Astron. Astrophys. 9, 79 7 (997) ASTRONOMY AND ASTROPHYSICS The anisotropy of the cosic background radiation due to ass clustering in a closed universe Daing Chen, Guoxuan Song, and Yougen Shen, Shanghai Observatory, Acadeia Sinica, Shanghai,, China China Center of Advanced Science and Technology (World Laboratory), P. O. Box 87, Beijing 8, China Received June 99 / Accepted Septeber 99 Abstract. We consider the effects on the Cosic Background Radiation (CBR) of a single, spherically syetric structure using the Tolan-Bondi solutions in a closed universe. The exact frequency shift of a photon passing through the center of a lup which is located between the last scattering surface and an observer is calculated. An approxiate expression of the frequency shift ν to the first order of δ(x) (the density contrast) is given. We find that there are several redshifts of the perturbing lup the net frequency shift is zero. With increasing Ω, the first positions of zero frequency shift ove toward lower redshift; the larger the value of Ω, the further they ove. At the vicinity of the last scattering surface, the universe is close to flat, the frequency shift will always be the sae as that for a flat universe. Key words: cosic icrowave background cosology: theory large-scale structure of the universe. Introduction Since the COBE satellite discovered the anisotropy of the Cosic Background Radiation (CBR) (Soot et al. 99; Bennett et al. 99; Wright et al. 99), a new era has been open in cosology. As a atter of fact, soon after the discovery of the CBR, it was noted that the CBR teperature distortion would provide a sensitive probe of large-scale density inhoogeneities in the universe(sachs & Wolfe,97;Rees & Sciaa,98). Large-scale density perturbations in the universe will induce CBR anisotropy; the relationship between the anisotropy and the perturbation depends on how the inhoogeneity is produced. However, there has been a renewed interest in studying anisotropy of the CBR produced by the late tie evolution of density inhoogeneities (Martinez-González & Sanz 99; Send offprint requests to: Daing Chen Supported by the National Natural Science Foundation of China and the Young Astronoer Laboratory of Shanghai Observatory Martinez-González, Sanz, & Silk 99, 99; Panek 99; Fang & Wu 99; Meśzarós 99; Meśzarós-González,Sanz, & silk 99; Hu & Sugiyaa 99). Most of the studies considered the effects on the CBR of a single, spherically syetric structure using a Swiss cheese odel, thin-shell approxiation, or the Tolan-Bondi solution (ore e.g., Rees & Sciaa 98; Dyer 97; Kaiser 98; Thopson &Vishniac 987; Dyer & Ip 988; Nottale 98). If the universe is hoogeneous at last scattering, then gravitational field perturbations ust subsequently grow fro zero to their present value in order to account for the observed structure. This tie-varying gravitational field will induce a CBR anisotropy which ay not be very sall copared with the anisotropy produced in the priordial instability condition(jaffe, Stebbins & Friean, 99; Fang & Wu,99). On the other hand, the CBR anisotropy depends priarily on the geoetry of the universe, which in a atter- doinated universe is deterined by Ω, and on the optical depth to the surface of last scattering. Therefore, the CBR anisotropy ay also provide inforation on the value of Ω (Kaionkowski, Spergel & Sugiyaa, 99). One of the ost robust predictions of inflationary cosology is that the universe after inflation becoes extreely flat, which corresponds to Ω =. However, it is also possible to construct inflationary odels with Ω > (Linde,995). Fro observational cosology we know that the density paraeter of the universe has not been finally deterined, the accepted range is believed to be... Recently, White and Scott (99) considered structure foration and CBR anisotropy in a closed universe, both with and without cosological constant. Fang and Wu (99) studied in detail the CBR anisotropy due to spherical clustering located between the last scattering surface and the observer. They considered only the flat universe case(ω = ). In this paper, we shall investigate the anisotropy due to collapsing spherical clustering located fro the last scattering surface to the observer in a closed universe (Ω > ). Like Kaionkowski et al., we are interested in the effect of Ω on CBR anisotropy. Although it is difficult to quantitatively deterine the value of Ω with this ethod at present tie, it can provide iportant inforation. Furtherore, if the CBR anisotropies
