aa r X i v : . [ g r- q c ] F e b Emergent gravity and Dark Energy
T. PadmanabhanIUCAA, Pune University Campus,Ganeshkhind, Pune 411 007, INDIAemail: [email protected]
Given the expansion rate of the universe in terms of the Hubble constant H =( ˙ a/a ) , one can define a critical energy density ρ c = 3 H / πG which is requiredto make the spatial sections of the universe compact. It is convenient to measurethe energy densities of the different species, which drive the expansion of theuniverse, in terms of this critical density using the dimensionless parametersΩ i = ρ i /ρ c (with i denoting the different components like baryons, dark matter,radiation, etc.) The simplest possible universe one could imagine would havejust baryons and radiation. However, host of astronomical observations availablesince mid-70s indicated that the bulk of the matter in the universe is nonbaryonicand dark. Around the same time, the theoretical prejudice for Ω tot = 1 gainedmomentum, largely led by the inflationary paradigm. During the eighties, thisled many theoreticians to push (wrongly!) for a model of the with Ω tot ≈ Ω DM ≈ DM ≃ . − . w = p/ρ . − .
78 and contributes Ω DE ∼ = 0 . − . dark energy with negative pressure is the cos-mological constant which is a term that can be added to Einstein’s equations.This term acts like a fluid with an equation of state p DE = − ρ DE . Combiningthis with all other observations [4, 5, 6], we end up with a weird compositionfor the universe with 0 . . Ω tot . .
08 in which radiation (R), baryons (B),dark matter, made of weakly interacting massive particles (DM) and dark en-ergy (DE) contributes Ω R ≃ × − , Ω B ≃ . , Ω DM ≃ . , Ω DE ≃ . , respectively. So the bulk of the energy density in the universe is contributed by dark energy , which is theme of this article.The remarkably successful paradigm of conventional cosmology is based on1hese numbers and works [7] as follows: The key idea is that if there existedsmall fluctuations in the energy density in the early universe, then gravitationalinstability can amplify them leading to structures like galaxies etc. today. Thepopular procedure for generating these fluctuations is based on the idea that ifthe very early universe went through an inflationary phase [8], then the quan-tum fluctuations of the field driving the inflation can lead to energy densityfluctuations [9, 10]. While the inflationary models are far from unique andhence lacks predictive power, it is certainly possible to construct models of in-flation such that these fluctuations are described by a Gaussian random fieldand are characterized by a power spectrum of the form P ( k ) = Ak n with n ≃ A in an un-ambiguous manner. But it can be determined from CMBR observations andthe inflationary model parameters can be fine-tuned to reproduce the observedvalue. The CMBR observations are consistent with the inflationary model forthe generation of perturbations and gives A ≃ (28 . h − M pc ) and n .
1. (Thefirst results were from COBE [11] and WMAP has re-confirmed them with fargreater accuracy). One can evolve the initial perturbations by linear perturba-tion theory when the perturbation is small. But when δ ≈ ( δρ/ρ ) is comparableto unity the perturbation theory breaks down and one has to resort to numeri-cal simulations [12] or theoretical models based on approximate ansatz [13, 14]to understand their evolution — especially the baryonic part, that leads to ob-served structures in the universe. This rapid summary shows that modelingthe universe and comparing the theory with observations is a rather involvedaffair; but the results obtained from all these attempts are broadly consistentwith observations.To the zeroth order, the universe is characterized by just seven numbers: h ≈ . DE ≃ . , Ω DM ≃ . , Ω B ≃ . , Ω R ≃ × − giving the composition of the universe; the amplitude A ≃ (28 . h − M pc ) and the index n ≃ The remaining challenge, of course, is to make some sense out of these numbersthemselves from a more fundamental point of view. Among all these compo-nents, the dark energy, which exerts negative pressure, is probably the weirdestone and has attracted most of the attention.The key observational feature of dark energy is that, when treated as afluid with a stress tensor T ab = dia ( ρ, − p, − p, − p ), it has an equation state p = wρ with w . − . g of the geodesicacceleration (which measures the relative acceleration of two geodesics in thespacetime) satisfies an exact equation in general relativity given by: ∇ · g = − πG ( ρ + 3 p ) (1)2his shows that the source of geodesic acceleration is ( ρ +3 p ) and not ρ . As longas ( ρ + 3 p ) >
0, gravity remains attractive while ( ρ + 3 p ) < w = − , ρ = − p = constant (for a fewof the recent reviews, see ref. [16]). The cosmological constant introduces afundamental length scale in the theory L Λ ≡ H − , related to the constantdark energy density ρ DE by H ≡ (8 πGρ DE / G, c and L Λ , it is not possible to construct any dimensionlesscombination from these constants, when one introduces the Planck constant, ~ ,it is possible to form the dimensionless combination λ = H ( G ~ /c ) ≡ ( L P /L ).Observations then require ( L P /L ) . − requiring enormous fine tuning.In the early days, this was considered puzzling but most people believed thatthis number λ is actually zero. The cosmological constant problem in those dayswas to understand why it is strictly zero. Usually, the vanishing of a constant(which could have appeared in the low energy sector of the theory) indicatesan underlying symmetry of the theory. For example, the vanishing of the massof the photon is closely related to the gauge invariance of electromagnetism.No such symmetry principle is known to operate at low energies which madethis problem very puzzling. There is a symmetry — called supersymmetry —which does ensure that λ = 0 but it is known that supersymmetry is broken atsufficiently high energies and hence cannot explain the observed value of λ .Given the observational evidence for dark energy in the universe and the factthat the simplest candidate for dark energy, consistent with all observationstoday, is a cosmological constant with λ ≈ − the cosmological constantproblem has got linked to the problem of dark energy in the universe. So, ifwe accept the simplest interpretation of the current observations, we need toexplain why cosmological constant is non zero and has this small value. Itshould, however, be stressed that these are logically independent issues. Even ifall the observational evidence for dark energy goes away we still have a problem— viz., explaining why λ is zero. There is another, related, aspect to cosmological constant problem whichneed to be stressed. In conventional approach to gravity, one derives the equa-tions of motion from a Lagrangian L tot = L grav ( g ) + L matt ( g, φ ) where L grav is the gravitational Lagrangian dependent on the metric and its derivative and L matt is the matter Lagrangian which depends on both the metric and thematter fields, symbolically denoted as φ . In such an approach, the cosmologi-cal constant can be introduced via two different routes which are conceptuallydifferent but operationally the same. First, one may decide to take the gravita-tional Lagrangian