Hartree approximation in curved spacetimes revisited II: The semiclassical Einstein equations and de Sitter self-consistent solutions

# Hartree approximation in curved spacetimes revisited II: The semiclassical Einstein equations and de Sitter self-consistent solutions

## Abstract

We consider the semiclassical Einstein equations (SEE) in the presence of a quantum scalar field with self-interaction . Working in the Hartree truncation of the two-particle irreducible (2PI) effective action, we compute the vacuum expectation value of the energy-momentum tensor of the scalar field, which act as a source of the SEE. We obtain the renormalized SEE by implementing a consistent renormalization procedure. We apply our results to find self-consistent de Sitter solutions to the SEE in situations with or without spontaneous breaking of the -symmetry.

###### pacs:
03.70.+k; 03.65.Yz
\date

## I Introduction

Quantum field theory in curved spacetimes (1); (2); (3); (4) is the natural framework for the study of quantum phenomena in situations where the gravitation itself can be treated classically. Of special interest is quantum field theory in de Sitter spacetime. In fact, de Sitter spacetime plays a central role in most of inflationary models of the early Universe (5); (6); (7), where the energy density and pressure of the inflaton field act approximately as a cosmological constant. Moreover, the amplification of quantum fluctuation during an inflationary period with an approximately de Sitter background metric, gives a natural mechanism for generating nearly scale-invariant spectrum of primordial inhomogeneities, which can successfully explain the observed CMB anisotropies (9); (8). De Sitter spacetime is also potentially important for understanding the final fate of the Universe if the current accelerated expansion is due to a small cosmological constant, which nowadays is a possibility that is compatible with observations (9); (10); (11); (12). On the other hand, previous studies of interacting quantum scalar fields in de Sitter spacetime have revealed that the standard perturbative expansion gives rise to corrections that secularly grow with time and/or infrared divergences (13); (18); (19); (14); (15); (16); (17); (20), signaling a possible deficiency of the perturbative approach. This has motivated several authors to consider alternative techniques (see for instance (18); (19); (21); (22); (23); (24); (20); (25); (26); (27)) and in particular, to use nonperturbative resummation schemes (28); (29); (30); (31); (32); (33); (34).

In the above situations, it is important to study not only test fields evolving on a fixed background, but also to take into account the backreaction of the quantum fields on the dynamics of the spacetime geometry. The backreaction problem has been explored by a number of authors in the context of semiclassical gravity (see for instance (35); (36); (37); (38); (39)), where the dynamics of the classical metric is governed by the so-called Semiclasical Einstein Equations (SEE). The SEE are a generalization of the Einstein equations that contain as a source the expectation value of the energy-momentum tensor of the quantum matter fields, (1); (2); (3); (4). Self-consistent de Sitter solutions have been found for the case of free quantum fields (40); (41); (42); (43); (44). The influence of the initial state of the quantum field on the semiclassical solutions has been studied in Refs. (45); (46).

Since is formally a divergent quantity, in order to address the backreaction problem it is necessary to analyze the renormalization process. For free and interacting quantum fields in the one-loop approximation, there are well known covariant renormalization methods (1); (2); (3); (4). Our main goal in this work is to improve the current understanding of these methods in the case in which the quantum effects are taken into account nonperturbatively. For this, we consider a quantum self-interacting scalar field in the Hartree approximation, which corresponds to the simplest nonperturbative truncation to the two-particle irreducible effective action (2PI EA), introduced by Cornwall, Jackiw and Tomboulis (47). The Hartree (or Gaussian) approximation involves the resummation of a particular type of Feynman diagrams which are called superdaisy (see for instance (48)) to an infinite perturbative order. This approximation can also be introduced by means of a variational principle (49); (50). However, the use of the 2PI EA is advantageous for at least two reasons. First, it provides a framework for resumming classes of diagrams that can be systematically improved. Second, for any truncation of the EA, it implies certain consistency relations between different counterterms that allow a renormalization procedure that is consistent with the standard perturbative (loop-by-loop) renormalization of the bare coupling constants (51). The latter is crucial for the consistent renormalization procedure developed in Ref. (51) for Minkowski spacetime, which in (34) (from now on paper I), using the same model considered here, we have extended to general curved background metrics.

