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Magnitude-number count relation of galaxies in an inhomogeneous universe
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Vistas in Astronomy, Vol. 37, pp. 531-534, 1993 Printed in Great Britain. All rights resa'ved.
MAGNITUDE-NUMBER COUNT RELATION OF GALAXIES IN AN INHOMOGENEOUS UNIVERSE K a z u y a Watanabe Physics Department, Tokyo Institute of Technology, Oh-Okayama, Meguro 152, Japan
INTRODUCTION The magnitude-number count (m-N) relationof galaxies is a cosmological test to determine the density p~ameter, f/0,and cosmologicalconstant, A0, in the universe. Let k~(M) be the number density of galaxies with absolute magnitude, M.
The
number of galaxies in a solid angle element, ~f~,is then given by
8N(< m) = 8f~
f ~P(M)dA -~zdZdM, 2d t
where dA, t, and z denote, respectively, the angular diameter distance, look back time, and redshift. Note that the factor, ~f~d42dt, is a volume element in the universe. The apparent magnitude, m, of a galaxy with M is, up to several correction factors, given by m - M = 51og(dL/lOpc), where dL is the luminosity distance. In this paper we investigate the gravitational lens effect on the m-N relation of galaxies and then examine whether the conventional relation in the Friedma~n-Robertson-Walker (FRW) model universe is significantly changed by cosmological inhomogeneities represented by galaxies and cluster of galaxies. The main effects of inhomogeneities on the m-N relation are: (1) Deformation of volmne element, that is, perturbations on the angular diameter distance, dA. (2) Gravitational (de-)amplification of flux of galaxies, that is, perturbations on the
K. Watanabe
532 luminosity distance, dL.
Omote & Yoshida (1990) calculated the N-z relation in an inhomogeneous universe under the following assumptions. (a) The perturbations on dA is negligibly small. (b) The gravitational amplification is approximated by the statistical amplification, that is, they assume that the gravitational amplification occurs with the probability,
P(p), where p is the amplification factor. Since the validity of their assumptions has not yet been verified, we first examine their validity via numerical simulations and then calculate the m-N relation using the formalism developed in Omote & Yoshida. (We omit any equation and figure in the next section. The details are found in Watanabe 1992.) RESULTS AND DISCUSSION We first perform the ray-shootings, numerically solve the optical equations (Sachs 1961, Watanabe & Tomita 1990), and then compare the numerical amplification probability with the analytic ones (Ehlers & Schneider 1986, Futamase & Sasaki
1989). Ehlers & Schneider derived the amplification probability, PEs(P), under the thin lens approximation where the lens equation is used as a basic tool. Recently, the validity of the lens equation and uncertainty of distance factor in the lens equation is discussed by Watanabe, Sasaki, & Tomita
(1992).
On the other hand, taking into
account multiple scattering by lenses, Futamase & Sasaki derived their probability function, PFS(P), under the statistical consideration using the linear approximation for dL (Sasaki 1988). Futamase & Sasaki also showed that the amplification probability is characterized by a single parameter, that is, a dimensionless surface mass density of lens objects, ~2/~. In the case of spherically symmetric lens whose mass and size are .M and t, we have ¢2/~; = GjVi/Hol2, where H0 = 100hkm/s is the Hubble parameter. We found that when 5f~ is as large as that in Tyson's deep survey (Tyson 1988), dA is well approximated by that in the FRW model universe. We also found that the
Magnitude-NumberCountRelation numerical amplification probability is well approximated by PFs(I~) (PEs(/~))when ~2/~ << 1(~ 1). Since the parameter, ¢2/~ is approximately inversely proportional to the optical depth of lensing (Futamase & Sasaki), this obtained results is very reasonable. On the basis of these preparations, we calculate the m-N relation in an inhomogeneous universe. As for the luminosity function of galaxies, we use the same one in Fukugita et al., and h = 1 is assumed. Here, let us briefly summarize the expected changes in the m-N relation due to the gravitational lens effect. Since all galaxies are, in a statistical sense, always amplified relative to the FRW model universe, we will have excess of galaxy counts for all m. However, most of galaxies corresponding to small m ( ~ 20) are nearby galaxies, and the amplification effect for them is very small. On the other hand, galaxies corresponding to larger m(~ 20) may be affected much more than nearby galaxies. We therefore expect that we may have much excess of galaxy count for m ~ 20, while the count for ~ 20 remains unchanged. If we axe ignorant of the cosmological inhomogeneities, and the gravitational lens effect plays an important role in amplification, we will underestimate f~0, or interpret that we have a non-vanishing cosmological term, because the evaluated comoving volume is larger. If this is the true case, the universe with ~0 '~ 1, A0 "~ 0 may be consistent with the present observational data, in contrast to the recent analysis by Fukugita et al. (1990). We found that for both choice of the probability function, i.e., PFS(P) and
PES(P), and for any wlue of (no, A0), the gravitational lens effect on the relation is comparatively weak. Therefore we conclude that the m-N relation is not significantly changed by inhomogeneities. However, it must be noted that when ~2/~ = 0.5 and (f~0, A0)=(0.1,0.9), the apparent magnitude of galaxies is systematically shifted about +0.5 mag. due to gravitational lensing, and this can explain the analysis of Fukugita et al., where the similar feature was found. We also calculate the number count-redshift (N-z) relation of galaxies and found that the N-z relation in an inhomogeneous universe for ~2/~ = 0.5 is consistent with the observational data by Broadhurst, Ellis, and Shanks (1988).
