BOSE-EINSTEIN CONDENSATION AND BLACK
HOLES IN DARK MATTER AND DARK ENERGY

A Thesis Submitted to
the Graduate School of Engineering and Sciences of
İzmir Institute of Technology
in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in Physics
by Kemal GÜLTEKİN

Supervisor: Prof. Dr. Recai ERDEM

July 2023
İzmir

## Acknowledgements

I would like to express my deepest gratitude and endless respect to my
supervisor Prof. Dr. Recai ERDEM for his inestimable support, guidance,
patience, and encouragement in this thesis study. He was always welcoming me.
It was an honour and a privilege to be supervised by him.

I would like to thank Prof. Dr. Kadri YAKUT, Assoc. Prof. Dr. Suat DENGİZ,
Prof. Dr. Alev Devrim GÜÇLÜ, and Assoc. Prof. Dr. Fatih ERMAN as the thesis
defence jury members. They carefully read my thesis and provided useful
comments that improved the quality of the thesis. It was a great pleasure for
me to meet and discuss with them.

I also would like to thank Dr. Betül DEMİRKAYA for her contributions and
supports. I am also very indebted to Hemza AZRİ for his endless friendship and
encouragement during my Ph.D period. Special thanks to my nearest friend
Bahadır PİLTEN who was there when I needed help. I always feel lucky to have a
friend like him.

Finally, my most important and heartfelt thanks to my family for their non-
stop support, quidance, encouragements and patience. They have shown me
unconditional love in all situations. I would like to dedicate this thesis to
them.

## Abstract

BOSE-EINSTEIN CONDENSATION AND BLACK HOLES

IN DARK MATTER AND DARK ENERGY

The main aim of this study is to reveal curved space and particle physics
effects on the formation of Bose-Einstein condensate scalar fields in
cosmology and around a black hole. Cosmological scalar fields for dark energy
and dark matter may be considered as a result of Bose-Einstein condensation.
In this regard, our main attention will be devoted to Bose-Einstein
condensates in curved space. By considering the dynamics of a scalar Bose-
Einstein condensation at a microscopic level, we first study the initial phase
of the formation of condensation in cosmology. To this end, we initially
introduce an effective Minkowski space formulation that enables considering
only the effect of particle physics processes, excluding the effect of
gravitational particle production and enabling us to see cosmological
evolution more easily. Then, by using this formulation, we study a model with
a trilinear coupling that induces the processes. After considering the phase
evolution of the produced particles, we find that they evolve towards the
formation of a Bose-Einstein condensate if some specific conditions are
satisfied. In principle, the effective Minkowski space formulation introduced
in this study can be applied to particle physics processes in any sufficiently
smooth spacetime. In this regard, we also analyse if a condensate scalar field
is realized in the spacetime around a Reissner - Nordstrøm black hole. We find
that the produced particles of particle physics processes are localized in a
region around the black hole and have a tendency toward condensation if the
emerged particles are much heavier than ingoing particles. We also find that
such a configuration is phenomenologically viable only if the scalars and the
black hole have dark electric charges. Finally, we consider gravitational
collapse around Schwarzschild black holes and form a first step towards a
study in future about the effects of gravitational collapse on Bose-Einstein
condensation.

## Publications

  1. 1.

R. Erdem and K. Gültekin, A mechanism for formation of Bose-Einstein
condensation in cosmology, JCAP 10 (2019), 061
doi:10.1088/1475-7516/2019/10/061, [arXiv:1908.08784 [gr-qc]].

  2. 2.

R. Erdem and K. Gultekin, Particle physics processes in cosmology through an
effective Minkowski space formulation and the limitations of the method, Eur.
Phys. J. C 81 (2021) no.8, 726 doi:10.1140/epjc/s10052-021-09524-8,
[arXiv:2102.05587 [gr-qc]].

  3. 3.

R. Erdem, B. Demirkaya, and K. Gültekin, Curved space and particle physics
effects on the formation of Bose–Einstein condensation around a
Reissner–Nordstrøm black hole, Eur. Phys. J. Plus 136 (2021) no.9, 972
[erratum: Eur. Phys. J. Plus 137 (2022) no.7, 822],
doi:10.1140/epjp/s13360-021-01973-0, [arXiv:2110.00799 [gr-qc]].

  4. 4.

R. Erdem, B. Demirkaya and K. Gültekin, A metric for gravitational collapse
around a Schwarzschild black hole, Mod. Phys. Lett. A 38 (2023) no.07,
2350048, doi:10.1142/S0217732323500487, [arXiv:2007.04672 [gr-qc]].

###### Contents

  1. 1 Introduction
  2. 2 The Standard Model of Cosmology
    1. 2.1 A Brief Overview
    2. 2.2 Problems of the Standard Model
  3. 3 Cosmological Scalar Field Models as Alternatives to the Standard Model of Cosmology
    1. 3.1 Quintessence
    2. 3.2 k-essence
    3. 3.3 Phantom
    4. 3.4 Tachyon
    5. 3.5 Coupled Scalar Field Models
  4. 4 Bose-Einstein Condensation
    1. 4.1 A Brief Overview of Bose-Einstein Condensation in Condensed Matter Physics
    2. 4.2 Derivation of Gross-Pitaevskii Equation from $\phi^{4}$ Theories
      1. 4.2.1 Reducing Klein-Gordon Equation in Flat Spacetime to Gross-Pitaevskii Equation
      2. 4.2.2 Reducing Klein-Gordon Equation in Curved Spacetime to Gross-Pitaevskii Equation
    3. 4.3 Some Cosmological Bose-Einstein Condensate Scalar Field Models
  5. 5 A Mechanism for the Formation of Bose-Einstein Condensation in Cosmology
    1. 5.1 An Effective Minkowski Space Formulation
    2. 5.2 Realizing the Conditions for Condensation
      1. 5.2.1 Achieving Coherence and Correlation
      2. 5.2.2 Achieving Finite Number Density of $\phi$ Particles
  6. 6 Curved Space and Particle Physics Effects on the Formation of Bose-Einstein Condensation around a Reissner - Nordstrøm Black Hole
    1. 6.1 Framework
      1. 6.1.1 Motion in Radial Direction
      2. 6.1.2 The Field Equations for the Scalars
    2. 6.2 A Special Solution
      1. 6.2.1 Derivation
      2. 6.2.2 Viability of the Solution
  7. 7 A Metric for Gravitational Collapse around a Schwarzschild Black Hole
    1. 7.1 Framework
      1. 7.1.1 Preliminaries
      2. 7.1.2 Derivations of a Condition that Insures Robertson-Walker and Schwarzschild Metrics being Particular Solutions
    2. 7.2 Derivation of a Metric for Gravitational Collapse
    3. 7.3 Some General Implications of the Metric
      1. 7.3.1 General Behaviour of the Solution
      2. 7.3.2 A Brief Overview of the Apparent Horizons and the Singularities of the Solution
    4. 7.4 Some Additional Comments about the Energy Density and the Pressures in the Dust Collapse
  8. 8 Concluding Remarks
  9. A Condition (5.6) in the Case of Radiation, Matter, Cosmological Constant, and Stiff Matter Dominated Era
  10. B Formal Derivation of Equation (5.8)
  11. C The Evolution of the Number Densities in the Case of $s>2$ in Equation (A.1)
  12. D The Scalar Field Equation in 1+1 Dimensions
  13. E Derivation of Equations (6.21)-(6.25)
  14. F Derivation of Equation (6.36)
  15. G Incompatibility of the Conditions $G^{0}_{\;1}\,=\,0$, (7.17), (7.20)

###### List of Figures

  1. 5.1 The leading order Feynman diagrams contributing to the production of $\phi$ particles. Here $\tilde{q}_{t}=\tilde{p}_{1}-\tilde{p}_{3}=\tilde{p}_{4}-\tilde{p}_{2},\ \tilde{q}_{u}=\tilde{p}_{1}-\tilde{p}_{4}=\tilde{p}_{3}-\tilde{p}_{2}$ are the 4-momenta carried in the internal lines. Note that the s-channel is forbidden in this case by kinematics.
  2. 6.1 $|\psi_{\omega}|$ versus $\bar{r}$ graph for $\bar{M}=10$, $\bar{Q}=17$, $\bar{r}_{0}=20$
  3. 6.2 $|\psi_{\omega}|$ versus $\bar{Q}$ graph for $\bar{M}=10$, $\bar{r}=20$, $\bar{r}_{0}=30$
  4. 7.1 $\gamma=\frac{t}{t_{0}}$ (where $t$ is the time of any epoch of the universe and $t_{0}$ is the present time) versus the location of apparent horizon for $M=10M_{\odot}$, where $M_{\odot}$ is the solar mass.
  5. 7.2 Time versus $G^{0}_{\;1}$ (blue) and $G^{1}_{\;1}$ (dashed red) graphs for $c=1$, $M=0.1$, $r=3$.
  6. 7.3 Time versus $G^{1}_{\;1}$ (dashed red), $G^{0}_{\;0}$ (blue), $G^{2}_{\;2}$ (dotted green) graphs for $c=1$, $M=1$, $r=3$.
  7. 7.4 Time versus $G^{1}_{\;1}$ (dashed red), $G^{0}_{\;0}$ (dotted blue), $G^{2}_{\;2}$ (green) graphs for $c=0.1$, $M=1$, $r=30$.
  8. 7.5 Figure 2 with the function $H=\left(\frac{\dot{a}}{a}\right)^{2}\,a^{3}\,r^{3}-a\,r+2M$ is included. The zeros of $H$ correspond to Eq.(7.43) (where $\left(\frac{\dot{a}}{a}\right)^{2}\,a^{3}\,=\,\frac{8\pi}{3}m\,n_{0}$ is used). The green solid line here corresponds to $\frac{H}{10^{4}}$.