The renormalization problem of the SEE in the Hartree approximation has been considered previously in (52); (32). However, it has not been analyzed using the consistent renormalization procedure (51) that we extended to curved spacetimes in paper I in order to renormalize the field and gap equations. Our focus in this paper is to prove that the same set of renormalized parameters leads to SEE that can be made finite, and independent on the arbitrary scale introduced by the regularization scheme (which for the field and gap equations was explicitly shown in paper I), by suitable renormalizations of the bare gravitational constants.

The paper is organized as follows. In Sec. II we introduce the 2PI EA in curved spacetimes. In Sec. III we present our model and summarize the main relevant results of paper I for the renormalization of the mass and coupling constant of the field. The reader acquainted with paper I may skip this section. In Sec. IV we show that the same counterterms that make finite the field and gap equations can also be used to absorb the non-geometric divergences in the SEE, extending the consistent renormalization procedure to the gravitational sector. The geometric divergences can be absorbed into the usual gravitational counterterms. In Sec. V we analyze the field, gap and SEE in de Sitter spacetimes. The high symmetry of these spacetimes allows us to compute explicitly the two point function and the energy-momentum tensor, to end with a set of algebraic equations that determine self-consistently the mean value of the field and the de Sitter curvature. We will present some numerical solutions to these equations. In Sec. VI we include our conclusions. Throughout the paper we set and adopt the mostly plus sign convention.

## Ii The 2PI effective action

A detailed description to the 2PI EA formalism can be found in several papers and textbooks, such as (53); (47); (54). In this section, in order to make this work as self-contained as possible and to set the notation, we briefly summarize the main relevant aspects of the formalism applied to a self-interacting scalar field in a general curved spacetime.

The 2PI generating functional can be written as (51)

 Γ2PI[ϕ0,G,gμν]=S0[ϕ0,gμν]+i2\textupTrln(G−1)+i2\textupTr(G−10G)+Γint[ϕ0,G,gμν], (1)

where is quadratic part of the classical action without any counterterms,

 iGab0(x,x′)=1√−gδ2S0[ϕ0,gμν]δϕa(x)δϕb(x′)1√−g′, (2)

and

 Γint[ϕ0,G,gμν]=Sint[ϕ0,gμν]+12Tr[δ2Sintδϕ0δϕ0G]+Γ2[ϕ0,G,gμν], (3)

where the functional is times the sum of all two-particle-irreducible vacuum-to-vacuum diagrams with lines given by and vertices obtained from the shifted action , which comes from expanding and collecting all terms higher than quadratic in the fluctuating field . Here are time branch indices (with index set in the usual notation) corresponding to the ordering on the contour in the “closed-time-path”(CTP) or Schwinger-Keldysh (53) formalism.

The equations of motion for the field and propagator are obtained by

 δΓ2PIδϕ0∣∣ϕ+=ϕ−=ϕ;gμν+=gμν−=gμν =0, (4a) δΓ2PIδG∣∣ϕ+=ϕ−=ϕ;gμν+=gμν−=gμν =0. (4b)

To arrive at the SEE we extremize the combination with respect to the metric,

 δ(Sg[gμν]+Γ2PI[ϕ0,G,gμν])δgμν∣∣ϕ+=ϕ−=ϕ;gμν+=gμν−=gμν=0, (5)

where is the gravitational action. As it is well known (1); (2); (3), this equation is formally divergent, with the divergences contained in the vacuum expectation value of the energy-momentum tensor , defined by

 ⟨Tμν⟩=−2√−gδΓ2PI[ϕ0,G,gμν]δgμν∣∣ϕ+=ϕ−=ϕ;gμν+=gμν−=gμν. (6)

It is also well known (1); (2); (3) that the renormalization procedure requires the inclusion of terms quadratic in the curvature in the gravitational action, so that

 Sg=12∫d4x√−g{κ−1B(R−2ΛB)−α1BR2−α2BRμνRμν−α3BRμνρσRμνρσ}, (7)

where is the curvature tensor, , and , , () are bare parameters which are to be appropriately chosen to cancel the divergences in .

