Phenomenological constraints on a scale-dependent gravitational coupling

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Phenomenological constraints

on a scale-dependent gravitational coupling∗

Orfeu Bertolamia and Juan Garcıa–Bellidob

aDepartamento de Fısica, Instituto Superior Tecnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal

bAstronomy Centre, University of Sussex, Brighton BN1 9QH, UK

We investigate the astrophysical and cosmological implications of the recently proposed idea of a running

gravitational coupling on macroscopic scales. We find that when applied to the rotation curves of galaxies, their

flatness requires still the presence of dark matter. Bounds on the variation of the gravitational coupling from

primordial nucleosynthesis, change of the period of binary pulsars, gravitational lensing and the cosmic virial

theorem are analysed.

1. INTRODUCTION

The flatness of the rotation curves of galaxiesand the large structure of the Universe indicatethat either the Universe is predominantly madeup of dark matter of exotic nature, i.e. non-baryonic, and/or that on large scales gravity isdistinctively different from that on solar systemscales, where Newtonian and post-Newtonian ap-proximations are valid. The former possibilityhas been thoroughly investigated (see Ref. [1]for a review) and is an active subject of researchin astroparticle physics. The second possibility,however relevant, has drawn less attention. Thisis essentially due to the fact that until recentlyno consistent and appealing modification of New-tonian and post-Newtonian dynamics has beenput foward. Many of these attempts [2], althoughconsistent with observations, were most often un-satisfactory from the theoretical point of view.Actually, it has been recently shown that undercertain fairly general conditions it is unlikely thatrelativistic gravity theories can explain the flat-ness of the rotation curves of galaxies [3]. Theseconditions however do not exclude the class ofgeneralizations of General Relativity that involvehigher-derivatives. Quantum versions of these

∗Talk presented by one of us (O.B.) at the Workshopon Theoretical and Phenomenological Aspects of Under-ground Physics (TAUP 95), Toledo, Spain, September1995; to appear in Nucl. Phys. B. supplement.

theories were shown to exhibit asymptotic free-dom in the gravitational coupling [4] and onewould expect this property to manifest itself onlarge scales. This possibility would surprisinglyimply that quantum effects could mimic the pres-ence of dark matter [5], as well as induce othercosmological phenomena [6,7]. One striking im-plication of these ideas is the prediction [6,7] thatthe power spectrum on large scales would havemore power than the one predicted by the Ω = 1Cold Dark Matter (CDM) Model, in agreementwith what is observed by IRAS [8]. Furthermore,due to the increase in the gravitational constanton large scales one finds that the energy densityfluctuations grow quicker than in usual matterdominated Friedmann-Robertson-Walker models[6,7]. Moreover, one can explain with a scale-dependent G the discrepancy between determina-tions of the Hubble’s parameter made at differentscales, as suggested in [6], and studied in [9].

Nevertheless, independently of the possiblerunning of the gravitational constant in a higherderivative theory of gravity, it is worthwhileanalysing the constraints on the scale-dependenceof G from astrophysical and cosmological phe-nomena, where such an effect would be dominant.On the other hand, in the last few years there hasbeen a revival of Brans-Dicke like theories, withvariable gravitational coupling, that has led toa number of constraints on possible time varia-tions of G. Of course, some of the constraints

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on G can be written as constraints on ∆G overscales in which a graviton took a time ∆t to prop-agate. For instance, during nucleosynthesis thelargest distance that a graviton could have tra-versed is the horizon distance at that time, i.e. afew ligh-seconds to a few light-minutes, approxi-mately the Earth-Moon distance. Such a distanceis too small for quantum effects to become ap-preciable, as we discuss below. However, thoseeffects become important at kiloparsec (kpc) dis-tances and therefore could be relevant for dis-cussing the rotation curves of galaxies. We shallactually show, for a particular theory [5,6], thatthe rotation curves of spiral galaxies cannot beentirely explained by the running of G, so someamount of dark matter is required, which is stillconsistent with the upper bound on baryonic mat-ter coming from primordial nucleosynthesis. Onthe other hand, we could impose bounds on apossible variation of G from a plethora of cos-mological and astrophysical phenomena at largescales, although the lack of precise observationsat those scales make the bounds rather weak [10].Of course, a difficulty in examining constraintson the variation of G is that in all gravitationalphenomena the gravitational coupling appears inthe factor GM , and hence we cannot distinguisha variation in G from the existence of some typeof dark matter.