##  Chapter 1 Introduction

The general theory of relativity, introduced by Albert Einstein in 1915, is
the most comprehensive and standard framework for gravity. One of the main
applications of general relativity (GR) is cosmology, i.e., the study of the
universe at the largest (i.e. IR) scales. Observations suggest that the
average distribution of matter at cosmological scales, i.e., at scales larger
than 100 Mpc ($\approx 3.3\times 10^{21}\ \text{km}$) is homogeneous and
isotropic. The corresponding general relativistic description of the universe
is formulated by Robertson-Walker-Friedmann-Lemaitre universe [1, 2].

Observations imply that the amount of the ordinary matter called baryonic
matter (i.e., stars, interstellar, intergalactic dust) is not enough to
account for the whole matter in the universe. Therefore, there must be some
non-luminous matter, called dark matter (DM), in addition to ordinary matter.
Further, the observed accelerated expansion of the universe, when formulated
in the Friedmann-Lemaitre-Robertson-Walker model requires additional energy
density called dark energy (DE). In this context, $\sim$ 95 $\%$ of the
universe consists of non-luminous energy densities known as DE ($\sim$ 72
$\%$) and DM ($\sim$ 23 $\%$), and the remaining $\sim$ 5 $\%$ consists of
visible matter known as baryonic matter ($\sim$ 4 $\%$) and radiation ($\sim
8\times 10^{-5}\ \%$). The simplest framework for DM is non-luminous
cosmological dust, which is cold dark matter (CDM), while the simplest
explanation for DE is the cosmological constant ($\Lambda$) of Einstein’s
general theory of relativity in addition to the baryonic matter and the
radiation (photons and neutrinos) [3]. This model is called the standard model
of cosmology, namely the $\Lambda\text{CDM}$ model [1, 2, 3].

$\Lambda\text{CDM}$ model agrees with observational data very well. However,
this model has some crucial shortcomings. One of the main shortcomings of
$\Lambda\text{CDM}$ is the cosmological constant (CC) problem [3, 4, 5]. The
potential theoretical contributions to CC are huge when compared to DE’s
energy density. For example, the contribution of zero point energies to CC is
about $10^{121}$ times the observed energy density of DE. This is called a CC
problem. CDM part of $\Lambda\text{CDM}$ also has some problems at a scale
smaller than the galactic scale, such as the problem of predicting too dense
cores for the galaxies (i.e., core-cusp or cuspy halo problem) and too many
dwarf galaxies when compared with observations [3, 6, 7]. There are also two
potential problems for the $\Lambda\text{CDM}$ model, which are $H_{0}$
tension between its values determined locally and the ones derived at
cosmological scales [8, 9, 10], and $\sigma_{8}-S8$ problems [11, 12, 13].

To cope with these problems, alternative theories to $\Lambda\text{CDM}$ are
considered. In literature, the most popular alternatives to
$\Lambda\text{CDM}$ are scalar field models for DE and DM, e.g. quintessence
models [14], k-essence, phantom scalar field, and coupled scalar field models,
etc. [14, 15, 16]. In these types of models, the scalar fields are assumed to
depend only on time because of the homogeneity and the isotropy of the
universe at the background level. Such scalar fields are called cosmological
scalar fields and may be considered to be the result of Bose-Einstein
condensation [17, 18].

The dynamics of Bose-Einstein condensates are described by the Gross-
Pitaevskii (GP) equation [19, 20]. GP equation may be considered the non-
relativistic limit of $\phi^{4}$ theories [21, 22, 23]. Many researches exist
on Bose-Einstein condensate (BEC) scalar field models for DE and DM [21, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. They are usually studied at a
macroscopic level (i.e., at the level of number densities or distribution
functions) in cosmology, while its microscopic nature at the level of particle
physics is not considered sufficiently. In this context, we have formulated a
new mechanism for the formation of scalar field Bose-Einstein condensation in
cosmology with particular emphasis on its microscopic description in particle
physics in [35, 36].

Besides the cosmological framework, there are also some studies [37, 38, 39,
40, 41, 42, 44, 43, 45, 46, 47, 48, 49], where the applicability of BEC at
galactic scale, in particular around black holes, is considered. Galactic
halos formed by bosons, either in the context of Bose-Einstein condensation or
in an appropriate isothermal distribution, was first suggested by Baldeschi et
al. [37] in 1983. In 1994, this idea was improved by Sin [38], who explained
the problem of galaxy rotation curves by self-gravitating BECs. In 2000,
Goodman suggested that a BEC interacting with gravity and with itself via an
appropriate potential may play the role of dark matter [39]. Arbey et al. [40]
in 2003 showed that the BEC of a self-interacting charged scalar field may
account for the dark matter and explain the problem of the rotation curve of
the dwarf spiral galaxies. In 2007, Böhmer and Harko [41] reinforced the idea
that dark matter can be a BEC and they compared the predictions of their model
with the observational data. There are also some recent studies pursuing the
idea of BEC dark matter. The problem of predicting DM distributions at cores
of the galaxies may be well-explained in the context of BEC in [42]. The
observed rotational curves of galaxies is realised in studies [44, 43] if the
BEC plays the role of DM halos. The dynamics of rotating BEC dark matter halos
consisting of ultralight spinless bosons around a galaxy and the effect of the
rotating halo on the galactic rotation curves are examined in [45]. The
formation of supermassive black holes via the collapsing BEC dark matter
distributions is studied in [46]. In [47, 48], the mechanisms of the
gravitational collapse of the BEC dark matter halos with repulsive self-
interactions of the particles are examined. The collapse of a self-gravitating
BEC with attractive self-interaction is studied in [49]. In a wholly general
relativistic framework, we have studied in [50] if condensation could be
formed around a Reissner-Nordstrøm (RN) black hole [51], where we consider the
curved space and particle physics effects on the formation of the condensate.

Besides, to set up a framework toward a future investigation into how
gravitational collapse can affect Bose-Einstein condensation, we derive a new
metric in [52] that describes the gravitational collapse of a cosmological
fluid, in particular, dust around a primordial Schwarzschild black hole. The
mechanism for forming stars, galaxies, and clusters of galaxies out of small
inhomogeneities is based on gravitational collapse. In this regard, there are
some significant studies [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67] examining gravitational collapse. One of them is the McVittie
spacetime [53] that studies, in 1933, Schwarzschild black holes embedded in a
cosmological background. It studied the effects of cosmological expansion on
the dynamics of black holes and considers the case of non-accretion, i.e.,
$G^{0}_{\;1}=0$ which corresponds to the zero radial power flux of the fluid.
Tolman [54] in 1934 studied the effect of inhomogenities on cosmological
models that consist of dust particles distributed around a central object
without an accretion, i.e., the pressure of the dust is negligible. Another
work by Oppenheimer and Synder in 1939 [55] examined a star that collapses by
the effect of its gravitational field. In 1947 Bondi [56] proposed a metric to
understand some characteristics of the spherically symmetric systems with zero
pressures. This metric can also be reduced to other type of metrics (such as
Schwarzschild, Robertson-Walker, Oppenheimer and Synder) except the McVittie
metric after various coordinate transformations specifying some particular
value [57]. However, all these models we mentioned do not consider the
accretion of the black holes [57, 58]. On the other hand, in 1951, Vaidya [59,
60] proposed a completely different metric that allows the absorption or
radiation of massless dust by a black hole so that the black hole is accreting
in case of null dust. All the other metrics that describe the gravitational
collapse are proposed as extensions of the above metrics. For instance, a
generalized metric of Oppenheimer-Snyder [61] explains the way of the
gravitational collapse of stars into regular black holes. Thakurta [62] and
McClur-Dyer [63] extend the McVittie metric. Still it has been shown [64, 65]
that this metric does not correspond to the cosmological black holes. In
contrast, Sultana-Dyer [66] metric, as an extension to McVittie metric,
studies a black hole in the presence of cosmological constant, i.e., a black
hole in de Sitter space [57, 58]. As another example, Faraoni-Jacques [67]
extend the McVittie metric to explain an accreting black hole in the presence
of an imperfect fluid with heat flow [57]. In other words, only Faraoni-
Jacques black holes among McVitie type metrics can define accreting black
holes. The Faraoni-Jacques solution in that form, however, cannot represent
accreting black holes in common fluids, such as cosmic dust.

