The Hubble Parameter in Void Universe

Transcription

1 Department of Physics Kyoto University KUNS-1323 OU-TAP-15 YITP/K-1100 astro-ph/??????? Feruary 1995 The Hule Parameter in Void Universe -Effect of the Peculiar Velocity- arxiv:astro-ph/ v1 10 Fe Ken-ichi Nakao, 2 Naoteru Gouda, 1 Takeshi Chia, 2 Satoru Ikeuchi 3 Takashi Nakamura and 2 Masaru Shiata 1 Department of Physics, Kyoto University Sakyo-ku, Kyoto , Japan 2 Department of Earth and Space Science, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan 3 Yukawa Institute for Theoretical Physics Kyoto University, Sakyo-ku Kyoto , Japan We investigate the distance-redshift relation in the simple void model. As discussed y Moffat and Tatarski, if the oserver stays at the center of the void, the oserved Hule parameter is not so different from the ackground Hule parameter. However, if the position of oserver is off center of the void, we must consider the peculiar velocity correction which is measured y the oserved dipole anisotropy of cosmic microwave ackground. This peculiar velocity correction for the redshift is crucial to determine the Hule parameter and we shall discuss this effect. Further the results of Turner et al y the N-ody simulation will e also considered. Recent oservation suggests that Hule parameter is large one, that is, 80 ± 17km/sec/Mpc (Freedman et al. 1994). The low Hule universe, however, is favored since the small value of Hule parameter is consistent with almost all oservations except for that of the Hule parameter itself (Bartlett et al. 1994). One of the theoretical ases for the possiility of smaller Hule parameter than that determined y local oservation is given y Turner, Cen and Ostriker (Turner et al. 1992). They performed very large scale N-ody simulations and constructed the ensemle of universe filled with the galaxies which, roughly speaking, are defined y density peaks of collisionless particles. Then, one of those galaxies is identified as our galaxy and they investigate the relation etween the distance of the other galaxies from our galaxy and the relative velocity with the correction aout the peculiar velocity only of our galaxy. Their result suggests that the Hule parameter determined y such oservations has the scale dependent variance. In order to otain the correct Hule parameter, we need the oservation of galaxies over the very wide region. On the other hand, Moffat and Tatarski considered the void universe in which the oserver is assumed to e at the center of void and investigated the effect of the void on the Hule parameter determined through the redshift and distance relation(moffat & Tatarski 1994). Their result reveals that when the oserver is at the center of void, the Hule parameter is not so different from the true value as long as the oserved region is smaller than the curvature radius within the void. This seems to contradict with the results of Turner et al. In this paper, we investigate the void universe, ut shall not restrict the position of the oserver to e the center of the void. Our void model is more simplified one than that of Moffat and Tatarski, ut will clarify the effect of the inhomogeneities on the oservation of the Hule parameter. We assume that the inside of the void is approximated y the Friedmann-Roertson- Walker (FRW) universe with the present density parameter Ω 0 < 1 while the outside of the void is also the FRW universe ut with Ω 0 = 1. The oundary of the void can e ignored as long as we exist within the void and oserve only inside of that. Here we will assume such a situation. Further we assume that the age of oth inside and outside of the void is the same and hence the time coordinate is common cosmic time t to oth the inside and

