On the microwave background anisotropy produced by big voids in open universes

Transcription

1 Mn. Nt. R. Astrn. Sc. 80, (1996) On the micrwave backgrund anistrpy prduced by big vids in pen universes M. J. Fullana,t J. V. Arnau and D. Saez 1 * [Departament d'astrnmia i Astrftsica, Universitat de Valencia, Burjasst, Valencia, Spain Departament de Matemiitica Aplicada, Universitat de Valencia, Burjasst, Valencia, Spain Accepted 1996 January t. Received 1995 December 1; in riginal frm 1995 August 9 ABSTRACT The Tlman-Bndi slutin f the Einstein equatins is used in rder t mdel the time evlutin f the vid bserved in Btes. The present density cntrast f the central regin ( '" ) and its radius ( '" 0 h -1 Mpc) are fixed, while the density parameter f the Universe, the amplitude f the density cntrast inside the vid wall, the width f this wall and the distance frm the vid centre t the Lcal Grup are apprpriately varied. The micrwave backgrund anistrpy prduced by Bteslike vids is estimated fr a significant set f lcatins. All the vids are placed far frm the last scattering surface. t is shwn that the anistrpy generated by these vids strngly depends n the density parameter, the wall structure and the vid lcatin. The Dppler diple and quadruple are subtracted and the residual anistrpy is calculated. n the case f sme islated Btes-like vids placed at redshifts between 1 and 10 in an pen universe with density parameter 'l = 0., the residual anistrpy appears t be a few times 10-6 n scales f a few degrees. This anistrpy is abut ne rder f magnitude greater than previus estimates crrespnding t ther cases. The anistrpy prduced by a distributin f vids is qualitatively studied in the light f this result. Cmparisns with previus estimates are discussed. Key wrds: methds: numerical - csmic micrwave backgrund - large-scale structure f Universe. 1 NTRODUCTON Tw methds have been used t estimate the anistrpies prduced by the nn-liriear vids f the galaxy distributin in pen universes. One f these methds is based n the scalled Swiss-cheese mdel (Rees & Sciama 1968). The riginal versin f this mdel applies t the case f verdensities surrunded by underdensities, but suitable mdificatins lead t a mdel fr underdensities surrunded by verdensities (Thmpsn & Vishniac 1987). Estimates f the anistrpies prduced by Swiss-cheese vids were btained by Thmpsn & Vishniac (1987), and Martinez-Gnzalez & Sanz (1990). The main elements f these spherical vids are an abslute vacuum in the vid cre, a unifrm verdense shell cmpensating the vacuum, and a general Friedmann Rbertsn-Walker backgrund utside this shell; hence * (Decnet) 16444::SAEZ, (nternet) DEGO.SAEZ@UV.ES the density prfile f the resulting structure is very particular, while the backgrund is general and the cmpensatin is exact. The secnd methd is based n the Tlman-Bndi slutin (TBS) f the Einstein equatins (Tlman 194; Bndi 1947). A general assympttic Friedmann-Rbertsn-Walker backgrund and a general spherically symmetric energy density prfile are cmpatible with this slutin; such a general prfile can be used t mdel bth a partial vacuum in the vid cre and a sharped wall. Tw different cdes based n the TBS were built up by Panek (199) and Arnau et al. (199). Panek's cde was used (Panek 199) t estimate the anistrpy prduced by vids with small cmpensating walls, while the cde due t Arnau et al. was used (Arnau et al. 199) in the case f vids withut walls; here, the vid walls are mdelled in sme detail, taking int accunt bservatinal data. n flat universes, anther pwerful methd is being used t estimate the anistrpy prduced by nn-linear csmlgical structures. This methd des nt require any sym RAS Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

2 118 M.l Fullana et al. metry; it is based n the ptential apprximatin develped by Martinez-Gnzalez, Sanz & Silk (1990), which applies beynd the linear regime. n the flat case, the methd was applied by Annins et al. (1991), Martinez-Gnzalez, Sanz & Silk (199, 1994), Tuluie & Laguna (1995) and Quilis, banez & Saez (1995); these authrs used varius cmplementary techniques and cnditins (N-bdy simulatins, high-reslutin shck capturing methds, and particular spectra and statistics). Recently, Arnau, Fullana & Saez (1994) prved that sme Great Attractr-like structures - evlving in pen universes (0 0 < 0.4) and placed at redshifts between and 0 - prduce anistrpies f the rder f 10-5 n angular scales f a few degrees. These structures had a density cntrast f the rder f 1 when they influenced the micrwave phtns; hence the resulting nn-linear gravitatinal anistrpies are prduced in the