Obtaining the spacetime metric from cosmological observations

Recent galaxy redshift surveys have brought in a large amount of accurate cosmological data out to redshift 0.3, and future surveys are expected to achieve a high degree of completeness out to a redshift exceeding 1. Consequently, a numerical programme for determining the metric of the universe from observational data will soon become practical; and thereby realise the ultimate application of Einstein's equations. Apart from detailing the cosmic geometry, this would allow us to verify and quantify homogeneity, rather than assuming it, as has been necessary up to now, and to do that on a metric level, and not merely at the mass distribution level. This paper is the beginning of a project aimed at such a numerical implementation. The primary observational data from our past light cone consists of galaxy redshifts, apparent luminosities, angular diameters and number densities, together with source evolution functions, absolute luminosities, true diameters and masses of sources. Here we start with the simplest case, that of spherical symmetry and a dust equation of state, and execute an algorithm that determines the unknown metric functions from this data. We discuss the challenges of turning the theoretical algorithm into a workable numerical procedure, particularly addressing the origin and the maximum in the area distance. Our numerical method is tested with several artificial data sets for homogeneous and inhomogeneous models, successfully reproducing the original models. This demonstrates the basic viability of such a scheme. Although current surveys don't have sufficient completeness or accuracy, we expect this situation to change in the near future, and in the meantime there are many refinements and generalisations to be added.Comment: 26 pages, 10 figures. Minor changes to match the published versio

On perfect fluid models in non-comoving observational spherical coordinates

We use null spherical (observational) coordinates to describe a class of inhomogeneous cosmological models. The proposed cosmological construction is based on the observer past null cone. A known difficulty in using inhomogeneous models is that the null geodesic equation is not integrable in general. Our choice of null coordinates solves the radial ingoing null geodesic by construction. Furthermore, we use an approach where the velocity field is uniquely calculated from the metric rather than put in by hand. Conveniently, this allows us to explore models in a non-comoving frame of reference. In this frame, we find that the velocity field has shear, acceleration and expansion rate in general. We show that a comoving frame is not compatible with expanding perfect fluid models in the coordinates proposed and dust models are simply not possible. We describe the models in a non-comoving frame. We use the dust models in a non-comoving frame to outline a fitting procedure.Comment: 8 pages, 1 figure. To appear in Phys.Rev.

Finding a Spherically Symmetric Cosmology from Observations in Observational Coordinates -- Advantages and Challenges

One of the continuing challenges in cosmology has been to determine the large-scale space-time metric from observations with a minimum of assumptions -- without, for instance, assuming that the universe is almost Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW). If we are lucky enough this would be a way of demonstrating that our universe is FLRW, instead of presupposing it or simply showing that the observations are consistent with FLRW. Showing how to do this within the more general spherically symmetric, inhomogeneous space-time framework takes us a long way towards fulfilling this goal. In recent work researchers have shown how this can be done both in the traditional Lema\^{i}tre-Tolman-Bondi (LTB) 3 + 1 coordinate framework, and in the observational coordinate (OC) framework. In this paper we investigate the stability of solutions, and the use of data in the OC field equations including their time evolution and compare both approaches with respect to the singularity problem at the maximum of the angular-diameter distance, the stability of solutions, and the use of data in the field equations. This allows a more detailed account and assessment of the OC integration procedure, and enables a comparison of the relative advantages of the two equivalent solution frameworks. Both formulations and integration procedures should, in principle, lead to the same results. However, as we show in this paper, the OC procedure manifests certain advantages, particularly in the avoidance of coordinate singularities at the maximum of the angular-diameter distance, and in the stability of the solutions obtained. This particular feature is what allows us to do the best fitting of the data to smooth data functions and the possibility of constructing analytic solutions to the field equations.Comment: 31 page

The Lemaitre Model and the Generalisation of the Cosmic Mass

We consider the spherically symmetric metric with a comoving perfect fluid and non-zero pressure -- the Lemaitre metric -- and present it in the form of a calculational algorithm. We use it to review the definition of mass, and to look at the apparent horizon relations on the observer's past null cone. We show that the introduction of pressure makes it difficult to separate the mass from other physical parameters in an invariant way. Under the usual mass definition, the apparent horizon relation, that relates the diameter distance to the cosmic mass, remains the same as in the Lemaitre-Tolman case.Comment: latex, 16 pages, Revision has minor changes due to referee's comments

Perturbations of a Universe Filled with Dust and Radiation

A first-order perturbation approach to $k=0$ Friedmann cosmologies filled with dust and radiation is developed. Adopting the coordinate gauge comoving with the perturbed matter, and neglecting the vorticity of the radiation, a pair of coupled equations is obtained for the trace $h$ of the metric perturbations and for the velocity potential $v$. A power series solution with upwards cutoff exists such that the leading terms for large values of the dimensionless time $\xi$ agree with the relatively growing terms of the dust solution of Sachs and Wolfe.Comment: 9 pp, typeset in late

