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Cosmology with Nonminimal Derivative Couplings

arXiv:gr-qc/9302010v1 10 Feb 1993gr-qc/9302010Cosmology with Nonminimal Derivative CouplingsLUCA AMENDOLAOsservatorio Astronomico di RomaViale del Parco Mellini, 84I-00136 Rome, ItalyABSTRACTWe study a theory which generalizes the nonminimal coupling of matter to gravityby including derivative couplings. This leads to several interesting new dynam-ical phenomena in cosmology.

In particular, the range of parameters in whichinflationary attractors exist is greatly expanded. We also numerically integratethe field equations and draw the phase space of the model in second order approx-imation.

The model introduced here may display different inflationary epochs,generating a non-scale-invariant fluctuation spectrum without the need of twoor more fields. Finally, we comment on the bubble spectrum arising during afirst-order phase transition occurring in our model.1IntroductionScalar fields in General Relativity has been a topic of great interest in the latest years,mainly because the scalar field dynamics allows to investigate the detailed features of theearly Universe.Without the need of a specific equation of state, a Universe filled by aphenomenological scalar field leads to an accelerated phase of expansion, the inflation, inalmost any kind of self-interaction potential.

As it has been shown in chaotic models ofinflation1, the accelerated phase, either power-law2 or quasi-exponential, is a phase-spaceattractor for most of the initial conditions, and this result also extends to a large classof inhomogeneous and anisotropic space-times3. The class of successful models has beenenlarged to scalar fields with a nonminimal coupling (NMC) to the curvature scalar R inthe gravity Lagrangian4, commonly in the form of a term √−gf(φ)R, where f(φ) is afunction of the scalar field φ.

The motivations for this step are manyfold: the idea that thefundamental constants are not constant, the “machian” theory of gravitation embodied inthe Jordan-Brans-Dicke theory5, the renormalizing term arising in quantum field theory incurved space6, the possibility to have gravity as a spontaneous symmetry-breaking effect7,the Kaluza-Klein compactification scheme8, the low-energy limit of the superstring theory9.Moreover, the NMC term has been employed to produce an oscillating Universe,10 to reconcilecosmic strings production with inflation,11 to generate a modified Newtonian dynamics able1

to model flat rotation curves in galaxies.12 Last but not least, an NMC term allows to solvethe “graceful exit” problem of the old inflation by slowing the false-vacuum expansion, as inextended inflation13. Recently, the NMC models have been generalized in various directions:models with different coupling functions14, with generalized coupling to the inflaton sector15,with fourth-order gravity16, and with a coupling to a dark matter sector17.In this Letter we wish to further explore the influence of nonminimal couplings in cosmol-ogy by introducing the nonminimal derivative coupling (NMDC) to gravity.

In this class ofmodels, the coupling function is also function of the derivatives of φ: f = f(φ, φ;µ, φ;µν, ...).Derivative couplings, although rares, are not a novelty in physics: the field theory of scalarquantum electro-dynamics includes derivative couplings between the electro-magnetic vectorAµ and the scalar field φ; this kind of interaction is indeed required by the U(1) gauge-invariance of the theory. In addition, if the idea underlying the nonminimal coupling theoryis that the Newton constant G itself depends on the gravitational field source mass, it seemsmore natural to couple the curvature to the energy-momentum tensor of the matter, intro-ducing the terms TR and TµνRµν, which in fact contain derivative couplings.

The NMDCmodel can be thought of as the scalar field formulation of the hypothesis that G = G(ρ),where ρ is the energy density of the gravitational field source.One of the aims of this Letter is to see whether the presence of NMDCs allows to findinflationary attractors in models where otherwise they are not present. We studied a largeclass of NMC models in Ref.

(18). For future reference, let us summarize the main resultsof Ref.

(18). Hereinafter, we will use f = f(φ) referring to the non-derivative terms coupledto the curvature scalar, and f1 = f1(φ, φ;α) when we refer to all the terms, including thederivative terms, coupled to the curvature scalar.

In Ref. (18) the coupling function f(φ)is left as general as possible, and the potential is assumed to be V = λf M.It is alsoassumed that f(φ) is semi-positive definite and that, for large |φ|, it grows monotonicallyfaster than φ2.