2 outside of the void. This assumption corresponds to the fact that the void structure comes from purely growing mode of the initial density perturation since the density contrast etween the inside and outside of the void vanishes as t 0, i.e., at the initial singularity. The metric within the void is written as ds 2 = dt 2 + a 2 v (t) 1 + (R v /R c ) 2 dr2 v + a 2 v(t)r 2 vds 2, (1) where R c is the comoving curvature radius and ds 2 = dθ 2 + sin 2 θdϕ 2 is the line element on the unit sphere. We should note that the center of the void agrees with the origin R v = 0 and hence, as for the time coordinate t, ds 2 is common to the inside and outside of the void. As is well known, the scale factor a v is given as the parametric form y the conformal time η, a v a v0 = H v0 t = Ω v0 (cosh η 1), 2(1 Ω v0 ) (2) Ω v0 (sinhη η), 2(1 Ω v0 ) 3/2 (3) where H v0, a v0 and Ω v0 are, respectively, the present Hule parameter, the present scale factor and the present value of the density parameter, within the void. On the other hand, we assume that the outside of the void is the flat FRW universe and hence its metric outside the void is given y ds 2 = dt 2 + a 2 (t)(dr2 + R2 ds2 ), (4) and the scale factor is written as a ( 9 = 0 4 H2 0 t2) 1/3, (5) where 0 and H 0 are, respectively, the present scale factor and the present Hule parameter, outside the void. As discussed y Bartlett et al. (1994), the ratio, H v0 /H 0, varies over the range 3/2 to 1 as Ω v0 varies form 0 to 1. Hence the maximum Hule parameter within the void is at most 3/2 times the ackground Hule parameter H 0. However, it should e noted that H v0 is not oserved directly. The oserved Hule parameter is determined through the relation etween the distance and redshift with the correction aout the peculiar velocity of oth the oserver and the oserved source. Here we define the peculiar velocity which is crucial to estimate the true redshift. Assuming that the cosmic microwave ackground (CMB) radiation is homogeneous and isotropic, the peculiar velocity is defined as that against the frame in which the CMB is oserved to e isotropic. Since the comoving oserver outside the void just oserves the isotropic CMB, we first define the new radial coordinate R for inside of the void as a v (t)r v = (t) R. It should e noted that the oserver along R=constant curve looks just isotropic CMB. The transformation matrix is given y d t = dt, (6) d R = a v (H v H )R v dt + a v dr v, (7) d S 2 = ds 2. (8) In the original coordinate (1), the comoving oserver and comoving oserved source move along R v =constant lines and hence the components of those 4-velocities are given y the common u µ = (1, 0, 0, 0). On the other hand, in the aove tilde coordinate system, the components are given y u t = t t ut = 1, (9) u R = R t ut = a v (H v H )R v, (10) u θ = 0 = u ϕ, (11) and the radial component u R corresponds to the peculiar velocity of the comoving oserver in the void. In order to otain the relation etween the redshift and the distance, it is sufficient to approximate the light ray y a null geodesic, i.e., to treat the propagation of light ray y the geometric optics (Misner, Thorn & Wheeler 1973). By virtue of the spherical symmetry of this system, without loss of generality, we focus only on the null geodesic within the equatorial plane θ = π/2. The solution for the null geodesic tangent k µ is then given y k t = a v0 a v (t) ω v0, (12) k Rv = ± a v0 a 2 v (t) [1 + (R v /R c ) 2 ][ωv0 2 (L 0/R v ) 2 ], (13) k ϕ = a v0l 0 a 2, (14) v (t)r2 v and k θ = 0. The radial trajectory of the null geodesic is otained as R v = R k (η) R c F 2 (η) 1, (15) with ( Lv0 ) 2 [ F(η) = 1 + cosh cosh 1{ ( Rv0 ) ω v0 R c R c ( Lv0 ) 2 } ] / 1 + ± (η η 0 ), (16) ω v0 R c where R v0, L v0 and ω v0 are the integration constants and η 0 is the present conformal time. It should e noted that, at η = η 0, (R v, ϕ) = (R v0, 0) and this corresponds to the position of the comoving oserver at the moment of oservation. L v0 is the conserved angular momentum of the light ray while, ω v0 is the angular frequency of that for the comoving oserver. Together with ω v0, L v0 2

