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Taylor expansion of luminosity distance in Szekeres cosmological models: Effects of local structures evolution on cosmographic parameters

We consider the Goode-Wainwright representation of the Szekeres cosmological models and calculate the Taylor expansion of the luminosity distance in order to study the effects of the inhomogeneities on cosmographic parameters. Without making a particular choice for the arbitrary functions defining the metric, we Taylor expand up to the second order in redshift for Family I and up to the third order for Family II Szekeres metrics under the hypotesis, based on observation, that local structure formation is over. In a conservative fashion, we also allow for the existence of a non null cosmological constant.


Introduction
At the end of the nineties, it has been proved that the Universe expansion is accelerating [1,2].
Assuming that General Relativity (GR) is the theory of gravitation (valid at every scale, from the Solar System to galaxy clusters), in order to explain this phenomenon one needs a new form of energy with negative pressure able to contrast gravitational attraction which, otherwise, would slow down expansion. The most successful model proposed in order to explain observations is ΛCDM: the Universe is assumed (at large scales) homogeneous and isotropic; it is described by flat Friedmann-Lamaître-Robertson-Walker (FLRW) metric; along with baryonic matter it contains a component of cold dark matter (in order to explain spiral galaxies rotation curves and structures fortmation) and a component of dark energy described by the cosmological constant Λ. At present time the dark energy is about 68% of the content of the Universe [5].
Even if ΛCDM is in very good agreement with observations (see [5] for recent results), it has some flaws (see [7,50], for example): the cosmological constant is usually interpreted in terms of vacuum energy, but its value is way too small: it is known, in fact, that there is a difference of over 100 orders of magnitude between the theoretical prediction and the actual value of Λ; furthermore the acceleration starts at redshifts close to 0 (the coincidence problem) and it is very difficult to explain why in ΛCDM model.
In the litterature, numerous different approaches have been proposed in order to explain the acceleration and overcome ΛCDM problems: some try to devise a new theory of gravity, like f (R) theories (see for example [6]); other consider new forms of energy or fields that have the same effects as the cosmological constant.
Another approach is to consider the effects of inhomogeneities: it is evident that, at least at small scales, the universe is not homogeneous and isotropic, therefore it is natural to ask whether inhomogeneities can have some effect on the expansion. Among these approaches we mention the cosmological backreaction (see for example [11] - [20]), and the use of non homogeneous metrics solution of the Einstein's equations. In the former, one studies the effects of local structure formation and evolution on the cosmic expansion. In the latter, one studies the effect of inhomogeneities on observations.
A good review of all these approaches is [7]. In this paper we follow the last approach and use Szekers models, exact non homogeneous solution of the Einstein's equations with non rotational dust. In a conservative fashion, we consider a non null cosmological constant.
In particular we work with the Goode-Wainwright representation, in which it is evident that the models describe non-linear perturbation of a FLWR background.
Our aim is to calculate the Taylor expansion of luminosity distance of Szekeres models in order to study the effects of inhomogeneities and structure evolution on the deacceleration and jerk parameters. This paper is organized as follows: in section 2, we briefly review the properties and the definition of the Szekeres models and define the redshift; in section 3 we calculate the Taylor expansion of the luminosity distance; in section 4 we discuss the direction dependance of our cosmographic parameters; in sections 5 and 6 we compare our effective parameters with the observed ones and discuss the effects that local inhomogeneities and local structures evolution have on the acceleration; in the appendix A, we report some relations needed in our calculation; in the appendix B, we report the Christoffel symbols of the considered metrics; and in appendix C we calculate null geodesics.

