where M denotes a tyical energy of the seed and is a random variable. We assume that ensemble averages and and satial averaging coincide, the usual er

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1 Cosmic Microwave Background Anisotroies from Scaling Seeds: Generic Proerties of the Correlation Functions R. Durrer and M. Kunz Deartement de Physique Theorique, Universite de Geneve, 24 quai Ernest Ansermet, CH-2 Geneve 4, Switzerland In this work we resent a artially new method to analyze uctuations which are induced by causal scaling seeds. We show that the ower sectra due to this kind of seed erturbations are determined by ve analytic functions, which we determine numerically for a secial examle. We ut forward the view that, even if recent work disfavors the models with cosmic strings and global O(4) texture, causal scaling seed erturbations merit a more thorough and general analysis, which we initiate in this aer. I. INTRODUCTION At resent, two ideas to exlain the origin of large scale structure in the universe are rimarily investigated: Perturbations in the energy density of matter may have emerged from quantum uctuations, which, during a hase of inationary exansion, are stretched beyond the Hubble scale and 'freeze in' as classical uctuations in the energy density of the cosmic uid []. These small initial uctuations then evolve according to linear cosmological erturbation theory. A hase transition in the early universe may lead to toological defects which seed uctuations in the cosmic uid [2]. In this case, the uid uctuations vanish initially and evolve according to inhomogeneous cosmic erturbation equations. The stress energy of the toological defects lays the role of the source term. In order for the gravitational eld of the defects to be suciently strong to seed cosmic structure, we have to require = 4G 2 0 5, where denotes the energy scale of the hase transition. This yields 0 6 GeV, a tyical GUT scale. Toological defects which neither over-close the universe nor die out are either cosmic strings or global defects. The anisotroies in the cosmic microwave background (CMB) rovide an imortant tool to discriminate between dierent models. On very large angular scales both classes of models lead to a Harrison Zel'dovich sectrum of uctuations. For inationary models this can easily be derived analytically. For defect models, the sectra were originally found numerically [3{5]. Analytical arguments for this behavior are given in [6]. On intermediate scales, inationary models redict a series of eaks due to acoustic oscillations in the baryon/hoton uid rior to recombination [7]. Present observations seem to conrm these eaks even though the error bars are still considerable [8]. Recently, several investigations led to the conclusion that cosmic strings [9,0] and global O(N) defects [{3] do not reroduce the acoustic eaks indicated by resent data. This led [2] and [0] to the conclusion that models of cosmic structure formation with scaling causal defects are ruled out. However, in a very simle arameterization of two families of more general scaling causal seed models, we were able to t resent data very well [3]. We are thus convinced that it is too early yet to comletely abandon scaling causal seeds as a mechanism for structure formation. In contrary, we think it is extremely useful to study them in full generality, ignoring in a rst ste the hysical origin of the seeds. This urely henomenological oint of view is analogous to inationary models where one sometimes manufactures the inationary otential to yield the required sectrum of initial erturbations. To determine the ower sectrum of the radiation and matter erturbations induced by seeds, we need to know the two oint