7 D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe on all angular scales are apped out with great precision, the ethod presented here can at least provide reference values of Ω to be checked by other ways. In Sect., we calculate in detail the linearized result of the CBR teperature distortion. The analytical solution is seeingly coplex but actually clear. In Sect., the nuerical result are given, and we discuss the results.. Teperature distortion As shown by Fang and Wu (99)(FW, hereafter), the etric for a spherically syetric overdense perturbation in an expanding universe can be generally written as: ds = e λ(x,t) dx + r (x, t)(dθ + sin θdφ ) dt. () Writing r = S(x, t)x () and S(x, t i )= () one can obtain the dynaical equation of S(x, t)as Ṡ H i ρ(x, t i ) ρ ci S = H i [ ρ(x, t i) ρ ci ] () H i =(Ṡ/S) t=ti and ρ ci =Hi /8πare the Hubble paraeter and critical density of the universe at t = t i (the decoupling epoch; for convenience, we assue in this paper that the decoupling epoch is at the sae tie when the last scattering takes place.), respectively. r ρ(x, t i )= ρr dr r (5) r dr is the ean density within the shell of radius x. As did FW, we consider an initial over-density perturbation region which is located at the origin(x = ) and at t = t i : { ρ(x, ti )=ρ i ( + δ ), x x ρ(x, t i )=ρ i, x > x () then ρ(x, t i )=ρ i [ + δ(x)] (7) Eq.() can be rewritten as Ṡ = H i { [ + δ(x)]( S )+ } = ρi ρ ci is the density paraeter of the universe at t = t i. The dynaical solution of Eq.(9) is [ + δ(x)] t + t (x) = (η sin η) () H i [ ( + δ(x)) ] [ + δ(x)] S = [ + δ(x)] cos η the integral constant t (x)is t (x) [ + δ(x)] = t i H i [ ( + δ(x)) ] (η / i sin η i ) When t t i, δ(x) =, one finds (9) () t = (η sin η) () H i ( ) S (t) = cos η () Setting η = η i when t = t i in Eqs.(), () and (), noting that S(t i ) =, we obtain cos η i = () sin η i = Ωi (5) t i H i = [η ( ) i Ωi ] () In order to obtain nuerical results of the CBR teperature distortion, one has to find a linear approxiation solution of S(x, t). We set S(x, t) =S (t)+δ(x)s (t) (7) Fro Eqs.(9) and (7) one has ds dη + sin η S = [ tan(η ) ( ) tan η η sin ] (8) The solution of Eq.(8) is δ(x) = { δ, x x δ (x /x), x > x (8) S =[ ( ) ]( η / tan η )
D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe 7 + ( ) (η sin η)/ tan η + C/tan η (9) or C = {[ ( ) ] ( η i / tan η i ) tan η i + ( ) (η i sin η i )} () Considering the photons which pass through the center of the collapsing lup, the photon trajectory equation in the zeroorder approxiation is dx dt = ±e λ(x,t)/ = ± x S (t) Hi ( ) () the signs inus and plus describe a photon oving toward and going away fro the center of the lup, respectively. The solution of Eq.() can be read as H i Ωi x = ± sin(η η ) () η is the paraeter corresponding to a photon that is passing through the center of a lup. The solution of the zero coponent of the null geodesic equation gives (FW), dλ dη = ds dη () S Fro Eqs.() and (), one obtains the position of last scattering surface: ν i η ds = exp[ η i dη dη] S = S (η ) S (η i ) = cos η cos η i =+z. (7) Here z = νi ν is the redshift of a photon eitted fro the last scattering surface, and is the observed frequency of a photon in the unperturbed case. If a photon passes through the center of a perturbing lup and is observed with the frequency, there is an extra frequency shift copared with photons in the unperturbed case. So we rewrite Eq.(7) as +z= ν i. (8) Eqs.() and (8) give the frequency shift k exp( λe λ/ dx) dλ(η) = exp[ dη] () dη thus the cooving observed frequency at tie t (the present tie) is related to the frequency ν i eitted at the last scattering surface by ν η i = exp[ η i dλ(η) dη] () dη the paraeter η is deterined by the present tie t. The solution of Einstein equation gives = η +z exp[ dλ dη] (9) η i dη In the first order approxiation, by using Eqs.(7) and (9), one can find fro Eq.(5) that dλ dη = sin η cos η + cos η [xδ (x)+δ(x)] ( ds dη sin η cos η S ).() One thus has the expression of frequency shift to the first order of δ(x): λ = ṙ r = xṡ + Ṡ xs + S the prie denotes derivative with respect to x. In the case δ(x) =, λ=ṡ, S (5) = η η i [xδ (x)+δ(x)] cos η ( ds dη sin η cos η )dη.()