533
If,. Watanabe
534
If this discrepancy found by Fukugita et al. is due to cosmological inhomogeneities (c2/~ ,,~ 0.5), the Hubble expansion may significantly affected by inhomogeneities and we may he able to observe the anisotropy of the Hubble parameter. This issue is left to be carefully investigated in future work. ACKNOWLEDGMENTS The author thanks Prof. K. Tomita for his continuous encouragement. The author also thanks Prof. M. ()mote for giving a lecture on the gravitational lens effect on observational relations at Uji Research Center, Yukawa Institute for Theoretical Physics, and thanks Prof. Suto for suggesting the possibility of this work. Finally, it was the great pleasure of the author to collaborate on several works concerning with the gravitational lens effect with Prof. Tomita and Prof. Sasaki, because without these collaboration works, the author could not complete this work. REFERENCES T. J. Broadhurst, R. S. Ellis, ~ T. Shanks, Mon. Not. R. astr. Soc. 235 (1988), 827. J. Ehlers & P. Schneider, Astron. Astrophys. 168 (1986), 57. M. Fukugita, T. Takahara, K. Yamashita & Y. Yoshii, Astrophys. J. Letters 361 (1990), L1. T. Futamase & M. Sasaki, Phys. Rev. D40 (1989), 2502. M. Omote ~ H. Yoshida, Astrophys. J. 361 (1990), 27. R. K. Sachs, Proc. Roy. Soc. London A264 (1961), 309. M. Sasaki, Mon. Not. R. astr. Soc. 240 (1988), 415. J. A. Tyson, Astron. J. 96 (1988), 1. K. Watanabe, Progr. Theor. Phys. 87 (1992), 367. K. Watanabe M. Sasaki, ~ K. Tomita, Astrophys. J. 394 (1992), 38. K. Watanabe & K. Tomita, Astrophys. J. 355 (1990), 1.
Vistas in Astronomy, Vol. 37, pp. 531-534, 1993 Printed in Great Britain. All rights resa'ved.
MAGNITUDE-NUMBER COUNT RELATION OF GALAXIES IN AN INHOMOGENEOUS UNIVERSE K a z u y a Watanabe Physics Department, Tokyo Institute of Technology, Oh-Okayama, Meguro 152, Japan
INTRODUCTION The magnitude-number count (m-N) relationof galaxies is a cosmological test to determine the density p~ameter, f/0,and cosmologicalconstant, A0, in the universe. Let k~(M) be the number density of galaxies with absolute magnitude, M.
The
number of galaxies in a solid angle element, ~f~,is then given by
8N(< m) = 8f~
f ~P(M)dA -~zdZdM, 2d t
where dA, t, and z denote, respectively, the angular diameter distance, look back time, and redshift. Note that the factor, ~f~d42dt, is a volume element in the universe. The apparent magnitude, m, of a galaxy with M is, up to several correction factors, given by m - M = 51og(dL/lOpc), where dL is the luminosity distance. In this paper we investigate the gravitational lens effect on the m-N relation of galaxies and then examine whether the conventional relation in the Friedma~n-Robertson-Walker (FRW) model universe is significantly changed by cosmological inhomogeneities represented by galaxies and cluster of galaxies. The main effects of inhomogeneities on the m-N relation are: (1) Deformation of volmne element, that is, perturbations on the angular diameter distance, dA. (2) Gravitational (de-)amplification of flux of galaxies, that is, perturbations on the
K. Watanabe
532 luminosity distance, dL.
Omote & Yoshida (1990) calculated the N-z relation in an inhomogeneous universe under the following assumptions. (a) The perturbations on dA is negligibly small. (b) The gravitational amplification is approximated by the statistical amplification, that is, they assume that the gravitational amplification occurs with the probability,
P(p), where p is the amplification factor. Since the validity of their assumptions has not yet been verified, we first examine their validity via numerical simulations and then calculate the m-N relation using the formalism developed in Omote & Yoshida. (We omit any equation and figure in the next section. The details are found in Watanabe 1992.) RESULTS AND DISCUSSION We first perform the ray-shootings, numerically solve the optical equations (Sachs 1961, Watanabe & Tomita 1990), and then compare the numerical amplification probability with the analytic ones (Ehlers & Schneider 1986, Futamase & Sasaki
1989). Ehlers & Schneider derived the amplification probability, PEs(P), under the thin lens approximation where the lens equation is used as a basic tool. Recently, the validity of the lens equation and uncertainty of distance factor in the lens equation is discussed by Watanabe, Sasaki, & Tomita
(1992).
On the other hand, taking into
account multiple scattering by lenses, Futamase & Sasaki derived their probability function, PFS(P), under the statistical consideration using the linear approximation for dL (Sasaki 1988). Futamase & Sasaki also showed that the amplification probability is characterized by a single parameter, that is, a dimensionless surface mass density of lens objects, ~2/~. In the case of spherically symmetric lens whose mass and size are .M and t, we have ¢2/~; = GjVi/Hol2, where H0 = 100hkm/s is the Hubble parameter. We found that when 5f~ is as large as that in Tyson's deep survey (Tyson 1988), dA is well approximated by that in the FRW model universe. We also found that the
Magnitude-NumberCountRelation numerical amplification probability is well approximated by PFs(I~) (PEs(/~))when ~2/~ << 1(~ 1). Since the parameter, ¢2/~ is approximately inversely proportional to the optical depth of lensing (Futamase & Sasaki), this obtained results is very reasonable. On the basis of these preparations, we calculate the m-N relation in an inhomogeneous universe. As for the luminosity function of galaxies, we use the same one in Fukugita et al., and h = 1 is assumed. Here, let us briefly summarize the expected changes in the m-N relation due to the gravitational lens effect. Since all galaxies are, in a statistical sense, always amplified relative to the FRW model universe, we will have excess of galaxy counts for all m. However, most of galaxies corresponding to small m ( ~ 20) are nearby galaxies, and the amplification effect for them is very small. On the other hand, galaxies corresponding to larger m(~ 20) may be affected much more than nearby galaxies. We therefore expect that we may have much excess of galaxy count for m ~ 20, while the count for ~ 20 remains unchanged. If we axe ignorant of the cosmological inhomogeneities, and the gravitational lens effect plays an important role in amplification, we will underestimate f~0, or interpret that we have a non-vanishing cosmological term, because the evaluated comoving volume is larger. If this is the true case, the universe with ~0 '~ 1, A0 "~ 0 may be consistent with the present observational data, in contrast to the recent analysis by Fukugita et al. (1990). We found that for both choice of the probability function, i.e., PFS(P) and
PES(P), and for any wlue of (no, A0), the gravitational lens effect on the relation is comparatively weak. Therefore we conclude that the m-N relation is not significantly changed by inhomogeneities. However, it must be noted that when ~2/~ = 0.5 and (f~0, A0)=(0.1,0.9), the apparent magnitude of galaxies is systematically shifted about +0.5 mag. due to gravitational lensing, and this can explain the analysis of Fukugita et al., where the similar feature was found. We also calculate the number count-redshift (N-z) relation of galaxies and found that the N-z relation in an inhomogeneous universe for ~2/~ = 0.5 is consistent with the observational data by Broadhurst, Ellis, and Shanks (1988).
533
If,. Watanabe
534
If this discrepancy found by Fukugita et al. is due to cosmological inhomogeneities (c2/~ ,,~ 0.5), the Hubble expansion may significantly affected by inhomogeneities and we may he able to observe the anisotropy of the Hubble parameter. This issue is left to be carefully investigated in future work. ACKNOWLEDGMENTS The author thanks Prof. K. Tomita for his continuous encouragement. The author also thanks Prof. M. ()mote for giving a lecture on the gravitational lens effect on observational relations at Uji Research Center, Yukawa Institute for Theoretical Physics, and thanks Prof. Suto for suggesting the possibility of this work. Finally, it was the great pleasure of the author to collaborate on several works concerning with the gravitational lens effect with Prof. Tomita and Prof. Sasaki, because without these collaboration works, the author could not complete this work. REFERENCES T. J. Broadhurst, R. S. Ellis, ~ T. Shanks, Mon. Not. R. astr. Soc. 235 (1988), 827. J. Ehlers & P. Schneider, Astron. Astrophys. 168 (1986), 57. M. Fukugita, T. Takahara, K. Yamashita & Y. Yoshii, Astrophys. J. Letters 361 (1990), L1. T. Futamase & M. Sasaki, Phys. Rev. D40 (1989), 2502. M. Omote ~ H. Yoshida, Astrophys. J. 361 (1990), 27. R. K. Sachs, Proc. Roy. Soc. London A264 (1961), 309. M. Sasaki, Mon. Not. R. astr. Soc. 240 (1988), 415. J. A. Tyson, Astron. J. 96 (1988), 1. K. Watanabe, Progr. Theor. Phys. 87 (1992), 367. K. Watanabe M. Sasaki, ~ K. Tomita, Astrophys. J. 394 (1992), 38. K. Watanabe & K. Tomita, Astrophys. J. 355 (1990), 1.