Growth of density perturbations with a cosmological constant

1997MNRAS.285..811O Mon. Not. R. Astron. Soc. 285, 811-819 (1997) Growth of density perturbations with a cosmological constant Reuven Opher, Nilza Pires* and Jose Carlos N. de Araujo Instituto Astronomico e Geofisico, Universidade de Sao Paulo, All. Miguel Stefano, 4200, 04301-904, Sao Paulo, Sp, Brazil Accepted 1996 October 20. Received 1996 August 28; in original form 1996 March 1 We investigate the influence both of the cosmological constant and of the physical processes that take place during the evolution of the Universe on the primordial cloud collapse. We study the evolution of a cloud with a density perturbation, b, from the recombination era to the present. As an example, we study an initial perturbation bj = 10- 6 for a cloud of mass 10\ 105 and 106 M 0 • In particular, we study the influence of the physical processes on the evolution of the peculiar velocity factor, f=d In bid Ina, where 'a' is the scalefactor ofthe Universe. We compare our results with the analytic expression for f obtained by Lahav et al. Our results show that the physical mechanisms (photon drag, photon cooling, recombination, etc.) amplify perturbations, and the cosmological constant is particularly important in a Universe dominated by it with a small matter density. Key words: cosmology: theory - early Universe - large-scale structure of Universe. 1 INTRODUCTION In a Friedmann-Robertson-Walker Universe, we can write (1) QT=Q+A=l- Qk' constant A (Davis et al. 1980). It is related to the peculiar velocity factor, t, that gives the relation between the peculiar velocity and the peculiar acceleration of the density perturbation in linear theory (Peebles 1980, section 14). Likewise, it has been shown (e.g. Peebles 1984; Lightman & Schechter 1990; Martel 1991 ) that the present infall velocity is almost entirely determined by with a very weak dependence on A. However, Labav et al. (1991, hereafter LLPR), as well as Barrow & Saich (1993), showed that this is true at the present epoch, but for redshifts in the range 0.5-2.0, the peculiar velocity is much more sensitive to A than to It is important to note that all studies (spherical infall: Peebles 1984, Martel 1991 and LLPR; non-spherical infall: Barrow & Saich 1993) without taking into account dark matter, were made considering only the linear theory of the perturbations and did not take into account the various physical processes (except gravitation) that were significant in the evolution of the perturbation. The main motivation for this paper is to compare the peculiar velocity factor, f, deduced by the linear theory, first by Peebles (1980) for z=o and then by LLPR for any z, A, and with the exact calculations of the hydrodynamical equations, taking into account the effects of a non-zero cosmological constant and the physical mechanisms that were present during and after the recombination era upon the peculiar velocity field evolution, as well as the amplification of the density perturbations since the recombination era. The physical processes that we take into account are no, where 81tG Q=-p 3IP ' A A=- 3m no. and H =a/a, a and a being the scalefactor and the time derivative of the scalefactor, respectively, with a/a o= (1 + Z)-l. z is the redshift, p = po(a o/a)3 is the ambient matter density, and A (or A) is called the cosmological constant. We have k = - 1 for an open universe, k = 0 for a flat universe and k = + 1 for a closed universe. The zero subscript designates the present-day value. The infall velocity of a supercluster has been used to compute the density parameter and the cosmological no *On leave from the Departamento de Fisica Te6rica e Experimental, UFRN, Campus Universitario, 59072-970, Natal, RN, Brazil no, © 1997 RAS © Royal Astronomical Society • Provided by the NASA Astrophysics Data System ABSTRACT 1997MNRAS.285..811O 812 R. Opher, N. Pires and 1. C. N. de Araujo For the cooling function of the baryonic matter we have: which includes recombination, photoionization and collisional ionization (first term), photon cooling (heating, second term), molecular hydrogen cooling and Lyman-IX cooling (third and fourth terms, respectively). i.e is the time derivative of ~. The hydrogen molecular cooling function (L Hj, taken from Lepp & Shull (1983), is good for densities n > 0.1 cm- 3 and temperatures 100 K < Tm < 106 K. For Lyman-IX cooling, L1,,' we use the expression of Carlberg (1981). With the dimensions, densities, and temperatures in the calculations made in this paper, radiative transfer is not important. All the calculations herein have small optical depth. In order to solve the above equations, we use a top-hat perturbation of mass Me, with a density Pc, Pc = P + PI (t) = p[l + b(t)], (9) where b is the contrast density, and a linear dependence for the velocity (which is consistent with the density profile) 2 THE BASIC EQUATIONS AND THE MODEL (10) The basic equations used (cf. de Araujo & Opher 1988, 1989, 1990, 1994) are op -+ V"(pv)=O, (2) ot the continuity equation, and dv -= dt 1 ubT:~ P mpc -V¢--VP---(V-Hr), (3) the equation of motion, where u is the Thomson crosssection, b=4/c the Stefan-Boltzmann constant, and Tr the radiation temperature. The last term in equation (2) is the term of photon drag arising from the background cosmic radiation. The degree of ionization Xe is given by Peebles (1968), modified to include collisional ionization. The field equation (Poisson's equation) is VZ¢=41tGp-A. (4) The equation of state is given by P=NkB P(l +~) Tm , (5) where Tm is the matter temperature, kB the Boltzmann constant, and N the Avogadro number. For the energy equation, we have dU Pdp -=-L+-dt p2 dt ' (6) where L is the cooling function and U is the energy density per unit mass of baryonic matter, given by 3 U=-NkB Tm(1 2 + Xe), neglecting the contribution of hydrogen molecules. (7) with rc being the radius of the cloud (or perturbation). The system of equations (equations 2-9) was numerically integrated using the sub-routine IMSL-DVERK (this uses the Runge-Kutta method - see e.g. Press et al. 1992), using HP-Apollo 900 and SUN SPARC2 workstations. We started the calculations at the beginning of the recombination era TR =4000 K (ZR ~ 1482). We studied five models, parametrized by 00 and A, which we refer to as models A-E. They are listed in Table 1. These models were suggested by Carroll, Press & Turner (1992). For each model we analyse clouds of masses 104, lOS and 106 MG. As an example, we study an initial perturbation bi = 10- 6 , small enough in order to compare with the linear approach. 3 THE PECULIAR VELOCITY FIELD AND THE PECULIAR VELOCITY FACTOR The peculiar velocity, Vi> is related to the peculiar acceleration, g, by the peculiar velocity factor f (Peebles 1980, section 14): . Hfg 2fg 41tGpb - 3HQ ' v-----I - (11) Table 1. The models for nT> no and l, utilized in the calculations (nr=ilo + A). Model QT Qo ,\ A B C D E 1 0.1 1 0.01 1 (...truncated)