From the emergence of cosmic space to horizon thermodynamics in Barrow entropy-based Cosmology

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Physics Letters B

journalhomepage:www.elsevier.com/locate/physletb

From the emergence of cosmic space to horizon thermodynamics in Barrow entropy-based Cosmology

G.G. Luciano

AppliedPhysicsSectionofEnvironmentalScienceDepartment,UniversitatdeLleida,Av.JaumeII, 69,25001Lleida,Spain

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received21December2022 Accepted21January2023 Availableonline25January2023 Editor: R.Gregory

Padmanabhan’sparadigm statesthat thespatial expansionofour Universe canbe understoodas the consequenceoftheemergenceofspacewiththeprogressofcosmictime.Basedonthisargument,here weextract thefirstFriedmannequationforacurved(n+¹)-dimensionalFriedmann-Robertson-Walker Universe and analyzethe consistencyofPadmanabhan’sproposal with horizonentropy maximization intheframeworkofBarrowentropy.Thelatterisaone-parameterdeformationofBekenstein-Hawking entropyinducedbyquantum-gravitationaleffects.WeshowthattheviabilityofPadmanabhan’sparadigm and its relationship with horizon thermodynamics are well supported in Barrow model, providing preliminaryindicationsonhowtheemergentgravityperspectiveshouldappearinaquantumgravity-like scenario.

©^{2023TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Indications from different research lines, such as black hole physics [1–5], holographic scenarios [6,7] and Verlinde’s formal- ism [8], allconvergetotheviewpointthat gravityandthermody- namics are deeply connected. Such a picture is further strength- enedbyJacobson’sconjecture [9],whichstatesthatEinstein’sfield equationscan bederived fromthecombinedusageofBekenstein bound and Clausius relation on a local Rindler causal horizon.

These arguments place gravitational field equations on an equal footing withequationsgoverningthedynamics ofemergentphe- nomena,likeelasticityorfluidmechanics.

While supporting the emergentgravity perspective, theabove studies treatspacetimeasafixed(i.e.a priorigiven)background.

ThepossibilitythatspacebeemergentwasfirstsuggestedbyPad- manabhaninCosmology [10,11].Inthisapproachtheexpansionof theUniverse isunderstoodastheemergence ofspacedueto the differencebetweenthenumberofdegreesoffreedomNsur f onthe boundarysurfaceoftheUniverseandtheoneN_bulk initsemerged bulk,i.e.

dV

dt

= 

^N

= 

N_{sur f}

−

^Nbulk

 ,

(1)

E-mailaddress:[email protected].

where V and dV denote the cosmic volume and its increasing in the time interval dt, respectively.¹ More precisely, t is the propertimemeasuredbyageodesicobserverwhoseesthecosmic microwave background radiation as homogeneous and isotropic.

Starting from Eq. (1), Friedmannequations for a flat Friedmann- Robertson-Walker(FRW)UniversecanbenaturallyderivedinGen- eralRelativity [10,11].

Overthe years, implications ofPadmanabhan’s paradigm (and related variants) have been considered in a variety of contexts.

Forinstance,in [12] Eq. (1) hasbeenexploitedtoinferFriedmann equationsforaflat(n+¹)-dimensionalFRWUniverseinEinstein, Gauss-BonnetandLovelockgravities,whileextensiontothecurved casehavebeensubsequentlyaddressedin [13].Asimilar analysis hasbeencarriedoutin [14,15] byassuminga changeofthe cos- mic volumein theform V = ^f(N,N_{sur f}), forsuitable choices ofthefunction f .Recently,Eq. (1) hasalsobeenappliedtoBIonic systems [16],brane scenarios [17–19] andminimallength frame- works [20] (see [21] and referencestherein for more studies on thesubject).

Ontheempiricalbasisofthesecondlawofthermodynamics,it isafactthatisolatedmacroscopicsystemsevolvetowardtheequi- libriumstate ofmaximum entropy [22] (seealso [23] for similar considerations in Thermal Quantum Field Theories). Remarkably, in [24] ithasbeenshownthatsuchaprinciplelikelyappliestoour Universetoo,the maximizedentropybeinginthiscasethe hori-

1 Unlessstatedotherwise,hereandhenceforthweusenaturalunits¯^h=^c=^G= kB=^1.

https://doi.org/10.1016/j.physletb.2023.137721

0370-2693/©^{2023TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

zonentropy.Consequencesofthisresulthavelaterbeenexplored in [25],where maximizationof horizonentropy hasbeenproven tobeimplied bytheholographicequipartitionlaw,andin [26] in connectionwiththeemergentgravityparadigm.

