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Nonlocal Cosmology
S. Deser ∗ California Institute of Technology, Pasadena, CA 91125 andDepartment of Physics, Brandeis University, Waltham, MA 02254 andR. P. Woodard † Department of Physics, University of Florida, Gainesville, FL 32611
ABSTRACTWe explore nonlocally modified models of gravity, inspired by quantum loopcorrections, as a mechanism for explaining current cosmic acceleration. Thesetheories enjoy two major advantages: they allow a delayed response to cosmicevents, here the transition from radiation to matter dominance, and theyavoid the usual level of fine tuning; instead, emulating Dirac’s dictum, therequired large numbers come from the large time scales involved. Their solarsystem effects are safely negligible, and they may even prove useful to theblack hole information problem.PACS numbers: 95.36.+x, 98.80.Cq ∗ e-mail: [email protected] † e-mail: [email protected]fl.edu Introduction
A variety of complementary data sets [1] have led to general agreement thatthe universe is accelerating as if it had critical density, comprised of about30% matter and 70% cosmological constant [2]. There is, however, no currentcompelling explanation for either the smallness of Λ, or for its recent domi-nance in cosmological history [3]. Two existing classes of models, scalars [4]and “ f ( R )” modifications of gravity [5], can be arranged to reproduce theobserved (or any other) expansion history [6, 7, 8, 5]. However, neither hasan underlying rationale nor do they avoid fine tuning [9]. Quantum scalareffects, depending on a very small mass have also been proposed [10].In this work, we account for the current phase of acceleration throughnonlocal additions to general relativity. Such corrections arise naturally asquantum loop effects and have of course been studied, though in other con-texts [6, 11, 12]. As we will see, even the simple models we explore here canboth generate large numbers without major fine tuning and deliver a delayedresponse to cosmic transitions, in particular to that from radiation to mat-ter dominance at z ∼ z ∼ z ∼ .
7, without recourse to large parameters. Large numbers comein our models precisely from the long time lags themselves, a mechanismreminiscent of some old ideas of Dirac.
For simplicity, we deal with homogeneous, isotropic and spatially flat geome-tries ds = − dt + a ( t ) d~x · d~x , (1)These correspond the following Hubble and deceleration parameters H ( t ) ≡ ˙ aa , q ( t ) ≡ − a ¨ a ˙ a = − − ˙ HH , (2)and to Ricci scalar R = 6(1 − q ) H . (3) Our conventions are R ≡ g µν R µν and R µν ≡ ∂ ρ Γ ρνµ − ∂ ν Γ ρρµ + Γ ρµσ Γ σνρ − Γ ρνσ Γ σρµ . a ( t ) grows as a power of time a ( t ) ∼ t s = ⇒ H ( t ) = st , q ( t ) = 1 − ss . (4)Perfect radiation dominance corresponds to s = , and perfect matter domi-nance to s = . The Ricci scalar of course vanishes for s = and is positivefor s = . It is the lowest dimension curvature invariant, and the only simplecurvature invariant to vanish at finite s , so we concentrate here on R -basedmodels.We seek the inverse of some differential operator to provide the requiredtime lag between the transition from radiation dominance to matter domi-nance at t eq ∼ years. The simplest choice is the scalar wave operator,suggested also by the fact that, for our background (1), dynamical gravitonsobey the scalar wave equation [13] with ≡ √− g ∂ ρ (cid:16) √− g g ρσ ∂ σ (cid:17) −→ − a ddt (cid:16) a ddt (cid:17) . (5)Acting on any function of time f ( t ), its retarded inverse reduces to simpleintegrations: h f i ( t ) ≡ G [ f ]( t ) = − Z t dt ′ a ( t ′ ) Z t ′ dt ′′ a ( t ′′ ) f ( t ′′ ) . (6)If we make the simplifying (and numerically justified) assumption thatthe power changes from s = to some other value at t = t eq , the integrals in(6) are easily carried out for our choice of f = R G [ R ]( t ) (cid:12)(cid:12)(cid:12) s = − s (2 s − s − ( ln (cid:16) tt eq (cid:17) − s − s − (cid:16) t eq t (cid:17) s − ) . (7)For the matter dominance value of s = 2 /
3, and at the present time of t ∼ years, this yields G [ R ]( t ) (cid:12)(cid:12)(cid:12) s = ≃ − . . (8)If we think of correcting the field equations by this term (apart from smalladditions that enforce conservation, and whose form we will shortly exhibit)times the Einstein tensor, this result already illustrates how nonlocality al-lows simple time evolution to generate large numbers without fine tuning.2uch larger values can be obtained through other operators, for example,the Paneitz operator arising in the context of conformal anomalies [14]. Whenspecialized to our geometry (1) it takes the form1 √− g ∆ P ≡ + 2 D µ (cid:16) R µν − g µν R (cid:17) D ν −→ a ddt (cid:16) a ddt a ddt a ddt (cid:17) . (9)One gets about 10 from the dimensionless combination of the inverse of thisoperator acting on R . Here we evaluate the consequences of the simplest alteration of the Einsteinaction, ∆
