BRX-TH 589

UFIFT-QG-07-03

Nonlocal Cosmology

S. Deser

California Institute of Technology, Pasadena, CA 91125 and

Department of Physics, Brandeis University, Waltham, MA 02254

and

R. P. Woodard

Department of Physics, University of Florida, Gainesville, FL 32611

ABSTRACT

We explore nonlocally modified models of gravity, inspired by quantum loop corrections, as a mechanism for explaining current cosmic acceleration. These theories enjoy two major advantages: they allow a delayed response to cosmic events, here the transition from radiation to matter dominance, and they avoid the usual level of fine tuning; instead, emulating Dirac’s dictum, the required large numbers come from the large time scales involved. Their solar system effects are safely negligible, and they may even prove useful to the black hole information problem.

PACS numbers: 95.36.+x, 98.80.Cq

e-mail:

e-mail:

## 1 Introduction

A variety of complementary data sets [1] have led to general agreement that the universe is accelerating as if it had critical density, comprised of about matter and cosmological constant [2]. There is, however, no current compelling explanation for either the smallness of , or for its recent dominance in cosmological history [3]. Two existing classes of models, scalars [4] and “” modifications of gravity [5], can be arranged to reproduce the observed (or any other) expansion history [6, 7, 8, 5]. However, neither has an underlying rationale nor do they avoid fine tuning [9]. Quantum scalar effects, depending on a very small mass have also been proposed [10].

In this work, we account for the current phase of acceleration through nonlocal additions to general relativity. Such corrections arise naturally as quantum loop effects and have of course been studied, though in other contexts [6, 11, 12]. As we will see, even the simple models we explore here can both generate large numbers without major fine tuning and deliver a delayed response to cosmic transitions, in particular to that from radiation to matter dominance at . We will neither attempt to derive our models from loop corrections nor to survey generic candidates here. Instead, we will show that natural nonlocal operators such as the inverse d‘Alembertian can explain the time lag between and the onset of acceleration at redshift , without recourse to large parameters. Large numbers come in our models precisely from the long time lags themselves, a mechanism reminiscent of some old ideas of Dirac.

## 2 Nonlocal Triggers

For simplicity, we deal with homogeneous, isotropic and spatially flat geometries

(1) |

These correspond the following Hubble and deceleration parameters

(2) |

and to Ricci scalar^{1}^{1}1Our conventions are and

(3) |

For much of cosmic history grows as a power of time

(4) |

Perfect radiation dominance corresponds to , and perfect matter dominance to . The Ricci scalar of course vanishes for and is positive for . It is the lowest dimension curvature invariant, and the only simple curvature invariant to vanish at finite , so we concentrate here on -based models.

We seek the inverse of some differential operator to provide the required time lag between the transition from radiation dominance to matter dominance at years. The simplest choice is the scalar wave operator, suggested also by the fact that, for our background (1), dynamical gravitons obey the scalar wave equation [13] with

(5) |

Acting on any function of time , its retarded inverse reduces to simple integrations:

(6) |

If we make the simplifying (and numerically justified) assumption that the power changes from to some other value at , the integrals in (6) are easily carried out for our choice of

(7) |

For the matter dominance value of , and at the present time of years, this yields

(8) |

If we think of correcting the field equations by this term (apart from small additions that enforce conservation, and whose form we will shortly exhibit) times the Einstein tensor, this result already illustrates how nonlocality allows simple time evolution to generate large numbers without fine tuning.

## 3 Specific Models

Here we evaluate the consequences of the simplest alteration of the Einstein action,

(10) |

One could modify the cosmological term in a similar way, but that turns out to require fine tuning to delay the onset of acceleration sufficiently.

Naively varying a nonlocal action such as (10) would result in
advanced Green’s functions as well as the retarded ones (6)
we desire. However, because conservation only depends on the Green’s
function being the inverse of a differential operator, one gets causal
and conserved equations by simply replacing the advanced Green’s
functions by the retarded ones [12].^{2}^{2}2To derive causal and
conserved field equations from quantum field theory one uses the
Schwinger-Keldysh formalism [15]. This will generally result in
dependence upon the real part of the propagator, as well as the retarded
Green’s function, which, if anything, may lead to even stronger effects than
those we consider. The resulting correction to the Einstein tensor is

(11) | |||||

As promised, it takes the form of a nonlocal distortion of the Einstein tensor, plus additional terms which enforce the Bianchi identity for any . The additional terms involve derivatives, so they are typically small when varies slowly. Note also that, except for the very special case of , no model of this form can be obtained from integrating out a scalar. Whatever these models’ origin, then, they are not scalar-tensor gravities in disguise.

Now note from (7) that is small for a long time after the onset of matter dominance. During this period we may think of as a perturbation of the stress tensor source, with and . Our corrections will tend to induce acceleration if evolution during matter domination carries us to the point where

(12) |

Naturally, once our corrections exceed the Einstein range, they are no longer perturbations and numerical integration of the field equations is required.

One illustrative class of models has

(13) |

The resulting modification gives

(14) |

Note that the right-hand side is positive for in the range is needed to make the correction term grow. Our results do depend on two dimensionless coupling constants, and , but neither need be very different from unity to provide a suitable delay for the onset of acceleration. For example, taking and would result in about the right onset time, in accord with the usual meaning of no fine tuning as involving parameters . ; actually, the range

As stated earlier, it is possible to construct the scalar potential to support an arbitrary expansion history obeying [6, 7], and a similar construction exists for theories [5, 8]. The same possibility is of course present in our models, and indeed a procedure has recently been worked out for reconstructing the nonlocal distortion function which would support an arbitrary expansion history [16]. Hence there are certainly models of the type (10) that fit the supernova data. Nor must one even resort to exotic choices of . As might have been guessed from viewing these models as effective nonlocal distortions of Newton’s constant, quiescence at recombination requires that be small for near zero, whereas obtaining de Sitter expansion at asymptotically late times requires that approach from above for large, negative . The onset of acceleration is controlled by the range of at which becomes of order .

## 4 Conclusions

We have explored the cosmological effects of some very simple nonlocally modified Einstein models inspired by loop corrections. Since their actual derivations from realistic quantum effects are likely to require nonperturbative summations, we regard them as purely phenomenological for now. Their two – equally important – main virtues are (unlike local variants): they naturally incorporate a delayed response to the transition from radiation to matter dominance, yet avoid major fine tuning. There are of course many other open questions raised by the present proposal, such as finding optimal candidate actions while ensuring that nonlocality has no negative unintended consequences. Some apparent worries, such as (unwanted) solar system effects, are easily allayed. There, is a small number. Although a single power of is observable — and constrains Brans-Dicke theory tightly [17] — higher powers, such as occur here, are negligible.

It should also be mentioned that nonlocality may have a positive use in the black hole information problem [18]: The infalling matter that creates or accretes to a black hole is imprinted on the external geometry through its stress tensor. Nonlocal dependence on the Einstein tensor will retain that information; while does not completely subsume the matter’s internal structure, it is a significant repository thereof; furthermore, is singular on null surfaces such as the event horizon.

Acknowledgements

This work was partially supported by National Science Foundation grants PHY04-01667, PHY-0244714 and PHY-0653085, and by the Institute for Fundamental Theory at the University of Florida.

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