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Cosmological history in York time: inflation and perturbations

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Abstract

The constant mean extrinsic curvature on a spacelike slice may constitute a physically preferred time coordinate, ‘York time’. One line of enquiry to probe this idea is to understand processes in our cosmological history in terms of York time. Following a review of the theoretical motivations, we focus on slow-roll inflation and the freezing and Hubble re-entry of cosmological perturbations. While the physics is, of course, observationally equivalent, we show how the mathematical account of these processes is distinct from the conventional account in terms of standard cosmological or conformal time. We also consider the cosmological York-timeline more broadly and contrast it with the conventional cosmological timeline.

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Notes

  1. Not all solutions of general relativity permit a global York-time slicing. Therefore, the proposal that York time be the physically fundamental time parameter rules out certain solutions of general relativity. However, for the most part these are arguably physically pathological, involving for example closed time-like lines [14].

  2. The relation between the momenta and the extrinsic curvature is given by \(\pi ^{ab}=\sqrt{g}(Kg^{ab}-K^{ab})\).

  3. In a forever expanding scenario, such as spatially flat cosmologies with conventional matter content (where the equation of state parameter \(w>-1\)) York time remains negative throughout cosmological history. Values \(T>0\) correspond to a possible symmetric extension of this history. In fact, such an extension may be necessary for consistent quantisation [28] given the behaviour of the constant-mean-curvature foliation around singularities.

  4. One could repeat the procedure iteratively but in general this is not necessary for an approximate description of the homogeneous cosmological background and would in any case not yield higher degrees of accuracy due to our neglect of the contribution of other matter. In the example below, the nth order term in this process introduces to the solution \(\phi _n\) a term proportional to \(t^n\), as well as corrections to lower-order terms in the polynomial. Each term is small (for appropriately small values of t falling within the presumed inflationary period) compared to previous terms because \(M_{Pl}mt\ll \phi _0\), which is guaranteed by the slow-roll conditions.

  5. For a closed cosmology an analogous (although algebraically more complicated) expression can be derived [18].

  6. Despite its broad appeal, there are rival theories to inflation aiming to explain our cosmological observations, such as, for example, bouncing cosmologies [see [32] for a recent review].

  7. The fact that physically the regimes are identified by \(\tilde{k}\ll 1\) and \(\tilde{k}\gg 1\) while mathematically the critical value is \(\sqrt{2}\) rather than 1 is of no significance provided we restrict analysis to modes sufficiently far into the sub-Hubble or super-Hubble regime.

  8. The possibility of critical damping may be neglected since the coefficients are time dependent and therefore no mode is critically damped for more than an instant.

  9. Any explicit relationship between t and T can be modified by translation in t since the cosmological equations in t are time-translation invariant. However, the York-time theory is not T-translation invariant since T has a physical meaning and the equations are explicitly T-dependent. The relationship given here therefore depends on the appropriate choice of time origin, namely that \(t=0\) corresponds to \(T=-\infty \).

  10. Changing units to, say, Planck units leads to absurdly large numbers for later eras instead.

  11. At least the ‘shape’ degrees of freedom are quantised. The local notion of scale, that is, the volume element takes the role of Hamiltonian density in the reduced-Hamiltonian formalism rather than that of a canonical variable and is therefore treated differently when quantising. At least for a minisuperspace model ‘volume’ actually shows classical-like behaviour [18].

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  1. Kinard Laboratory, Department of Physics and Astronomy, Clemson University, Clemson, SC, 29631-0978, USA

    Philipp Roser & Antony Valentini

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  1. Philipp Roser
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  2. Antony Valentini
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Correspondence to Philipp Roser.

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Roser, P., Valentini, A. Cosmological history in York time: inflation and perturbations. Gen Relativ Gravit 49, 13 (2017). https://doi.org/10.1007/s10714-016-2180-9

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