Distortions of Robertson-Walker metric in perturbative cosmology and interpretation as dark matter and cosmological constant

Using the instruments of perturbative cosmology, it is possible to study not only the development of structures, but also the resultant gravitational potentials. These are retarded potentials, depending not on the actual matter density, but on the past one, which is greater. This magnification effect can generate a contribution to dark matter. The expansion rate of the universe is perturbed as well, maybe of a not negligible rate although matter inhomogeneities are little, because the same magnification effect. The resultant distortion of deceleration parameter, with respect to a homogeneous model, could be interpreted with some dark matter, or a cosmological constant, or both. We applied this idea for a simplified case, obtaining cheering results.


Introduction
We know the matter distribution in the universe is almost homogeneous, for enough large scales. This approximation was assumed in the first cosmological models [11] and is confirmed by recent observation [13]. It allows us to study the inhomogeneities with linearized Einstein Equations, as is usually done for models of structure formation [16] [12] [26] [4] [15] [18], obtaining their growing rates. The same linearized equations can give the development of metric distortion, which means the gravitational potentials due to matter inhomogeneities.
The General Relativity states that the gravity is not an instant action, but an interaction with finite speed [10]. It was recently confirmed by the discovery of gravitational waves [1]. This means the potentials from linearized Einstein Equations can be expressed as retarded potentials, choosing the suitable gauge. The contribution from these far, past fields decreases with time and distance, but however it could be dominant if the decreasing of near matter density with time is greater. This would generate a bigger gravitation than what we imagine observing the only mean density, or the near one as well. If all the unexpected gravitation effects are interpreted with a presence of matter, as usual, this would explain why the actual cosmological models need to work some invisible matter, the dark matter. On the other hand, if the inhomogeneous distribution of matter gives less gravitation than a homogeneous one, it could be interpreted as a cosmological constant, that is an unexpected gravitational effect of repulsion. These ideas was yet proposed in a publication [6] and now we apply them with a General Relativity formalism.
At the present days, the search of dark matter didn't give conclusive results. The hot dark matter, even if exists, cannot explains local effects as galaxy rotation curve, since its high velocity forbids links to structures [14]. MaCHOs give a contribution to dark matter, but only for a little, according to recent estimates [25]. The search of WIMPs, conjectured as supersimmetric partners or other kinds of particles, produced no results at the moment[9] [7]. The state of art suggests that the dark matter doesn't exist, or that it is less than what we expect from astrophysical phenomena, because our Cosmological Standard Model should be improved in some way. This possibility is studied by the MOND theories, but no attempt to modify the present theory of gravitation obtained a good fit with data, currently [23] [17]. This situation allows us to propose a new kind of answer, which doesn't need some form of unobservable matter, neither modifications to Einstein's gravitation, but explores unusual consequences of the usual theory.
We can rudely divide the "dark matter phenomena" in two categories: global dark matter, which consists in unexpected values of cosmological parameters, and local dark matter, which comes from observations of astronomical objects. Notice that the dark energy, or cosmological constant, appears only as global phenomena. Indeed, the measures of deceleration parameter [19] [22], together with a zero curvature of space, is the only proof of existence of cosmological constant, and the most important global effect of dark matter. If retarded potentials of matter inhomogeneities modifies enough the universe expansion, it could explains as apparent one or the other, or both. Almost all the other global dark matter effects, as the deuterium abundance and the power spectrum of CMB anisotropies, are referable to a distortion of universe expansion law.
Local dark matter phenomena are only gravitational phenomena, as well, and can be explained with any mechanism that modify the space-time metric, even if it is without local presence of matter. It is self-evident for gravitational lensing [24], but also the unexpected magnitude of galaxy rotation [27] [8] and of the virial in clusters [3] are due to gravitational potential greater than what that the visible matter should generate. The influence of far matter on these phenomena is usually neglected, because a consequence of Birkhoff Theorem [5] states that an isotropic distribution gives no influence to its center, and anyway this influence decrease rapidly with the distance. That's why the basic ingredients of our hypothesis are matter inhomogeneity and retarded potentials: the first one gives anisotropies that avoids Birkhoff Theorem; the second one compensates the decreasing for distance with the magnification for past matter density.
The most problematic aspect for our hypothesis is the structure formation, which models requires dark matter to provide gravitational collapse fast enough to obtain galaxies and clusters nowadays. If dark matter is due to inhomogeneities, it would be both cause and effect of structure formation, maybe retarding too much the development. However, once we have gravitational holes from the far fields, matter would fall into them, producing automatically galaxies concentric with "dark matter halo", what we observe from rotation curve.
In this article, we obtain the linearized equations for retarded potentials, in the flat space case, defining the mean perturbation of the metric and the mean perturbed expansion law. An exact solution for potentials and deceleration parameter is calculated for a particularly simple universe, chosen such that the unperturbed Hubble parameter is constant. The apparent dark matter and/or cosmological constant are so approximately expressed, depending by matter inhomogeneity.
Notations. In this article is used the signature (+1; −1; −1; −1) and units of measurement such that c = 1. The Ricci tensor is defined as with allowed values (τ I ; τ F ) := {τ ∈ R|a(τ ) > 0}. H(τ ) is the Hubble parameter and R(τ ) = τ − τ I the radius of the observable universe, both of them for the unperturbed model.V is a divergenceless vector.

