Über-Gravity and the Cosmological Constant Problem
aa r X i v : . [ g r- q c ] M a y ¨Uber-Gravity and the Cosmological Constant Problem Nima Khosravi ∗ Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran (Dated: May 30, 2018)Recently, the idea of taking ensemble average over gravity models has been introduced. Basedon this idea, we study the ensemble average over (effectively) all the gravity models (constructedfrom Ricci scalar) dubbing the name ¨uber-gravity which is a fixed point in the model space. The¨uber-gravity has interesting universal properties, independent from the choice of basis: i ) it mimicsEinstein-Hilbert gravity for high-curvature regime, ii ) it predicts stronger gravitational force for anintermediate-curvature regime, iii ) surprisingly, for low-curvature regime, i.e. R < R where R isRicci scalar and R is a given scale, the Lagrangian vanishes automatically and iiii ) there is a sharptransition between low- and intermediate-curvature regimes at R = R . We show that the ¨uber-gravity response is robust to all values of vacuum energy, ρ vac when there is no other matter. So asa toy model, ¨uber-gravity, gives a way to think about the hierarchy problems e.g. the cosmologicalconstant problem. Due to the transition at R = R there is a chance for ¨uber-gravity to bypassWeinberg’s no-go theorem. The cosmology of this model is also promising because of its non-trivialpredictions for small curvature scales in comparison to ΛCDM model. I. INTRODUCTION:
A century ago Einstein introduced the cosmologicalconstant (CC) to address static universe [1] which be-came his biggest blunder after Hubble’s discovery of ex-panding universe. On the other hand, from the viewpointof particle physics it is well-known that there is a non-vanishing vacuum energy, ρ vac , which has no effect onmost of particle physics’ calculations. But in presence ofgravity, it predicts an inflating universe which is not com-patible with the observations before 1998. Accordingly itraised a question: why the vacuum energy has no effecton gravity? which is known as old CC-problem. Dataacquired by Sueprnovae observations in 1998 [2] and re-cent Plank data [3] implies a tiny value for CC, whichshall be 120 orders of magnitude smaller than ρ vac ; thisprediction sometimes will refer to as “the worst theoreti-cal prediction in the history of physics” [4]. To solve thisdiscrepancy a fine-tuning is required which is known asthe new CC-problem (CCP) [5].There are three different approaches to solve the CCP: i ) modifying the Einstein-Hilbert (EH) model in a waythat gravity becomes insensitive to ρ vac [6], ii ) revisingfield theory calculation of ρ vac [7] and iii ) connectingthe CCP (which is in IR regime) to UV-completion ofgravity [8]. An idea in the context of modifying gravityis degravitation which proposes switching off the gravityfor very large wavelengths and consequently filters ρ vac [9]. It is worth mentioning that some believes old CCPshall be addressed before moving to new CCP. This ideais supported by ’tHooft conjecture: if the gravitationaleffects of ρ vac can be canceled by a symmetry then atiny fluctuation from this symmetric situation is natural.Supersymmetry is an idea in this direction assuming thepresence of a boson particle for each fermion consequently ∗ Electronic address: [email protected] paving the way for a mechanism to eliminate ρ vac [10].In this paper, we will study the CCP within thecontext of ¨uber-modeling introduced in [11]. We try toshow that ¨uber-modeling of (effectively) all gravitationalmodels eliminates gravity for low-curvature regimeswhich can be interpreted as degravitation. Our modelcoincides with the EH model in high-curvature regimealthough, there is an intermediate-curvature regimewhere gravity is stronger than the standard EH model.It will be shown that our model is not sensitive to valueof vacuum energy, ρ vac , thanks to a sharp (but con-tinuous) transition from low- to intermediate-curvatureregime. Interestingly, this means there is no need offine-tuning and the CC is “natural”[22]. II. ¨UBER-GRAVITY:
In [11], we introduced an idea based on ensemble aver-age of models within the context of gravity. According tothis idea, we start with the space of all consistent modelsof gravity, M , and then take an ensemble average overall models. This idea is inspired by statistical mechan-ics which employed in a very different context. In [12],Arkani-Hamed et al. employed a similar idea to addressthe hierarchy problem in particle physics. They mentionthat in principle an average should be taken on all pos-sible models but for simplicity, they just considered thestandard model with different Higgs masses. The mainidea behind addressing the hierarchy problem in both [11]and [12] is a dynamical mechanism which can make ourcurrent model dominant. In [12] this mechanism is real-ized by introducing a new field, named reheaton, which“deposits a majority of the total energy density into thelightest sector” (which is our observed standard model ofparticle physics). In our ¨uber-modeling this mechanismis given by the assigned probability to each model whichis introduced by hand at this step. On the other hand ouridea can be seen as a realization of the Tegmark’s math-ematical universe idea [13], specially when he argues “alllogically acceptable worlds exist”. In [11] we assumedall the theoretically possible (gravity) models play a rolein the final model (of gravity). To make ¨uber-modellingidea applicable, we assigned a Lagrangian to each modeland define (ensemble) average of all the Lagrangians asfollowing: L = (cid:18) N X i =1 L i e − β L i (cid:19)(cid:30)(cid:18) N X i =1 e − β L i (cid:19) , (1)where β is a free parameter and model space is repre-sented by M = {L i | i ∈ { , N }} while N is number ofall possible models. We emphasize that the above for-mulation is inspired by ensemble average procedure instatistical mechanics. However our suggested probabili-ties are fundamentally different with what is in statisticalmechanics. As it is obvious from (1) that we use the La-grangian in the exponent while in statistical mechanics itis E i which is energy of each state. The above Lagrangiancan be beautifully re-written in a more compact form as L = − ddβ ln Z , Z = N X n =1 e − β L n (2)which reminds us of the partition function and its rela-tion to energy. In [11] we assumed M = { R, G } where R is the Ricci scalar and G is the Gauss-Bonnet term. Inthis paper we generalize the model space to (effectively)all the gravity models based on curvature scalar: all an-alytic f ( R ). Schematically we can write correspondingpartition function as Z = X f ( R ) e − βf ( R ) . (3)Here we deal with analytic functions of f ( R ) and we canarbitrarily choose the basis. We are working with M = { R n | ∀ n ∈ N } . The ensemble averaged Lagrangian takesthe following form L = (cid:18) ∞ X n =1 ¯ R n e − β ¯ R n (cid:19)(cid:30)(cid:18) ∞ X n =1 e − β ¯ R n (cid:19) , (4)where ¯ R = R/R . This model, which belongs to f ( R )family, has two free parameters: R and dimensionless β . In FIG. 1, the above Lagrangian is plotted for β = 1.The above Lagrangian belongs to the f ( R ) family andeffectively is ensemble average of all possible models ofgravity based on the curvature tensor. There is a pos-sibility to add a constant to each f ( R ), i.e. workingwith R n − λ n as our basis. We plotted its Lagrangianin FIG. 2 for λ n = λ , which mimics GR plus a cosmolog-ical constant. This model with additional constant hasbeen studied in [14] with very interesting observationalconsequences. But in this work we focus on (4) to studythe theoretical properties of the model. Note that in a FIG. 1: Blue line is our Lagrangian (4) where we do sum up to N = 1000 (It is easy to see that for larger N ’s the plot is prac-tically the same.) and yellow dashed line shows the EH actionfor comparison. The universal behavior of our model is obvi-ous: i) in high-curvature regime our model coincides with theEH model, ii) in intermediate-curvature regime where grav-ity is stronger than the EH model, iii) for R < R gravityvanishes and iiii) there is a sharp transition at R = R .FIG. 2: We plotted the ¨uber-gravity Lagrangian with R n − λ n as our basis. We assumed λ n = R n and obviously our modelmimics GR plus cosmological constant for high-curvatureregime. general case we could work with all possible linear com-binations e.g. L = α R + α R with two constants α and α . It is easy to observe that adding such terms donot change the interesting aspects of the model, so with-out loss of generality we focus on the above Lagrangian.Only difference will be in the form of the Lagrangian overthe intermediate-curvature regime while for both high-and low-curvature regimes nothing is changed. Even inthe intermediate-curvature regime the general predictionis a stronger gravity compared to the EH model. How-ever, the form of our model in the intermediate-curvatureregime is sensitive to the parameter β . As an exampleFIG.3 shows the Lagrangian (4) for β = 0 .