7 D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe. Nuerical results Fro Eqs.(8),(9), () and (), one can obtain an analytical result for the frequency shift, in the linearized approxiation. However, there is a ter ln( + )dη, = tan η η, = tan η, for which no analytical solution is available. One thus needs an expansion in series for the expression, which has different fors in three different regions. The results can be written as: = A (η c+,η ) A (η,η ) +A (η i,η ) A (η c+,η ), (for η < sin [X cw Ωi ]) A (η c,η ) A (η i,η )+A (η c+,η ) A (η,η ) A (η,η )+A (η,η )+A (η c,η ) A (η c+,η ), () (for sin [X cw Ωi ] <η < η ) A (η c+,η ) A (η,η )+A (η c,η ) A (η i,η ) +A (η c,η ) A (η c+,η ), η (for <η <η ) η = cos [ +z ( ) ] () +z Ω i η c = η sin [X c W Ωi ] () η c+ = η + sin [X c W Ωi ] (5) η = cos [ ( + z) ( ) ] () W = ( ) / [cos ( ) Ωi ] (7) X c = x H i W (8) The functions A,A and A are given in Appendix A. The paraeters η c and η c+ denote, respectively, the ties when the photons are just entering and leaving the collapsing lup. It is not difficult to show that the relationship between the teperature distortion and the frequency shift is: T T = ν. T T = ν versus the different locations z for =.(Ω =.) is given in Fig.. Fro this plot we can see that for relatively sall sizes of the perturbing lup there several positions z of the lup is located, for which the frequency shift is zero. When the position of the perturbing lup goes through one of these positions, the frequency shift changes its sign. Hereafter, these positions are called zero positions. For coparison, the frequency shift is derived also for a flat universe in Appendix B. It turns out that the teperature distortions as a function of the location of the clusterings are qualitatively siilar to those in Fig.. Fig.. CBR teperature distortion versus the location of a spherical clustering in an alost flat universe, =., corresponding to Ω =.. X c =(H i/w )x is a diensionless quantity representing the size of the perturber, H i is the initial Hubble constant, W is given in Eq.(7) depending on and x is the size of the perturber. For a sall size of the perturber, there are several locations zero CBR teperature distortion will arise. In this plot only positive teperature distortions are indicated. By the way, it can easily find the difference between Fig. in this paper and Fig. in FW, the sae odel as in this paper is adopted and the frequency shift ν to the first order is given by Eq.() in FW. In Eq.() of FW, three functions G(T ), F + (T ) and F (T ) are included. We recalculated the result ν for the flat universe fro Eq.(8) in FW and found that F + (T ) and F (T ) ust be replaced by a single function F (T ), which could be read: F (T )=F (y) (9) F (y) = y 8a 9a ( y + 5 y y y +5y y )() i.e., the only difference between Eq.() in FW and our Eq.() is the signs + and before the ter y. Eq.() in FW will be correctly rewritten as = 5 δ X c[f (T ) +F () F (T c ) F (T c+ )] + 5 δ [G(T c+ ) G(T c )]() G(T ) is as sae as Eq.(7) in FW. The detailed calculation is given in Appendix B. Although it sees that the difference in the expression between FW and ours is tiny, the influence on the result is very iportant. As in a closed universe, in a flat universe there are
D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe 7 several zero positions when X c is sall enough (see Fig. ). How to interpret the extra zero positions is one question, and whether or not there exist several zero positions is another. We adopt the interpretation given by FW for the first zero position, according to which, when a photon passes through a collapsing lup, the gravitational field is weaker for a photon travelling toward its center, which results in a positive shift of the CBR frequency, and stronger for a photon leaving fro its center, which results in a negative shift of the CBR frequency. In a general way, therefore, the negative shift effect always wins out over the positive one except when the location of the collapsing lup is close enough to the observer. therefore, the net contribution of a collapsing lup to the frequency shift of the CBR is always negative except in a certain position, the location of the lup is so close to the observer that there is only a part of the axiu negative effect of frequency shift that can cancel the positive one. That is to say, there ust exist a location of the collapsing lup the net contribution to the frequency shift of the CBR is zero. However, one can see fro Fig. that the nuber of positions of perturber (the collapsing lup) giving a zero frequency shift is either or, which ensures that when the perburber is located near the last scattering surface, the teperature distortion (the frequency shift) is negative. Clearly, we can not