The present thesis is organised as follows: In Chapter 2, we briefly review
the $\Lambda$CDM model and its potential problems. In Chapter 3, we briefly
examine cosmological scalar field models, particularly the quintessence scalar
field model. We also mention other types of scalar fields. In Chapter 4, we
study the BEC scalar field at the levels of condensed matter physics and
relativistic scalar field models. We conclude the section after examining some
cosmological BEC scalar field models. In Chapter 5, we formulate a mechanism
for the formation of the BEC scalar field in cosmology. To this end, we first
introduce an effective Minkowski space formulation and then we proceed to
realize the condensation. In Chapter 6, we study the curved space and particle
physics effects on the formation of Bose-Einstein condensation around an RN
black hole. In Chapter 7, we derive a new metric for the gravitational
collapse around a black hole. This metric may lead to a condensate around an
accreting black hole, which is the more realistic point to be addressed in
future studies. We finally give concluding remarks in Chapter 8.

##  Chapter 2 The Standard Model of Cosmology

###  2.1 A Brief Overview

At the beginning of modern cosmology, without any data, the scientists (just
for mathematical simplicity) came up with the idea that we are not at the
center of the universe, the universe looks the same at every point and in all
directions, and we do not have a distinguished position in the universe. This
hypothesis is called the ‘cosmological principle’ (CP). CP implies that the
universe is homogeneous (i.e., the universe appears the same at every point)
and isotropic (i.e., the universe appears the same in all directions) on large
scales $>$ 100Mpc. The cosmic microwave background (CMB) data [68, 69, 70] in
the last decades seem to confirm CP.

The homogeneity and the isotropy of the universe on large scales imply that
the universe should be described by Friedman-Robertson-Walker (FRW) metric [1,
2]:

$ds^{2}=-dt^{2}+a(t)^{2}\bigg{[}\frac{dr^{2}}{1-kr^{2}}+r^{2}\big{(}d\theta^{2}+\text{sin}^{2}\theta
d\varphi^{2}\big{)}\bigg{]}.$ (2.1)

Here $ds$ measures the proper distance between two distinct locations in
spacetime when they are separated by $dx^{\mu}$. $a(t)$ is the cosmic scale
factor and measures how physical distances are expanding by time. $k$ is the
constant measures the curvature of the spatial part of the spacetime and it
may be rescaled to take the values +1 , 0, -1 for the spherical, flat, and
hyperbolic geometries, respectively.

The metric tensor of the spacetime $g_{\mu\nu}$ (in
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$), namely the gravitational field, is
considered as the fundamental dynamical variable in the general theory of
relativity, and it satisfies Einstein’s field equation:

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-g_{\mu\nu}\Lambda=8\pi G_{N}T_{\mu\nu},$
(2.2)

where both $R$ (Ricci curvature scalar), $R_{\mu\nu}$ (Ricci curvature
tensor), and $\Lambda$ (cosmological constant) are the geometrical objects
that describe the curvature of the spacetime and the right-hand side is
related to the energy-momentum tensor $T_{\mu\nu}$ of any energy distribution.
(2.2) tells us how the curvature of the spacetime and the matter affect each
other. $g_{\mu\nu}\Lambda$ term may be absorbed in the redefinition of
$T_{\mu\nu}$.

For the homogeneity and isotropy of the universe, the appropriate energy-
momentum tensor is that of a perfect fluid, namely,
$T^{\mu}_{\nu}=\text{diag}(-\rho,p,p,p)$ [1, 2], where $\rho$ is the mass
density and $p$ is the pressure of the fluid (A perfect fluid is a fluid with
no heat flow nor viscosity). By considering the metric in Eq.(2.1) together
with the perfect fluid form for the energy-momentum tensor, the Einstein field
equations (2.2) result in the following Friedmann equations:

$H^{2}\equiv\bigg{(}\frac{\dot{a}}{a}\bigg{)}^{2}=\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}$ (2.3) $\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho+3p),$ (2.4)

where $\dot{a}=\frac{da}{dt}$, $\ddot{a}=\frac{d^{2}a}{dt^{2}}$, and the rate
of the expansion of the universe is described by the Hubble parameter
$H=\frac{\dot{a}}{a}$. The first Friedmann equation (2.3) describes how the
universe’s energy drives its expansion. The second Friedmann equation (2.4)
gives the rate of the acceleration of the expansion. These two Friedmann
equations can be combined to obtain the continuity equation

$\dot{\rho}+3H(\rho+p)=0,$ (2.5)

which can be also derived by the conservation law of the energy-momentum
tensor, i.e., $\nabla_{\nu}T^{\mu\nu}=0$. $\rho$ and $p$ may be related by the
equation of state parameter $w=\frac{p}{\rho}$ which is constant for each
energy density in $\Lambda\text{CDM}$. For a constant $w$, Eq.(2.5) can be
solved to obtain the relation

$\rho\propto a^{-3(1+w)},$ (2.6)

which after plugging into Eq.(2.3) with $k\simeq 0$ by the observations [71],
gives the explicit time dependence of the scale factor:

$\displaystyle a(t)\propto t^{2/(3(1+w))}\ \ \ \text{for}\ \ \ w\neq 1$ (2.7)
$\displaystyle a(t)\propto e^{Ht}\ \ \ \text{for}\ \ \ w=-1.$ (2.8)

The universe consists of radiation, non-relativistic matter, and DE [1, 2].
For each fluid, the equation of state parameter can be written separately as
$w_{i}=\frac{p_{i}}{\rho_{i}}$. Extremely relativistic matter is included in
radiation, which at this time consists of only photons and neutrinos and we
have $w_{\text{rad}}=1/3$ for radiation. Hence $\rho_{\text{rad}}\propto
a^{-4}$ and the scale factor evolves as $a_{\text{rad}}\propto t^{1/2}$ for
the radiation dominated universe. For non-relativistic matter consisting of
baryons and DM, the pressure may be negligible compared to their mass density
so that we have $w_{\text{mat}}=0$. Thus $\rho_{\text{mat}}\propto a^{-3}$ and
the scale factor evolves as $a_{\text{mat}}\propto t^{2/3}$ for the matter
dominated universe. For the cosmological constant (i.e., vacuum energy), we
have $w_{\Lambda}=-1$. Thus, $\rho_{\Lambda}$ is constant and the scale factor
grows as $a_{\Lambda}\propto e^{Ht}$, which implies an exponentially
accelerated expansion for the universe. Radiation controlled the dynamics of
the early universe until around 47,000 years after the Big Bang. The energy
density of matter overtook those of radiation (and vacuum energy) about 47,000
years after the Big Bang. The cosmological constant dominated epoch is
regarded as the final of the known universe’s three eras and it started after
the matter dominated epoch, when the universe was approximately 9.8 billion
years old.

If we introduce a parameter $\Omega_{i}=\rho_{i}/\rho_{c}$ where
$\rho_{c}=3H^{2}/8\pi G$ is the critical density, Eq.(2.3) can be rewritten as

$\Omega=\Omega_{rad}+\Omega_{mat}+\Omega_{\Lambda}=1+\frac{k}{(aH)^{2}}.$
(2.9)

The observed present values of the quantities for each type fluid are [1, 2]

$\Omega_{\text{rad}}\sim 8\times 10^{-5},\ \ \ \
\Omega^{\text{Baryonic}}_{\text{mat}}\sim 0.04,\ \ \ \
\Omega^{\text{Dark}}_{\text{mat}}\sim 0.23,\ \ \ \ \Omega_{\Lambda}\sim 0.72$
(2.10)

with $k\simeq 0$ (i.e., approximately flat). As clearly can be seen from
Eq.(2.10), the present universe is dominated by the cosmological constant
$\Lambda$ adopted as dark or vacuum energy.

###  2.2 Problems of the Standard Model

Having introduced the standard model of cosmology ($\Lambda$CDM) briefly, we
are now able to mention its problems in short. The problem of cosmological
constant arises from corrections to the vacuum energy density of the energy-
momentum tensor in Eq.(2.2), e.g. as arising from zero point energies of
quantum theory. The contribution of zero point energies to vacuum energy
density is [72, 73, 74, 75]

$\rho_{\text{vac}}=\ <0\mid\delta\hat{T}_{00}\mid 0>\
=\frac{1}{2}\int_{0}^{\infty}\frac{d^{3}\textbf{k}}{(2\pi)^{3}}\sqrt{k^{2}+m^{2}},$
(2.11)

which consists of all the vacuum energy (i.e., lowest energy state)
contributions of quantum fields of mass $m$. Because there is an upper limit
for the energy scale of quantum field theory, the infinity in the upper limit
of the integral can be replaced by some cut-off scale $k_{\text{max}}$ [72,
73, 74, 75]. This gives

$\rho_{\text{vac}}\approx\frac{k_{\text{max}}^{4}}{16\pi^{2}}.$ (2.12)

Since the General Relativity still hold true slightly below the Planck scale,
one may use for the cut-off $k_{\text{max}}=m_{\text{pl}}=1,22\times
10^{19}\text{GeV}$ [4, 5, 76], which results in the vacuum energy density

$\rho_{\text{vac}}\approx 10^{74}\text{GeV}^{4}.$ (2.13)

This contributes to the bare energy density of $\Lambda$ and the effective
vacuum energy density is obtained as

$\rho^{\text{theo}}_{\text{eff}}\sim\rho_{\Lambda}+10^{74}\text{GeV}^{4}.$
(2.14)

However, the present cosmological observations based on the standard model of
cosmology shows that [4, 5, 76]

$\rho^{\text{obs}}_{\text{eff}}\lesssim 10^{-47}\text{GeV}^{4}.$ (2.15)

Therefore, $\rho^{\text{theo}}_{\text{eff}}$ predicted from the theory is
about $10^{121}$ times larger than $\rho^{\text{obs}}_{\text{eff}}$ based on
observations. This huge difference gives rise to the so-called cosmological
constant problem and this extreme discrepancy is needed to be explained and
resolved.

For DM, the problems arise from the discrepancies between the observations and
the simulations based on the $\Lambda$CDM, at small scales. One of the well-
known problems is the rotational curves of galaxies. The observations show
that in a wide variety of distances around a galaxy, the rotating stars orbit
in circles at approximately constant tangential speeds as opposed to the
theory where these tangential speeds are expected to be decreased. Therefore,
there must be additional matter distributions to keep the rotational speeds of
the stars constant. Another disagreement is called a core-cusp problem. The
simulations based on $\Lambda$CDM form DM distributions in galaxies with inner
radial density $\rho\propto r^{-1}$, which shows the cuspy density profile at
small radii. However, observations show the constant-density DM profiles at
cores as $\rho\propto r^{0}$. Another problem is the missing satellites
problem (or dwarf galaxy problem). Compared to the measured number of
satellite galaxies around the Milky Way, the simulations forecast excessive
dwarf galaxies orbiting Milky Way-sized galaxies. There is also an issue with
the local voids at small scales, which consist of only a few galaxies in their
vast spaces. However, simulations predict more dense regions in these voids. A
detailed review of these and additional problems at small scales are given in
[6, 7, 77]. Understanding these problems is needed for the applicability of
standard cosmology at small scales.

The Hubble tension is another intriguing issue in cosmology at the moment [8,
9, 10]. Local measurements of the universe’s current expansion rate that don’t
substantially rely on cosmological model assumptions, such as those obtained
from supernovae, provide numbers clustering around $73\ \text{km}\
\text{s}^{-1}\ \text{Mpc}^{-1}$. $\Lambda$CDM, which examines the early
universe through the CMB, offers a second, indirect method for measuring what
we assume to be the same amount. Using this approach, the Hubble constant is
estimated to be approximately $67\ \text{km}\ \text{s}^{-1}\ \text{Mpc}^{-1}$.
Thus the difference between the readings taken locally and at cosmological
scales is around 4$\sigma$, the standard deviation between the two
measurements.

In addition to the Hubble tension, there is another main anomaly called
$\sigma_{8}$ tension [11, 12, 13]. On a scale of $8\ \text{h}^{-1}\
\text{Mpc}$ ($\text{h}=H_{0}/(100\ \text{km}\ \text{s}^{-1}\
\text{Mpc}^{-1})$), $\sigma_{8}$ represents the amount of current matter
fluctuation. There is a discrepancy in this amount between the study of recent
data on the CMB observation and observations of galaxy clusters. According to
CMB estimates $\sigma_{8}\sim 0.82$ in contrast to lower estimates
$\sigma_{8}\sim 0.75$ based on the detection of galaxy clusters.

##  Chapter 3 Cosmological Scalar Field Models as Alternatives to the Standard
Model of Cosmology

As discussed in the previous chapter, the evolution of the universe is
described by the Einstein field equations (2.2). These equations are obtained
by considering the Einstein-Hilbert action [78, 79]

$S_{EH}=\int d^{4}x\sqrt{|g|}\bigg{[}\frac{c^{4}}{16\pi
G_{N}}\big{(}R(g)-2\Lambda\big{)}+\mathcal{L}_{M}\bigg{]},$ (3.1)

where $g\equiv\text{Det}[g_{\mu\nu}]$ is the determinant of the metric tensor,
$R(g)=g^{\mu\nu}R_{\mu\nu}(g)$ is the curvature scalar of metric, $\Lambda$ is
the cosmological constant, $\mathcal{L}_{M}$ is the Lagrangian density
describing matter fields, $G_{N}$ is the gravitational Newtonian constant and
$c$ is the speed of light in vacuum. The energy-momentum tensor is given by
$T_{\mu\nu}\equiv\frac{-2}{\sqrt{|g|}}\frac{\delta S_{M}}{\delta g^{\mu\nu}}$,
where $S_{M}=\int d^{4}x\sqrt{|g|}\mathcal{L}_{M}$ is the matter part of the
action (3.1). As we pointed out, $\Lambda\text{CDM}$ has some problems related
to DE and DM. Therefore, some alternative models to $\Lambda\text{CDM}$ are
proposed in the literature. These models may be classified broadly as
‘modified gravity models’ and the ‘modified energy-momentum models’. In this
chapter, some alternative scalar field models to $\Lambda\text{CDM}$ are
discussed in the context of ‘modified energy-momentum models’. In these
models, DE and DM are provided by scalar fields with a negative pressure for
DE and zero pressure for DM. This may be done by replacing $\Lambda$ and/or
CDM by the contribution of a scalar field. In this chapter, we shall be
interested in the modified $T_{\mu\nu}$ models, particularly cosmological
scalar models. Although we particularly consider some pure scalar field models
in this chapter, there are also some recent intriguing models such as [80, 81,
82] concerning Weyl’s local invariant gravity theories where the local
conformal gauge symmetry is achieved with the help of additional extra scalar
and gauge fields, and the masses of the excitations are generated through a
specific spontaneous symmetry breaking mechanism as in the standard model. It
may be interesting in future studying these extra fields in the context of
dark matter and/or dark energy.

In the following section, we shall start with the quintessence model, the most
preferred scalar field model. Then, in the following sections, we shall
continue with brief discussions on the other preferred scalar field models,
such as k-essence, phantom, tachyon and coupled scalar field models [14].

###  3.1 Quintessence

Quintessence model is associated with a canonical scalar field, say $\phi$,
which has the following form of the Lagrangian density [83, 84]

$\mathcal{L}_{\phi}=X-V(\phi),$ (3.2)

where $X=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ and
$V(\phi)$ are the kinetic energy and the potential of the field, respectively.
The corresponding action is obtained by replacing the cosmological constant
$\Lambda$ in Eq.(3.1) with the lagrangian density of the quintessence field,
$\mathcal{L}_{\phi}$:

$S=\int d^{4}x\sqrt{|g|}\bigg{[}\frac{c^{4}}{16\pi
G_{N}}R(g)+\mathcal{L}_{\phi}+\mathcal{L}_{M}\bigg{]},$ (3.3)

where the main dynamical fields are $g_{\mu\nu}$ and $\phi$. The condition of
stationarity of the action under variations of $g_{\mu\nu}$ results in

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi
G_{N}\big{(}T^{M}_{\mu\nu}+T^{\phi}_{\mu\nu}\big{)}.$ (3.4)

Here $T_{\mu\nu}\equiv\frac{-2}{\sqrt{|g|}}\frac{\delta S_{M}}{\delta
g^{\mu\nu}}$, where $S_{M}=\int d^{4}x\sqrt{|g|}\mathcal{L}_{M}$ and
$T^{\phi}_{\mu\nu}\equiv\frac{-2}{\sqrt{|g|}}\frac{\delta S_{\phi}}{\delta
g^{\mu\nu}}=\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\big{[}\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+V(\phi)\big{]}$,
where $S_{\phi}=\int d^{4}x\sqrt{|g|}\mathcal{L}_{\phi}$. The equation of
state parameters for $T^{M}_{\mu\nu}$ (for a general perfect fluid) and for
$T^{\phi}_{\mu\nu}$ (for quintessence field) are $p_{M}=w_{M}\rho_{M}$ and
$p_{\phi}=w_{\phi}\rho_{\phi}$, respectively. Here $w_{M}$ is introduced for
the cases of the radiation and non-relativistic matter and $w_{\phi}$ is
introduced in place of $w_{\Lambda}$. Bu using the FRW metric with the above
form for $T^{\phi}_{\mu\nu}$, the energy density and the pressure of the
quintessence field are obtained as

$\displaystyle\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V(\phi),$ (3.5)
$\displaystyle p_{\phi}=\frac{\dot{\phi}^{2}}{2}-V(\phi).$ (3.6)

Then, the corresponding equation of state of the field is given by

$w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{\dot{\phi}^{2}-2V(\phi)}{\dot{\phi}^{2}+2V(\phi)},$
(3.7)

where we have assumed that $\phi=\phi(t)$, i.e. $\nabla\phi\rightarrow 0$, by
considering the average homogeneity and the isotropy of the universe. Eq.(3.7)
implies that the equation of state parameter of $\phi$ may change dynamically
with time for the different epochs of the universe in contrast to the constant
equation of state parameter of $\Lambda$. The field equations (3.4)
corresponding to the FLRW universe are given by

