!emergence and expansion of cosmic space as due to the quest for holographic equipartition 1206.4916

7/27/2019 !Emergence and Expansion of Cosmic Space as Due to the Quest for Holographic Equipartition 1206.4916

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arXiv:1206

.4916v1

[hep-th]

21Jun2012

Emergence and Expansion of Cosmic Space as due to the Quest for Holographic

Equipartition

T. Padmanabhan

IUCAA, Post Bag 4, Ganeshkhind,Pune University Campus, Pune - 411 007, India

(Dated: June 22, 2012)

One possible interpretation of the holographic principle is the equality of the number of degreesof freedom in a bulk region of space and the number of degrees of freedom on the boundary surface.It is known that such an equality is maintained on equipotential surfaces in any static spacetimein the form of an equipartition law Nbulk [E/(1/2)kBT] = Nsur. In the cosmologicalcontext, the de Sitter universe obeys the same holographic equipartition. I argue that the differencebetween the surface degrees of freedom and the bulk degrees of freedom in a region of space (whichhas already emerged) drives the accelerated expansion of the universe through a simple equationV = t(Nsur Nbulk) where V is the Hubble volume in Planck units and t is the cosmic timein Planck units. This equation reproduces the standard evolution of the universe. This approachprovides a novel paradigm to study the emergence of space and cosmology, and has far reachingimplications.

PACS numbers:

Recent research supports the point of view that gravi-tational field equations in a wide class of theories shouldhave the same status as the equations of emergent phe-nomena like fluid mechanics or elasticity. (For a review,see [1]; for a small sample of work in the same spirit, see[2].) In my previous work on the emergent paradigm, Ihave treated only the field equations as emergent whileassuming the existence of a spacetime manifold, metric,curvature etc. as given structures. The field equationsthen arise as certain consistency conditions obeyed bythe background spacetime.

The question arises as to whether one can think ofthe spacetime itself as an emergent structure from

some suitably defined pre-geometric variables. While thephrase in quotes above sounds attractive, it is not ofmuch use unless the idea can be translated into mathe-matical expressions consistent with what we know aboutspacetime. There are (at least) two key issues which needto be satisfactorily addressed in this context.

To begin with, it appears conceptually very difficult tothink of time as being emergent from some pre-geometricvariable and yet provide a natural way of describing theevolution of dynamical variables. Secondly, it seems face-tious if not downright incorrect to argue that spaceis emergent around finite gravitating systems like theEarth, Sun, Milky Way, etc. Any emergent description

of the gravitational fields of such systems must acceptspace around them as pre-existing, just based on phys-ical evidence. Thus, in the study of finite gravitatingsystems without any special status to a time variable, itis extremely difficult to come up with a conceptually con-sistent formulation for spacetime itself as an emergentstructure.

Electronic address: [email protected]

Interestingly enough, both these difficulties disappearwhen we consider cosmology! Observations indicate thatthere is indeed a special choice of time variable avail-able in our universe, viz., the proper time of the geodesicobservers to whom CMBR appears homogeneous andisotropic. So, one has some justification in treating timedifferently from space in (and possibly only in) the cosmiccontext. Second, the spatial expansion of the universecan certainly be thought of as emergence of space as thecosmic time progresses. Thus, in the case of cosmology,we have a well defined setting to make concrete the ideathat cosmic space is emergent as cosmic time progresses.

To understand why cosmic space emerges or, equiv-alently, to understand why the universe is expanding

we can use a specific version of holographic principle. Tomotivate this, let us begin by considering a pure de Sit-ter universe with a Hubble constant H. It is known thatsuch a universe obeys the holographic principle in theform

Nsur = Nbulk (1)

Here the Nsur is the number of degrees of freedom on thespherical surface of Hubble radius H1 given by

Nsur =4

L2P

H2(2)

if we attribute one degree of freedom per Planck area.