Conditions for spontaneous homogenization of the Universe

The present-day Universe appears to be homogeneous on very large scales. Yet when the casual structure of the early Universe is considered, it becomes apparent that the early Universe must have been highly inhomogeneous. The current paradigm attempts to answer this problem by postulating the inflation mechanism However, inflation in order to start requires a homogeneous patch of at least the horizon size. This paper examines if dynamical processes of the early Universe could lead to homogenization. In the past similar studies seem to imply that the set of initial conditions that leads to homogenization is of measure zero. This essay proves contrary: a set of initial conditions for spontaneous homogenization of cosmological models can form a set of non-zero measure.Comment: 7 pages. Fifth Award in the 2010 Gravity Research Foundation essay competitio

The Sachs-Wolfe Effect: Gauge Independence and a General Expression

In this paper we address two points concerning the Sachs-Wolfe effect: (i) the gauge independence of the observable temperature anisotropy, and (ii) a gauge-invariant expression of the effect considering the most general situation of hydrodynamic perturbations. The first result follows because the gauge transformation of the temperature fluctuation at the observation event only contributes to the isotropic temperature change which, in practice, is absorbed into the definition of the background temperature. Thus, we proceed without fixing the gauge condition, and express the Sachs-Wolfe effect using the gauge-invariant variables.Comment: 5 pages, closer to published versio

Differential Density Statistics of Galaxy Distribution and the Luminosity Function

This paper uses data obtained from the galaxy luminosity function (LF) to calculate two types of radial number densities statistics of the galaxy distribution as discussed in Ribeiro (2005), namely the differential density $\gamma$ and the integral differential density $\gamma^\ast$. By applying the theory advanced by Ribeiro and Stoeger (2003), which connects the relativistic cosmology number counts with the astronomically derived LF, the differential number counts $dN/dz$ are extracted from the LF and used to calculate both $\gamma$ and $\gamma^\ast$ with various cosmological distance definitions, namely the area distance, luminosity distance, galaxy area distance and redshift distance. LF data are taken from the CNOC2 galaxy redshift survey and $\gamma$ and $\gamma^\ast$ are calculated for two cosmological models: Einstein-de Sitter and an $\Omega_{m_0}=0.3$, $\Omega_{\Lambda_0}=0.7$ standard cosmology. The results confirm the strong dependency of both statistics on the distance definition, as predicted in Ribeiro (2005), as well as showing that plots of $\gamma$ and $\gamma^\ast$ against the luminosity and redshift distances indicate that the CNOC2 galaxy distribution follows a power law pattern for redshifts higher than 0.1. These findings bring support to Ribeiro's (2005) theoretical proposition that using different cosmological distance measures in statistical analyses of galaxy surveys can lead to significant ambiguity in drawing conclusions about the behavior of the observed large scale distribution of galaxies.Comment: LaTeX, 37 pages, 6 tables, 10 figures. Accepted for publication in "The Astrophysical Journal

A fully covariant description of CMB anisotropies

Starting from the exact non-linear description of matter and radiation, a fully covariant and gauge-invariant formula for the observed temperature anisotropy of the cosmic microwave background (CBR) radiation, expressed in terms of the electric ($E_{ab}$) and magnetic ($H_{ab}$) parts of the Weyl tensor, is obtained by integrating photon geodesics from last scattering to the point of observation today. This improves and extends earlier work by Russ et al where a similar formula was obtained by taking first order variations of the redshift. In the case of scalar (density) perturbations, $E_{ab}$ is related to the harmonic components of the gravitational potential $\Phi_k$ and the usual dominant Sachs-Wolfe contribution $\delta T_R/\bar{T}_R\sim\Phi_k$ to the temperature anisotropy is recovered, together with contributions due to the time variation of the potential (Rees-Sciama effect), entropy and velocity perturbations at last scattering and a pressure suppression term important in low density universes. We also explicitly demonstrate the validity of assuming that the perturbations are adiabatic at decoupling and show that if the surface of last scattering is correctly placed and the background universe model is taken to be a flat dust dominated Friedmann-Robertson-Walker model (FRW), then the large scale temperature anisotropy can be interpreted as being due to the motion of the matter relative to the surface of constant temperature which defines the surface of last scattering on those scales.Comment: 18 pages LaTeX, 1 figure. Submitted to Classical and Quantum Gravity. Also available at http://shiva.mth.uct.ac.za/preprints/9705.htm

Dynamics of Inflationary Universes with Positive Spatial Curvature

If the spatial curvature of the universe is positive, then the curvature term will always dominate at early enough times in a slow-rolling inflationary epoch. This enhances inflationary effects and hence puts limits on the possible number of e-foldings that can have occurred, independently of what happened before inflation began and in particular without regard for what may have happened in the Planck era. We use a simple multi-stage model to examine this limit as a function of the present density parameter $\Omega_0$ and the epoch when inflation ends.Comment: 9 Pages RevTex4. Revised and update