Szekeres cosmological models
Szekeres cosmological models are solutions of the Einstein's Field equations of the type: with irrotational dust as a source (see [34] - [41]). One can also include a cosmological constant as a source (see for example [40] and [43]) 1 . In their most general form they have no killing vectors [42] and are therefore inhomogeneous. 2 There are two families of this metric: the one with β, r = 0 (Family I) and the one with β, r = 0 (Family II). The function α always depends on all of the variables.
Goode and Wainwright gave a representation of the metric for the case Λ = 0 in which it is evident that the models describe non linear perturbations of the FLRW metrics. It also encompasses properties of both families [38]. The metric is given by: (2. 2) The function S(t,r) is the solution of the equation: where M(r) is an arbitrary function linked to the matter density (see later); in the case a cosmological constant is present, we have [43]: With or without a cosmological constant, we have: where f ± (t) are independent solutions of the (Raychaudhuri) equation: 3 and the β ± (r) functions are arbitrary: choosing β ± ≡ 0 leads to Robertson-Walker models, written in an unusual coordinate system (see [40]). The two families are defined as follows: 2.1 Family I (β, r = 0) M, 2 r +T, 2 r = 0, and S = S(t, r). 4 Moreover: f is arbitrary and: Functions f , a, b, c, d, M and T are arbitrary, but one should remember that there is the freedom to make a change of coordinates of the form r → g(r), so, keeping in mind condition (2.8), there are actually five arbitrary functions (see also [38]).
2.2 Family II (β, r = 0) M and T are constants, S = S(t) and W = 1. Moreover: a, b, c are arbitrary. In [43] it is shown that for the case κ = 0 and Λ = 0, we have: Function a, b, c, β ± are arbitrary, but with the coordinate freedom reported above, one can see that there are actually four arbitrary functions (see [38]).

Density contrast and shear
In both families the density is given by: (2.14) The density contrast is given by: for both families (but see [46]). Since the density is non negative, one needs δ > −1.
The non-null components of the observer's rate of shear tensor are given by [44]: where we defined σ := 2 3δ 1 + δ the first eigenvalue of the rate of the shear tensor.
We emphasize that it depends onδ, the rate of structure growth.

Cosmographic parameters
We define the Hubble parameter in the usual fashion as: 5 where we defined: It is easy to prove that we have also the usual relation for the present time cosmological parameters: It is not difficult to prove that, if we define the deacceleration parameter associated to H as: we have, at the present epoch (S ≡ 1): One can also define a jerk parameter: It is easy to show that, at present time (with S = 1), we have j O ≡ 1.

Null geodesics and redshift
Null geodesics are defined by the condition: where k α = dx α dλ , and λ is an affine parameter: λ O is the position of the observer, λ S is the position of the source. The geodesics equation is: We give their expression in appendix C (the Christoffel symbols needed for the calculation are reported in appendix B).
The redshift z is defined as: (2.26)

Luminosity distance
Since there is no evidence for local structure growth, we make the assumption thatδ(0) = 0; from equation (2.16), this means that We base our work on Sachs equations [48], and calculate angular diameter distance D A from [49]: where: Σ is the shear of the light bundle, R αβ is the Ricci tensor, C αβµν is the Weyl tensor, k µ is the null vector, µ is space-like and orthogonal to k µ , and, finally, λ is the affine parameter.
In terms of redshift, we have, for the first equation: The initial conditions are [50]: We impose also Σ(0) = 0 (see [43]). The term depending on Ricci tensor, is given by (we use the definition of the redshift, eqn. (2.26)): The coefficients in front of derivatives can be calculated from the definition of redshift given in equation (2.26) and the geodetic equations given in appendix C: where we remind that W ≡ 1 for Family II models and, using the definition of k α in section given 2.5: (3.8) 6 In the following a dot˙stands for a time derivative, while a prime stands for a redshift derivative.
It is easy to show that in the first line of equations (3.8), one has: so, in order to have a definite expression for this term, one has to give an expression at least for the arbitrary functions M(r) and T (r), since these are the arbitrary functions defining S (see section 2 and [40]). We are interested in the effects of the inhomogeneities on the acceleration of the expansion and on the possible effects on the jerk parameter which contains information on the dark energy equation of state. Therefore we only need the Taylor expansion of D A (z) up to the third order: and susbstitute it in (3.3) with Taylor expanded coefficients (3.6) and (3.7), in order to get A e B.
In the following two subsection we will give an expression for A and B for the two families.

