Degravitation, Inflation and the Cosmological Constant as an Afterglow
aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - HYPER VERSION
HU-EP-07/60
Degravitation, Inflation and the CosmologicalConstant as an Afterglow
Subodh P. Patil
Humboldt Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstraße 15, D-12489 Berlin,Germanyemail: [email protected]
Abstract:
In this report, we adopt the phenomenological approach of taking the degravi-tation paradigm seriously as a consistent modification of gravity in the IR, and investigateits consequences for various cosmological situations. We motivate degravitation– whereNetwon’s constant is promoted to a scale dependent filter function– as arising from eithera small (resonant) mass for the graviton, or as an effect in semi-classical gravity. After ad-dressing how the Bianchi identities are to be satisfied in such a set up, we turn our attentiontowards the cosmological consequences of degravitation. By considering the example filterfunction corresponding to a resonantly massive graviton (with a filter scale larger than thepresent horizon scale), we show that slow roll inflation, hybrid inflation and old inflationremain quantitatively unchanged. We also find that the degravitation mechanism inheritsa memory of past energy densities in the present epoch in such a way that is likely signif-icant for present cosmological evolution. For example, if the universe underwent inflationin the past due to it having tunneled out of some false vacuum, we find that degravitationimplies a remnant ‘afterglow’ cosmological constant, whose scale immediately afterwardsis parametrically suppressed by the filter scale ( L ) in Planck units Λ ∼ l pl /L . We discusscircumstances through which this scenario reasonably yields the presently observed valuefor Λ ∼ O (10 − ). We also find that in a universe still currently trapped in some falsevacuum state, resonance graviton models of degravitation only degravitate initially Planckor GUT scale energy densities down to the presently observed value over timescales compa-rable to the filter scale. We argue that different functional forms for the filter function willyield similar conclusions. In this way, we argue that although the degravitation models westudy have the potential to explain why the cosmological constant is not large in additionto why it is not zero, it does not satisfactorily address the co-incidence problem withoutadditional tuning. ∗ ∗ Dedicated to the memory of Kris Bellamkonda ontents
1. Introduction 1 G N
2. The Bianchi Identities and the Modified Einstein Equations 73. Inflation (v.s. the Cosmological Constant) 104. An Afterglow Cosmological Constant? 165. The Coincidence Problem 206. Conclusions 237. Acknowledgements 23
1. Introduction
The degravitation paradigm [1][2][3] is a phenomenological proposal designed to solve thecosmological constant (CC) problem directly. Rather than positing some symmetry whichsets the CC to be vanishing, or invoking anthropic arguments such that the problem is para-phrased away, the degravitation paradigm invokes an IR modification to general relativityat the level of the equations of motion, such that Newton’s constant becomes dependent onthe wavelength of the source. Specifically, taking λ as a characteristic length scale of somesource, and taking L to be some IR length scale to be specified later, degravitation requiresthat Newton’s constant be promoted to some function G N ( λ, L ) such that the followingproperties are satisfied: G N ( λ, L ) → λ ≫ L, (1.1) G N ( λ, L ) → G N λ ≪ L. In this way we see that sources with wavelengths greater than the IR scale introducedby L effectively degravitate as their gravitational coupling vanishes, whereas sources withwavelengths less than L gravitate normally . Such scale dependent coupling, although Requiring that gravity be unaffected at presently accessible scales requires us to set this IR scale to belarger than the present horizon scale. – 1 –eemingly odd in the context of gravity, has many precedents in gauge theory. As examplefrom electromagnetism [1], we note that in a linear dielectric medium, the electric fieldresponds to a source as ∇ · E = ρ π ¯ ǫ ¯ ǫ = ǫ (1 + κ ) , (1.2)where κ is the dielectric constant of the ambient medium. For a frequency dependentdielectric, we see that if the medium is such that ¯ ǫ ≫ ω ≫ /T (where T is some time scale), then all such sources will effectively’de-electrify’ and not source the electric field. This is effectively the underlying physics ofa high pass filter.As proposed in [1] and further developed in [2][3], a scale dependent in Newton’sconstant (1.1) can be implemented by promoting it to a differential operator: G N → G N ( L (cid:3) ) , (1.3)where (cid:3) is the covariant D’Alembertian operator, and L is taken to be a large but finite IRcutoff scale, typically much larger than any scale presently accessible to cosmology. As aself-adjoint differential operator, (cid:3) is guaranteed to have a complete set of mode functionswith real eigenvalues. Expanding any source in terms of these mode functions, we see thatmode by mode, the effect of acting on a source with the operator (1.3) is to make thereplacement G N → G N ( L λ − ) , (1.4)where λ − is the eigenvalue of the corresponding mode function being acted on. If forheuristic purposes we were to consider the example of Minkowski space, then the modefunctions of the D’Alembertian are plane waves, and the effect of acting on a mode describedby the vector k µ , is that Newtons constant becomes the scale dependent function G N ( L k ).Hence degravitating all sources with wavelengths larger than L requires that G N ( α ) → α ≪ . (1.5)As a source which is constant in both space and time, the cosmological constant corre-sponds to the zero mode k µ = 0, with the corresponding eigenvalue λ − = 0, and is henceannihilated by the action of the differential operator (1.3) by virtue of (1.5). In this way,we see that any source of the form T µν = Λ δ µν (1.6)effectively degravitates. At this rather heuristic level, it is trivial to observe that degrav-itation kills more than just the spacetime zero mode, as plane wave sources (also witheigenvalues λ − = k = 0) degravitate as well. In fact, in a general background, any sourcewith an energy density which lies in the null eigenspace of the D’Alembertian degravitates.For the purposes of cosmology, this is only of passing interest as most cosmological sources– 2 –ther than the cosmological constant gravitate normally . It remains then to realize (1.3)from some underlying physical model. There are several distinct ways through which thismight occur, and we discuss them presently. In [2], and further developed in [3], a class of models in which degravitation is effectedby massive or resonance gravitons was shown to result in a filter function for Newton’sconstant of the form: 8 πG N → πG N m (cid:3) ) − α , ≤ α < , (1.7)where α = 0 corresponds to a theory of massive gravity, and α = 1 / . One obtains such a filterfunction by the argument provided in [2], which we sketch here. We begin with the Einsteinequations, linearized around Minkowski space:Ω αβµν h αβ := (cid:3) h µν − η µν (cid:3) h − ∂ µ ∂ α h αν − ∂ ν ∂ α h αµ + η µν ∂ α ∂ β h αβ + ∂ µ ∂ ν h = − πG N T µν . (1.8)By adding the Fierz-Pauli mass term [5], which is the only ghost free form for the propagatorof a massive spin two field, we obtain the equation of motion:Ω αβµν h αβ − m [ h µν − η µν h ] = − πG N T µν . (1.9)Group theoretic considerations tell us that a massive spin two particle has five polarizations,as opposed to the usual two for a massless spin two particle. As a way of book keeping theextra polarizations, we perform the St¨uckelberg decomposition, and write h µν as h µν = ˜ h µν + ∂ µ A ν + ∂ ν A µ , (1.10)with ˜ h µν being the helicity two piece, and with A µ carrying the extra polarizations. Wenote that this decomposition introduces the following gauge symmetry:˜ h µν → ˜ h µν + ∂ µ ξ ν + ∂ ν ξ µ , A µ → A µ − ξ µ . (1.11)Taking the divergence of (1.9) once we have St¨uckelberg decomposed h µν , we end up withthe equation of motion for A µ : ∂ µ F µν = − ∂ µ (cid:16) ˜ h µν − η µν ˜ h (cid:17) , (1.12)with F µν = ∂ µ A ν − ∂ ν A µ . Solving for A µ using this equation, and substituting this backinto (1.9), one can show after accounting for residual gauge invariances [2] that the resultingequations for the helicity two component is It is straightforward to show that although plane waves degravitate, radiation gases gravitate normallydue to decoherent superposition, and source the usual FRW radiation dominated expansion. G N is the usual 4-dimensional Newtons constant in the above. – 3 – m (cid:3) (cid:17) Ω αβµν ˜ h αβ = − πG N T µν . (1.13)One can generalize the Fierz-Pauli mass term to allow for a resonance graviton, where m → m ( (cid:3) ), such that (cid:16) m ( (cid:3) ) (cid:3) (cid:17) Ω αβµν ˜ h αβ = − πG N T µν (1.14)results. This is the basis for proposing (1.7) as filter function for Netwon’s constant.However, there are some immediate caveats to take note of here. Certainly (1.13) wasderived