Inflation and Fractional Quantum Cosmology

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arXiv:2210.00909v2 [gr-qc] 13 Apr 2023

Inﬂation and Fractional Quantum Cosmology

S. M. M. Rasoulia,b, Emanuel W. de Oliveira Costac, Paulo Vargas Moniza, Shahram Jalalzadehc

aDepartamento de F´ısica, Centro de Matem´atica e Aplica¸c˜oes (CMA-UBI), Universidade da Beira Interior, Rua

Marquˆes d’Avila e Bolama, 6200-001 Covilh˜a, Portugal.

bDepartment of Physics, Qazvin Branch, Islamic Azad University, Qazvin, Iran

cDepartamento de F´ısica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil

Abstract

The Wheeler–DeWitt equation for a ﬂat and compact Friedmann–Lemaˆıtre–Robertson–Walker

cosmology at the pre-inﬂation epoch is studied in the contexts of the standard and fractional quan-

tum cosmology. Working within the semiclassical regime and applying the WKB approximation,

we show that some fascinating consequences are obtained for our simple fractional scenario that

are completely diﬀerent from their corresponding standard counterparts: (i) The conventional

de Sitter behavior of the inﬂationary universe for constant potential is replaced by a power-law

inﬂation. (ii) The non-locality of the Riesz’s fractional derivative produces a power-law inﬂation

that depends on the fractal dimension of the compact spatial section of space-time, independent

of the energy scale of the inﬂaton.

Keywords:

inﬂation, fractional quantum cosmology, Wheeler–DeWitt equation, fractional calculus,

non-locality

1. Introduction

Inﬂation is widely accepted as a formal solution to cosmological problems such as horizon

and ﬂatness [1, 2, 3]. The conventional approach to controlling inﬂation is that a scalar ﬁeld with

an appropriate potential, such as the Coleman–Weinberg potential [4], dominates the energy

density of the universe from the start. In early theories of inﬂation, this energy density promotes

a rapid expansion for the scale factor of the universe [5, 6]. This accelerated expansion would be

exponential, similar to de Sitter space, or power law, according to which the scalar ﬁeld gradually

decreases to the global minimum of its potential.

In addition to the well-known standard models of inﬂation, recent scenarios using non-

commutative cosmological models, Generalized Uncertainty Principle (GUP) and other modiﬁed

gravity models have shown that some problems with the standard models (such as graceful exit

and the Hubble constant problem) can be solved, see, for instance [7, 8, 9, 10, 11, 12].

It should be noted that quantum cosmology is a suitable method to study the essential initial

conditions for the emergence of an inﬂationary phase. Tryon proposed in 1973 that a closed

universe [13] could emerge spontaneously as a quantum ﬂuctuation. He realized that in a spatially

closed universe, all conserved charges are zero. Consequently, no conservation law prevents

such a universe from forming spontaneously. Furthermore, according to general relativity, at the

instant of creation of such a universe, not only matter ﬁelds, but also space-time itself is created,

and there has not been anything earlier than that. In fact, if we assume that the universe is

Preprint submitted to Elsevier Apr-13- 2023

spatially homogeneous, isotropic, compact, and simply compact, the only feasible choice to have

a space without a boundary is a closed space. The volume of open and ﬂat spaces that are simply

connected is inﬁnite. By removing the restriction of simple connectivity, more sophisticated yet

ﬁnite volume spaces for all possible ﬂat, open, and close homogeneous and isotropic spaces can

be obtained (please see [14] and references therein). In this work, we assume that the cosmic

manifold is spatially compact since we are trying to understand the universe as a whole and

it is implicitly inconceivable that the universe has a spatial boundary. This implies that the

three-dimensional spacelike hypersurfaces are compact (in mathematical language, a compact

manifold without a boundary is called a closed manifold).

Intuitively, a universe formed by a quantum ﬂuctuation should be exceedingly small, having

a Planck length linear size. This was initially a major challenge since it was unclear how to

create the enormous universe we live in from a tiny compact quantum cosmological model. With

the emergence of inﬂationary theories in which the universe goes through a de Sitter phase of

exponential growth, the dilemma has vanished. All scales in the universe are expanded by a

gigantic factor exp(Ht) as a result of inﬂation, where His the constant expansion rate and tis the

duration of the inﬂationary phase or tαwhere αmust be greater than one and ideally at least of

the order of ten.