01 which repre-sents a very different behavior in intermediate-curvatureregime.In summary, the ¨uber-gravity model has the following
FIG. 3: Blue line shows our Lagrangian (4) where β = 0 . β . However we emphasize that the universal fea-tures are the same as β = 1 case, see FIG. 1. However we arenot interested in β < . universal properties (independent to the choice of basisi.e. M ): • for high-curvature regime it reduces to the EH ac-tion, • for intermediate-curvature regime it predicts astronger gravity than the EH model, • it is vanishing for low-curvature regime ( R < R ), • there is a sharp transition at R .It is worth mentioning that adding the ¨uber-gravity(4) to M and re-employing the ¨uber-modelling procedurecannot affect above universal features [23]. This is a verysignificant property since it means ¨uber-gravity is a fixedpoint the model space of f ( R ) models and this makes itremarkable. In addition above properties are shared forall values of β .It is crucial to discuss about the stability of our model(4) which belongs to f ( R ) models. A dark energy f ( R )model is viable if it satisfies f ′ ( R ) > f ′′ ( R ) > R ≥ R T > R T is the today value of Ricciscalar [15]. In our scenario, R will be the late timecosmological constant, so R T → R +0 in the presence ofmatter fields. So to show the stability of our model (4) weneed to show that f ′ ( R ) > f ′′ ( R ) > R > R .It is obvious from FIG. 4 and FIG. 5 that there is alwaysan R > R where stability conditions can be satisfied.The interesting point is that for β > . f ′′ ( R ) > R > R (see solid line in FIG. 5) otherwise the conditionis satisfied for a larger value than R . It is importantto mention that the sharp transition at R = R maybehaves like a discontinuity which should be take intoaccount very seriously. FIG. 4: f ′ ( R ) is shown for our model (4) where β = 1 thoughthis is not sensitive to β ’s value too much.FIG. 5: f ′′ ( R ) is shown for β = 1 . β = 1 in solid blueline and yellow dashed line respectively. It is obvious thatsatisfaction of stability condition, i.e. f ′′ ( R ) >
0, dependson the value of β . For β > . f ′′ ( R ) is always positivefor R > R . However in the presence of the matter fieldwe expect to have R t > R where R plays the role of thelate time cosmological constant. So one can imagine that ourmodel should satisfy the stability conditions even for β ∼ III. ¨UBERGRAVITY AND THE CCP:
In this section we study properties of ¨uber-gravitymodel (4). We will show this model is not sensitive tothe value of the vacuum energy under a specific circum-stances.
A. Equations of Motion:
For a general f ( R ) model the equation of motion is asfollow [15]: Σ µν = κ T µν whereΣ µν = F ( R ) R µν − f ( R ) g µν + (cid:18) g µν (cid:3) − ∇ µ ∇ ν (cid:19) F ( R ) , F ( R ) = ∂f∂R and T µν is the energy-momentum tensor. Inour case F ( R ) can be written as: F ( R ) = ∞ X n =1 R n e − βR n ! ∞ X n =1 βnR n − e − βR n ∞ X n =1 e − βR n ! − ∞ X n =1 (cid:16) βnR n − e − βR n − nR n − e − βR n (cid:17) ∞ X n =1 e − βR n . It is obvious from the above relations, our model isvery complicated for analytical calculations. In the nextsection we introduce a simplified model which shares allthe interesting properties of our model.
