Quantum Scalar Corrections to the Gravitational Potentials on de Sitter Background

We employ the graviton self-energy induced by a massless, minimally coupled (MMC) scalar on de Sitter background to compute the quantum corrections to the gravitational potentials of a static point particle with a mass $M$. The Schwinger-Keldysh formalism is used to derive real and causal effective field equations. When evaluated at the one-loop order, the gravitational potentials exhibit a secular decrease in the observed gravitational coupling $G$. This can also be interpreted as a (time dependent) anti-screening of the mass $M$.


Introduction
Studies of quantum loop corrections to the gravitational potentials in flat space background have a long history [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These studies are typically based on computing the scattering amplitude for two massive particles and then solving the inverse scattering problem to reconstruct a Newtonian potential which would produce the same scattering amplitude in quantum mechanics. This technique is well-tested and has the tremendous advantage of being independent of the choice of gauge and of field variable. However, it seems artificially restricted to asymptotic scattering problems, as compared with the time-dependent effects which can be explored using the classical field equations. And its application to cosmology seems inappropriate because the formal S-matrix which can be defined for massive scalars on de Sitter [17] is not observable.
A more suitable technique for time-dependent sources and cosmological backgrounds is the Schwinger-Keldysh, or in-in, formalism [18][19][20][21][22], which provides expectation values of operators instead of in-out matrix elements. The authors have previously solved the Schwinger-Keldysh effective field equations to work out quantum corrections (from a massless, minimally coupled (MMC) scalar) to the two potentials of a point mass on flat space background [23][24][25]. When graviton and gauge particles start to appear in loops the problem of gauge dependence must be faced, but that is not an issue here. And it should be noted that the Schwinger-Keldysh results are consistent with those derived using conventional scattering techniques. They also furnish an essential correspondence limit for the current computation in de Sitter.

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De Sitter space is of particular interest in cosmology as a paradigm for the background of primordial inflation. A generic prediction of inflation is that the quantum fluctuations of MMC scalars and gravitons are amplified and preserved to late times so that they seed large scale structure formation [26][27][28][29][30][31][32][33]. This is a tree order effect, but the same quantum fluctuations inevitably give rise to loop effects which have been studied in recent years . The purpose of this paper is to learn how a loop of MMC scalars changes the gravitational potentials of a point mass on de Sitter. This involves three tasks: 1. Compute and renormalize the one-loop contribution to the graviton self-energy −i[ µν Σ ρσ ](x; x ) from a MMC scalar on de Sitter background; 2. Convert the in-out self-energy to the retarded one of the Schwinger-Keldysh formalism, µν Σ ρσ (x; x ) → µν Σ ρσ Ret (x; x ) ; (1.1) 3. Solve the quantum corrected, linearized Einstein field equation Here D µνρσ κh ρσ (x) is derived by expanding the gravitational side of the Einstein equation, (R µν + Λg µν − 1 2 g µν R) √ −g, about de Sitter background to first order in the metric perturbation, δg ρσ (x) ≡ a 2 (t)κh ρσ (x), where a = a(t) is the scale factor, κ 2 = 16πG is the quantum gravitational loop counting parameter, G is the Newton constant, and T µν lin (x) is the linearized stress-energy tensor density.
The first two steps have been already performed in the ref. [55] and we summarize the results in section II. Section III is devoted to the last step, that is to solving the Schwinger-Keldysh effective field equations for the graviton field sourced by a static point mass. Our discussion comprises section IV, and some tedious technical details from section III have been subsumed to an appendix.

Schwinger-Keldysh effective field equations
The point of this section is to present the Schwinger-Keldysh effective field equations which we will solve in the next section. We first set up the background geometry and define the graviton field as a perturbation around this background. We then give the inout effective field equations derived in [55] and discuss how to solve them perturbatively. Finally, we explain why it is more appropriate to convert to in-in equations for cosmological backgrounds such as de Sitter, and we make the conversion.

Preliminaries
We consider the Lagrangian of gravity plus a MMC scalar,

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where G is Newton's constant, R is the Ricci scalar and Λ is the cosmological constant. Our computation is based on perturbation theory in the Poincaré patch of de Sitter space The coordinate ranges are Here the scale factor depends on conformal time η as, a = −1/Hη and the Hubble param- It is also useful to employ the de Sitter length function, where a ≡ a(η) and a ≡ a(η ). Note that y(x; x ) is related to geodesic distance on de Sitter (x; x ) as, y(x; x )| =0 = 4 sin 2 (H (x; x )/2). We define the graviton field h µν by subtracting the background from the full metric and then conformally rescaling, where κ 2 ≡ 16πG is the loop-counting parameter of quantum gravity.