to be L grav = (2 κ ) − ( R − g ) where Λ g is a parameter in the(low energy effective) action just like the Newtonian gravitational constant κ .The second route is by shifting the matter Lagrangian by L matt → L matt − λ m .3uch a shift is clearly equivalent to adding a cosmological constant 2 κλ m to the L grav . In general, what can be observed through gravitational interaction is thecombination Λ tot = Λ g + 2 κλ m .It is now clear that there are two distinct aspects to the cosmological constantproblem. The first question is why Λ tot is very small when expressed in naturalunits. Second, since Λ tot could have had two separate contributions from thegravitational and matter sectors, why does the sum remain so fine tuned? Thisquestion is particularly relevant because it is believed that our universe wentthrough several phase transitions in the course of its evolution, each of whichshifts the energy momentum tensor of matter by T ab → T ab + L − δ ab where L is the scale characterizing the transition. For example, the GUT and WeakInteraction scales are about L GUT ≈ − cm, L SW ≈ − cm respectivelywhich are tiny compared to L Λ . Even if we take a more pragmatic approach, theobservation of Casimir effect in the lab sets a bound that L < O (1) nanometer,leading to a ρ which is about 10 times the observed value [17].Finally, I will comment on two other issues related to cosmological constantwhich appear frequently in the literature. The first one is what could be calledthe “why now” problem of the cosmological constant. How come the energydensity contributed by the cosmological constant (treated as the dark energy) iscomparable to the energy density of the rest of the matter at the current epoch of the universe? I do not believe this is an independent problem; if we havea viable theory predicting a particular numerical value for λ , then the energydensity due to this cosmological constant will be comparable to the rest of theenergy density at some epoch. So the real problem is in understanding thenumerical value of λ ; once that problem is solved the ‘why now’ issue will takecare of itself. In fact, we do not have a viable theory to predict the currentenergy densities of any component which populates the universe, let alone thedark energy!. For example, the energy density of radiation today is computedfrom its temperature which is an observed parameter — there is no theory whichtells us that this temperature has to be 2.73 K when, say, galaxy formation hastaken place for certain billion number of years.One also notices in the literature a discussion of the contribution of the zeropoint energies of the quantum fields to the cosmological constant which is oftenmisleading, if not incorrect. What is usually done is to attribute a zero-point-energy (1 / ~ ω to each mode of the field and add up all these energies with anultra violet cut-off. For an electromagnetic field, for example, this will lead toan integral proportional to ρ = Z k max dk k ~ k ∝ k max (2)which will give ρ ∝ L − P if we invoke a Planck scale cut-off with k max = L − P . Itis then claimed that, this ρ will contribute to the cosmological constant. Thereare several problems with such a naive analysis. First, the ρ computed abovecan be easily eliminated by the normal ordering prescription in quantum fieldtheory and what one really should compute is the fluctuations in the vacuum4nergy — not the vacuum energy itself. Second, even if we take the nonzerovalue of ρ seriously, it is not clear this has anything to do with a cosmologicalconstant. The energy momentum tensor due to the cosmological constant has avery specific form T ab ∝ δ ab and its trace is nonzero. The electromagnetic field,for example, has a stress tensor with zero trace, T aa = 0; hence in the vacuumstate the expectation value of the trace, h vac | T aa | vac i , will vanish, showing thatthe equation of state of the bulk electromagnetic vacuum is still ρ = 3 p whichdoes not lead to a cosmological constant . (The trace anomaly will not workin the case of electromagnetic field.) So the naive calculation of vacuum energydensity with a cutoff and the claim that it contributes to cosmological constantis not an accurate statement in many cases. A nice possibility would be to postulate that λ = 0 and come up with a symme-try principle which will explain why this is the case. One probably has a greaterchance of success in such an attempt than in coming up with an explanation for λ ≈ − . But then, one needs to provide an alternative explanation for thedark energy observations. We shall now discuss two classes of such explanations,one which uses conventional physics and the other which is totally speculative— and conclude that both are not viable! One of the least esoteric ideas regarding the dark energy is that the cosmo-logical constant term in the equations arises because we have not calculatedthe energy density driving the expansion of the universe correctly. This ideaarises as follows: The energy momentum tensor of the real universe, T ab ( t, x )is inhomogeneous and anisotropic. If we could solve the exact Einstein’s equa-tions G ab [ g ] = κT ab with it as the source we will be led to a complicated metric g ab . The metric describing the large scale structure of the universe should beobtained by averaging this exact solution over a large enough scale, leading to h g ab i . But since we cannot solve exact Einstein’s equations, what we actually dois to average the stress tensor first to get h T ab i and then solve Einstein’s equa-tions. But since G ab [ g ] is nonlinear function of the metric, h G ab [ g ] i 6 = G ab [ h g i ]and there is a discrepancy. This is most easily seen by writing G ab [ h g i ] = κ [ h T ab i + κ − ( G ab [ h g i ] − h G ab [ g ] i )] ≡ κ [ h T ab i + T corrab ] (3)If — based on observations — we take the h g ab i to be the standard Friedmanmetric, this equation shows that it has, as its source, two terms: The first is thestandard average stress tensor and the second is a purely geometrical correctionterm T corrab = κ − ( G ab [ h g i ] − h G ab [ g ] i ) which arises because of nonlinearities inthe Einstein’s theory that leads to h G ab [ g ] i 6 = G ab [ h g i ]. If this term can mimic5he cosmological constant at large scales there will be no need for dark energyand — as a bonus — one will solve the “why now” problem!To make this idea concrete, we have to identify an effective expansion factor a eff ( t ) of an inhomogeneous universe (after suitable averaging), and determinethe equation of motion satisfied by it. The hope is that it will be sourced byterms so as to have ¨ a eff ( t ) > ρ + 3 p ) >
0) leads to deceleration of standard expansion factor a ( t ). Since any correctaveraging of positive quantities in ( ρ + 3 p ) will not lead to a negative quantity,the real hope is in defining a eff ( t ) and obtaining its dynamical equation suchthat ¨ a eff ( t ) >