mildly nn-linear regime. The fllwing questin arises: are there vid-like bjects - suitable structures, lcatins and backgrunds - prducing significant anistrpies as in the case f the Great Attractr-like bjects studied by Arnau et al. (1994)? n the case f nn-linear structures (density cntrasts b>o.) evlving in pen universes, the TBS and the Swiss-cheese mdel can be used t answer this questin. The use f the TBS seems t be preferable, because this slutin invlves apprpriate density prfiles. n rder t mdel vid-like bjects, sme bservatinal data must be taken int accunt. n the 1980s, there were many bservatins n the spatial distributin f galaxies; in these bservatins, sme vids with walls have been detected (Kirshner et al. 1981; Davis et al. 198; Vettlani et al. 1985; de Lapparent, Geller & Huchra 1986; Rd 1988; Dey, Strauss & Huchra 1990); amng them, the Btes Vid (BV) seems t be the greatest ne. This vid was first described by Kirshner et al. (1981). Frm the data given by these authrs and thse due t de Lapparent et al. (1986) it fllws that the vid in Btes is a big quasispherical regin with a defect f galaxies; its centre is lcated at ~ 150 h- Mpc frm the Lcal Grup, and its radius is ~ 0 h - Mpc. This regin nly cntains the 5 per cent f the galaxies expected in the same vlume f the backgrund; hence the energy density cntrast f galaxies inside the vid is ~ Surrunding this regin, there is an irregular shell having an excess f galaxies, in ther wrds, there is an inhmgeneus sharped wall. Althugh the bservatins are nt very accurate (Kirshner et al. 1981; de Lapparent et al. 1986), current data suggest that the amplitude f the density cntrast f galaxies inside the wall is ~ 4, and that the mean width f this wall is ~ 5 h- Mpc. The abve bservatinal data must be cmplemented with apprpriate assumptins abut the dark matter distributin. Since the nn-linear anistrpy prduced by vids lcated far frm the last scattering surface is a gravitatinal effect, this anistrpy is prduced by the ttal energy density cntrast p/ p. This cntrast can be btained frm the bservatinal value f the density cntrast prduced by galaxies (llp/ p )gal and the value f the s-called linear bias parameter b. This parameter is defined by the relatin (lp/p)gal=b(llp/p). n this paper, it is assumed that luminus galaxies trace the mass distributin. This means that the parameter b is assumed t be unity and, cnsequently, the relative defects (excesses) f galaxies and dark matter are identical inside the vid (wall). The case b < 1 has neither theretical nr bservatinal supprt; it crrespnds t a vid (wall) with an amunt f dark matter smaller (greater) than that f the case b = 1, and s the anistrpy f the case b < 1 is expected t be greater than that estimated in the case b = 1. Similar arguments lead t the cnclusin that, in the case b > 1, the anistrpies are smaller than thse f the case b = 1. Since the cnditin b = 1 is used alng the paper, we can state that ur cmputatins give upper limits t the anistrpy prduced by Btes-like bjects, except in the unlikely case b < 1. n this paper, the wall is described by tw parameters, the amplitude, (llp/ p )max, f the ttal density cntrast inside the wall and the distance, d w, between the tw pints f the wall in which p/ p is the 0 per cent f (llp/ p )max; the distance d w is called the wall width. n the case b = 1, current data suggest that the value f (llp/ p )max is ~ 4 and the wall width d w is ~ 5 h - Mpc. t can be easily verified (see Sectin ) that a wall having these features vercmpensates the central underdensity described abve. The mass excess f this wall is abut twice the mass defect f the underdense regin. This is nt a mdel-dependent cnclusin, but a direct cnsequence f the bservatins. n this task, the anistrpy prduced by an islated structure is estimated. The chsen vercmpensated structure is frmed by a vid and all the matter surrunding it. Accrding t the abve bservatinal evidences, vercmpensated structures f this kind are present in the Universe. As required by the csmlgical principle, each f these structures shuld be cmpensated by ther structures in large vlumes cntaining varius vids. n rder t imagine this cmpensatin, it is useful t take int accunt the fact that the energy excess surrunding a certain underdensity is shared by the neighburing nes and, cnsequently, this excess als cntributes t the cmpensatin f ther neighburing underdensities; in ther wrds, nly a part f this excess must cmpensate the central underdensity. The anistrpy prduced by a realistic distributin f irregular vids and walls cannt be calculated frm the