The case f ∼φ2 is the standard choice, and has been extensively treatedin the literature4. The relation between the potential and the coupling function assumedin Ref.

(18) allows to determine the class of models with successful inflationary attractors;here “successful” means that the inflation eventually stops and a Friedmann stage begins.In Ref. (18) and in the present work the inflationary attractors are all asymptotic, in thelimit of large |φ|; in this limit the influence of the coupling is clearly larger.Moreover,the spirit of chaotic inflation is that whatever the initial conditions of the Universe are, thedynamical trajectory falls onto an inflationary attractor.

This is verified only if the attractoritself extends to very large values of the field variables, until the classical description losesmeaning (i.e. near the Planck boundary).

In Ref. (18) it is shown that such asymptoticattractors exist only if 2 ≤M < 2 +√3, whatever functional form of f(φ) is considered.In this range, it is found for the cosmic scale factor a(t) a power-law inflation a ∼tp withexponentp =3 + (M −2)3(M −2)2(M −1) .

(1)In the conformally rescaled frame the power-law exponent has a much simpler form, ˜p =3/(M −2)2. If M = 2 a deSitter exponential inflation is found.

Further, in the cases inwhich a successful inflation exists, the spectrum of primordial fluctuations is independent of2

f(φ) and depends on the wavenumber k according to the power-law k1/(1−˜p). Finally, in Ref.

(18) it is shown that the fluctuations have a Gaussian distribution. In the present work weextend some of the cited results to derivative couplings.

We assume as in Ref. (18) thatV = λf M, but now we will freeze the functional form of f(φ) to a power-law, f(φ) = φ2m.We will show that the narrow range in which the parameter M is confined in NMC modelsis greatly expanded when one includes a derivative coupling.It rises natural the question of whether the conformal transformation19, which allows oneto put complicated unconventional gravity theory in pure Einsteinian form with one or morescalar fields, works also when derivative couplings are present.

We will show schematicallythat a conformal rescaling ˜gµν = e2ωgµν cannot recast our theory in Einsteinian form. In orderto recover the Einstein field equations the metric transformation should be generalized to aLegendre transformation, as in Ref.

(20). This would introduce in the equations additionaltensor fields, instead of additional scalar fields as in non-derivative cases.

Due to this reason,we avoided here the use of metric transformations.The plan of this paper is as follows: in the next section we introduce the derivativecouplings, in Sect. 3 we derive the equations of motion for the fields, in Sect.

4 we performthe second-order approximation, in Sect. 5 we discuss the phase space of some models andfinally in Sect.

6 we present the conclusions.2Derivative couplingsTerms with NMDCs generalize the Einstein-Hilbert Lagrangian. The most general gravityLagrangian linear in the curvature scalar R, quadratic in φ, containing terms with fourderivatives includes all of the following terms:L1=µφ;αφ;αR ;L2 = τφ;αφ;βRαβ;L3 = ηφ✷φR ;L4=θφφ;αβRαβ ;L5 = νφφ;αR;α;L6 = σφ2✷R .

(2)The constants µ, τ, ... have the same dimensions in natural units as the Newton constant G,namely mass−2. Here and in the following the conventions are: signature (+ −−−) ; units8πG = c = 1 .

Due to the following relations0=Zd4x √−g✷(φ2R) =Zd4x √−gh2φ✷φR + 2φ;αφ;αR + φ2✷Ri,0=Zd4x √−g∇µ(R∇µφ2) =Zd4x √−g [2φφ;αR;α + 2φ✷φR + 2φ;αφ;αR] ,0=Zd4x √−g∇µ(φφ;νRµν) =Zd4x √−ghφ;µφ;νRµν + φφ;µνRµν + φφ;νRµν;µi,(3)three of the six terms L1 −L6 can be neglected (also taking into account the Bianchi iden-tities). Other total divergences, like (φ2Rµν;ν );µ and (φ2Rµν);µν are linear combinations ofEqs.

(3). Nevertheless, the equations remain hopelessly complicated.

