Abstract
In the previous chapter, and in particular in Sect. 2.1, we discussed an approach to modifying gravity in which its force-carrier particle, the graviton, is given a small mass. In particular, by specialising to the dRGT interaction potentials (2.22) we ensure that the notorious Boulware-Deser ghost mode is absent, and by allowing both metrics to be dynamical and taking the graviton mass to be of the order of the present-day Hubble rate, we can obtain cosmological solutions which agree well with observations of the cosmic expansion history. These solutions are self-accelerating: the Hubble parameter goes to a constant at late times even in the absence of a cosmological constant. The action of this bimetric theory, or bigravity, is given by Eq. (2.30), and the associated modified gravitational field equations were presented as Eqs. (2.39) and (2.40).
There is nothing stable in the world; uproar’s your only music.
John Keats, Letters
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- 1.
This should not be confused with the Higuchi ghost instability, which affects most massive gravity cosmologies and some in bigravity, but is, however, absent from the simplest bimetric models which produce \(\Lambda \)CDM-like backgrounds [7].
- 2.
By leaving the lapse N in the g metric general, we retain the freedom to later work in cosmic or conformal time. There is a further practical benefit: since this choice makes the symmetry between the two metrics manifest, and the action is symmetric between g and f as described in Sect. 2.1.2, this means the f field equations can easily be derived from the g equations by judicious use of ctrl-f.
- 3.
We thank Shinji Mukhoyama for discussions on this point.
- 4.
The discussion in this section is indebted to useful conversations with Macarena Lagos and Pedro Ferreira.
- 5.
Specifically, they appear without time derivatives. Recall that we are working in Fourier space where spatial derivatives effectively amount to multiplicative factors of ik.
- 6.
We recognise that the number of unused synonyms for “these equations are very long” is growing short as this chapter progresses.
- 7.
With the exception of the \(\beta _{0}\) model, which is simply \(\Lambda \)CDM.
- 8.
We frequently discuss the viability of various models in this section; all such results were derived in Ref. [3].
- 9.
We do not have the freedom to include nonzero \(\beta _2\) or \(\beta _3\); in either case the background evolution would not be viable [3]. We can see this from the expressions (2.59) and (2.60) for \(\rho (y)\) and \(H^2(y)\). If \(\beta _3\) were nonzero, then \(\Omega _{\text {m}} = \rho /(3M_{g}^{2}H^{2})\) would diverge as y at early times. Setting \(\beta _{3=0}\), we find \(\Omega _{\text {m}} \rightarrow 1-3\beta _2/\beta _{4}\) as \(y\rightarrow \infty \). If we demand a matter-dominated history, then \(\beta _2\) must at the very least be small compared to \(\beta _{4}\).
- 10.
Moreover, the square root of this, \(\dot{Y}/\mathscr {H}\), appears in the mass terms. We choose branches of the square root such that this quantity starts off negative at early times and then becomes positive.
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Solomon, A.R. (2017). Cosmological Stability of Massive Bigravity. In: Cosmology Beyond Einstein. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-46621-7_3
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