Abstract
Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential. Ultimately, this inflaton must be identified with the expectation value of a quantum field, evolving in a quantum effective potential. The shape of this potential is determined by the underlying tree-level potential, dressed by quantum corrections from the scalar field itself and the metric perturbations. Following [1], we compute the effective scalar field equations and the corrected Friedmann equations to quadratic order in both scalar field, scalar metric and tensor perturbations. We identify the quantum corrections from different sources at leading order in slow-roll, and estimate their magnitude in benchmark models of inflation. We comment on the implications of non-minimal coupling to gravity in this context.
Highlights
Background and field equations inNewtonian gaugeWe consider a single inflaton field φ non-minimally coupled to gravity with action of the form (following the sign convention (+++) of [25]) S= d4 √ x −g MP2lF (φ)R − 1 2 gμν φ,μφ,ν V (φ) (2.1)where MP2l ≡ (8πG)−1 is the reduced Planck mass and where for the moment F (φ) and the tree-level potential V (φ) are kept general.3 The scalar field φ lives in a spacetime defined by the metric gμν, and we will assume that field and metric may be written as φ(x) = φ(t) + δφ(t, xi), gμν (x) = gμν (t) + δgμν (t, xi)
Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential
This inflaton must be identified with the expectation value of a quantum field, evolving in a quantum effective potential
Summary
We consider a single inflaton field φ non-minimally coupled to gravity with action of the form (following the sign convention (+++) of [25]). The metric perturbations may be decomposed into scalar, vector, and tensor modes. Where Φ and Ψ are scalar potentials and Eij is a traceless, transverse matrix containing the tensor perturbations, i.e. The Einstein field equations are obtained by variation of the action in eq (2.1) with respect to the metric, which yields. (2.4)–(2.7) and extract equations to zeroth order in perturbations (the “classical” Friedmann and inflaton equations); to first order in perturbations (the mode equations for the field and metric scalar and tensor perturbations in the background FLRW metric); and to quadratic order in perturbations (the “quantum-corrected” Friedmann and inflaton equations). The zeroth-order equations will provide a slow-roll background in which to solve the first-order (linear) mode equations These will in turn be quantised and inserted into the quadratic-order equations, in order to explicitly compute the quantum corrections to the cosmological evolution
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