## Iii λϕ4 theory in the Hartree approximation: renormalization of the field and gap equations

We consider a nonminimally coupled scalar field with quartic self-coupling in a curved background with metric . The corresponding classical action reads

 Sm[ϕ,gμν]=−∫d4x√−g[12ϕ(−□+m2B+ξBR)ϕ+14!λBϕ4], (8)

where , .

In the Hartree approximation, which corresponds to the inclusion of only the double-bubble diagram shown in Fig. 1, the 2PI effective action is given by

 Γ2PI[ϕ0,G,gμν] = −∫d4x√−g[12ϕ0(−□+m2B2+ξB2R)ϕ0+14!λB4ϕ40]+i2Trln(G−1) (9) −12∫d4x√−g[−□+m2B0+ξB0R+12λB2ϕ20]G(x,x) −λB08∫d4x√−gG2(x,x),

where, for the sake of simplicity, we drop the time branch indices, since for the Hartree approximation it is known that the CTP formalism gives the same equations of motion than the usual in-out formalism (54).

Taking the variation with respect to and we obtain equations of motion for the mean field and the propagator:

 (−□+m2B2+ξB2R+λB46ϕ20+λB22[G])ϕ0(x) = 0, (10) (−□+m2B0+ξB0R+λB22ϕ20+λB02[G])G(x,x′) = −iδ(x−x′)√−g′, (11)

with the coincidence limit of the propagator .

It is important to note that here we are taking into account the possibility of having different counterterms for a given parameter of the classical action Eq. (8). These are denoted using different subscripts in the bare parameters that refer to the power of in the corresponding term of the action. In the Hartree approximation, this point turns out to be crucial for the implementation of the consistent renormalization procedure described in (51) . Indeed, as shown in (51) (see also Appendix A of paper I), there are various possible -point functions that can be obtained from functionally differentiating with respect to and , which in the exact theory must satisfy certain consistency conditions. On the other hand, for any truncation of the 2PI EA, the validity of such consistency conditions is not guarantee. However, one can find a relation between the different counterterms by imposing the consistency conditions at a given renormalization point. Doing this, any possible deviation of the consistency conditions is finite and under perturbative control. In other words, had we not allowed for different counterterms, the diagrams contributing to the consistency conditions could contain perturbative divergent contributions which could not be absorbed anywhere.

In our case, the consistency conditions for the two- and four-point functions, evaluated at , are given by

 δ2Γintδϕ1δϕ2∣∣∣ϕ=0=2δΓintδG12∣∣∣ϕ=0, (12)

and

 δ4Γ1PI[ϕ0]δϕ1δϕ2δϕ3δϕ4∣∣∣ϕ0=0=2[δ2ΓintδG12δG34∣∣∣¯G,ϕ0=0+perms(2,3,4)]−12δ4Γintδϕ1δϕ2δϕ3δϕ4∣∣∣¯G,ϕ0=0, (13)

where

 Γ1PI[ϕ0,gμν]=Γ2PI[ϕ0,¯G[ϕ0],gμν]. (14)

In what follows we consider two different parametrizations of the bare couplings:

 m2Bi =m2+δm2i=m2R+δ~m2i(i=0,2), (15a) ξBi =ξ+δξi=ξR+δ~ξi(i=0,2), (15b) λBi =λ+δλi=λR+δ~λi,(i=0,2,4). (15c)

The first separation corresponds to the MS scheme (i.e., the counterterms , and (,) contain only divergences and no finite part), while in the second separation , and are chosen to be the renormalized parameters as defined from the effective potential (see below).

