Two Loop Scalar Bilinears for Inflationary SQED

We evaluate the one and two loop contributions to the expectation values of two coincident and gauge invariant scalar bilinears in the theory of massless, minimally coupled scalar quantum electrodynamics on a locally de Sitter background. One of these bilinears is the product of two covariantly differentiated scalars, the other is the product of two undifferentiated scalars. The computations are done using dimensional regularization and the Schwinger-Keldysh formalism. Our results are in perfect agreement with the stochastic predictions at this order.Comment: 43 pages, LaTeX 2epsilon, 5 figures (using axodraw.sty) Version 2 has updated references and important corrections to Tables 3-5 and to eqns (139-141), (145-146), (153-155), (158) and (160

A Simple Operator Check of the Effective Fermion Mode Function during Inflation

We present a relatively simple operator formalism which reproduces the leading infrared logarithm of the one loop quantum gravitational correction to the fermion mode function on a locally de Sitter background. This rule may serve as the basis for an eventual stochastic formulation of quantum gravity during inflation. Such a formalism would not only effect a vast simplification in obtaining the leading powers of $\ln(a)$ at fixed loop orders, it would also permit us to sum the series of leading logarithms. A potentially important point is that our rule does not seem to be consistent with any simple infrared truncation of the fields. Our analysis also highlights the importance of spin as a gravitational interaction that persists even when kinetic energy has redshifted to zero.Comment: 39 pages, no figuire.(1) New version has clarified the ultimate motivation by adding sentences to the abstract and to the penultimate paragraph of the introduction. (2) By combining a number of references and equations we have managed to reduce the length by 2 page

A Simplified Quantum Gravitational Model of Inflation

Inflationary quantum gravity simplifies drastically in the leading logarithm approximation. We show that the only counterterm which contributes in this limit is the 1-loop renormalization of the cosmological constant. We go further to make a simplifying assumption about the operator dynamics at leading logarithm order. This assumption is explicitly implemented at 1- and 2-loop orders, and we describe how it can be implemented nonperturbatively. We also compute the expectation value of an invariant observable designed to quantify the quantum gravitational back-reaction on inflation. Although our dynamical assumption may not prove to be completely correct, it does have the right time dependence, it can naturally produce primordial perturbations of the right strength, and it illustrates how a rigorous application of the leading logarithm approximation might work in quantum gravity. It also serves as a partial test of the "null hypothesis" that there are no significant effects from infrared gravitons.Comment: 41 pages, version 2 revised for publication in Classical and Quantum Gravity. The principal change is the discussion in section 3 of concerns about spatial inhomogeneities and the validity of expectation value

Infrared Propagator Corrections for Constant Deceleration

We derive the propagator for a massless, minimally coupled scalar on a $D$-dimensional, spatially flat, homogeneous and isotropic background with arbitrary constant deceleration parameter. Our construction uses the operator formalism, by integrating the Fourier mode sum. We give special attention to infrared corrections from the nonzero lower limit associated with working on finite spatial sections. These corrections eliminate infrared divergences that would otherwise be incorrectly treated by dimensional regularization, resulting in off-coincidence divergences for those special values of the deceleration parameter at which the infrared divergence is logarithmic. As an application we compute the expectation value of the scalar stress-energy tensor.Comment: 34 pages, uses LaTeX 2 epsilo

A graviton propagator for inflation

We construct the scalar and graviton propagator in quasi de Sitter space up to first order in the slow roll parameter $\epsilon\equiv -\dot{H}/H^2$. After a rescaling, the propagators are similar to those in de Sitter space with an $\epsilon$ correction to the effective mass. The limit $\epsilon\to 0$ corresponds to the E(3) vacuum that breaks de Sitter symmetry, but does not break spatial isotropy and homogeneity. The new propagators allow for a self-consistent, dynamical study of quantum back-reaction effects during inflation.Comment: 23 page

Fermion Propagator in Cosmological Spaces with Constant Deceleration

We calculate the fermion propagator in FLRW spacetimes with constant deceleration $q=\epsilon-1$, $\epsilon=-\dot{H}/H^{2}$ for excited states. For fermions whose mass is generated by a scalar field through a Yukawa coupling $m=g_{\mathrm{\scriptscriptstyle{Y}}} \phi$, we assume $\phi \propto H$. We first solve for the mode functions by splitting the spinor into a direct product of helicity and chirality spinors. We also allow for non-vacuum states. We normalise the spinors using a consistent canonical quantisation and by requiring orthogonality of particle and anti-particle spinors. We apply our propagator to calculate the one loop effective action and renormalise using dimensional regularisation. Since the Hubble parameter is now treated dynamically, this paves the way to study the dynamical backreaction of fermions on the background spacetime.Comment: 18 pages, 1 figure, published versio

Unruh response functions for scalar fields in de Sitter space

We calculate the response functions of a freely falling Unruh detector in de Sitter space coupled to scalar fields of different coupling to the curvature, including the minimally coupled massless case. Although the responses differ strongly in the infrared as a consequence of the amplification of superhorizon modes, the energy levels of the detector are thermally populated.Comment: 16 pages, 1 figure, accepted for publication by Classical and Quantum Gravit

Stress Tensor Correlators in the Schwinger-Keldysh Formalism

We express stress tensor correlators using the Schwinger-Keldysh formalism. The absence of off-diagonal counterterms in this formalism ensures that the +- and -+ correlators are free of primitive divergences. We use dimensional regularization in position space to explicitly check this at one loop order for a massless scalar on a flat space background. We use the same procedure to show that the ++ correlator contains the divergences first computed by `t Hooft and Veltman for the scalar contribution to the graviton self-energy.Comment: 14 pages, LaTeX 2epsilon, no figures, revised for publicatio

The Fermion Self-Energy during Inflation

We compute the one loop fermion self-energy for massless Dirac + Einstein in the presence of a locally de Sitter background. We employ dimensional regularization and obtain a fully renormalized result by absorbing all divergences with BPHZ counterterms. An interesting technical aspect of this computation is the need for a noninvariant counterterm owing to the breaking of de Sitter invariance by our gauge condition. Our result can be used in the quantum-corrected Dirac equation to search for inflation-enhanced quantum effects from gravitons, analogous to those which have been found for massless, minimally coupled scalars.Comment: 63 pages, 3 figures (uses axodraw.sty), LaTeX 2epsilon. Revised version (to appear in Classical and Quantum Gravity) corrects some typoes and contains some new reference

Issues Concerning Loop Corrections to the Primordial Power Spectra

We expound ten principles in an attempt to clarify the debate over infrared loop corrections to the primordial scalar and tensor power spectra from inflation. Among other things we note that existing proposals for nonlinear extensions of the scalar fluctuation field $\zeta$ introduce new ultraviolet divergences which no one understands how to renormalize. Loop corrections and higher correlators of these putative observables would also be enhanced by inverse powers of the slow roll parameter $\epsilon$. We propose an extension which should be better behaved.Comment: 36 pages, uses LaTeX2e, version 3 revised for publication with a much expanded section 4, proving that our proposed extension of the zeta-zeta correlator absorbs the one loop infrared divergences from graviton