correlation functions of the seed energy momentum tensor. In this aer we resent a simle arameterization of these functions and oint out an error frequently made in the literature. We then exemlify our ndings with the large N limit of global scalar elds [4,5]. For simlicity, we work in a satially at Friedman universe. The metric is thus given by ds 2 = a(t) 2 (dt 2 ij dx i dx j ) ; where t denotes conformal time. II. CORRELATION FUNCTIONS OF CAUSAL SCALING SEEDS We dene seeds to be any non-uniformly distributed form of energy, which contributes only a small fraction to the total energy density of the universe and which interacts with the cosmic uid only gravitationally. We arameterize the energy momentum tensor of the seed by T (seed) = M 2 ; ()

2 where M denotes a tyical energy of the seed and is a random variable. We assume that ensemble averages and and satial averaging coincide, the usual ergodicity hyothesis. Furthermore, we assume the random rocess to be satially homogeneous and isotroic, so that the two oint correlation function Z h (x;t) (x + y;t 0 )i= (x;t) (x + y;t 0 )d 3 xc (y;t;t 0 ) (2) V is a function of the dierence y only. Due to the exansion of the universe, which breaks time translation symmetry, we however exect C to deend on t and t 0 and not just on the dierence t t 0. We consider causal seeds. Causality requires C (y;t)=0; if jyj >t+t 0 : (3) The two oint function in osition sace thus has comact suort which imlies that its Fourier transform is analytic. We dene a seed to be scaling, if the Fourier transform, bc (k;t;t 0 )=hb (k;t)b (k;t 0 )i ; (4) contains no dimensional arameter other than t; t 0 and k. This imlies that the Ricci curvature induced by the seeds is a function of k and t only, multilied by the dimensionless arameter = 4GM 2. Since we dene Fourier transforms with the normalization Z ^f(k) = d 3 xf(x) ex(ikx) ; (5) V V Z f(x) = d 3 k (2) ^f(k) ex( ikx) ; (6) 3 and since (x;t) has the dimension of (length) 2, b C has the dimension of an inverse length. From scaling we therefore conclude that for urely dimensional reasons, we can write the correlations functions in the form ^C (k;t;t 0 )= tt 0 F ( tt 0 k;t 0 =t) ; (7) where F is a dimensionless function of the four variables z i t 0 tk i and r t 0 =t, which is analytic in z i. We also require the seed to decay inside the horizon, which imlies We neglect the transition from a radiation to a matter dominated universe, which actually breaks scaling, since the scale t eq, the time of equal matter and radiation density enters the roblem. In numerical examles, we have found that this transition in general leads to somewhat dierent decay laws for the correlation functions at large kt, but it will not alter our main conclusions ^C (k;t;t 0 )! kt! = ^C (k;t;t 0 )! kt 0! =0: (8) Furthermore, since the seeds interact with the cosmic uid only gravitationally, satises covariant energy momentum conservation, ; =0: (9) With the hel of these four equations, we can, for examle, exress the temoral comonents, 0 in terms of the satial ones, ij. The seed correlations are thus fully determined by the satial correlation functions b Cijlm. Statistical isotroy, scaling and symmetry in i; j and l; m as well as under the transformation i; j; k; t! l; m; k; t 0 require the following form for the satial correlation functions: bc ijlm (k;t;t 0 )= tt 0 [z iz j z l z m F (z 2 ;r)+ (z i z l jm + z i z m jl + z j z l im + z j z m il )F 2 (z 2 ;r)+ z i z j lm F 3 (z 2 ;r)=r + z l z m ij F 3 (z 2 ; =r)r + + ij lm F 4 (z 2 ;r)+( il jm + im jl )F 5 (z 2 ;r)] ; (0) where the functions F a are analytical functions of z 2 tt 0 k 2, and for a 6= 3 they are invariant under the transformation r! =r, F a (z 2 ;r)=f a (z 2 ;=r). The