By imposing the conditions (12) and (13), one can obtain the following relation between the different counterterms (34):

 δm20=δm22≡δm2, (16a) δξ0=δξ2≡δξ, (16b) δλ0=δλ2, (16c) δλ4−3δλ2=2(λ−λR), (16d)

with

 δ4Γ1PI[ϕ0]δϕ1δϕ2δϕ3δϕ4∣∣∣ϕ0=0=−λRδ12δ13δ14, (17)

where we used as a notational shorthand. Recalling that the effective potential is proportional to the effective action at a constant value of , the renormalized self-interaction coupling can be also written as

 λR=d4Veffdϕ40∣∣∣0. (18)

With the use of these relations, one can recast Eqs. (10) and (11) as

 (−□+m2ph+ξRR−13λRϕ20)ϕ0(x) = 0, (19) (−□+m2ph+ξRR)G1(x,x′) = 0, (20)

where is identified with the physical mass of the fluctuations and satisfies a self-consistent equation (i.e., the gap equation) that reads

 m2ph+ξRR=m2+δm2+(ξ+δξ)R+12(λ+δλ2)ϕ20+14(λ+δλ2)[G1]. (21)

A point that is worth emphasizing here is that these relations cannot be imposed in an arbitrary spacetime metric, since the renormalized parameters must be constant, while the fourth derivative of 1PI EA in Eq. (17) might not. However, in order to define the renormalized parameters, one can choose a particular fixed background metric with constant curvature invariants as the renormalization point at which the consistency conditions are imposed. In paper I we considered both Minkowski and de Sitter spacetimes. Here, for the sake of generality, we will also consider both renormalization points. Therefore, we define the renormalized parameters as those derived from the effective potential and evaluated for a fixed de Sitter spacetime with ,

 M2R ≡d2Veffdϕ20∣∣∣ϕ0=0,R=R0=M2ph(ϕ0=0,R=R0), (22a) ξR ≡d3VeffdRdϕ20∣∣∣ϕ0=0,R=R0=dM2phdR∣∣∣ϕ0=0,R=R0, (22b) λR ≡d4Veffdϕ40∣∣∣ϕ0=0,R=R0=3d2M2phdϕ20∣∣∣ϕ0=0,R=R0−2λR, (22c)

where we are using the notation . In particular, the limit could be taken to recover the usual renormalized parameters defined in Minkowski spacetime.

In order to obtain the renormalized gap equation it is useful to consider the adiabatic expansion of the propagator at the coincidence limit:

 [G1] = 18π2(m2phμ2)ϵ/2∑j≥0[Ωj](m2ph)1−jΓ(j−1−ϵ2) (23) ≡ 14π2ϵ[m2ph+(ξR−16)R]+2TF(m2ph,ξR,R,~μ),

where , is the Gamma function, and the Schwinger-DeWitt coefficients are scalars of adiabatic order built from the metric and its derivatives and satisfy certain recurrence relations. In the second line, we have used the explicit expressions for the coefficients and , given in (55), we have expanded for and we have redefined to absorb some constant terms, defining

 TF(m2ph,ξR,R,~μ) = 116π2{[m2ph+(ξR−16)R]ln(m2ph~μ2)+(ξR−16)R (24) − 2F(m2ph,{R})},

where the function contains the adiabatic orders higher than two, is independent of and , and satisfies the following properties:

 F(m2ph,{R})∣∣∣Rμνρσ=0=0, (25a) dF(m2ph,{R})dm2ph∣∣∣Rμνρσ=0=0, (25b) dF(m2ph,{R})dR∣∣∣Rμνρσ=0,ϕ0=0=0. (25c)

Taking into account the relations in Eq. (16) between the counterterms, the gap equation can be made finite with the use of the following MS counterterms:

 δm2 =−λ16π2ϵm21+λ16π2ϵ, (26a) δξ =−λ16π2ϵ(ξ−16)1+λ16π2ϵ, (26b) δλ2 =−λ16π2ϵλ1+λ16π2ϵ. (26c)

Once made finite and written in terms of the MS parameters, it reads

 m2ph+ξRR = m2+ξR+12λϕ20+λ32π2{[m2ph+(ξR−16)R]ln(m2ph~μ2) (27) + (ξR−16)R−2F(m2ph,{R})}.

