903 Progress of Theoretical Physics, Vol. 101, No. 4, April 1999 Anisotropies in Luminosity Distance Norimasa Sugiura, 1, ∗) Naoshi Sugiyama 1, ∗∗) and Misao Sasaki 2, ∗∗∗) 1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2 Department of Earth and Space Science, Oasaka University Toyonaka 560-0043, Japan (Received October 28, 1998) Anisotropies in the luminosity distance-redshift (d L -z) relation caused by the large- scale structure (LSS) of the universe are studied. We solve the Raychaudhuri equation in Friedmann-Robertson-Walker models, taking account of the LSS by the linear perturbation method. Numerical calculations to evaluate the amplitude of the anisotropies are carried out in flat models with the cosmological constant and in open models, employing the cold dark matter model and the COBE-normalization for the power spectrum of the density perturba- tion. The implications of our calculations for observation are discussed. These anisotropies in d L may cause uncertainties in determining cosmological parameters, e.g., the deceleration parameter q 0 , via the magnitude-redshift relation. We found that the effects on the d L -z relation of the LSS are divided into three types: the peculiar velocity effect, gravitational lensing and the Sachs-Wolfe effect. We show that, for lower redshifts, the peculiar velocity effect is dominant, while around z 0.5, the gravi- tational lensing is dominant, though the amplitude is rather small, affecting the estimate of q 0 by at most about 5%. §1. Introduction The large-scale structure of the universe has been a recent topic of great in- terest both in observational and theoretical cosmology. For two decades, we have accumulated a large amount of observational data on the distribution of galaxies in the nearby universe by redshift surveys. 1) - 3) We also have recently succeeded in obtaining data on the cosmic microwave background (CMB) anisotropies using COBE. 4) By analyzing these data, great progress has been made in our under- standing of the structure of the universe, and it has been revealed that our universe can be described well by a homogeneous and isotropic Friedmann-Robertson-Walker (FRW) model with small inhomogeneities. With regard to the parameters necessary to specify the model, however, no definitive conclusion has been submitted so far in spite of a great amount of research employing various approaches. One reason for this is the lack of our observational information concerning the universe at rather high redshifts, but there are currently two plans for further redshift surveys to over- come this difficulty: 5), 6) the Sloan Digital Sky Survey (SDSS), which will obtain redshifts for 10 6 galaxies and 10 5 quasars, and the Two-Degree Field Survey (2dF), which will measure 250,000 galaxies at even deeper redshifts over a limited region of the sky. With these studies, it is certain that we will obtain in the near future ∗) E-mail: sugiura@tap.scphys.kyoto-u.ac.jp ∗∗) E-mail: naoshi@tap.scphys.kyoto-u.ac.jp ∗∗∗) E-mail: sasaki@vega.ess.sci.osaka-u.ac.jp Downloaded from https://academic.oup.com/ptp/article/101/4/903/1902493 by guest on 01 September 2022