$\displaystyle H^{2}=\frac{8\pi
G}{3}\bigg{[}\frac{\dot{\phi}^{2}}{2}+V(\phi)+\rho_{M}\bigg{]},$ (3.8)
$\displaystyle\dot{H}=-\frac{8\pi
G}{2}\bigg{[}\dot{\phi}^{2}+\rho_{M}+p_{M}\bigg{]}.$ (3.9)

By use of these equations, we have two continuity equations for both the fluid
and the quintessence field, as given below:

$\displaystyle\dot{\rho}_{M}+3H(\rho_{M}+p_{M})=0,$ (3.10)
$\displaystyle\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=0.$ (3.11)

During the radiation and matter dominated epochs of the universe, it is
assumed that $\rho_{\phi}\ll\rho_{M}$ if we assume that the only function of
$\phi$ is to accelerate the expansion, i.e., $\phi$ plays the role of DE. The
first condition for this is to make $\dot{\phi}^{2}/2\gg V(\phi)$ in Eq.(3.7).
In this case, we obtain $w_{\phi}\simeq 1$ for the quintessence field in these
eras and hence the continuity equation results in the density evolution of the
field $\rho_{\phi}\propto a^{-6}$. This result is enough for $\phi$ to
decrease much faster than the background fluid density, namely
$\rho_{\text{rad}}\propto a^{-4}$ and $\rho_{\text{mat}}\propto a^{-3}$ for
the radiation and matter dominated epochs, respectively. However, for the
explanation of the late-time acceleration of the universe caused by DE
density, $\rho_{\phi}$ should track $\rho_{M}$. This requires
$\dot{\phi}^{2}<V(\phi)$ and equivalently $w_{\phi}<-1/3$. Therefore, the
evolution of the field $\phi$ by time should be slow enough in comparison with
the potential $V(\phi)$, which is called the slow roll condition [85]. The
expected present DE dominated epoch is provided by the condition
$\dot{\phi}^{2}\ll V(\phi)$ for which $w_{\phi}\simeq-1$ as in the case of
$\Lambda$.

To make $\phi$ play the role of cosmological constant, it must mimic
cosmological constant at late times. This may be provided by considering
slowly varying potentials [83]. In this respect, some quintessence models are
depending on the different kinds of quintessence potentials such as freezing
models, thawing models, etc. Detailed discussions on these models are given in
[14].Using the quintessence potentials, the source of DE is then provided by
the dynamical quintessence field $\phi$ in place of the constant $\Lambda$.

Further, by imposing the condition $\dot{\phi}^{2}\approx 2V(\phi)$, which
gives the zero pressure for $\phi$, i.e., $w_{\phi}=0$, one may assume that
$\phi$ may play the role of DM. An example of this model is given in [86],
where a physically reasonable more general potential for $\phi$ is introduced
as $V(\phi)=V_{0}(\text{cosh}\ \lambda\phi-1)^{\kappa_{\phi}}$, where
$\kappa_{\phi}=\frac{1+w_{\phi}}{1-w_{\phi}}$. The choice of this potential
makes the dynamics of $\phi$ more adaptable for cosmology so that one may
employ it for DM and DE. Here the field may play the role of only DM for
$w_{\phi}=0$, i.e., $\kappa_{\phi}=1$ while it may play the role of only DE
for $w_{\phi}\leq-1/3$, i.e., $\kappa_{\phi}\leq 1/2$. In the same study, the
possibility of a unified model of DE and DM is also discussed by considering
the potential $V(\phi,\psi)=V_{\phi}(\text{cosh}\
\lambda_{\phi}\phi-1)^{\kappa_{\phi}}+V_{\psi}(\text{cosh}\
\lambda_{\psi}\psi-1)^{\kappa_{\psi}}$, where $\phi$ and $\psi$ are employed
for DE (with $\kappa_{\phi}\leq 1/2$) and DM (with $\kappa_{\psi}=1$ ),
respectively. The quintessence field obeys the relation $-1<w<-1/3$ at the
cosmological scale if the field plays the role of DE. Another study [87] shows
that $-1<w<-1/3$ also holds at a galactic scale if the field plays the role of
DM, which is called quintessence-like or exotic DM. More examples of DM in the
context of quintessence are given in [88, 89, 90, 91].

###  3.2 k-essence

Besides the quintessence models, there are also other modified matter models
known as k-essence as alternatives to the DE model of $\Lambda$CDM [92, 93].
In these models, the Lagrangian density contains a non-canonical kinetic term
of the field, which is a function of both the kinetic energy of the field and
the field itself. The typical action for these models is

$S=\int d^{4}x\sqrt{|g|}\bigg{[}\frac{c^{4}}{16\pi
G_{N}}R(g)+P({\phi},X)+\mathcal{L}_{M}\bigg{]},$ (3.12)

where $P({\phi},X)$ is a function of the k-essence field $\phi$ and the
kinetic energy $X=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$.
The corresponding energy-momentum tensor of the field is given by

$T^{\phi}_{\mu\nu}=P_{,X}\partial_{\mu}\phi\partial_{\nu}\phi+g_{\mu\nu}P,$
(3.13)

where $P_{,X}\equiv\frac{\partial P}{\partial X}$. Then the most general
equation of state for such models is obtained as

$w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{P}{2XP_{,X}-P}.$ (3.14)

In the case of $|2XP_{,X}|\ll|P|$, the requirement $w_{\phi}\simeq-1$ for DE
is realized. There are some models belonging to the k-essence family such as
the ghost condensate, tachyon, and Dirac-Born-Infeld (DBI) models [14]; the
Lagrangian densities for these models are given by $P=-X+X^{2}/M^{4}$ (with
constant $M$ having dimension of mass) for ghost condensate,
$P=-V(\phi)\sqrt{-\text{det}(g_{\mu\nu}+\partial_{\mu}\phi\partial_{\nu}\phi)}$
for tachyon, and $P=-f(\phi)^{-1}\sqrt{1-2f(\phi)X}+f(\phi)^{-1}-V(\phi)$ for
Dirac-Born-Infeld (DBI). With some reasonable choices of $f(\phi)$ and
$V(\phi)$ [14], the fields of the models behave as the sources of the DE.

There are also some studies on employing the non-canonical scalar field for
both DE and DM [94, 15, 95, 16]. As an example, one may consider a Lagrangian
density $P({\phi},X)=X\left(\frac{X}{M^{4}}\right)^{\alpha-1}-V(\phi)$ [94],
which is the generalised form of the Lagrangian density (3.2) of the canonical
scalar field. Then the energy density and the pressure are obtained as
$\rho_{\phi}=(2\alpha-1)X\left(\frac{X}{M^{4}}\right)^{\alpha-1}+V(\phi)$ and
$p_{\phi}=X\left(\frac{X}{M^{4}}\right)^{\alpha-1}-V(\phi)$, respectively.
Here with some reasonable choices for the potential $V(\phi)$ [94], the
equation of state of the kinetic term, $\rho_{X}$, of the density
$\rho_{\phi}$, is given by $w_{X}=\frac{1}{2\alpha-1}$, where $w_{X}\simeq 0$
for $\alpha\gg 1$. We therefore conclude that for the proper potentials and
high values of $\alpha$, the kinetic term $\rho_{X}$ may behave as DM while
the potential term $V({\phi})$ may play the role of DE.

###  3.3 Phantom

The recent reported values [96, 97] for the equation of state of DE lead to
the possibility that $w_{\text{DE}}<-1$, generally called phantom DE. Scalar
field models of this type may be considered a specific k-essence case .
Considering the model with a positive energy density $\rho_{\phi}$, the only
condition to obtain $w_{\text{DE}}<-1$ is to get $P_{,X}<0$ in Eq.(3.14). This
may be provided with the simplest choice of the Lagrangian density with a
negative kinetic energy [98, 99]:

$P(X,\phi)=-X-V(\phi).$ (3.15)

The corresponding energy density and the pressure of a phantom scalar field
are given by $\rho_{\phi}=-\dot{\phi}^{2}/2+V(\phi)$ and
$p_{\phi}=-\dot{\phi}^{2}/2-V(\phi)$. The equation of state of the field is
then obtained as

$w_{\phi}=\frac{-\dot{\phi}^{2}/2-V(\phi)}{-\dot{\phi}^{2}/2+V(\phi)},$ (3.16)

which results in $w_{\phi}<-1$ for $\dot{\phi}^{2}/2<V(\phi)$. Some detailed
discussion on the cosmological dynamics of phantom field with different
choices of the potential $V(\phi)$ are given in the papers [100, 101, 102].

###  3.4 Tachyon

The string theory tachyon model is associated with the action [103, 104]

$S_{\phi}=-\int d^{4}x\
V(\phi)\sqrt{-\text{det}(g_{\mu\nu}+\partial_{\mu}\phi\partial_{\nu}\phi)},$
(3.17)

where $V(\phi)$ is the potential of the field. The corresponding energy
density and pressure by use of the FRW metric is then obtained as

$\rho_{\phi}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^{2}}},$ (3.18)
$p_{\phi}=-V(\phi)\sqrt{1-\dot{\phi}^{2}},$ (3.19)

which result in the equation of state parameter

$w_{\phi}=-1+\dot{\phi}^{2}.$ (3.20)

If we here assume that the tachyon field plays the role of DE, we then expect
that $w_{\phi}\approx-1$, which requires the condition $\dot{\phi}^{2}\ll 1$.
Tachyon field is designated in k-essence family because it has an action of a
type Eq.(3.12). However, this differs from k-essence in the sense that the
kinetic energy of the tachyon must be here suppressed in order to achieve
accelerated expansion of the universe. Some examples on tachyon models are
given in [105, 106, 107, 108].