0. In spite of some recent attention this idea has received [18] itis doubtful whether it will lead to the correct result when implemented properly.The reasons for my skepticism are the following: • It is obvious that T corrab is — mathematically speaking — non-zero (for anexplicit computation, in a completely different context of electromagneticplane wave, see [19]); the real question is how big is it compared to T ab . Itseems unlikely that when properly done, we will get a large effect for thesimple reason that the amount of mass which is contained in the nonlinearregimes in the universe today is subdominant. • Any calculation in linear theory or any calculation in which special sym-metries are invoked will be inconclusive in settling this issue. The keyquestion, of identifying a suitable analogue of expansion factor from anaveraged geometry, is nontrivial and it is not clear that the answer willbe unique. To illustrate this point by an extreme example, suppose wedecide to call a ( t ) n with, say n > a eff ( t ) = a ( t ) n ; obviously ¨ a eff can be positive (‘accelerating universe’) evenwith ¨ a being negative. So, unless one has a unique procedure to identifythe expansion factor of the average universe, it is difficult to settle theissue. • This approach is strongly linked to explaining the acceleration as observedby SN. Even if we decide to completely ignore all SN data, we still havereasonable evidence for dark energy and it is not clear how this approachcan tackle such evidence.Another equally conservative explanation for the cosmic acceleration will bethat we are located in a large underdense region in the universe; so that, locally,the underdensity acts like negative mass and produces a repulsive force. Whilethere has been some discussion in the literature [20] as to whether observationsindicate such a local ‘Hubble bubble’, this does not seem to be a tenable ex-planation that one can take seriously at this stage. Again, CMBR observationsindicating dark energy, for example, will not be directly affected by this featurethough one does need to take into account the effect of the local void.6 .2 Dark Energy from scalar fields
The most popular alternative to the cosmological constant uses a scalar field φ with a suitably chosen potential V ( φ ) so as to make the vacuum energy varywith time. The hope then is that, one can find a model in which the currentvalue can be explained naturally without any fine tuning. The scalar fieldscome in different shades and hues like quintessence, K-essence, tachyonic fieldsamongst others. For a small sample of recent [ & V ( φ ), it is possible to choose this function in order to produce a givenexpansion history of the universe characterized by the function H ( a ) = ˙ a/a expressed in terms of a . To see this explicitly, let us assume that the universehas two forms of energy density with ρ ( a ) = ρ known ( a ) + ρ φ ( a ) where ρ known ( a )arises from all known forms of source (matter, radiation, ...) and ρ φ ( a ) is dueto a scalar field. Let us first consider quintessence models with the Lagrangian: L quin = 12 ∂ a φ∂ a φ − V ( φ ) (4)Here, the potential is given implicitly by the form [23, 24] V ( a ) = 116 πG H (1 − Q ) (cid:20) H + 2 aH ′ − aHQ ′ − Q (cid:21) (5) φ ( a ) = (cid:20) πG (cid:21) / Z daa (cid:20) aQ ′ − (1 − Q ) d ln H d ln a (cid:21) / (6)where Q ( a ) ≡ [8 πGρ known ( a ) / H ( a )] and prime denotes differentiation withrespect to a . Given any H ( a ) , Q ( a ), these equations determine V ( a ) and φ ( a )and thus the potential V ( φ ). Every quintessence model studied in the literaturecan be obtained from these equations.
Similar results exists for the tachyonic scalar field as well [23] which has theLagrangian: L tach = − V ( φ )[1 − ∂ a φ∂ a φ ] / (7)Given any H ( a ), one can construct a tachyonic potential V ( φ ) so that the scalarfield is the source for the cosmology. The equations determining V ( φ ) are nowgiven by: φ ( a ) = Z daaH (cid:18) aQ ′ − Q ) − aH ′ H (cid:19) / (8) V ( a ) = 3 H πG (1 − Q ) (cid:18) aH ′ H − aQ ′ − Q ) (cid:19) / (9)Equations (8) and (9) completely solve the problem. Given any H ( a ), theseequations determine V ( a ) and φ ( a ) and thus the potential V ( φ ). A wide varietyof phenomenological models with time dependent cosmological constant havebeen considered in the literature; all of these can be mapped to a scalar fieldmodel with a suitable V ( φ ). 7t is very doubtful whether this — rather popular — approach, based onscalar fields, has helped us to understand the nature of the dark energy at anydeeper level. These models, viewed objectively, suffer from several shortcomings: • The most serious problem with them is that they have no predictive power.As explicitly demonstrated above, virtually every form of a ( t ) can be mod-eled by a suitable “designer” V ( φ ). • We see from the above discussion that even when w ( a ) is determinedby observations, it is not possible to proceed further and determine thenature of the scalar field Lagrangian. The explicit examples given aboveshow that there are at least two different forms of scalar field Lagrangians— corresponding to the quintessence or the tachyonic field — which couldlead to the same w ( a ). (See the first paper in ref.[3] for an explicit exampleof such a construction.) • By and large, the potentials used in the literature have no natural fieldtheoretical justification. All of them are non-renormalisable in the conven-tional sense and have to be interpreted as a low energy effective potentialin an ad hoc manner. • One key difference between cosmological constant and scalar field modelsis that the latter lead to a ( p/ρ ) ≡ w ( a ) which varies with time. So theyare worth considering if the observations have suggested a varying w , orif observations have ruled out w = − w = − w is strongly constrained [25].As an aside, let us note that in drawing conclusions from the observationaldata, one should be careful about the hidden assumptions in the statisticalanalysis. Claims regarding w depends crucially on the data sets used, priorswhich are assumed and possible parameterizations which are adopted. (Formore details related to these issues, see the last reference in [25].) It is fair tosay that all currently available data is consistent with w = −
1. Further, thereis some amount of tension between WMAP and SN-Gold data with the recentSNLS data [15] being more concordant with WMAP than the SN Gold data.One also needs to remember that, for the scalar field models to work, we firstneed to find a mechanism which will make the cosmological constant vanish. Inother words, all the scalar field potentials require fine tuning of the parametersin order to be viable. This is obvious in the quintessence models in which addinga constant to the potential is the same as invoking a cosmological constant. Ifwe shift
L → L matt − λ m in an otherwise successful scalar field model fordark energy, we end up ‘switching on’ the cosmological constant and raising theproblems again. 8 Cosmological Constant as dark energy
Even if all the evidence for dark energy disappears within a decade, we stillneed to understand why cosmological constant is zero and much of what I haveto say in the sequel will remain relevant. I stress this because there is a recenttendency to forget the fact that the problem of the cosmological constant existed(and was recognized as a problem) long before the observational evidence fordark energy, accelerating universe etc cropped up. In this sense, cosmologicalconstant problem has an important theoretical dimension which is distinct fromwhat has been introduced by the observational evidence for dark energy.Though invoking the cosmological constant as the candidate for dark energyleads to well known problems mentioned earlier, it is also the most economical(just one number) explanation for all the observations. Therefore it is worthexamining this idea in detail and ask how these problems can be tackled.If the cosmological constant is nonzero, then classical gravity will be de-scribed by the three constants