anistrpy prduced by an islated vid with walls; nevertheless, if the chsen islated structure prduces large enugh anistrpies and its spatial distributin is apprpriate, the true distributin f vids culd prduce relevant effects; in this case, the infrmatin btained frm the study f islated structures strngly mtivates further research based n suitable appraches. Since the effects f islated vids with walls placed far frm the last scattering surface are expected t be small, the greatest bserved vid, namely the BV, has been selected. Several BV realizatins evlving in varius backgrunds have been cnsidered in rder t undertake an exhaustive study f the anistrpies prduced by islated vids. Observatinal evidence is taken int accunt in rder t select these realizatins and backgrunds. Hencefrth, a is the scalefactr, t is the csmlgical time, an verdt stands fr a derivative with respect t t, H is the rati a/a, and n is the density parameter. The subscripts D, 0 and B indicate that a quantity has been valued at decupling, at the present time and in the backgrund, respectively; fr instance, H is the Hubble cnstant. f H is given in units f km S-1 Mpc-l, the parameter h =H/1OO is Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

3 the reduced dimensinless Hubble cnstant. The plan f this paper is as fllws. Several BV mdels cmpatible with current bservatins are defined in Sectin. The initial cnditins - at decupling - leading t these mdels are derived in Sectin. The anistrpies prduced by the BV mdels f Sectin are presented in Sectin 4; apprpriate cmparisns with previus cmputatins are als given in this sectin. Finally, the main cnclusins are summarized and discussed in Sectin 5. BV MODELS A BV realizatin is defined by the present density cntrast inside the vid (/1p/ p)" the present radius f the underdense regin R" the present amplitude f the density cntrast inside the wall (/1p/ P )max, and the present wall width d w' Any present cnfiguratin f the vid in Btes is a BV realizatin. Each BV realizatin is the final state f an evlutinary prcess, which takes place in a certain Friedmann-Rbertsn-Walker universe. Hereafter, a BV realizatin and a backgrund define a BV mdel. Any mdel describes the time evlutin f the vid in a certain backgrund. This evlutin leads t a present state (a realizatin). f the csmlgical cnstant vanishes, h and n are the free parameters f the backgrund. The numerical cdes need a fixed value f h; nevertheless, if the distances are given in units f h- Mpc, the final results d nt depend n the chsen value f h; thus nly the backgrund parameter n is a physically significant free parameter t be varied. Six BV realizatins have been selected t be cnsidered in the next sectins. One f the chsen BV realizatins crrespnds t Micrwave backgrund anistrpy due t vids 118 (/1p/p)v~ -1, Rv~0 h- Mpc, (/1P/P)max~.5 and dw~ h - Mpc. This realizatin is studied nly in the case n = 1. The resulting mdel is cnsidered with the essential aim f testing ur cdes and cmparing ur results with previus nes (Panek 199). The remaining five BV realizatins are btained as fllws. The quantities (/1p/ p)v and Rv are fixed; their values are assumed t be ~ and 0 h- Mpc, respectively. The amplitude (/1P/P)max is varied frm t 6, and the wall width d w is varied frm t 7 h- Mpc. Taking int accunt the fact that the values f these parameters suggested by the bservatins (de Lapparent et al. 1986) are (/1P/P)max~4 and d w ~ 5 h - Mpc, we prceed as fllws: in a first step, the wall width d w = 5 h - Mpc is fixed and the amplitudes, 4 and 6 are cnsidered and, in a secnd step, the amplitude (/1P/P)max=4 is fixed and the wall widths are assumed t be, 5 and 7 h - Mpc. Nte that the realizatin (/1p/ P )max ~ 4 and d w ~ 5 h - Mpc is cnsidered in each f the abve steps. Hereafter, any f these five realizatins is identified by the values f the quantities (/1p/ P )max and dw' Each f these realizatins has been studied in several cases crrespnding t n values ranging frm 0. t 1; nevertheless, fr brevity, nly sme apprpriate mdels crrespnding t n = 0. and 1 are presented. Fr each BV realizatin, the wall cmpensates the central underdensity at a certain distance frm the vid centre. This distance is called the cmpensatin radius, Re. Tables 1 and (seventh clumn) shw these radii fr all the mdels cnsidered in this paper. n the cases f the first and furth rws f Table 1 and the secnd rw f Table, the cmpensatin ccurs utside the wall, while in the remaining cases, it ccurs inside the wall; fr example, in the case Table 1. BV mdels. n = 0.. (h-1mpc) (h- Mpc) (h- 1 Mpc) (h-1 Mpc) Table. BV mdels. n = 1. (7)max dw El X 10 E X 10 (h-1 Mpc) Rxl X 10 Rx X 10 Rc (h-1mpc) (h-1mpc) (h-1 Mpc) Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

