Abstract
Contracting cosmologies are known to be flawed with a shear instability, where the contribution from the anisotropic stress to the overall energy density grows as a^{-6}, with a the scale factor. Classically, whether or not this contribution becomes important before the bounce depends on its initial value, which can always be sufficiently fine tuned to make it irrelevant. However, vacuum quantum fluctuations inevitably provide a non-vanishing source of anisotropic stress. In this work, we compute the minimum amount of shear that is obtained if one assumes that it vanishes initially, but lets quantum fluctuations build it up. In practice, we consider a massless test scalar field, and describe its quantum fluctuations by means of the stochastic “inflation” (though here applied to a contracting phase) formalism. We find that, if the equation-of-state parameter of the contraction satisfies w>-1/9, regardless of when the contracting phase is initiated, the time at which the shear becomes sizeable is always when the Hubble scale approaches the Planck mass (which is also where the bounce is expected to take place). However, if w<-1/9, the shear backreaction becomes important much earlier, at a point that depends on the overall amount of contraction.
Highlights
Introduction3, we derive a stochastic formalism, analogous to the stochastic-inflation formalism, to describe these quantum fluctuations in an inflating or a contracting cosmology, and to compute the expectation value of the shear they generate
We find that, when w > −1/9, the shear produced out of massless scalar field fluctuations is always negligible until the energy density of the universe reaches the Planck scale, regardless of initial conditions; while for w < −1/9, it becomes sizeable much before at a point that depends on the time at which the contraction is initiated
If w > −1/9, the shear contribution to the overall energy density reaches an attractor that remains negligible until the energy density of the universe reaches the Planck scale, where a bounce is expected to take place
Summary
3, we derive a stochastic formalism, analogous to the stochastic-inflation formalism, to describe these quantum fluctuations in an inflating or a contracting cosmology, and to compute the expectation value of the shear they generate. We find that, when w > −1/9, the shear produced out of massless scalar field fluctuations is always negligible until the energy density of the universe reaches the Planck scale, regardless of initial conditions; while for w < −1/9, it becomes sizeable much before at a point that depends on the time at which the contraction is initiated. 5, we compare the shear backreaction with the direct contribution of the test scalar field fluctuations to the energy density of the universe, and conclude that the later is always more important than the former. MP2l which correspond respectively to the “00” component, the trace and the traceless part of the “i j” component, and where
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