Effective field equations
Varying the one-particle irreducible (1PI) effective action corresponding to the Lagrangian (2.1) with respect to the graviton field h µν , and retaining only the linear terms gives, Here the Lichnerowicz operator for de Sitter is [55,67], Quantum corrections come from the graviton self-energy whose general form is, The four projection operators F µν , G µν , F µνρσ and G µνρσ and one loop results for the corresponding structure functions F 0 , G 0 , F 2 , and G 2 are given in ref. [55].
It is convenient to re-express the action of the Lichnerowicz operator on the graviton by extracting the scale factor a, 1 1 Due to an error, only the first term in (2.9) is found in ref. [55].

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where the Lichnerowicz tensor factor L µνρσαβ is For the quantum correction, we extract the unprimed derivatives from the x µ integration and partially integrate the primed derivatives, bringing the effective field equation (2.6) to the form Here R lin and C αβγδ lin are the linearized Ricci scalar and Weyl tensor of the conformally rescaled metric. R lin and C αβγδ lin are their purely spatial parts, respectively.

Perturbative solution
Because we only possess one loop results for the structure functions, we must solve (2.11) perturbatively by expanding the graviton field and the structure functions in loop orders, 0,2 (x; x ) + . . . (2.14) By substituting (2.12)-(2.14) into (2.11), we obtain equations for the tree order field h µν , .
ρσ . Note that in (2.15) we regard the matter source as 0th order, assuming the stress tensor includes no loop corrections from the 1PI 1-point function. The solution of the 0th order equation h

Schwinger-Keldysh formalism
The perturbative effective field equation (2.16) seems to be ready for use, but if one were to interpret it in the spirit of the in-out formalism it would possess two disturbing features:

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• Acausality: the in-out effective field equation at x µ receives influence from points x µ which lie in the future of x µ , and at spacelike separation from it.
• Imaginary parts: the in-out effective field develops an imaginary part if there is particle production.
Neither of these features prevents one from describing flat space scattering problems, but they would be problematic for cosmological settings in which we do not know what happens in the asymptotic future and the more natural question is how the fields evolve when released at finite time in some prepared state. That question is answered by the Schwinger-Keldysh formalism [18][19][20][21][22]. This technique produces true expectation values, rather than in-out matrix elements, so the effective field equations at x µ depend only on points x µ on or within its past light-cone, and the effective fields associated with Hermitian operators are real. Because excellent reviews on the Schwinger-Keldysh formalism exist [68][69][70][71], and the current authors have described it before in [55], we merely comment that the linearized Schwinger-Keldysh effective field equation is obtained by replacing the in-out self-energy with its retarded counterpart, In this expression, we obtain µν Σ ρσ ++ and µν Σ ρσ +− from (2.8) by replacing the de Sitter length function y(x; x ) by y ++ (x; x ) and y +− (x; x ), respectively, where This converts the nonzero structure functions in (2.16) to the retarded ones of the Schwinger-Keldysh formalism [55], 0 (x; x ) is zero for the MMC scalar at one loop.) Here the symbol Θ stands for the Heaviside step function which ensures causality, Also note that the structure functions are pure imaginary, which makes the effective field equation (2.16) manifestly real. Therefore, the resulting Schwinger-Keldysh effective field equation is causal and real as promised.

Quantum corrected gravitational potentials
In this section we solve the effective field equations (2.15) and (2.16) which we repeat below, with the retarded structure functions (2.20)-(2.22) for the graviton field. We are interested in quantum loop corrections to the gravitational response of a static point mass M . The 0th order equation (3.1) determines the classical response to a point particle. The 1st order equation (3.2) leads to the one-loop correction to the classical gravitational potentials.

Classical solutions
The linearized stress tensor density T µν lin (x) in (3.1) for a static point mass M on the de Sitter background takes the form [72], The symmetries of this system imply a solution of the form, where r ≡ x . It is convenient to choose the longitudinal (Newtonian) gauge f 2 = 0 and f 4 = 0. 2 In terms of these variables the E µν of expressions (3.1), (3.2) take the form,

Computing the one-loop source integrals
The one-loop source terms on the right hand side of (3.2) in the 3 + 1 decomposition are where 14) One can check that the left hand side of the effective equation with arbitrary functions f 1,3 obey a conservation identity, 2 . These represent a nontrivial check of our equations. Moreover, these tell us that the four equations are not independent. One can solve any two equations; the other two follow from the conservation identities. (Had one proceeded with the gauge invariant formulation, one would need to cleverly combine the four equations into two gauge invariant equations, resulting in the two equations for gauge invariant scalar potentials.) There is one ugly term on the right hand side of the second line of the (ij) equation. All other terms contain only time integrations, but that term requires a three dimensional spatial integration. It is hence worth spending some effort and analyse all four equations, to see whether we can get rid of the spatial integration when evaluating the one-loop corrected f 1,3 . These equations can be easily obtained from the (00), (0i) and (ij) equations given in eqs. . (3.20) The third equation (3.19) tell us that in order to determine the gravitational slip (defined as the difference of the two potentials) one ought to perform both the integrals over time