The Nbulk = |E|/[(1/2)kBT] is the effective number ofdegrees of freedom which are in equipartition at the hori-zon temperature T = (H/2) if we take |E| to be theKomar energy |( + 3p)|V contained inside the Hubblevolume V = (4/3H3). That is,

Nbulk = E

(1/2)kBT=

2( + 3p)V

kBT(3)

If we substitute p = , then Eq. (1) reduces to thestandard result H2 = 8L2

P/3, showing the consistency.
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The ( + 3p) is the proper Komar energy density andV = 4/3H3 is the proper volume of the Hubble sphere.The corresponding co-moving expressions will differ bya3 factors in both, which will cancel out.

The condition in Eq. (1) can also be expressed in theform |E| = (1/2)NsurkBT which is the standard equipar-tition law that I have obtained for static spacetimes inGR in 2004 (see, Ref. [3]) and later generalized to a much

wider class of models [4]. It is therefore appropriate tocall the condition Eq. (1) as holographic equipartition be-cause it relates the effective degrees of freedom residingin the bulk, determined by the equipartition condition,to the degrees of freedom on the boundary surface.

Our universe, of course, is not exactly de Sitter butthere is considerable evidence that it is asymptoticallyde Sitter. This would suggest that the expansion of theuniverse which I consider as conceptually equivalent tothe emergence of space is being driven towards holo-graphic equipartition. Then the basic law governing theemergence of space must relate the emergence of space (inthe form of availability of greater and greater volumes ofspace) to the difference (NsurNbulk). The most naturaland simplest form of such a law will be

V = t(Nsur Nbulk) (4)

where V is the Hubble volume in Planck units and t isthe cosmic time in Planck units. More generally, onewould have expected (V/t) to be some function of(Nsur Nbulk) which vanishes when the latter does. Wecould then imagine Eq. (4) as a Taylor series expansionof this function truncated at the first order.

I will now elevate this relation to the status of a postu-late and show that it is equivalent to the standard Fried-mann equation. Reintroducing the Planck scale and set-

ting (V/t) = dV/dt, this equation becomesdV

dt= L2P(Nsur Nbulk) (5)

Substituting V = (4/3H3), Nsur = (4/L2PH2), T =

H/2 and using Nbulk in Eq. (3), this equation simplifiesto the relation:

a

a=

4L2P

3( + 3p) (6)

which is the standard dynamical equation for the Fried-mann model. Using the energy conservation for matterd(a3) = pda3 and the de Sitter boundary condition

at late times, one gets back the standard acceleratinguniverse scenario.

The definition of Nbulk given in Eq. (3) makes senseonly in the accelerating phase of the universe where ( +3p) < 0 thereby making Nbulk > 0. For normal matterwe would not like to have the negative sign in Eq. ( 3).This is easily taken care of by using appropriate signs forthe two different cases and writing:

dV

dt= L2P(Nsur Nbulk); (7)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1V = t(Nsur Nbulk)Degrees of freedom whichhave already emerged

Surface degrees of freedom

Cosmic space whichhas already emerged

Degrees of freedom and spacewhich are yet to emerge

Nbulk

Nsur

FIG. 1: Illustration of the ideas described in this paper. Theshaded region is the cosmic space which has already emergedby the cosmic time t, along with (a) the surface degrees of free-dom (Nsur) and (b) the bulk degrees of freedom (Nbulk) thathave reached equipartition at the Hubble temperature. Thefurther emergence of cosmic space, measured by the increasein the volume of the Hubble sphere with respect to cosmictime, is driven by the holographic discrepancy (NsurNbulk)

between these two as indicated by the equation in the figure.This equation correctly reproduces standard cosmic evolution.

and redefining

Nbulk = 2( + 3p)V

kBT(8)

with = +1 if ( + 3p) < 0 and = 1 if ( + 3p) > 0.(One could have, of course, used the opposite conventionfor and omitted the minus sign in Eq. (8); I prefer tomaintain the form of Eq. (4) for the accelerating phaseof the universe.) Since only the combination +2( +3p) (+3p) occurs in (dV/dt), the derivation of Eq. (6)remains unaffected and we also maintain Nbulk > 0 in allsituations. (See Fig. 1.)

