## 3. Example: Lemaître-Tolman-Bondi model with zero cosmological constantThe complexity of the redshift-distance relation in inhomogeneous
models and their deviation from the Friedmann relation have recently
been stressed by Kurki-Suonio and Liang (1992) and Mustapha et al.
(1998). In a previous very interesting paper, Partovi and Mashhoon
(1984) have shown that the luminosity distance-redshift relation, in
local models with radial inhomogeneities and the barotropic equation
of state for the source of gravitational energy, cannot be
distinguished from the FLRW one, at least to second order in A class of spatially spherically symmetrical solutions of Einstein's equations, with dust (pressureless ideal gas) as a source of gravitational energy, was first proposed by Lemaître (1933). It was later on discussed by Tolman (1934) and Bondi (1947), and became popular as the "Tolman-Bondi" model. In the following, it will be refered to as LTB model. Where the Cosmological Principle could to be ruled out, the LTB solution would appear as a good tool for the study of the observed universe in the matter dominated region (Célérier & Schneider 1998, Schneider 1999). It is used here as an example to show that a non vanishing cosmological constant in a FLRW universe can be replaced by inhomogeneity with a zero cosmological constant to fit the SNIa data. The LTB line-element, in comoving coordinates
() and proper time in units . Einstein's equations, with 0 and the stress-energy tensor of dust, imply the following constraints upon the metric coefficients: where a dot denotes differentiation with respect to One easily verifies that Eq. (16) possesses solutions for , which differ owing to the sign of function and run as follows. where is another arbitrary
function of The physical interpretation of must also be discussed. Bondi (1947) presents it as the radial luminosity distance of a radiating source. But, with the definition used here (above Eq. (1)) and a corrected expression for apparent luminosity, it corresponds in fact to the luminosity distance divided by a factor , as is shown below. Assuming that the light wavelength is much smaller than any reasonably defined radius of curvature for the universe (geometrical optics approximation), Kristian & Sachs (1965) established that the apparent intensity of a source, as measured at any point of one of its emitted light rays by an observer with proper velocity , is where From the definition of redshift, one obtains (Ellis 1971) where is tangent to the null
geodesics on which light travels, the subscripts For a measure realised at the source by an observer motionless in the rest frame of the source which gives Assuming that the electromagnetic tensor for the light emitted by a distant source verifies Maxwell's equations for vacuum (Kristian & Sachs 1966, Bondi 1947), it follows, for a radial measurement realised on any light ray by an observer located at the symmetry center (), that The definition of luminosity distance given by Eq. (1) yields the expression retained by Partovi & Mashhoon (1984), namely One thus sees that the luminosity distance
is a function of the redshift
A light ray issued from a radiating source with coordinates and radially directed towards an observer located at the symmetry center of the model, satisfies, from Eqs. (14) and (15), For a given function , Eq. (27) possesses an infinite number of solutions of the form , depending on the initial conditions at the source or on the final conditions at the observer. One can thus consider two rays emitted by the same source at slightly different times separated by . The equation for the first ray can be written The equation for the second ray is therefore Assuming: for all which gives the equation of a ray (Eq. (30)) and the equation for the variation of along this ray (Eq. (31)). If one considers as the period of oscillation of some spectral line emitted by the source and , its period as measured by the observer at , the definition of the redshift is From this equation, another way of writing the rate of variation of along a ray is One can always choose Eqs. (34) and (35) form a system of two partial differential
equations of which each null geodesic is a solution starting with
Successive partial derivatives of One thus easily sees that Expressions for the coefficients of the luminosity distance expansion naturally follow; after some calculations, one obtains with implicit evaluation of the partial derivatives at the observer. Following Humphreys et al. (1997), one can adopt a covariant definition for the Hubble and deceleration parameters of a spherically symmetric inhomogeneous universe. These authors (see also Partovi & Mashhoon (1984)) derive expressions for and at the observer, in units : Substituting into Eqs. (38) to (40), it is easy to see that the
expressions for the FLRW coefficients in the expansion of
in powers of © European Southern Observatory (ESO) 2000 Online publication: December 8, 1999 |