Family I Szekeres cosmological models
Since our aim here is not to construct a phenomenological model, we will not try to give a definite expression for the arbitrary functions, instead we will stop at the second order for Family I and study the effects of the dishomogeneities only on the deacceleration parameter. 7 In Family II we do not need to specify a model and we will go up to the third order and study the effects of dishomogeneities on the jerk parameter.
It is easy to get: where q O is given in (2.22). Luminosity distance is given by: where: D where we defined the effective deacceleration parameter for Family I: It depends on the radial derivative of the Hubble parameter given in (3.9), on the direction of observation (see section 4) and on σ (0), which, we remind, is linked to the inhomogeneities growth (see equation (2.16)).

Family II Szekeres cosmological models
As previously said, in the case of Family II cosmological models, we can easily Taylor expand the angular diameter distance up to the third order without specifying a particular model. Using equations (C.1), (C.4), we find: for the second order, while for the third order, we have:

(3.17)
Luminosity distance is given, as usual, by D L (z) = (1 + z) 2 D A (z), and we find: where: D In the previous equation, we defined the parameter Ω O = 1 − Ω k (see also [6]), and the effective deacceleration parameter for Family II models, q II ef f : (3.20)

Angular dependance and averages
The terms k i k 0 O that appear in the previous equations are direction dependent.
Following, for example, [43], we define, at the observer position, α, the angle formed by the k µ with the r axis, and β, the angle formed with the r − x plane; in this way we have: (4.1) In usual supernova surveys, one does not consider the direction of observation (this is because of the assumed homogeneity and isotropy), so in order to confront our results with the ones present in the litterature, we will need to average over all directions; the formula for the average is simply: In the following two sections we will calculate the averaged parameters and confront them with the ones present in litterature.
We stress that in case one wanted to fit our luminosity distance with data, one should not use the averaged parameters, but q I ef f or q II ef f as given in (3.15) and (3.20) respectively.

The effective deacceleration parameters
The usual expression for the second order of the luminosity distance is [47]: where q obs is the observed deacceleration parameter. We defined the deacceleration parameteres in equation (3.15) for Family I and in equation (3.20). Using equations (4.1)-(4.3), we rewrite them respectively as: Using equation (4.4), we see that the average gives the same result for both families: where, in the last line, we used equations (2.22). From the last equality, we can see the effects of the shear and of the local structure evolution on the dark energy: if the σ (0) term in the square brackets is positive, less dark energy is needed in order to explain the observed accelerated expansion.
From equation (5.5), we see that in order to have a positive σ (0) we need a negative density contrast, so the observer has to be in a void.

Comparison with observations
The idea that we might live in a void is not new: see for example references [22]- [33] and section 10.1 of [7].
In particular, in [23] the authors use 44 close SNe Ia (20-300h −1 Mpc) to study the monopole of peculiar velocities of the host galaxies: they find that their data are consistent with a local void with an underdensity of about 20%, so δ(0) ≈ −0.20.
In [26], the authors fit 3-yr WMAP data and SNe Ia with the hypothesis that we live in a spherical-symmetric void described by a LTB metric: they find that they need δ(0) ≈ −0.4 and that the observer must be close to the center of the sphere.
More recently, in [33] the authors use a LTB void metric to describe a local void and fit their model using CMB, SN, BAO and H 0 allowing for a non null space curvature: they find that their model fit the data with a δ(0) = −0.65 void that extends up to z ≈ 1 (gigaparsec scale); they also find Ω k = −0.2 and that the model is consistent with the CMB dipole if the observer is displaced by about 15 Mpc from the center.
The fact that the observer must be close to the center of the void in order to be consistent with the CMB data is a violation of the Cosmological Principle, since that position is obviously a particular one. Even if the void is non-symmetric (so that there is not an actual center), there is still a weak violation of the Principle, since usually galaxies are found in clusters not in voids [50].
With these issues in mind, we can try to use the reported values for δ(0) and see how much the value of Ω Λ is affected by the void assumption in our case.
Using the equation, derived in appendix A: we can reformulate the equation (5.4) in terms of the local density contrast: If the Universe is flat, with Ω m = 0.315 and q obs = −0.527 (see [5]), we find that Ω Λ = 0.66, 0.64, respectively for the two considered cases: corresponding to a reduction of 5-9% of dark energy. In the third case we find a reduction of about 10%.
We also notice that we cannot have acceleration considering only the σ (0) term: in fact, if we put Ω Λ ≡ 0 and considered the value δ(0) = −1, we would get q ef f = 0. 8