###  3.5 Coupled Scalar Field Models

As mentioned in Eq.(2.10), DM and DE have the same order of densities.
Therefore, there may be a relationship between them. In this regard, some
models such as coupled quintessence field, coupled k-essence field, etc. [109,
110, 111], where the scalar fields adopted as DE couple to the non-
relativistic matter, are studied in this context.

As an example of these types of models, we consider the coupled quintessence
field [14, 110]. Here, the Lagrangian density given in Eq.(3.3) is modified
with the interaction Lagrangian density $\mathcal{L}_{\text{int}}$:

$S=\int d^{4}x\sqrt{|g|}\bigg{[}\frac{c^{4}}{16\pi
G_{N}}R(g)+\mathcal{L}_{\phi}+\mathcal{L}_{M}+\mathcal{L}_{\text{int}}\bigg{]},$
(3.21)

where $\mathcal{L}_{\text{int}}$ causes interaction between the quintessence
field and the non-relativistic matter through a coupling between the energy-
momentum tensors of quintessence and matter [14, 110]:

$\nabla_{\mu}T^{\mu}_{\nu(\phi)}=-QT_{M}\nabla_{\nu}\phi,\ \ \ \ \
\nabla_{\mu}T^{\mu}_{\nu(M)}=QT_{M}\nabla_{\nu}\phi.$ (3.22)

Here $T_{M}=-\rho_{M}+3p_{M}$ is the trace of the energy-momentum tensor for
the matter fluids (radiation and non-relativistic matter) and $Q$ is the
coupling constant. Since $w_{\text{rad}}=1/3$, the trace vanishes for the
radiation. Consequently, only the non-relativistic matter, i.e., dark and
baryonic matters, couple to the quintessence field $\phi$. This coupling with
the suitable choice of the potential $V(\phi)$ [14, 110, 112] brings about the
desired explanations for DM and the universe’s accelerated expansion.

##  Chapter 4 Bose-Einstein Condensation

Bose-Einstein condensation (BEC) is one of the most interesting physical
phenomena explored by Bose [17] and Einstein [18] in the 1920’s. It was
theoretically shown that under some conditions, large numbers of particles
obeying the Bose statistics could collapse into the lowest quantum state to
realize a macroscopic quantum phenomenon known as BEC, which was, after 75
years, experimentally verified [113, 114, 115]. The macroscopic dynamics of a
BEC is described by the GP equation, which was independently derived by Gross
[19] and Pitaevskii [20] in 1961. In the first section of this chapter, we
shall study the conditions to form a BEC and derive the GP equation in detail.

Besides, BEC can also be considered of the cause of spatial independence
cosmological scalar fields given in Chapter 3. If a scalar field forms a Bose-
Einstein condensate, the field then naturally depends only on time, as in the
case of the scalar fields for DE and DM. In this regard, the formation of a
condensate scalar field has physical importance at the cosmic level. Further,
to explore the implications of a Bose-Einstein scalar field condensation in
cosmology, one should also study the relations between the $\phi^{4}$ theories
and the GP equation. To this end, in the Section 4.2 we shall derive the GP
equations from $\phi^{4}$ theories by considering both flat and curved
spacetimes.

In the last section, we shall study some cosmological Bose-Einstein condensate
scalar field models.

###  4.1 A Brief Overview of Bose-Einstein Condensation in Condensed Matter
Physics

To understand BEC, the simplest case is to consider an ideal non-relativistic
Bose gas, where we shall assume N undistinguishable, non-interacting, non-
relativistic quantum particles confined in a box of volume $L^{3}$ [116, 117].
To this end, we need the quantum statistics of bosons represented by the
grand-canonical ensemble, where the system is considered in thermal
equilibrium at temperature $T$ with a reservoir with chemical potential $\mu$.
Thus, the total occupation number of particles over all states, denoted by s,
is given by

$N=\sum_{s\geq
0}\frac{1}{e^{\beta(\epsilon_{s}-\mu)}-1}=\frac{1}{e^{\beta(\epsilon_{0}-\mu)}-1}+\sum_{s>0}\frac{1}{e^{\beta(\epsilon_{s}-\mu)}-1}\equiv
N_{0}+N_{\text{excited}},$ (4.1)

where $\beta=(k_{B}T)^{-1}$ with Boltzmann’s constant $k_{B}$, and $N_{0}$ and
$N_{\text{excited}}$ are the occupation numbers for the ground state with the
energy $\epsilon_{0}$ and the excited states with the energy $\epsilon_{s}$,
respectively. For the ideal gas confined in the box, the allowed eigenvalues
$\epsilon$ of the Hamiltonian $H^{(1)}=\hat{\textbf{p}}^{2}/2m$ (where the
momentum operator $\hat{\textbf{p}}=-i\hbar\nabla$) for the particles in the
gas are given by $\epsilon_{n}=p^{2}_{n}/2m$, where
$\textbf{p}_{n}=2\pi\hbar\textbf{n}/L$ by the de Broglie relation with n being
a vector with either $0$ or $\pm$ integer components $n_{x}$, $n_{y}$,
$n_{z}$. In this regard, Eq.(4.1) can be rewritten as

$\displaystyle
N=\frac{1}{e^{\beta(\epsilon_{0}-\mu)}-1}+V\bigg{(}\frac{2\pi}{\hbar}\bigg{)}^{3}\int\frac{1}{e^{\beta(\frac{p^{2}}{2m}-\mu)}-1}d\textbf{p}$
$\displaystyle=\frac{1}{e^{\beta(\epsilon_{0}-\mu)}-1}+\frac{V}{\lambda^{3}}g_{3/2}(e^{\beta\mu})$
$\displaystyle\equiv N_{0}+N_{\text{excited}},$ (4.2)

where

$\lambda=\sqrt{\frac{2\pi\hbar^{2}}{mk_{B}T}}$ (4.3)

is the thermal wavelength and $g_{3/2}(e^{\beta\mu})=\frac{2}{\sqrt{\pi}}\int
dxx^{1/2}\frac{1}{e^{(x-\beta\mu)}-1}$. Here, the chemical potential of the
ideal Bose gas, either in the grand canonical ensemble or confined in the box,
has the physical constraint $\mu<\epsilon_{0}$ [116] and $N_{0}$ of the lowest
energy state becomes increasingly large as $\mu\rightarrow\epsilon_{0}$. For
the ideal Bose gas confined in the box, the lowest energy is $\epsilon_{0}=0$
(i.e., $\mu$ must be always negative) [116]. In the derivation of Eq.(4.2), we
have used the integral $V\bigg{(}\frac{2\pi}{\hbar}\bigg{)}^{3}\bigintss
d\textbf{p}$ in place of summation $\sum_{\textbf{p}}$ with the transormation
$p^{2}=2mk_{B}Tx$. By using the total number density $n=N/V$, Eq.(4.2) can be
written as

$\lambda^{3}n_{0}=\lambda^{3}n-g_{3/2}(e^{\beta\mu}).$ (4.4)

As clearly seen from the last equation, its left-hand side should always be
positive, i.e., $\lambda^{3}n>g_{3/2}(e^{\beta\mu})$. The transition between a
gas phase and a condensed phase starts when $\mu\rightarrow 0$ [116] (since
$\epsilon_{0}\rightarrow 0$ in this case), which implies

$\lambda>\lambda_{c}\equiv n^{-1/3}\ (2.612)^{1/3}\ \ \ \ \text{or}\ \ \ \
T<T_{c}\equiv\frac{2\pi\hbar^{2}}{mk_{B}\lambda^{2}_{c}},$ (4.5)

where $g_{3/2}(1)=2.612$. The interpretation of BEC may be better clarified by
considering the fraction of the particle numbers as follows. Using (4.5) for
Eq.(4.2), we get

$\frac{N_{0}}{N}=1-\bigg{(}\frac{\lambda_{c}}{\lambda}\bigg{)}^{3}\ \ \ \
\text{or}\ \ \ \ \frac{N_{0}}{N}=1-\bigg{(}\frac{T}{T_{c}}\bigg{)}^{3/2}.$
(4.6)

Here, when $\lambda>\lambda_{c}$ or $T<T_{c}$ (which implies that $N_{0}\neq
0$), the bosons start to form a BEC and when $\lambda\gg\lambda_{c}$ or $T\ll
T_{c}$ (which implies that $N_{0}\approx N$), approximately all the particles
accumulate on the ground state with the energy $\epsilon_{0}=0$ or
equivalently with the momentum $\textbf{p}=0$. The latter case is the
realization of BEC and the pure condensation of the gas phase occurs at $T=0$.
Thus the state of BEC can be described by a unique condensate or macroscopic
wave function.