We take here a firststep. We keep only the derivative term L1, together with the NMC correction L0 = ξf(φ)R.This particular choice turns out to be the simplest one.

Moreover, the terms studied hereare those appearing in the simplest “natural” coupling, namely TR. In this way we may3

draw an useful comparison between Ref. (18), where L0 was the only correction to Einsteingravity, and the present work.

We are then left with the following Lagrangian densityL=−R + f1(φ, φ;α)R + φ;αφ;α −2V (φ) ,f1(φ, φ;α)=ξf(φ) + µφ;αφ;α . (4)3Field equationsThe set of equations derived from (4) isGµν(1 −f1)=gµν✷f1 −f1;µν + µφ;µφ;νR + Tµν ,Tµν=φ;µφ;ν −12gµνφ;αφ;α + V (φ)gµν ,(5)while the scalar field equation is✷φ(1 + µR) + µφ;αR;α −ξf ′R/2 + V ′ = 0 .

(6)It can already be seen that, even if the new terms introduced by the derivative coupling (letus call them µ-terms) do not determine the critical points of the theory, they do contributeto the asymptotic dynamical properties of Eq. (6).

Indeed we will see that in the limit oflarge R and R;α (in other words, in the early Universe), the µ-terms will sensibly modify thebehavior of the non-derivative theory.The trace of Eqs. (5) reads simplyR(ξf −1) = T + 3✷f1 ,(7)where T = −φ;αφ;α + 4V .

The derivative of the Ricci scalar, to be used in Eq. (6), isR;α =1ξf −1 [T;α + 3✷f1,α −ξf ′φ;αR] .

(8)In a Friedmann-Robertson-Walker (FRW) spatially flat metric with scale factor a(t), the(0, 0) component of (5) isH2(1 −f1) = H ˙f1 + µ3 R ˙φ2 + 1312˙φ2 + V,(9)where H = ˙a/a and where our conventions are such that R = −6 ˙H −12H2. From now on,we confine ourselves to a FRW spatially flat metric.

The system (6-9) is closed. We havefive degrees of freedom H, φ, ˙φ, ¨φ, φIII and indeed we have one equations of fourth order inφ [Eq.

(6)] and one of first order in H [Eq. (9)].

The solution of this system appears verydifficult. In the next section we will make use of a second-order approximation to derive theasymptotic properties of our model and to sketch its phase space.Let us conclude this section showing how a conformal transformation operates on theNMDC theory.

Ordinary NMC theories are recast in Einstein form by the metric rescaling19˜gµν = e2ωgµν, withe2ω = |∂L/∂R| . (10)4

Adopting the same formula, in our case we get e2ω = |1−f1|. Now, we have that the Einsteintensor Gµν = Rµν −Rgµν/2 trasforms according toGµν = ˜Gµν −2ω;µω;ν + 12gµνω;αω;α + gµν✷ω −ω;µν.

(11)Inserting (11) into (5), with the choice (10), one has˜Gµν = 3e−4ωf1;µf1;ν −12gµνf1;αf ;α1+ µe−2ωφ;µφ;νR + e−2ωTµν ,(12)which is not in Einsteinian form. Any other choice of the conformal factor would not cancelthe second derivatives of f1 which are present in (5).

Then, the right-hand-side of (12) couldnot be written as a scalar field energy-momentum tensor. Notice that R at the right-hand-side is not trasformed.

If one also trasforms R in (12), additional terms like (✷ω)φ;µφ;νarise.4Second-order approximationIn slow-rolling motion one neglects all terms with more than one time derivative in thefield equations. In this section we are more general, and we examine the field equations byneglecting all terms of order higher than the second one.

This gives us three advantages: theequations are notably simplified, we can calculate analytically some inflationary solutions,and we can plot the phase-space portrait of the model. Neglecting all terms higher thansecond order in (6) we have✷φ(1 + µR(0)) + µ ˙φ ˙R(1) −ξf ′R(2)/2 + V ′ = 0 .

(13)The indexes in parentheses denote the order to which R and ˙R are to be calculated. FromEqs.