interpret the extra zero positions with the sae way that used to interpret the first zero positions, the detailed research on this proble will be given in a successive paper. In order to investigate the influence of on the CBR anisotropy, the CBR teperature distortion (anisotropy) as a function of the location of the collapsing lup is plotted in Fig. for three different values of the density paraeter of the universe. The values of these density paraeters of the universe fall within the range accepted by current observational cosology (i.e.,. Ω.). Coon characteristics in Fig. a, Fig. b and Fig. c are: () when X c is large enough (in our odel, when X c.), the CBR anisotropy induced by a collapsing lup near the observer has the sae order of agnitude as that near the last scattering surface, as with a saller size of the lup (e.g., when X c.), the CBR anisotropy near the observer is one order of agnitude less than that near the last scattering surface. () when the size of the lup is sall enough (in our odel, when X c.), there exist ore than one position the CBR frequency shift induced by the lup located there is zero, and when z crosses a zero position, the sign of the CBR teperature distortion (i.e., the CBR anisotropy) changes. () The first zero positions will ove to the left with increasing, or with decreasing X c for afixed. The larger the value of or the saller the size of the collapsing lup, the ore the zero positions will ove. On the contrary, no atter what the value of the density paraeter is, the distribution of the CBR teperature distortion induced by the lup with different sizes reains alost unchanged near the last scattering surface, which, as a atter of fact, akes no difference with that in a flat universe. () In all the figures, the dotted lines denote the region that the collapsing lup situated there would cross with the last scattering surface. The connecting point of the two different style of lines corresponding to a certain value of X c reains unchanged with the increasing values of. In fact, the position of each connecting point in closed universes is just the sae as that in a flat universe. On the other hand, one can see clearly the difference of the influence on the CBR anisotropy due to the different values of the density paraeter of the universe. As entioned above, a connecting point is a critical position (which will be refered as critical position(s), hereafter.) which deterines whether the lup will cross the last scattering surface. There exist siilar critical positions ( critical position(s) hereafter.) near the observer which deterine whether the observer is situated inside the lup. In our figures, the CBR teperature distortions induced by the lups situated between the observer and the critical positions are not plotted. In contrast to the fact that the critical position corresponding to any definite size of a lup reains the sae in closed universes deterined by various values of as well as in flat universe, critical positions will ove to the left with increasing. When Ω =.5or Ω =., as long as X c is large enough (in our odel when X c.), one can find the critical position in the range. z. (for which our results are plotted). As it does, in the universe defined by larger Ω ( or Ω ), the distance between the observer and the center of a lup is larger, provided the redshift of the photons coing fro the center of a lup is the sae. This iplies that the influence of the density paraeter of the universe on CBR anisotropy varies with the evolution of the universe: it is stronger in late ties than that in early ties. Copared with the case of a flat universe, when Ω =. each of the first zero positions corresponding to different sizes of the lup oves to the left, as the second zero positions corresponding to X c =.and X c =.respectively, ove to the right (see Fig. a). When Ω =.5, each first zero position oves to the left as in the case Ω =., while the first zero position corresponding to X c =.oves to the range of. z.; at the sae tie, each second zero position has oved further to the right copared with that in Fig. a (see Fig. b). When Ω =., each first zero position corresponding to X c =.,.,.oves to the range. z., while each second zero position is shifted further to the right (see Fig. c). Actually, it is easy to show that if the universe is closed, in the decoupling epoch, one has: = Ω +zω = Ω z+ Ω < z. () Ω is the present value of the density paraeter of the universe. If z, one finds that ( ) upliit.
