aa r X i v : . [ a s t r o - ph ] S e p Structure formation and the origin of dark energy
Golam Mortuza Hossain ∗ Institute for Gravitation and the Cosmos, The Pennsylvania State University,104 Davey Lab, University Park, PA 16802, USA
Cosmological constant a.k.a. dark energy problem is considered to be one major challenge inmodern cosmology. Here we present a model where large scale structure formation causes spatially-flat FRW universe to fragment into numerous ‘FRW islands’ surrounded by vacuum. We show thatthis mechanism can explain the origin of dark energy as well as the late time cosmic acceleration.This explanation of dark energy does not require any exotic matter source nor an extremely fine-tuned cosmological constant. This explanation is given within classical general relativity and relieson the fact that our universe has been undergoing structure formation since its recent past.
PACS numbers: 95.36.+x,98.80.-k,98.80.Jk
Several recent experimental observations [1] seem tostrongly suggest that we live in the universe whose en-ergy budget is dominated by contribution from a mys-terious source which is otherwise invisible or missing indirect observations. This energy component is generallyreferred as dark energy . Furthermore, the dark energycomponent appears to have negative pressure. Resultfrom supernova observations [2] that universe is under-going a recent acceleration seems to confirm such peculiarbehavior of dark energy.Many attempts have been made in literature to under-stand the origin of dark energy (see [3] for some reviews).Arguably the most economical one is to introduce a non-zero cosmological constant in Einstein equation. How-ever, there are several conceptual difficulties associatedwith it. Firstly, experimentally required value of cosmo-logical constant turns out to be ∼ − in Planck unit.Such an extremely fine-tuned value of cosmological con-stant appears to defy any hope of possible explanationfrom a fundamental theory. Second conceptual problemis the so-called cosmic coincidence problem: why does en-ergy density due to cosmological constant which remainsunchanged, become comparable to changing matter en-ergy density only at current epoch?There have been other attempts where it is anticipatedthat late time acceleration could be due to a dynamicalscalar field that evolves in a suitably engineered potential[4], due to super-horizon perturbations [5], from averag-ing of LTB models [6] or due to back-reaction effects ofinhomogeneous structures in the universe [7, 8, 9]. Inparticular, using Buchert equations one can get accel-eration by incorporating inhomogeneous back-reaction.However, same back-reaction term contributes negatively [8] to energy density thus fails to explain missing energyunless one also introduces strong negative spatial curva-ture. In brief, none of the current attempts can explainthe origin of dark energy satisfactorily.According to standard model of cosmology, continuedexpansion of the universe causes radiation to eventu-ally decouple from matter. Subsequently, largely homo-geneous matter distribution with small inhomogeneity FIG. 1: A simple illustration of late time universe wherestructure formation causes FRW universe to fragment intonumerous ‘FRW islands’. Each FRW island (shaded region)is surrounded by vacuum metric. starts collapsing to give rise current large scale struc-tures. Initial phase of structure formation can be de-scribed by linear perturbation theory around homoge-neous background. However, in later phase when struc-ture formation enters non-linear regime such descriptionsare insufficient. In this era, matter distribution consistsof numerous dense regions surrounded by relatively rarerregions. For simplicity here we make sharp-boundary ap-proximation for the matter distribution during later pe-riod of structure formation such that matter is containedwithin spherical, homogeneous regions surrounded byempty space. Boundary of such spherical regions, how-ever, needs to be continuously refined as on-going struc-ture formation continues to cause dense regions to be-come denser. In other words, in this model structure for-mation along with sharp-boundary approximation causesFRW universe to fragment into numerous ‘FRW islands’with shrinking boundary that are surrounded by vacuum(see Fig.1). We will show that this mechanism can ex-plain the origin of dark energy as well as the late timecosmic acceleration.In the standard model of cosmology, spacetime describ-ing our universe is assumed to be foliated by homoge-neous and isotropic spatial hyper-surfaces parametrizedby a global time. We imagine such an observer who treats
FIG. 2: A spherical patch of universe of coordinate diameter L which was homogeneous at the beginning of structure for-mation. In a later period the matter distribution, by means ofsharp-boundary approximation, is contained within a spheri-cal region (shaded) of coordinate diameter l i . the spatial hyper-surfaces as homogeneous and isotropicduring entire evolution of universe and accordingly mea-sures distance using an average spatially-flat Friedman-Robertson-Walker (FRW) metric ds = − dt + a ( t ) d x , (1)where a ( t ) is the scale factor . We refer this observer asobserver A. We consider another observer, say observerB, who uses same time parametrization of the spatialhyper-surfaces as observer A, but is careful to considerthe effects of structure formation. In particular, to mea-sure distance during structure formation observer B usesflat FRW metric g iµν = diag ( − , a i , a i , a i ) inside thespherical regions containing homogeneous matter distri-bution otherwise uses vacuum metric g oµν . To illustratethis, let’s consider a spherical patch of universe of coor-dinate (also co-moving for observer A) diameter L whichis homogeneous at the beginning of structure formation(see Fig.2). The patch then begins to undergo structureformation such that at a given time collapsing matter dis-tribution, by means of sharp-boundary approximation, iscontained within a spherical region of coordinate diam-eter l i ( t ) with 0 < l i ≤ L . We assume that the vacuummetric g oµν describing the spherically symmetric emptyspace created due to structure formation, is static . (Thiswould be the case if we ignore the presence of neigh-bouring patches, as then Birkhoff ’s theorem would implyspherically symmetric vacuum metric is static.) The vac-uum metric g oµν however is inhomogeneous as can be seenfrom continuity of metric at the boundary of inside re-gion.To have a precise notion we define average FRW met-ric (1) such that the proper distance between the points P and P (see Fig.2) as measured by observer A is equal to the proper distance measured by observer B. With-out loss of generality we assume that the points lie on x -axis. Given elementary proper distance in a spatialhyper-surface is | p g jk dx j dx k | where j , k represent spa-tial coordinates, the definition leads to aL = a i l i + 2 Z L/ l i / q g o ( x , x , x ) dx . (2)Using the definition, we can also compute relation be-tween expansion rates of the points as measured by bothobservers. In particular,˙ aL = ddt Z L/ − L/ p g ( x , x , x ) dx = ˙ a i l i + a i ˙ l i − ˙ l i q g o ( l i / , ,
0) = ˙ a i l i , (3)where over-dot denotes derivative w.r.t. to time t . Inlast line, we have used continuity of metric solution i.e. g o ( l i / , ,
0) = g i = a i . We may note that to havean expanding average metric, the metric of the insideregion must also be expanding. However, proper volumeof the inside region can still decrease as its coordinatediameter shrinks. We now define average energy density ρ for observer A, by requiring that at any given time totalenergy contained within the patch of coordinate diameter L is same as measured by observer B. In particular, ifobserver B measures energy density of inside region tobe ρ i and of outside region to be zero then ρ = (cid:18) a i l i aL (cid:19) ρ i =: n ρ i . (4)As defined, n is the fraction of total proper volume oc-cupied by matter distribution. Physically, parameter n isa measure of amount of structure formation and satisfies0 < n ≤
1. In particular, n = 1 implies there are nounderlying structures. This can be seen from the equa-tion (2). As we will see, equations (3) and (4) form thebackbone of arguments presented here.Let’s imagine that both observers want to confronttheir respective Einstein equations with experimentaldata. Observer A performs separate observations to mea-sure expansion rate as well as energy density. However, itturns out that to make a right balance, observer A needsto postulate an extra invisible component in Friedmannequation for the average metric i.e. (cid:18) ˙ aa (cid:19) = 8 πG ( ρ + ρ DE ) , (5)where G is Newton’s constant and ρ DE denotes dark en-ergy component. On the other hand observer B usesstandard Friedmann equation for the region containinghomogeneous matter distribution and uses vacuum Ein-stein equation for the remaining region. In particular,Friedmann equation for observer B is3 (cid:18) ˙ a i a i (cid:19) = 8 πGρ i . (6)Equipped with the details of underlying structures i.e. using equations (3), (4) and Friedmann equation (6), ob-server B can derive Friedmann equation for average FRWmetric (1), given by3 (cid:18) ˙ aa (cid:19) = 8 πG (cid:20) ρ + (cid:18) n − (cid:19) ρ (cid:21) . (7)One may note that right hand side of equation (7) hasan extra energy density component apart from averageenergy density ρ . Thus, comparing equations (5) and (7),observer B can derive the expression of dark energy thatobserver A should perceive ρ DE = (cid:18) n − (cid:19) ρ . (8)In the situation when n = 1, dark energy component dis-appears. In other words, observer A wouldn’t have per-ceived any dark energy component if there were no under-lying structures in universe. Since existence of underlyingstructures requires parameter values to be 0 < n < positive . This is in contrast with back-reaction models such as [7, 8] whereback-reaction term contributes negatively to energy den-sity. As evident, dark energy component is comparableto the magnitude of average energy density ρ . So thismodel can naturally explain cosmic coincidence prob-lem. Another crucial property of dark energy component(8) is that it may appear as constant even though it isnaively proportional to decreasing average energy density ρ . With the beginning of structure formation, the valueof parameter n starts decreasing from unity. So duringstructure formation proportionality factor (1 /n −