4 1996MNRAS F 1184 M. 1. Fullana et al. (Api p )max - 4 and d w - 5 h -, Mpc, the cmpensa~in radius is Rc =.7 h -, Mpc and, cnsequently, the cmpensatin f the central underdensity takes place near the wall centre; this means that abut ne-half f the wall cmpensates the central vid, while the remainder f the wall must cmpensate ther underdensities. Fr a given n value, the quantities defining any BV realizatin (present cntrasts and distances) must be numerically btained - after evlutin - frm apprpriated initial cnditins; the chice f the initial cnditins crrespnding t a given mdel (a BV realizatin evlving in a defined backgrund) is nw discussed. NTAL CONDTONS The main gal f this task is the estimatin f the secndary gravitatinal anistrpies prduced by big vids lcated far frm the last scattering surface. Since the evlutin f the micrwave phtns must be studied frm the last scattering surface t ur psitin in the Universe, the initial cnditins fr the vid evlutin are set at decupling time (redshift Zdec = 1000). The initial prfiles f the ttal energy density and the peculiar velcity field fix the tw arbitrary functins invlved in the TBS (see Arnau et al. 199); hence these initial prfiles fix the time evlutin f the resulting vid frm decupling t present time. n this paper, the initial density prfile is assumed t be where R is a radial crdinate and the cnditins 8, > 0, 8 < 0, 18,1 < 181 and Rx, > Rx are satisfied. The initial peculiar velcity is v, = _ ~ H 1) (P - PBD) n.6 D 1)-" D, PBD where the angular brackets dente a mean value frm R = 0 ~R. Since the csmlgical cnstant vanishes, the backgrund parameters invlved in equatins (1) and () can be written in terms f n and h. Only the chice f the prfile (1) is arbitrary. The prfile () is btained frm equatin (1). t crrespnds t vanishing, nn-grwing mdes (Peebles 1980). The frm f the initial density prfile (1) has n theretical justificatin. This frm nly gives a certain parametrizatin f the initial cnditins; this parametrizatin is expected t be suitable fr describing vids with walls as a result f tw facts: (1) fr small values f RRx, the quantities (RRx,)6 and (RRx)6 becme very small and p becmes quasi-cnstant; this means that the prfile (1) describes a central underdensity with a density cntrast - 8, + 8 < 0 and, () the central underdensity is initially surrunded by an verdensity, which is the rigin f the present vid wall. t has been verified that the expnent 6 is suitable fr btaining the required amplitudes and wall widths at the present time, but ther expnents culd be als tested. A realizatin is defined by the quantities (Api p )v, Rv, (1) () (Api p )max and d w' The questin is: which are the values f the parameters 8" 8, R x ' and R x leading t a given realizatin in a fixed backgrund? Fr each n value, a numerical cde based n the TBS plus equatins (1) and () calculates the quantities (Aplp)v, Rv, (Api p )max and d w frm initial values f 8" 8, RX and Rx; this means that the quantities defining a BV realizatin are nt initial cnditins fr the numerical cde, but quantities derived frm it. Given a BV mdel, the crrespnding initial cnditins are btained as fllws: arbitrary values f 8" 8, Rx' and Rx are assumed, and the resulting values f (Aplp)v, Rv, (Api p )max and d w are cmpared with thse f the chsen mdel; if they are different, the parameters 81, 8, RX and Rx are varied and the results are cmpared again. These calculatins and cmparisns are carried ut by a numerical cde based n the 'gradient methd' (as in Saez, Arnau & Fullana 199). This cde mdifies the initial values f the parameters 8" 8, Rx' and Rx in such a way that the new parameters lead t a BV mdel better than the previus ne. This cde repeats the mdificatin f the parameters until the resulting BV mdel is sufficiently similar t the required ne. This prcess requires nn-linear techniques because the final BV mdel is a nn-linear ne (see Fig. 1); in ther wrds, nn-linear methds are necessary as a result f ur BV nrmalizatin, which is based n present nnlinear bservatinal data. Table 1 gives the initial values f the parameters 8" 8, R x ' and Rx fr each f the n = 0. mdels studied in this paper. Table gives the same infrmatin fr the n = 1 mdels. The present Api p prfiles crrespnding t ur BV mdels are shwn in Fig. 1. Fr a given realizatin, the present energy density prfiles f the mdels n = 0. and 1 are indistinguishable because the same values f (Aplp)v, R" (Api p )max and d w have been chsen in bth cases (see Tables 1 and ). t is well knwn that the TBS nly applies befre shell crssing. Hellaby & Lake (1985) gave the necessary and sufficient cnditins fr the presence f shell crssing; these cnditins are satisfied in the cases studied in this paper; hence the shell crssing is unavidable. The time at which this phenmenn takes place is nt given by the Hellaby & Lake cnditins; this time must be determined in each particular case. n the cases studied in this paper, it has been verified that the shell crssing des nt take place befre present time; hence the TBS can be used in ur cmputatins. 