around a linearized approximation around Minkowski space. Although massivegravity can be defined around other backgrounds [6], one can only make sense of it in thelinearized approximation. In fact, its non-linear generalization is problematic and resultsin the instability of Minkowski space [7]. Hence to take (1.7) as it stands literally would bea mistake, not least because in addition, satisfying the Bianchi identities would impose newconstraints on top of the usual covariant conservation (used to derive (1.12)). Althoughthe spirit of the investigation that follows is to only consider filter functions of the form(1.7) phenomenologically, such an investigation would only be meaningful if our formalismis self-consistent. Hence it is neccesary to address the issue of the non-linear completionof (1.7) as well as the issue of satisfying the Bianchi identities. As we demonstrate next,there is another possibility in realizing a non-trivial filtering of gravity as in (1.3) whichaddresses both of these anxieties. G N The running of coupling constants is a well understood phenomenon in quantum fieldtheory. The nature of the running is determined by the renormalization group equations(RGE) of the theory in consideration. For a theory with a single coupling parameter, suchas quantum electrodynamics, the RGE takes the form µ dαdµ = β ( α ) , (1.15)where µ is the energy scale of interest (the external momenta flowing into a given process).In a typical gauge theory, solving for the differential equation that is the RGE results inthe following expression for the running couplings: α i ( µ ) = α i ( µ ) + c i ln ( µ/µ ) , (1.16)where µ is some reference energy scale where the coupling strength is experimentallymeasured, and c i is determined by the RGE at hand. When computing physical quantitiesderived from scattering amplitudes, we usually work in momentum space. However, if forsome reason we wanted to implement the running of couplings in configuration space, wecould do so via the replacement [8][9][10]: α → α ( (cid:3) ) = α ( µ ) + c ln [ (cid:3) /µ ] , (1.17)– 4 –here µ is some reference energy scale such that α ( µ ) = α , and c depends on the RGEof the theory in question. A simple context in which one can illustrate this explicitly isfurnished by QED, where the interaction Lagrangian is given by: L int = e ¯Ψ γ µ A µ Ψ( x ) . (1.18)In Fourier space, the net effect of running our coupling, is that physical processes becomefunctions of the scale of the process at hand, set by the external momenta p , i.e.: e → e ( p /µ ). From the properties of Fourier transforms, it is easy to see that taking thecoefficient of a product of local operators to now also depend on momentum, ‘de-localizes’the vertex. This is captured by the prescription of rewriting the coupling as e ( (cid:3) /µ ), wherethe delta function constraint on the vertex effects the replacement (cid:3) → p . This is to say,in position space, (1.17) encodes (typically non-local) quantum corrections to the classicalequations of motion. We illustrate this in context. In QED, we know that corrections tothe photon propagator from virtual fermion loops results in the running α → α ( k ) = α h α π ln − k m e + O ( e ) i ; k /m e >> , (1.19)with α ≈ /
137 being the low energy fine structure constant. In configuration space, thisresults in the replacement α → α h α π ln [ − (cid:3) /m e ] + O ( e ) i , (1.20)which effects the observed quantum corrections to the classical equations of motion. Forexample, we know that the electrostatic potential between an electron and an infinitelyheavy point charge (with momentum transfer k µ = (0 , ~k )) is obtained from the inverseFourier transform of the scattering amplitude: V ( r ) = Z d k (2 π ) e i~k · ~r πα ( − k ) k . (1.21)Taking the fine structure constant to be truly constant (4 πα = e ) results in the usualclassical electrostatic potential: V ( r ) = e πr . (1.22)However, we know that the coupling does in fact run, and the net effect of this is thatquantum effects effectively smear out the charge into a charge distribution given by [11] ρ ( r ) = Z d k (2 π ) e i~k · ~r πα ( − k ) , (1.23)which can be implemented in (1.21) through the prescription (1.17): V ( r ) = Z d k (2 π ) πα ( − (cid:3) ) e i~k · ~r k . (1.24)– 5 –n the limit of a vanishing electron mass ( m e r ≪ V ( r ) = e πr h α π (cid:16) ln m e r − γ (cid:17)i , (1.25)with γ being the Euler constant ( − γ ≈ − . (cid:3) . We now ask, isthis possible in gravity? We begin with a suggestive toy example.Quantum gravity in two dimensions is a renormalizable theory. This is due to the factthat Newton’s constant in 2-d is dimensionless. We could couple gravity to any number ofmatter and gauge fields, and compute the one loop beta function for Newton’s constant insuch a setting, as was done by Christensen and Duff in [12]. There one finds (working in d = 2 + ǫ dimensions) β ( G ) = ( d − G − β G − ..., (1.26)where β can be computed as β = 23 [1 − n / + n / − n − N + N / − N ] , (1.27)where our field content consists of n s massless spin s particles and N s massive spin sparticles. From (1.26) we see immediately the celebrated UV fixed point for gravity in2 + ǫ dimensions at G c = ( d − /β , for positive β . The differential equation defining thebeta function: µ dG ( µ ) dµ = β ( G ) (1.28)can be solved to give us the running of G : G ( k ) = G c ± ( m /k ) d − , (1.29)where m is now an integration constant which corresponds to the scale where we experi-mentally determine G . In configuration space, this corresponds to the filter function G ( (cid:3) ) = G c ± ( m / (cid:3) ) d − , (1.30)which is to be compared to the one parameter family of filter functions proposed in [2](1.7): G ( (cid:3) ) = G c m / (cid:3) ) − α , (1.31)where 0 ≤ α <
1. The question of how one would generalize this to four dimensionsto obtain a suitably degravitating filter function for G N is presently the subject of aninvestigation [13]. For the present purposes however, we only wish to draw from thisexample the lesson that the effects of a running G N would be to promote it to a function of– 6 – at the level of the field equations. Although the precise functional form might differ from(1.7), the objectives of degravitation will be met provided the condition (1.5) is satisfied.In fact, although we shall stick to (1.7) in much of what follows as a phenomenologicalexample through which to study degravitation, we argue further on by example that theconclusions we present in what follows depends only on the filter scale L ( = m − inthe above), and not the precise functional form of the filter. In addition, as we shall seein the next section, understanding degravitation as a semi-classical effect also allows usnew perspectives on how the Bianchi identities are to be satisfied. We now outline theinvestigation to follow. In this report, we study the various cosmological consequences of promoting Newton’sconstant to an operator valued filter function. We adopt a phenomenological approach inthis report, and take (1.7) as an example filter function which might have arisen from anynumber of ways– integrating out extra polarizations to the graviton arising from either asmall mass or through extra dimensional effects, or from the running of Newton’s constantas illustrated above. We argue further that the conclusions we draw from this example filterfunction generalize to other functional forms of suitably degravitating filter functions.We begin by setting up the formalism and addressing the issue of defining and solvingthe modified Einstein’s equations with a filter function for Newton’s constant, taking careto satisfy the Bianchi identities in this context. We then proceed to consider our samplefilter in the context of various cosmological scenarios, namely slow roll inflation, hybridinflation and old inflation, and show that in spite of neutralizing the cosmological constant,degravitation leaves these scenarios qualitatively unchanged. We then uncover the resultthat in a universe which is the end result of having tunneled out of some false vacuum(vacua) in the past, there exists a present day ‘after-glow’ cosmological constant whose scaleis parametrically suppressed by the square of the filter scale in Planck units. Given that weshould set the filter scale such that it should be larger than the present day horizon scale soas not to conflict with cosmological observations, we find Λ ∼ l pl /L < l pl H ∼ − inreduced Planck units. We then uncover the result that in the scenario where our universeremains stuck in some false vacuum state, degravitation only suppresses this false vacuumdensity enough to yield a cosmological constant similar to the one observed today withadditional fine tuning. We discuss this last result in the context of the coincidence problem,after which we offer our conclusions. In the following we work in natural units, and unlessotherwise states, take our metric tensor to be diag [1 , − a ( t ) , − a ( t ) , − a ( t ) ] . (1.32)The 4-dimensional Planck mass is taken to be M pl = (8 πG N ) − , where G N is the 4-dimensional Newton’s constant.
