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Cosmology Beyond Einstein pp 71–101Cite as

Linear Structure Growth in Massive Bigravity

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Abstract

We have, to this point, reviewed the background FLRW solutions of massive bigravity in Chap. 2 and begun to analyse linear perturbations around these solutions in Chap. 3. In addition to introducing the formalism for cosmological perturbation theory in bigravity, the specific aim of the previous chapter was to identify which models are stable at the linear level and which are not. The natural next step is to use perturbation theory to derive observable predictions for the stable models.

The wonder is, not that the field of the stars is so vast, but that man has measured it.

Anatole France, The Garden of Epicurus

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Notes

  1. 1.

    As discussed in Chap. 3, this model is linearly unstable. However, if the instability is cured at higher orders in perturbation theory before the background solution is spoiled, then at the background level this is a perfectly viable model.

  2. 2.

    Given two separate diffeomorphisms for the g and f metrics, only the diagonal subgroup of the two preserves the mass term. In practice, this means that we have a single coordinate system which we may transform by infinitesimal diffeomorphisms, exactly as in GR.

  3. 3.

    In the approach of Ref. [6], where this limit is taken by dropping all derivative terms, this step is crucial for the results to be consistent; in our case it is simply useful for rewriting derivatives of \(\dot{A}_g\) and \(\ddot{A}_g\) in terms of \(E_g\), so that the equation is manifestly algebraic in the perturbations.

  4. 4.

    This equation holds beyond the subhorizon limit, in a particular gauge; see Appendix A.

  5. 5.

    Not to be confused with the background quantity defined in Eq. (3.9), \(Q \equiv \beta _1 + \left( x + y\right) \beta _2 + xy\beta _3\).

  6. 6.

    After gauge fixing there are six metric perturbations, but once we substitute the 0–i equations into the trace i–j equations, \(F_f\) drops out of our system. In a gauge where \(F_g=F_f=0\), as was used in Ref. [6], the equivalent statement is that the \(B_g\) and \(B_f\) parameters are only determined up to their difference, \(B_f - B_g\), which is gauge invariant.

  7. 7.

    We focus on these simpler models to illustrate bigravity effects on growth. Current growth data are not able to significantly constrain these models, so we would not gain anything by adding more free parameters.

  8. 8.

    These differ slightly from the best-fit \(B_1=1.38\pm 0.03\) reported by Ref. [5], also based on the Union2.1 supernovae compilation.

  9. 9.

    This does not need to coincide with the value of \(\Omega _{\Lambda }\) derived in the context of \(\Lambda \)CDM models. For the \(B_1\)-only model and hence all the two-parameter finite-branch models, they happen to be similar in value, although this was not a priori guaranteed, while in the infinite-branch \(B_1B_4\) model, the best-fit value to the background data is \(\Omega _\Lambda ^\mathrm {eff}=0.84_{-0.02}^{+0.03}\) [5].

  10. 10.

    For \(B_2=(5,50,500)\), and \(B_3\) chosen to give an effective \(\Omega _\Lambda ^\mathrm {eff}\approx 0.7\) today, the pole occurs at \(z\approx (1.99,8.19,27.95)\).

  11. 11.

    A similar singular evolution of linear perturbations in a smooth background has been observed in the cosmology of Gauss-Bonnet gravity [23, 24]. This instability is different from the early-time instabilities discussed in Chap. 3, as those do not arise in the quasistatic limit which we are now taking.

  12. 12.

    There is also a positive root, but this is not physical. When \(B_3<0\), that root yields \(y_0<0\). When \(B_3>0\), which is only the case for a small range of parameters, then the positive root of \(y_0\) is greater than the far-future value \(y_c\) and hence is also not physical.

  13. 13.

    Note, per Eq. (4.22), that this is equivalent to simply imposing \(\Omega _\Lambda ^\mathrm {eff}<1\), which must be true since we have chosen a spatially-flat universe a priori.

  14. 14.

    In the singly-coupled version of massive bigravity we are studying, matter loops only contribute to the g-metric cosmological constant, \(B_0\).

  15. 15.

    Up to the addition of a cosmological constant, which is uninteresting.

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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). Linear Structure Growth in Massive Bigravity. In: Cosmology Beyond Einstein. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-46621-7_4

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