Entropic corrections to Friedmann equations and bouncing universe due to the zero-point length

2025-04-22 0 0 207.88KB 7 页 10玖币

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Kimet Jusuﬁ1, ∗and Ahmad Sheykhi2, †

1Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, North Macedonia

2Physics Department and Biruni Observatory, College of Sciences,

Shiraz University, Shiraz 71454, Iran

We employ Verlinde’s entropic force scenario to extract the modiﬁed Friedmann equations by taking into

account the zero-point length correction to the gravitational potential. Starting from the modiﬁed gravitational

potential due to the zero-point length, we ﬁrst ﬁnd the logarithmic corrections to the entropy expression and

then we derive the modiﬁed Friedman equations. Interestingly enough, we observe that the corrected Fried-

mann equations are similar to the Friedmann equations in braneworld scenario. In addition, from the corrected

Friedmann equations, we pointed out a possible connection to the GUP principle which might have implications

on the Hubble tension. To this end, we discuss the evolution of the scale factor under the effect of zero-point

length. Finally, having in mind that the minimal length is of the Planck order, we obtain the critical density and

the bouncing behavior of the universe with a critical density and a minimal scale factor of the order of Planck

length.

I. INTRODUCTION

Since the discovery of black holes thermodynamics in

1970’s [1], physicists have been speculating that there should

be a profound connection between the gravitational ﬁeld equa-

tions and the law of thermodynamics. This is due to the fact

that thermodynamic quantities such as entropy and temper-

ature are, respectively, proportional to the horizon area and

surface gravity, which are pure geometrical quantities. Jacob-

son was the ﬁrst who disclosed that Einstein ﬁeld equation of

gravity is indeed an equation of state for the spacetime [2].

According to Jacobson’s argument, one can derive the hyper-

bolic second order partial differential equations of general rel-

ativity by starting from the Clausius relation δQ =T δS, to-

gether with the relation between horizon area and entropy [2].

Jacobson’s discovery was a great step toward understanding

the nature of gravity and supports the idea that gravitational

ﬁeld equations are nothing but the ﬁrst law of thermodynam-

ics for the spacetime. The profound connection between the

ﬁrst law of thermodynamics and the gravitational ﬁeld equa-

tions were also generalized to other gravity theories includ-

ing f(R)gravity [3], Gauss-Bonnet gravity, the scalar-tensor

gravity, and more general Lovelock gravity [4,5]. In the cos-

mological background, it has been shown that the differential

form of the Friedmann equation on the apparent horizon can

be rewritten in the form of the ﬁrst law of thermodynamics and

vice versa [6–14]. Although Jacobson’s derivation is logically

clear and theoretically sound, the statistical mechanical origin

of the thermodynamic nature of gravity remains obscure.

The next great step towards understanding the nature of

gravity put forwarded by Verlinde who claimed that gravity

is not a fundamental force and can be regarded as an entropic

force [15]. Verlinde proposal based on two principles. The

equipartition law of energy for the degrees of freedom of the

system and the holographic principle. Using these two princi-

ples, he derived the Newton’s law of gravity, the Poisson equa-

∗kimet.jusuﬁ@unite.edu.mk

†asheykhi@shirazu.ac.ir

tions and in the relativistic regime, the Einstein ﬁeld equations

of general relativity. Similar discoveries were also made by

Padmanabhan [16] who observed that the equipartition law

for horizon degrees of freedom combined with the Smarr for-

mula leads to the Newton’s law of gravity. This may imply

that the entropy is to link general relativity with the statisti-

cal description of unknown spacetime microscopic structure

when the horizon is present.

It is important to note that Verlinde’s proposal changed our

understanding on the origin and nature of gravity, but it con-

siders the gravitational ﬁeld equations as the equations of an

emergent phenomenon and leave the spacetime as a back-

ground geometry which already exists. In line with studies

to understand the nature of gravity, Padmanabhan [17] argued

that the spacial expansion of our Universe can be understood

as a consequence of the emergence of space. Equating the dif-

ference between the number of degrees of freedom in the bulk

and on the boundary with the volume change, he extracted the

Friedmann equation describing the evolution of the Universe

[17]. The idea of emergence spacetime was also extended to

Gauss-Bonnet, Lovelock gravities [18–20].

In the present work, we adopt the viewpoint of Verlinde and

consider gravity as an entropic force caused by the changes

in the information of the system. Using this scenario, we

shall investigate the effects of zero-point length corrections to

the gravitational potential on the cosmological ﬁeld equations.

The concept of duality and zero point length was derived on

the framework of quantum gravity by Padmanabhan [21]. In

his reasoning, the spacetime manifold can be taken as a large

distance limit of the main quantum spacetime. In this manner,

the discrete to continuum transition should have a memory

of the ﬂuctuations of the quantum spacetime. It was recently

used to obtain black hole solutions [22–25]. In order to in-

corporate such quantum gravity effects we need to know the

modiﬁed entropy. Toward this goal, we shall use Verlinde’s

entropic force scenario [15] to obtain the corrected entropy.

