A unied model of ination and dark energy based on the holographic spacetime foam Daniel Jim enez-Aguilar Department of Physics UPVEHU 48080 Bilbao Spain

2025-04-27 0 0 956.2KB 6 页 10玖币

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A uniﬁed model of inﬂation and dark energy based on the holographic spacetime foam

Daniel Jim´enez-Aguilar∗

Department of Physics, UPV/EHU, 48080, Bilbao, Spain

I present a model of inﬂation and dark energy in which the inﬂaton potential is constructed by

imposing that a scalar ﬁeld representing the classical energy of the spacetime foam inside the Hubble

horizon is an exact solution to the cosmological equations. The resulting potential has the right

properties to describe both the early and late expansion epochs of the universe in a uniﬁed picture.

1. INTRODUCTION

Recent cosmological observations indicate that the

universe is currently experiencing an accelerated ex-

pansion [1–4]. This fact can be attributed to the

existence of some form of energy (dubbed dark en-

ergy) with negative pressure. Our unawareness of

the nature of dark energy is certainly at the root of

one of the greatest unknowns in theoretical physics:

the cosmological constant problem [5]. On the other

hand, it is widely accepted today that the very early

universe underwent a period of quasi-exponential ex-

pansion called inﬂation, which can address many of

the problems of the standard Big Bang cosmology [6–11].

Both expansion epochs may be traced to a com-

mon cause by interpreting dark energy as a dynamical

scalar ﬁeld (the inﬂaton) that slowly rolls down its

potential to drive inﬂation in the early universe and

ﬁnally resembles a cosmological constant at the present

time, at a much lower energy scale. Depending on the

shape of the potential, the ﬁeld will end up oscillating

about a minimum (as in the model presented in this

paper) or rolling down an inﬁnite tail as quintessence.

The latter scenario is called quintessential inﬂation [12].

Another chance of uncovering the nature of dark

energy may be found in the structure of spacetime itself.

Due to its quantum nature, spacetime is foamy on scales

of the order of the Planck length [13]. The quantum

ﬂuctuations of the metric are responsible for a perpetual

change in the geometry of spacetime, and consequently,

any measurement of space and time intervals acquires

some uncertainty. In a region of spatial extent L, this

uncertainty is given by the K´arolyh´azy relation [14],

which was derived independently by other authors

[15–18]:

δL ∼L2/3

pL1/3,(1)

where Lpis the Planck length. Note that δL ∼L

precisely at the Planck scale. This picture of the small

scale structure of spacetime is called spacetime foam.

An appealing idea that has been proposed is that the

energy density associated to the spacetime foam is the

one that drives both the early and late expansions of the

universe [19].

The K´arolyh´azy uncertainty relation (1) is closely

related to the holographic principle [20, 21], as it

establishes a connection between the ultraviolet (δL)

and infrarred (L) cut-oﬀ scales of the system. Indeed,

expression (1) is also known as the holographic space-

time foam model, as it suggests that the number of

degrees of freedom or bits of information in that region

is proportional to its surface area: (L/δL)3∝L2. The

ultraviolet cut-oﬀ scale is related to the energy density

of the vacuum, and one is led to a holographic dark

energy of the form

ρ∼1

pL2.(2)

The usual approach in holographic dark energy models

is to propose an ansatz for the infrarred cut-oﬀ L(some

natural choices are the Hubble radius [22, 23], the

event horizon [24], the age of the universe [25] and the

Ricci length [26]) and combine equation (2) with the

Friedmann equation in order to extract the Hubble rate

as a function of time. Then, one can see whether this

particular choice leads to an accelerated expansion, and

also constrain the model with the observational data.

While these models have been intensively applied to the

late universe (see [27] for a review), only a few authors

have extended these ideas to the early universe (for

instance, see [28]).

Finally, and more on the line of thought of this

work, there have been several studies aimed at estab-

lishing a connection between holographic dark energy

and scalar ﬁeld models [29–32], although the explicit

reconstruction of the scalar ﬁeld potential has proved to

be challenging (however, see [33, 34]).

In this paper, I present a uniﬁed model of inﬂation

and dark energy in which the scalar potential is recon-

structed by making a speciﬁc ansatz for the ﬁeld in the

cosmological equations. This ansatz will be rooted at

the holographic model of spacetime foam.

This letter is structured as follows: in section 2, I

expose the main idea behind the construction of the

inﬂaton potential. This idea is applied in section 3

under the assumption that the scalar ﬁeld is real. In

arXiv:2210.02556v2 [gr-qc] 18 Apr 2023

this case, the potential obtained is not satisfactory.

The calculation is redone in section 4 for a complex

scalar ﬁeld, obtaining a family of acceptable potentials.

In section 5, I pick up the potential that has direct

connection with the spacetime foam and I discuss some

of its properties. Finally, some concluding remarks are

made in section 6.

Natural units (c= ¯h≡1) are used throughout

the letter. In particular, this implies that Newton’s

gravitational constant is given by G=L2

p=M−2

p, where

Mpis the Planck mass.

2. THE MAIN IDEA

Regardless of the initial ﬁeld conﬁguration in the

patch of the universe that is going to inﬂate, the ﬁeld

evolves under the inﬂuence of a potential V(φ). The

Friedmann and Klein-Gordon equations in a spatially

ﬂat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW)

universe establish a correspondence between the poten-

tial and a homogeneous function of time φ(t). Therefore,

one can obtain the scalar potential by imposing that a

particular φs(t) is an exact solution to these equations

(the subscript sstands for solution). For instance, V(φ)

could be determined by imposing that φs(t)∝H(t)

is a solution to the equations (here and henceforth,

Hdenotes the Hubble rate). Whatever our ansatz

is, φs(t) is not the actual inﬂaton ﬁeld φ(t, ~x) in the

universe, but just a tool to determine the potential.

Indeed, a natural initial state for the ﬁeld is arbitrary

and inhomogeneous. Since φ(t= 0, ~x)6=φs(t= 0) and

φ(t, ~x)6=˙

φs(t= 0), the inﬂaton ﬁeld would simply end

up settling in the vacuum of the potential, possibly

resembling the cosmological constant.

Here we will obtain a family of scalar ﬁeld poten-

tials by considering the simplest ansatz with dimensions

of energy that one can make out of Mpand H:

φs(H) = AMγ

pH1−γ,(3)

where Ais a dimensionless constant and γis a generic

exponent that will label the diﬀerent elements of the

family of solutions. Some of them will not only have the

right shape to allow for inﬂation in the early universe

and cosmological constant at the present time, but also a

connection with the spacetime foam, as we point out now.

Note that ansatz (3) for γ= 2/3 is essentially the

inverse of the K´arolyh´azy uncertainty relation (1)

evaluated at the Hubble scale. One can show that 1/δL

is proportional to the energy of the spacetime foam

inside the Hubble horizon. The classical energy density

associated to the metric ﬂuctuations is given by [35]

ρfoam ∼1

L2/3

pL10/3,(4)

and although this expression was derived in Minkowski

spacetime, we will assume that the powers of Lpand L

remain unchanged in a spatially ﬂat FLRW universe. If

the scale Lin equation (4) is chosen to be the Hubble

radius, the classical energy of the spacetime foam in that

volume is

Efoam ∼M2/3

pH1/3.(5)

This expression for the energy can also be derived from

the Margolus-Levitin theorem in quantum computation

[36]. The dynamical evolution of any physical system

can be thought of as a succession of orthogonal quantum

states, and each step between consecutive states can be

understood as an elementary operation. It can be eas-

ily shown that each of these steps takes at least time

δt ∼E−1, where Eis the average energy of the sys-

tem. Therefore, E∼δt−1, which yields expression (5) if

we impose that δt corresponds to the uncertainty given

by the K´arolyh´azy relation (1) evaluated at the Hubble

scale.

3. REAL SCALAR FIELD

The simplest possibility is to consider a real scalar

ﬁeld. The ﬁrst Friedmann equation and the Klein-

Gordon equation read

H2=8π

3M2

p"˙

φ2

2+V(φ)#,(6)

φ+ 3H˙

φ+dV

dφ = 0.(7)

Taking the time derivative of (6) and combining it with

(7) yields

H=−4π

φ2.(8)

Since φ=φ(t) and H=H(t), one can take H=H(φ).

Taking this into account, and using equations (6) and

(8), one can express V(φ) as

V(φ) = M2

8π"3H2(φ)−M2

4πdH

dφ 2#.(9)

Now we determine V(φ) by imposing that φ=φs(H),

given by equation (3). Inverting to get H(φ) and substi-

tuting in (9) yields

V(φ) = M

2(1−2γ)

1−γ

8πA 2

1−γ"3φ2

1−γ−M2

4π(1 −γ)2φ2γ

1−γ#.(10)

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AuniedmodelofinationanddarkenergybasedontheholographicspacetimefoamDanielJimenez-AguilarDepartmentofPhysics,UPV/EHU,48080,Bilbao,SpainIpresentamodelofinationanddarkenergyinwhichtheinatonpotentialisconstructedbyimposingthatascalareldrepresentingtheclassicalenergyofthespacetimefoaminsidetheHubbleh...

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A unied model of ination and dark energy based on the holographic spacetime foam Daniel Jim enez-Aguilar Department of Physics UPVEHU 48080 Bilbao Spain

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