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Astron. Astrophys. 353, 63-71 (2000)

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3. Example: Lemaître-Tolman-Bondi model with zero cosmological constant

The 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 z. A special case of the solutions studied by these authors is used here to explore to which extent such inhomogeneous models can mimic homogeneity.

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 ([FORMULA]) and proper time t, is

[EQUATION]

in units [FORMULA].

Einstein's equations, with [FORMULA]0 and the stress-energy tensor of dust, imply the following constraints upon the metric coefficients:

[EQUATION]

where a dot denotes differentiation with respect to t and a prime with respect to r, and [FORMULA] is the energy density of the matter. [FORMULA] and [FORMULA] are arbitrary functions of r. [FORMULA] can be interpreted as the total energy per unit mass and [FORMULA] as the mass within the sphere of comoving radial coordinate r.

One easily verifies that Eq. (16) possesses solutions for [FORMULA], which differ owing to the sign of function [FORMULA] and run as follows.

  1. with [FORMULA], for all r

    [EQUATION]

  2. with [FORMULA], for all r

    [EQUATION]

  3. with [FORMULA], for all r

    [EQUATION]

where [FORMULA] is another arbitrary function of r, usually interpreted, for cosmological use, as a Big-Bang singularity surface 3, and for which [FORMULA]. One can choose [FORMULA] at the symmetry center [FORMULA] by an appropriate translation of the [FORMULA] const. surfaces and describe the universe by the [FORMULA] part of the [FORMULA] plane, increasing t corresponding to going from the past to the future.

The physical interpretation of [FORMULA] 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 [FORMULA], 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 [FORMULA], is

[EQUATION]

where µ is a scalar, corresponding to the magnitude of the electromagnetic tensor components as measured by the observer.

From the definition of redshift, one obtains (Ellis 1971)

[EQUATION]

where [FORMULA] is tangent to the null geodesics on which light travels, the subscripts s and o denoting values at the source and at the observer, respectively.

For a measure realised at the source by an observer motionless in the rest frame of the source

[EQUATION]

which gives 4

[EQUATION]

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 ([FORMULA]), that

[EQUATION]

The definition of luminosity distance given by Eq. (1) yields the expression retained by Partovi & Mashhoon (1984), namely

[EQUATION]

One thus sees that the luminosity distance [FORMULA] is a function of the redshift z and through R, of the parameters of the model: [FORMULA], [FORMULA] and [FORMULA].

A light ray issued from a radiating source with coordinates [FORMULA] and radially directed towards an observer located at the symmetry center of the model, satisfies, from Eqs. (14) and (15),

[EQUATION]

For a given function [FORMULA], Eq. (27) possesses an infinite number of solutions of the form [FORMULA], 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 [FORMULA]. The equation for the first ray can be written

[EQUATION]

The equation for the second ray is therefore

[EQUATION]

Assuming: [FORMULA] for all r, Eq. (27) implies

[EQUATION]

which gives the equation of a ray (Eq. (30)) and the equation for the variation of [FORMULA] along this ray (Eq. (31)).

If one considers [FORMULA] as the period of oscillation of some spectral line emitted by the source and [FORMULA], its period as measured by the observer at [FORMULA], the definition of the redshift is

[EQUATION]

From this equation, another way of writing the rate of variation of [FORMULA] along a ray is

[EQUATION]

One can always choose z as a parameter along the null geodesics of the rays and obtain from the above equations

[EQUATION]

Eq. (27) therefore becomes

[EQUATION]

Eqs. (34) and (35) form a system of two partial differential equations of which each null geodesic is a solution starting with z at the source and finishing at the observer with [FORMULA].

Successive partial derivatives of R with respect to r and t, and derivatives of [FORMULA] with respect to r, evaluated at the observer, contribute to the expression of the coefficients of the luminosity distance (Eq. (26)) expansion in powers of z. It is therefore interesting to note the behaviour of [FORMULA] and [FORMULA] near the symmetry center of the model, i.e. near the observer (Humphreys et al. 1997):

[EQUATION]

One thus easily sees that R, [FORMULA], and higher order derivatives of R with respect to t alone, vanish at the observer, as do E and [FORMULA].

Expressions for the coefficients of the luminosity distance expansion naturally follow; after some calculations, one obtains

[EQUATION]

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 [FORMULA] and [FORMULA] at the observer, in units [FORMULA]:

[EQUATION]

Substituting into Eqs. (38) to (40), it is easy to see that the expressions for the FLRW coefficients in the expansion of [FORMULA] in powers of z can mimic LTB with [FORMULA] ones, at least to third order. This is straightforward for [FORMULA]. The case of [FORMULA] is discussed at length in Partovi and Mashhoon (1984). For higher order terms, it implies constraints on the LTB parameters, which will be illustrated below on the peculiar example of flat models. In fact, owing to the appearance of higher order derivatives of the parameter functions in each higher order coefficient, LTB models are completly degenerate with respect to any magnitude-redshift relation, while FLRW ones, of which the parameters are constants, are more rapidly constrained and cannot thus fit any given relation, when tested at sufficiently high redshifts.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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