Here, the explicit dependence on the renormalization scale should be compensated with an implicit -dependence on the finite MS parameters , and . Indeed, the invariance of this equation under changes of becomes manifest when we express it in terms of the renormalized quantities , and . The latter are related to the former ones by

 m2R =m2+λ16π2[R0dFdSdR∣∣m2R,R0−FdS(m2R,R0)][1−λ32π2ln(m2R~μ2)], (28a) (ξR−16) =(ξ−16)−λ16π2dFdSdR∣∣m2R,R0[1−λ32π2−λ32π2ln(m2R~μ2)], (28b) λR =λ[1−λ32π2−λ32π2ln(m2R~μ2)−λ32π2((ξR−16)R0m2R−2dFdSdm2ph∣∣m2R,R0)]. (28c)

Two useful -independent combinations follow immediately from these relations:

 m2BλB2=m2λ=m2Rλ∗R+(ξR−16)R032π2 (29)

and

 (ξB−16)λB = (ξ−16)λ (30) = (ξR−16)λR+(ξR−16)32π2[(ξR−16)R0m2R−2dFdSdm2ph∣∣m2R,R0]+116π2dFdSdR∣∣m2R,R0 ≡ (ξR−16)λR+J(R0,m2R,ξR).

where is defined by

 1λ∗R≡1λR+132π2. (31)

Using these parameters, the self-consistent equation for can be written as

 m2ph = m2R+λ∗R2ϕ20+λ∗R32π2{[m2ph+(ξR−16)R]ln(m2phm2R) (32) + (m2ph−m2R)⎡⎣2dFdSdm2ph∣∣m2R,R0−(ξR−16)R0m2R⎤⎦ + 2[FdS(m2R,R0)+dFdSdR∣∣m2R,R0(R−R0)−F(m2ph,R)]}.

Finally, as will be needed for the renormalization of the energy-momentum tensor in next section, we write the results for the counterterms associated to the non-MS renormalized parameters defined in Eq. (15):

 δ~m2 ≡ Missing or unrecognized delimiter for \right (33) δ~ξ ≡ Missing or unrecognized delimiter for \Big (34) δ~λ ≡ Missing or unrecognized delimiter for \Big (35)

Note that the well known one-loop results can be recovered from these expressions, making the replacements , , , and on the right-hand-sides.

## Iv Renormalization of the semiclassical Einstein equations

So far we have dealt with Eqs. (19) and (20), that give the dynamics of and for a given choice of metric . However these equations do not take into account the effect of the quantum field on the background geometry. In order to assess whether this backreaction is important or not, we must deal with the SEE, obtained from the stationarity condition given in Eq. (5) with the gravitational action Eq. (7) and the definition of the vacuum expectation value of the energy-momentum tensor given in Eq. (6). The resulting equations are

 κ−1BGμν+ΛBκ−1Bgμν+α1B(1)Hμν+α2B(2)Hμν+α3BHμν=⟨Tμν⟩, (36)

where . An explicit expression for the tensors and can be found for instance in (55).

The renormalization procedure then involves the calculation of and the regularization of its divergences. The divergences can be of either one of two types, independent of the field and therefore only geometrical, or otherwise -dependent either explicitly or implicitly through . The SEE are renormalizable if, with the same choice of counterterms as for the field and gap equations, the non-geometrical divergences can be completely dealt with. In order to absorb the geometrical divergences in the renormalization of the parameters of the gravitational part of the action, , and , these divergences must be proportional to the tensors that appear on the left-hand side of Eq. (36) (note that in four spacetime dimensions the tensors and are not all independent).

We will follow the usual procedure and define the renormalized energy-momentum tensor as

where the fourth adiabatic order is understood as the expansion containing up to four derivatives of the metric and up to two derivatives of the mean field (55). Our goal in this section is to show that with the choice of the counterterms for the field and gap equations, only contains geometric divergences, that can be absorbed into the bare gravitational constants.

The expectation value can be formally computed from the definition Eq. (6). One can show that (54)

 ⟨Tμν⟩=Tμν(ϕ0)+⟨Tfμν⟩+λB232[G1]2gμν, (38)

where the first term is the classical energy-momentum tensor evaluated at

 Tμν(ϕ0)=−2√−gδSmδgμν = (1−2ξB)ϕ0,μϕ0,ν−2ξBϕ0;μνϕ0+2ξBgμνϕ0□ϕ0+ξBϕ20Gμν (39) +(2ξB−12)gμνϕ,λ0ϕ0,λ−m2B2gμνϕ20−λB44!gμνϕ40.