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η and space x . It is convenient to define the two source integrals, the one-loop spin zero S (1) 0 and spin two, S 2 , as follows,

(3.24)
The equation for the gravitational slip (3.19) then becomes, 2 (x) . The solutions for f where

Discussion
We have included one loop effects from MMC scalars to derive quantum loop corrections to the potentials associated with a static point mass. Each of the full potentials, φ and ψ, can be presented as its classical value times a series of quantum corrections we have obtained in (3.29)- (3.30), which are at late times (when a 1), These can be compared with the corresponding flat space results which have been previously computed in [23][24][25],

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From this comparison, one can see that the first quantum correction terms in (4.1)-(4.2) represent the de Sitter version of the flat space correction and the terms proportional to GH 2 are the intrinsic de Sitter corrections. Note that every factor of the co-moving distance r which appears in expressions (4.1)-(4.2) is multiplied by a scale factor a(η) so that their product gives the physical distance from source to observation point. The remaining factor of ln(a) multiplies a term of the same form as the classical potential. Because these secular terms contribute equally to both potentials, they can be reinterpreted as a time dependent renormalization of the mass term, 5) or equivalently a time dependent renormalization of the Newton's constant, Even though the secular terms are suppressed by the loop counting parameter, GH 2 /c 5 , whose value is less than 10 −10 for primordial inflation, they are growing in time, and can eventually become large. Indeed, when the number of e-foldings, ln(a) = Ht becomes of the order c 5 /( GH 2 ), the correction becomes large, signifying a breakdown of perturbation theory in the sense that, when [ GH 2 /c 5 ] ln(a) ∼ 1, all orders contribute significantly. To understand what happens at very late times, one would have to sum these higher loop contributions, which is a major unsolved problem [44-46, 73, 74].
The secular screening effect we have just described is fascinating. It might represent the seed of an explanation for why the Newton constant seems so much smaller than any other length scale of fundamental theory. However, there is no avoiding the sense of strangeness. If we adopt the perspective of an observer at fixed co-moving position, whose physical distance to the source increases exponentially in co-moving time, then quantum scalar fluctuations are erasing the gravitational imprint of a point source faster than its classical redshift, in precisely the region where the source has almost no effect. From the perspective of an observer at fixed physical distance (in static coordinates) one wonders why anything is changing at all.
The sense of strangeness is even stronger when we compare with the recent result [67] for one loop corrections to the gravitational potentials from virtual photons. Unlike our case of MMC scalars, conformal invariance means that photons behave the same, in de Sitter conformal coordinates, as they do in flat space. Yet they also induce secular screening [67]. From this we can infer that the screening effect originates not so much from the way quantum fluctuations are affected by inflation but rather from the different way that gravity responds to sources on de Sitter space as opposed to flat space.
Finally, it is interesting to speculate that quantum corrections to gravity from the epoch of primordial inflation might modify late time gravity in observable ways, for example, as regards explaining the current phase of cosmic acceleration. These type of questions have been investigated in the context of Einstein's gravity endowed with a non-minimally coupled, light scalar [75][76][77], as well as in the context of some simple non-local extensions of gravity [78][79][80][81].

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Acknowledgments We are grateful for conversations on this subject with M. Fröb and E. Verdaguer. This work was partially supported by KASI, by the D-ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW), by NSF grants PHY-1205591 and PHY-1506513, and by the Institute for Fundamental Theory at the University of Florida.

A Evaluating the source integral
In this appendix we give some details on how to perform the source integrals S where I 1b +I 2 +I 3a = − H 2 a r ln 1+ 1 Har . (A.8)

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Now, it is convenient to re-express the scalar gravitational fields f ln 1+ 1 Har + Har ln 1+ 1 Har To fully reconstruct the gravitational potentials (A.9)-(A.10) we also need to evaluate 2 , which can be broken into three parts, There are three pieces on which the inverse Laplace operator act. One can check that the first evaluates to, where, for convenience, we fixed the integration constant to be − ln(H)/r. 4 Strictly speaking, when ∇ 2 acts on − ln(H)/r it generates a delta function, ∝ δ 3 ( x ), but that term can be subsumed in a (finite) renormalization of the Newton constant. To evaluate the other two terms in (A.12) the following integral representation can be used, case the final answer depends on the physical distance ar only. However, an inspection of the constraint equation (3.20) shows that this choice is illegitimate. Indeed, since J1 contributes equally to both f1 and f3, but it does not contribute to the right hand side, J1 must satisfy, (∂ 2 0 + 3aH∂0 + 3a 2 H 2 )J1 = a 2 (∂ 2

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where [r 2 g(r)] r→0 must vanish. Applying this to the other two terms in (A.12) yields, In terms of these integrals, ∇ −2 S 2 is simply, According to the expressions for the scalar potentials f    Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.