2. ASYMPTOTIC FREEDOM OF THE

GRAVITATIONAL COUPLING

The main idea behind the results of Refs. [5,6]is the scale depedence of the gravitational cou-pling. The inspiration for this comes from theproperty of asymptotic freedom exhibited by 1-loop higher–derivative quantum gravity models[4]. Since there exists no screening mechanismfor gravity, asymptotic freedom may imply thatquantum gravitational effects act on macroscopicand even on cosmological scales, a fact which hasof course some bearing on the dark matter prob-lem [5] and on the large scale structure of the Uni-verse [6,7]. It is in this framework that a powerspectrum which is consistent with the observa-tions of IRAS [8] and COBE [11] can be obtained[6,7].

We briefly outline this proposal. Removing theinfinities generated by quantum fluctuations andensuring renormalizability of a quantum field the-ory requires a scale–dependent redefinition of thephysical parameters. Furthermore, the removalof those infinities still leave the physical param-eters with some dependence on finite quantitieswhose particular values are arbitrary. These canbe assigned by specifying the value of the phys-ical parameters at some momentum or lengthscale; once this is performed, variations on scaleare accounted for by appropriate changes in thevalues of the physical parameters via the renor-malization group equations (RGEs). Thus, theequations of motion in the quantum field theoryof gravity should be similar to the ones of theclassical theory, but with their parameters re-placed by the corresponding ‘improved’ values,that are solutions of the corresponding RGEs.However, since gravity couples coherently to mat-ter and exhibits no screening mechanism, quan-tum fluctuations of the gravitational degrees offreedom contribute on all scales. One must there-fore include the effect of these quantum correc-tions into the gravitational coupling, G, promot-ing it into a scale–dependent quantity. One-loopquantum gravity models indicate that the cou-pling G(µ2/µ2

∗∼ r2

∗/r2) is asymptotically free,

where µ∗ is a reference momentum, meaning thatG grows with scale [4]. A typical solution forG(r2

∗/r2) was obtained in Ref. [5] setting the β-

functions of matter to vanish and integrating theremaining RGEs:

G(r2∗/r2) = Glab δ(r, rlab) , (1)

where Glab is the value of G measured in the lab-oratory at a length scale rlab, and δ(r, rlab) is agrowing function of r. In order for the asymp-totic freedom of G(µ2/µ2

∗) to have an effect on

for instance the dynamics of galaxies and theirrotation curves, the function δ(r, rlab) should beclose to one for r < 1 kpc, growing significantlyonly for r ≥ 1 kpc. A convenient parametrizationfor δ(r, rlab) from the fit of Ref. [5] in the kpcrange is the following:

δ(r, rlab) = 1.485

[

1 + β

(

r

r0

)γ

ln(r

r0

)

]

, (2)

3

where β ≃ 1/30, γ ≃ 1/10 and r0 = 10 kpc.We mention that a scale dependence of the

gravitational constant also arises from completelydifferent reasons in the so-called stochastic infla-tion formalism [12] and that the scaling behaviourand screening of the cosmological constant wasalso discussed in the context of the quantum the-ory of the conformal factor in four dimensions[13].

In what follows we shall use the fit (1),(2) inour analysis of the rotation curves of galaxies, andextract a prediction for the distribution of darkmatter. However, before we pursue this discus-sion let us present some of the ideas developedin Refs. [5-7]. As discussed above, the classi-cal equations have to be ‘improved’ by introduc-ing the scale dependence of the gravitational cou-pling. This method suggests that the presence ofcosmological dark matter could be replaced by anasymptotically free gravitational coupling. As-suming that the Friedmann equation describingthe evolution of a flat Universe is the improvedone, then:

H2(ℓ) =8π

3G(

a20ℓ

2∗

a2ℓ2)ρm , (3)

where a = a(t) is the scale factor, H = a/a is theHubble parameter, ρm is the density of matter,ℓ is the comoving distance and ℓ∗ is some conve-nient length scale.

From Eq. (2) one sees that the present phys-

ical density parameter, Ωphys0 , is by construction

equal to one. However, the quantity which is usu-ally referred to as density parameter is actually:

Ω0 =8π

3

Gρm0

H20∗

, (4)

where H0∗is the present Hubble parameter for

a given large scale distance, r = r∗. This leadsto Ω0 as a growing function of scale, which is inagreement with observations for a constant ρm0

.Furthermore, from Eq. (3) one can clearly see

the scale dependence of the Hubble parameter[6,7,9]. Moreover, as shown in Refs. [6] and [7],the power spectrum resulting from these consid-erations is similar to that of a low density ColdDark Matter model with a large cosmological con-stant [14].

