Low red-shift effects of local structure on the Hubble parameter in presence of a cosmological constant

In order to estimate the effects of local structure on the Hubble parameter we calculate the low-redshift expansion for $H(z)$ and $\frac{\delta H}{H}$ for an observer at the center of a spherically symmetric matter distribution in presence of a cosmological constant. We then test the accuracy of the formulae comparing them with fully relativistic non perturbative numerical calculations for different cases for the density profile. The low red-shift expansion we obtain gives results more precise than perturbation theory since is based on the use of an exact solution of Einstein's field equations. For larger density contrasts the low red-shift formulae accuracy improves respect to the perturbation theory accuracy because the latter is based on the assumption of a small density contrast, while the former does not rely on such assumption. The formulae can be used to take into account the effects on the Hubble expansion parameter due to the monopole component of the local structure. If the $H(z)$ observations will show deviations from the $\Lambda CDM$ prediction compatible with the formulae we have derived, this could be considered an independent evidence of the existence of a local inhomogeneity, and the formulae could be used to determine the characteristics of this local structure.


I. INTRODUCTION
Cosmic microwave background (CMB) radiation [1] observations have pointed to an apparent discrepancy between the value of the Hubble parameter H 0 deduced from cosmological data and the value inferred from local astrophysical observations [2]. Spatial homogeneity and isotropy are some of the main assumptions of the standard cosmological model used in fitting these data. Nevertheless we cannot exclude the presence of a local inhomogeneity around us which could affect our interpretation of cosmological data [3][4][5]. As shown in [6] such a local inhomogeneity could have a negligible effect on the CMB radiation since it would not significantly change the distance to the last scattering surface but it could have much larger effects on local measurements.
The effects of a local inhomogeneity on cosmological observations have been studied already for different cases [3,4, such as equation of state of dark energy or the luminosity distance [3,4,20]. It has been shown for example that the value of the cosmological constant could be affected, significantly by the presence of local inhomogeneity seeded by primordial curvature perturbations [6], which could also lead to the wrong conclusion of a varying equation of state for dark energy while only a cosmological constant is present. The origin of these effects is that spatial inhomogeneities can change the energy of the propagating photons, contaminating the cosmological red-shift due to the universe expansion, and consequently introducing some errors in the estimation of parameters based on cosmological models which ignore the effects of the inhomogeneities.
In this paper we will focus on the low red-shift effects of inhomogeneities on the Hubble expansion parameter, adopting a non pertubative analytical approach based on the use of an exact solution of Einstein's equations to model the local structure. We first derive the red-shift expansion of the geodesics equations , and use it to obtain the expansion of H(z). We then compute a formula for δH H ,the relative difference between the ΛCDM and the inhomogeneous case. Finally we compare the formulae with the numerical calculations based on the integration of the Einstein's equation and the geodesics equations, finding a good agreement.

II. MODELING THE LOCAL UNIVERSE
We use the LTB solution to model the monopole component of the local structure [27][28][29] where R is a function of the time coordinate t and the radial coordinate r, E(r) is an arbitrary function of r, and R ′ (t, r) = ∂ r R(t, r). The Einstein's equations imply that where M(r) is an arbitrary function of r,Ṙ = ∂ t R(t, r) and we choose a system of units in which c = 8πG = 1.
To compute H(z) we need to solve the radial null geodesics [30] dr dz and then we substitute in the formula for the Hubble parameter in a LTB space [31,32] where The analytical solution can be derived [33,34] introducing a new coordinate η = η(t, r), and new functions ρ 0 (r) and k(r) given by the so called FLRW gauge. We can then express Eq. (2) in the form The coordinate η, which can be considered a generalization of the conformal time in a homogeneous FLRW universe, is defined implicitly by Eq. (8). The relation between t and η is obtained by integrating of Eq. (10) and is given by [23] t(η, r) where t b (r) is a functional constant of integration called bang function, since it corresponds to the fact that in these models the scalar factor can vanish at different times at different locations.
We will consider models with t b (r) = 0. The solution of eq.(13) can then be written in the form where ℘(x; g 2 , g 3 ) is the Weierstrass elliptic function and In terms of η and a(η, r) the radial null geodesics and the Hubble parameter are given by [10] dη dz = − ∂ r t(η, r) + G(η, r) H(η, r) = H(t(η, r), r) , where The function G(η, r) has an explicit analytical form, making the above geodesics equations particularly suitable for a low-red-shift expansion.