D
(3) L and the effective jerk parameter j ef f The usual expression for D (3) L in ΛCDM models (for arbitrary values of κ) is given by (see for example [6,51]): D where one expects j obs ≡ 1 [10,51]. In order to compare our D At the end, we get:
In some recent works ( [51,52], for example) it has been reported that the jerk parameter might be less than 1.
In the reference [51], the authors consider 109 GRBs (0.033 < z < 8.26) and 557 SNe Ia from the Union 2 data set (0.015 < z < 1.55) and fit the luminosity distance up to the 4 th order with a Markov Chain Monte Carlo (MCMC); they find for the jerk parameter the following interval of confidence: so they conclude that at 2σ of confidence level the jerk parameter is less than 1.
In [52], authors use H O , SNe Ia, BAO and GRBs to fit the luminosity distance, using again a MCMC: they also find that the jerk parameter is less than 1.
On the other side, there are other works that confirm the ΛCDM model (see for example [54,55]), while in [56] the authors find j O = 2.16 +0.86 −0.75 .

Conclusions
Szekeres cosmological models are inhomogeneus and anisotropic solutions of the Einstein's Field equations with irrotational dust and a cosmological constant as a source. Starting from the Sachs equations, we have calculated the Taylor expansion of the Szekeres luminosity distance for both Family I and Family II allowing for the existence of a cosmological constant. Relying on observations, we make the hypotesis that there is no structure growth at the observer position:δ(0) = 0.
Confronting with the usual expansion of the ΛCDM luminosity distance, we have found corrections proportional to σ (0) that are also direction dependent. σ is the eigenvalue of the observer's rate of the shear tensor σ = σ rr and is linked to the evolution of the density contrast (see equation (2.16)). Therefore, inhomogeneities and local structure evolution affect the way an observer sees the expansion.
In Family I models, at second order, there is also a correction proportional to ∂H ∂r Usual supernova surveys do not consider angular dependance because of the assumed homogeneity and isotropy, so we had to perform an average over all directions before we could compare our parameters with the observed ones.
Comparing with some results reported in litterature, we have found that if we lived in a void, we could reduce of about 5-10% the quantity of dark energy needed to explain the accelerated expansion and also why in some recent papers jerk parameter is found to be less than 1.

A Useful relations
In this appendix we give a derivation of equation (5.5), which relates σ (0) and the density contrast δ(0) at the observer's position, and we give an analogue expression also for σ (0).
Since, in Family II models, we can impose that H O = S O = 1, we have: Since for Family I, we have (HW ) O = 1, we have, in this case: Using the definition of redshift and of k 0 given in 2.5, the first derivative of the σ with respect to the redshift calculated at the position of the observer is: where in the third line we used equation (2.26) and the definition of k 0 given in 2.5. In the last line, the second term is zero because of equation (A.1). We now use the Raychaudhury equation (2.6), so we get: In the second line we used again (A.1) and (A.2) and also the definition of Ω m . Simplifying, we get: which is the expression given in (5.5).
On the same line, we find, for Family II: where in the second line we used (A.6). We also report, without proving them, two relations involving the first and second time derivatives of the Hubble parameter and the deacceleration and the jerk parameters needed in our calculations: 9Ḣ

B Christoffel symbols
In this appendix we report the non-null Christoffel symbols for the Goode-Wainwright representation for both families needed for calculating the geodesics.