2.13 Dark-matter surrogates in theories of modified gravity

2.13.1 Extra fields in modified gravity

The idea that the dark universe may be a signal of modified gravity has led to the development of a plethora of theories. From polynomials in curvature invariants, preferred reference frames, UV and IR modifications and extra dimensions, all lead to significant modifications to the gravitational sector. A universal feature that seems to emerge in such theories is the existence of fields that may serve as a proxy to dark matter. This should not be unexpected. On a case by case basis, one can see that modifications to gravity generically lead to extra degrees of freedom.

For example, polynomials in curvature invariants lead to higher-derivative theories which inevitably imply extra (often unstable) solutions that can play the role of dark matter. This can be made patently obvious when mapping such theories onto the Einstein frame with an addition scalar field (Scalar-Tensor theories). Einstein-Aether theories [989Jump To The Next Citation Point] explicitly introduce an extra time-like vector field. The time-like constraint locks the background, leading to modifications to the background expansion; perturbations in the vector field can, under certain conditions, lead to growth of structure, mimicking the effect of pressureless dark matter. The vector field plays the same role in TeVeS [117], where two extra fields are introduced to modify the gravitational dynamics. And the same effects come into play in bigravity models [83Jump To The Next Citation Point] where two metrics are explicitly introduced – the scalar modes of the second metric can play the role of dark matter.

In what follows we briefly focus on three of the above cases where extra gravitational degrees of freedom play the role of dark matter: Einstein-Aether models, TeVeS models and bigravity models. We will look at the Einstein-Aether model more carefully and then briefly discuss the other two cases.

2.13.2 Vector dark matter in Einstein-Aether models

As we have seen in a previous section, Einstein-Aether models introduce a time-like vector field Aa into gravitational dynamics. The four vector a A can be expanded as μ j 𝜖- A = (1 + 𝜖X, 𝜖∂ Z ) = (1 + 𝜖X, a2∂jZ) [989]. In Fourier space we have A μ = (1 − 𝜖Ψ,i𝜖kjV ) a, where, for computational convenience, we have defined V ≡ Z∕a and have used the fact that the constraint fixes X = − Ψ.

The evolution equation for the perturbation in the vector field becomes (where primes denote derivatives with respect to conformal time)


The perturbation in the vector field is sourced by the two gravitational potentials Φ and Ψ and will in turn source them through Einstein’s equations. The Poisson equation takes the form


To understand why the vector field can play the role of dark matter it is instructive to study the effect of the vector field during matter domination. It should give us a sense of how in the generalized Einstein-Aether case, the growth of structure is affected. Let us consider the simplest case in which the the dominant remaining contribution to the energy density is baryonic, treated as a pressureless perfect fluid with energy-momentum tensor T and let us introduce the variable V′ ≡ E. For ease of illustration we will initially consider only the case where V is described by a growing monomial, i.e. V = V0(η∕η0)p. During the matter era we have


with lE ≡ − (c1(2 + p)n + 2α (1 − 2n )n), lS ≡ − (c1 + c3)n(6n − p − 10), and

[ ( )2 ]n−1 ξ(k ) ∼ γV0 (k)η−pk6− 6n 3α Ωm H0- , (2.13.5 ) 0 hub M
where khub ≡ 1∕ηtoday. Hence, the vector field affects our evolution equations for the matter and metric perturbations only through its contribution to the energy density and its anisotropic stress. On large scales, kη ≪ 1, and assuming adiabatic initial conditions for the fields δ, Φ and 𝜃, this leads to
---6lS-ξ(k-)----5+p−6n δ = C1 (k ) + (10 + p − 6n )η , (2.13.6 )
where C1 is a constant of integration and we have omitted the decaying mode. Therefore, even before horizon crossing, the anisotropic stress term due to the vector field can influence the time evolution of the baryon density contrast.

On small scales (k η ≫ 1), we find

(1lE + lS) δ(k,η) = C2 (k )η2 + --------2----------------ξ(k)(kη)2η5+p−6n, (2.13.7 ) (5 + p − 6n )(10 + p − 6n )
where C2(k) is a constant of integration. Hence, for sub-horizon modes, the influence of the vector field on the evolution of δ is a combination of the effect of the energy density and anisotropic stress contributions though both, in this limit, result in the same contributions to the scale dependence and time evolution of the density contrast. The net effect is that, for particular choices of parameters in the action, the perturbations in the vector field can enhance the growth of the baryon density contrast, very much along the lines of dark matter in the dark matter dominated scenario.

2.13.3 Scalar and tensors in TeVeS

We have already come across the effect of the extra fields of TeVeS. Recall that, in TeVeS, as well as a metric (tensor) field, there is a time-like vector field and a scalar field both of which map the two frames on to each other. While at the background level the extra fields contribute to modifying the overall dynamics, they do not contribute significantly to the overall energy density. This is not so at the perturbative level. The field equations for the scalar modes of all three fields can be found in the conformal Newtonian gauge in [841]. While the perturbations in the scalar field will have a negligible effect, the space-like perturbation in the vector field has an intriguing property: it leads to growth. [318] have shown that the growing vector field feeds into the Einstein equations and gives rise to a growing mode in the gravitational potentials and in the baryon density. Thus, baryons will be aided by the vector field leading to an effect akin to that of pressureless dark matter. The effect is very much akin to that of the vector field in Einstein-Aether models – in fact it is possible to map TeVeS models onto a specific subclass of Einstein-Aether models. Hence the discussion above for Einstein-Aether scenarios can be used in the case of TeVeS.

2.13.4 Tensor dark matter in models of bigravity

In bigravity theories [83], one considers two metrics: a dynamical metric gμν and a background metric, &tidle;g αβ. As in TeVeS, the dynamical metric is used to construct the energy-momentum tensor of the non-gravitational fields and is what is used to define the geodesic equations of test particles. The equations that define its evolution are usually not the Einstein field equations but may be defined in terms of the background metric.

Often one has that &tidle;gαβ is dynamical, with a corresponding term in the gravitational action. It then becomes necessary to link &tidle;gαβ to gμν with the background metric determining the field equations of the dynamical metric through a set of interlinked field equations. In bigravity models both metrics are used to build the Einstein–Hilbert action even though only one of them couples to the matter content. A complete action is of the form

∫ [ ] --1--- 4 √ --- ∘ --- &tidle; &tidle; ∘ ----1 − 1αβ S = 16πG d x − g(R − 2 Λ) + − &tidle;g(R − 2Λ ) − − &tidle;gℓ2(&tidle;g ) gα β , (2.13.8 )
where Λ and &tidle;Λ are two cosmological constant terms and 2 ℓ defines the strength of the linking term between the two actions. The cosmological evolution of perturbations in these theories has been worked out in some detail. It turns out that perturbations in the auxiliary field can be rewritten in the form of a generalized dark matter fluid [453] with fluid density, momentum, pressure and shear that obey evolution equations which are tied to the background evolution. As a result, it is possible to work out cosmological observables such as perturbations in the CMB and large scale structure. If we restrict ourselves to a regime in which &tidle;ρ simply behaves as dark matter, then the best-fit bimetric model will be entirely indistinguishable from the standard CDM scenario.
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