3. ROTATION CURVES OF GALAXIES

Let us now turn to the discussion of the im-plications of the fit (2) for the rotation curvesof galaxies. It is a quite well established observa-tional fact that the rotation curves of spiral galax-ies flatten after about 10 to 20 kpc from theircentre, which of course is a strong dynamical ev-idence for the presence of dark matter and/or ofnon-Newtonian physics. The rotation velocity ofthe galaxy is given by the non-relativistic relation,

v2 =G(r)M(r)

r, (5)

which approaches a constant value some distancefrom the centre, e.g. v2

0 = 220 km/s for the MilkyWay. Assuming that the gravitational couplingis precisely Newton’s constant GN and imposingthat the rotation velocity is constant, using theVirial Theorem at r = R ≡ 500 kpc, one finds thestandard expression for the mass distribution ofdark matter:

MN (r) = MN(R)r

R. (6)

Assuming instead a running gravitational cou-pling (2), the condition that the rotation velocityis constant yields:

M(r) =0.673

[

1 + β( rr0

)γ ln( rr0

)]MN(r) . (7)

Equation (7) reveals after simple computationthat the running of the gravitational coupling re-duces the amount of dark matter required to ex-plain the flatness of the rotation curves of galaxiesby about 44%, assuming that galaxies stretch upto a distance of about 500 kpc. This result [10](see also Ref. [15]) is a clear prediction of the de-pendence of the gravitational coupling with scaleand, in particular, of the fit (2). Furthermore,since the possibility that the Galactic halo is en-tirely made up of baryonic dark matter is barelyconsistent with the nucleosynthesis bounds on theamount of baryons [16], the running of G is quitewelcome since it reduces the required amount ofbaryonic dark matter in the halo. An entertain-ing hypothesis could be that precisely this effectis responsible for the reduction in the microlens-ing event rates across the halo in the direction of

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the Large Magellanic Cloud [17,18] with respectto those along the bulge of our galaxy, as reportedby [19].

4. BOUNDS ON THE VARIATION OF GWITH SCALE

In this section we constrain the variation of thegravitational coupling given by the fit (2) withbounds from primordial nucleosynthesis, binarypulsars and gravitational lensing and also discussthe effect that a scale-dependent G has on thepeculiar velocity field [10].

4.1. Primordial nucleosynthesis

As mentioned in the introduction, one couldobtain bounds on the variation of the gravita-tional coupling from observations of the light el-ements’ abundances in the Universe. Such obser-vations are in agreement with the standard pri-mordial nucleosynthesis scenario, but there is stillsome room for variations in the effective numberof neutrinos, the baryon fraction of the universeand also in the value of the gravitational constant.For instance, the predicted mass fraction of pri-mordial 4He can be parametrised, in theories witha variable gravitational coupling, in the followingway [20],

Yp = 0.228 + 0.010 lnη10 + 0.327 log ξ , (8)

where η10 is the baryon to photon ratio in unitsof 10−10 and ξ is the ratio of the Hubble param-eter at nucleosynthesis and its present value, it-self proportional to the square root of the corre-sponding gravitational constant. In the fit (8) itis assumed that the effective number of light neu-trinos is Nν = 3 and that the neutron lifetime isτn = 887 seconds.

By running the nucleosynthesis codes for dif-ferent values of G, it was shown in Ref. [21] thata variation of ∆G/G = 0.2 on the values of thegravitational coupling was compatible with theobservations of the primordial D, 3He, 4He and7Li abundances at 1σ level.

This result will now be used to constrain therunning of G in an asymptotically free theory ofgravity. In a theory with a scale-dependent grav-itational constant, the maximum value of G at

a given time is the one that corresponds to thephysical horizon distance at that time. Duringprimordial nucleosynthesis, the horizon distancegrows from a few light-seconds to a few light-minutes, i.e. less than a few milliparsecs. Atthat scale we find ∆G/G = 0.07, see Eq. (2),which is much less than the allowed variation of Ggiven in [21]. Therefore, primordial nucleosynthe-sis does not rule out the possibility of an asymp-totically free gravitational coupling. Of course, alight-second is about the distance to the Moon,and there are similar constraints on a variation ofG at this scale coming from lunar laser ranging,∆G/G < 0.6 [22].

4.2. Binary pulsars

The precise timing of the orbital period of bi-nary pulsars and, in particular, of the pulsar PSR1913+16, provides another way of obtaining amodel-independent bound on the variation of thegravitational coupling [23]. Since the semima-jor axis of that system is just about a few light-seconds, the resulting limits on the variation ofG can be readily compared with the ones arisingfrom nucleosynthesis. The observational limits onthe rate of change of the orbital period, mainlydue to gravitational radiation damping, togetherwith the knowledge of the relevant Keplerian andpost-Keplerian orbiting parameters, allows one toobtain the following limit [23]:

σ ≡∆G

G< 0.08 h−1 , (9)

where h is the value of the Hubble parameter inunits of 100 km/s/Mpc. For h = 0.8, [24] one ob-tains σ = 0.1 which is more stringent than the nu-cleosynthesis bound, but is still compatible withthe fit (2).