The method applied above is for the ideal Bose gas. For a more general and
realistic case, one should, however consider the interacting Bose gases [116,
117]. To understand the interacting picture, we use dilute Bose gas
approximation and it is given by the condition $r_{0}\ll n^{-1/3}$, where
$r_{0}$ is the range of the interatomic forces. By this condition, the
interaction among particles can be described to be a two-body scattering
process. We also impose $r_{0}\ll\lambda$ which states the very low momenta of
the incoming Bose particles. In this case, we can consider s-wave ($l=0$)
scattering length $a$ with $a\ll n^{-1/3}$; that is, the gas is weakly
interacting. Thus the corresponding Hamiltonian operator for this
configuration is given by

$\hat{H}=\int\bigg{[}\frac{\hbar^{2}}{2m}|\nabla\hat{\Psi}(\textbf{r})|^{2}+V_{\text{ext}}(\textbf{r})|\hat{\Psi}(\textbf{r})|^{2}+\frac{1}{2}\int\hat{\Psi}^{\dagger}(\textbf{r}^{\prime})\hat{\Psi}^{\dagger}(\textbf{r})V(\textbf{r}^{\prime}-\textbf{r})\hat{\Psi}(\textbf{r}^{\prime})\hat{\Psi}(\textbf{r})d\textbf{r}^{\prime}\bigg{]}d(\textbf{r}),$
(4.7)

where $V_{\text{ext}}(\textbf{r})$ and $V(\textbf{r})$ are the external and
the two-body potentials, respectively.

The study of BEC in the case of the interacting approach is grounded on the
Bogoliubov approximation as follows. The field operator, which is given in the
form of $\hat{\Psi}(\textbf{r})=\sum_{i}\varphi_{i}\hat{a}_{i}$, can be
separated into two as
$\hat{\Psi}(\textbf{r})=\hat{\Psi}_{0}(\textbf{r})+\hat{\Psi}_{\text{NC}}(\textbf{r})$,
where the first term represents the ground or the condensate state and the
second term is for the non-condensate states of the system. The macroscopic
nature of the ground state of the system in the case of BEC implies
$N_{0}=|\hat{a}_{0}|^{2}\gg 1$. In this respect, Bogoliubov considers the
macroscopic component of the field operator $\hat{\Psi}_{0}(\textbf{r})$ a
classical field and the other component $\hat{\Psi}_{\text{NC}}(\textbf{r})$ a
perturbative field, i.e.,
$\hat{\Psi}(\textbf{r})=\Psi_{0}(\textbf{r})+\delta\hat{\Psi}(\textbf{r})$.
Further, in the case of very low temperatures, we can also ignore the second
term and consider the field operator completely classical, i.e.,
$\hat{\Psi}(\textbf{r})=\Psi_{0}(\textbf{r})$, which means that the system
completely behaves as a classical object in the case of BEC.

However, the Bogoliubov approximation is not applicable in the case of the
realistic potentials $V(\textbf{r})$. In this regard, we additionally
reconsider the potential in the third term of $\hat{H}$ as the effective or
pseudo one $V_{\text{eff}}(\textbf{r})$ in the sense that these two potentials
reproduce the same low-energy scattering properties. In the same term, we can
also assume that most of the fields are in a condensate state, so
$\Psi(\textbf{r}^{\prime},t)\approx\Psi_{0}(\textbf{r},t)$ at the order of the
range of the interatomic force.

With the assumptions introduced above, the form of the Hamiltonian (4.7)
changes to

$\hat{H}=\int\bigg{[}\frac{\hbar^{2}}{2m}|\nabla\Psi_{0}(\textbf{r},t)|^{2}+V_{\text{ext}}(\textbf{r})|\Psi_{0}(\textbf{r},t)|^{2}+\frac{1}{2}g|\Psi_{0}(\textbf{r},t)|^{4}\bigg{]}d\textbf{r},$
(4.8)

where we consider the time dependence of the wave function and time-
independent external potential and $g=\int
V_{\text{eff}}(\textbf{r})d\textbf{r}=4\pi\hbar^{2}a/m$ with the s-wave
scattering length $a$. Thus, using the Hamiltonian (4.8) in the Heisenberg
equation of motion, $i\hbar\frac{\partial}{\partial
t}\hat{\Psi}(\textbf{r},t)=[\hat{\Psi}(\textbf{r},t),\hat{H}]$, we finally
obtain the GP equation for the time evolution of the condensate field
$\Psi_{0}(\textbf{r},t)$:

$i\hbar\frac{\partial}{\partial
t}\Psi_{0}(\textbf{r},t)=\bigg{(}-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{\text{ext}}(\textbf{r})+g|\Psi(\textbf{r},t)|^{2}\bigg{)}\Psi_{0}(\textbf{r},t).$
(4.9)

One may proceed with the GP equation (4.9) and obtain its static form by
considering static solutions $\tilde{\Psi}_{0}(\textbf{r})$ so that one may
consider harmonic time-dependent wave function

$\Psi(\textbf{r},t)=\tilde{\Psi}_{0}(\textbf{r})exp\left(\frac{-i\mu
t}{\hbar}\right),$ (4.10)

where $\mu$ is the chemical potential defined as $\frac{\partial E}{\partial
N}$. After plugging Eq.(4.10) into Eq.(4.9), the form of GP reduces to

$\bigg{(}-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{\text{ext}}(\textbf{r})+g|\tilde{\Psi}_{0}(\textbf{r})|^{2}-\mu\bigg{)}\tilde{\Psi}_{0}(\textbf{r})=0.$
(4.11)

###  4.2 Derivation of Gross-Pitaevskii Equation from $\phi^{4}$ Theories

The theories based on the relativistic dynamics of a scalar field, say $\phi$,
with its self-interacting potential energy $\frac{\lambda}{4}|\phi|^{4}$,
where $\lambda$ is the dimensionless coupling constant, are called $\phi$-four
theories [72, 73, 74, 75]. The dynamics of the scalar field is described by
the Klein-Gordon (KG) equation:

$\bigg{[}-\square+m^{2}+\lambda|\phi|^{2}\bigg{]}\phi=0,$ (4.12)

where
$\square\phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi]$
in curved space-time [72] (i.e in the presence of a gravitational background)
and it reduces to the form
$\square\phi=g^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi$ in flat space-time
(i.e., in the absence of a gravitational background), and we use $\hbar=c=1$
by convention. The Klein-Gordon Eq.(4.12) is obtained by the variation of the
action

$S_{\phi}=\int
dx^{4}\sqrt{-g}\bigg{(}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi^{\star}\partial_{\nu}\phi-\frac{1}{2}m^{2}|\phi|^{2}-\frac{\lambda}{4}|\phi|^{4}\bigg{)}$
(4.13)

with respect to the dynamical complex scalar field $\phi^{\star}$.

To study the implications of the BEC in the context of cosmology (i.e., in the
presence of the gravitational background), one should understand the analogy
between the KG equation for the scalar field and the GP equation for Bose-
Einstein condensates. To this end, we first begin with the scalar field
satisfying KG equation in the flat spacetime background and see how this KG
equation corresponds to the GP equation. Then, we continue with the same
analysis for the curved spacetime.

####  4.2.1 Reducing Klein-Gordon Equation in Flat Spacetime to Gross-
Pitaevskii Equation

We begin with analysing a scalar field described by a KG equation in the
absence of a gravitational background. Considering metric in flat spacetime

$ds^{2}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}$ (4.14)

and $\square\phi=g^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi$, the KG equation
(4.12) is written as

$\ddot{\phi}-\triangle\phi+m^{2}\phi+\lambda|\phi|^{2}\phi=0,$ (4.15)

where the dot denotes a derivative with respect to $t$ and $\triangle$ is the
3-dimensional Laplace operator. If we consider the transformation [23]:

$\phi=e^{-iwt}\chi,$ (4.16)

where $w=\sqrt{\vec{p}^{2}+m_{\phi}^{2}}$ is the energy of the field $\phi$,
Eq.(4.15) then becomes

$\ddot{\chi}-2iw\dot{\chi}-\triangle\chi+(m_{\phi}^{2}-w^{2})\chi+\lambda|\chi|^{2}\chi=0.$
(4.17)

Since the GP equation is considered as the non-relativistic limit of
$\phi^{4}$ theories, we here assume that $\phi$ is non-relativistic for a long
time then its time evolution should be slow. This implies that
$|\frac{\ddot{\chi}}{\dot{\chi}}|\ll 2m_{\phi}$ and $w^{2}\simeq
m_{\phi}^{2}$. Then Eq.(4.17) reduces to

$i\dot{\chi}=(-\frac{\triangle}{2m_{\phi}}+\frac{\lambda}{m_{\phi}}|\chi|^{2})\chi,$
(4.18)

which is the GP equation (4.9) without external potential.