G, c and Λ ≡ L − . Since Λ( G ~ /c ) ≡ ( L P /L Λ ) ≈ − , it is obvious that the cosmological constant is actually telling us some-thing regarding quantum gravity , indicated by the combination G ~ . An acid testfor any quantum gravity model will be its ability to explain this value; needlessto say, all the currently available models — strings, loops etc. — flunk this test.While the occurrence of ~ in Λ( G ~ /c ) shows that it is a relic of a quan-tum gravitational effect (or principle) of unknown nature, cosmological constantproblem is an infrared problem par excellence in terms of the energy scaleswhich are involved. This is a somewhat unusual possibility of a high energyphenomenon leaving a low energy relic and an analogy will be helpful to illus-trate this idea [26]. Suppose we solve the Schrodinger equation for the Heliumatom for the quantum states of the two electrons ψ ( x , x ). When the result iscompared with observations, we will find that only half the states — those inwhich ψ ( x , x ) is antisymmetric under x ←→ x interchange — are realizedin nature. But the low energy Hamiltonian for electrons in the Helium atomhas no information about this effect! Here is a low energy (IR) effect which is arelic of relativistic quantum field theory (spin-statistics theorem) that is totallynon perturbative, in the sense that writing corrections to the Hamiltonian ofthe Helium atom in some (1 /c ) expansion will not reproduce this result. I sus-pect the current value of cosmological constant is related to quantum gravityin a similar spirit. There must exist a deep principle in quantum gravity whichleaves its non-perturbative trace even in the low energy limit that appears asthe cosmological constant. We shall now attempt a more quantitative discussionof these possibilities. Given the theory with two length scales L P and L Λ , one can construct two en-ergy scales ρ UV = 1 /L P and ρ IR = 1 /L in natural units ( c = ~ = 1). There issufficient amount of justification from different theoretical perspectives to treat L P as the zero point length of spacetime [27], giving a natural interpretation to9 UV . The second one, ρ IR also has a natural interpretation. Since the universedominated by a cosmological constant at late times will be asymptotically De-Sitter with a ( t ) ∝ exp( t/L Λ ) at late times, it will have a horizon and associatedthermodynamics [28] with a temperature T = H Λ / π . The corresponding ther-mal energy density is ρ thermal ∝ T ∝ /L = ρ IR . Thus L P determines the highest possible energy density in the universe while L Λ determines the lowest possible energy density in this universe. As the energy density of normal matterdrops below this value, ρ IR , the thermal ambiance of the DeSitter phase willremain constant and provide the irreducible ‘vacuum noise’. The observed darkenergy density is the the geometric mean ρ DE = √ ρ IR ρ UV = 1 L P L (10)of these two energy densities. If we define a dark energy length scale L DE suchthat ρ DE = 1 /L DE then L DE = √ L P L Λ is the geometric mean of the two lengthscales in the universe.It is possible to interpret this relation along the following lines: Consider a3-dimensional region of size L with a bounding area which scales as L . Letus assume that we associate with this region N microscopic cells of size L P each having a Poissonian fluctuation in energy of amount E P ≈ /L P . Thenthe mean square fluctuation of energy in this region will be (∆ E ) ≈ N L − P corresponding to the energy density ρ = ∆ E/L = √ N /L P L . If we make theusual assumption that N = N vol ≈ ( L/L P ) , this will give ρ = √ N vol L P L = 1 L P (cid:18) L P L (cid:19) / (bulk fluctuations) (11)On the other hand, if we assume that (for reasons which are unknown), therelevant degrees of freedom scale as the surface area of the region, then N = N sur ≈ ( L/L P ) and the relevant energy density is ρ = √ N sur L P L = 1 L P (cid:18) L P L (cid:19) = 1 L P L (surface fluctuations) (12)If we take L ≈ L Λ , the surface fluctuations in Eq. (12) give precisely the geomet-ric mean in Eq. (10) which is observed. On the other hand, the bulk fluctuations lead to an energy density which is larger by a factor ( L/L P ) / . Of course, if wedo not take fluctuations in energy but coherently add them, we will get N/L P L which is 1 /L P for the bulk and (1 /L P ) ( L P /L ) for the surface. In summary,we have the following hierarchy: ρ = 1 L P × " , (cid:18) L P L (cid:19) , (cid:18) L P L (cid:19) / , (cid:18) L P L (cid:19) , (cid:18) L P L (cid:19) ..... (13)in which the first one arises by coherently adding energies (1 /L P ) per cell with N vol = ( L/L P ) cells; the second arises from coherently adding energies (1 /L P )10er cell with N sur = ( L/L P ) cells; the third one is obtained by taking fluctu-ations in energy and using N vol cells; the fourth from energy fluctuations with N sur cells; and finally the last one is the thermal energy of the DeSitter spaceif we take L ≈ L Λ ; clearly the further terms are irrelevant due to this vacuumnoise.Of all these, the only viable possibility is what arises if we assume that:(a) The number of active degrees of freedom in a region of size L scales as N sur = ( L/L P ) . (b) It is the fluctuations in the energy that contributes to thecosmological constant [29, 30] and the bulk energy does not gravitate.It has been demonstrated recently [31] that it is possible to obtain classicalrelativity from purely thermodynamic considerations in which the surface termof the gravitational action plays a crucial role. The area scaling is familiar fromthe usual result that entropy of horizons scale as area. (Further, in cases likeSchwarzschild black hole, one cannot even properly define the volume inside ahorizon.) In fact, one can argue from general considerations that the entropyassociated with any null surface should be (1 /
4) per unit area and will beobserver dependent. A null surface, obtained as a limit of a sequence of timelikesurfaces (like the r = 2 M obtained from r = 2 M + k surfaces with k → + in thecase of the Schwarzschild black hole), ‘loses’ one dimension in the process (e.g., r = 2 M + k is 3-dimensional and timelike for k > k = 0) suggesting that the scaling of degrees of freedom has to changeappropriately. It is difficult to imagine that these features are unconnected andaccidental and we will discuss these ideas further in the next Section. I will now describe an alternative perspective in which gravity is treated as anemergent phenomenon – like elasticity – and argue that such a perspective isindeed necessary to succeed in solving the cosmological constant problem. To dothis, I will first identify the key ingredient of the cosmological constant problemand try to address it head on.
The equations of motion of gravity is obtained in the conventional approachto gravity from a Lagrangian L tot = L grav ( g ) + L matt ( g, φ ) where L grav is thegravitational Lagrangian dependent on the metric and its derivative and L matt is the matter Lagrangian which depends on both the metric and the matterfields, symbolically denoted as φ . This total Lagrangian is integrated over thespacetime volume with the covariant measure √− gd x to obtain the action.Suppose we now add a constant ( − λ m ) to the matter Lagrangian therebyinducing the change L matt → L matt − λ m . The equations of motion for matterare invariant under such a transformation which implies that — in the absenceof gravity — we cannot determine the value of λ m . The transformation L → matt − λ m is a symmetry of the matter sector (at least at scales below the scaleof supersymmetry breaking; we shall ignore supersymmetry in what follows).But, in the conventional approach, gravity breaks this symmetry. This is theroot cause of the cosmological constant problem.
As long as gravitational fieldequations are of the form E ab = κT ab where E ab is some geometrical quantity(which is G ab in Einstein’s theory) the theory cannot be invariant under theshifts of the form T ab → T ab + ρδ ab . Since such shifts are allowed by the mattersector, it is very difficult to imagine a definitive solution to cosmological constantproblem within the conventional approach to gravity.If metric represents the gravitational degree of freedom that is varied inthe action and we demand full general covariance, we cannot avoid L matter √− g coupling and cannot obtain of the equations of motion which are invariant underthe shift T ab → T ab + Λ g ab . Clearly a new, drastically different, approach togravity is required. We need to look for an approach which has the followingingredients [32]:To begin with, the field equations must remain invariant under the shift L matt → L matt + λ m of the matter Lagrangian L matt by a constant λ m . Thatis, we need to have some kind of ‘gauge freedom’ to absorb any λ m . Generalcovariance requires using the integration measure √− gd D x in actions. Since wedo not want to restrict general covariance but at the same time do not wantthis coupling to metric tensor via √− g , it follows that Metric cannot be thedynamical variable in our theory.
Secondly, even if we manage to obtain a theoryin which gravitational action is invariant under the shift T ab → T ab + Λ g ab , wewould have only succeeded in making gravity decouple from the bulk vacuumenergy. While this is considerable progress, there still remains the second issueof explaining the observed value of the cosmological constant. Once the bulkvalue of the cosmological constant (or vacuum energy) decouples from gravity, classical gravity becomes immune to cosmological constant; that is, the bulkclassical cosmological constant can be gauged away. Any observed value of thecosmological constant has to be necessarily a quantum phenomenon arising asa relic of microscopic spacetime fluctuations. The discussion in section 4.1,especially Eq. (12), shows that the relevant degrees of freedom should be linkedto surfaces in spacetime rather than bulk regions. The observed cosmologicalconstant is a relic of quantum gravitational physics and should arise from degreesof freedom which scale as the surface area.Thus, in an approach in which the surface degrees of freedom play the domi-nant role, rather than bulk degrees of freedom, we have a hope for obtaining thecorrect value for the cosmological constant. One should then obtain a theory ofgravity which is more general than Einstein’s theory with the latter emergingas a low energy approximation. For reasons described above, we abandon the usual picture of treating the metricas the fundamental dynamical degrees of freedom of the theory and treat it asproviding a coarse grained description of the spacetime at macroscopic scales,12omewhat like the density of a solid — which has no meaning at atomic scales[33]. The unknown, microscopic degrees of freedom of spacetime (which shouldbe analogous to the atoms in the case of solids), will play a role only whenspacetime is probed at Planck scales (which would be analogous to the latticespacing of a solid [27]).Some further key insight can be obtained by noticing that in the study ofordinary solids, one can distinguish between three levels of description. At themacroscopic level, we have the theory of elasticity which has a life of its ownand can be developed purely phenomenologically. At the other extreme, themicroscopic description of a solid will be in terms of the statistical mechanicsof a lattice of atoms and their interaction.Both of these are well known; but interpolating between these two limits isthe thermodynamic description of a solid at finite temperature which providesa crucial window into the existence of the corpuscular substructure of solids.
AsBoltzmann told us, heat is a form of motion and we will not have the thermo-dynamic layer of description if matter is a continuum all the way to the finestscales and atoms did not exist!
The mere existence of a thermodynamic layer inthe description is proof enough that there are microscopic degrees of freedom.
