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## The backreaction of inhomogeneities

**The backreaction problem in cosmology**

The cosmological principle asserts that there exists a scale above which the Universe can be considered as homogeneous and isotropic. The SMC then assumes that the inhomogeneities under this scale do not aﬀect the dynamics of domains of size bigger than the homogeneity scale. This implies that the expansion of such domains is given by the (homogeneous) FLRW solution of general relativity, and the Friedmann expansion laws. That is why the dynamics of structure, under the SMC, is generally solved as a deviation around a FLRW background expansion.

This assumption is not a consequence of the cosmological principle, and is an additional hypothesis. In reality the inhomogeneities (with scale necessarily smaller than the homogeneity scale) might aﬀect the expansion at large scales. This eﬀect is called the backreaction. The main question surrounding this phenomenon is to know whether or not it could be important enough to explain the recent acceleration of the local scale factor, and if the dark energy could be entirely explained by the inhomogeneities. This is the problem of backreaction in cosmology (see Buchert, 2008).

A way to tackle this problem is by averaging the 3+1-Einstein equations on a spatial domain supposedly bigger than the homogeneity scale, and derive the expansion law of this domain. This approach was followed by Buchert (2000) for irrotational dust fluids, Buchert (2001) for irrotational perfect fluids, and by Buchert et al. (2020) for general fluids. In the following sections we present the case of an irrotational dust fluid.

**Averaging procedure**

As we seek for information on the average properties of the fluid, this requires the use of equations featuring the kinematical variables related to this fluid. These equations correspond to the 1+3-Einstein equations. We also want to perform an average over a spatial domain, which requires the introduction of a foliation and the use of the 3+1-Einstein equations. A way to deal with both of these systems is to assume that the fluid is a Cauchy fluid, i.e. irrotational and such that the foliation it defines is a Cauchy foliation. We will then perform the spatial average on this foliation. We also take a dust fluid, which is generally assumed for the description of the Universe after the surface of last scattering. Therefore, we have N = 1, n = u, K = −Θ , Ω = 0, and ǫ = ρ the rest mass density.

Let us consider a compact domain D on the fluid orthogonal spatial hypersur-faces, which is propagated between each slice along the fluid flow. It is defined as a time independent domain in the comoving class of coordinates X0u, i.e. a Lagrangian domain. This ensures that the total fluid rest mass in D is conserved through time.

where VD (t) := D det(hab)d3x is the volume of D. This definition ensures that for a spatially constant scalar ψ(t), we have ψ D = ψ.

The averaging procedure (1.79) is only well defined for scalar fields. For tensor fields, it would require the comparison of the components of the tensor at diﬀerent points in Σt, which would depend on the coordinates, and therefore would not be covariant. The problem of averaging procedures on tensors, see Zalaletdinov (1992) for a proposed formalism, will not be addressed in this thesis.

**Averaged expansion laws and the backreaction**

**Commutation rule**

A key point in the emergence of backreaction is the non-commutativity of averaging and dynamics (Ellis, 1984; Ellis & Buchert, 2005): the averaging operator • D do not commute with the Lagrangian evolution operator u∂t|0 (this notation is defined in section 1.1.2.6). The commutation rule is (Buchert & Ehlers, 1997; Buchert, 2000) Therefore θ D corresponds to the expansion rate of the volume of the domain D.

**Caveats of this approach to the backreaction problem**

One of the caveats of the above approach to the backreaction problem at large scales, is the arbitrariness of the domain D. A natural way to deal with this problem is to consider a spatially closed universe Σ and to take the domain D to be Σ. In this case, the backreaction variables QΣ and WΣ quantify the eﬀect of all the inhomogeneities of this universe on its expansion. In this thesis, we will assume that our Universe is closed, and will only study the backreaction on its whole volume. Note that this might be a restriction to the study of the backreaction as the current acceleration of the scale factor could be a local eﬀect, only needing a backreaction on medium scales.

Because the average procedure can only be done on the scalar equations, part of the dynamics is lost in the average process. As a consequence, the equations (1.83)– (1.85) are not closed: there are four variables (aD , ρ D , QD , R D ) for three equa-tions. Therefore closure conditions are generally assumed using cosmological mod-els. Another way to deal with this problem would be to assume that our Universe is well described by Newtonian gravitation, and study the backreaction under the Newtonian cosmology. This is presented in the next section.

**Table of contents :**

**1 Introduction **

1.1 The 3+1 and 1+3 formalisms of general relativity

1.2 The standard model of cosmology

1.3 The backreaction of inhomogeneities

1.4 Newtonian cosmology

1.5 Strategy of the thesis

References

**2 1+3-Newton-Cartan system and Newton-Cartan cosmology **

2.1 Why Newton-Cartan?

2.2 Galilean spacetimes

2.3 The 1+3-Newton-Cartan equations

2.4 Space expansion in Newton-Cartan

2.5 Gravitational field and cosmological equations in Newton-Cartan

2.6 Observers in the Newton-Cartan theory

2.7 Conclusion

References

**3 On non-Euclidean Newtonian theories and their global backreaction**

3.1 Introduction of a spatial Ricci curvature

3.2 Poisson equation approach

3.3 The Newton-Cartan approach

3.4 Lorentzian manifold from a solution of the Newton’s equations

3.5 Conclusion

References

**4 Galilean limit of general relativity**

4.1 Motivations

4.2 Galilean limit of Lorentzian structures

4.3 The (Euclidean) Newtonian limit

4.4 The non-Euclidean Newtonian limit

4.5 Backreaction from post-Newtonian terms

4.6 Conclusion

References

**5 Towards non-Euclidean relativistic simulations**

5.1 Lattice cosmology

5.2 Relativistic numerical cosmology with fluid models

5.3 Probing the effect of topology with the BSSN formalism

5.4 Which topology to choose?

5.5 Conclusion

References

**6 Conclusion and perspectives**