So far, horizon entropy maximization has been examined as- suming thesemiclassical Bekenstein-Hawking area law S=^A/A0 for the horizon entropy, where A and A0 denote the surface area of the Universe and Planck area, respectively. Bekenstein- Hawking entropy notoriouslyarises asthe blackhole application ofBoltzmann-Gibbs-Shannonentropy,whichisframedinthecon- textofclassicalMaxwell-Boltzmannstatistics.However,aspointed out in [27],long-rangeinteractingsystems - suchasgravitational ones- shouldobeynon-extensiveTsallisstatistics [28–34] inorder fortheirpartitionfunctionnottobedivergent.Furthercorrections areexpectedtocomeintoplaywhenthepresumablyquantumna- tureofgravityistakenintoaccount [35–39].

Among thevarious proposalsofquantum-like deformationsof Bekenstein-Hawking arealaw, theone introducedby Barrow [40]

lendsitselfparticularlywell foruseincosmologicalcontexts [41–

53],leadingtomodifiedFriedmannequationsthatpredictaricher phenomenologycomparingtothestandardtheory.Barrowentropy hastheform

S

=



A

A0



1+/²

,

(2)

where 0≤ ≤1 quantifies departure from Bekenstein-Hawking arealaw.Inparticular,=^{0 givesthestandardlimit,while}=¹ corresponds toits maximal deformation. In passing, we mention that observational constraints on this parameter have been ob- tainedin [41–50].

Motivatedbytherecentefforttoanalyzethecosmologicalim- plicationsof quantum gravitymodels for thedynamics ofspace- time andtheensuing departuresfromclassical GeneralRelativity, in the present work we examine the emergent gravity perspec- tive andits connection withhorizonthermodynamics within the framework of Barrow entropy. Specifically, we derive the Fried- mann equation for a (n+¹)-dimensional Friedmann-Robertson- WalkerUniversewithnon-vanishingspatialcurvatureanduseitas atestbench tostudy theconsistencyofPadmanabhan’sparadigm with horizon entropy maximization. We emphasize that the us- age ofa (n+¹)-dimensional backgroundshould not be regarded as a bare technicality, but is rather dictated by arguments from different quantum gravity approaches. Forinstance, stringtheory predicts that spacetimehasmorethan fourdimensions, withthe extra dimensions being curled up into microscopic spaces. Sim- ilarly, gauge/gravity duality formalism connects certain strongly coupled non-gravitational theories to higher-dimensional models includinggravity [54].Therefore, higher-dimensionalspacetimeis expectedtobe a prominentingredient ofmodelsthat attemptto embedquantumgravityeffectsintoCosmology.Inthissetting,we showthatthelawofemergenceiscompatiblewiththemaximiza- tion of horizon entropy, thus allowing to retain the view of the emergenceofcosmicspaceasatendencytoextremizehorizonen- tropyeveninquantumgravity(Barrow-like)scenarios.

The manuscript is structured as follows: in the next Section we obtain the modified first Friedmann equation for a (n+¹)- dimensional FRW Universe with non-vanishing spatial curvature in Barrow model. Following [29], we resort both to the law of emergence andfirst lawof thermodynamics, commentingon the consistency of the two approaches. Section 3 is devoted to con- strain the maximization of Barrow entropy, while in Sec. 4 we explorecompatibilityofourresultswithPadmanabhan’sparadigm.

ConclusionsandoutlookarefinallysummarizedinSec.5.

2. ModifiedCosmologyfromBarrowentropy

LetusconsideraspatiallyhomogenousandisotropicFRWback- groundofconstantspatialcurvaturek.Inordertofixnotationand tools,wefirstfollow [52] andfocuson(3+¹)-dimensions.Results are thenextended to themore general(n+¹)-dimensional case, withn≥^3.

Thespacetimemetricofa(3+¹)-dimensionalFRWUniverseis givenby

ds²

=

^hbcdx^bdx^c

+ ˜

^r2



d

θ

²

+

^sin2

θ

d

φ

²



,

(3)

where h_bc=^diag[−¹,a²/(1−^kr2)] ^{is the metric of the} (1+¹)- dimensionalsubspace,x^b= (^t,r),r˜=^a(t)r,a(t)isthescalefactor andr thecomovingradius.