3 determines the angle θ k etween the radial direction and the propagation direction of the light ray as, (see Fig.1), ( Lv0 ) 2. cosθ k = 1 (17) ω v0 R v0 Next, we consider the effect of the peculiar velocity on the angular frequency of the light ray. The comoving oserver (comoving oserved source) detects (emits) the light ray k µ with the angular frequency, ω v k µ u µ = k t On the other hand, the oserver and oserved source moving along R =constant curve, have 4-velocity w µ = (1, 0, 0, 0) in the tilde coordinate and hence the angular frequency for those is given y ω c k µ w µ = k t = ω v + k Ru R = ω v + (H v H )R v k Rv. (18) It should e noted that ω c corresponds to the angular frequency with the correction for the peculiar velocity. Oservationally, we can consider the effect only of our own peculiar velocity and hence hereafter we focus on the quantities with the correction aout the peculiar velocity only of the oserver and those without any corrections for the peculiar velocity. Then we define the following two kinds of the redshift as z = ω v ω v0 1, and z co = ω v ω co 1, (19) where ω v is the angular frequency of the light ray at the oserved source while ω co is given y ω co = ω v0 + (H v0 H 0 )R v0 k Rv (η 0 ). (20) Hence, z is the ear oserved redshift and z co is the redshift with the correction aout the peculiar velocity only of the oserver. We shall employ the luminosity distance D L as the distance measure etween the oserver and oserved source. Here, the luminosity distance D L is given y well-known relation in the FRW universe with (1) as 1 D L = H v0 qv0 2 [zq v0 + (q v0 1)( 1 + 2q v0 z + 1)], (21) where q v0 = Ω v0 /2. It should e noted that the luminosity distance D L is just the oserved quantity which is determined y, for example, Tully-Fisher relation. Then, using D L, we define the oserved Hule parameter H co with the correction for the peculiar velocity only of the oserver with the assumption that oservers regard their own universe as the flat FRW space-time, H co = 2 D L [z co + 1 z co + 1]. (22) In fact, we can measure H co instead of H 0 in the real oservations. In Fig.2, H co is depicted for θ k = 0, π/2 and π. In this figure, the density parameter inside the void, Ω v0 is equal to 0.1 and the radial position of the oserver is fixed as a v0 R v0 = H h 1 Mpc. We find that, for H v0 D L 1, H co strongly depends on the oserved direction along which the light ray propagates. This comes from the wrong peculiar velocity correction and it should e noted that the Hule parameter defined y Turner et al. is a volume average of just H co. To understand the direction dependence of H co, we investigate that only for H v0 D L 1. In this case, H co z co /D L H v0 (z co /z) and, assuming the case of Ω v0 = 0.1, we otain H v0 /H Further, the distance, a v0 R v0, of the oserver from the center of the void is assumed to e less than aout 100h 1 Mpc, i.e., a v0 H 0 R v0 < Hence, we otain z co H v0 D L (H v0 D L + 1)k Rv (η 0 ) ω v0 ( Rv0 100Mpc ). (23) Since a v0 R c = Hv0 1 (1 Ω v0) 1/2 Hv0 1, R v0/r c is much less than unity and hence we can see that k Rv (η 0 ) ω v0 cosθ k. Then we get H co H vo ( cosθ k H 0 H 0 H v0 D L Rv0 100Mpc ). (24) From the aove equation, when the distance of the oserver from the center of void is 30h 1 Mpc and when such an oserver looks to the direction θ k = 0 and the oserved distance is D L = Hv0 1 7h 1 Mpc, the oserver may estimate H co to e factor two times larger than H 0. On the other hand, if that oserver looks to the opposite direction θ k = π, the oserver may otain almost vanishing H co. This is just the dipole anisotropy due to the wrong correction for the peculiar velocity. Here we shall consider the relation etween our simple void model and the results y Turner et al. In our case, the averaged H co agrees with H v0 as < H co >= 1 π π 0 dθ k H co = H v0. (25) It should e noted that we assume the uniform distriution of oserved source, i.e., galaxy when we perform the aove averaging. However, in the N-ody simulation, the galaxy is not uniformly distriuted in contrast with our model and the integral of the second term in R.H.S. of Eq.(24) may remain. Fig.1 shows an example in which the numer of galaxies on the direction θ k = 0 is larger than that on θ k = π direction. In such a case, the averaged H co is greater than H v0. Therefore it may e a reason why the variance of the Hule parameter depends on the scale of the oservational regions and there appears the large variance of the Hule parameter in the small scale oservation in the results of Turner et al. Of course, in order to confirm this expectation, the detailed investigation y the N-ody simulation is needed (Gouda et al. 1995). 3