4 ANSOTROPY The initial cnditins discussed in Sectin define the space-time structure and, cnsequently, these cnditins fix the differential equatins f the phtn trajectries. These differential equatins must be integrated in rder t estimate the anistrpy prduced by a BV mdel; this integratin is carried ut by using the cde due t Arnau et al. (199) and Saez et al. (199) plus the initial prfiles (1) and (). The centre f the BV pen mdels (n=0.) has been lcated at a significant set f distances frm the bserver, which crrespnds t redshifts between 0.05 ( -150 h-' Mpc) and 100 (860 h-' Mpc); nevertheless, nly the results Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

5 '. '. ", ~.",, "-. '. "-, :' -: :: Micrwave backgrund anistrpy due t vids 1185 crrespnding t a few apprpriate distances are displayed in the figures. The centre f the BV flat mdel crrespnding t the first rw f Table has been placed at the same redshifts, while the mdel used fr cmparisns with previus cmputatins (secnd rw f Table ) has been lcated nly at 100 h -1 Mpc in rder t facilitate these cmparisns; hence all the selected structures are lcated far frm the last scattering surface and, cnsequently, they prduce negligible temperature fluctuatins and Dppler shifts n this surface, where the temperature is assumed t be cnstant. Our cde (Arnau et al. 199; Saez et al. 199) numerically cmputes the temperature T f the micrwave backgrund as a functin f the bservatin angle t{!; this is the angle frmed by the line f sight and the line jining the bserver and the inhmgeneity centre. The functin T(t{!) is then used t calculate the mean temperature <T) = (1/) f ~ T (t{!) sin t{! dt{! and the ttal temperature cntrast 6 T (t{!) = [T(t{!) - <T)]/<T). n the expansin f 6 T in spherical harmnics, 6 T ( t{!) = D cs t{! + Q ( cs t{!- 1) + higher rder multiples; D and Q are the ttal diple and quadruple, respectively. The diple D is assumed t be a Dppler effect appearing as a result f the present peculiar velcity f the bserver prduced by the Btes-like bject; in ther wrds, any gravitatinal cntributin t the diple is neglected; thus the relativistic Dppler quadruple is D/. The ttal Dppler effect (diple and quadruple) prduced by the peculiar mtin f the bserver is subtracted frm 6 T ( t{!) t btain the residual anistrpy 6 R (t{!) = 6 T ( t{!) - D cs t{!- D/ ( cs t{!- 1); therefre, n accunt f the large distance separating the chsen vids frm the last scattering surface, this anistrpy is a pure gravitatinal effect. The rigrus cmputatin f this effect requires nnlinear techniques when the amplitude f the density cntrast reaches values greater than ~ 0.1 in sme regin f the structure. The residual anistrpies prduced by the BV mdels f Sectin - fr apprpriate lcatins - are nw presented and discussed.,-.. -' ~1 Q.. <J RO RO 4.1 Open universe, n=o. The upper panels f Fig. shw the residual anistrpy prduced by three BV mdels. The backgrund is pen (n = 0.), the wall width is d w = 5 h -1 Mpc in all the cases, and the values f the amplitude (l1p/ P )max are, 4 and 6. The present energy density cntrasts f these mdels are presented in the upper panel f Fig. 1. The initial values f the free parameters are given in Table 1. n the upper left (right) panel f Fig., the vid is centred at z=0.005 (z =.4). The first f these redshifts crrespnds t the lcatin f the true BV, and the secnd ne t the psitin leading t the maximum anistrpy (see belw). As is Figure 1. Present density cntrast Ap/p(t) as a functin f the present radial distance R in units f h - Mpc. Upper panel crrespnds t three BV realizatins with the same wall width d w = 5 h 1 Mpc and three different amplitudes displayed inside the panel. The BV realizatins f the intermediate panel crrespnd t the fixed amplitude (AP/P)m,,=4, and the wall widths are shwn inside the panel in units f h- Mpc. Bttm panel crrespnds t (AP/P)max =.5, d w = h- Mpc. Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