7 D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe. Conclusion and discussion One can see fro the nuerical results that: () In a closed universe as well as in a flat universe, an initial perturbation located at the region z < can cause an anisotropy as large as that caused by perturbations located on the last scattering surface; () near the observer, the density paraeter plays a ore iportant role than near the last scattering surface: e.g., when z <, with ncreasing, the position of zero frequency shift oves toward lower redshifts, while the situation is alost unchanged in the region z >. The appearance of extra zero positions of frequency shift ay iply that the first order approxiation for δ(x) istoo crude, the ethod provided here can only describe the ain characters of the frequency shift. Appendix A The functions in Eq.() are: A (η, η ) = Q(P +F ) A (η, η )=δ S /sin η A (η, η ) = Q(P +F ) Q = ( ) / δ (X cw ) F = ( S ) sin (η η ) sin η P = F W Z C Z + F + F + W(ηZ Z Z ) P = F W Z C Z + F + F + W(ηZ Z Z ) W C = ( ) = C F = cos(η η) sin (η η ) [W/ tan η + C / tan η ] Setting =/tan η η, =/tan η,wehave: Z = ( + + )+ + {( + + ) + + +[ 5 + (+) Fig. a c. Sae as Fig., but with three larger values of. The first zero positions ove to the range of. z. for saller sizes of the perturber. a =.9, Ω =.b =., Ω =.5 c =.5, Ω =. 7(+) + (+) ] + + 5 ( + ) (+ ) + + ( + ) ( + )ln + }
D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe 75 Z = (+ ) ( + 5 ) 5 + 8 (7+ + + 5 ) + ( + 5 ) +( + 5 + ) + ( + ) 8 (+5 + ) + ( + ) 7 ( + ) (+ ) (+5)ln + (+ ) ( + 5 5 ) + (+ ) (+ ) (+ ) (+ ) (+ ) + We eet in the calculation a ter Z dη, which can be expressed as { Z Z dη =, < Z, > in Z, one has : ln( + )=ln + ( ) + ( ) ( ) ;inz, one has: ln( + )=ln u + ( ) + ( ) ( ). One thus has Z = (+ ) ( + 5 )[ 5 ln( + )] + (7+ + + 5 ) ( + tan ) + + ( + 5 )[ + ln( + )] +( + 5 8 + ) [ + tan ] ( + ) ln( + )+ (+5 + ) ( tan ) (+ ) [ ln( + )] + 8 ( + tan ) 7 ( + )η [ ln( + )] + ( + tan )} Z = (+ ) (+5 )[ ln( + )] + ( +7 + +5 8 ) ( + tan ) + + ( + + 5 +5 7 )[ + ln( + )] + 8 (+ 7 8 ) ( + + tan ) ( + ) ln( + ) + (+5 + ) ( tan ) (+ ) [ ln( + )] + 8 ( + tan ) 7 ( + )η + (+ ) (+5 ){η η η [η + η η η ]+ (η η ) + 7 9 8 (η η ) 5 } + (+ ) {[ + 5(+) (+) ] + + } + (+ ) [ + ln( + )+ tan ] Z = 8 [ + (+ ) (+) 8 5 (+) ] + + ] + { [ + (+) ] + 9 (+ ) + } + + 8 { [ + } [ + + (+ ) + ]+ 8+ + + + 8 + ( + + 5 )ln + 8 (+7 (+ ) + + + + ) + 8 ( + ) (+ ) ( + ) + (+ ) ( + 5 ) {η η η [η + η η η ]+ (η η ) + 7 9 8 (η η ) 5 } F = ( Ωi ) sin (η η ) + (+ ) {[ + 5(+) (+) ] + + } + (+ ) [ + ln( + ) + tan ]+ (+ ) ( + 5 5 ) {η ln( + )+ ( tan ) F = W η cos(η η ) sin (η η ) tan η Appendix B Fro Eq.(8) in FW, we have = 5 T [xδ (x)+δ(x)]( e T/ +e T )dt, (B)