The situation is similar in the case of the spacetime [34]. Again we shouldhave three levels of description. The macroscopic level is the smooth spacetimecontinuum with a metric tensor g ab ( x i ) and the equations governing the metrichave the same status as the phenomenological equations of elasticity. At themicroscopic level, we expect a quantum description in terms of the ‘atoms ofspacetime’ and some associated degrees of freedom q A which are still elusive.But what is crucial is the existence of an interpolating layer of thermal phe-nomenon associated with null surfaces in the spacetime. Just as a solid cannotexhibit thermal phenomenon if it does not have microstructure, thermal natureof horizon, for example, cannot arise without the spacetime having a microstruc-ture. In such a picture, we normally expect the microscopic structure of spacetimeto manifest itself only at Planck scales or near singularities of the classical theory.However, in a manner which is not fully understood, the horizons — whichblock information from certain classes of observers — link [35] certain aspectsof microscopic physics with the bulk dynamics, just as thermodynamics canprovide a link between statistical mechanics and (zero temperature) dynamicsof a solid. The reason is probably related to the fact that horizons lead toinfinite redshift, which probes virtual high energy processes.The following three results, showing a fundamental relationship between thedynamics of gravity and thermodynamics of horizons [36] strongly support theabove point of view: • The dynamical equations governing the metric can be interpreted as athermodynamic relation closely related to the thermodynamics of hori-zons. An explicit example was provided in ref. [37], in the case ofspherically symmetric horizons in four dimensions in which it was shownthat, Einstein’s equations can be interpreted as a thermodynamic relation13 dS = dE + P dV arising out of virtual radial displacements of the horizon.Further work showed that this result is valid in all the cases for which ex-plicit computation can be carried out — like in the Friedmann models [38]as well as for rotating and time dependent horizons in Einstein’s theory[39]. • The standard Lagrangian in Einstein’s theory has the structure L EH ∝ R ∼ ( ∂g ) + ∂ g . In the usual approach the surface term arising from L sur ∝ ∂ g has to be ignored or canceled to get Einstein’s equations from L bulk ∝ ( ∂g ) . But there is a peculiar (unexplained) relationship [31]between L bulk and L sur : √− g L sur = − ∂ a (cid:18) g ij ∂ √− g L bulk ∂ ( ∂ a g ij ) (cid:19) (14)This shows that the gravitational action is ‘holographic’ with the sameinformation being coded in both the bulk and surface terms and one ofthem should be sufficient. • One can indeed obtain Einstein’s equations from an action principle whichuses only the surface term and the virtual displacements of horizons [40,32]. It is possible to determine the form of this surface term from generalconsiderations. If we now demand that the action should not receive con-tributions for radial displacements of the horizons, defined in a particularmanner using local Rindler horizons, one can obtain — at the lowest order— the equations ( G ab − κT ab ) ξ a ξ b = 0 (15)where ξ a is a null vector. Demanding the validity of Eq.(15) in all localRindler frames then leads to Einstein’s theory with the cosmological con-stant emerging as an integration constant. Note that Eq.(15) is invariantunder the constant shift of matter Lagrangian, making gravity immuneto bulk cosmological constant. Since the surface term has the thermody-namic interpretation as the entropy of horizons, this establishes a directconnection between spacetime dynamics and horizon thermodynamics. • Further work has shown that all the above results extend beyond Einstein’stheory.
The connection between field equations and the thermodynamicrelation
T dS = dE + P dV is not restricted to Einstein’s theory alone, butis in fact true for the case of the generalized, higher derivative Lanczos-Lovelock gravitational theory in D dimensions as well [41, 42]. The sameis true [43] for the holographic structure of the action functional: theLanczos-Lovelock action has the same structure and — again — the en-tropy of the horizons is related to the surface term of the action. Theseresults show that the thermodynamic description is far more general thanjust Einstein’s theory and occurs in a wide class of theories in which themetric determines the structure of the light cones and null surfaces existblocking the information. 14he conventional approach to gravity fails to provide any clue regardingthe thermodynamic aspects of gravity just as Newtonian continuum mechanics— without corpuscular, discrete, substructure for matter — cannot explainthermodynamic phenomena. A natural explanation for these results requiresa different approach to spacetime dynamics which I will now outline. (Moredetails can be found in ref. [44] )
In obtaining the relation between gravitational dynamics and horizon thermo-dynamics, one treats the null surfaces (which act as horizons) as the limit ofa sequence of, say, timelike surfaces. The virtual displacements of the horizonin the direction normal to the surfaces will be used in the action principle. Allthese suggest that one may be able to obtain a more formal description of thetheory in terms of deformation of surfaces in spacetime. I will now describe onesuch model which is unreasonably successful.To set the stage, let us suppose there are certain microscopic — as yet un-known — degrees of freedom q A , analogous to the atoms in the case of solids,described by some microscopic action functional A micro [ q A ]. In the case of asolid, the relevant long-wavelength elastic dynamics is captured by the displace-ment vector field which occurs in the equation x a → x a + ξ a ( x ) which is onlyvery indirectly connected with the microscopic degrees of freedom. Similarly,in the case of spacetime, we need to introduce some other degrees of freedom,analogous to ξ a in the case of elasticity, and an effective action functional basedon it. (As explained above, we do not want to use the metric as a dynamicalvariable.) Normally, varying an action functional with respect to certain degreesof freedom will lead to equations of motion determining those degrees of free-dom. But we now make an unusual demand that varying our action principlewith respect to some (non-metric) degrees of freedom should lead to an equationof motion determining the background metric which remains non-dynamical.Based on the role expected to be played by surfaces in spacetime, we shalltake the relevant degrees of freedom to be the normalized vector fields n i ( x ) inthe spacetime [44] with a norm that is fixed at every event but might vary fromevent to event: (i.e., n i n i ≡ ǫ ( x ) with ǫ ( x ) being a fixed function which takesthe values 0 , ± ξ a captures themacro-description in case of solids, the normalized vectors (e.g., local normalsto surfaces) capture the essential macro-description in case of gravity in termsof an effective action S [ n a ]. More formally, we expect the coarse graining of mi-croscopic degrees of freedom to lead to an effective action in the long wavelengthlimit: X q A exp( − A micro [ q A ]) −→ exp( − S [ n a ]) (16)To proceed further we need to determine the nature of S [ n a ]. The general form15f S [ n a ] in such an effective description, at the quadratic order, will be: S [ n a ] = Z V d D x √− g (cid:0) P cdab ∇ c n a ∇ d n b − T ab n a n b (cid:1) , (17)where P cdab and T ab are two tensors and the signs, notation etc. are chosenwith hindsight. (We will see later that T ab can be identified with the matterstress-tensor.) The full action for gravity plus matter will be taken to be S tot = S [ n a ] + S matt with: S tot = Z V d D x √− g (cid:0) P cdab ∇ c n a ∇ d n b − T ab n a n b (cid:1) + Z V d D x √− g L matt (18)with an important extra prescription: Since the gravitational sector is related tospacetime microstructure, we must first vary the n a and then vary the matterdegrees of freedom. In the language of path integrals, we should integrate outthe gravitational degrees of freedom n a first and use the resulting action for thematter sector.We next address one crucial difference between the dynamics in gravity andsay, elasticity, which we mentioned earlier. In the case of solids, one will writea similar functional for thermodynamic potentials in terms of the displacementvector ξ a and extremising it will lead to an equation which determines ξ a . In thecase of spacetime, we expect the variational principle to hold for all vectors n a with a fixed norm and lead to a condition on the background metric. Obviously,the action functional in Eq.(17) must be rather special to accomplish this andone need to impose two restrictions on the coefficients P cdab and T ab to achievethis. First, the tensor P abcd should have the algebraic symmetries similar to theRiemann tensor R abcd of the D -dimensional spacetime. Second, we need: ∇ a P abcd = 0 = ∇ a T ab . (19)In a complete theory, the explicit form of P abcd will be determined by the longwavelength limit of the microscopic theory just as the elastic constants can —in principle — be determined from the microscopic theory of the lattice. Inthe absence of such a theory, we can take a cue from the renormalization grouptheory and expand P abcd in powers of derivatives of the metric [40, 44]. Thatis, we expect, P abcd ( g ij , R ijkl ) = c P abcd ( g ij ) + c P abcd ( g ij , R ijkl ) + · · · , (20)where c , c , · · · are coupling constants and the successive terms progressivelyprobe smaller and smaller scales. The lowest order term must clearly dependonly on the metric with no derivatives. The next term depends (in additionto metric) linearly on curvature tensor and the next one will be quadratic incurvature etc. It can be shown that the m-th order term which satisfies ourconstraints is unique and is given by ( m ) P cdab ∝ δ cda ...a m abb ...b m R b b a a · · · R b m − b m a m − a m = ∂ L ( D ) m ∂R abcd . (21)16here δ cda ...a m abb ...b m is the alternating tensor and the last equality shows that itcan be expressed as a derivative of the m th order Lanczos-Lovelock Lagrangian[40, 44, 45], given by L ( D ) = K X m =1 c m L ( D ) m ; L ( D ) m = 116 π − m δ a a ...a m b b ...b m R b b a a R b m − b m a m − a m , (22)where the c m are arbitrary constants and L ( D ) m is the m -th order Lanczos-Lovelock term and we assume D ≥ K + 1. The lowest order term (whichleads to Einstein’s theory) is (1) P abcd = 116 π δ a a b b = 132 π ( δ ac δ bd − δ ad δ bc ) . (23)while the first order term gives the Gauss-Bonnet correction. All higher ordersterms are obtained in a similar manner.In our paradigm based on Eq.(16), the field equations for gravity arise fromextremising S with respect to variations of the vector field n a , with the con-straint δ ( n a n a ) = 0, and demanding that the resulting condition holds for allnormalized vector fields . One can show [40, 44] that this leads to the fieldequations 16 π (cid:20) P ijkb R aijk − δ ab L ( D ) m (cid:21) = 8 πT ab + Λ δ ab (24)where Λ is an undetermined integration constant. These are identical to thefield equations for the Lanczos-Lovelock gravity with a cosmological constantarising as an undetermined integration constant. To the lowest order, when weuse Eq.(23) for P ijkb , the Eq.(24) reproduces Einstein’s theory. More generally,we get Einstein’s equations with higher order corrections which are to be inter-preted as emerging from the derivative expansion of the action functional as weprobe smaller and smaller scales. Remarkably enough, we can derive not onlyEinstein’s theory but even Lanczos-Lovelock theory from a dual description interms on the normalized vectors in spacetime, without varying g ab in an actionfunctional! The crucial feature of the coupling between matter and gravity through T ab n a n b in Eq. (18) is that, under the shift T ab → T ab + ρ g ab , the ρ term inthe action in Eq.(17) decouples from n a and becomes irrelevant: Z V d D x √− gT ab n a n b → Z V d D x √− gT ab n a n b + Z V d D x √− gǫρ (25)Since ǫ = n a n a is not varied when n a is varied there is no coupling between ρ and the dynamical variables n a and the theory is invariant under the shift T ab → T ab + ρ g ab . We see that the condition n a n a = constant on the dynami-cal variables have led to a ‘gauge freedom’ which allows an arbitrary integrationconstant to appear in the theory which can absorb the bulk cosmological con-stant. 17o gain a bit more insight into what is going on, let us consider the on-shellvalue of the action functional in Eq. (18). It can be shown that the on-shell value is given by a surface term which will lead to the entropy of the horizons(which will be 1/4 per unit transverse area in the case of general relativity).Even in the case of a theory with a general P abcd it can be shown that the on-shell value of the action reduces to [44] the entropy of the horizons. The generalexpression is: S | H = K X m =1 πmc m Z H d D − x ⊥ √ σ L ( D − m − = 14 [Area] ⊥ + corrections (26)where x ⊥ denotes the transverse coordinates on the horizon H , σ is the deter-minant of the intrinsic metric on H and we have restored a summation over m thereby giving the result for the most general Lanczos-Lovelock case obtained asa sum of individual Lanczos-Lovelock Lagrangian’s. The expression in Eq.(26) is precisely the entropy of a general Killing horizon in Lanczos-Lovelock grav-ity based on the general prescription given by Wald and computed by severalauthors.This result shows that, in the semiclassical limit, in which the action canpossibly be related to entropy, we reproduce the conventional entropy whichscales as the area in Einstein’s theory. Since the entropy counts the relevantdegrees of freedom, this shows that the degrees of freedom which survives andcontributes in the long wave length limit scales as the area. The quantumfluctuations in these degrees of freedom can then lead to the correct, observed,value of the cosmological constant. We will discuss this aspect briefly in thenext section.Our action principle is somewhat peculiar compared to the usual actionprinciples in the sense that we have varied n a and demanded that the resultingequations hold for all vector fields of constant norm. Our action principle actu-ally stands for an infinite number of action principles, one for each vector fieldof constant norm! This class of all n i allows an effective, coarse grained, de-scription of some (unknown) aspects of spacetime micro physics. This is why weneed to first vary n a , obtain the equations constraining the background metricand then use the reduced action to obtain the equations of motion for matter.Of course, in most contexts, ∇ a T ab = 0 will take care of the dynamical equationsfor matter and these issues are irrelevant.At this stage, it