Anisotropy and inhomogeneity of the universe from ∆ T / T

A recent paper (Martinez–Gonzalez & Sanz 1995) showed that if the universe is homogeneous but anisotropic, then the small quadrupole anisotropy in the cosmic microwave background radiation implies that the spacetime anisotropy is very small. We point out that more general results may be established, without assuming a priori homogeneity. We have proved that small anisotropies in the microwave background imply that the universe is almost Friedmann–Robertson–Walker. Furthermore, the quadrupole and octopole place direct and explicit limits on the degree of anisotropy and inhomogeneity, as measured by the shear, vorticity, Weyl tensor and density gradients. In the presence of inhomogeneity, it is only possible to set a much weaker limit on the shear than that given by Martinez–Gonzalez & Sanz. School of Mathematical Studies, Portsmouth University, Portsmouth PO1 2EG, England Member of Centre for Nonlinear Studies, Witwatersrand University, South Africa Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town 7700, South Africa Vatican Observatory Research Group, Steward Observatory, University of Arizona, Tucson AZ 85721, USA 1 Martinez–Gonzalez & Sanz (1995) (hereafter MS) point out that part of the foundation for the standard Friedmann–Robertson–Walker (FRW) model of the universe is to prove that the small anisotropies in the cosmic microwave background radiation (CBR) imply that only small deviations from homogeneity and isotropy of the universe are possible. They prove a particular special case of such a general theorem: if the universe is homogeneous and flat (i.e. a Bianchi I model), and if the dynamical effects of radiation are neglected, then the small quadrupole moment of the CBR implies that the anisotropy of the universe (i.e. deviation from an FRW model) is very small. In fact, the general theorem has been proved by Stoeger et al. (1995). This theorem generalises the exact–isotropy theorem of Ehlers et al. (1968) to the case of almost– isotropy. It follows without making assumptions about the spacetime inhomogeneity and anisotropy, and without neglecting the dynamical effect of radiation: Theorem: if all fundamental observers measure the CBR to be almost isotropic in an expanding universe, then that universe is locally almost spatially homogeneous and isotropic (i.e. it is almost FRW) after last scattering. This result provides a consistent theoretical foundation for the standard analyses of the CBR based on the Sachs–Wolfe effect (see e.g. Hu & Sugiyama 1995), which assume that the universe is almost FRW. Note that the theorem incorporates the Copernican Principle, i.e. if the CBR is almost isotropic for our galaxy, then it is almost isotropic for all galaxies, since we do not occupy a privileged position. The proof of the theorem is based on a covariant and gauge–invariant analysis of the Einstein–Boltzmann equations governing dust and radiation after last scattering. This formalism is then applied by Maartens et al. (1995a,b) (hereafter MESa,b) to a quantitative investigation of the relationship between temperature anisotropies and the inhomogeneity and anisotropy of the universe. Before we describe the general limits on spacetime inhomogeneity and anisotropy that are imposed by CBR anisotropies, we situate the special result of MS within the general results of MESa,b. MS make the non–observational assumption that the universe has exact Bianchi I symmetry. Strictly, this rules out density perturbations, vorticity and gravitational wave perturbations, and also excludes the cases where the FRW background has non– The Copernican principle is in fact partially testable via the Sunyaev–Zeldovich effect (Maartens et al. 1995b; Goodman 1995). 2 critical matter density (i.e. is not flat). However, the model of MS is clearly intended as a first step towards the general case. As such, we can provide an observational basis for their model via one of the results of MESa (p. 1532): if the residual dipole of the CBR temperature anisotropy vanishes to first order, and if the quadrupole and octopole are spatially homogeneous to first order, then the spacetime is locally Bianchi I to first order. Indeed this special case is still more general than the MS model, since the spacetime is not exactly Bianchi I, but only to first order. With this qualification, the result of MS may be interpreted as the special case of the general theorem of Stoeger et al. (1995) which applies if the quadrupole and octopole of the temperature anisotropy are almost spatially homogeneous, and if the residual dipole vanishes to first order. By using exact solutions for Bianchi I dust models (and therefore ignoring the radiation energy density after last scattering), MS deduce the following limit imposed by COBE observations on the relative distortion at the current time: