Radiative Corrections in the (Varying Power)-Law Modified Gravity
RRadiative Corrections in the (Varying Power)-Law Modified Gravity
Fay¸cal Hammad ∗ Physics Department & STAR Research Cluster, Bishop’s University,& Physics Department, Champlain College,2600 College Street, Sherbrooke,Qu´ebec J1M 1Z7, Canada.
Although the (varying power)-law modified gravity toy model has the attractive feature of unifyingthe early and late-time expansions of the Universe, thanks to the peculiar dependence of the scalarfield’s potential on the scalar curvature, the model still suffers from the fine-tuning problem whenused to explain the actually observed Hubble parameter. Indeed, a more correct estimate of themass of the scalar field needed to comply with actual observations gives an unnaturally small value.On the other hand, for a massless scalar field the potential would have no minimum and hencethe field would always remain massless. What solves these issues are the radiative corrections thatmodify the field’s effective potential. These corrections raise the field’s effective mass rendering themodel free from fine-tuning, immune against positive fifth-force tests, and better suited to tacklethe dark matter sector.
PACS numbers: 04.50.-h, 98.80.Es, 95.35.+d, 03.70.+k
I. INTRODUCTION
Modified gravity theories play a major role in cosmol-ogy [1]. In the absence of any guiding principle one usu-ally begins by writing down a simple toy model of mod-ified gravity and uses it to tackle cosmological problems.One then tries to tailor one’s model in such a way tomake it capable of explaining, not only the early and thelate-time expansions of the Universe (see e.g., [2, 3]), butalso dark matter (see e.g., [4–7]).Among the simplest and most attractive classes ofmodified gravity theories are the so-called power-law toymodels. By adding extra terms with different powers ofthe scalar curvature inside the gravitational Lagrangianone makes gravity evolve with the Universe because theseextra terms manifest themselves differently at differentepochs in the evolution of the Universe [8, 9]. However,when trying to fit many of the plausible models with ob-servation one finds that the required powers of the Ricciscalar are not necessarily integers or rational numbersbut might be real numbers that span a finite intervaldepending on the cosmological observation used in thefitting [4, 10, 11].The (varying power)-law modified gravity toy modelhas been proposed in Ref. [12] as a compromise betweena power-law model and a scalar-tensor model [13, 14].The power of the Ricci scalar in this model need not befixed by hand each time one confronts the model withobservation but rather left to be selected naturally bythe surrounding environment. In Ref. [12], such a modelwas examined and found to be able to unify both theearly and late-time expansions of the Universe as well asto eventually incorporate dark matter. The early expan-sion of the Universe might easily be incorporated within ∗ [email protected]; [email protected] the model thanks to one of its two free mass parameters.However, when the model is used to represent the late-time expansion of the Universe, a fine-tuning issue arisesbecause the second free mass parameter, which is sup-posed to represent the mass of the scalar field, is requiredto be unnaturally small in order to make the model ableto reproduce the actually observed rate of cosmic expan-sion.Now, although the scalar field is present in the modelas the power of the Ricci scalar, which makes the modelhighly non-linear, quantum corrections due to the quan-tum fluctuations of the scalar field could easily be incor-porated. Remarkably enough, it turns out that thanks tothe peculiar form of the scalar field’s potential, quantumfluctuations of the scalar field make the latter acquire aneffective mass through its coupling with geometry suchthat no fine-tuning is required on the initial mass of thescalar field. Moreover, the mass thus acquired is so bigthat the model becomes immune against the fifth-forcetests [15].The outline of the paper is as follows. In Sec. II, wegive a more correct estimate of the scalar field’s massneeded, before the quantum corrections are taken intoaccount, to produce the currently observed Hubble pa-rameter. We show that it should be, contrary to whathas been suggested in Ref. [12], exceedingly small. Wethen argue why a completely massless field couldn’t beused satisfactorily either. In Sec. III, we show that de-spite the high non-linearity of the model, the applicationof the regular techniques in perturbation theory to takeinto account the quantum fluctuations of the scalar fieldis justified. We then compute the radiative corrections tothe potential at one-loop order, find the needed counterterms, and finally write down the effective potential thatwould result from these radiative contributions to theclassical potential. Using the obtained effective poten-tial we deduce the expression of the effective mass of thescalar field. In Sec. IV, we show how the behavior of the a r X i v : . [ g r- q c ] D ec effective potential is affected at late-times, making it pos-sible to remedy the fine-tuning issue. We show that atlow curvatures a massive as well as a massless scalar fieldboth acquire a very big mass. The dynamical equationfor the Hubble parameter is then given and the latter isfound to be compatible with its actually observed valuewithout imposing any kind of fine-tuning. We end thispaper by a brief discussion section. II. THE MODEL AND ITS FINE-TUNINGISSUE
Unless otherwise stated, the natural units (cid:126) = c = 1will be used through out the paper. Then, the gravita-tional action of the model studied in Ref. [12] is M (cid:90) d x √− g (cid:34) R − ∂ µ φ∂ µ φ − m φ − µ (cid:18) RR (cid:19) φ (cid:35) , (1)where M stands for the Planck mass and, for definite-ness, we have chosen the scalar field’s kinetic energy pa-rameter used in Ref. [12] to be equal to one. The scalarfield’s mass is m whereas µ is a mass parameter, takenin [12] to be of the order ∼ ∼ GeV, but which wetake here to be of the order of the GUT scale, namely ∼ GeV for reasons to be given below. The scalar cur-vature R has been identified with the Planck curvature M ∼ (GeV) .A very important step before examining the conse-quences of having such a peculiar model is to isolate thescalar field’s potential, which can be read off from (1) as, V = 12 m φ + 12 µ (cid:18) RR (cid:19) φ . (2)Then a straightforward computation shows that the min-imum of this potential occurs for a value φ of the scalarfield given by [12] (cid:18) RR (cid:19) φ = − m φ µ ln RR . (3)With this identity at hand, we can go on to exam-ine what Hubble flow would be produced, both in theearly and late-time evolution of the Universe, wheneverthe field φ settles down at the bottom of its potentialcurve specified by this identity. For that purpose weshall take up the Friedmann equation given in Eq. (13)of Ref. [12] for a flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) Universe: µ φR (cid:18) RR (cid:19) φ ˙ H + (cid:34) µ φR (cid:18) RR (cid:19) φ (cid:35) H = m φ µ (cid:18) RR (cid:19) φ + ˙ φ µ H dd t (cid:20) φR ( R/R ) φ (cid:21) , (4) where H = ˙ a/a is the Hubble parameter correspondingto the scale factor a ( t ) of the flat FLRW metric. Giventhe high non-linearity of this dynamical equation, how-ever, we will focus mainely on the orders of magnitude.In order therefore to make the analysis more tractablewe shall simplify the above equation and write it in thefollowing form, µ φR (cid:18) RR (cid:19) φ ˙ H + (cid:34) α + µ φR (cid:18) RR (cid:19) φ (cid:35) H = m φ βµ (cid:18) RR (cid:19) φ , (5)where α and β are dimensionless constants of order unity.To obtain the latter form, we have transposed the term˙ φ / φ is of the order H , and hence, thequadratic term ˙ φ / H /
6. Also, theterm µ H dd t [ φR ( R/R ) φ ] being of the same order as theterm µ ( R/R ) φ /
6, both present in the right-hand sideof (4), we have approximated the former by the latterand multiplied by the factor of order unity β . We shallnow examine the consequences of equations (3) and (5)successively for the early and late-time expansion of theUniverse.As for the early Universe, due to the prevailing highcurvatures, the Ricci scalar R might be taken to be ofthe order of the Planck curvature R making the ratio R/R very close to unity and hence the logarithm in (3)very small. Furthermore, as argued in [12], the scalarfield starts out very close to the origin, φ (cid:28)
1, makingthe term (
R/R ) φ in the left-hand side of (3) very closeto unity. This could only be consistent with the right-hand side however if the order of magnitude of the mass m does not exceed the order of magnitude of µ . Keep-ing only the dominant terms in (5) we would then beleft with H ∼ µ , which implies that the Hubble flowsquared H at the inflationary period would be of theorder µ as already showed in [12]. Therefore the natu-ral order of magnitude of µ could reach up to the GUTscale ∼ GeV according to what is required from theinflationary scales [16, 17].Let us now use the two previous equations to exam-ine the late-time expansion of the Universe, for which weshall take the Ricci scalar R of the order of the actu-ally observed Hubble parameter H ∼ − (eV) . Inthis case the ratio R/R becomes of the order ∼ − .Therefore, for values of φ of the order ∼ − , as givenin [12] in Planck unites, the required mass m to satisfy(3) should be absurdly big (of the order ∼ eV [12])but moreover this would be in conflict with the late-timeexpansion. Indeed, keeping only the leading terms in (5)in this case would imply again that H ∼ µ . In orderto find the right estimate of φ at late times, and hence,the required mass m let us substitute (3) into (5). Thelatter then reads µ φ R (cid:18) RR (cid:19) φ ˙ H + (cid:34) α + µ φ R (cid:18) RR (cid:19) φ (cid:35) H = µ (cid:18) RR (cid:19) φ (cid:18) β − φ RR (cid:19) . (6)Now it is clear that, given the orders of magnitude of µ and R chosen above, what is needed is to have the term( R/R ) φ tame the order of magnitude in the right-handside of the above equation coming from the factor µ .This could be achieved only if φ ∼
1. When substitutingthis back into (3) we find in fact a very small but non-vanishing value for the scalar field’s mass: m ∼ − eV.This value is unnaturally small for a scalar field and hencethe model is plagued with the usual fine-tuning problem.Actually, a massless scalar field could very well be com-patible with (5). In that case, the resulting Hubble pa-rameter for values of φ very close to the origin wouldagain be given by H ∼ µ . Then, as φ departs fromthe origin, the term µ φ ( R/R ) φ /R becomes graduallynegligible and (5) would eventually reduce to H ∼ µ (cid:18) RR (cid:19) φ . (7)This would give the observed value H for the value φ ∼ .
95 of the scalar field. The issue in this case how-ever lies in the fact that for a massless scalar field φ , thepotential (2) would not have any minimum for finite val-ues of φ . Therefore, the scalar field would always remainmassless and the model becomes vulnerable against thefifth-force problem, unless one imposes a non-couplingbetween the scalar field φ and matter, a property whichis not desirable if one wishes to explain the creation ofmatter and radiation at the end of inflation.A possible solution to this issue would be to modify thescalar field’s potential in such a way that the minimumcondition (3) combined with (5) would allow us either (i)to have a reasonable and not fine-tuned mass m for thefield φ if the latter is chosen to be a massive scalar field,or (ii) to make the scalar field acquire a mass throughit’s coupling with gravity even if it was initially chosento be massless. Such a modification, as we shall see inthe next section, arises naturally by taking into accountthe quantum corrections. III. THE EFFECTIVE POTENTIAL FROMONE-LOOP QUANTUM CORRECTIONS
In this section we shall examine how radiative correc-tions modify the effective potential of the model. First,recall that quantum corrections to the potential insidea Lagrangian L are obtained from the following formalexpansion around the classical value φ c ( x ) of the scalarfield given by the vacuum expectation value φ c ( x ) = ( (cid:104) | φ ( x ) | (cid:105) / (cid:104) | (cid:105) ) J of the operator φ ( x ) in the presenceof the source J ( x ) (see e.g. [18]), (cid:90) ( L [ φ ] + Jφ ) = (cid:90) ( L [ φ c ] + Jφ c ) + (cid:90) η (cid:18) δ L δφ + J (cid:19) φ c + 12 (cid:90) η δ L δφ (cid:12)(cid:12)(cid:12) φ c + 13! (cid:90) η δ L δφ (cid:12)(cid:12)(cid:12) φ c + 14! (cid:90) η δ L δφ (cid:12)(cid:12)(cid:12) φ c + ..., (8)where η ( x ) is the perturbation of φ ( x ) = φ c ( x ) + η ( x )around the classical value φ c ( x ) and all the functionalderivatives of the Lagrangian with respect to the field φ are evaluated at the classical configuration φ c . There-fore, before we proceed with the usual method of findingthe effective potential based on the effective action [18]we should first ascertain that such expansion would beconvergent for the specific Lagrangian we have.At first sight though it seems that the high non-linearity of the non-minimal coupling in the potential (2)would render the expansion badly divergent and henceunjustified. When substituting the classical potential (2)inside L in the above expansion, however, one finds sim-ply the following formal expansion (cid:90) ( L [ φ ] + Jφ ) = (cid:90) ( L [ φ c ] + Jφ c ) + (cid:90) η (cid:18) δ L δφ + J (cid:19) φ c + 12 (cid:90) η (cid:34) − ∂ + m + µ (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35) + µ (cid:90) η (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c + µ (cid:90) η (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c + ... (9)Then it is clear that, by redefining η → η (cid:48) = η ln( R/R ),the expression becomes a convergent series in the field η (cid:48) .So the above expansion could be truncated as usual at thequadratic order in η (cid:48) making the use of the definition ofthe effective action, and hence of the radiative potentialwith contributions restricted to the one-loop diagrams,amply justified.The vertex Feynman diagrams giving rise to such anon-minimal coupling of the scalar field φ with gravityare depicted in the figure below. + + + … = ( ) ( ) ( ) ( ) ( ) + + + … FIG. 1. Vertex Feynman diagrams giving rise to the non-minimal coupling of the scalar field φ with gravity. Using these vertex diagrams, the one-loop radiativecorrections to the effective potential, at which we restrainourselves in this paper, are given by the following dia-grammatic expansion: + + + … = ( ) ( ) ( ) ( ) ( ) + + + … FIG. 2. Diagrammatic expansion of the one-loop radiativecontributions to the scalar field’s potential.
Therefore, the radiative potential coming from thisone-loop expansion would be given by the following inte-gral (see e.g. [19]) V = iM (cid:90) d k (2 π ) ∞ (cid:88) n =1 n (cid:34) µ (cid:18) ln RR (cid:19) ( R/R ) φ c k − m + i(cid:15) (cid:35) n (10)in which we have used the fact that the Feynman propa-gator is massive with mass m . After a Wick rotation, thesum may be transformed into a logarithm and the aboveintegral becomes116 π M (cid:90) Λ0 d k E k E ln (cid:34) µ (cid:18) ln RR (cid:19) ( R/R ) φ c k E + m (cid:35) , (11)where Λ is the ultraviolet cut-off and k E is the Euclideanmomentum. The evaluation of this last integral gives theradiative potential as, V = µ π M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c × (cid:32) Λ − ln Λ M (cid:34) m + µ (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35)(cid:33) + 164 π M (cid:34) m + µ (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35) × ln (cid:34) m M + µ M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35) − m π M ln m M . (12)Therefore, the effective potential when quantum cor-rections are included becomes V eff = m φ + µ (cid:18) RR (cid:19) φ c + V + δV, (13)where δV represents the contribution of the counterterms that should be added in order to cancel the diver-gences that rise in V when the limit Λ → ∞ is performed.Comparing (12) and (13), we deduce that the counterterms contribution should be of the following form, δV = A (cid:18) RR (cid:19) φ c + B (cid:18) RR (cid:19) φ c , (14)where, A = − µ π M (cid:18) ln RR (cid:19) (cid:18) Λ − m ln Λ M (cid:19) , (15) B = µ π M (cid:18) ln RR (cid:19) ln Λ M . (16)Note that although the form of the potential rising fromthe quantum corrections has a new form which we do notfind in the original classical potential (2), the number ofcounter terms required to cancel the divergences agreeswith the number of free parameters the original potentialhad, namely, the two mass parameters m and µ .Substituting these counter terms back inside the cor-rected potential (13), the effective potential V eff becomesfinite and reads V eff = m φ c + µ (cid:18) RR (cid:19) φ c − m π M ln m M + 164 π M (cid:34) m + µ (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35) × ln (cid:34) m M + µ M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35) . (17)We can see straightaway from (16) that the extra termsbrought by the radiative modifications to the classical po-tential could be as high as the original terms we startedwith thanks to this highly non-linear coupling of thescalar field with curvature. This has been made possi-ble by making the second mass parameter µ contributesubstantially and in varying strengths depending on thedifferent gravitational environments. Before examiningin detail in the next section the consequences of such de-pendence on the environment for the early and late-timeevolution of the Universe, we shall first write down herethe condition that gives the minimum of this effectivepotential and determine the expression of the resultingeffective mass.The minimum of the effective potential (17) occurs at( ∂V eff /∂φ ) φ = 0. Differentiating expression (17) oncewith respect to φ yields the following constraint, m φ + µ RR (cid:18) RR (cid:19) φ + µ π M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ × (cid:34) m + µ (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ (cid:35) × (cid:32)
12 + ln (cid:34) m M + µ M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ c (cid:35)(cid:33) = 0 . (18)The effective mass of the scalar field can be found by writ-ing m = (cid:0) ∂ V eff /∂φ (cid:1) φ . Differentiating twice the ef-fective potential (17) and then using the constraint (18),gives m = m (cid:18) − φ ln RR (cid:19) + µ π M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ × (cid:32)
32 + ln (cid:34) m M + µ M (cid:18) ln RR (cid:19) (cid:18) RR (cid:19) φ (cid:35)(cid:33) . (19) IV. CONSEQUENCES OF THE EFFECTIVEPOTENTIAL
Now that we have obtained all the necessary ingredi-ents, we can look again at the consequences this effectivepotential brings to the dynamics of the Universe. Tobegin with, we shall first examine in which way the min-imum condition (3), found for a classical filed, would bemodified here. Using (18), we learn that the analogue of(3) is given by (cid:18) RR (cid:19) φ = − m φ ξµ ln RR , (20)where the quantity ξ is given by the following identity ξ = 1 + (ln r ) π M (cid:18) m + µ (ln r ) x (cid:19) × (cid:18)
12 + ln (cid:20) m M + µ (ln r ) M x (cid:21)(cid:19) , (21)in which we set r = R/R and defined x = ( R/R ) φ .First, for very high curvatures r ∼
1, as might be thecase during the early Universe, we have ln r → ξ ∼
1, making thereby (20) reduce to its clas-sical counterpart (3) whereas the effective potential (17)recovers at this very high curvature its classical form (2).For very low curvatures r (cid:28)
1, however, thanks tothe presence of this new factor ξ , the denominator in theright-hand side of (20) could become big enough that themass in the numerator would not have to be as small aswhat we found using the condition (3). Therefore thisnew factor ξ is exactly what would allow us to avoidthe fine-tuning problem we found in the classical regimewhere ξ reduces to unity. Moreover, thanks to this addi-tional factor the scalar field could consistently be mass-less in (20) provided only that ξ = 0. Let us then firstdiscuss the consequences of this change for the case of amassless scalar field and then examine the massive scalarfield case. A. A Massless Scalar Field
In the case m = 0, the minimum condition (20) yields ξ = 0 which, in turn, imposes the following constraint12 + ln (cid:20) xµ (ln r ) M (cid:21) = − π M µ (ln r ) x . (22)Substituting this in the effective mass expression (19),gives m = x µ (ln r ) π M (cid:20) − π M µ (ln r ) x (cid:21) . (23)Therefore, although being massless initially, the scalarfield acquires a huge mass coming from the mass param-eter µ through geometry. Indeed, for low curvatures, byassuming R ∼ H we have r ∼ − . Substitutingthis into (22) and solving numerically for x gives us ac-tually two solutions, x ∼ .
014 as well as x ∼
15. Thefirst solution is achieved for φ ∼ .
01 whereas the sec-ond solution is achieved for φ ∼ − .
01. Only the secondsolution gives a positive effective mass when using (23)though. The first one would give a tachyon which meansthat the vacuum becomes unstable whenever φ departsfrom the origin towards the positive values. However,substituting the second solution into (23) we find theremarkable fact that the mass acquired by the initiallymassless field when the latter departs from the origintowards the left of the vertical axis is of the order of ∼ GeV.Having a different effective potential, the dynamicalequation for the Hubble parameter around the minimum φ of the effective potential will also be modified. Inorder to see this, first note that in terms of the effectivepotential, equation (5) actually reads,2 ∂V eff ∂R (cid:12)(cid:12)(cid:12) φ ˙ H + (cid:18) α + 2 ∂V eff ∂R (cid:12)(cid:12)(cid:12) φ (cid:19) H ∼ V eff (cid:12)(cid:12)(cid:12) φ . (24)Next, substituting the above constraint (22) into the ef-fective potential V eff given by (17) as well as into itsderivative ∂V eff /∂R , the dynamical equation (24) readsmore explicitly as follows, − xµ R ln r ˙ H + (cid:18) α − xµ R ln r (cid:19) H ∼ xµ (cid:20) − xµ (ln r ) π M (cid:21) . (25)Keeping only the leading terms in the left-hand side,this equation acquires for x ∼ .
014 and x ∼ H + H ∼ ∓ R ln r , respec-tively; the upper-sign corresponding to the first solu-tion. Using the fact that in the flat FLRW geometry R = 6 ˙ H + 12 H , these two possibilities are equivalentrespectively to H ∼ ± R ln r . Thus, we see that onlythe second solution corresponding to the minus sign,which is also the one that gives a stable vacuum as wesaw, is acceptable. This, in turn, would give the ob-served Hubble parameter H ∼ − (eV) providedthat R ∼ − (eV) . Note that although the solutions x and x obtained above for equation (22) were foundusing the assumption R ∼ H , injecting the aboveslightly different estimate for R again inside (22) doesnot alter significantly the solutions x and x previouslyobtained because R intervenes in (22) only inside a log-arithm. Therefore the order of magnitude for H thatwill be obtained from (25) with these new solutions willbe the same. B. A Massive Scalar Field
In the case m (cid:54) = 0, the full minimum condition (20),with ξ given by (21), also admits the previous two so-lutions x ∼ .
014 and x ∼
15. This being the casefor all values of m up to the GUT scale. This is due tothe fact that the presence of µ multiplied by ln r over-shadows the effect of m inside ξ . When substituting thefull minimum condition (20) into expression (19) of theeffective mass, the latter becomes, m = m (1 − φ ln r ) + x µ (ln r ) π M × (cid:20) − π M xµ (ln r ) · m φ + xµ (ln r )2 m + xµ (ln r ) (cid:21) . (26)First, we see that for the solution x ∼ .
014 the vac-uum will be unstable because the effective mass squaredcomes out negative again, whereas for the second solution x ∼
15 the vacuum will be stable and the effective masswill be of the order ∼ GeV, as found previously forthe massless case. These results being true for any initialmass m below the GUT scale. Inspecting the contentof the square brackets in the above expression, however,we find that this time there is the possibility of havinga stable vacuum for both positive and negative values of φ , though at the price of starting with an initial mass m that exceeds the GUT scale by one order of magnitudeat least.Finally, computing the effective potential V eff as wellas the derivative ∂V eff /∂R using the minimum condition(20) we find, respectively, V eff (cid:12)(cid:12)(cid:12) φ = m φ xµ − m π M ln m M − m xµ (ln r ) (cid:20) m φ + xµ ln r (1 + φ ln r )2 (cid:21) − π M (cid:20) m + xµ (ln r ) (cid:21) , (27)and ∂V eff ∂R (cid:12)(cid:12)(cid:12) φ = − xµ R ln r − (2 + φ ln r ) m φ R (ln r ) . (28)When inserting these new expressions inside (24) we ob-tain again, after keeping only the leading terms, the same approximate dynamical equation (25) for all values of theinitial mass m below the GUT scale. Then, the analy-sis and the conclusions drawn previously for a masslessscalar field are also valid here. For an initially massivescalar field whose mass exceeds the GUT scale by at leastone order of magnitude, for which the vacuum becomesstable on both side of the vertical axis, the terms con-taining m dominate in both (27) and (28). Therefore,the latter expressions reduce to V eff (cid:12)(cid:12) φ ∼ m φ / ∂V eff /∂R (cid:12)(cid:12) φ = − m φ / ( R ln r ), respectively. Insertingthese inside (24) we learn that the result H ∼ − R ln r will be valid for both positive and negative values of φ . V. DISCUSSION
The presence of the scalar field inside the power of theRicci scalar in the (varying power)-law modified gravitymodel provides the latter with the main attractive fea-ture of usual power-law models, namely, the unification ofthe early and late-time expansions of the Universe. Un-fortunately though, the model also suffers from the usualfine-tuning problem familiar to anyone relying on scalarfields to explain the order of magnitude of the cosmo-logical constant. The peculiar form of the scalar field’spotential in this model, however, gives rise to substantialdifferences in the physics when the scalar field is treatedquantum mechanically. The major modification broughtby implementing the radiative corrections is the fact thatnot only the mass of the scalar field need not be finelyadjusted to produce the effects of a tiny cosmologicalconstant, but the mass of the field itself vary substan-tially, depending on the curvature of the surrounding en-vironment. For low curvatures, the mass acquired by thescalar field is so big that the model could not in principlefail any fifth-force test. These conclusions being true forwhatever initial mass the scalar field happens to have,nothing a priori prevents us from identifying this scalarfield with the fundamental Higgs field.Another new feature found here, though, is the factthat, in contrast to the case one finds when the fieldis treated classically, the vacuum is not symmetric withrespect to the vertical axis anymore. An asymmetry be-tween the positive and negative values of the scalar fieldarises whenever the radiative corrections are taken intoaccount. This asymmetry makes the vacuum unstable inthe positive direction and stable in the negative direc-tion. This asymmetry arises for all values of the initialmass of the scalar field that are below the GUT scale.For values exceeding the GUT scale by one order of mag-nitude and beyond, though, the vacuum becomes stablefor both directions. Nevertheless, just by choosing minusthe absolute value of the scalar field for the power of theRicci scalar we could make the vacuum stable again forboth directions.In our investigation of the consequences of combiningvacuum fluctuations of the scalar field with its peculiarnon-minimal coupling with gravity we have completelyignored in this paper the matter sector and the eventualcoupling of the matter fields with the scalar field. An im-portant question arises then, which is: What differencewould a coupling of the scalar field with matter bring toour results and conclusions. Although a rigorous analy-sis of this question lies beyond the scope of the presentpaper, hints for a qualitative answer could be found byrecalling that whenever radiative corrections are takeninto account, one obtains effective parameters dependingon the precise structure of the interactions present in themodel [18, 19]. Since in our model only the scalar fieldcouples to gravity, the effects of vacuum fluctuations ofthe former due to its non-minimal coupling with the lat-ter will appear simply as additive contributions to theusual calculations one performs in the absence of gravity.Indeed, taking into consideration a coupling of φ withmatter fields ψ within our model amounts to adding aninteraction term I ( φ, ψ ) and the corresponding counterterm δI ( φ, ψ ) in the sum on the right-hand side of expres-sion (13), giving the contributions of radiative correctionsto the effective potential in the absence of gravity. This,in turn, would simply introduce in the right-hand sideof (17) an effective interaction potential I eff ( φ, ψ ). Theconsequences of having such additional term could befound by examining how our main equations (20), (26),(27) and (28) would be modified. First, the minimumcondition for the new effective potential could simply befound by performing inside (20) the following substitu-tion m φ → m φ + I (cid:48) eff ( φ , ψ ), where the prime denotespartial differentiation with respect to φ . Therefore, dueto the presence of the huge parameter µ , as explainedin Sec. IV B, the solutions x and x found there for thevariable x would still be valid after performing the pre-vious substitution. Similarly, the effective mass (26), theeffective potential (27), and the partial derivative (28) will acquire, respectively, the following new expressions: m = I (cid:48)(cid:48) eff ( φ , ψ ) + (cid:2) m φ → m φ + I (cid:48) eff ( φ , ψ ) (cid:3) ,V eff (cid:12)(cid:12)(cid:12) φ = I eff ( φ , ψ ) + (cid:2) m φ → m φ + I (cid:48) eff ( φ , ψ ) (cid:3) ,∂V eff ∂R (cid:12)(cid:12)(cid:12) φ = (cid:2) m φ → m φ + I (cid:48) eff ( φ , ψ ) (cid:3) , where the square brackets mean: the same expressionsas those found before but rewritten with the indicatedsubstitution. Now, since, as we saw, the solutions x and x one obtains for the variable x from the new min-imum condition will remain the same, the effect the ra-diative corrections will induce on the effective mass willalso remain the same as those discussed in Sec. IV B.Indeed, as explained below Eq. (26), the latter formulais not sensitive to the initial mass m of the scalar fieldone starts with, and hence, will also not be sensitive tothe substitution m φ → m φ + I (cid:48) eff ( φ , ψ ). Next, theabove new expressions for V eff (cid:12)(cid:12) φ and ∂V eff /∂R (cid:12)(cid:12) φ are theones that will determine via (24) the dynamical equa-tion for the Hubble parameter in the presence of φ -fieldcouplings with matter. However, since the substitution m φ → m φ + I (cid:48) eff ( φ , ψ ) inside (27) and (28) will noteffect the terms containing µ (the second and the lasttwo terms in the case of (27) and the first term in thecase of (28)), and since only such terms dominate, usingthe above new expressions for V eff (cid:12)(cid:12) φ and ∂V eff /∂R (cid:12)(cid:12) φ will not effect our physical conclusions drawn from thedynamical equation (25) of the Hubble parameter in theabsence of matter couplings with the scalar field φ .Finally, the fact that not only the mass acquired by thescalar field due to its non-minimal coupling with gravityis huge but the acquired value itself depends on the en-vironment, does not exclude the scalar field from beingamong the serious candidates for the dark matter sector. ACKNOWLEDGMENTS
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