(7,8) and for ξf ≫1 we haveR(0)=4Vξf ,(14)R(2)=1ξfh−˙φ2 + 4V + 3ξ✷f)i,(15)˙R(1)=4 ˙φξf 2 [V ′f −V f ′] . (16)We now explicitely adopt the functional relation V = λf M. This will simplify our equations.Eq.

(13) then writes✷φ"f + 4µλξ f M −32ξf ′2#+ f ′ ˙φ2"4µλξ f M−1(M −1) + 12(1 −3ξf ′′)#+λf ′f M(M −2) = 0 . (17)Let us call A the coefficient of ✷φ and B the coefficient of f ′ ˙φ2.

In order to avoid singularitiesin Eq. (17) one should have a nonvanishing coefficient A for ξf ≫1 (in the opposite limit5

the equations reduce to the minimally coupled model and the regularity follows automat-ically). A sufficient condition, although not a necessary one, is that ξ, µ ≤0, along withthe already imposed condition f ≥0.

This assumption greatly simplifies the investigationof the dynamical properties of the model under study and will be adopted in the following.It also makes clearer the comparison with previous work in ordinary NMC theories, wherethe condition ξ < 0 is often adopted to get a regular metric rescaling. Notice that ξ, µ ≤0ensures the positivity of the effective Newton constant in the Lagrangian density (4).The d’Alambertian operator reads ✷φ = ¨φ + 3H ˙φ in FRW metric, where the Hubblefunction H is to be taken from Eq.

(9) (to the first order):ξH2f + ξH ˙f + 4µV3ξf˙φ2 + 1312˙φ2 + V= 0 . (18)The asymptotic behavior of Eq.

(17) can be now easily discussed in the limit in whichsome of the terms in A and B can be neglected. A first consideration is however immediate:for M < 2 the second derivative of the effective potential in Eq.

(17) is negative, in thehypothesis that f grows monotonically with |φ|. As a consequence, any attractor solution ofEq.

(17) leads to ever growing f; the Friedmann behavior cannot be reached. Let us makean example.

If we assume f(φ) →φ2m for large |φ|, it turns out that if M < 2 −1/m onehas that f ′2 dominates in A and f ′′ in B. Then Eq.

(17) simplifies to✷f −[2λ(M −2)/3ξ]f M = 0(19)(where ✷f = f ′✷φ + f ′′ ˙φ2). In this case the influence of the µ-terms disappears; the modelreduces to the ordinary NMC theory, already discussed in Ref.

(18). There it was found thatfor M < 2 no successful inflation was allowed.

Indeed, in Eq. (19) the effective potentialderivative is negative definite (since ξ < 0 and M < 2).

As already discussed, the attractorsolutions of Eq. (19) are directed toward increasing f; the Friedmann behavior will never berecovered.

The same conclusion holds for any f(φ), provided that f ′2 ≫f M, f.In the special case M = 2 one sees that ¨φ = ˙φ = 0 is a solution of (17). One has thenan asymptotic deSitter inflation, analogously to what occurs in NMC models.

We will notgive details of this case, since the dynamics depends on the behavior of V and f at small φ,while here we are interested mainly in the asymptotic properties.Let us come to the third, more interesting, case, namely M > 2. Now f M dominates inA and f M−1 in B. Eq.

(17) then reads✷φ + f ′f˙φ2(M −1) + ξ4µf ′(M −2) = 0 . (20)It worths remarking that the effective potential is U(φ) = ξ(M −2)(4µ)−1f(φ); the couplingfunction f(φ) reveals itself as an effective potential for the theory.

Notice also that, contraryto the previous case, the constant µ plays here an important role. Let us put now f(φ) = xm,where x ≡φ2.

It then follows✷x −˙x22x[1 −2N −2m] + ξNµ xm = 0 ,(21)6

where N = m(M −2), for x ≥0. To have a positive-definite potential derivative is sufficientto have M > 2 (provided that ξ/µ > 0) .To find explicit solutions to Eq.

(21) we need also H = H(φ, ˙φ).From Eq. (18),neglecting all terms higher than first order and in the limit ξf ≫1, we get for M > 2 −1/m(i.e.