2. The Bianchi Identities and the Modified Einstein Equations
As first proposed in [1], we consider the following formal modification of Einstein’s equa-– 7 –ions: G µν = 8 πG N ( L (cid:3) ) T µν . (2.1)One encounters the immediate curiosity that since in general [ ∇ µ , (cid:3) ] = 0, either the Bianchiidentities are not satisfied in the above, or new constraints are imposed on a covariantlyconserved energy momentum tensor. We wish to demonstrate that it is in fact the latterthat is true, and that the correct Bianchi identity is indeed ∇ µ [ G ( L (cid:3) ) T µν ] = 0.In the context of obtaining a filtering G N as the configuration space expression of itsrunning, this is to be expected considering the fact that in reality, the equations (2.1) areactually the equations G µν = 1 M pl h T µν i (2.2)in disguise. As we encountered with the previous examples, just as quantum effects modifythe classical equations of motion so as to introduce non-local operators (1.17) in placeof coupling constants (so that point sources get smeared out (1.23)), so we expect thequantum mechanical running of G N to give us (2.1) as an paraphrasal of (2.2), the righthand side of which is to be obtained from the effective action as: h T µν i = 2 √− g δWδg µν . (2.3)We observe that the effective Lagrangian density (we take for illustrative purposes, a mas-sive degree of freedom with mass m ) is given by the expression [14]: W = Z d x p − g ( x ) L eff ( x ) (2.4) L eff = i x ′ → x Z ∞ m d ¯ m G DSF ( x, x ′ ; ¯ m ) , (2.5)where G FDS ( x, x ′ ; ¯ m ) is the Feynman propagator for a field with mass ¯ m in the SchwingerDe-Witt representation. We note that it is the part of L eff which remains finite in thelimit x ′ → x which is sensitive to the large scale structure of the manifold (i.e. is non-local)and to the quantum state defining the expectation value [14].In this way, it should not be surprising that the true quantum corrected equations ofmotion should be ∇ µ [ G ( L (cid:3) ) T µν ] = 0, as ∇ µ T µν = 0 implies only the classical equationsof motion, derived from the classical Lagrangian density L c as opposed to L eff . Thuswe understand the modified Bianchi identities as actually encoding quantum correctionsto our classical equations of motion. This can also be understood as coming from thenon-commutativity of taking the expectation value of an operator, and taking its covariantderivative one we account for the fact that the perturbative vacuum differs from the freevacuum. We flesh this argument out with the example of a free scalar field, which has theenergy momentum tensor: T µν = ∇ µ φ ∇ ν φ − g µν ∇ κ φ ∇ κ φ + m g µν φ . (2.6)– 8 –e define the above bilinear quantity as T µν = T µν [ φ, φ ]. If the field admits the modeexpansion φ ( x, t ) = Z d k (cid:16) u ~k a ~k + u ∗ ~k a † ~k (cid:17) , (2.7)where u k is the appropriate mode function, indexed by ~k such that the vacuum is definedas a ~k | i = 0 ∀ ~k . It is easy to check [14] that h | T µν | i = Z d k T µν [ u ~k , u ∗ ~k ] , (2.8)whereas for non vacuum expectation values: h n ( k ) , n ( k ) ... | T µν | n ( k ) , n ( k ) ... i = Z d k T µν [ u ~k , u ∗ ~k ] + 2 Z d k X i n i ( k ) T µν [ u ~k , u ∗ ~k ] . (2.9)We note from the form of (2.6), that the graviton propagator will receive quantum correc-tions from scalar field loops. In addition, any scalar self couplings will also run, and thevacuum of the interacting theory will differ from the vacuum of the free theory. We canunderstand how quantum effects alter the classical equations of motion by the followingargument. Although it is clear from (2.8) that ∇ µ h | T µν | i = 0 (as on the right handside we simply have the mode decomposition of ∇ µ T µν = 0, where by construction, themode functions satisfy the equations of motion), it should also be clear that the presenceof non zero occupation numbers n i ( k ) makes the inverse transform of the divergence ofthe second term on the right hand side of (2.9) no longer vanish. Considering that thevacuum defined by the mode functions used above is not the true perturbative vacuum forthe interacting theory, the interacting vacuum expectation value h I | T µν | I i is in generalgoing to contain non-zero particle occupation numbers of the free vacuum (i.e. terms ofthe form (2.9) will result). Hence non-trivial vacuum structure results in non-satisfactionof the classical Bianchi identities as a result of (2.9), which implies quantum correctedequations of motion. Thus the result of inverting the mode expansion of the vev of theenergy momentum tensor in the interacting vacuum back to configuration space (in flatspace, we’d simply Fourier transform), is that we would obtain the modified equation ofmotion (3.1): G µν = 1 M pl h I | T µν | I i = 8 πG N ( (cid:3) ) T µν (0) (2.10)where T µν (0) is the classical energy momentum tensor corresponding to the non-interactingvacuum.In the event that our gravitational filter arises from a small resonant mass to thegraviton (perhaps through extra dimensional effects), the new constraints imposed by theBianchi identities are to be understood through the following argument proposed in [2].The filter function in (2.1) is obtained through integrating out the extra polarizations of amassive spin two theory, which contains five propagating degrees of freedom as opposed tothe usual two of Einstein gravity. However the Einstein tensor in (2.1) is constructed onlyfrom the spin two sector of this theory, therefore the energy momentum tensor should not– 9 –e conserved with respect to covariant differentiation with the spin two metric, as it is onlyconserved with respect to the full metric which contains the extra polarizations. In thisway, the non-conservation of the right hand side of the above is accounted for by the everpresent extra polarizations [2]. With this in mind, we proceed to study the cosmologicalconsequences of the modified Einstein equations, to be viewed as the phenomenologicalmanifestation of whatever the physics might be that underlies the filtering of gravity.Before we continue however, we wish to note that in general, the action of (cid:3) ona rank two tensor mixes up its components non-trivially. However, since the covariantd’Alembertian always commutes with the metric tensor, we can avoid this concern bysimply considering the trace of the above: R = − πG N ( L (cid:3) ) T, (2.11)In this form, we do not even have to make a specific ansatz for the form of the metrictensor to write down a formal expression for curvature as a function of a matter source.Having set up the bare basics of the degravitation framework, we turn our attentiontowards inflationary cosmology in this setup. As articulated in [15], a long standing concerndirected at any proposed direct solution to the cosmological constant problem is that itshould not also render inflation null. Since inflation depends on the time independent pieceof a scalar field potential, this is a valid concern and as we will shortly see, one that thedegravitation paradigm answers in a rather appealing manner.
