Question

1) a) List the assumptions that underpin all of the Friedmann-Robertson–Walker models. b) Describe in words the FRW model that is currently considered to best represent the Universe. c) Describe in detail the observational data that forms the basis of the current model.

Answer 1

(a) The FRW metric model starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is

where ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. {\displaystyle \mathrm {d} \mathbf {\Sigma } } does not depend on t — all of the time dependence is in the function a(t), known as the "scale factor".

(b) The Friedmann-Robertson-Walker model metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected.The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Howard P. Robertson and Arthur Geoffrey Walker – are customarily grouped as Friedmann-Robertson-Walker (FRW). This model is sometimes called the Standard Model of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model.

(c) The difference between the absolute and apparent luminosity of a distance object is given
by, µ(z) = 25 + 5 log10 dL(z).
With numerical computation, we solve the system of dynamical equations for Ωφ˙, Ωk, Ωmf
and ΩV . While best fitting the model parameters and initial conditions with the most recent
observational data, the Type Ia supernovea (SNe Ia), in order to accomplish the mission, we need the two auxiliary equations for the luminosity distance and the hubble
parameter.
To best fit the model for the parameters α, β and the initial conditions Ωφ˙(0), Ωmf (0),
ΩV (0) and H(0) with the observational data, SNe Ia, we employe the χ^2 methodmethod.