Microwave background anisotropies arising from non-linear structures in open and Λ universes

Abstract

A new method arising from a gauge-theoretic approach to general relativity is applied to the formation of clusters in an expanding universe. The three cosmological models (Ω 0 =1, Ω Λ =0), (Ω 0 =0.3, Ω Λ =0.7), (Ω 0 =0.3, Ω Λ =0) are considered, which extends our previous application . A simple initial velocity and density perturbation of finite extent is imposed at the epoch z = 1000 and we investigate the subsequent evolution of the density and velocity fields for clusters observed at redshifts z = 1, z = 2 and z = 3. Photon geodesics and redshifts are also calculated so that the Cosmic Microwave Background (CMB) anisotropies due to collapsing clusters can be estimated. We find that the central CMB temperature decrement is slightly stronger and extends to larger angular scales in the non-zero Ω Λ case. This effect is strongly enhanced in the open case. Gravitational lensing effects are also considered and we apply our model to the reported microwave decrement observed towards the quasar pair PC 1643+4631 A&B.Key words: Gravitation -cosmology: theory -cosmology: gravitational lensingcosmic microwave background -quasars: individual: PC1643+4631 A&B -galaxies: clustering SECONDARY GRAVITATIONAL EFFECTRees and Sciama (1968) suggested that the presence of an evolving structure on the line of sight of a Cosmic Microwave Background (CMB) photon could significantly affect its observed temperature. This secondary gravitational effect is often described in terms of the potential well experienced by the CMB photon. For example, the potential well of a collapsing cluster becomes deeper over time so that the CMB photon has to climb out of a well deeper than that into which it fell, suffering a net loss of energy. As suggested in Rees and Sciama (1968) there is however a competing effect due to the extra time delay encountered by the photon. The overall effect can therefore be of either sign. The photon accumulates a redshift along its geodesic, resulting in a net temperature perturbation. In the weak-field approximation we have (see Martínez-González, Sanz & Silk 1990) where Φ is the gravitational potential of the perturbation and t is the cosmic time. A rough estimate of an upper limit to this effect can be estimated by assuming that the potential well varies from Φ = 0 to Φ = Φc during the time ⋆ Email: youri@mrao.cam.ac.uk that the photon traverses it, where Φc is the gravitational potential of a rich Abell cluster. In this case, we haveThe potential of the cluster can be related to the velocity dispersion σ through σ 2 = GM/R = Φc, where M and R are the mass and radius of the cluster. Assuming σ = 1000 km s −1 givesIn most cases, this anisotropy should therefore be small compared with other secondary anisotropies caused by the interaction of CMB photons with non-linear structures, such as the thermal Sunyaev-Zel'dovich (SZ) effect. However, equation (1) is only valid in the linear regime and a fully relativistic treatment in the non-linear regime is necessary to estimate accurately the Rees-Sciama effect. Seve...

Halo models of large scale structure

“…The opposite limit is also very interesting. As V → ∞, M → ∞ as well: in this limit, σ 2 (M) → 0 and |δ| → 0, and so equation (63) reduces to n(m, z 1 ), as expected.…”

“…As V → ∞, δ → ∞ and δ 0 → δ sc (z 0 ) independent of the value of M. A region of small size which contains mass M, however, is what we call a halo, with mass M. Thus, if we are given a halo of mass M at z 0 , then N(m, z 1 |M, V = 0, z 0 ) is the average number of subclumps of mass m it contained at the earlier time when z 1 ≥ z 0 . This limit of equation (63) gives what is often called the conditional or progenitor mass function [23,19,163,255]. The opposite limit is also very interesting.…”

An alternative approach to modelling a cosmic void and its effect on the cosmic microwave background

“…The equations ( 4) and ( 10) can be solved analytically using elliptic integrals to obtain the position r at some time t of a fluid particle, given its position r i at some initial time t i . It is often simpler, however, instead to solve numerically the system of first-order ordinary differential equations ( 4) and (9), namely (Dabrowski et al 1999):…”

“…At such early epochs, it is valid to linearise the field equations around a homogeneous cosmology, which yields two solutions: a growing mode and a decaying mode. By demanding that the decaying mode is absent, and assuming a flat-Λ background cosmology, one finds that the initial velocity and mass distributions are related by (Dabrowski et al 1999)…”

“…Indeed, the gauge choices employed lead to dynamical equations that are essentially Newtonian in form. Assuming a pressureless fluid throughout, we have already applied the method to modelling the evolution of a finitesize, spherically-symmetric cluster, with continuous radial density and velocity profiles, that is embedded in an expanding background universe and compensated so that it does not exert any gravitational influence on the exterior universe (Lasenby et al 1999;Dabrowski et al 1998Dabrowski et al , 1999. In our approach, one considers an initial velocity profile from which the initial density profile is determined uniquely by the constraints that there are no decaying modes present and that the density distribution is compensated.…”