Gauge-invariant Curvature Perturbation Research Articles

Quantum initial conditions for inflation and canonical invariance

We investigate the transformation of initial conditions for primordial curvature perturbations under two types of transformations of the associated action: simultaneous redefinition of time and the field to be quantised, and the addition of surface terms. The latter encompasses all canonical transformations, whilst the time- and field-redefinition is a distinct, non-canonical transformation since the initial and destination systems use different times. Actions related to each other via such transformations yield identical equations of motion and preserve the commutator structure. They further preserve the time-evolution of expectation values of quantum operators unless the vacuum state also changes under the transformation. These properties suggest that it is of interest to investigate vacuum prescriptions that also remain unchanged under canonical transformations. We find that initial conditions derived via minimising the vacuum expectation value of the Hamiltonian and those obtained using the Danielsson vacuum prescription are not invariant under these transformations, whereas those obtained by minimising the local energy density are. We derive the range of physically distinct initial conditions obtainable by Hamiltonian diagonalisation, and illustrate their effect on the scalar primordial power spectrum and the Cosmic Microwave Background under the just enough inflation model. We also generalise the analogy between the dynamics of a quantum scalar field on a curved, time-dependent spacetime and the gauge-invariant curvature perturbation. We argue that the invariance of the vacuum prescription obtained by minimising the renormalised stress--energy tensor should make it the preferred procedure for setting initial conditions for primordial perturbations. All other procedures reviewed in this work yield ambiguous initial conditions, which is problematic both in theory and practice.

Vector and tensor contributions to the curvature perturbation at second order

We derive the evolution equation for the second order curvature perturbation using standard techniques of cosmological perturbation theory. We do this for different definitions of the gauge invariant curvature perturbation, arising from different splits of the spatial metric, and compare the expressions. The results are valid at all scales and include all contributions from scalar, vector and tensor perturbations, as well as anisotropic stress, with all our results written purely in terms of gauge invariant quantities. Taking the large-scale approximation, we find that a conserved quantity exists only if, in addition to the non-adiabatic pressure, the transverse traceless part of the anisotropic stress tensor is also negligible. We also find that the version of the gauge invariant curvature perturbation which is exactly conserved is the one defined with the determinant of the spatial part of the inverse metric.

A short note on the curvature perturbation at second order

Working with perturbations about a Friedmann–Lemaître–Robertson–Walker spacetime, we compute the gauge-invariant curvature perturbation to second order solely in terms of scalar field fluctuations. Using the curvature perturbation on uniform density hypersurfaces as our starting point, we give our results in terms of field fluctuations in the flat gauge, incorporating both large and small scale behaviour. For ease of future numerical implementation, we give our result in terms of the scalar field fluctuations and their time derivatives.

Uniqueness of the gauge invariant action for cosmological perturbations

In second order perturbation theory different definitions are known of gauge invariant perturbations in single field inflationary models. Consequently the corresponding gauge invariant cubic actions do not have the same form. Here we show that the cubic action for one choice of gauge invariant variables is unique in the following sense: the action for any other, non-linearly related variable can be brought to the same bulk action, plus additional boundary terms. These boundary terms correspond to the choice of hypersurface and generate extra, disconnected contributions to the bispectrum. We also discuss uniqueness of the action with respect to conformal frames. When expressed in terms of the gauge invariant curvature perturbation on uniform field hypersurfaces the action for cosmological perturbations has a unique form, independent of the original Einstein or Jordan frame. Crucial is that the gauge invariant comoving curvature perturbation is frame independent, which makes it extremely helpful in showing the quantum equivalence of the two frames, and therefore in calculating quantum effects in nonminimally coupled theories such as Higgs inflation.

Infrared divergence does not affect the gauge-invariant curvature perturbation

We address the infrared(IR) divergence problem during inflation that appears in the loop corrections to the primordial perturbations. In our previous paper, we claimed that, at least in single field models, the IR divergence is originating from the gauge artifact. Namely, diverging IR corrections should not appear in genuine gauge-invariant observables. We propose here one simple but explicit example of such gauge-invariant quantities. Then, we explicitly calculate such a quantity to find that the IR divergence is absent at the leading order in the slow-roll approximation for the usual scale invariant vacuum state. At the same time we notice that there is a subtle issue on the gauge-invariance in how to specify the initial vacuum state.

Non-Gaussian and nonscale-invariant perturbations from tachyonic preheating in hybrid inflation

We show that in hybrid inflation it is possible to generate large second-order perturbations in the cosmic microwave background due to the instability of the tachyonic field during preheating. We carefully calculate this effect from the tachyon contribution to the gauge-invariant curvature perturbation, clarifying some confusion in the literature concerning nonlocal terms in the tachyon curvature perturbation; we show explicitly that such terms are absent. We quantitatively compute the non-Gaussianity generated by the tachyon field during the preheating phase and translate the experimental constraints on the nonlinearity parameter ${f}_{NL}$ into constraints on the parameters of the model. We also show that nonscale-invariant second-order perturbations from the tachyon field with spectral index $n=4$ can become larger than the inflaton-generated first-order perturbations, leading to stronger constraints than those coming from non-Gaussianity. The width of the excluded region in terms of the logarithm of the dimensionless coupling $g$, grows linearly with the log of the ratio of the Planck mass to the tachyon VEV, $\mathrm{log}({M}_{p}/v)$; hence very large regions are ruled out if the inflationary scale $v$ is small. We apply these results to string-theoretic brane-antibrane inflation, and find a stringent upper bound on the string coupling, ${g}_{s}<{10}^{\ensuremath{-}4.5}$.

Gauge-invariant second-order perturbations and non-Gaussianity from inflation

We present the first computation of the cosmological perturbations generated during inflation up to second order in deviations from the homogeneous background solution. Our results, which fully account for the inflaton self-interactions as well as for the second-order fluctuations of the background metric, provide the exact expression for the gauge-invariant curvature perturbation bispectrum produced during inflation in terms of the slow-roll parameters. The bispectrum represents a specific non-Gaussian signature of fluctuations generated by quantum oscillations during slow-roll inflation. However, our findings indicate that detecting the non-Gaussianity in the cosmic microwave background anisotropies emerging from the second-order calculation will be a challenge for the forthcoming satellite experiments.