3. Inflation (v.s. the Cosmological Constant)
Consider a universe that is dominated by matter with an equation of state parameter w = −
1. 8 πG N ( L (cid:3) ) ρ = 8 πG N ( L (cid:3) ) V ( φ ) . (3.1)Focussing on a filter function of the form:8 πG N ( L (cid:3) ) = 8 πG N m (cid:3) , (3.2)we can immediately prove that such a filter function does in fact degravitate sources withwavelengths larger than the filter scale. Consider a mode expansion of source in terms ofhomogeneous plane waves , and consider the action of (3.2) on such a mode. Were we toexpand (3.2) in terms of its power series, we have to take care of the fact that we mustexpand within the radius of convergence. That is, if (cid:3) acts on such a mode function witheigenvalue − k , then we have two possible expansions:11 + m (cid:3) = X n =0 (cid:16) − m (cid:3) (cid:17) n = X n =0 (cid:16) m k (cid:17) n ; m < k (3.3) Although these are not technically the eigenfunctions of the D’Alembertian in a de Sitter spacetime(which are hypergeometric functions), for homogeneous sources they do satisfy (cid:3) e iωt = λe iωt , where λ = − ω + 3 iHω is a complex ‘eigenvalue’. – 10 –1 + m (cid:3) = (cid:3) m
11 + (cid:3) m = X n =0 (cid:16) − (cid:3) m (cid:17) n (cid:3) m = X n =1 − (cid:16) k m (cid:17) n ; k < m . (3.4)Where we draw attention to the fact that the sum in (3.4) commences at n = 1 whereas thesum in (3.3) commences at n = 0. From all this we see that for all sources with wavelengthsmuch greater than the filter scale ( L = m − ) effectively degravitate: G eff = G N O ( k /m ) k ≪ m , (3.5)whereas all sources with wavelengths much smaller than the filter scale gravitate almostnormally: G eff = G N (1 + O ( m /k )) m ≪ k . (3.6)In other words, because the leading order term in (3.4) is proportional to (cid:3) , any bona fidespacetime zero mode is annihilated by the degravitation filter. One might then wonder howthis is consistent with causality, for how is the degravitation filter to know immediatelywhether an energy density is a legitimate zero mode, or whether it might change in thefuture? The answer to this lies in the fact that the filter is in fact sensitive to localchanges in the energy density, as evidenced by the continuous dependence of (3.4) on thewavenumber of the source. Any local variation, no matter how small will result in a non-zero Newton’s constant in a manner that continuously approaches zero for a spacetime zeromode. In fact, as we shall see shortly, if our energy density were to suddenly jump fromone constant in the present to another in the future, then the filter does not annihilate thesource, except in the far future once the source has had enough time to look like a zeromode to an asymptotic observer.Since we are interested in cosmological applications, we first restrict ourselves to theeffect of the degravitation filter on homogeneous sources (with respect to the spatial coor-dinates). We can express such a source in terms of its inverse Laplace transform: ρ ( t ) = 12 π Z ∞−∞ dωe iωt ρ ( iω ) , (3.7)where the Laplace transform is defined as ρ ( ω ) = Z ∞ dt ′ e − ωt ′ ρ ( t ′ ) , = Z ∞−∞ dt ′ e − ωt ′ ˜ ρ ( t ′ ) , with the definition ˜ ρ ( t ) = ρ ( t ) t >
0= 0 t < . – 11 –he use of the Laplace transform is relevant for a universe with a beginning in the finitepast. To model an eternal universe we simply set ˜ ρ ( t ) = ρ ( t ) for all times and understandthe above as a Fourier transform. We now consider the effect of the filter (3.2) on ahomogeneous source :8 πG N m (cid:3) ρ ( t ) = 8 πG N Z ∞−∞ dω e iωt ( ω − iHω ) ω − iHω − m ρ ( iω )2 π . (3.8)In order to study the effect of degravitation on various inflationary scenarios, it suffices toconsider energy densities corresponding to two special cases– a slowly varying potential,and a potential which is constant everywhere except for a finite jump/ drop at a specifictime (i.e. a step function).In frequency space, a slow rolling potential can be modeled as the real part of V ( ω ) = V δ ( ω ) + V δ ( ω − ǫ ) , (3.9)which corresponds to the potential V ( t ) = V + V e iǫt , (3.10)where ǫ is taken to be some very small frequency (relative to the degravitation and Hubblescales). Evaluating (3.8) in this context is rather straight forward: V deg ( t ) = V ˜ δ ( m ) + V e iǫt − m ǫ − iHǫ , (3.11)where V deg ( t ) is the degravitated potential, and ˜ δ ( m ) is formally defined as˜ δ ( m ) = 1 , m = 0 (3.12)= 0 , m = 0 . In this way, we again see how degravitation immediately annihilates any pure spacetimezero mode for any finite filter scale. For the slowly varying piece, for ǫ ≫ m and H ≫ m (which corresponds to the field velocity and the Hubble scales being much less than the timescale associated with the filter scale), we see then that (3.11) reduces to (3.10) immediately.In fact for chaotic slow roll inflation, we find that ǫ is typically given by ǫ ∼ m φ / ˜ ǫ, (3.13)where m φ is the mass of the inflaton and ˜ ǫ is the standard dimensionless slow roll parameter.From this, we see that the condition m/ǫ = ˜ ǫm/m φ ≪ For the moment, we make the approximation that in an inflationary universe, the time scale set by thefilter scale τ = m − is far greater than at least one scale in the problem (e.g. the Hubble scale H − , or thetime scale over which we are considering cosmological evolution). We will attempt to relax this requirementfurther on, but for the time being it allows us to simply act through with the D’Alembertian on (3.8), as itensures that any time dependence in H (inferred from the 00 modified Einstein equation) will be negligible.This is certainly justified up to terms of order m . – 12 –he filter scale is to be larger than the present Hubble scale in order not to effect gravity atpresently accessible scales ( m − > H − ), and that the mass of the inflaton is bound to betens of orders of magnitude greater than this. Thus we conclude that aside from the noncontribution of the zero mode (which is typically set to zero by hand anyway), slow rollinflation is virtually unaffected by the degravitation filter once we account for the relevantscales.The next example we turn to is that of a step function (in time) change in energydensity, which is of interest to us for its utility in modeling various scenarios of interest.We represent the step function as θ ( t − t ) = Z ∞−∞ dω e iω ( t − t ) h δ ( ω )2 + 12 πiω i . (3.14)By applying the degravitation filter to the above, we obtain (cid:16) m (cid:3) (cid:17) − θ ( t − t ) = Z ∞−∞ dω e iω ( t − t ) ( ω − iHω ) ω − iHω − m h δ ( ω )2 + 12 πiω i , (3.15)which evaluates to (cid:16) m (cid:3) (cid:17) − θ ( t − t ) = ˜ δ ( m )2 + 12 πi Z ∞−∞ dω e iω ( t − t ) ( ω − iH )( ω − ω + )( ω − ω − ) , (3.16)with ˜ δ ( m ) defined in (3.12), and where the denominator factors into the roots ω ± = 3 iH (cid:16) ± r − m H (cid:17) . (3.17)we see that (3.16) is readily evaluated in light of the pole structure at the points ω ± . Wesee that for non zero values of m , both poles lie above the real axis, and when m vanishes, ω − lies on the real axis. Hence for t < t , in closing the contour in (3.16) in the lower halfplane, we encounter no poles unless m = 0, in which case we get a contribution of − / t < t we get (cid:16) m (cid:3) (cid:17) − θ ( t − t ) = ˜ δ ( m )2 − ˜ δ ( m )2 = 0 , t < t (3.18)which is exactly what we would have got in the undegravitated case. This should not besurprising, as degravitation is engineered to preserve causality, and up to t < t should notbe able to tell the