R. P. Woodard

Avoiding Dark Energy with 1/R Modifications of Gravity

Scalar quintessence seems epicyclic because one can choose the potential to reproduce any cosmology (I review the construction) and because the properties of this scalar seem to raise more questions than they answer. This is why there has been so much recent interest in modified gravity. I review the powerful theorem of Ostrogradski which demonstrates that the only potentially stable, local modification of general relativity is to make the Lagrangian an arbitrary function of the Ricci scalar. Such a theory can certainly reproduce the current phase of cosmic acceleration without Dark Energy. However, this explanation again seems epicyclic in that one can construct a function of the Ricci scalar to support any cosmology (I give the technique). Models of this form are also liable to problems in the way they couple to matter, both in terms of matter’s impact upon them and in terms of the long range gravitational force they predict. Because of these problems my own preference for avoiding Dark Energy is to bypass Ostrogradski’s theorem by considering the fully nonlocal effective action built up by quantum gravitational processes during the epoch of primordial inflation.

Quantum gravity slows inflation

Abstract We consider the quantum gravitational back-reaction on an initially inflating, homogeneous and isotropic universe whose topology is T 3 x ℛ. Although there is no secular effect at one loop, an explicit calculation shows that two-loop processes act to slow the rate of expansion by an amount which becomes non-perturbatively large at late times. By exploiting Feynmans tree theorem we show that all higher loops act in the same sense.

The Quantum gravitational back reaction on inflation

Abstract We describe our recent calculation of two-loop corrections to the expansion rate of an initially inflating universe on the manifoldT3× R . If correct, our result proves that quantum gravitational effects slow the rate of inflation by an amount which becomes non-perturbatively large at late times. In a preliminary discussion of basic issues we show that the expansion rate is a gauge invariant and that our ultraviolet regulator does not introduce spurious time dependence. We also derive a sharp bound on the maximum strength of higher loop effects.

Stochastic inflationary scalar electrodynamics

Abstract We stochastically formulate the theory of scalar quantum electrodynamics on a de Sitter background. This reproduces the leading infrared logarithms at each loop order. It also allows one to sum the series of leading infrared logarithms to obtain explicit, nonperturbative results about the late time behavior of the system. One consequence is confirmation of the conjecture by Davis, Dimopoulos, Prokopec and Tornkvist that super-horizon photons acquire mass during inflation. We compute M γ 2 ≃ 3.2991 × H 2 . The scalar stays perturbatively light with M φ 2 ≃ 0.8961 × 3 e 2 H 2 / 8 π 2 . Interestingly, the induced change in the cosmological constant is negative, δΛxa0≃xa0−0.6551xa0×xa03GH4/π.

Photon mass from inflation

We consider vacuum polarization from massless scalar electrodynamics in de Sitter inflation. The theory exhibits a 3+1 dimensional analog of the Schwinger mechanism in which a photon mass is dynamically generated. The mechanism is generic for light scalar fields that couple minimally to gravity. The nonvanishing of the photon mass during inflation may result in magnetic fields on cosmological scales.

Relaxing the cosmological constant

A non-zero cosmological constant allows massless gravitons to self-interact via a coupling of dimension three. If the cosmological constant is positive then the background geometry will subject gravitons to an enormous redshift. These two facts are together responsible for the severe infrared divergences which detailed, explicit calculations [N.C. Tsamis and R.P. Woodard, Strong infrared effects in quantum gravity, preprint CRETE-92-17, UFIFT-92-24] reveal in loop corrections to the gravitational force law. We argue that, as a result, the cosmological interaction has a finite lifetime. Furthermore, this lifetime is consistent with inflation.

Stochastic quantum gravitational inflation

Abstract During inflation explicit perturbative computations of quantum field theories which contain massless, nonconformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskiĭs technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features.

Reconstructing the Distortion Function for Nonlocal Cosmology

We consider the cosmology of modified gravity models in which Newtons constant is distorted by a function of the inverse dAlembertian acting on the Ricci scalar. We derive a technique for choosing the distortion function so as to fit an arbitrary expansion history. This technique is applied numerically to the case of ΛCDM cosmology, and the result agrees well with a simple hyperbolic tangent.

Leading log solution for inflationary Yukawa theory

We generalize Starobinskiis stochastic technique to the theory of a massless, minimally coupled scalar interacting with a massless fermion in a locally de Sitter geometry. The scalar is an active field that can engender infrared logarithms. The fermion is a passive field that cannot cause infrared logarithms but which can carry them, and which can also induce new interactions between the active fields. The procedure for dealing with passive fields is to integrate them out, then stochastically simplify the resulting effective action following Starobinskii. Because Yukawa theory is quadratic in the fermion this can be done explicitly using the classic solution of Candelas and Raine. We check the resulting stochastic formulation against an explicit two loop computation. We also derive a nonperturbative, leading log result for the stress tensor. Because the scalar effective potential induced by fermions is unbounded below, backreaction from this model might dynamically cancel an arbitrarily large cosmological constant.

A Nonlocal metric formulation of MOND

We study a class of nonlocal, but causal, covariant and conserved field equations for the metric. Although nonlocal, these equations do not seem to possess extra graviton solutions in weak field perturbation theory. Indeed, the equations reduce to those of general relativity when the Ricci scalar vanishes throughout spacetime. When a static matter source is present, we show how these equations can be adjusted to reproduce Milgroms modified Newtonian dynamics in the weak field regime, while reducing to general relativity for strong fields. We compute the angular deflection of light in the weak field regime and demonstrate that it is the same as for general relativity, resulting in far too little lensing if no dark matter is present. We also study the field equations for a general Robertson–Walker geometry. An interesting feature of our equations is that they become conformally invariant in the MOND limit.