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Cosmology Beyond Einstein pp 117–134Cite as

Cosmological Implications of Doubly-Coupled Massive Bigravity

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Abstract

So far we have studied the cosmological solutions of massive bigravity in Chaps. 3 and 4 with matter coupled only to one metric, and discussed some of the theoretical issues with extending to a bimetric matter coupling in Chap. 5. As emphasised in the introduction of Chap. 5, the singly-coupled theory spoils the metric interchange symmetry present in vacuum; the kinetic and mass terms treat the metrics on equal footing, but this is broken when one couples matter to only one metric. It is therefore compelling to investigate other types of matter coupling that extend this metric-interchange symmetry to the entire theory. Moreover, as demonstrated in Chap. 3, cosmological background viability and linear stability rule out all but a small handful of the parameter space of the singly-coupled theory. By extending the matter coupling, we may be able to open up the space of observationally-allowed bimetric theories.

The universe is full of magical things patiently waiting for our wits to grow sharper.

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Notes

  1. 1.

    We will denote the effective metric with “eff” written as a superscript or subscript interchangeably.

  2. 2.

    In Ref. [3] the effective metric is given in an explicitly symmetric form, but this is not needed since \(g_{\mu \alpha }\mathbb X^{\alpha }{}_{\nu } = g_{\nu \alpha }\mathbb X^{\alpha }{}_{\mu }\), as first shown in Ref. [1]; see also Appendix C.

  3. 3.

    See also Sect. 2.1.2 for the redundancy of the Planck masses in the singly-coupled theory.

  4. 4.

    In the singly-coupled theory, Eq. (6.22) would be a constraint equation arising from the Bianchi identity and stress-energy conservation. When using the effective coupling, the stress-energy conservation holds with respect to the effective metric, rather than \(g_{\mu \nu }\) or \(f_{\mu \nu }\). This gives rise to the pressure-dependent term in the left bracket. Due to this term, both branches—obtained by setting either bracket to zero—can be regarded as dynamical. We choose to adopt the terminology from the singly-coupled case here, however.

  5. 5.

    These are not, however, \(\Lambda \)CDM cosmologies for the effective metric due to the nontrivial coupling to \(\rho \).

  6. 6.

    It is not difficult to see that there are no cases in which the two metrics are related by a dynamical conformal factor; from Eq. (6.31) any conformal relation means that \(da_f/da_g=a_f/a_g\), but this implies \(a_f/a_g=\mathrm {const.}\)

  7. 7.

    Note that \(\beta <0\) leads to instabilities in the case of doubly-coupled dRGT massive gravity, in which one of the metrics is nondynamical [8].

  8. 8.

    So that a partially-massless graviton has four polarisations rather than the five of a massive graviton, hence the name.

  9. 9.

    If the case described in Sect. 6.4.1 is truly partially massless, this may be an exception, as there is a new gauge symmetry to protect against quantum corrections.

  10. 10.

    Indeed, the fact that a small graviton mass is stable against quantum corrections is one of the main motivations for studying massive (bi)gravity, particularly as a candidate to explain the accelerating Universe.

  11. 11.

    Vacuum solutions for this model were previously studied in Ref. [30].

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  1. Center for Particle Cosmology, University of Pennsylvania, Philadelphia, USA

    Adam Ross Solomon

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Solomon, A.R. (2017). Cosmological Implications of Doubly-Coupled Massive Bigravity. In: Cosmology Beyond Einstein. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-46621-7_6

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