Asstatedin [55],thedynamicalapparenthorizonisfixedby

h^bc

∂

br

˜ ∂

cr

˜ =

0

,

(4)

whichgivesfortheFRWUniverse

˜

r_A

= 

¹

H²

+

^k

/

a²

.

(5)

Here,H= ˙^a(t)/a(t)istheHubbleparameterandwehavedenoted theordinaryspatialderivativeby∂.²

The temperature associated to the apparent horizon follows fromthedefinitionofsurfacegravity [55]

κ =

¹ 2

√

−

^h

∂

a

 −

^hhab

∂

b

˜

^r



= −

¹

˜

r_A

1

−

r

˙˜

A

2H^r

˜

A

,

(6)

whichleadsto T

= κ

2

π ^{= −}

1 2

π

r

˜

A

1

−

^r

˙˜

A

2H

˜

rA

.

(7)

Clearly,for^r˙˜A≤^2Hr˜A,wehaveT ≤^{0.Negativevaluesoftemper-} aturecanbeavoidedbydefining T= |

κ

|/²

π

.Moreover,withinan infinitesimalintervaldt,onemayassume^r˙˜A^2Hr˜A,whichphys- icallyimpliesthat the apparenthorizonradius is keptfixed (this amounts to considering an equilibrium-likesystem). In turn, the approximation

T



¹ 2

π

r

˜

A

(8)

holdstrue [29,56].

WenowassumetheUniversetobefilledupwithaperfectfluid ofstress-energytensor

T_μν

= ( ρ +

^p

)

u_μu_ν

+

^pgμν

,

(9) whereuμ,

ρ

and p arethefour-velocity, densityandpressure of the fluid, respectively (we neglect tiny cosmological constant ef- fects).Fromtheconservationequation

∇

μT^μν

=

⁰

,

(10)

wethengetthecontinuityequation

˙

ρ _{= −}

3H

( ρ ₊

p

) .

(11)

Letus also define the effective area of the apparent horizon, the numberof surface degrees offreedom andthe increasing in theeffectivevolumeoftheUniverseinBarrowframeworkas [52]

2 Inournotationtheoverheaddotindicatesordinaryderivativewithrespectto thecosmictimet.

A

˜ =

4

π

r

˜

²_A^+

,

(12) N_{sur f}

=

A

˜

G_{e f f}

=

⁴

π

^r

˜

²_A^+

G_{e f f}

,

(13)

d_V

˜

dt

= ˜

rA

2 d_A

˜

dt

=

²

π (

2

+ ) ˜

^r2_A^+r

˙˜

A

,

(14)

respectively.Ontheotherhand,thenumberofdegreesoffreedom inthebulkisgivenby

Nbulk

=

^{2EK omar}

T

= −

¹⁶

π

²

3

( ρ +

3p

) ˜

r⁴_A

.

(15) Here

E_{K omar}

= |( ρ +

^3p

) |

^V

,

(16)

istheKomarenergycontainedinsidethesphereofvolume V and wehavetakenintoaccountthat

ρ

+^3p<0,consistentlywiththe conditionofanacceleratedUniverse [10].

Combining all the above ingredients, one can show that Pad- manabhan’s emergence of gravityandthe first lawofthermody- namicsbothleadtothemodifiedFriedmannequation [52]



H²

+

^k

a²



_1−/2

=

⁸

π

G_{e f f}

3

ρ ,

(17)

wheretheeffectivegravitationalconstantG_{e f f} reads G_{e f f}

=

^A0

4



2

− 

2

+ 

 

A₀ 4

π



_/2

.

(18)

Notice that in the →^{0 limit, G}e f f →^A0/4, thus returning backthestandardFriedmannequation.

2.1. ModifiedFriedmannequationfromthelawofemergencein (n+¹)-dimensions

Wenowextendtheaboveconsiderationsto(n+¹)dimensions.

In this setting the effectivearea (12) of the holographic surface shouldbegeneralizedto

A

˜ =

ⁿ

n

˜

r⁽_A^{n−1)(1+/2)}

,

(19) where

_n

≡ π

^n/2

(

n

/

2

+

¹

) ,

(20)

is the angularpart of then-dimensional sphere and is Euler’s function. Forn=^{3,we have}3=⁴

π

/3 andEq. (12) iscorrectly restored.

FromEq. (19),theincreasingintheeffectivevolumeoftheUni- verseisnow

dV

˜

dt

=

^r

˜

A

n

−

¹ dA

˜

dt

=

ⁿ

n

2

(

2

+ ) ˜

^r(_A^{n−1)(1+/2)}r

˙˜

A

,

(21) whichisconvenienttorecastintheequivalentform

dV

˜

dt

=

ⁿ

n

(

2

+ ) ˜

^rn_A⁺² 2 [

(

n

−

¹

)(

1

+ /

²

) −

ⁿ

−

^1]

d

dtr

˜

⁽_A^{n−1)(1+/2)−n−1}

.

(22) Similarly to [21],the definition (13) of numberofsurface de- greesoffreedombecomes³

3 We notice that the number ofdegrees offreedom as defined in Eqs. (13) and (23) arecorrectlydimensionlessforanyvalueofn and.Ontheotherhand, itisnotclearhowthecorrespondingdefinitionsin [21] canbesuchforTsallispa- rameterβ=^1.

N_{sur f}

= α (

2

+ )

ⁿ

n

[n

+

¹

− (

ⁿ

−

¹

)(

1

+ /

²

)

]

˜

r⁽_A^{n−1)(1+/2)}

G⁽_{e f f}^n−1)/2

.

(23) Again,it iseasy to check that Eq. (13) is recoveredforn=^3, providedthat

α ₌

²

^{− }

2

+  .

(24)

ThegeneralizationofKomarformula (16) to(n+¹)dimensions takestheform [57]

E_{K omar}

= |(

n

−

2

) ρ +

np

|

n

−

^{2 V}

= |(

ⁿ

−

²

) ρ +

^np

|

n

−

²

n

˜

rⁿ_A

,

(25)

where

ρ

andp mustnowberegardedasenergyandpressureper unit of n-volume, respectively. In turn, the number (15) of bulk degreesoffreedomis [52]

N_bulk

= −

⁴

π

n

(

n

−

2

) ρ +

np

n

−

^{2 r}

˜

ⁿ_A⁺¹

,

(26)

wherewehaveconsideredagain(n−²)

ρ

+^np<0 (seethediscus- sionbelowEq. (16)).

Additionally,weobservethattheconservation (9) ofthestress- energytensorinthe(n+¹)-dimensionalcaseyieldsthe general- izedcontinuityequation

˙

ρ _{= −}

nH

( ρ ₊

p

) .

(27)

Inorder to derive the modified Friedmann equation fromthe emergence law,we now need theproper formof Eq. (1). In this regard, we note that in [13] the following generalization ofPad- manabhan’s paradigm has been proposed for a curved (3+¹)- dimensionalFRWUniverseintheBekenstein-Hawkingholographic context

dV

dt

=

G r

˜

A

H⁻¹



Nsur f

−

Nbulk

 ,

(28)

where we have temporarily restored G for reasons that will be clearbelow.ComparisonwithEq. (1) pointsoutthatinanon-flat Universe,thecosmicvolumestillincreaseswiththedifferencebe- tweenthenumberofdegreesoffreedomontheapparenthorizon andtheonein thebulk. However,theproportionality functionis notaconstantanymore,butitisequaltotheratiooftheapparent horizonandtheinverseoftheHubbleparameter.Clearly,foraflat Universer˜A=^{H−1 andEq. (1) isrecovered.}

ThemostnaturalextensionofEq. (1) to Barrowentropy-based CosmologyisobtainedunderthereplacementsG→^Ge f f andV → V in˜ Eq. (28).Thisleadsto [52]

α

^dV

^˜

dt

=

^Ge f f

˜

rA

H⁻¹



N_{sur f}

−

^Nbulk

 ,

(29)

wherethefactor

α

hasbeenintroduced forlater convenience.In (n+¹)-dimensions,theaboverelationbecomes

α

^dV

^˜

dt

=

G⁽_{e f f}^n−1)/2 r

˜

A

H⁻¹



Nsur f

−

Nbulk



(30)

Asarguedin [52],theassumption (29) (and,consequently, (30)) canbeconceivedasasortofpostulate,whosevalidityischecked a posteriori by verifying that it correctly reproduces Friedmann equationdescribingthedynamicsoftheUniverse.

The proposal (30) is the key assumption of the presentanal- ysis, asit allowsusto inferFriedmannequation fromemergence scenario.Specifically,bypluggingEqs. (22), (23) and (26) into (30)

andmakinguseofthe generalizedcontinuityequation (27),after somealgebraonearrivesto

G⁽_{e f f}^n−1)/2 4

π

n

−

² d dt



a²

ρ ^

= α

n

(

2

+ )

2 [

(

n

+

1

) − (

n

−

1

) (

1

+ /

2

)

]

×

^d

dt



a²r

˜

⁽_A^{n−1)(1+/2)−n−1}



.

(31)

Integrationofbothsidesyields

r⁽_A^{n−1)(1+/2)−n−1} (32)

=

^G(_{e f f}^{n−1)/2 8}

π

n

−

²

[n

+

1

− (

n

−

1

) (

1

+ /

2

)

]

α

n

(

2

+ ) ρ ,

where the integration constant has been fixed by imposing the boundarycondition8

πρ

vac= ^0.

Asafinalstep,byusingEq. (5) weobtain



H²

+

^k

a²



1−(ⁿ−¹)/4

=

⁸

π

G⁽_{e f f}^n−1)/2

3

σ ρ ,

(33)

wherewehavedefined

σ ≡

³ n

−

²

[n

+

¹

− (

ⁿ

−

¹

) (

1

+ /

²

)

]

α

n

(

2

+ )

=

³

n

−

2

[n

+

¹

− (

ⁿ

−

¹

) (

1

+ /

²

)

]

n

(

2

− ) .

(34)

Therelation (33) providesthemodifiedFriedmannequation in acurved(n+¹)-dimensionalFRWUniversewithinBarrowframe- work.Somecommentsareinorderhere:first,weemphasizethat, forn=^{3,we have}

σ

=^1,sothatEq. (17) iscorrectlyreproduced.

Additionally,Eqs. (33) and (34) implytheconstraint n

+

¹

− (

ⁿ

−

¹

) (

1

+ /

²

) >

0

=⇒  <

⁴

n

−

¹

.

(35)

For n=^{3, this entails} <2, which is clearly satisfied because of the intrinsic limit ≤^{1 on Barrow exponent.More stringent} boundson canbe derivedforincreasing dimensionsn.Forin- stance,theconstraint^{1 isobtainedforn}^{5.Thus,strong(i.e.}

^{1)quantumgravityeffectsinBarrowframeworkarecompati-} blewithquantumtheoriesin(5+¹)-dimensionalspace.Pertinent examples ofthesemodels are those explored in [59] in thecon- text of particles and strings and in [60] inconnection with the one-looprenormalizationofcubicgravity.Finally,wewouldliketo note that, in orderfor the→^{0 limit of}Eq. (33) to be consis- tent withthe(n+¹)-dimensionalFriedmannequation derived in literature(seeforinstance [58]),onecouldsuitablyredefinetheef- fectivegravitationalconstantG_{e f f} inn+1 dimensions.Thissimply resultsinanextramultiplicativefactorinEq. (33).

2.2. ModifiedFriedmannequationfromthefirstlawof thermodynamicsin(n+¹)-dimensions

Further arguments supporting the consistency of the above computationscanbeprovidedbyderivingthemodifiedFriedmann equation (33) fromthefirstlawofthermodynamics

dE

=

^{T dS}

+

^{W dV (36)}

appliedtotheapparenthorizonofa(n+¹)-dimensionalFRWUni- verse embedded with Barrow entropy (2). Here, E=

ρ

V is the totalenergyofmatterinsidethen-dimensionalvolume V = nr˜ⁿ_A enclosed by the horizonsurface (notto be confusedwithKomar energy (25) usedintheemergent gravityperspective, whichalso

takes account of pressure contribute). Since the Universe is ex- panding,aworkdensity

W

=

¹

2

( ρ −

^p

)

(37)

isassociatedtothechangeofvolumedV . Now,differentiationofE gives dE

=

^{V d}

ρ ₊ ρ

dV

=

nr

˜

ⁿ_A

ρ ˙

dt

+ ρ

nnr

˜

ⁿ_A⁻¹dr

˜

A

,

(38) whichcanbe furthermanipulatedby useofthecontinuityequa- tion (27) togive

dE

= −

nr

˜

ⁿ_AnH

( ρ +

^p

)

dt

+ ρ

nn

˜

rⁿ_A⁻¹dr

˜

_A

.