N > −1)[H(1)]2 =λ3|ξ|xN−1"3µξ ˙x2 + xm+1#. (22)Let us put a trial solution ˙x = bxp in (21) and (22).

In the limit of large x we find theattractorp=(m −N)/2 ,b=−N√3 ξ3λµ!1/2. (23)It can be easily proved that this really is an attractor; a graphic evidence is provided byour phase-space portraits, Figs (1,2).

Along the solution (23) the term ¨x is negligible; thisjustifies the second order approximation previously performed. Integrating ˙x = bxp one getsx = x0(1 + t/τ)1/(1−p) ,(24)where τ −1 = b(1 −p)xp−10and x(t = 0) = x0.

For p < 1 the time constant τ is negative;clearly our approximations break down for t →τ. The inflationary regime we will find alsobreaks for t →τ.

Along the attractor we have the Hubble functionH = (λ/3|ξ|)1/2 x(N+m)/2 ,(25)and the cosmic scale factora(t) = a0 expnEh1 −(1 + t/τ)kio,(26)whereE = 3(|µ/ξ|)xN+10N(N + 1),k = 2(N + 1)2 + N −m . (27)The scale factor expansion is always inflationary for t < τ.

For t ≪τ the expansion followsthe deSitter law a ∼exp[(E|k/τ|)t], (k/τ is always negative). The constant E sets the totalnumber of e-foldings of the inflationary stage.

If, for instance, ξ and µ are approximativelyequal, values of φ0 = x1/20around unity in Planck units are required to have E > 60, the samerequirement commonly found in chaotic inflation. This kind of behavior is often called quasi-deSitter inflation.

The narrow range in which viable inflation is found in NMC theories18,2 ≤M < 2 +√3, is here expanded to M ≥2, without upper bound. The effect of includinga derivative coupling is then to substantially enlarge the class of viable inflationary models.This is the main result of this paper.7

It can be checked that H(t) and x(t) decrease as the time increases. As |φ| decreases, thesolution crosses to the NMC region, where the derivative coupling is negligible compared tothe non-derivative one.

In particular, the attractor enters the NMC region forx < x1 ="3ξ2m22|µ|λ#1/(N+1). (28)As |φ| further decreases, ξf(φ) becomes smaller than unity for x < x2 = ξ−1/m and theattractor enters the minimally coupled central region.

For instance, in a model in which|µ|λ = 3/2 and M = 3, m = 1, one has x1 = ξ and x2 = 1/ξ. If x2 > x1 the NMC phasedisappears.In Figs.

(1,2) we show the numerical phase space of the model for some parameter values.The plots are obtained integrating Eq. (17) for several initial conditions, and then performinga Poincar´e projection onto the unitary circle.

For any trajectory we can identify four stages,or cosmological epochs. All trajectories start ideally from the initial singularity at φ, ˙φ →∞,and they fall after a short transient on the inflationary attractors.

In this first stage thehigher-order terms that we are neglecting in this section can be important. The secondstage is represented by the outer part of the attractors, where the µ-terms are dominant;now the trajectories follow Eq.

(23), and the cosmic expansion is given by (26). The thirdstage begins when x < x1, where the attractors enter the NMC region.

Now the behaviorwill be as described in the introduction, with the power-law expansion (1) (inflationary if2 ≤M < 2+√3). Finally, the coupling terms become negligible (for x < x2) and the theoryreduces to ordinary gravity.

The details of this last stage depend on the specific form ofV (φ) for small φ, but eventually (perhaps after a last quasi-deSitter inflationary episode)the mass term will dominate, and a series of damped oscillations around the potential groundstate will occur. A Friedmannian expansion then takes place.

The basin of attraction ofthe attractors extends to almost all of the phase space, and all initial conditions lead to thecentral Friedmann region. Since the higher-order terms do not introduce new critical pointsin the model, the qualitative picture given here should be of general validity.

Notice thatFig. (2) displays a model with M = 4, outside the successful range in NMC models, butinflationary in the NMDC model.5ConclusionsWe have shown that nonminimal derivative couplings are an interesting source of new cos-mological dynamics.

