Low-redshift formula for the luminosity distance in a LTB model with cosmological constant
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We calculate the low-redshift Taylor expansion for the luminosity distance for an observer at the center of a spherically symmetric matter inhomogeneity with a non-vanishing cosmological constant. We then test the accuracy of the formulas comparing them to the numerical calculation for different cases for both the luminosity distance and the radial coordinate. The formulas can be used as a starting point to understand the general non-linear effects of a local inhomogeneity in the presence of a cosmological constant, without making any special assumption as regards the inhomogeneity profile.
KeywordsCosmological Constant Elliptic Function Geodesic Equation Local Inhomogeneity Luminosity Distance
Modern cosmological observations such as the luminosity distance [1, 2, 3, 4, 5, 6] and the WMAP measurements [7, 8] of the cosmic microwave background radiation (CMBR) have provided strong evidence for the presence of dark energy. One of the main assumptions of the standard cosmological model used in fitting these observational data is spatial homogeneity of the Universe. We cannot nevertheless exclude the presence of a local inhomogeneity around us which could affect our interpretation of the cosmological data [9, 10, 11].
So far most of the efforts in estimating these effects have consisted in using some ansatz for the profile of the inhomogeneity and then calculating numerically the effects on cosmological observables. Such an approach has the limitation of depending on the particular functional form chosen to model the local inhomogeneity, and of relying completely on numerical calculations. In order to provide a more general study of this effects we approach the problem analytically and we derive a low-redshift formula for the luminosity distance for an observer at the center of a matter inhomogeneity in the presence of a cosmological constant modeled by a LTB solution.
The paper is organized as follows. We first calculate the low-redshift expansion of the null radial geodesics for a central observer and then use it to obtain the luminosity distance. The calculation is based on using the analytical solution and the geodesic equation expressed in the same coordinates of the analytical solution. The formula obtained is then compared to the numerical calculation of the luminosity distance to test its accuracy. In the appendix we give details of the derivation and the simplified formulas in the limit in which the inhomogeneity can be treated perturbatively.
2 LTB solution with a cosmological constant
3 Geodesic equations and luminosity distance
We will solve  the null geodesic equation written in terms of the coordinates \((\eta ,r)\). We then perform a local expansion of the solution around \(z=0\) corresponding to the point \((t_0,0)\), or equivalently \((\eta _0,0)\), where \(t_0=t(\eta _0,0)\).
4 Formula for the luminosity distance
We now find a local Taylor expansion in redshift for the geodesics equations , and we then calculate the luminosity distance. The general expression is rather cumbersome, and is given in the appendix. Here we will report only the result assuming \(K_0=0\), which still shows the general nature of the effect. From a physical point of view fixing \(K_0\) does not affect the value of \(H_0\), but it does affect the age of the Universe as shown in . Yet, using the freedom in the choice of the bang function, it is possible to obtain any age, by appropriately fixing it to a constant value \(t_b(r)=t_0\), while since \(t'_b(r)=0\) there would not be any problem related with the compatibility with early universe perturbations which should not contain decaying modes.
5 Testing the accuracy of the formula
We have derived the analytical low-redshift expansion of the luminosity distance for a central observer at the center of a spherically symmetric matter inhomogeneity in the presence of a cosmological constant. We have first solved the null radial geodesic equation and calculated the local redshift for \(r(z)\) and \(\eta (z)\), and we have then used these to calculate the expansion of the luminosity distance. The formulas obtained take a simpler form in the case in which \(K_0=0\), while in general they are rather long and complicated, but they can be reduced to a more tractable form in the limit in which the deviation from homogeneity can be treated perturbatively.
The formulas we have derived can be used to understand the physical effects of local inhomogeneities in the presence of a cosmological constant. It has the advantage, contrary to previous numerical studies, of not depending on any functional ansatz for the profile of the local inhomogeneity. This makes it particularly useful to study possible low-redshift inhomogeneities in a model independent way in the regime in which perturbation theory cannot be applied.
Chen and Romano are supported by the Taiwan NSC under Project No. NSC97-2112-M-002-026-MY3, by Taiwan’s National Center for Theoretical Sciences (NCTS). Chen is also supported by the US Department of Energy under Contract No. DE-AC03-76SF00515. AER is also supported by the UDEA Dedicacion Exclusiva and GFIF Sostenibilidad program, and the CODI project IN10219CE.
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