(39) Onthe other hand,in theright handside of Eq. (36) we use the(n+¹)-dimensionalgeneralizationofBarrowentropy-arealaw

S

=

^c

A

A⁽₀^n−1)/2

1+/2

,

(40)

whereA=ⁿnr˜ⁿ_A⁻¹ and c

≡ π

^(n−1)/4

2

(

n

n

)

^/24^{(1+/2)(1−n)/2}

(

n

−

²

) (

n

−

¹

)



2

− 

2

+ 



(3−ⁿ)/2

(41) isaconstant thatplays thesamerole as

γ

inthe definition (37) of [21].Noticethatc→^{1 forn}=^{3,sothatEq. (2) isrecoveredin} thislimit.

FromEq. (40) itfollowsthat

dS

=

^c

1

A⁽₀^n−1)/2

1+/²

n

n

(

1

+ /

²

) (

n

−

¹

)

× 

n

nr

˜

ⁿ_A⁻¹



/2

˜

rⁿ_A⁻²d

˜

rA

.

(42) SubstitutionofEqs. (37)-(42) into (36) leadsto

H

( ρ +

p

)

dt

=

^c

(

n

−

1

) (

1

+ /

2

)



n

˜

rⁿ_A⁻¹



/2

2

π

r

˜

³_A

×

1

A⁽₀^n−1)/2

1+/²

dr

,

(43)

which,afteruseofthecontinuityequation (27),becomes

−

²

π



A⁽₀^n−1)/2



_1+/2

c n

(

n

−

¹

) (

1

+ /

²

) (

n

)

^/2d

ρ _{= ˜}

r⁽_A^{n−1)/2−3}dr

˜

_A

.

(44) Integratingtheaboverelation,weobtain

˜

r⁽_A^{n−1)(1+/2)−n−1}

= π

[4

− (

ⁿ

−

¹

) 

]



A⁽₀^n−1)/2



_1+/2

c n

(

n

−

¹

) (

1

+ /

²

) (

n

)

^/2

ρ .

(45) ByusingEq. (5),wearriveto



H²

+

^k

a²



1−(n−1)/4

= π

[4

− (

ⁿ

−

¹

) 

]



A⁽₀^n−1)/2



1+/²

c n

(

n

−

¹

) (

1

+ /

²

) (

n

)

^/2

ρ .

(46) WiththehelpofEq. (41), itiseasy toshow that thisrelation coincides with the modified Friedmann equation (33) extracted

fromthe emergence of cosmic space. Giventhe quantumgravity argumentsunderpinningBarrowentropy,theobtainedconsistency suggeststhatadeepconnectionbetweenPadmanabhan’sperspec- tive andthefirst lawofthermodynamicsattheapparenthorizon of aFRWUniverse should be rootedin quantumgravity theories too.Inturn,thismightprovide valuablehintsonhowFriedmann equation wouldbe correctedinaquantumgravity-basedCosmol- ogy.

3. MaximizationofhorizonentropyinBarrowscenario

Based onthe second lawof thermodynamics,it is empirically established that anyisolated macroscopicsystem evolves toward themaximumentropystate,consistentlywiththeconstraints [22]

S

˙ ≥

⁰

,

(47)

S

¨ <

0

.

(48)

Thefirstcondition alwaysholdstrue,whilethesecondoneisen- suredatleastasymptotically(i.e.fort→ ∞^).

Recently, in the framework of Einstein’s gravity it has been argued that our Universe behaves as a macroscopic-like system, wherethe maximizedentropy isthehorizonentropyandderiva- tives arecomputed withrespect tothe cosmic time [24]. Such a resulthasbeenlater proven tohavebroadervalidity, since itex- tendstoGauss-BonnetandLovelockgravities,andalsotonon-flat Universe [25,26].

The above models all consider the semiclassical Bekenstein- HawkingentropyforthehorizonoftheUniverse.Inviewofgradu- allyapproachingafullyquantumtheoryofgravity,hereweintend toexplorewhetherBarrowentropyisstillmaximizedasourUni- verseevolves.ThisisanessentialconditionforBarrowproposalto beacandidategeneralizationofBekenstein-Hawkingarealaw.

InordertocheckthevalidityofEq. (47),weresorttoEq. (40).