We can gain more insights about Eq. (7) by separat-ing out the matter component, which causes deceleration,from the dark energy which causes acceleration. Let usassume, for simplicity, that the universe has two compo-nents (matter and dark energy) with ( + 3p) > 0 formatter and ( + 3p) < 0 for dark energy. Then, Eq. (7)can be written in an equivalent form as

dV

dt= L2P(Nsur + Nm Nde) (9)

with each ofNsur, Nm, Nde being positive (as they shouldbe) with (Nm Nde) = (2V/kBT)( + 3p)tot. We seethat if we want dV/dt 0 asymptotically, we need acomponent in the universe with ( + 3p) < 0. A universewithout a dark energy component has no hope of reach-ing holographic equipartition. It is also important thatthe equation of state for the dark energy component isnot too different from that of a cosmological constant sothat the universe does not go into a super exponentialexpansion.


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Stated in this manner, the existence of a cosmologicalconstant in the universe is required for asymptotic holo-graphic equipartition. While these arguments, of course,cannot determine the value of the cosmological constant,the demand of holographic equipartition makes a strongcase for its existence. As far as I know, this is more thanwhat any other model has achieved.

Writing the basic equation in the form of Eq. (9), sug-

gests that there must be an intimate relationship betweenthe matter degrees of freedom and dark energy degrees offreedom. In any fundamental theory of quantum gravity,we expect matter and gravitational degrees of freedomto emerge together and hence such a relationship is ex-pected. Discovering this relationship will be a primarychallenge for any pre-geometric model since it might evenallow us to determine the value of the cosmological con-stant.

The most obvious and important extension of this workwill be to study the growth of perturbations in the uni-verse using a perturbed version of Eq. (9). To do thisformally in a gauge invariant manner, one will need to

express the full gravitational field equation as an evolu-tion equation towards holographic equipartition. (I hopeto discuss this in detail in a future publication.) But it isfairly straightforward to work out perturbation theory ina specific gauge even without the full formalism. Giventhe fact that Eq. (9) is identical to Eq. (6), we do notexpect any surprises in the perturbation theory exceptfor the following feature.

These ideas suggest that we should think of the uni-verse at any given time as being endowed with tempera-ture T = H/2 as far as the emergence of space is con-cerned. This is the temperature related to the degrees offreedom which have already become emergent in the cos-

mos (from the pre-geometric variables) and should notbe confused with the normal kinetic temperature of mat-ter. But since the mathematics is identical, we will ex-pect thermal fluctuations at the temperature T = H/2to be imprinted on any sub-system which has achievedequipartition. This effect will lead to some corrections tothe cosmological perturbation theory in the late universewhen we do a thermal averaging in a canonical ensemble.This is somewhat similar in spirit to the thermal fluctu-ations at the de Sitter temperature leaving their imprinton the density fluctuations which were generated duringinflation.

To avoid possible misunderstanding, I stress that thisis in addition to (and not instead of) any imprint ofthe current Hubble constant H0 on the sub-horizon cos-mic structures due to standard structure formation pro-cesses. There are straightforward dynamical effects invarious aspects of structure formation (like formation ofdark matter halos, cooling of gas, formation of galax-ies with flat rotation curves arising due to interplay ofbaryonic and DM components, etc.) all of which de-pend on the background expansion and thus, on H0.For example, the standard cosmic structure formationscenario does suggest that (see e.g., [5]) a preferred ac-

celeration scale a0 = cH0 is imprinted on galactic scalestructures. (This is sometimes presented as evidence forMOND, etc. which I believe is unfounded.) One cantake any standard result in structure formation whichdepends on H0, and rewrite it in terms of the horizontemperature using H = 2T, and try to present it in anemergent/thermodynamic language. This is unnecessaryand does not add to our insight. It is therefore important

to distinguish between (a) rewriting standard results interms of the horizon temperature, and (b) deriving gen-uine effects which arise due to the emergence of cosmicspace and the evolution of degrees of freedom towardsholographic equipartition. I hope to return to this issuein a future publication.