ositivity of the ower sectra ^Cijij (k;t;t) = hj ij j 2 i leads to a series of ositivity conditions for the functions F a : 0 F 5 (z 2 ; ) ; () 0 F 4 (z 2 ; )+2F 5 (z 2 ;) ; (2) 0 z 2 F 2 (z 2 ; ) + F 5 (z 2 ; ) ; (3) 0 z 4 F (z 2 ; )+4z 2 F 2 (z 2 ;) + 3F 5 (z 2 ; ) ; (4) 0 z 4 F (z 2 ; )+2z 2 (F 3 (z 2 ;)+2F 2 (z 2 ;)) +F 4 (z 2 ; )+2F 5 (z 2 ;) : (5) Since ^Cijlm is the Fourier transform of a real function, ^C ijlm (k;t;t 0 ) = ^Cijlm ( k;t;t 0 ) ; (6) and thus the ansatz (0) imlies that the functions F a (z 2 ;r) are real. Furthermore, decay inside the horizon (condition (8)) yields lim F a(z 2 ;r)= lim F a(z 2 ;r)=0: (7) z 2 r! z 2 =r! In addition, analyticity imlies that the functions F a do not diverge in the limit z! 0, thus with lim F a(z; r) =A a (r) z!0 A a (r) =A a (=r) for all a 6= 3: 2

3 As an examle, we haveworked out the functions F to F 5 in the large N limit of global scalar eld seeds. A discussion of this simle model of scaling causal seeds ands its relation to the texture model of structure formation can be found in [4] and [5]. In Figs. and 2 we resent the functions F 5 (z 2 ;r) and F 2 (z 2 ;r). FIG. 2. The same as Fig. for the function F 2(z; r). FIG.. The function F 5(z; r), linear (to) and jf 5(z; r)j logarithmic are shown. Because of their small amlitude, the oscillations are virtually invisible in the linear lot. To show the symmetry r! =r, the r-axis is chosen logarithmic in both lots. The symmetry under the transition r! =r is clearly visible. Also the conditions that F a! 0 if either z! or r! 0 or r! which follows from Eq. (7) is evidently satised. For xed z the functions oscillate with a frequency which grows with z. Since the amlitude decays raidly, these oscillations are only visible in the log-lots. The correlations always decay like ower laws, never exonentially. The equal time correlation functions, F (z 2 ; ) to F 5 (z 2 ; ) are lotted in Fig. 3. In Fig. 4 we show A a (r). All the functions, excet F 5 which is constrained by Eq. () go through 0 (For F the assage through 0 is not visible since it occurs only at z ' 30). We have also veried the ositivity constraints Eq. (2) to Eq. (5). The asymtotic behavior of the functions can be obtained analytically. The same is true for the functions A to A 5. As argued above, all the functions A i excet A 3 are symmetric under r! =r. The techniques emloyed to calculate the functions F i are analogous to those exlained in [5]. III. SCALAR, VECTOR AND TENSOR DECOMPOSITION AND CMB ANISOTROPIES The energy momentum tensor of seeds is often slit into scalar, vector and tensor erturbations, since the time evolution of each of these comonents is indeendent. Furthermore, due to statistical isotroy, the scalar, vector and tensor modes are uncorrelated. A suitable arameterization of this decomosition is ij = ij f (k i k j k 2 3 ij)f + 2 (w ik j + w j k i )+ ij ; (8) where f ; f are arbitrary random functions of k; w is a transverse vector, w k = 0; and ij is a symmet- 3

4 FIG. 3. The functions jf i(z; )j are shown. The zeros are visible as sikes in the log-lot. (Further below, at z 30, also F asses through zero.) FIG. 4. The functions ja i(r)j = jf i(0;r)j are shown. As discussed in the text, all of them excet A 3 are symmetrical under the transition r! =r. ric, traceless, transverse tensor, i i = ij k j = 0. The variables f: ;w and ij reresent the scalar, vector and tensor degrees of freedom of ij. The functions F to F 5 determine the correlations: To work out the correlation functions we use P j i f = 3 i i = 3 ij ij ; (9) f = w i = 2 k ( l i ^k m ij =(P il P jm 3 2k (^k i^kj 2 3 ij) ij = 3 2k S ij ij ; (20) 2 ^ki^kl^km ) lm = V lm i lm ; (2) (=2)P ij P lm )P ma P lb ab = T ab ij ab ; (22) where P ij = ij ^ki^kj ; ^ki = k i =k (23) is the rojection oerator onto the sace orthogonal to k and S ij, Vi lm and Tij ab are the rojection oerators to the scalar vector and tensor arts of ij. Using these identities and our ansatz (0), one easily veries hf (t)w i (t 0 )i = hf (t)w i (t 0 )i =0 (24) hf (t)ij(t 0 )i = hf i(t)ij(t 0 )i =0 (25) hw i (t) jl (t 0 )i =0 (26) hf (t)f (t 0 )i = 3 tt 0 [2F 5(z 2 ;r)+3f 4 (z 2 ;r) +z 2 (F 3 (z 2 ;r)=r + F 3 (z 2 ; =r)r)+ 4 3 z2 F 2 (z 2 ;r)+ 3 z4 F (z 2 ;r)] (27) hf (t)f(t 0 )i = tt 0 k [3F z 2 F 2 +z 4 F ] (28) hf (t)f (t 0 )i = tt 0 [ 3 z2 F (z 2 ;r)+ 4 3 F 2(z 2 ;r)+rf 3 (z 2 ; =r)] (29) hw i (t)w j (t 0 )i = 4 k 4 tt 0 [F 5 + z 2 F 2 ](k 2 ij k i k j ) (30) h ij (t)lm(t 0 )i = tt 0 F 5[ il jm + im jl ij lm + k 2 ( ij k l k m + lm k i k j il k j k m im k l k j jl k i k m jm k l k i )+ k 4 k i k j k l k m ]: (3) It is interesting to note that although Cijlm b is analytic, the correlation functions of the scalar vector and tensor comonents, in general, are not. The reason for that is that the rojection oerators S; V and T are not analytic. This is imortant. It imlies, e.g., that the anisotroic stresses in general have a white noise and not a k 4 sectrum as erroneously concluded in [6{8]. The scalar anisotroic stress otential thus diverges on large scales, hjf j 2 i/=(tk 4 ) for kt. A result which we also have obtained in the large-n limit and in numerical simulations of O(N) models. The ower sectrum of the scalar anisotroic stress otential f is analytic if and only if vector and tensor erturbations are absent, F 5 = F 2 =0. In the generic situation, F 5 (z =0;r =)=A 5 () 6= 0. Wethus exect the following relation between scalar vector and tensor erturbations of the gravitational eld on suer-horizon scales, x kt : (The equations for the scalar, vector and tensor gravitational otentials in terms of f:, w and ij can be found in [9] and [20].) 4

5 hj j 2 i 22 tk 4 A 5() (32) hj i j 2 i 62 t k A 5() 2 (33) hjh ij j 2 i4 2 t 3 A 5 () ; (34) where and are the Bardeen otentials, is the vector contribution to the shear of the t =const. slices and h ij are tensor erturbations of the metric. If it would be solely suer horizon erturbations which induce the large scale CMB anisotroies, this could be translated into a ratio between the scalar, vector and tensor contributions to the C`'s on large scales, ` < 50. However, since the main contribution to the CMB anisotroies is induced at horizon crossing, x = (see below) the above relations cannot be translated directly and we can just learn that one exects, in general, contributions of the same order of magnitude from scalar, vector and tensor erturbations. Finally, we want to discuss in some detail the CMB anisotroies induced from scalar erturbations. In this case, the gravitational erturbation equations (see e.g. [9,20]) imly + = 2f : (35) Esecially, if has a white noise sectrum due to 'comensation' [2], this leads to a k 4 sectrum for and for the combination which enters in Eq. (36). This nding is in contradiction with [6,7], which redict a white noise sectrum for, but it is not in con- ict with the Harrison Zel'dovich sectrum of CMB uctuations which has been obtained numerically in [3{5]. This can be seen by the following simle argument: Since toological defects decay inside the horizon, the Bardeen otentials on sub-horizon scales are dominated by the contribution from dark matter and thus roughly constant. The integrated Sachs Wolfe term then contributes only u to horizon scales. Therefore, using the fact that for defect models D g and V are much smaller than the Bardeen otentials on suer-horizon scales (see [2]), we obtain (T=T)`(k)j SW ( )(k; x dec )j`(x 0 x dec ) + Z x dec ( 0 0 )(k; x)j`(x 0 x)dx ; (36) where x = kt and rime stands for derivative w.r.t. x. The lower boundary of the integrated term roughly cancels the ordinary Sachs Wolfe contribution and the uer boundary leads, to hj(t=t)`(k)j 2 ij SW 2 k [3F 5() + 4F 3 2 () + F 3 ()]j 2` (x 0 ) ; (37) a Harrison-Zel'dovich sectrum. The main ingredients for this result are the decay of the sources inside the horizon as well as scaling, the rest follows for urely dimensional reasons. IV. CONCLUSIONS In this aer, we outline a rocedure to investigate causal scaling seed erturbations. We show, that the large number of seed correlations, which determine fully the induced ower sectra of dark matter and CMB hotons, can be cast in only ve analytic functions with certain well dened roerties. We schematically estimate the large scale CMB anisotroies induced. However, we are convinced that the relative amlitudes of large scale CMB anisotroies and the acoustic eaks as well as the dark matter ower sectrum deend on details of the model and thus scaling causal seeds cannot be ruled out from studies of secic models. This nding is also con- rmed in [3]. Our work is just the beginning of a rogram to be carried out which goes in dierent directions and to which we invite researchers in the eld to articiate. Some of the questions which should be exlored are the following: Are there further general restrictions for the correlation functions which have not been mentioned here? Given the functions F to F 5 what is the exact exression for the C`'s and the dark matter ower sectrum? What are good aroximations? Are there simle conditions which the functions F to F 5 have to satisfy in order to lead to ower sectra which are in agreement with data. Are there hysically lausible causal scaling seed models other than toological defects? Acknowledgment This work is artially suorted by the Swiss NSF. Numerical simulations have been erformed at the Centro Svizzero di Calcolo Scientico (CSCS). It is a leasure to thank P. Ferreira, M. Sakellariadou and N. Deruelle for stimulating discussions. [] J.M. Bardeen, P.J. Steinhardt and M.S. Turner Phys. Rev. D28, 679 (983). [2] T.W.B. Kibble, J. Phys. A9, 387 (976). [3] U. Pen, D. Sergel and N. Turok, Phys. Rev. D49, 692 (994). [4] R. Durrer and Z. Zhou, Phys. Rev. D53, 5394 (996). [5] B. Allen at al., \Large Angular Scale CMB Anisotroy Induced by Cosmic Strings", astro-h/ (996). 5

6 [6] R. Durrer, in Proceedings of the Worksho on Toological Defects in Cosmology, astro-h/ (997). [7] W. Hu, N. Sugiyama and J. Silk Nature 386, 37 (995). [8] M. Tegmark and A. Hamilton, astro-h/ and refs. therein. [9] B. Allen, R.R. Caldwell, S. Dodelson, L. Knox E.P.S. Shellard and A. Stebbins, rerint archived under astroh/ (997). [0] A. Albrecht, R.A. Battye and J. Robinson, \The case against scaling defect models of cosmic structure formation", rerint archived under astro-h/ (997). [] R. Durrer, A. Gangui and M. Sakellariadou, Phys. Rev. Lett. 76, 579 (996). [2] U. Pen, U. Seljak and N. Turok, \Power Sectra in Global Defect Theories of Cosmic Structure Formation", astroh/ (997). [3] R. Durrer, M. Kunz, C. Lineweaver and M. Sakellariadou, rerint archived under astro-h/ (997). [4] N. Turok and D. Sergel Phys. Rev. Lett. 66, 3093 (99). [5] M. Kunz and R. Durrer, Phys. Rev. D55, 456 (997). [6] J. Magueijo, A. Albrecht, P. Ferreira and D. Coulson Phys. Rev. D54, 3727 (996). [7] W. Hu. D.N. Sergel and M. White, Phys. Rev. D55, 3288 (997). [8] N. Deruelle, D. Langlois and J.P. Uzan, \Cosmological Perturbations seeded by Toological Defects", rerint archived under gr-qc/ (997). [9] R. Durrer, Phys. Rev. D42, 2533 (990). [20] R. Durrer, Fund. of Cosmic Physics 5, 209 (994). [2] R. Durrer and M. Sakellariadou, Phys. Rev. D56, 4480 (997). 6