Derivativewithrespecttothecosmictimegivesthecondition^r˙˜A≥ 0.FromEq. (45),thisamountstosaying

˙˜

r_A

=

^{2n H}

(

1

+ ω )

4

− (

ⁿ

−

¹

) 

^r

˜

A

≥

⁰

,

(49)

where

ω

=^p/

ρ

is the so called Equation of State parameter of the perfectfluid that fills up our Universe.Since recentobserva- tions point out that our Universe is evolving toward a pure de Sitter state with

ω

≥ −^{1 [61] andbecause ofthe} condition (35), theconstraint (49) issatisfied.Accordingly,Eq. (47) remainsguar- anteedwithinBarrowframework.

LetusturnourattentiontoEq. (48).Bycomputingthesecond order derivative ofEq. (40) and imposing S¨<0 in thelong-time regime,wegetthecondition

[

−

⁴

−  +

ⁿ

(

2

+ )

^]r

˙˜

²_A

< −

²

˜

rA

¨˜

rA

.

(50) Sinceinthefinal deSitterUniverse

ω

→ −^1,fromEq. (49) we have˙˜^rA→^{0 fort}→ ∞^{.Hence,theaboveconstraintismetasymp-} totically,providedthat¨˜^rA<0.FromEq. (49),onehas

¨˜

r_A

=

²ⁿ 4

− (

ⁿ

−

¹

) 

 (

1

+ ω )

 ˜

^rAH

˙ +

^H

˙˜

^rA

 +

^Hr

˜

A

ω ˙ 

=

²ⁿ

˜

r_A 4

− (

ⁿ

−

¹

) 

(

1

+ ω )



H

˙ +

^{2n H2}

(

1

+ ω )

4

− (

ⁿ

−

¹

) 



+

H

ω ˙



,

(51)

whichgives tlim→∞r

¨˜

A

=

²ⁿr

˜

A

4

− (

ⁿ

−

¹

) 

^H

ω ˙ .

(52)

By exploitingagain Eq. (35) along with the fact that

ω

˙ <0, we inferthatEq. (48) isverifiedasymptotically.

Therefore, we conclude that Barrow entropy in a (n+1)- dimensionalFRWUniversewithnon-vanishingspatialcurvatureis stillmaximizedinthelong-timeevolution.

4. Connectingthelawofemergenceandhorizonentropy maximization

Letusnow explore whetherthe emergentgravity perspective isconsistentwiththemaximizationofhorizonentropyinBarrow Cosmology.Towardthisend,wecan useEqs. (21) and (40) tore- latetheincreasingintheeffectivevolumeoftheUniverseandthe entropyvariationas

dV

˜

dt

=



A⁽₀^n−1)/2



1+/²

c

(

n

−

¹

) (

n

)

^/2r

˜

A

˙

S

.

(53) FromEq. (30),thisinturngives

˙

S

=

^H

2

(

n

−

²

) 

N_{sur f}

−

^Nbulk

 .

(54)

Inorderforthecondition (47) tobesatisfied,wemusthave

N_{sur f}

−

^Nbulk

≥

⁰

.

(55)

Toshow whether thisinequality is true,we employ thedefi- nitions of N_{sur f} and N_bulk in Eqs. (23) and (26), respectively. By furtherresortingtothecontinuityequation (27) andthemodified Friedmannequation (33),weobtain

N_{sur f}

−

^Nbulk

=

¹ G⁽_{e f f}^n−1)/2

(

2

− )

ⁿ

2H

˜

rⁿ_A^{−2+(n−1)/2}r

˙˜

A

.

(56) Since our Universe is assumed to be asymptotically de Sitter, wehave˙˜^rA≥^{0 (seethediscussionbelowEq. (49)),whichensures} thevalidityofEq. (55).

On the other hand, as regards the condition (48), we obtain fromEq. (54)

¨

S

= (

n

−

2

)

2



H

˙ 

N_{sur f}

−

^Nbulk

 +

^Hd dt



N_{sur f}

−

^Nbulk

 .

(57)

Wenownoticethat,astheUniverseevolves, thegapbetween N_{sur f} andN_bulk tendstobereduced.Consequently,weget

d dt



N_{sur f}

−

^Nbulk

 <

0

.

(58)

InthefinaldeSitterstate,wewillhave

N_{sur f}

→

^Nbulk

,

(59)

sothat thefirstterminEq. (57) isasymptoticallyvanishing.This conditionalongwithEq. (58) thenimpliesthat S¨<0 inthelong- timeevolution.