Linearised Einstein Equations
So we apply the perturbative approach on a Robertson-Walker metric, looking for a wave equation for the perturbation.
As usual [20], we consider an unperturbed metric with little perturbation. It is assumed a flat space, so that the calculation are simpler: Similarly: where ρ = ρ +ρ, p = p +p and U µ = aδ µτ +Ũ µ , in general. The unperturbed Ricci tensor and the unperturbed Einstein tensor are well known: The unperturbed Einstein Equations are nothing more the Friedman equations For any superposition of cosmic components ρ = w ρ w , p = w p w = w wρ w and choosing as variables , so w Ω w (0) = 1 we obtain the usual differential equation for the universe expansion: Performing the scalar-vector-tensor decomposition we can express the perturbation on Ricci tensor as: . We can use the geometric condition g µν U µ U ν = 1 = g µν U µ U ν , from which the perturbation of velocities isŨ and so, remembering the inhomogeneities are due only to matter, for whichp = 0, the perturbation of stress-energy tensor is where v(τ ; x) is the field of spatial velocities, and we defined q := (ρ + p) v.
From the Field Equations, we want PDEs for retarded potentials, or gravitational waves. That's why the gauge we fix won't be the Newtonian one neither the synchronous one [20], but the harmonic gauge, what is used to study gravitational waves [26]. Abstracting from the background metric, the harmonic condition on the perturbation of connection isΓ λµ µ = 0.
We obtain a scalar and a vector condition on A, B, h ij : It is exactly the gauge such that, in the perturbation of Ricci tensor, the second order part is a d'alambertian, as we want. Indeed, equation (7) can be rewritten as Following again the gravitational waves formalism [26], we express the Einstein Field Equations as The perturbation on S tensor is and so the linearized Einstein Equations, simplified using equations (4), are From them, we can derive the linearized conservation of energy and momentum:

Retarded potentials and mean perturbed metric
The homogeneous solutions of the PDE system (15) describe gravitational waves on an expanding space-time. For a given distribution of matter and velocities as source, the PDEs return the correspondent space-time metric. For a bounded distribution of matter, the inhomogeneous solution without gravitational waves is such that the metric is asymptotically minkowskian, and we choose this solution as gravitational potential. Similarly to the usual wave equation, the characteristic curves are light rays, and so the potentials are retarded with the speed of light.
First of all, we observe h ij has no traceless source, hence we can choose an inhomogeneous solution h ij = 2Cδ ij . Moreover, the divergenceless partB can be set to zero. Decomposing q := ∇q +q as well, the system remains These all have the PDE form Let G(τ, x; τ ′ , x ′ ) be its Green function, i.e. the asymptotically zero solution for a source δ(τ − τ ′ )δ (3) (x − x ′ ). It will be zero for |x − x ′ | > τ − τ ′ , because causality. It is also homogeneous and isotropic in space: Assuming it is possible to separate variables for a generic source S(τ ; x) = T (τ )S 0 (x), we can express the retarded potential as For a study of global effects, we take the average of these potentials on all the space, obtaining a function depending only by the time. It results For B it is necessary a different procedure, since it is not a component of the metric, but ∇B is. Its mean is zero, because is an integral of an odd function: What we find is a diagonal average metric, which we can set equal to a Robertson-Walker metric with a perturbed time t and a perturbed expansion law a(t): s.t. dt := a(τ ) 1 + 2 A (τ )dτ, a := aã := a(τ ) 1 − 2 C (τ ). (22) A cosmologist living inside our model, could measure the expansion rate and acceleration, evaluating them with a Friedman model (5), assuming perfectly homogeneous components.
To justify the distortion due to A and C , with respect to the observed matter and energy, he can introduce adequate dark matter and/or cosmological constant, obtaining an expected expansion law a D s.t.
Calling Ω B := Ω w | w=0 the baryonic matter, the total matter for this cosmologist would be Ω M := Ω B + Ω D . He would estimate these quantities setting . (24) 4 Constant coefficient case: solution, matter source and perturbed deceleration parameter It is not easy to solve a PDE like (18), since has not constant coefficients. To obtain some explicit results at least for a simplified case, from now on we will choose a universe with constant Hubble parameter, so that Moreover, in this case the system is completely decupled. We obtain such a situation if there is only one dominating component Ω w , with w = − 1 3 . It is the same expansion as in [6], but treated in General Relativity.
Through Fourier transformation, we obtain the Green function for (18) with constant coefficients: (26) where we defined K := −K 0 − H 0 2 2 the discriminant, and J 0 is the zeroth order Bessel function, which appears derived. It is easy to recognize the causality in the potential, since the first term has the speed of light and the second one is slower.
For A and B it is K = 3H 2 0 , whence it comes a factor √ 3. For C it is K = −H 2 0 , for which in the Green function J 0 is replaced by I 0 , the modified Bessel function.
In our case w = − 1 3 , the wave equation for B and the energy-momentum conservation become From these, remembering ρ = 3H 2 0 8πG a −2 , we can obtain a condition on acceptableρ: .
Define δ M :=ρ ρ M ∝ρa 3 the density contrast for matter, and assume that variables are separable for it. It results from (28): This means X is an eigenfunction of the laplacian, so it must have negative eigenvalues. Substituting inside p 4 and p 2 , this condition on n becomes − Φ < n < −1 ∨ ϕ < n < 1 (30) where we call ϕ = Φ −1 :=  (20), remembering R(τ ) ≡ +∞ in our case. For A: Similarly, for C: Here we defined the integrals Since I 0 grows exponentially, M is divergent for n ≥ 1, as requested by condition (30). For n < 1 they are The divergence could be interpreted as an infinite quantity of apparent dark matter, for constant coefficient case, due to the expansion law near the first instant τ I . However, any physical theory fails near the Big Bang, and so we should put a cut-off on integrals N , M which made them finite. In our simplified treatment, we will assume the formulas for n > 1 are valid for any n, performing a renormalization with analytical continuation. The quantities of apparent dark matter and cosmological constant can be derived from system (24). In our case, the model (23) becomes Deriving it: Deriving a two times and remembering (24): Here H 0 and t 0 depend from the observable cosmic parameters H D0 , ρ 0 by the algebraic system Equations (38)