Given the centrality of inﬂation in our present understanding of cosmology, it is reasonable

and vital to try to grasp its details within the context of fractional quantum cosmology. The

history of fractional derivatives is as long as that of classical calculus. A fractional deriva-

tive is a generalization of the integer-order derivative. It originated in the letter regarding the

meaning of half-order derivative from L’ Hˆopital to Leibnitz in 1695 and is a promising tool

for explaining various phenomena. Various deﬁnitions of fractional derivatives exist in the lit-

erature, including Riesz, Riemann–Liouville, Caputo, Hadamard, Marchand, and Griinwald–

Letnikov, among others [15, 16]. Recent quantum gravity conclusions have provided an essen-

tial push for the increased use of fractional calculus in quantum theory. Various approaches

to quantum gravity, such as asymptotically safe quantum gravity [17, 18, 19, 20], causal dy-

namical triangulations [21, 22, 23, 24], loop quantum gravity, and spin foams [25, 26, 27],

Hoˇrava–Lifshitz gravity [28, 29], non-local quantum gravity [30, 31, 32], and others, all lead

to the same result: the dimension of space-time changes with scale. As a result of the abnor-

mal scaling of the space-time dimension, all known theories of quantum gravity are multiscale.

There are numerous applications for fractional calculus and active research in this ﬁeld. Frac-

tional quantum mechanics has been used to model fractional space-time in gravity and cosmol-

ogy [33, 34, 35, 36, 37, 38, 39, 40], and the fractional quantum ﬁeld theory [41, 42, 43].

The main objective of this work is to reexamine the standard inﬂation in the context of frac-

tional quantum cosmology (FQC) as a novel underlying scenario for studying the early state of

the universe. Concretely, this study aims to employ the FQC with a minimally coupled scalar

ﬁeld and a compact 3-space as a probe model. We are aware of the limited breadth. However,

fascinating aspects can be found. In the next section, we present a review of the model in the

context of the canonical quantum cosmology and obtain a semi-classical wavefunction. Section

3 focuses on the fractional counterpart of the model mentioned above. Then, by analyzing the

implications of fractional quantum cosmology, we examine why they may be more intriguing

than those reached in Section 2.

2. Wheeler-DeWitt Equation in Slow Roll Regime

Let us consider the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric

ds2=−N2(t)dt2+a2(t)dr2

1− Kr2+r2(dθ2+sin2θdϕ2),(1)

where N(t) is the lapse function, a(t) is the scale factor, and we will assume that the space-time

manifold, M, is a spatially compact and globally hyperbolic Lorentzian manifold. We claimed

that Mis spatially closed since we want to investigate the cosmological model as a whole. It

is basically impossible to imagine that the universe has a spatial boundary. Consequently, the

spacelike sections of the space-time manifold, (Σ,h), have a ﬁnite volume with the normalized

curvature index Ktaking values 0,±1. Diﬀerent ranges of variation for the coordinates (r, θ, ϕ)

determine diﬀerent topologies. Let us rewrite the metric as

ds2=gµνdxµdxν=−N2(t)dt2+a2(t)hi jdxidxj.(2)

Then, we can deﬁne the volume of the spacelike hypersurfaces as

VK≡Zd3x√h.(3)

Therefore, scalar curvature of the spacelike hypersurface (Σ,h) of the above metric is constant

and equal to 6K/a2. One can show that any compact Riemannian three-manifold with constant

curvature is homeomorphic to ˜

Σ/Γ, where Γdenotes the group of the covering transformations

and ˜

Σis the universal covering space, which is either R3(3-dimensional Euclidean space), S3

(3-sphere), or H3(3-dimensional hyperbolic space) regarding the sign of K(K=0, K=1,

or K=−1, respectively). Therefore, in three dimensions, an oriented compact and without

boundary space is now a polyhedron whose faces are identiﬁed in pairs. For an overview, see

Ref. [14].

As we would like to investigate a new scenario regarding an inﬂationary universe, let us

consider the same action corresponding to standard inﬂationary theories:

S=Zd4x√−g"R

16πG−1

2gµν∇µφ∇νφ−V(φ)#,(4)

where φis a scalar ﬁeld minimally coupled to the Ricci scalar R, and we used the units where

c=1=~. As during inﬂation only, the inﬂaton φdominates the dynamics, the action does not

contain any other matter ﬁelds.

By substituting the Ricci scalar corresponding to the metric (1), i.e.,

R=6

N2"¨a

a+˙a

a2

− ˙

N!˙a

a+KN

a2#,(5)

into action functional (4), we obtain Arnowitt–Deser–Misner (ADM) action of the model

SADM =3VK

8πGZdtNa3"K

a2−1

N2˙a

a2#− VKZdtNa3"−1

2N2˙

φ2+V(φ)#,(6)

where an overdot denotes diﬀerentiation with respect to t.

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摘要：

arXiv:2210.00909v2[gr-qc]13Apr2023InﬂationandFractionalQuantumCosmologyS.M.M.Rasoulia,b,EmanuelW.deOliveiraCostac,PauloVargasMoniza,ShahramJalalzadehcaDepartamentodeF´ısica,CentrodeMatem´aticaeAplica¸c˜oes(CMA-UBI),UniversidadedaBeiraInterior,RuaMarquˆesd’AvilaeBolama,6200-001Covilh˜a,Portugal.bDepa...

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