7 D. Chen et al.: The anisotropy of the cosic background radiation due to ass clustering in a closed universe [xδ (x)+δ(x)] = δ X c X, T T c δ, T c T T c+ δ X c X, T c+ T T (B) Eq.(B.) should be divided into three parts Let F (T )= 7 K (T )dt, G(T) = K (T)dT. then (B7) = Tc 5 5 5 Tc+ T c T T c+ δ Xc X ( e T/ +e T )dt δ ( e T/ +e T )dt δ X c X ( e T/ +e T )dt. (B) X and T has the relation as expressed in Eq.(9)(FW), which can be rewritten as X = { (e T/ +a), T T c (e T/ +a), T c+ T T (B) a = exp T = 5 7 δ Xc 5 δ Tc+ + 5 7 δ Xc = 5 7 δ Xc 5 δ (Fro Eq.() in FW). thus Tc T c ( e T/ +e T )dt Tc+ T c + 5 7 δ Xc = 5 δ Xc[ 7 + 7 T K (T )= et/ e T (e T/ +a), K (T)= e T/ +e T. T T c+ Tc K (T )dt T e T/ +e T (e T/ +a) dt e T/ +e T (e T/ +a) dt [ K (T )]dt T c+ [ K (T )]dt Tc K (T )dt T c+ K (T )dt ]+ 5 δ [ Tc+ T c K (T )dt ] (B5) (B) Fro Eqs.(B.5) and (B.), it can be clearly found that there is only one function F (T ) other than the two functions F ± (T ), and that the sybol before the ter y should be unique. F (T )=F (y), y= e T/ e T/ +a. F (y) = e T/ e T 7 (e T/ +a) dt = e T/ 7 (e T/ +a) dt 7 = ydy ( y) 5 9a 9a y = 8a y 9a ( y + 5 y e T (e T/ +a) dt dy (B8) y +5y y y ) (B9) G(T) given by Eq.(B.7) is as the sae as that in FW. Finally, fro Eqs.(B.5), (B.)and (B.7) one can obtain eq.() in this paper: = 5 δ X c[f (T )+F() F (T c ) F (T c+ )] + 5 δ [G(T c+ ) G(T c )] References Bennett C. L. et al., 99, ApJ 9, L7 Dyer C. C.,97, MNRAS 75, 9 Dyer C. C. & Ip P. S. S., 988, MNRAS 5, 895 Fang L. Z. and Wu X. P., 99, ApJ 8, 5 (FW) Hu W., & Sugiyaa N. 99, Phys. Rev. D 5, 7 Jaffe A. H., Stebbins A. & Friean J. A., 99, ApJ, 9 Kaionkowski M., Spergel D. & Sugiyaa N., 99, ApJ, L57 Kaiser N.,98, MNRAS 98, Linde A., 995, Phys. Lett. B 5, 99 Linde A., 995, Phys. Rev. D 5, 789 Martinez-González E., & Sanz J. L. 99, MNRAS 7, 7 Martinez-González E., Sanz J. L., & Silk J., 99, ApJ 55, L5 Martinez-González E., Sanz J. L., & Silk J., 99, Phys. Rev. D, 9 Martinez-González E., Sanz J. L., & Silk J., 99, ApJ, L Meśzarós A., 99, ApJ, 9 Nottale L., 98, MNRAS, 7 Panek M., 99, ApJ 88, 5 Rees M.,& Sciaa D., 98, Nature 7, 5 Sachs R. G., & Wolfe A. M., 97, ApJ 7, 7 Soot G. F. et al., 99, ApJ ]9, L Thopson K. L., & Vishniac E. T., 987, ApJ, 57 White M., & Scott D., 99, ApJ 59, 5 Wright E. L. et al., 99, ApJ 9, L