9.1 Expansion history

A viable theory of modified gravity, including dark fields or not, should not only be able to reproduce observations in quasi-stationary galactic and extragalactic systems, but also to reproduce all of the major probes of observational cosmology, including (i) the Hubble diagram out to large z, (ii) the anisotropies in the cosmic microwave background (CMB), and (iii) the matter power spectrum on large scales. The first requires a detailed knowledge of FLRW cosmology, and the last two a knowledge of cosmological perturbations on a FLRW background.

Concerning the first point, the FLRW solutions have been extensively studied for TeVeS (Sections 7.3 and 7.4, see, e.g., [70Jump To The Next Citation Point]) and GEA (Section 7.7, see, e.g., [515]) theories, for BIMOND (Section 7.8, see, e.g., [101Jump To The Next Citation Point]), and for theories based on dipolar dark matter (Section 7.9, see, e.g., [60Jump To The Next Citation Point]). In the latter case, the theory [58, 60] has been shown to be strictly equivalent to ΛCDM out to first-order cosmological perturbations (but very different in the galaxy formation regime), together with a natural explanation for Λ ∼ a2 0. For the other theories, it has been shown that the contribution of the extra fields to the overall expansion is subdominant to the baryonic mass and does not affect the overall expansion [151Jump To The Next Citation Point]. Such theories can predict an extremely wide range of cosmological behavior, ranging from accelerated expansion to contraction on a finite time scale [70]. The key point is that the expansion history mainly depends on the form of the “MOND function” f(X ) for the unconstrained domain X < 0 in any of these theories.

View Image

Figure 44: In solid blue, the Zhao–Famaey [508] &tidle;μ(s)-function (Eq. 79View Equation) of TeVeS (Section 7.3 and 7.4), compared to the original Bekenstein one (dashed green) with a discontinuity at s = 0 [33Jump To The Next Citation Point]. The ZF function provides a more natural transition from static systems (the positive side) to cosmology (the negative side).

For instance, in TeVeS, 2 X ∝ (∇ ϕ) > 0 in static configurations (see Eq. 85View Equation), and 2 X ∝ − 2(∂ϕ∕∂t) in evolving homogeneous and isotropic configurations such as the expanding universe. The form of f(X ) is clearly constrained from the MOND phenomenology only for X > 0, meaning that a lot of freedom exists for X < 0. Exactly the same is true in GEA and BIMOND theories [101]. For instance, Bekenstein [33] originally proposed for TeVeS an ′ f-function (corresponding to μ&tidle;, see Eq. 79View Equation) with a discontinuity at X = 0 (the B04 function on Figure 44View Image) not enabling galaxies to collapse continuously out of the Hubble expansion. Afterwards, Zhao & Famaey proposed an improved “mirror-function” f′(X ) such that the corresponding μ&tidle;-function reproduces the simple μ-function (α = 1 in Eq. 46View Equation) for X > 0, and f (X ) = f(− X ) for the cosmological regime X < 0 (see Figure 44View Image, leading to an acceptable expansion history. However, when connecting a static galaxy to the expanding universe, the limit &tidle;μ(0) = 0 would predict the existence of a singular surface around each galaxies on which the scalar degree of freedom does not propagate, meaning that it is better to reconnect the two sides at &tidle;μ(0) = 𝜀 (see Section 6.2). In addition, the integration constant f (0 ) can play the role of the cosmological constant [184] to drive accelerated expansion, but even some f(0) = 0 models can drive late-time acceleration [125], which is not surprising since k-essence scalar fields were also introduced to address the dark energy problem. In the case of BIMOND (see Section 7.8), a symmetric matter-twin matter early universe yields a cosmological constant through the zero-point of the MOND function, thereby naturally leading to Λ ∼ a2 0.

All in all, with the additional freedom of a hypothetical dark component in the matter sector, in the form of, e.g., ordinary or sterile neutrinos, playing with the form of f(X ) for X < 0 in TeVeS, GEA and BIMOND always allows one to reproduce an expansion history and a Hubble diagram almost precisely identical to ΛCDM, justifying the assumption made in Section 8 of an expansion history for gravitational lensing in relativistic MOND. However, it is important to note that MOND theories are not providing a unique prediction on this.

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