely accepted theorelacal prejuchce that the mflatton can only be of standard type Abbott and Wise [14] have recently analysed the large-scale anlsotroples of the cosimc background rachatlon due to the grawtatxonal waves generated m power-law inflation (a kind of sub-mflataonary model), thereby obtmnmg constramts on the reheatang temperature, Trh, at the end of the mflattonary phase 49

Volume 166B, number 1

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The purpose of thas letter is to perform a more general analysis of the contnbuUons to the dipole and quadrupole amsotroples of the background radiataon arising from both scalar and tensor waves with powerlaw spectra Taking into account the observed lunlts on the large-scale amsotroples, we deduce constramts on Trh for both sub-mflattonary and super-mflataonary models In the generalmed inflationary models winch we consider, neglectmg the curvature term, as it is evidently self-consistent, the cosrmc scale-factor turns out to be [16]

a(t)

~a(t*)[1 + ~(1 + w)H(t*)(t-

t*)] 2/3(1+w),

(1)

where t* is an arbitrary tune a n d H = a/a is the Hubble parameter One can distmguL~h m (1) two cases 0) - 1 / 3 > w > - 1 , which rapidly gwes a power-law mflatlon Tlus kind of expansion can be found m many models, e g the wall-dominated inflation [14,18] and some reduced-gravity models [19], (u) - 1 > w, which gives the "pole-inflation" ,2, a kind of super-mflataon that occurs m many Kaluza-Klem cosmologles [20] By following the general method g/yen m ref [14], one finds that the variance AT of the gravltons, produced dunng the mflatlonary era, is gwen by

AT(k,

tHC) ~

(21r2)-l/2mp1H(tl)

(2)

(mp is the Planck mass) In (2) tHC is the tune when the scale corresponding to the comovmg wave-number k, which left the horizon at tl during the accelerated phase, reenters the Hubble radius. It is easy to work out [16] that, for the models descnbed by (1), the spectral dependence atAT is

A T =AOT(k/21r) ~ ,

(3)

2 January 1986

responds to the mode crossing the honzon at the present time to The discussion concemmg the scalar vanance A S at the horizon is more revolved, the amplitude depending on the physics underlying mflalaon In the power-law inflation analyzed m ref [15] It was found that the gauge-mvanant density perturbations amplitude Is gwen by

As(k,

tHC) ~

(2rr)-l bmp l(H2/ IHIl/2)t I

(6)

(b = 4 or 2/5 if tHC < teq or tHC > teq, teq being the equivalence tune) Eq (6) lmphes t h a t A s stncfly follows the AT spectral dependence of eq (3), with a correspondmgA0s given by A0S =AoTb(3[1 + wl) -1/2

(7)

It seems quite natural to ascribe some generahty to eq (6), so that (6) and (7) can be used m the pole-mflatton case (w < - 1 ) too [16] It is however worth noticing that, even if the above formulae do not have a complete generality, a power-law dependence of AS will typically be a reasonable approxamataon m the range of scales of cosmological mterest As we shall discuss below, some results of the present work will indeed be valid m general Although the previous formulae do not work for w = - 1 (standard inflation), our numerical results for such a case have been obtained with an mterpolatlon of the w ~ - 1 cases Next we need to consider the temporal evolutaon of the waves after horizon crossing The waves of interest for the large-scale nucrowave background amsotroples reentered the Hubble radius at t > teq, during tim era, as is well known, the density perturbataon 8 and the gravitataonal-wave field h~, m a fiat umverse, are respectavely

where A 0T ~ 2rtl/21 0-86 (l+w)/(1+3w)

× (Trh/mp)-

(1-3w)/(1+3w),

(4)

8 (x, n) = f d a k Us(k) exp(Uk • x) ~(k) CS(r/),

(8)

hj(x, n) = f d 3 k

(9)

UT(k) exp(uk

x)~](k)¢T(n),

and = 3(1 + w)/(1 + 3 w ) ,

(5)

a suitable matching of the inflationary and Fnedmann eras has been taken rote account In eq (3) the wavenumber was normahzed m such a way that k = 2~r cor,2 We are indebted to Kel-ichl Maeda for this terminology 50

where r/is the conformal tune related to the proper tune t by dr/= dt/a(t), the functions ~bS(7/) = */2/10