B. Simplified Model:
The following model has the same features in all thecurvature regimes [24] f ( R ) = ( ¯ R n R ≤ R ¯ R + e − ( ¯ R − . R < R (5)where for the limit n → ∞ the low-curvature regimeshares exactly the same feature with our model, seeFIG.1. Note that here the exponential term is addedphenomenologically in R < R region to recover theintermediate-curvature behavior which mimics (4) for β = 1 and λ n = 0. In practice by changing β one needs tore-calculate parameters in the exponent in simplified La-grangian (5). For our purpose, we focus on low-curvatureregime. It is easy to see that the equation of motion forthe low-curvature part is [16]:Σ µν = (6) n ¯ R n − ¯ R µν −
12 ¯ R n g µν + n R − (cid:18) g µν (cid:3) − ∇ µ ∇ ν (cid:19) ¯ R n − where ¯ R µν = R − R µν . Obviously for high-curvatureregime the model reduces to Einstein’s gravity. In thenext section based on this model we will show how thismodel can give us a proposal to resolve the CCP. C. An attempt to solve the CCP:
To address the CCP we need to take care of the vac-uum energy, ρ vac , in presence of gravity. To do this we need to recall that ρ vac is encoded in the trace of energy-momentum tensor, T . By looking at (6) it is easy to seethat the trace of equations of motion yields:( n −
2) ¯ R n + 3 n R − (cid:3) ¯ R n − = κ T. (7)We are interested in solutions like R = cte since for ourpurpose T is vacuum expectation value which is a con-stant. With this assumption the above equation reducesto RR = (cid:18) κ Tn − (cid:19) n . (8)For all T = 0; limits of equation (8) for n → ∞ results in R → R . This is a very interesting result which meansthe model’s response to the vacuum energy is robust i.e.gravity sector is not sensitive to ρ vac . In other words,the cosmological constant value R , shall be fixed onlyby observation without fine-tuning. Such will imply thatthe cosmological constant value is natural and the CCPcan be solved by this approach. More interestingly, thisresult is not valid for zero vacuum energy which meansparticle physics’ prediction for non-zero vacuum energyis crucial for our model. D. A subtlety:
Above argument contains a subtlety which we shallclarify herein. The point is that by definition T ∝ − ρ vac where ρ vac >
0; the negative sign is the origin ofproblem. In the EH model, trace of equation of motiongives − R = κ T hence R = κ ρ vac . But in our scenario(8) for any n > − ρ vac ) /n on the right handside of equation. Now the question is what is valueof ( − /n for n → ∞ ? For any given n there are n solutions in complex plane and none of them is exactlyone. For sure there is a solution which is as close aspossible to one but the infinitesimal difference alwayshas an imaginary part. This behavior is shared betweenboth simplified model (5) and the ¨uber-gravity (2). Toillustrate this fact, we plot the trace of equations ofmotion for ¨uber-gravity in FIG. 6. We believe that thisissue can be addressed by full analysis of the analyticalcontinuation of our model but its concrete study remainsopen for future investigations. Hereby we will try to givesome ideas which can resolve this problem and hopefullyguide us to a concrete proposal to solve the CCP. E. Towards a Proposal to Solve the CCP:
Here we will try to give two proposals to resolve theabove subtlety:I: Particle Physics approach: The negative sign in T = − ρ vac in standard particle physics is because of larger FIG. 6: Blue line is the trace of equation of motion in ¨uber-gravity where β = 1 and yellow dashed line shows the same forthe EH action. Obviously at R = R the trace goes asymp-totically to positive infinity. It is clear that for positive ρ vac there is only the EH’s solution while for any ρ vac < R = R is the solution. number of fermionic degrees of freedom compared tobosonic counterpart. String theory predicts new specieslike axions [17], which are candidates for dark matter [18].The axions are bosons which means if one calculates theaxion’s contribution to the vacuum energy then ρ vac canbe negative. As it is obvious from FIG. 6 for a negative ρ vac we have R = R as a solution. As mentioned above,this solution is not sensitive to any value of ρ vac whichmeans there is no need to fine-tuning.II: ¨Uber-Gravity approach: This approach suggests tomodify the gravity model. As it is clear from FIG. 6what we need is an asymptotic behavior with an oppositesign at R = R . For this purpose, we modify our model(2) phenomenologically by multiplying it by a hyperbolictangent function: L = tanh (cid:20) N ( ¯ R − (cid:21) × N X n =1 ¯ R n e − β ¯ R n N X n =1 e − β ¯ R n . (9)Obviously tanh function changes its sign at R = R andbehaves as a step function while N → ∞ . Using tanhfunction instead of step function makes our Lagrangiancontinuous which is useful for future purposes. For thismodel the trace of equation of motion is plotted in FIG.7. In this scenario for a positive ρ vac there are two dis-tinguishable solutions R = T and R = R . The R = T solution is exactly the EH solution which is not compati-ble with the late time observations where we have positiveacceleration (note that we have not assume an effectivecosmological constant in matter sector i.e. T µν ). But the R = R solution is the solution which is not sensitive to ρ vac ’s value and makes the CC natural. FIG. 7: Blue line is the trace of equation of motion in the mod-ified model (9) where β = 1 and yellow dashed line shows thesame for the EH action. Obviously, at R = R the trace goesto negative infinity and comes up to positive infinity asymp-totically. In this scenario for ρ vac > R = R and R = ρ vac . The R = ρ vac solution is same as theEH solution which is not in agreement with the observations.But, the solution R = R is the one which can be compatiblewith observations. This solution is not sensitive to the valueof ρ vac which means there is no need to fine-tuning and theCC is natural. F. A comment on Weinberg’s no-go theorem:
In [19], Weinberg shows that, under some general as-sumptions, to cancel the large vacuum density one needsa new fine-tuning (for a short review see [20]). ThoughWeinberg’s argument is very general but one should becareful to apply it to ¨uber-gravity model. In ¨uber-gravitywe have a very sharp transition from zero to a non-zerovalue at R = R and the Lagrangian is vanishing for R < R . These properties make ¨uber-gravity beyondWeinberg’s no-go theorem assumptions. It is easy toshow that in Einstein frame ¨uber-gravity in R < R willcause all the masses goes to zero[25] which means ourmodel bypasses Weinberg’s argument. Though Wein-berg’s no-go theorem argues this solution does not de-scribe our real world [20] but we should emphasize thatin ¨uber-gravity due to having two different regimes thereis a chance to resolve this issue. I.e. R < R regimesolves the CCP and R > R describes the real world.This needs more considerations which remains for futurework. IV. DISCUSSIONS AND CONCLUDINGREMARKS:
Based on the ¨uber-modelling idea [11] we calculatedensemble average of (effectively) all gravity models. Thisprocedure results in an effective Lagrangian (4) whichhas interesting features shown in FIG.1. The final La-grangian (4), ¨uber-gravity, is a “fixed point” in the modelspace of f ( R ) models which makes it very special in thisspace. Below we briefly summarize ¨uber-gravity proper-ties: • There is a universal prediction for our model whichdoes not depend on the choice of β : At high-curvature regime it is the EH gravity which can becrucial to address the local tests. There is alwaysa stronger gravity in an intermediate-curvatureregime. For low-curvature regime, R < R , thegravity is vanishing which can be interpreted asdegravitation. Consequently, a sharp transition oc-curs at R = R . • For any (non-zero) value of ρ vac our model predictsan exact deSitter solution i.e. R = R . This is aninteresting result since observations support deSit-ter background. • If ρ vac = 0 then our model gives a Minkowskispacetime instead of deSitter spacetime which isnot compatible with observations. • We assume our model works effectively up toPlanck mass scale i.e. quantum gravity scale. Thismeans, up to that scale, we do not need to takecare of quantum gravity corrections and may con-ceive our model as a classical field theory. Thismeans even loop corrections to ρ vac cannot changeour conclusions and there is no need for infiniteregularizing counter terms to keep R fixed.According to the above properties, we think ¨uber-gravity is a promising model to study in more details. In[14], we have introduced a cosmological model, ¨uΛCDM,based on ¨uber-gravity which is a promising solution for H tension (e.g. see [21]) while it fits background data(including SNe, BAO and first peak of CMB) slightlybetter than ΛCDM with very non-trivial predictions atthe perturbation level. There are several ways to pursuethis idea which are beyond the scope of this paper andremain open for further investigations: ⋆ Making the ¨uber-modelling idea more concrete byfocusing on its mathematical foundations. Spe- cially we need to address the (fundamental) originof the probability of each model. ⋆ One specific way to shed light on our model is tocalculate effective Newtonian constant. It is doableby a conformal transformation and going from f ( R )frame to Brans-Dicke frame. ⋆ Doing perturbation theory of our model and exam-ine the results by observations. This is a crucialtest for our model since at the level of background,the solution is exact deSitter which is same as theΛCDM. ⋆ Focusing on the intermediate-curvature regimewhich seems potentially attractive. Stronger grav-ity may address the production of massive blackholes in high-redshifts. In addition since the inter-mediate regime is very close to R (which shouldbe fixed by Hubble parameter) then we expect tohave features on e.g. CMB in scales very close toHubble’s scale. ⋆ We can extend our proposal to other kind of modelse.g. Horndeski Lagrangians, massive gravity andother healthy gravity models. ⋆ ¨Uber-gravity gives two different phases of gravitydepending on the value of Ricci scalar. This sug-gests that maybe one can think about a phase tran-sition in the cosmology which distinguish early andlate time eras. Acknowledgments:
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