Cosmology Beyond Einstein by Adam Ross Solomon

Author:Adam Ross Solomon
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

5.1 The Lack of a Physical Metric

We consider a doubly-coupled bimetric theory in which the action (2.​30) is extended by the addition of a minimal-coupling term between matter fields, , and ,

(5.2)

This extends the symmetry (5.1) to the entire action, as long as we also exchange and ,

(5.3)

The presence of the interaction term is crucial; if one were to couple two pure, noninteracting GR sectors to the same matter, the Bianchi identities would constrain that matter to be entirely nondynamical [9]. Note that and are not both necessary to fully specify the theory; only their ratio is physical, as can be seen by rescaling the action by . For the purposes of this chapter, we will find it useful to leave both in so as to keep the symmetry between the two metrics explicit.

An immediate concern is the violation of the equivalence principle. However, because the Vainshtein mechanism screens massive-gravity effects [12], it is not obvious how stringent the constraints from tests of GR in the solar system would be: the modifications might be hidden from local experiments while showing up at cosmological scales. The cosmology of this doubly-coupled theory has been studied and shown to produce viable late-time accelerating background expansion without an explicit cosmological constant term [9], and with a phenomenology which can be interestingly different from that of the singly-coupled theory [13]. We emphasise again that this model itself possesses the Boulware-Deser ghost and hence we cannot trust its cosmological solutions, but a ghost-free doubly-coupled theory may well have similar properties. Indeed, the cosmological phenomenology of this theory is quite similar to that of the healthier doubly-coupled theory introduced in Ref. [5], as we will show in Chap. 6.

We can readily confirm that no physical Riemannian metric exists in the sense that all matter species would minimally couple to it and thus follow its geodesics. Indeed, for some matter fields such a metric does not exist at all. Consider a massive scalar field. Its action is given by

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