4 From the oservational point of view, if the variance of Hule parameter comes from the dipole anisotropy such as aove, it is important to confirm the isotropy of Hule parameter. Lauer and Postman reported the highly isotropic Hule parameter y the rather large scale oservation z 0.05 (Lauer & Postman 1992). Hence, even if we stay in the void, we are near the center of that. In the case that we stay near the center of the void, the oserved Hule parameter is H v0 and this varies over the range H 0 to 1.5H 0. Since this is not so large variance, we can find almost the same Hule parameter as the ackground one. Of course, our model is too simple and more complicated situations may e imagined, which makes us to fail the true Hule parameter. Hence further theoretical investigation should e continued and deeper oservation over whole direction in the sky is very important. Finally, we should comment on the effect of the void on the anisotropy of CMB. Here we shall assume that the CMB is completely isotropic at the last scattering surface and that the anisotropy is caused only y the effect of one void. The dipole anisotropy of CMB is aout v/c where v is the peculiar velocity of the comoving oserver in the void and it is given roughly as (H v0 H 0 )a v0 R v0 y Eq.(10). If the density parameter inside the void is nearly zero, we otain v (a v0 R v0 /100h 1 Mpc)km/sec. Assuming that the oserved dipole anisotropy comes form the peculiar velocity of our local group, that is estimated as aout 600km/sec (Smoot et al 1991). If we live in such a void, then our position is 10h 1 Mpc apart from the center of the void. However our void considered here is nothing ut a toy model and it should not e seriously considered. The rather serious suject is the quadrupole or higher multi-pole anisotropies which come from the gravitational redshift. We consider the situation that the size of the void is sufficiently smaller than the horizon scale L of the ackground flat FRW universe and hence the Newtonian approximation is applicale. In this case, the metric is written as ds 2 = (1 2U)dt 2 + a 2 (t)(1 + 2U)(dR 2 + R 2 ds 2 ), (26) where U 1. Further we assume the step-function-like density configuration, ρ = { ρv (t) R < R void ρ (t) otherwise (27) where ρ corresponds to the critical density. Then the Newton potential U inside the void R < R void is otained as U = 2πδρl 2 2π 3 δρ(r) 2, (28) where l R void. Here, since we consider the case in which δρ ρ H 2 = L 2, we see that δρl 2 κ 2 (l/l) 2 1. Thus we can roughly estimate the Newtonian potential as U κ 2 κ 2 ( R/l) 2, t U H U and r U κ 2 ( /l) 2 R. Here we shall estimate the Sachs-Wolfe effect on the CMB y the aove Newtonian potential. The anisotropy of CMB is expressed y the integrated rightness temperature perturation Θ and the equation for Θ is written as d ( ) dt (Θ U) t + γi i (Θ U) = 2 t U, (29) where γ i is the direction cosine of the photon (Kodama & Sasaki 1986). Then, the difference etween the two opposite radial directions is roughly estimated as T T = 2 ( ) dt t U θk =0 dt t U θk =π κ 3( R ) o, l (30) where R o denotes the radial position of the oserver and the integration is performed along the path of the light ray. It should e noted that the aove result is consistent with the analysis y Meszaros (1994) for the case that the position of the oserver is outside of the void. Hence if we live in the 100h 1 Mpc scale void, since κ , the higher multi-pole anisotropy of CMB y the such a void does not conflict with COBE results (Smoot et al. 1992). However, this estimate is so rough that we need more detailed investigation and this is in progress. ACKNOWLEDGMENTS We would thank to H. Sato and M. Sasaki for their useful discussion. KN would like to thank T. Tanaka for his crucial suggestion on the peculiar velocity. REFERENCES Freedman W.L. et al., 1994, Nature, 371, 757 Bartlett J.G., Blanchard A., Silk J., Turner M., 1994, FERMILAB-Pu-94/173-A (1994) Turner E.L., Cen R., Ostriker J.P., 1992, Astron. J., 103, 1427 Moffat J.W., Tatarski D.C., 1994, preprint, UTPT Misner C.W., Thorne K.S., Wheeler J.A., 1973, Gravitation(Freeman) 4

5 Lauer T.R., Postman M., 1992, ApJ, 400, L47 Smoot G. F. et al, 1991, ApJ, 371, L1 G Kodama H., Sasaki M., 1986, Intern. J. Mod. Phys., A1, 256 Meszaros A., 1994, ApJ, 423, 19 Gouda N. et al., 1995, in preparation Smoot G.F et al, 1992, ApJ, 396, L1 G F G F G The schematic diagram of the position of oserver and the oserved direction. The angle θ k is defined in Eq.(17). The Hule parameter H co with the correction for the peculiar velocity only of the oserver is plotted against the luminosity distance D L for various direction. The density parameter Ω v0 within the void is

6 This figure "fig1-1.png" is availale in "png" format from:

7 This figure "fig1-2.png" is availale in "png" format from: