Vacuum energy sequestering and cosmic dynamics

P. P. Avelino*

Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas,PT4150-762 Porto, Portugal; Centro de Astrofísica da Universidade do Porto, Rua das Estrelas,PT4150-762 Porto, Portugal; and Departamento de Física e Astronomia, Faculdade de Ciências,

Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal(Received 16 October 2014; published 18 November 2014)

We explicitly compute the dynamics of closed homogeneous and isotropic universes permeated by asingle perfect fluid with a constant equation-of-state parameter w in the context of a recent reformulation ofgeneral relativity, proposed by Kaloper and Padilla [Phys. Rev. Lett. 112, 091304 (2014)], which preventsthe vacuum energy from acting as a gravitational source. This is done using an iterative algorithm, taking asan initial guess the background cosmological evolution obtained using standard general relativity in theabsence of a cosmological constant. We show that, in general, the impact of the vacuum energysequestering mechanism on the dynamics of the universe is significant, except for the w ¼ 1=3 case wherethe results are identical to those obtained in the context of general relativity with a null cosmologicalconstant. We also show that there are well-behaved models in general relativity that do not have a well-behaved counterpart in the vacuum energy sequestering paradigm studied in this paper, highlighting thespecific case of a quintessence scalar field with a linear potential.

DOI: 10.1103/PhysRevD.90.103523 PACS numbers: 98.80.-k, 95.36.+x

I. INTRODUCTION

Solving the cosmological constant problem constitutesone of the most ambitious challenges of fundamentalphysics [1]. The latest constraints [2–6] suggest that acosmological constant may be responsible for the observedacceleration of the Universe, assuming that gravity isdescribed by general relativity on cosmological scales.However, this interpretation of the data faces severalproblems: (i) why is the vacuum energy density about120 orders of magnitude smaller than the Planck density?,and (ii) why dowe seem to live at a very special epochwherethe fractional contribution of the cosmological constant tothe energy density of the Universe appears to be rapidlyevolving from 0 in the relatively recent past towards 1 in thenot too distant future? The answer to these questions may liein dynamical dark energy models [7–10], finite-lifetimecosmologies in which the matter and dark energy densitiescan be of the same order for most of the universe’s lifetime[11–16], and/or anthropic considerations [16–19].Another relatedproblemhas to dowith the fact that, unlike

the other fundamental interactions, general relativity is notinvariant under the shifting of the Lagrangian by a constant,implying that the vacuum energy density is a source for thegravitational field in general relativity. This has been amatterof debate for many years, with some authors arguing that asatisfactory solution to the cosmological constant problemrequires a modification of general relativity (see, for exam-ple, Ref. [20]). In Ref. [21] (see also Refs. [22,23]) a newmechanism was proposed which prevents the vacuumenergy from acting as a gravitational source, thus providing

a possible explanation for the huge discrepancy betweenthe estimation of the vacuum energy density from quantumzero-point fluctuations and the value inferred from cosmo-logical observations. In the context of this reformulation ofgeneral relativity the Universe should be finite in space andtime,with thepresent epochof accelerated expansionbeing atransient stage before the big crunch.In the present paper we shall investigate how the

cosmological dynamics is affected by the vacuum energysequestering mechanism. This paper is organized as fol-lows. In Sec. II, some of the key features of the theoryproposed in Ref. [21] are outlined. In Sec. III, we use aniterative algorithm to determine the impact of the vacuumenergy sequestering mechanism on the dynamics of closedhomogeneous and isotropic universes filled with a perfectfluid with a constant equation-of-state parameter. Theresults are then compared with those obtained in thecontext of general relativity with a null cosmologicalconstant. In this section we also explore the implicationsof the vacuum energy mechanism in the context of moregeneral models, with special emphasis on the case of aquintessence scalar field with a linear potential. We thenconclude in Sec. IV.Throughout this paper we use units such that

8πG ¼ c ¼ 1, where G is the gravitational constant andc is the value of the speed of light in vacuum. We adopt themetric signature ð−;þ;þ;þÞ.

II. THE MODEL

Here we shall consider the action defined in Ref. [21]which yields the following equations of motion for thegravitational field:*pedro.avelino@astro.up.pt

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Gμν ¼ Tμν − Λgμν; ð1Þ

where Gμν ≡ Rμν − gμνR=2 are the components of theEinstein tensor, gμν are the components of the metric,Rμν are the components of the Ricci curvature tensor, R≡Rμ

ν is the Ricci scalar curvature, Tμν are the components ofthe energy-momentum tensor and Λ is given by

4Λ ¼ hTμμi≡

Rd4x

ffiffiffiffiffiffi−gpTμ

μRd4x

ffiffiffiffiffiffi−gp ; ð2Þ

where g ¼ detðgμνÞ is the metric determinant. These equa-tions of motion for the gravitational field are invariantunder the transformation Tμν → Tμν þ Cgμν where C is anarbitrary real constant. Consequently, any bulk constantenergy density is effectively gauged away.In this paper we shall consider a closed homogeneous

and isotropic universe described by the Friedmann-Lemaître-Robertson-Walker metric. The line element isgiven by

ds2 ¼ −dt2 þ aðtÞ2�

dr2

1 − kr2þ r2ðdθ2 þ sin2θdϕ2Þ

�;

ð3Þ

where t is the physical time, ðr; θ;ϕÞ are comovingspherical coordinates and k > 0 is the constant curvatureof the three-dimensional space. In a homogeneous andisotropic spacetime the energy-momentum tensor of thebackground source must have a perfect fluid form

Tμν ¼ ðρþ pÞuμuν þ pgμν; ð4Þ

where ρ is the energy density, p is the pressure and uμ arethe components of the four-velocity of the fluid (u0 ¼ −1and ui ¼ 0) in comoving coordinates. The trace of theenergy-momentum tensor is

Tμμ ¼ −ρþ 3p ¼ ρð3w − 1Þ; ð5Þ

where w ¼ p=ρ is the equation-of-state parameter.The dynamics of the universe can be obtained from

Eqs. (1) and (2), considering the metric given by Eq. (3).The equations of motion are given by

H2 ¼ ρ

3þ Λ

3−

ka2

; ð6Þ

aa¼ −

ρð1þ 3wÞ6

þ Λ3; ð7Þ

with

Tμμ ¼ ρð3w − 1Þ ¼ 4Λ − 6

�H2 þ a

aþ ka2

�: ð8Þ

Equations (2) and (8) then imply the following restrictionon the overall dynamics of the universe:

�H2 þ a

aþ ka2

�¼ 0: ð9Þ

This restriction is a direct consequence of the vacuumenergy sequestering mechanism and it is not present ingeneral relativity.

III. COSMIC DYNAMICS

A. Constant-w models

Here we use an iterative algorithm to determine thecosmological evolution of universes permeated by a singleperfect fluid with a constant equation-of-state parameter w(in which case ρ ∝ a−3ðwþ1Þ). Starting from the w ¼ 0.3model the algorithm calculates the value of Λ for each w(considering fixed positive or negative steps of Δw). Foreach w the evolution of the universe is computed and thevalue of Λ is estimated iteratively using

Λ ¼ 1

4hTμ

μi ¼3w − 1

4

Rdta3ρRdta3

: ð10Þ

Note that for w ¼ 1=3 the universe dynamics is not affectedby the vacuum energy sequestering mechanism (Λ ¼ 0 inthat case). However, this is no longer the case for w ≠ 1=3(Λ < 0 for w < 1=3 and Λ > 0 for w > 1=3). In theiterative process Λ ¼ 0 is taken as an initial guess for0.3–3Δw < w < 0.3þ 3Δw. For w < 0.3–3Δw and w >0.3þ 3Δw the initial condition for ln jΛj is obtained fromthe previous two w steps using a linear fit. For large valuesof w (larger than 0.7), the iterative procedure does notalways converge and a few additional constraints arenecessary in order to ensure convergence. The numericalresults presented in this section have consideredwmin ¼ −0.32, wmax ¼ 0.90 and Δw ¼ 0.01.Figure 1 shows the evolution of the scale factor with

cosmic time for a model with w ¼ −0.1 considering i)general relativity with a null cosmological constant(model I, solid line), and ii) the reformulation of generalrelativity incorporating the vacuum energy sequesteringmechanism (model II, dashed line). The maximum value ofthe scale factor amax and of the universe’s lifetime tu inmodel I are both normalized to unity. Figure 1 illustrates thelarge impact that this vacuum energy sequestering mecha-nism has on the dynamics of the universe for values of wnot too far from −1=3. Close to this limit, the first and thelast terms on the rhs of Eq. (6) have a similar evolution withthe scale factor, and, consequently, in model I the stage withH ∼ 0 corresponds to a very large variation of the scalefactor. This implies that the introduction of the negative-Λterm (in model II) has a dramatic effect on the dynamics ofthe universe for values of w larger but close to −1=3,

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leading to much smaller universe lifetimes tu and maximumvalues of the scale factor amax compared to model I, asillustrated in Fig. 1.Figure 2 is analogous to Fig. 1 but now considering

w ¼ 0.9. Figure 2 illustrates the very large impact that thevacuum energy sequestering mechanism has on the dynam-ics of the universe for w → 1−. Although

Rdta3ρ diverges

near a ¼ 0 for w ¼ 1 (note that ρ ∝ a−6 and a ∝ t1=3 forw ¼ 1 near the big bang, an epoch in which the curvature ofthe universe has no impact on its dynamics), there is a well-behaved model II solution in the w → 1− limit. Close to thatlimit, the universe may spend an arbitrarily large amount oftime in a quasistatic state with H ∼ 0 and a=a ∼ 0, whichweighs down the contribution of the phase close to a ¼ 0 inthe calculation of Λ. Figure 2 illustrates this, showing thatthe universe’s lifetime is much larger for model II than formodel I.Figure 3 shows the ratio between the universe lifetimes

in models II and I (tIIu=tIu) as a function of w. As previouslydiscussed, this ratio tends to zero in the w → −1=3þ limitand to∞ in the w → 1− limit. Except for the w ¼ 1=3 case,

the reformulation of general relativity proposed in Ref. [21]to prevent the vacuum energy from sourcing the gravita-tional field has a significant impact on the dynamics of theuniverse. This can also be seen in Fig. 4 which shows theratio between the maximum values of the scale factor inmodels II and I (aIImax=aImax) as a function of w. As expected,this ratio becomes very small in the w → −1=3þ limit andtends to a constant in the w → 1− limit.Figure 5 shows the ratio between the last two terms on

the rhs of Eq. (6) for a ¼ amax (−Λa2max=ð3kÞ) in model IIas a function of w while the ratio between Λ and theminimum density ρðamaxÞ for the same model is plotted inFig. 6 also as a function of w. We conclude from the plotsthat jΛj is never larger than a few times the minimumdensity for this family of models. We shall see in the nextsection that this is not always the case in a more generalframework.The ratios shown in Figs. 5 and 6 can be found

analytically in the w → 1− limit by taking into accountthat, in this limit,

H2 ¼ ρ

3þ Λ

3−

ka2

¼ 0; ð11Þ

aa¼ −

2

3ρþ Λ

3¼ 0; ð12Þ

0 0.2 0.4 0.6 0.8 10

0.5

1

t/tuI

a/a

max

I

FIG. 1. Evolution of the scale factor with cosmic time for amodel with w ¼ −0.1 considering general relativity with a nullcosmological constant (model I, solid line) and the reformulationof general relativity incorporating the vacuum energy sequester-ing mechanism (model II, dashed line). The maximum value ofthe scale factor amax and the universe’s lifetime tu in model I areboth normalized to unity.

0 2 4 6 80

0.5

1

1.5

t/tuI

a/a

max

I

FIG. 2. The same as in Fig. 1 but with w ¼ 0.9.

FIG. 3. Ratio between the universe’s lifetimes in models II and Ias a function of w.

FIG. 4. Ratio between the maximum values of the scale factorin models II and I as a function of w.

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for a ¼ amax. Equation (12) implies that ρðamaxÞ ¼ Λ=2.Substituting this into Eq. (12), one obtains −Λa2max=ð3kÞ ¼−2=3.The value of −Λa2max=ð3kÞ may also be calculated

analytically in the w → −1=3þ limit. In this limit ρ ∝a−2 and Λ < 0. Consequently, Eq. (6) may be written as

H2 ¼ Λ3

�1 −

a2max

a2

�; ð13Þ

where we have taken into account thatH ¼ 0 for a ¼ amax,thus implying that

ρðamaxÞ3

−k

a2max¼ −

Λ3: ð14Þ

The solution to Eq. (13) is given by

a ¼ amax sinðffiffiffiffiffiffiffiffiffiffiffiffi−Λ=3

ptÞ; ð15Þ

and the average value of ρ may be calculated as

hρi ¼ a2maxρðamaxÞRdtaRdta3

¼ 3

2ρðamaxÞ; ð16Þ

taking into account that ρ ¼ ρðamaxÞðamax=aÞ2 in the w →−1=3þ limit. Using Eqs. (6), (7) and (9) one finds thathρi ¼ −2Λ, in the w → −1=3þ limit, implying thatρðamaxÞ ¼ −4Λ=3. We may now conclude that, in thislimit,

0 ¼ H2 ¼ ρ

3þ Λ

3−

ka2

¼ −Λ9−

ka2

; ð17Þ

for a ¼ amax, which implies that −Λa2max=ð3kÞ ¼ 3.In Figs. 5 and 6 these analytical constraints have been

used to extend the results to the w → −1=3þ and w → 1−

limits.

B. Scalar field with a linear potential

Here we shall consider a homogeneous and isotropicuniverse filled with matter and a standard quintessencescalar field ϕ. In the context of general relativity and in theabsence of a cosmological constant this model is fullydescribed by the equations

H2 ¼ 1

3ðρm þ _ϕ2=2 − VðϕÞÞ − k

a2; ð18Þ

ϕþ 3H _ϕ ¼ −dVdϕ

; ð19Þ

where ρm ∝ a−3 is the matter density. Here we shall assumethat the scalar field potential VðϕÞ is a linear function of ϕ,namely

VðϕÞ ¼ V0 þdVdϕ

ðϕ − ϕ0Þ; ð20Þ

where jdV=dϕj is a constant and the subscript “0” meansthat the variables are to be evaluated at the present time t0(see Ref. [15] for more details on this model in the contextof general relativity, including a possible solution to thecoincidence problem).In the dark-energy-dominated era the dark energy scalar

field ϕ is constrained to be in a slow-roll regime with

wϕ ¼_ϕ2=2 − VðϕÞ_ϕ2=2þ VðϕÞ ∼ −1; ð21Þ

and

3H _ϕ ∼ −dVdϕ

: ð22Þ

In this regime the evolution of ϕ is very slow and the maincontribution to the energy density of the universe comesfrom VðϕÞ. In this phase the kinetic energy of the scalarfield increases very slowly due to the corresponding veryslow decrease of V. However, no matter how small the

FIG. 5. Ratio between Λ=3 and −k=a2max as a function of w(model II).

FIG. 6. Ratio between Λ and the minimum density (obtainedfor a ¼ amax) as a function of w (model II).

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value of j dVdϕ j > 0 is, at some point the slow-roll regimeends, VðϕÞ turns negative and the universe collapses withthe energy density becoming dominated by the kineticenergy density of the scalar field ϕ until the big crunch.The conclusion that the universe eventually collapses

and that the late-time evolution of the universe is dominatedby the kinetic energy of the scalar field ϕ remains valideven if one allows for an additional finite Λ contributionassociated to the vacuum energy sequestering mechanism(this has also been shown in Ref. [23]). Also, near the bigcrunch the universe is nearly flat and the curvature may alsobe neglected.Equation (19) implies that

H2ϕ″ þ a

�aaþ 2H2

�ϕ0 ¼ −

dVdϕ

; ð23Þ

where a prime denotes a derivative with respect to ln a.Neglecting the curvature and Λ terms in Eq. (6), settingw ¼ 1 in Eq. (7) and taking into account that ρ ∝ a−6 forw ¼ 1 one obtains that

a−6ϕ00 ¼ −C1

dVdϕ

; ð24Þ

where C1 is a positive constant. The solution to Eq. (24) isthen given by

ϕ ¼ −C1

36

dVdϕ

a6 þ C2 þ C3 ln a: ð25Þ

Hence, the values of ϕ and VðϕÞ display only a relativelyslow change with a as the universe approaches the bigcrunch. On the other hand, the kinetic energy of the scalarfield is increasing proportionally to a−6, driving the valueof w closer and closer to unity. This results in an infinitevalue for Λ if the vacuum energy mechanism is applied,thus invalidating this quintessence model as a viable

cosmological scenario (in Ref. [23] the authors did notconsider the possibility of a divergent Λ which resulted in adifferent conclusion). Note that this only occurs for modelswhere w tends to unity sufficiently fast at the big crunch (orat the big bang) so as to make the integral in Eq. (2) diverge,which does not happen in general.

IV. CONCLUSIONS

In this paper we have quantified the impact of a recentlyproposed vacuum energy sequestering mechanism on thebackground evolution of the universe, using both analyticaland numerical analyses. We confirmed that, in general, thismechanism significantly modifies the dynamics of theuniverse with respect to the cosmological dynamicsobtained in the context of general relativity with a nullcosmological constant. We have shown that in some cases,in particular for values of w close to −1=3 and 1, thedynamical changes can be dramatic. We have also shownthat there are well-behaved quintessence models in thecontext of general relativity which do not have a well-behaved counterpart in the vacuum energy sequesteringparadigm studied in this paper. We have highlighted theparticular case of a quintessence scalar field with a linearpotential, which has been suggested as a possible solutionto the coincidence problem in the context of generalrelativity.

ACKNOWLEDGMENTS

This work was supported by Fundação para a Ciência e aTecnologia (FCT) through the Investigador FCT contract ofreference IF/00863/2012 and POPH/FSE (EC) by FEDERfunding through the program Programa Operacional deFactores de Competitividade - COMPETE. P. P. A. is alsopartially supported by Grant No. PTDC/FIS/111725/2009 (FCT).

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