There are three more comments I want to make re-garding Eq. (4). First, the simplicity of Eq. (4) is quitestriking and it is remarkable that the standard Fried-mann equation can be reinterpreted in this form as anevolution towards holographic equipartition. If the un-derlying ideas are not correct, it is very difficult to under-stand why Eq. (4) holds in our universe. For example,

consider the definition of horizon temperature in a non-deSitter universe. In obtaining Nbulk we have assumed thatthe relevant temperature is given by T = H/2. Defin-ing a horizon temperature when H is time-dependent isnot easy and there is some amount of controversy in theliterature regarding the correct choice. It is possible toobtain equations similar to Eq. (4) with other definitions(all of which agree in the de Sitter limit) but none of themseems to have the compelling naturalness which Eq. (4)possesses. The same is true as regards the volume ele-ment V which I took as the Hubble volume; other choicesare possible but then the resulting equation has no simpleinterpretation.

Second, this equation presents the emergence of cosmicspace as an evolution towards holographic equipartitionin a bit-by-bit increase in Planck units. When thecosmic time changes by one Planck unit, the amount ofextra cosmic space which becomes available changes by(Nsur Nbulk). This is reminiscent of ideas in whichone considers cosmic expansion as a computational pro-cess. Presented in Planck units, Eq. (4) has no ad-

justable parameters and opens the possibility for inter-pretation in combinatorial terms when we understand thepre-geometric variables better.

Third, we expect that when the relevant degrees offreedom like Nsur are of the order of unity, Eq. (4) mustbreak down. Studying possible modifications of this

equation when the degrees of freedom are few will helpus to study the evolution of the universe close to the bigbang in a quantum cosmological setting but withoutthe usual complications related to the time coordinate.In other words, postulating well motivated corrections tothe bit dynamics expressed by Eq. (4) may be a betterway of tackling the singularity problem, rather than byquantizing gravity in the usual manner.

To conclude, let me summarize how these ideas fit ina broader context.


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The degrees of freedom are the basic entities in physicsand the holographic principle suggests a deep relation-ship between the number of degrees of freedom residingin a bulk region of space and the number of degrees offreedom on the boundary of this region. Given a fun-damental area scale, L2

P, it makes sense to count the

surface degrees of freedom as A/L2P

where A is the areaof the surface. The non-trivial task is to come up with a

suitable measure for the bulk degrees of freedom whichmust depend on the matter residing in the bulk. (Thisnecessary dependence on the matter variables precludescounting the bulk degrees of freedom as V /L3

P.) It is here

that the idea of equipartition comes to our aid. If thesurface is endowed with a horizon temperature T, thenwe can think of E/(1/2)kBT as the number of effectivebulk degrees of freedom. The logic behind this identifi-cation is as follows. We assume that the bulk region islike the inside of a microwave oven with the temperatureset to the surface value, as far as the degrees of freedomwhich have already emerged (along with the space) areconcerned. Then, if these degrees of freedom accountfor an energy E, it follows that E/(1/2)kBT is indeedthe correct count for effective Nbulk. This temperatureT and Nbulk should not be confused with the normal ki-netic temperature of matter residing in the bulk and the

standard degrees of freedom we associate with matter. Itis more appropriate to think of these degrees of freedomas those which have already emerged, along with space,from some pre-geometric variables. I postulate that theemergence of cosmic space is driven by the holographicdiscrepancy (Nsur + Nm Nde) between the surface andbulk degrees of freedom where Nm is contributed by nor-mal matter with ( + 3p) > 0 and Nde is contributed by

the cosmological constant with all the degrees of freedombeing counted positive. It is obvious that in the absenceofNde, this expression can never be zero and holographicequipartition cannot be achieved. In the presence of thecosmological constant, the emergence of space will soonlead to Nde dominating over Nm with the universe com-mencing accelerated expansion. Asymptotically, Nde willapproach Nsur and the rate of emergence of space, dV/dt,will tend to zero in a de Sitter universe. The cosmoswould have found its peace.

The expansion which we witness (which is equivalent toemergence of space) is an evolution towards equilibriumof a system which is right now away from equilibrium.Such an evolution makes perfect thermodynamic sense.Needless to say, this approach provides a completely dif-ferent paradigm to study cosmology.

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!emergence and expansion of cosmic space as due to the quest for holographic equipartition 1206.4916

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