difference between a step function (with the step yet to come), and azero mode. Proceeding similarly for t > t , we find in closing the contour in the upper halfplane we will always enclose both roots, except that in the case where m = 0, the rootat ω − contributes half the residue it would otherwise as it then lies on the real axis ratherthan above it. With this in mind, we can evaluate (3.16) for t > t as (cid:16) m (cid:3) (cid:17) − θ ( t − t ) = ( ω + − iH ) e iω + ( t − t ) − ( ω − − iH ) e iω − ( t − t ) ω + − ω − , t > t (3.19)– 13 – e − H ( t − t ) h cosh (cid:16) H t − t ) r − m H (cid:17) (3.20)+ 1 q − m H sinh (cid:16) H t − t ) r − m H (cid:17)i , whose leading order behavior in the inverse filter scale m is given by: (cid:16) m (cid:3) (cid:17) − θ ( t − t ) = (cid:16) m H (cid:17) e − m H ( t − t ) − m H e − H ( t − t ) ∼ − m t − t ) , (3.21)which around t , mimics a Gaussian: (cid:16) m (cid:3) (cid:17) − θ ( t − t ) = e − m ( t − t ) . (3.22)That is, the degravitation filter appears to transform the step function into a function thatis zero before t = t , and decays exponentially afterwards with a characteristic time setby the inverse filter scale t = m − , which is typically larger than the present age of theuniverse.In this way, we see that although degravitation kills any ’bare’ cosmological constantterm, energy densities which condense at some finite time into the history of the universewill require a timescale parametrically dependent on the inverse filter scale to degravitate.We apply these considerations to hybrid inflation [16], which evidently relies on the space-time zero mode of the inflaton potential to drive inflation. The potential for a generichybrid inflation model is given by V ( φ, σ ) = 14 λ ( M − λσ ) + m φ φ + g φ σ , (3.23)where the mass of φ is tuned such that it rolls very slowly. Given that the effective massfor the field σ is equal to − M + g φ , for φ > φ c = M/g , there is only one minimafor the field σ at σ = 0. In this regime, the zero mode piece of the potential dominates( V = M /λ ) and drives inflation, which ends when φ rolls below φ c . Were we to applythe degravitation filter to (3.23), the zero mode piece of the potential would seem to beannihilated as evidenced from the expansion of the filter in (3.4). However we note thatone should really think of (3.23) as being multiplied by a step function in time whichmodels the fact that the energy condenses at some finite time, or alternatively, modelsthe beginning of the universe at some finite time in the past . Hence we conclude thatdegravitation leaves hybrid inflation unaffected provided it lasts for a duration less thanthe timescale associated with the filter scale, which it certainly does if m − > H − , where H is the present day Hubble parameter. To not do so would imply that this energy density has been around since the infinite past, by whichtime we expect degravitation to have naturally killed the constant part of the energy density. Hence in anon-eternal universe, we have to multiply all energy densities by a step function to model a universe witha beginning. – 14 –e next consider the effect of degravitation on the following potential: V ( t ) = V + N X i =1 ( V i − V i − ) θ ( t − t i ) , t i − < t i (3.24)which models a series of potential drops from V i − to V i at times t i , i.e. V ( t ) = V k if t k < t < t k +1 . Using the results from previous calculation, we find that the effect of thedegravitation filter on the potential at a time t k < t < t k +1 is : (cid:16) m (cid:3) (cid:17) V ( t ) = k X i =1 ( V i − V i − )2 h(cid:16) − q − m H (cid:17) e − H ( t − t i )(1+ q − m H ) i ; t k < t < t k +1 + k X i =1 ( V i − V i − )2 h(cid:16) q − m H (cid:17) e − H ( t − t i )(1 − q − m H ) i , (3.25)which to second order in the inverse filter scale m , is given by (cid:16) m (cid:3) (cid:17) V ( t ) = V k − m k X i =1 ( V i − V i − )( t i − t i − ) ; t k < t < t k +1 . (3.26)Here we again see how degravitation leaves the potential unaffected for all time scales muchsmaller than the degravitation scale. Hence we see that old inflation, which can heuristicallybe modeled as such a single step function potential drop is unaffected provided its durationis microscopic compared to the degravitation scale. We note here that the degravitatedpotential at any given time is always going to be slightly larger than the undegravitatedpotential for any series of cascades, as degravitation carries a memory of the previous vacua(suppressed by powers of m ) in the second term in the above. It is this observation thatsuggests that todays cosmological constant might be due to a similar such afterglow byconsidering the above in the context of some sort of potential (e.g. stringy) landscape.Supposing that our universe is presently in its true vacuum, but was once trapped in aseries of false vacua, modeled by the transitions V → V → ... → V N = 0. We find thatalthough the matter sector potential energy in the present epoch vanishes, degravitationimplies an inherited memory of previous false vacua such that immediately after the finaltunneling event, we still feel an remnant energy density: V rem ≤ m | ∆ V | ∆ T . (3.27)In the above, ∆ V is the total potential drop since the beginning of the universe and∆ T = t − t N , where we note that the longer the duration of the period we are stuck inthe false vacua, the stronger the afterglow we feel, as the non-locality of the degravitation Where we set V = 0 to model a universe where a potential energy of V = 0 appears at the beginningof the universe at time t = 0, and successively cascades down a series of steps. – 15 –lter has a longer duration source to act on. In the context of the Einstein equations3 H = 8 πG N V rem , this implies an apparent cosmological constantΛ = 3 H M pl = m M pl | ∆ V | ∆ T M pl , (3.28)whose scale is an energy density in dimensionless units parametrically suppressed by thesquare of the filter scale in Planck units. Recalling that in requiring m < H , we see that m M pl < l pl H ∼ l pl R , (3.29)where R is the size of the universe today. Hence we see that although degravitation killsany spacetime zero mode (i.e. the bare cosmological constant), this calculation impliesa remnant cosmological constant might still be observed today with a magnitude that issuppressed by the degravitation scale:Λ ∼ l pl L ∼ − , (3.30)if we take L ∼ H − . In the following sections, we will rederive this result through refor-mulating the problem in a manner which admits generalization to spatially inhomogeneoussources. We will also uncover the result that in the scenario where we remain stuck in somefalse vacuum state, although degravitation causes the vacuum energy density to decay overtime, it does so over a timescale that is typically too slow compared to the age of theuniverse.Before we continue however, we parenthetically note that all of the results and con-clusions we have just discussed generalize rather straightforwardly to other values of α in(1.7). For example, having accounted for subtleties involving branch cuts in the integral(3.8), we find in the context of (3.19) that in place of ω ± as being given by (3.17), we haveinstead (for 0 ≤ α < ω ± = 3 iH (cid:16) ± r m H e iπ ( − α ) (cid:17) , (3.31)with similarly straightforward generalizations for other results in the above. We thuscontinue to work with α = 0 in the next section for quantitative convenience, with theresults concerning values of α being similarly generalizable.