III. LOW RED-SHIFT EXPANSION OF THE HUBBLE PARAMETER H(z)
In order to obtain a low-redshift formula for the Hubble parameter we expand the function according to where we defined the function t 0 (r) by Using the properties of the Weierstrass elliptic functions ℘, ζ and σ [35] we can compute the integral in eq. (14), obtaining where ℘ ′ is the derivative of the Weierstrass' elliptic function ℘. ℘ −1 , ζ and σ are defined by the equations: The use of the exact expression for t(η, r) improves the accuracy for the expansion for the geodesics respect to previous calculations [20], which were based on a pertubative expansion of t 0 (r), rather than the use of the exact value.
Now we can find the low red-shift Taylor expansion for the geodesic equations [3], and then calculate the Hubble parameter. We expand the solution of the geodesic equations according to After substituting the above expansion in the geodesic equations we can map the solution of the system of differential equations into the solution of a system of algebraic equations for the coefficients of the expansion. The general expressions for the coefficients are rather complicated, and are given in details in appendix B. Here we give the formulae for the case in which k 0 = 0, which is enough to understand qualitatively the effects of the inhomogeneity. The term k 0 corresponds in fact to the homogeneous component of the curvature function, which in absence of inhomogeneities is simply the curvature of a FLRW model, and as such is not associated to any new physical effect not already known from standard cosmology.
For the geodesics we get: where Ω M , Ω Λ , T 0 and K n are dimensionless quantities given by [14] We have also used the following definitions where ζ is the Weierstrass Zeta Function. As we can see in the above formulae the effects of the inhomogeneity start to show respectively at first order for η(z) and second order for r(z).
In order to obtain a formula for the Hubble parameter as a function of the red-shift we need to substitute eq. (17)(18) in eq. (19) H(z) = H(η(z), r(z)) . After expanding up to second order in z we get: The procedure to reduce the analytical formula to this form is rather complicated since it involves to express wherever possible all the intermediate expressions in terms of physically meaningful quantities using the properties of the Weierestrass elliptic functions [35] and we give more details in appendix A.
As we can see from the first order coefficient H 1 , at leading order in t ′ 0 (0) determines the sign of the correction respect to the homogeneous case, and for this reason we plot t ′ 0 (0) as a function of K 1 in figure 1. At second order we have a more complicated dependency for H 2 , which involves also K 2 and t ′′ 0 (r). We can easily interpret the linear behavior shown fig. 1, applying the chain rule for the

IV. RELATIVE DIFFERENCE OF H(z) RESPECT TO THE HOMOGENEOUS CASE
For a flat FLRW solution the expansion rate is given by Since we want to compare the inhomogeneous case with the flat FLRW case we define the relative difference as δH(z) where we are denoting with H ΛLT B (z) the expansion rate defined in eq.(42).
We can now expand the above expression at low-redshift to get It is straightforward to verify that in the homogeneous limit, when K 0 = K 1 = K 2 = 0 , as expected, δH 1 = δH 2 = 0, since t ′ 0 (0) = t ′′ 0 (0) = 0 . From the expression for δH 1 H it is clear the crucial role played by t ′ 0 (0), which determines the sign of the relative difference at leading order in red shift, and according to eq.(46-47) is proportional to K 1 .
Using eq.(47) we can also see that for a fixed K 0 the sign of K 1 determines the sign of δH 1 H at very low red-shift when the second order contributions can be neglected. This in in good agreement with fig.(3-4), which show an approximate linear behavior with opposite slope, corresponding to different values of K 1 .

A. Testing the accuracy of the formulae
In order to test the accuracy the analytical formulae we consider inhomogeneities modeled by the following type of k(r): The parameter ǫ controls how large is the difference from a homogeneous case. As seen in figure   2 the analytical formula is in good agreement with H(z) obtained by integrating numerically the geodesics and background equations. The formula for δH H shows also a good agreement with numerical results as shown in fig. (3-4). While the expansion for r(z) depends on our choice of radial coordinate, the formulae for the H(z) are independent of it, since both H and the redshift are physically observable quantities, and as such are independent of the gauge choice. Since in the derivation of the formulae the inhomogeneity profile is determined by the coefficients of the expansion of the function K(r), there is still some coordinate dependency left in the way we parameterize the inhomogeneity. While our choice of gauge is quite natural, since in the F RW gauge, corresponding to ρ 0 (r) = const, the radial coordinates reduces to the radial F RW comoving coordinate in the limit in which the function k(r) goes to zero, a totally coordinate independent formula would still be preferable. For this reason in the future it will be interesting to derive formulae which are completely independent of the choice of the coordinate system, parameterizing the inhomogeneity in terms of the redshift expansion of the density ρ(z), which is a physical observable, instead of using the expansion of K(r).

V. CONCLUSIONS
We have derived a low-redshift analytical formula for the Hubble parameter for an observer at the center of a spherically symmetric matter distribution, using an exact solution of Einstein's field equations. Such a formula can capture the non linear low redshift effects since it does not rely on perturbation theory, and shows a good agreement with numerical calculations.
Since the observed apparent discrepancy in the H 0 value estimation is related to low red-shift observations, the obtained formulae could be used to account for the effects of the monopole component of the local structure.
In the future it will be interesting to derive formulae which do not depend on the choice of the coordinate system, parameterizing the inhomogeneity in terms of the density ρ(z) as a function of the redshift, which is an observable quantity completely independent from the coordinates.
It would be also important to study the directional dependence of the Hubble parameter, which could partially explain the H 0 tension [6]. This could be achieved by deriving other formulae similar to the ones obtained in this article, but using for example non spherically symmetric solutions of Einstein's equations to model the higher multipoles of the local structure, or considering off-center observers in spherically symmetric spaces. where By inverting the previous equations we obtain the following relations We can then substitute the above expressions everywhere ℘ and ℘ ′ appear, making the final formula only depending on physical quantities such as H 0 . In order to simplify the results we have also used the Einstein's equation at the center (η 0 , 0) Appendix B: General formulae In this appendix we will give the formulae for the cases in which K 0 is not zero.
For the expansion of the geodesics equations we obtain