This paper is outlined as follows. In Section II, we derive

the corrected entropy and the modiﬁed Friedmann equations.

In Section III, we discuss the evolution of the scale factor un-

der the effect of zero-point length. In Section IV, we explore

the bouncing behavior of the modiﬁed Friedmann equations.

arXiv:2210.01584v3 [gr-qc] 7 Nov 2023

We comment on our results in Section IV. Through the paper

we set G=c=ℏ= 1.

II. ENTROPIC CORRECTIONS TO FRIEDMANN

EQUATIONS

According to Verlinde’s argument, when a test particle or

excitation moves apart from the holographic screen, the mag-

nitude of the entropic force on this body has the form

F△x=T△S, (1)

in this equation △xgives the displacement of the particle

from the holographic screen, on the other hand Tand △Sare

the temperature and the entropy change on the screen, respec-

tively. Verlinde’s derivation of Newton’s law of gravitation at

the very least offers a strong analogy with a well understood

statistical mechanism. It is important to note that in Verlinde

discussion, the black hole entropy Splays a crucial role in the

derivation of Newton’s law of gravitation. Indeed, the deriva-

tion of Newton’s law of gravity depends on the entropy-area

relationship S=A/4of black holes in Einstein’s gravity,

where A= 4πR2represents the area of the horizon. However,

this deﬁnition can be modiﬁed from the inclusion of quantum

effects, here we shall use the following modiﬁcation [26,27]

S=A

4+S(A).(2)

It can be seen that if we take the second term zero, the stan-

dard Bekenstein-Hawking result is reproduced. Assuming a

test mass mwith a distance △x=ηλmaway from the surface

Σ, where λmis the reduced Compton wavelength of the par-

ticle given by λm= 1/m in natural units, ηis some constant

of proportionality. then the entropy of the surface changes by

one fundamental unit △Sﬁxed by the discrete spectrum of

the area of the surface via the relation

dS =∂S

∂A dA =1

4+∂S

∂A dA. (3)

Here we note that the energy of the surface Sis identiﬁed with

the relativistic rest mass Mof the source mass:

E=M. (4)

On the surface Σ, we can relate the area of the surface to the

number of bytes according to

A=QN, (5)

where Qis a fundamental constant and Nis the number of

bytes. Let us assume tha the temperature on the surface is T,

by means of the equipartition law of energy [28], we get the

total energy on the surface via

E=1

2NkBT. (6)

We also need the force, which, according to this picture, it is

the entropic force obtained from the thermodynamic equation

of state

F=T△S

△x,(7)

where △Sis one fundamental unit of entropy when |△x|=

ηλm, and the entropy gradient points radially from the out-

side of the surface to inside. Note that we have △N= 1,

hence from (5) we have △A=Q. Now, we are in a posi-

tion to derive the entropic-corrected Newton’s law of gravity.

Combining Eqs. (3)- (7), we can get Fusing ∆S≃dS, fur-

thermore as we noted ∆x=ηλm=η/m and Tis found

from Eq. (6), where N=A/Q and T=M Q/(2πkBR2),

we get

F=−Mm

R2Q2

2πkBη1

4+∂S

∂A A=4πR2

,(8)

This is nothing but the Newton’s law of gravitation to the ﬁrst

order provided we deﬁne η= 1/8πkB, we get Q2= 1. Thus

we reach

F=−Mm

R21+4∂S

∂A A=4πR2

,(9)

In T-duality, it was argued that the gravitational potential due

to the zero-point length is modiﬁed as [22,24]

ϕ(r) = −M

pr2+l2

0|r=R,(10)

where l0is the zero-point length and its value is expected to

be of Planck length (see, [22]). Now by using the relation

⃗

F=−m∇ϕ(r)|r=R, we obtain the modiﬁed Newton’s law at

of gravitation as

F=−Mm

R21 + l2

R2−3/2

.(11)

Thus, with the correction in the entropy expression, we see

that

1 + 1

2πR dS

dR =1 + l2

R2−3/2

(12)

Solving the above equation, for the entropy we obtain

S=πR2+S=πR21 + l2

R2−1/2

+ 3πl2

01 + l2

R2−1/2

−3πl2

0ln(R+qR2+l2

0),(13)

It is very interesting to see that we obtained the log corrections

to the entropy in accordance with quantum effects. This is

the ﬁrst important results in the present work. Let us now

extend our discussion to the cosmological setup. Assuming

the background spacetime to be spatially homogeneous and

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EntropiccorrectionstoFriedmannequationsandbouncinguniverseduetothezero-pointlengthKimetJusufi1,∗andAhmadSheykhi2,†1PhysicsDepartment,StateUniversityofTetovo,IlindenStreetnn,1200,Tetovo,NorthMacedonia2PhysicsDepartmentandBiruniObservatory,CollegeofSciences,ShirazUniversity,Shiraz71454,IranWeemployVer...

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Entropic corrections to Friedmann equations and bouncing universe due to the zero-point length

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