is not possible to proceed further and relate n i to somemicroscopic degrees of freedom q A . This issue is conceptually similar to askingone to identify the atomic degrees of freedom, given the description of an elasticsolid in terms of a displacement field ξ a — which we know is virtually impossible.However, the same analogy tells us that the relevant degree of freedom in thelong wavelength limit (viz. ξ a or n i ) can be completely different from themicroscopic degrees of freedom and it is best to proceed phenomenologically.18 .4 Gravity as detector of the vacuum fluctuations The description of gravity given above provides a natural back drop for gaug-ing away the bulk value of the cosmological constant since it decouples fromthe dynamical degrees of freedom in the theory. Once the bulk term is elim-inated, what is observable through gravitational effects, in the correct theoryof quantum gravity, should be the fluctuations in the vacuum energy. Thesefluctuations will be non-zero if the universe has a DeSitter horizon which pro-vides a confining volume. In this paradigm the vacuum structure can readjustto gauge away the bulk energy density ρ UV ≃ L − P while quantum fluctuations can generate the observed value ρ DE .The role of energy fluctuations contributing to gravity also arises, moreformally, when we study the question of detecting the energy density usinggravitational field as a probe. Recall that a detector with a linear coupling tothe field φ actually responds to h | φ ( x ) φ ( y ) | i rather than to the field itself [46].Similarly, one can use the gravitational field as a natural “detector” of energymomentum tensor T ab with the standard coupling L = κh ab T ab . Such a modelwas analyzed in detail in ref. [47] and it was shown that the gravitational fieldresponds to the two point function h | T ab ( x ) T cd ( y ) | i . In fact, it is essentiallythis fluctuations in the energy density which is computed in the inflationarymodels [8] as the source for gravitational field, as stressed in ref. [10]. All thesesuggest treating the energy fluctuations as the physical quantity “detected” bygravity, when one incorporates quantum effects.Quantum theory, especially the paradigm of renormalization group has taughtus that the concept of the vacuum state depends on the scale at which it isprobed. The vacuum state which we use to study the lattice vibrations in asolid, say, is not the same as vacuum state of the QED and it is not appropriateto ask questions about the vacuum without specifying the scale. If the cosmolog-ical constant arises due to the fluctuations in the energy density of the vacuum,then one needs to understand the structure of the quantum gravitational vac-uum at cosmological scales. If the spacetime has a cosmological horizon whichblocks information, the natural scale is provided by the size of the horizon, L Λ ,and we should use observables defined within the accessible region. The oper-ator H ( < L Λ ), corresponding to the total energy inside a region bounded by acosmological horizon, will exhibit fluctuations ∆ E since vacuum state is not aneigenstate of this operator. A rigorous calculation (see the first reference in [30])shows that the fluctuations in the energy density of the vacuum in a sphere ofradius L Λ is given by ∆ ρ vac = ∆ EL ∝ L − P L − (27)The numerical coefficient will depend on c as well as the precise nature ofinfrared cutoff radius; but it is a fact of life that a fluctuation of magnitude∆ ρ vac ≃ H /G will exist in the energy density inside a sphere of radius H − ifPlanck length is the UV cut off. On the other hand, since observations suggestthat there is a ρ vac of similar magnitude in the universe it seems natural toidentify the two. Our approach explains why there is a surviving cosmological19onstant which satisfies ρ DE = √ ρ IR ρ UV .Such a computation of energy fluctuations is completely meaningless in themodels of gravity in which the metric couples to the bulk energy density. Oncea UV cut-off at Planck scale is imposed, one will always get a bulk contribu-tion ρ UV ≈ L − P with the usual problems. It is only because we have a way ofdecoupling the bulk term from contributing to the dynamical equations that,we have a right to look at the subdominant term L − P ( L P /L Λ ) . Approachesin which the sub-dominant term is introduced by an ad hoc manner are techni-cally flawed since the bulk term cannot be ignored in these usual approaches togravity. Getting the correct value of the cosmological constant from the energyfluctuations is not as difficult as understanding why the bulk value (which islarger by 10 !) can be ignored. Our approach provides a natural backdropfor ignoring the bulk term — and as a bonus — we get the right value forthe cosmological constant from the fluctuations. Cosmological constant is smallbecause it is a quantum relic. The simplest choice for the negative pressure component in the universe is thecosmological constant; other models based on scalar fields (as well as those basedon branes etc. which I have not discussed) do not alleviate the difficulties facedby cosmological constant and — in fact — makes them worse. The cosmologicalconstant is most likely to be a low energy relic of a quantum gravitationaleffect or principle and its explanation will require a radical shift in our currentparadigm.A new approach to gravity described here could provide a possible broadparadigm to understand the cosmological constant. The conceptual basis forthis approach rests on the following logical ingredients. I have shown that it isimpossible to solve the cosmological constant problem unless the gravitationalsector of the theory is invariant under the shift T ab → T ab + λ m g ab . Any approachwhich does not address this issue cannot provide a comprehensive solution tothe cosmological constant problem. But general covariance requires us to usethe measure √− gd D x in D-dimensions in the action. This will couple the metric(through its determinant) to the matter sector. Hence, as long as we insist onmetric as the fundamental variable describing gravity, one cannot address thisissue. So we need to introduce some other degrees of freedom and an effectiveaction which, however, is capable of constraining the background metric.An action principle, based on the normalized vector fields in spacetime,satisfies all these criteria mentioned above. The new action does not coupleto the bulk energy density and maintains invariance under the shift T ab → T ab + λ m g ab . What is more, the on-shell value of the action is related to theentropy of horizons showing the relevant degrees of freedom scales as the area ofthe bounding surface. Since our formalism ensures that the bulk energy densitydoes not contribute to gravity — and only because of that — it makes senseto compute the next order correction due to fluctuations in the energy density.20his is impossible to do rigorously with the machinery available but a plausiblecase can be made as how this will lead to the correct, observed, value of thecosmological constant.An effective theory can capture the relevant physics at the long wavelengthlimit using the degrees of freedom contained in the fluctuations of the nor-malized vectors. The resulting theory is more general than Einstein gravitysince the thermodynamic interpretations should transcend classical considera-tions and incorporate some of the microscopic corrections. 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