6 1186 M. 1 Fullana et al. bserved in the plt, the greater (l1p/ P )max, the greater the amplitude f OR' n the upper left (right) panel, the maximum amplitude f the residual anistrpy is OR ~ 8 X 10-7 (OR ~ 4.5 x 10-6 ); this value is btained in the case (1P/P)max=6. Fr the mdel d w =5 h- 1 Mpc, (1P/P)max=4, the residual anistrpies are OR ~ 4 X 10-7 (left) and OR ~. X 10-6 (right). The bttm panels f Fig. als display the residual anistrpy prduced by three BV mdels. The backgrund is the same as in the upper panels, but the realizatins are different. The wall widths, are, 5 and 7 h- 1 Mpc, and the amplitude is (l1p/ P )max = 4 in all the cases. The present energy density cntrasts f these mdels are given in the intermediate panel f Fig. 1 and the initial cnditins can be fund in Table 1. The redshifts f the vid centres are the same as in the tp panels: z=o.05 (left) and z=.4 (right). n the bttm left (right) panel, the maximum amplitude f the residual anistrpy is OR ~ 6.8 X 10-7 (OR ~.8 x 10-6 ). These values crrespnd t the maximum wall width d w =7 h- 1 Mpc. The greater is d w, the greater is the amplitude f OR' All the cases f the left-hand panels f Fig. crrespnd , t x ~~~~~~~~~~~~~~~~~~~ /1 r-... ~ x t ~ x ~~~~~~~~~~~~~~~~~~~ / Figure. Left-hand panels shw the residual anistrpy DR x 10 7 as a functin f the bservatin angle t/ (in deg) fr several BV realizatins placed at z=0.05. The density parameter is 0,,=0.. Upper (bttm) left panel crrespnds t the same BV realizatins as in the upper (intermediate) panel f Fig. 1. Right-hand panels display the quantity DR x 10 6 fr the same mdels as in the left-hand panels. Vid centres are lcated at z =.4. Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

7 t x r-- 0 X r- X Z = _. Z =.4 Z = 1.5 Z = L-~~~~~~~~~~~~~==~~~ ,. '".... :... "....,, / / " ~ / : r r- ',. Z = _. Z =.4 Z = 1.5 Z = L-~~~~ L-~~~~~~~~~~~ Micrwave backgrund anistrpy due t vids 1187 t the same backgrunds and lcatins, and the central underdense regins have the same structure; hence the differences between the anistrpies f tw f these cases are due t the wall. The same can be stated fr the cases f the right-hand panels. n the cases crrespnding t the cntinuus and dashed lines, the anistrpy prduced by the wall dminates the ttal effect. The upper panel f Fig. shws the residual anistrpy prduced by the mdel il=0., (AP/P)max=4, and d w =5 h- 1 Mpc. Each curve crrespnds t a lcatin f the vid. The redshift defining this lcatin is given inside the panel. This mdel is als cnsidered in Fig.. As is shwn in this panel, the amplitude f the residual anistrpy is an increasing functin f the redshift z - defining the lcatin f the vid - frm z = 0.05 t z '".4, while it becmes a decreasing functin fr z >.4. The maximum amplitude is.6 x 10-6 ; it is fund at z '".4. As is pinted ut belw, this behaviur is nt bserved in the case f flat mdels. Fig. 4 shws the density prfiles f the three realizatins described in the tp panel f Fig. 1 fr z =.4 and n == 0.. As can be seen in Fig. 4, these structures were evlving in the mildly nn-linear regime when they prduced the maximum anistrpy. The amplitude f the density cntrast inside the underdensity is in all the cases. nside the wall, this amplitude takes n the values 0.48, 0.69 and 0.8 fr the amplitudes, 4 and 6, respectively; hence the standard Eulerian linear apprach des nt apply. 4. Flat universe, n = 1 Fr n = 1, the residual anistrpy crrespnding t the realizatin (AP/P)max=4, d w =5 h- 1 Mpc is shwn in the intermediate panel f Fig.. The vid centre is lcated at the redshifts displayed inside the panel. The initial cnditins are given in Table. Fr a flat backgrund, the amplitude f the anistrpy prduced by the chsen BV realizatin (the same as in the tp panel f Fig. ) is '" -. X 10-7 in the case z=0.05. The mdulus f this amplitude decreases as z increases frm z = 0.05 t z '".4, and it is a slwly increasing functin f z fr z >.4; hence, at redshifts between 1 and 10, the value f this mdulus is much smaller than the amplitude f the residual anistrpy crrespnding t the pen case (tp panel f Fig. ). At z =.4, the rati between this amplitude and the mentined mdulus is '" 8; we can therefre state that, at lw redshifts, the anistrpies crrespnding t the pen case are much greater than thse f the flat case. n the intermediate panel f Fig., it can be seen that the wall prduces a lcal effect. When the structure is lcated at z = 0.05, this effect appears between", = 8 and", = 1. As the redshift increases, the effect appears at smaller angles. n any case, the angular psitin f the feature cincides with that f the sharped wall. The phtns cming alng 1996 RAS, MNRAS 80, Figure. Same as Fig.. Tp panel crrespnds t the realizatin (fj,p/p)max=4, d w =5 h- 1 Mpc evlving in an pen universe with n = 0.. This realizatin is placed at the redshifts displayed inside the panel. n the intermediate panel the realizatin and the redshifts are identical t thse f the tp panel, but the backgrund is flat. The bttm panel crrespnds t the energy density prfile f the bttm panel f Fig. 1. The backgrund is flat. Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

8 1188 M. J. Fullana et al. ~ Q. < by this authr in his case (c). These results simultaneusly test ur cdes and thse used by Panek (199). Thmpsn & Vishniac (1987) and Martinez-Gnzalez & Sanz (1990) predicted BV anistrpies f a few times '" 10-7 fr Btes-like bjects lcated at z '" 0.05 in any admissible backgrund, and similar anistrpies fr ther redshifts in a flat universe. These authrs used the Swisscheese mdel. n the case f small cmpensating walls (bttm panel f Fig. ) and in the absence f walls (Arnau et al. 199), ur results essentially agree with these previus estimates; hwever, under the fllwing assumptins: (1) vercmpensating walls with the features suggested by the bservatinal data, () an pen universe with n = 0., and () lw redshifts ranging in the interval (1,10), previus predictins are magnified by a factr'" R Figure 4. Density cntrast Api p as a functin f the radial distance Rat redshift.4. R is given in units f h- 1 Mpc. The three BV realizatins have the same wall width d w =5 h- 1 Mpc, and three different amplitudes are displayed inside the panel. The density parameter is n = 0.. these directins crss a great part f the wall. The accuracy f ur cdes allws us t btain these small features. The sign f DR(/=O) is psitive fr 00=0. and negative fr 00= 1 (see the upper and intermediate panels f Fig. ). n the absence f walls (Arnau et al. 199) as well as in the case f small cmpensating walls (Panek 199), the sign f DR(/=O) is negative fr bth 00 values; therefre the sign change appears nly in the case f vercmpensated vids. 4. Cmparisns with previus calculatins Panek (199) studied fur vid realizatins evlving in a flat backgrund. The mdel ( c) f Panek's paper is a BV mdel having the fllwing features: 00= 1, (!'J.p/p)v'" -1, Rv '" 0 h - Mpc, (!'J.p/ p )max '".5 and d w '" h - Mpc. Our cde - based n the gradient methd - has been used in rder t find the initial cnditins crrespnding t this mdel. The values f 61> 6, Rxl and Rx are given in Table (secnd rw). The present density cntrast f this mdel is displayed in the bttm panel f Fig. 1. t has been verified that the wall cmpensates the central underdensity at Rc = 51.6 h - Mpc. As in Panek's paper, the vid centre is placed at 100 h- Mpc frm the bserver in rder t cmpute anistrpies. The residual anistrpy prduced by this mdel is pltted in the bttm panel f Fig.. This panel and the bttm panel f Fig. 1 are t be cmpared with figs 6 and 1 f Panek's paper, respectively. These panels have a special frmat - different frm that f the remaining nes - in rder t facilitate cmparisns with Panek's figures (Panek 199). These cmparisns clearly shw that ur cdes - based n the TBS and equatins (1) and () - have led t a BV mdel very similar t that f Panek (199); accrdingly, the residual anistrpy appears t be very similar t that predicted 5 CONCLUSONS AND DSCUSSON Equatin (1) defines a gd parametrizatin f the initial density prfiles in the case f vids with sharped walls. Our cde based n the gradient methd - plus equatins (1) and () - gives the initial cnditins leading t any BV mdel. As a result f the fact that the vid walls have been mdelled taking int accunt the bservatinal evidences, the cmpensatin f the central underdensity takes place at scales larger than that f a single vid. Varius vids cntribute t this- cmpensatin. The anistrpy prduced by a Btes-like vid strngly depends n the wall structure, the density parameter and the lcatin f the symmetry centre. Accrding t previus estimates, which are cnfirmed in this paper, the anistrpy prduced by cmpensated vids evlving in a flat universe has an amplitude f a few times 10-7 ; hwever, fr n = 0. and lcatins between z = 1 and 10, the vercmpensated vids suggested by the bservatins prduce anistrpies f a few times 10-6 n scales f a few degrees. A questin is relevant: what is the effect prduced by a distributin f vids in an pen universe? n the flat case 00 = 1, the anistrpies crrespnding t the same range f redshifts are much smaller. The value f the density parameter is f crucial imprtance. n the case f vercmpensating walls, the sign f the effect twards the central regin f the vid is psitive (negative) fr pen (flat) universes. f this effect is detected in future in the case f a single bservable structure, results shuld be used in rder t cnstraint the density parameter; nevertheless, it shuld be pinted ut that such a detectin is nt easy, in particular, in the flat case, where the resulting anistrpy is very small. Fr the abve interval f redshifts (1, 10) and 00=0., the present distances frm the vid centre t the bserver range frm '" 000 t '" 5900 h - Mpc. There are many vids lcated between these distances; nevertheless, nly sme rare vids wuld be Btes-like vids (r greater) prducing anistrpies f a few times 10-6 Given tw bservatin angles 1/11 and 1/1' the number f big vids nl and n crssed by the phtns travelling alng the chsen directins can be different. Fr nl - n1 > 1, the relative temperature difference crrespnding t 1/11 and 1/1 wuld be near 10-5 This pssibility cannt be rejected a priri. t must be either rejected r accepted after quantitative calculatins. The feasibility f the cnditin nl - n1 > 1 depends n the 1996 RAS, MNRAS 80, Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018

9 abundance f big vids. Since the anistrpy - n scales f a few degrees - bserved in experiments as COBE (Smt et al. 199) and Tenerife (Watsn et al. 199) is near 10-5, the cntributin f big vids at lw redshifts culd be imprtant. n the flat case, this cntributin is expected t be t small. n any case, it wuld appear superimpsed t the primary anistrpy prduced near the last scattering surface in the linear regime. Similar results were btained in the case f Great Attractr-like bjects (Arnau et al. 1994). Fr 00 < 0.4 and < z < 0, these structures prduce anistrpies f the rder f 10-5 n scales f a few degrees. n bth cases, either the universe is pen enugh r the anistrpy is negligible. Results abut Great Attractr-like bject enhance the interest f the abve questin, which shuld be rewritten as fllws: what is the anistrpy prduced by a distributin f vids and great verdensities in an pen universe? Althugh a mdel f vercmpensated islated vids based n the TBS is currently cmpetitive, it has sme imprtant limitatins related t the spherical symmetry. Even if the central underdense regin is quasi-spherical, the true wall is nt regular and the mtin f the matter cntained in this part f the structure is nt strictly radial. There are clusters and structures in the walls, which prduce lcal peculiar mtins tangent t the wall. These mtins wuld als prduce anistrpy (Tuluie & Laguna 1995). The anistrpy prduced by the substructures f the walls are expected t be imprtant n angular scales smaller than a few degrees and, cnsequently, the estimates f this paper shuld be admissible. ACKNOWLEDGMENTS This wrk has been supprted by the prject GV-07/94. The numerical cmputatins have been carried ut in the Cmputatinal Center f the 'Universitat de Valencia'. Micrwave backgrund anistrpy due t vids 1189 MJF thanks t the 'Cnselleria de Cultura, Educaci6 i Ciencia de la Generalitat Valenciana' fr a fellwship. REFERENCES Annins P., Matzner R A, Tuluie R, Centrella J., 1991, ApJ, 8, 71 Arnau J. V., Fullana M. J., Mnreal L., Saez D., 199, ApJ, 40, 59 Arnau J. V., Fullana M. J., Saez D., 1994, MNRAS, 68, L17 Bndi H., 1947, MNRAS, 107, 410 Buchet F. R, Juszkiewicz R, Clmbi S., Pellat R, 199, ApJ, 94, L5 Davis M., Huchra J., Latham D. W., Tnry J., 198, ApJ, 5, 4 de Lapparent V., Geller M. J., Huchra J. P., 1986, ApJ, 0, L1 Dey A, Strauss M. A, Huchra J., 1990, ApJ, 99, 46 Hellaby c., Lake K., 1985, ApJ, 90, 81 Kirshner R P., Oemler A, Jr, Schechter P. L., Schectman S. A, 1981, ApJ, 48, L57 Martinez-Gnzalez E., Sanz J. L., 1990, MNRAS, 47, 47 Martinez-Gnzalez E., Sanz J. L., Silk J., 1990, ApJ, 55, L5 Martinez-Gnzalez E., Sanz J. L., Silk J., 199, Phys. Rev. D, 46, 419 Martinez-Gnzalez E., Sanz J. L., Silk J., 1994, ApJ, 46,1 Panek M., 199, ApJ, 88, 5 Peebles P. J. E., 1980, The Large Scale Structure f the Universe. Princetn Univ. Press, Princetn Quilis V., banez J. M., Saez D., 1995, MNRAS, 77, 445 Rees M., Sciama D. W., 1968, Nat, 17, 511 Rd H. J., 1988, ARA&A, 6, 45 Saez D., Arnau J. V., Fullana M. J., 199, MNRAS, 6, 681 Smt G. F. et a., 199, ApJ, 96, L1 Thmpsn K. L., Vishniac E. T., 1987, ApJ, 1, 517 Tlman R c., 194, Prc. Natl. Acad. Sci., 0, 169 Tuluie R, Laguna P., 1995, ApJ, 445, L7 Vettlani G., de Suza R E., Maran B., Chincarini G., 1985, A&A, 144,506 Watsn R A, Gutierrez de la Cruz R D., Davies RD., Lasenby AN., Rebl R, Beckman J. E., Hancck S., 199, Nat, 57, 660 Ryal Astrnmical Sciety Prvided by the NASA Astrphysics Data System Dwnladed frm n 04 February 018