Evaluation of dark matter and dark energy magnitude
Since we assumed inhomogeneities are little, so that we could use the perturbative approach, here it is possible to approximate It results ρ ef f (0) = 0 and after derivation we can write Simplifying ρ 0 , it is: This formula gives the relative, apparent quantity of dark matter Ω D0 and cosmological constant Ω Λ0 , with respect to the real quantity of baryonic matter Ω B0 inhomogeneously distributed.
The magnification effect we imagined is confirmed if this ratio has absolute value bigger than 1. Remembering the condition (30), we can try to replace n ∼ = 2 3 ; this gives which proves the existence of a magnification effect, at least if H(τ ) ≡ H 0 . We can compare the formula (42) with the most recent measures of cosmological parameters in our universe [2], remembering they are not taken in a universe with linear expansion: Ω B0 ∼ = 0.043 ± 0.004; q 0 ∼ = −0.53 ± 0.01; Ω M 0 ∼ = 0.315 ± 0.007; Ω Λ0 ∼ = 0.685 ± 0.007; Returning to the theoretical formula (42), we can ask which value fits for n if we want to explain the dark matter, the cosmological constant, or both of them, observed in the real universe. If we consider only the dark matter effects: If we consider only the cosmological constant effects: If we consider both: ρ ef f ρ 0 ∼ = −24.65 ⇒ n ∼ = 0.279.

Conclusions
Our theoretical results don't fit with measures, but it was predictable, since the constant coefficient case we chose is different from the real universe. Our significant goal is to underline the not negligible distorting effect of inhomogeneous matter on deceleration parameter, which suggest to correct the values we believe for dark matter and cosmological constant. Repeating our method, inserting the real expansion law a(τ ) inside equations (15) and integrating numerically them, Ω M 0 and Ω Λ0 should be reduced by some factor. These contribute to dark matter and dark energy effects, eventually with future discoveries, could get to justify completely them. Another generalization of this article could be an evaluation of the gravitational forces due to retarded potentialsg µν . The comparison to the attraction of dark matter in galaxies or cluster could justify it completely or partially. These effects could be linked to the standard deviation ofρ, rather than the average. Since the standard deviation has not linear dependence, the linearized approach could be insufficient.
To obtain a theoretical prevision of the dark matter distribution, mathematical tools to describe the baryonic matter distribution would be necessary. Since the matter inhomogeneity seems to have a fractal shape [21], it could be useful a study of singular distributions as sources in General Relativity, and their application to cosmology.
The study of homogeneous solutions of (18) could be useful for the interpretation of ancient gravitational waves, since universe has expanded from their emission.