L ≡ πG R √− g × f (cid:16) G [ R ] (cid:17) . (10)One could modify the cosmological term in a similar way, but that turns outto require fine tuning to delay the onset of acceleration sufficiently.Naively varying a nonlocal action such as (10) would result in advancedGreen’s functions as well as the retarded ones (6) we desire. However, be-cause conservation only depends on the Green’s function being the inverseof a differential operator, one gets causal and conserved equations by simplyreplacing the advanced Green’s functions by the retarded ones [12]. Theresulting correction to the Einstein tensor is∆ G µν = h G µν + g µν − D µ D ν i( f (cid:16) G [ R ] (cid:17) + G h Rf ′ (cid:16) G [ R ] (cid:17)i) + h δ ( ρµ δ σ ) ν − g µν g ρσ i ∂ ρ (cid:16) G [ R ] (cid:17) ∂ σ G h Rf ′ (cid:16) G [ R ] (cid:17)i! . (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 g µν . Theadditional terms involve derivatives, so they are typically small when f ( x )varies slowly. Note also that, except for the very special case of f ( x ) = − x , no To derive causal and conserved field equations from quantum field theory one uses theSchwinger-Keldysh formalism [15]. This will generally result in dependence upon the realpart of the propagator, as well as the retarded Green’s function, which, if anything, maylead to even stronger effects than those we consider. G [ R ]( t ) is small for a long time after the on-set of matter dominance. During this period we may think of ∆ G µν asa perturbation of the stress tensor source, with ∆ G = − πG ∆ ρ and g ij ∆ G ij = − πG ∆ p . Our corrections will tend to induce acceleration ifevolution during matter domination carries us to the point where∆ G + g ij ∆ G ij = − πG (cid:16) ∆ ρ + 3∆ p (cid:17) > qH = 43 · t . (12)Naturally, once our corrections exceed the Einstein range, they are no longerperturbations and numerical integration of the field equations is required.One illustrative class of models has f ( x ) = Ce − kx . (13)The resulting modification ∆ G µν gives∆ G + g ij ∆ G ij ≃ · t × C (cid:16) k (cid:17)(cid:16) − k (cid:17)(cid:16) tt eq (cid:17) k . (14)Note that the right-hand side is positive for k in the range − < k < + ;actually, the range 0 < k < + is needed to make the correction term grow.Our results do depend on two dimensionless coupling constants, C and k ,but neither need be very different from unity to provide a suitable delay forthe onset of acceleration. For example, taking k = . C = . ∼ V ( φ ) tosupport an arbitrary expansion history a ( t ) obeying ˙ H > f ( R ) theories [5, 8]. The same possibility isof course present in our models, and indeed a procedure has recently beenworked out for reconstructing the nonlocal distortion function f ( x ) whichwould support an arbitrary expansion history [16]. Hence there are certainlymodels of the type (10) that fit the supernova data. Nor must one even resortto exotic choices of f ( x ). As might have been guessed from viewing thesemodels as effective nonlocal distortions of Newton’s constant, quiescence atrecombination requires that f ( x ) be small for x near zero, whereas obtainingde Sitter expansion at asymptotically late times requires that f ( x ) approach − x . The onset of acceleration is controlledby the range of x at which f ( x ) becomes of order − Conclusions
We have explored the cosmological effects of some very simple nonlocallymodified Einstein models inspired by loop corrections. Since their actualderivations from realistic quantum effects are likely to require nonpertur-bative 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 radia-tion to matter dominance, yet avoid major fine tuning. There are of coursemany other open questions raised by the present proposal, such as findingoptimal candidate actions while ensuring that nonlocality has no negativeunintended consequences. Some apparent worries, such as (unwanted) so-lar system effects, are easily allayed. There, G [ R ] ∼ GM/ ( c r ) is a smallnumber. Although a single power of G [ R ] is observable — and constrainsBrans-Dicke theory tightly [17] — higher powers, such as occur here, arenegligible.It should also be mentioned that nonlocality may have a positive use inthe black hole information problem [18]: The infalling matter that createsor accretes to a black hole is imprinted on the external geometry through itsstress tensor. Nonlocal dependence on the Einstein tensor will retain thatinformation; while T µν does not completely subsume the matter’s internalstructure, it is a significant repository thereof; furthermore, G is singular onnull surfaces such as the event horizon. Acknowledgements
This work was partially supported by National Science Foundation grantsPHY04-01667, PHY-0244714 and PHY-0653085, and by the Institute forFundamental Theory at the University of Florida.
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