(10)

and CT(r/) = (4//¢r/)(d/dk'r/) [sm(/c0)//cr/]

(11)

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allow for the time evolutmn of 5 and h~ [21] ,~(k) is a stochastic variable satasfymg (,~(k) ~ * ( k ' ) ) = 83(k - k ' ) ,

(12)

and (~] (k)is a random polarlzataon tensor defined as m ref [12] In eqs (8) and (9) the spectral functaons Us(k ) and UT(k) are easily found to be related t o A s and AT at the horizon by, respectwely,

2 January 1986

108l

sl

,.,q

T $2

U2(k) = --¢~Sk A2tkst , tHC) = ~ TrA2s(k/2rr) 2a+1, (13) U2 (k) = k-3A2 (k, tHC ) = (2rr)-3A2oT(k/2rO 2a'--3 (14) Once the perturbatmn spectra are specified, one can calculate the expected large-scale amsotropy of the cosrmc rachatlon More precisely, for comparison with experiment, it is useful to gwe the RMS contnbulaons o f h a r m o m c s of order l (see, e g , ref [14])

(zxr/r)2

l

_= 1

=lm~=_l(lalm 12>

(15)

In general the calculalaon of a/2 reqmres solving a transfer equatmn for the cosmic rachaUon [22] However, lmutmg ourselves to wavelengths larger than the last scattering interval 0 e k < kmax ~ 4007r, correspondmg to about 60 Mpc now, with a Hubble constant H 0 = 100 Km/s Mpc) the amsotroples are smaply given by the Sachs-Wolfe effect [23], and we can write a2 = (41r)2(21 + 1)

fJ, k-2U2(1Sl)2 ,

(16)

and a 2 = (47r)2(21 + 1)(1 + 2)(l + 1)l(l - 1)

x

fdk k2

(17)

for scalar and tensor waves respectively The coefficmnts IKl for both lands of waves were calculated m ref [24] (see also refs [8,14,25]) In the integrals m eqs (16) and (17) we should Impose an upper cutoff at k = kmax, the contnbutaons to the amsotropy of larger wave-numbers being somewhat suppressed by effects connected with the last scattering interval [26] However, our calculations should be considered quite accurate only when the integrals are practacally independent of kmax, otherwise

-

06

2

08

~

10

12

14 Iw[

Fig. 1 Normahzed values ofa~/4zr for/= 1, 2, $1 and $2 label the dipole and quadrupole scalar waves and T the quadrupole curve for tensor waves. they only prowde rough estunates We also note that, for certain values o f / a n d w, we may have "infrared" chvergences, so that we should impose a lower cutoff at k = kmm Since (16) and (17) are qmte accurate on the long-wave side of the spectrum down to k = 0, we need a cutoff m the perturbaUon spectrum, st is reasonable that such a cutoff does exist because perturbalaons on scales larger than the Hubble rachus at the onset of mflatlon (whxch are now much larger than the horizon) can be thought as a contnbutaon to the local cosmological background [27] The only theoretical reqmrement, however, is kmm ,~ 2rr In fig 1 we report normahzed values o f ( A T / T ) 2 for l ~< 2 and w E ( - 0 6, - 1 5) the former bound anses smce for smaller values of lw[ the mflataonary models are so httle accelerated to be uneffectlve [15], the latter one xs due to the fact that, as we shall see below, models wath larger values of I wl are unconstrained m the context of flus paper The quadrupole curves for both scalar and tensor waves are independent of kmax and kmm m the considered range as tested by us numerically (the plotted curves of figs 1 and 2 refer to kmm = 27r/10) On the other hand, the &pole curve depends only on the upper cutoff for Iwl > 2/3 In such a range it xs easy to see that eq (16) approxunately gwes loga 1 ~ 2[(2 + 3w)/(1 + 3w)] log kmax + const

(18)

We wish to stress that the results of fig 1 remain vahd also when the mflataonary model gives a density perturbatmn spectrum not related to the gravatataonal 51

Volume 166B, number 1 i

~

i

PHYSICS LETTERS i

f j

r

i

j.?;~-

J

$1

~ JJJ

2 January 1986

13], since however our formula (7) for A S formally diverges for w = - 1 , we cannot venfy Lyth's statement (see ref [10]) that the effect of gravitataonal waves Is less Important For the pole-mflataon models the constraints on Trh are less stringent than m power-law inflation In particular for Iwl/> 1.2 n o 1Lmlt IS set by amsotropy bounds

References

O5

o'6 -~7

0'8

o'9

1'o

1'1

1'2 I,,,I

Fig 2 The maxtmum allowed value of the reheating temperatare Trh as a function of the w parameter We mdlcate separately the hmlts obtamed from the chpole (S1), and from the quadrupole due to scalar ($2) and tensor (T) waves

one as m (6) and (7), m such a case the w parameter just defines the spectral Index a through eq (5) Moreover the results hold for any power-law gausslan perturbation spectra, m a fiat unwerse, whatever their origin In fig 2 we gwe the m a x i m u m allowed value of the reheatmg temperature as a function of the w parameter The constraints are obtained assurmng the experimental upper bounds [6] a l < 2 × 10 - 3 a n d a 2 < 10 - 4 we take the experimental dipole amphtude as an upper lmut for Its ongln is probably local [28] For low values of w (low acceleration models) the bounds derived from the dipole and quadrupole due to scalar waves and from the quadrupole due to tensor waves practically coincide. For more accelerated models, the dipole constraint Is more stringent roughly by one order of magrutude, tins being mainly due to the expression o f A 0 s denved m ref. [16] Moreover, a higher value of kmax would gwe a more stringent dipole constraint our assumed value of kmax is sufficiently conservatwe and does n o t reqmre the particular dark matter model considered by Abbott and Wise [9] The tensor wave curve m the power-law mflatlon case is m satisfactory agreement with the results [14] (within the uncertamty on the interpretation of the expenmental bounds) Such an agreement also holds for gravltons produced m standard inflation [10,12, 52

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[18] D Seckel, Fermdab Rep 84/92-A (1984) [19] B L Spokomy, Phys Lett. 147B (1984) 39, F S Accetta, D J Zoller and M S Turner, Phys Rev. D31 (1985) 3046, M D Pollock, Phys LetL 156B (1985) 301, F Lucchm, S Matarrese and M.D Pollock, Inflation vath a non-mmtrnally coupled scalar field, ISAS preprmt 49/85/EP (1985) [20] P G O Freund, Nucl Plays B209 (1982) 146, E Alvarez and M Belen-Gavela, Phys Rev Lett 51 (1983) 931, D Sahdev, Phys Lett 137B (1984)155, R.B Abbott, S M Barr and S D Ellis, Phys. Rev D30 (1984) 720, D31 (1985)673, E W Kolb, D Lmdley and D Seckel, Phys. Rev D30 (1984) 1205, Chen Shl and Yu Yunqlang, ISAS preprmt 56/85/A (1985), Q Shaft and C Wettench, Phys Lett. 152B (1985) 51 [21] E M. Lffshltz and I.M Khalalaukov, Adv Phys 12 (1963) 185

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[22] P.J E Peebles and J T Yu, Astrophys J 162 (1970) 815, M L Wilson and J Silk, Astrophys J 243 (1981) 14, S.A Bonometto, A Caldara and F Lucehm, Astrophys J 126 (1983) 377 [23] R K Sachs and A.M Wolfe, Astrophys. J. 147 (1967) 73 [24] R Fabbn, I Guldl and V Natale, Astron. Astrophys 122 (1983) 151, IL Fabbn, Astron. Astrophys. 135 (1983) 225 [25] R Fabbn, VI Cony Naz Rela~vtt~ generale e fislca delia gravltazlone (Ftrenze, October 1984), to be pubhshed [26] R A Sunyaev and Ya B Zel'dovlch, Astrophys Space Scl (1970) 20 [27] R.H Brandenberger, Nucl Phys B245 (1984) 328, S Y P~, Nucl Phys B252 (1985) 127, A,H Guth and S Y Pl, preprlnt CPT#1246 (1985) [28] L Hart and R D Davis, Nature 297 (1982) 191

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