4.3. Gravitational lensing

Gravitational lensing of distant quasars by in-tervening galaxies may provide, under certain as-sumptions, yet another method of constraining,on large scales, the variability of the gravitationalcoupling. The four observable parameters associ-ated with lensing, namely, image splittings, timedelays, relative amplification and optical depth dodepend on G, more precisely on the product GM ,where M is the mass of the lensing object. This

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dependence might suggest that limits on the vari-ability of G could not be obtained before an in-dependent determination of the mass of the lens-ing object. However, as the actual bending angleis not observed directly, the relevant quantitiesare the distance of the lensing galaxy and of thequasar. Since these quantities are inferred fromthe redshift of those objects, they depend on theirhand on G, on the Hubble constant, H0, and onthe density parameter, Ω0. However, as we havepreviuosly seen, a scale-dependent gravitationalcoupling implies also a dependence on scale of H0

and Ω0, see Eqs. (3) and (4). This involved de-pendence on scale makes it difficult to proceed asin Ref. [25], where gravitational lensing in a flat,homogeneous and isotropic cosmological model,in the context of a Brans-Dicke theory of gravity,was used to provide a limit on the variation of G:

∆G

G= 0.2 . (10)

Since for this limit Ω0 = 1 was assumed, whilein a scale-dependent model it is achieved via therunning of the gravitational coupling, the bound(10) contrains only residual variations of G thathave not been already taken into account whenconsidering the dependence on scale of H0 andΩ0. Of course, for models where the cosmologicalparameters are independent of scale, the bound(10) can be readily used to constrain the variabil-ity of G on intermediate cosmological scales. Itis worth stressing that this method, besides be-ing one of the few available where this variabil-ity is directly constrained at intermediate cosmo-logical times between the present epoch and thenucleosynthesis era, it is probably the only onewhich can realistically provide in the near futureeven more stringent bounds on even larger scalesby observing the lensing of light from far awayquasars caused by objects at redshifts of orderz ≥ 1.

4.4. Peculiar velocity field

Since we expect the effects of a running G to be-come important at very large scales, one could tryto explore distances of hundreds of Mpc, wherethe gravitational coupling is significantly differentfrom that of our local scales. That is the realm

of physical cosmology where of particular impor-tance is the study of the peculiar velocity field. Apossible signature of the running of G would be amismatch between the velocity fields and the ac-tual mass distribution, such that at large scalesthe same mass would pull more strongly. To bemore specific, in an expanding universe there is arelation between the kinetic and gravitational po-tential energy of density perturbations known asthe Layzer-Irvine equation (see eg.Ref. [26]) thatcan be written as a relation between the mass-weighted mean square velocity v2 and the massautocorrelation function ξ(r),

v2(r) = 2πG ρb J2(r) , (11)

where ρb is the mean local mass density andJ2(r) =

∫ r

0r dr ξ(r). The galaxy-galaxy corre-

lation function can be parametrized by ξ(r) ∼

(r/r0)−1.8 with r0 = 5h−1 Mpc, while the cluster-

cluster correlation function has the same expres-sion with r0 = 20h−1 Mpc. This means thatthe velocity field (11) should be proportional to(r/r0)

0.2, unless the gravitational constant hassome scale dependence. So far the relation seemsto be satisfied, under rather large observationalerrors (see Ref. [27] for a review). Unfortunately,the errors are so large that it would be prematureto infer from this a scale dependence of G. Evenworse, phenomenologically there is a proportion-ality constant between the galaxy-galaxy corre-lation function and the actual mass correlationfunction, the so- called biasing factor, which issupposed to be scale dependent and could mimica variable gravitational constant. However, fu-ture sky surveys might be able to constrain morestrongly the relation (11) by measuring peculiarvelocities with better accuracy at larger distancesand it might then be possible to extract the scale-dependence of G.

5. CONCLUSIONS

We have seen that the running of the gravi-tational coupling is compatible with the obser-vational fact that the rotation curves of galaxiesare constant provided some amount of baryonicdark matter is allowed, actually about 44% lessthan what is required for a constant G. This

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could also explain why we see less microlensingevents towards the halo than in the direction ofthe bulge of our galaxy. Failure in reproducingthe predicted distribution of baryonic dark mat-ter would signal either that the approach adoptedhere is unsuitable or that the fit (2) is inadequate.We have looked for possible bounds on variationsof G with scale from primordial nucleosynthesis,variations in the period of binary pulsars, macro-scopic gravitational lensing and deviations in thepeculiar velocity flows. Unfortunately, as obser-vational errors tend to increase with the scaleprobed, we cannot yet seriuosly constrain an in-crease of G with scale, as proposed by the asymp-totically free theories of gravity.

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