Now,byuseofEq. (56),Eq. (57) canberecastas

¨

S

= (

n

−

²

) (

2

− )

ⁿ

8G⁽_{e f f}^{n−1)/2 r}

˜

ⁿ_A^{−3+(n−1)/2}

× 

2r

˜

_A

¨˜

^rA

+

^[n

(

2

+ ) −

⁴

− 

^]

˙˜

^r2_A



.

(60)

ThenegativityofS is¨ thenobtained,providedthat

2

˜

rAr

¨˜

A

+

[n

(

2

+ ) −

4

− 

]r

˙˜

²_A

<

0

,

(61) whichpreciselycoincideswiththeconstraint (50).

Thus,weconcludethatPadmanabhan’sperspectiveofemergent gravity is still consistent with maximization of horizon entropy within Barrow framework. This further supports the non-trivial relationship betweenthe lawof emergence andhorizonthermo- dynamics,andprovidepreliminaryguidelinesonhowthegravity- thermodynamic connection could appear approaching quantum gravityregime.

5. Conclusionsandoutlook

Inspired by Covid-19 fractal structure, Barrow argued that Bekenstein-Hawking entropy-area law should be generalized to Eq. (2) to take into account quantum gravitational effects on the black hole horizon surface [40]. In the lines of the gravity- thermodynamicconjecture,thisparadigm hasbeenappliedtothe Universe horizon too, giving rise to Barrow entropy-based Cos- mology. Withinsuchaframework, byresortingtoPadmanabhan’s perspectiveofemergentgravity [10,11],wehavederivedthemod- ifiedFriedmannequation fora(n+¹)-dimensionalFRWUniverse with non-vanishing spatial curvature. To strengthen our finding, the sameequation hasbeen theninferred starting fromthe first lawofthermodynamicsontheapparenthorizonsurface.Thecon- sistency between thetwo approaches both supports the viability ofPadmanabhan’slawandrevealstheintimaterelationshipofthe latter with horizon thermodynamics (and in particular with the firstlawofthermodynamics)inBarrowCosmology.

Furthermore,motivatedby therecentpredictionthat ourUni- versebehaveslikeamacroscopicsystemevolvingtowardanequi- librium state of maximum entropy [24], we have investigated Padmanabhan’s paradigminconnectionwiththemaximization of horizonentropy.Asaresult, wehaveshownthat botharguments lead to the same constraint, thus allowing to interpret the law of emergence as a tendency of our Universe to extremize hori- zon entropy. It is worth mentioning that this outcome fits with results formerlyobtained in Einstein, Gauss-Bonnet andLovelock gravities [25,26] and in the context of non-extensive Tsallis en- tropy [21]. In thissense, ourwork provides an effort toward the extension ofthegravity-thermodynamicconnectiontoa quantum gravity-likescenario.

Further aspects remain to be investigated: first, following the arguments as [52],the (n+¹)-dimensionalgeneralization (30) of the increasing in the effective volume of the Universe has been takenasasortofpostulate,whosevalidityhasbeenprovedapos- terioribyverifyingthatitleadstothecorrectFriedmannequation.

It wouldbe interesting toelaborate moreon thedeeper physical origin of Eq. (30) to establish how it can be derived at a more fundamental level.Thisisinorderto consolidatethefoundations ofthe connectionbetweenPadmanabhan’sparadigm andhorizon thermodynamics in the most general case of a curved (n+¹)- dimensional FRW Universe examined above. Moreover, since our model attempts toembed quantum gravitationalcorrections into CosmologyviaBarrowentropy,itisimportanttostudythepresent results comparing to predictions from more fundamental candi- date theories of quantum gravity, such as String Theoryof Loop QuantumGravity.Finally,weaimatextendingouranalysistothe caseof other deformedentropies,in particularto Kaniadakisen- tropy [62,63],whicharisesfromacoherentandself-consistentrel- ativistic generalization of the classical Boltzmann-Gibbs-Shannon entropy.Workalongtheseandother directionsiscurrentlyunder investigationandwillbepresentedelsewhere.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Dataavailability

Nodatawasusedfortheresearchdescribedinthearticle.

Acknowledgements

The author acknowledges the Spanish “Ministerio de Univer- sidades” for the awarded Maria Zambrano fellowship and fund- ing received from the European Union - NextGenerationEU. He is also grateful for participation in the COST Association Action CA18108“Quantum Gravity Phenomenology in the Multimessen- gerApproach”andLISACosmologyWorkinggroup.

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