4. An Afterglow Cosmological Constant?
We can formally recast (3.2) as8 πG N m (cid:3) = 8 πG N (cid:3)(cid:3) + m = 8 πG N (cid:3) ( (cid:3) + m ) − , (4.1)so that its action on any source takes the form– 16 – πG N (cid:16) (cid:3) m (cid:17) ρ ( x ) = 8 πG N (cid:3) Z d x ′ p − g ( x ′ ) G ( x, x ′ ) ρ ( x ′ ) , (4.2)where G ( x, x ′ ) satisfies ( (cid:3) x + m ) G ( x, x ′ ) = δ ( x, x ′ ) p − g ( x ′ ) . (4.3)Acting through with (cid:3) in (4.2) and using (4.3), we obtain the result ρ degrav ( x ) = ρ ( x ) − m Z d x ′ p − g ( x ′ ) G ( x, x ′ ) ρ ( x ′ ) , (4.4)which makes manifest the causality (and non-locality) of degravitation for the appropriatechoice for the Greens function in the above. Furthermore, upon integrating (4.3) we find1 = m Z d x p − g ( x ) G ( x, y ) + Z ∂ M d ξ n a ∂ a G ( ξ, y ) , (4.5)where we note that the surface integral vanishes at coordinate infinity by virtue of the finitemass scale m (which results in exponentially damped asymptotic behavior of the Green’sfunction). Hence we find that the Green’s function is normalized as 1 = m R d x ′ p − g ( x ′ ) G ( x ′ , x ) ,which allows us to recast (4.4) as ρ degrav ( x ) = ρ ( x ) − R d x ′ p − g ( x ′ ) G ( x, x ′ ) ρ ( x ′ ) R d x ′ p − g ( x ′ ) G (0 , x ′ ) . (4.6)If we assume that we are in a de Sitter background, then we know that the retarded Greensfunction is[14] G ( x, x ′ ) = θ ( x − x ′ ) R e i H π Γ (cid:16)
32 + ν (cid:17) Γ (cid:16) − ν (cid:17) F (cid:16)
32 + ν, − ν, , z + iǫ (cid:17) , (4.7)with z being the de Sitter invariant geodesic distance between x and x ′ and ν =9 / p − m / H . This suggests an interpretation of (4.6) as the weighted average: ρ degrav ( x ) = ρ ( x ) − h ρ i x , (4.8)with the average defined by: h f i x := R d x ′ p − g ( x ′ ) G ( x, x ′ ) f ( x ′ ) R d x ′ p − g ( x ′ ) G (0 , x ′ ) . (4.9)We immediately see how a spatial zero mode (bare cosmological constant) is degravitated,as Λ degrav = Λ(1 − h i ) = 0 . (4.10) The integral is independent of the argument x , which we pick to be the origin in the following See [19] for this and other aspects of de Sitter space physics used here. – 17 –onsider now a source corresponding to a bubble of one vacuum nucleating in anotherstarting at time t = t at the origin: V ( x ) = V + ( V f − V ) θ [ t − t ] θ [ z ( x, − , (4.11)where we see that the product of step functions effects a potential of V everywhere exceptinside the future lightcone of the nucleation event, where it is V f . Consider the effect ofthe degravitation filter on the step function through (4.8) and (4.9): θ [ t − t ] θ [ z − degrav = θ [ t − t ] θ [ z − − R (cid:7) √− gG ( x ) R N √− gG , (4.12)where z is the de Sitter invariant geodesic distance to the nucleation event ( z = 1 implieszero or null separation), R (cid:7) indicates an integral over the causal diamond bounded by x in the present, and the nucleation event in the past and R N indicates an integral over theentire past light cone of any observer at the origin (see footnote associated with (4.6)). For t < x and the nucleation event as it is yet to happen(the theta function vanishes in the integrand), so we get: θ [ t − t ] θ [ z − degrav = θ [ t − t ] θ [ z −
1] = 0 t < t , (4.13)which is exactly as we had obtained in our previous example (3.18). For t > t , the integralover the causal diamond commences at zero and asymptotes towards the integral over theentire past light cone of the origin event ( R (cid:7) G → R N G ). Hence θ [ t − t ] θ [ z − degrav = 1 − f ( t ) ; f (0) = 0 → f ( ∞ ) = 1 . (4.14)Since the function G ( x, x ′ ) describes the propagation of a massive particle, we expect it tobe an exponentially decaying function of the geodesic distance between x and x ′ , with acharacteristic scale set by m . To see this (and to reproduce the results of the last section),we work in co-ordinates where ds = 1 H η ( dη − dx i dx i ) , (4.15)and consider for simplicity, a homogeneous source as in the previous section. In this case,the relevant source is θ ( x ) = θ ( η − η ). In this case, (4.4) becomes θ degrav ( η − η i ) − m Z dη ′ d x ′ p − g ( x ′ ) G ( x, x ′ ) θ ( η ′ − η i ) , (4.16)with G ( x, x ′ ) given as in (4.7). Since our source is now spatially homogeneous, we canintegrate the Green’s function over the spatial coordinates to give us [20] Z d x ′ p − g ( η ′ ) G ( x, x ′ ) = θ ( η − η ′ )( − η ) / H ν ( − η ′ ) / sinh [ νln ( η/η ′ )] (4.17)= θ ( η − η ′ )2 H ν (cid:16) η / − ν η ′ / − ν − η / ν η ′ / ν (cid:17) , – 18 –ence the integral we have to perform is: Z N d x ′ p − g ( x ′ ) G ( m ; x, x ′ ) = 12 H ν Z ηη i dη ′ (cid:16) η / − ν η ′ / − ν − η / ν η ′ / ν (cid:17) (4.18)= 1 m − νm h ( ν + 3 / η/η i ) / − ν + ( ν − / η/η i ) / ν i , from which we see that if we were to push the initial time η i → −∞ , the integral asymptotesto 1 /m , as implied by (4.5). When we rewrite conformal time in terms of cosmologicaltime: η = − e − Ht H , (4.19)we see that (4.18) becomes:1 m h − e − H ( t − t i ) / (cid:16) cosh [ νH ( t − t i )] + 32 ν sinh [ νH ( t − t i )] (cid:17)i , (4.20)so that the degravitated step function becomes: θ ( t − t i ) degrav = 0 ; t < t i (4.21)= e − H ( t − t i ) / (cid:16) cosh [ νH ( t − t i )] + 32 ν sinh [ νH ( t − t i )] (cid:17) = e − H ( t − t i ) / (cid:16) cosh [ r − m H H ( t − t i ) / q − m H sinh [ r − m H H ( t − t i ) / (cid:17) ; t > t i , which is exactly as in (3.19). From this, we can again follow the logic which led up to(3.27) and (3.30), allowing us the possibility that although the degravitation mechanismhas rendered the spacetime zero mode of the energy momentum tensor gravitationallynull, it does so in a way that preserves inflation, and in such a way that it can accountfor the presently observed value of the cosmological constant as a memory of previousenergy densities. The scales of our problem allow us the possibility that the value for thecosmological constant Λ ∼ − (4.22)is set through the hierarchy of scales that is the filter scale in Planck units (as in (3.28)).It might appear to us at this point that we have replaced one tuning for another, in thatnow the smallness of the cosmological constant depends on the extreme smallness of thefilter scale (albeit a smallness that is forced on us by the consistency of GR at most scalesaccessible to our observations). However in the context of the filter arising from a smallgraviton mass, this tuning is technically natural, in that the graviton mass is radiativelystable. The reason for this can be understood from the fact that the graviton mass is– 19 –rotected quantum mechanically (e.g. against gauge field loops) by the gravitational Wardidentities, hence any small mass causes the quantum mechanical corrections to the gravitonmass to be proportional to the mass itself, and hence can be neglected.We now turn our attention to the scenario that our universe is still stuck in some metastable vacuum state (as implied by the notion that we live in potential landscape). Wewish to explore whether or not degravitation can degravitate this false vacuum energy toa reasonably small value over the age of the universe, and whether or not it does so in away that addresses the coincidence problem. As we shall see, the class of degravitationmodels [2] that we study here act over time scales that are far too slow to degravitate falsevacuum energy densities in a way that addresses the coincidence problem .