Generally speaking, their presence allows inflationary attractors whereotherwise they would be absent. The quite narrow range of parameters that allows a viableinflation in NMC theories is here expanded to a semi-infinite range.

Let us remark thatthe succession of different inflationary epochs may be used to break the scale-invariance ofinflationary perturbation spectra21. In the model effectively considered, three inflationaryphases (quasi-deSitter, power-law, quasi-deSitter) may occur, depending on the parameter.This realizes a scenario of double (or triple) inflation21 without the need to introduce morefields.Before concluding, let us comment on another possible consequence of a NMDC cosmol-ogy.

The inflationary attractors found in the previous section can be used to implement8

a model of primordial phase transition along the scheme of extended inflation. Suppose asecond scalar field ψ with double-well self-interaction potential V2(ψ) is present in the the-ory.

During the inflationary stage driven by φ the new field may perform a first-order phasetransition from the more energetic false-vacuum state to the less energetic true vacuum. Thetransition proceeds through quantum and thermal nucleation of true-vacuum bubbles, whosesurface energy density contains the “latent heat” of the process.

The bubbles grow convert-ing the background false-vacuum energy density into surface “kinetic energy” and eventuallycoalesce and percolate, unless the false-vacuum inflationary expansion is too rapid. If Γ isthe nucleation rate, the (not normalized) probability that a point is still in the false-vacuumstate at the time t is22pF V (t) = a3(t)e−I(t) ,(29)whereI(t) = 4π/3Z t0 dt′Γa3(t′)"Z tt′dua(u)#3,(30)where t = 0 is the instant in which the nucleation begins.

(Let us remark that pF V should becalculated in an Einstein frame; for what concerns the following discussion this precisationis of secundary importance.) If pF V →0 the transition is completed.

It is well known thatin old inflation, where the expansion is exponential, the transition can never be completed;extended inflation cures this “graceful exit” problem by slowing down the inflation to apower-law. In our model the scale factor expands according to Eq.

(26) during the NMDCphase (the second stage) and according to Eq. (1) during the NMC phase (the third stage).It is the latter phase to be crucial.

During the quasi-deSitter NMDC expansion the falsevacuum grows so fast that the bubbles nucleated are rapidly diluted before they can coalesce,unless Γ is unrealistically large. This is the same problem one has in old inflation.

However,during the subsequent NMC epoch the false vacuum slows down to a power-law expansion(it does not matter here whether inflationary or not) and the phase transition can find anatural exit, just as one has in extended inflation. One should also ensure that the possiblelast inflationary phase in the minimally coupled regime, which depends on the details for|φ| →0 of V (φ), be negligible.

In this scenario, the transition occurs almost entirely duringthe NMC epoch, when the scale factor expands slowly enough. Also the bubble nucleationwill take place mostly in the NMC epoch.

Depending on the relative duration of the NMDCinflationary phase and the NMC phase one can have different bubble spectra. If the quasi-deSitter NMDC phase lasts for a large fraction of the 60 or so e-foldings one needs for theinflation to be successful, the bubbles produced during the subsequent power-law NMC phaseare very small, they rapidly thermalizes (perhaps with a production of gravitational waves 23)and do not affect neither the large scale structure, nor the cosmic microwave background24.If, on the contrary, the NMC phase takes, say, 55 e-foldings, then the bubbles would betoday very large and hardly thermalized.

The cosmic microwave background measurementsand the primordial nucleosynthesis would put very stringent limits on the model, but theintriguing possibility that the primordial bubbles do participate in the large-scale structureformation arises25.9

ACKNOWLEDGMENTSThe author thanks A. A. Starobinsky, H.-J.

Schmidt and G. Pollifrone for thoughtfulcomments on the manuscript.10

Figure Caption1. Poincar´e phase space for a model with m = 1, M = 3, ξ = µ = −1, λ = 3/2.

Theinflationary attractors are clearly seen. The trajectories start at t = 0 from the “north”and “south” poles and rapidly reach the attractors.

Eventually, all trajectories fall ontothe global stability point at ˙φ = φ = 0, after a number of damped oscillations. Thediscrete symmetry φ, ˙φ →−φ, −˙φ, also evident in the Lagrangian, shows up.2.

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