1) in-creases. This implies that for suitable rate of structureformation dark energy component (8) may appear as con-stant.For observer A total energy contained within the patchis given by E = V ρ where proper volume of the patch V = ( πa L / P = − ( ∂E/∂V ), one can derive the conservation equation forenergy density ρ as˙ ρ = − ˙ VV ! (cid:18) EV − ∂E∂V (cid:19) = − Hρ (1 + ω ) , (9)where H := ( ˙ a/a ) is Hubble parameter and ω := P/ρ isthe corresponding equation of state. Analogously, we candefine equation of state ω DE for dark energy componentsuch that ˙ ρ DE = − Hρ DE (1 + ω DE ). Using equations(8) and (9), we can compute equation of state for thedark energy ω DE = − (cid:20) (1 + ω ) − r n − n ) (cid:21) , (10) where ( ˙ n/n ) =: − r n H . Parameter r n is a measure of rate of structure formation. Dark energy expression (8)along with its equation of state (10) can mimic a cosmo-logical constant at current epoch for suitable values ofstructure formation parameters n and r n . In particular,the values of structure formation parameters such that r n = 3(1 − n )(1 + ω ) will lead to ω DE = − (cid:18) ¨ aa (cid:19) = 4 πG (cid:20) r n n − (1 + 3 ω ) n (cid:21) ρ . (11)From equation (11) it can be seen that the values of struc-ture formation parameter such that r n > (1 + 3 ω ) willlead to an accelerating phase for observer A.The modifications to average dynamics of the givenpatch due to underlying structures, depend on the valuesof parameters n and r n . However, the modifications do not depend explicitly on coordinate diameters L and l i .Thus, if one considers a different patch but with same values of parameters n and r n , then one will get same Friedmann and Raychaudhuri equations. Given one canpack R space very closely using 3-spheres of arbitrary diameters hence the modified Friedman equation (7) andmodified Raychaudhuri equation (11) can be consideredas good approximation of equations that describe averagedynamics of the model universe with underlying struc-tures. Observed contribution from the dark energy com-ponent at current epoch is about 70% of the critical en-ergy density. The equation (8) with the parameter value n = 0 . i.e. ω = 0 then ω DE = − r n = 2 . ω and equationof state for inside region ω i := P i /ρ i where pressure P i = − ( ∂E i /∂V i ). E i and V i are total energy and propervolume of the inside region respectively. As earlier wecan derive the conservation equation for inside region,given by ˙ ρ i = − ˙ V i V i ! (cid:18) E i V i − ∂E i ∂V i (cid:19) = − ˙ a i a i + ˙ l i l i ! ρ i (1 + ω i ) . (12)Coordinate diameter of inside region is time-dependentand it is reflected in conservation equation (12) with itsexplicit dependence on ( ˙ l i /l i ). Using the relation betweenenergy density (4), their conservation equations (9) and(12), one can compute relation between the equation ofstates ω = (1 − r n ) ω i . (13)For gravitational collapse to occur with ordinary matter,the corresponding matter distribution must be pressure-less. So for observer B, matter distribution of inside re-gion should be pressure-less i.e. ω i = 0. On the otherhand observer A finds average matter also to be pressure-less i.e. ω = 0. The equation (13) ensures that physicalrequirement of observer B and observed fact for observerA can be consistently met. We should mention here thatto derive equation (11), one can also use Raychaudhuriequation for observer B. However, in doing so one shouldbe careful to include the additional pressure componentcoming from the shrinking boundary.Experimental observations seem to also imply that to-tal energy of the universe has another dark component,the so-called dark matter which is pressure-less. Fromequation (10), one may note that if ω = 0 and r n = 0 i.e. if ( − ˙ l i /l i ) = (1 /n − H then the equation of statemimics a pressure-less energy component. It is conceiv-able that some ‘FRW islands’ may have different values ofstructure formation parameters leading to such behavior.However, whether such scenario can explain phenomenaascribed to the presence of dark matter such as galaxyrotation curves, remains to be explored.To summarize, we have argued that the origin of darkenergy can be understood as a consequence of large scalestructure formation. This explanation of dark energydoes not require any exotic matter source nor a fine-tuned cosmological constant. However, presented modelin its current form has several deficiencies. Firstly, weassume that structure formation leads to creation of voidaround each FRW island. However, we know cosmic mi-crowave background (CMB) photons fills up entire uni-verse. Thus, even though CMB contribution to averageenergy density is negligible during structure formationbut for an accurate description one should consider theirpresence. In this model net effects of structure formationon dynamics of average metric can be summarized by in-troducing just two characteristic parameters n and r n .However, the model itself does not shed any light on thevalues of the parameters n and r n . We may recall thatthe model is based on sharp-boundary approximation ofthe matter distribution which is under-going structureformation. Thus, to compute relation between the val-ues of the parameters one needs to perform a detailedsimulation of structure formation with a matter distri-bution which should be then successively approximated by sharp-boundary approximation. Finally, we have notaddressed the issue: why is cosmological constant zero inour universe? Acknowledgments:
Author thanks Abhay Ashtekar fordiscussions and Martin Bojowald for comments on themanuscript. This work was supported in part by NSFgrant PHY0456913. ∗ Electronic address: [email protected][1] P. de Bernardis et al.
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