5. The Coincidence Problem
To offer us further perspective on the degravitation mechanism, and to study degravitationin other contexts, we reformulate the problem in yet another manner. We begin by actingon both sides of (3.2) with appropriate powers of (cid:3) and (cid:3) + m , to rewrite the modifiedEinstein’s equations as 8 πG N T µν = G µν + m (cid:3) − G µν , (5.1)whence the 00 equation becomes8 πG N ρ ( x ) = 3 H ( x ) + m Z p − g ( x ) G ( x, x ′ )3 H ( x ′ ) , (5.2)and where G ( x, x ′ ) satisfies (cid:3) x G ( x, x ′ ) = δ ( x, x ′ ) p − g ( x ) . (5.3)Rewriting the above as3 H ( x ′ ) = 8 πG N ρ ( x ′ ) − m Z p − g ( x ′′ ) dx ′′ G ( x ′ , x ′′ )3 H ( x ′′ ) , (5.4)and substituting in the integrand in (5.2) and iterating, for a step function potential drop ρ = ∆ V θ [ t − t ] θ [ z −
1] := θ , we arrive at the expression3 H = ∆ VM pl h θ − m ( G , θ ) + m ( G , ( G , θ )) − m ( G , ( G , ( G , θ ))) + ... i , (5.5)where G is the retarded Greens function for a minimally coupled massless scalar field inthe fully degravitated spacetime (for its expression in pure de Sitter space, see for example[21]), and ( G , θ ) := R (cid:7) G is the integral over the domain (cid:7) , again defined as the causaldiamond bounded by the nucleation event and the observation event. Clearly the right side However it might be that there are models that act over much faster time scales [1] which might workin this regard, we postpone this to a future study. – 20 –f the above vanishes for t < t . For t > t , realizing the ordered nature of the multipleintegrals in the above, for strictly homogeneous sources (i.e. ones which are only functionsof time) we can re-express (5.5) as3 H = ∆ VM pl h − m ∆ + m ∆ − m ∆
3! + ... i , (5.6)where ∆ is given by ∆ = Z N G , (5.7)and N denotes the backward lightcone of the observation event. This expression easilyresums to the following expression for the degravitated step function θ ( t − t i ) degrav = e − m R N G ; t > t i . (5.8)In the context of (5.6), this implies the rather complicated integral equation3 H = ∆ VM pl e − m R N G [ H ] , (5.9)where the implicit dependence on H in the integrand is highlighted. This equation is exact,and in principle (though not in practice) solvable. We can make progress by invoking anadiabatic approximation, namely that we take the scale that sets spacetime curvature H to be a very slowly varying function of co-ordinate time. This implies at the very leastthat m << H . For convenience, we also consider the modified filter function πG N m (cid:3) + µ , (5.10)where µ is some other IR length scale that is taken to be much less than m , and take theappropriate limit at the end of our calculations. In this case we find that (5.10) results inthe expression θ ( t − t i ) degrav = e − m R tti R d xG ( x,x ′ ) , (5.11)with G ( x, x ′ ) given by (4.7), but with κ in place of ν , with κ given by κ = 9 / − µ /H . (5.12)As in the previous section, we can evaluate the integrand in the above to yield θ ( t − t i ) degrav = e − m µ (cid:16) − κ { ( κ +3 / exp [ − H (3 / − κ )( t − t i )]+( κ − / exp [ − H ( κ +3 / t − t i )] } (cid:17) , (5.13) the corresponding expression for a nucleating bubble (which is manifestly inhomogeneous) is not soeasily resummable. In general, one might even expect a filter function of the form (5.10) as powers of (cid:3) and multiples ofthe identity operator mix readily under renormalization group transformations. – 21 –hich in the limit µ → θ ( t − t i ) degrav = e − m ( t − t i ) , (5.14)exactly as in (3.22). Hence we can infer in a slightly different context, that the relevanttimescale for degravitation to effect itself is set by the inverse filter scale τ = m − . Wethus reason that we were to require degravitation to degravitate Planck or GUT scaleenergy densities (as one might expect if we were stuck in some metastable vacuum statein the landscape), down to the presently observed value, we then require that a time ofat least an order of magnitude larger than m − to have elapsed. That is, from (5.14) wecan estimate that we require a time interval of ∆ t ∼ m − √
276 to elapse . Typically, werequire m − > H − , where H is the presently observed Hubble scale. Hence we find thatunless the universe began much earlier than we infer from current measurements of H (asis entirely likely if we live in an eternally inflating landscape), that degravitation is unlikelyto have degravitated primordial string or Planck scale energy densities sufficiently.However, the main observation of this section, is that regardless of whether or notdegravitation has had enough time to act in order to degravitate potentially high scaleprimordial energy densities, if it is to satisfactorily address the coincidence problem, it hasto act on a timescale comparable to the present Hubble scale:∆ t ∼ H − , (5.15)which we have already demanded not be the case so as not to conflict with other cosmolog-ical observations. Hence it appears as if the degravitation filter (1.7) acts too slowly to beof any help in addressing the coincidence problem without additional tuning. To concludethis report, we rework the calculation for how the degravitation filter degravitates a stepfunction source for a different functional form for the filter function. Consider the samplefilter function: 8 πG N → πG N [ ln (cid:16) e (cid:3) + µ µ (cid:17) − ln (cid:16) (cid:3) + µ µ (cid:17)i (5.16)= 8 πG N [1 + ln (cid:16) (cid:3) + µ /e (cid:3) + µ (cid:17)i (5.17)which is easily checked to yield a degravitating filter function. We chose this functionalform only for illustrative purposes. Using the representation[8][10] ln [ (cid:3) /µ ] = Z ∞ dκ (cid:16) µ + κ − (cid:3) + κ (cid:17) , (5.18)we find that the action of such a filter function on a step function potential would result in θ degrav ( t − t ) = Chi (cid:16) H ∆ t − p − µ / H ] (cid:17) − Shi (cid:16) H ∆ t − p − µ / H ] (cid:17) + Chi (cid:16) H ∆ t p − µ / H ] (cid:17) − Shi (cid:16) H ∆ t − p µ / H ] (cid:17) − [ ... ] µ → µ /e (5.19) The factor 276 comes from the fact that e − = 10 − . – 22 –here Chi ( x ) and Shi ( x ) are the hyperbolic cosine integral and hyperbolic sine integralfunctions respectively. We compare this expression to (3.20) to infer the same characteristicdependence on the inverse filter scale, such that the timescale associated with degravitationis given by τ = µ − . In this way, we infer that the conclusions we drew in the previoussections would be true of this filter function as well, implying that degravitation appearsto be somewhat insensitive to the precise functional form of the filter function. We nowoffer our concluding thoughts.