The second term is formally the mean value of the energy-momentum tensor of a free field, constructed with the two-point function . More explicitly, it can be written as (56); (55)

 ⟨Tfμν⟩=−12[G1;μν]+(1−2ξB)4[G1];μν+(ξB−14)gμν2□[G1]+ξBRμν[G1]2. (40)

As a side point, we mention that one could also derive Eq. (38) using a different approach: take the classical energy-momentum tensor for the action Eq. (8), evaluate for and then expand on the fluctuation . Afterwards take the expectation value and recall that in the Hartree approximation one can write the expectation values of products of fields in terms of and (and derivatives), using that

 ⟨φ3⟩ =0, (41a) ⟨φ4⟩ =34[G1]2. (41b)

For the renormalization it is useful to separate, in the expressions for and , the bare couplings into the corresponding renormalized parts and the nonminimal subtraction counterterms

 Tμν(ϕ0) = Missing or unrecognized delimiter for \right (42) ⟨Tfμν⟩ = ⟨Tfμν⟩∣∣∣B=R+δ~ξ2(−[G1];μν+gμν□[G1]+Rμν[G1]), (43)

where is a notational shorthand to indicate a replacement of the bare couplings with the renormalized ones. It will be also useful to write separately the interaction term in the classical energy momentum tensor

 Tμν(ϕ0)∣∣∣B=R=Tμν(ϕ0)∣∣∣B=R,free−λB44!ϕ40gμν. (44)

Note that while there are no divergences in , the quantity still has divergences that arise from the coincidence limit of and of its derivatives. Recall Eq. (20), which implies that in our case the two-point function is that of a field of mass and curvature coupling .

We are now ready to show that the counterterms already chosen to renormalize the mean field and gap equations also cancel the non-geometrical divergences in . The third term of Eq. (38) as well as the terms that were isolated in Eq. (43) involve and its derivatives, and therefore they can be expressed in terms of and the bare couplings by using that the physical mass is defined by the equality of Eqs. (11) and (20), which in a more convenient form reads

 λB24[G1]=m2ph−~δξR−m2B−λB22ϕ20. (45)

With this replacement we have

 ⟨Tμν⟩ = Tμν(ϕ0)∣∣∣B=R,free+⟨Tfμν⟩∣∣∣B=R+(3λB2−λB4)4!ϕ40gμν (46) +2δ~ξλB2[−m2ph;μν+gμν□m2ph+Gμνm2ph]+m4ph2λB2gμν−m2phm2BλBgμν +δ~ξ2λB2(1)Hμν−2δ~ξm2BλB2Gμν+m2B2m2BλBgμν +(m2R−m2ph)ϕ202gμν.

Here the term proportional to is already finite because of the relation Eq. (16d) between the counterterms, and thus equal to . The fourth, fifth and sixth terms contain the non-geometrical divergences that will have to be cancelled by those from . The remaining terms contain purely geometrical divergences.

It is worth to emphasize that the divergences in Eq. (46) are proportional to simple poles in . Indeed, from the definition of and the relations (30) it is straightforward to see that

 δ~ξλB2 =(1λR−1λB2)(ξR−16)+J, (47a) δ~ξ2λB2 =λB2⎡⎢ ⎢⎣(ξR−16)λR+J⎤⎥ ⎥⎦2−2(ξR−16)⎡⎢ ⎢⎣(ξR−16)λR+J⎤⎥ ⎥⎦ +(ξR−16)2λB2, (47b)

which are exact expressions. Note that contains just a simple pole,

 1λB2=1λ+116π2ϵ. (48)

We now expand up to the fourth adiabatic order. We will use the explicit expressions for the coincidence limit of and its derivatives that are given in Ref. (55). The fourth adiabatic order expansion for is

 ⟨~Tμν⟩ad4 = 116π2(m2phμ2)ϵ/2[12m4phgμνΓ(−2−ϵ2)+m2ph{12[Ω1]gμν+(ξR−16)Rμν} (49) × Γ(−1−ϵ2)+{12[Ω2]gμν+(ξR−16)Rμν[Ω1]−[Ω1;μν] + (12−ξR)[Ω1];μν+(ξR−14)g