6. Conclusions
In this report, we have seen in detail how degravitation works in annihilating the bare cos-mological constant whilst preserving inflation in all of its forms. We have also demonstratedhow degravitation inherits a memory of previous energy densities in such a way that evenif our universe were to exist in its true vacuum state today, degravitation would imply anafterglow cosmological constant which can naturally be arranged to mimic the dark energythat we infer today. The key physics of this observation is that such an afterglow is sup-pressed by the square of the inverse filter scale in Planck units m /M pl = l pl /L < − .We then showed that if we exist in a universe which is still trapped in some false vacuumstate, degravitation can degravitate the energy of such a false vacuum into a remnantenergy density of the order Λ ∼ − , however this typically occurs over timescales atleast an order of magnitude larger than the age of the universe. In this way, althoughdegravitation answers why the cosmological constant is not large (the bare cosmologicalconstant problem), as well as why it is not zero (that the cosmological constant might bethe dark energy that we observe today)[22], it does not satisfactorily explain the coinci-dence problem, or why it only begins to dominate now (although this is not to say thatother models of degravitation may not have something to say about this). However, anymechanism that contains hints of being able to address all three aspects of the cosmolog-ical constant problem if further developed (while preserving the successes of inflationarycosmology) certainly warrants closer attention, and we hope that the findings of this reportwill motivate further investigation into this promising paradigm.
7. Acknowledgements
I am grateful to Gia Dvali and the CCPP at NYU for hospitality during the time inwhich this work was initiated, and for illuminating discussions on the sidelines of variousconferences throughout the year. Thanks to Justin Khoury and the Perimeter Institute forthe invitation to attend a stimulating conference and many useful exchanges. Thanks toGregory Gabadadze for many useful discussions, perspectives and a copy of the preprint[23]. I remain indebted to Robert Brandenberger for his continued support and frequentdialogue. Special thanks to Zoe Greenberg, whose presence in my life has left a wake In the context of massive gravity, it is interesting to note how our results appear as the reverse inter-pretation of the bound m ≥ Λ / – 23 –f inspiration that persists. The portion of this work undertaken whilst still at McGillUniversity was supported in part by an NSERC discovery grant. This work is supportedat the Humboldt University by funds from project B5 of the Sonderforschungsbereich 647(Raum Zeit Materie) grant, for which I am grateful to Alan Rendall at Albert EinsteinInstitute and Jan Plefka at the Humboldt University, whom I also thank for his continuedsupport and encouragement. References [1] N. Arkani-Hamed, S. Dimopoulos, G. Dvali and G. Gabadadze, “Non-local modification ofgravity and the cosmological constant problem,” arXiv:hep-th/0209227.[2] G. Dvali, S. Hofmann and J. Khoury, “Degravitation of the cosmological constant andgraviton width,” arXiv:hep-th/0703027.[3] C. de Rham, S. Hofmann, J. Khoury and A. J. Tolley, “Cascading Gravity andDegravitation,” arXiv:0712.2821 [hep-th].[4] G. R. Dvali, G. Gabadadze and M. Porrati, “4D gravity on a brane in 5D Minkowski space,”Phys. Lett. B , 208 (2000) [arXiv:hep-th/0005016].[5] M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in anelectromagnetic field,” Proc. Roy. Soc. Lond. A , 211 (1939).[6] S. Deser and A. Waldron, “Stability of massive cosmological gravitons,” Phys. Lett. B ,347 (2001) [arXiv:hep-th/0103255].[7] G. Gabadadze and A. Gruzinov, “Graviton mass or cosmological constant?,” Phys. Rev. D ,124007 (2005) [arXiv:hep-th/0312074].[8] A. O. Barvinsky and G. A. Vilkovisky, “Covariant perturbation theory. 2: Second order in thecurvature. General algorithms,” Nucl. Phys. B , 471 (1990).[9] A. O. Barvinsky, “Nonlocal action for long-distance modifications of gravity theory,” Phys.Lett. B , 109 (2003) [arXiv:hep-th/0304229].[10] D. Lopez Nacir and F. D. Mazzitelli, “Running of Newton’s constant and non integer powersof the d’Alembertian,” Phys. Rev. D , 024003 (2007) [arXiv:hep-th/0610031].[11] K. Huang, “Quantum field theory: From operators to path integrals,” New York, USA: Wiley(1998) 426 p [12] S. M. Christensen and M. J. Duff, “Quantum Gravity In Two + Epsilon Dimensions,” Phys.Lett. B , 213 (1978).[13] G.Gabadadze, S.Hofmann and S.P.Patil; in preparation [14] N. D. Birrell and P. C. W. Davies, “Quantum Fields In Curved Space,” Cambridge, Uk:Univ. Pr. ( 1982) [15] R. H. Brandenberger, “Inflationary cosmology: Progress and problems,”arXiv:hep-ph/9910410.[16] A. D. Linde, “Hybrid inflation,” Phys. Rev. D , 748 (1994) [arXiv:astro-ph/9307002].[17] H. S. Yang, “Emergent Gravity And The Cosmological Constant Problem,” arXiv:0711.2797[hep-th]. – 24 –
18] A. A. Tseytlin, “Duality symmetric string theory and the cosmological constant problem,”Phys. Rev. Lett. , 545 (1991).[19] M. Spradlin, A. Strominger and A. Volovich, “Les Houches lectures on de Sitter space,”arXiv:hep-th/0110007.[20] D. Lopez Nacir and F. D. Mazzitelli, “Running of Newton’s constant and non integer powersof the d’Alembertian,” Phys. Rev. D , 024003 (2007) [arXiv:hep-th/0610031].[21] K. Kirsten and J. Garriga, “Massless minimally coupled fields in de Sitter space: O(4)symmetric states versus de Sitter invariant vacuum,” Phys. Rev. D , 567 (1993)[arXiv:gr-qc/9305013].[22] J. Polchinski, “The cosmological constant and the string landscape,” arXiv:hep-th/0603249.[23] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, G. Gabadadze, A. Linde, “Self-TerminatingInflation,” Unpublished preprint CERN-TH/2003-191[24] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, “Theory of cosmologicalperturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part3. Extensions,” Phys. Rept. , 203 (1992)., 203 (1992).