We now give an example [22] where a specific solution of $\phi$ in one-
dimensional case (for simplicity) is considered:

$\phi(t,x)=e^{iwt}\chi(x),$ (4.19)

which is the harmonic time solution of $\phi$. The field here is considered to
be confined in an infinite potential well. Plugging the solution (4.19) into
Eq.(4.15), we get

$\chi^{\prime\prime}-(m^{2}-w^{2})\chi-\lambda|\chi|^{2}\chi=0,$ (4.20)

where the prime denotes a derivative with respect to $x$. Eq.(4.20) is in the
specific form of the GP equation given in Eq.(4.11) (the three-dimensional
analysis of the study can be found in [23, 122, 123, 124, 125]).

Further, one may search for a solution to the KG equation for the field
$\chi(x)$. The integration of Eq.(4.20) leads to

$\frac{1}{2}\chi^{\prime
2}-\frac{\lambda}{4}\left(\chi^{2}+\frac{m^{2}-w^{2}}{\lambda}\right)^{2}=C,$
(4.21)

where $C$ is a constant. The last equation can be solved by a kink solution
[22, 126]

$\chi(x)=|\chi_{0}|\text{tanh}\left(x\sqrt{\frac{w^{2}-m^{2}}{2}}\right),$
(4.22)

with a constant $\chi_{0}$, which represents the wave function far away from
the wall of the potential well. The solution (4.22) has the same form as the
one derived for a Bose-Einstein condensate confined in a box [127].

Hence, it is shown that in the absence of a gravitational background, one may
relate the classical scalar field (satisfying the KG equation) to a Bose-
Einstein condensate (satisfying the GP equation).

####  4.2.2 Reducing Klein-Gordon Equation in Curved Spacetime to Gross-
Pitaevskii Equation

We now proceed with the analysis for a scalar field satisfying the KG equation
in the presence of a gravitational background [23]. In this respect, we
consider the FRW metric (2.1). Using the definition
$\square\phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi]$
in curved space-time with the metric (2.1), the KG equation (4.12) takes the
following form:

$\ddot{\phi}+3H\dot{\phi}-\frac{1}{a^{2}}\triangle\phi+m^{2}\phi+\lambda|\phi|^{2}\phi=0.$
(4.23)

Using the transformation (4.19) in Eq.(4.23), we get

$\ddot{\chi}-2iw\dot{\chi}+3H\dot{\chi}-\triangle\chi+(m_{\phi}^{2}-w^{2})\chi-3iHw\chi+\lambda|\chi|^{2}\chi=0.$
(4.24)

We know that the GP equation is the equation for a non-relativistic particle
in a condensate at atomic scale, where the time evolution of the cosmic scale
factor $a$ is negligible [23]. Therefore, the terms with the Hubble parameter
in Eq.(4.24) are negligible compared to the other terms. Further, considering
the non-relativistic limit, one may assume (as we have noticed in the previous
subsection) $|\frac{\ddot{\chi}}{\dot{\chi}}|\ll 2m_{\phi}$ and $w^{2}\simeq
m_{\phi}^{2}$. Then Eq.(4.24) reduces to

$i\dot{\chi}=(-\frac{\triangle}{2a^{2}m_{\phi}}+\frac{\lambda}{m_{\phi}}|\chi|^{2})\chi,$
(4.25)

which is the GP equation (4.9) without external potential.

As an example for the GP equation in curved spacetime [22], we now consider
the spherically symmetric and static metric

$ds^{2}=-A(r)dt^{2}+\frac{1}{A(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\varphi^{2}.$ (4.26)

Using the definition
$\square\phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi]$
in curved space-time with the metric (4.26), the KG equation (4.12) takes the
following form:

$\frac{\ddot{\phi}}{A}-A\phi^{\prime\prime}-\left(\frac{2A}{r}+A^{\prime}\right)\phi^{\prime}+m^{2}\phi+\lambda|\phi|^{2}\phi=0,$
(4.27)

where the prime denotes a derivative with respect to $r$. If we particularly
use the monopolar and the harmonic time solution of the scalar field

$\phi(t,r)=e^{iwt}\frac{u(r)}{r},$ (4.28)

Eq.(4.27) becomes

$u^{\prime\prime}-\left[A\left(m^{2}+\frac{A^{\prime}}{r^{\star}}\right)-w^{2}\right]u-\lambda\frac{A}{r^{2}}|u|^{2}u=0,$
(4.29)

which is in the specific form of the GP equation (4.11). Here, we introduce a
new coordinate $r^{\star}=\int\frac{dr}{A}$ and the prime denotes a derivative
with respect to $r^{\star}$. The field is here trapped by the effective
potential

$V_{\text{eff}}=A\left(m^{2}+\frac{A^{\prime}}{r^{\star}}\right),$ (4.30)

which is identified with the external (or trapping) potential
$V_{\text{ext}}(\textbf{r})$ of the GP equation (4.11). In contrast to the
case of flat spacetime, we here do not need to introduce an external trapping
potential. The curvature of a spacetime associated with some $A(r)$ can
naturally provide trapping potential by Eq.(4.30) to confine the scalar field
in some region of gravitational background. As examples of such backgrounds,
Schwarzschild spacetime with $A(r)=1-\frac{2MG}{r}$ and Schwarzschild-de
Sitter spacetime with $A(r)=1-\frac{2MG}{r}-\frac{\Lambda}{3}$ (where $M$ is
the mass of the blackhole and $\Lambda$ is the cosmological constant) may be
given. With some conditions, they can confine the scalar field in some region
of their backgrounds so that stationary (or quasi-stationary) scalar field
distributions can be obtained. In this regard, such distributions reinforce
the idea that there may be close relations between such scalar fields in
cosmology and Bose-Einstein condensates in atomic physics. Detailed
discussions on these examples are given in [22, 128, 129].

###  4.3 Some Cosmological Bose-Einstein Condensate Scalar Field Models

As we have clarified in the previous chapter, Bose-Einstein condensates (BEC)
of scalar fields at the cosmological level may be good candidates for DM and
DE. In this regard, there are some studies on BEC scalar field models for DE
and DM [21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In one of them [30] it is
shown that if a gas of bosons all in the condensate phase is identified with
DM, the critical temperature (4.5) (below which the bosons start to form a
BEC) reduces to

$T_{c}=\frac{6\times 10^{-12}}{m^{1/3}a}\ K,$ (4.31)

where $a$ is the scale factor and m is in the unit of $kg$. Thus, for
$m<1\text{eV}$, $T_{c}>2.7/a$. Because $2.7/a$ is the background temperature
of the universe at all times, the BEC phase of a tiny mass of bosons can start
at the very early era of the universe. Further, the bosons in this phase have
approximately zero momenta and zero pressure, meaning that they can be
considered CDM.

Moreover, we have seen that a BEC is a macroscopic quantum state, which is
represented by a scalar field in the form of $\phi=Re^{iS}$, where $R$ and $S$
are the real functions of space and time, as clarified in the previous
sections. Considering quantal (Bohmian) trajectories [130, 131, 132] where the
velocity field is defined by $u_{a}=\hslash\partial_{\alpha}S/m$ and also
defining the induced metric $h_{ab}=g_{ab}-u_{a}u_{b}$, the quantum mechanical
contribution to the cosmological constant in the Friedmann equation is
obtained by [133, 134, 135]

$\Lambda_{Q}=\frac{\hslash^{2}}{m^{2}c^{2}}h^{ab}\nabla_{a}\nabla_{b}\left(\frac{\square
R}{R}\right).$ (4.32)

Here $\Lambda_{Q}$ is related to the relativistic quantum potential
$V_{Q}=\frac{\hslash^{2}}{m^{2}c^{2}}\left(\frac{\square R}{R}\right)$ [133].
Because our universe is homogeneous and isotropic, the wavefunction $R$ is
expected to disperse uniformly over the range of the Hubble radius, say
$L_{0}$, of the universe. Due to the condensate nature of the field $\phi$,
$R$ is also taken to be time-independent. With all these physical points of
view, the most suitable choices of $R$ may be given as
$R=R_{0}e^{-r^{2}/L_{0}^{2}}$ [136] (which describes the ground state of the
harmonic oscillator) or $R=R_{0}\text{tanh}(r/L_{0}\sqrt{2})$ for $g>0$ or
$R=\sqrt{2}R_{0}\text{sech}(r/L_{0})$ for $g<0$ [137, 138], where $g$ is the
interaction of strength. All these choices give rise to the same expression as
$\nabla_{a}\nabla_{b}\left(\frac{\square R}{R}\right)\simeq 1/L_{0}^{4}$.
Moreover, $L_{0}$ may also be considered as the Compton wavelength of the
bosons of mass $m$, that is, $L_{0}=h/mc$ [139]. Then by using the present
value of Hubble radius $L_{0}=1.4\times 10^{26}\text{meter}$, the mass of the
bosons is obtained as $m\simeq 10^{-68}kg$ or $10^{-32}eV$, which gives rise
to the observed value of DE

$\Lambda_{Q}=10^{-123}l_{\text{Pl}}^{-2},$ (4.33)

where $l_{\text{Pl}}$ denotes the Planck units. In conclusion, it is shown
that DM can be explained by the density of the bosons of mass $m$ all in BEC,
while DE can be explained by the quantum potential of the BEC macroscopic
wavefunction.

There is another study [28] on BEC employed as DE and DM. In the model, it is
first introduced that at the very beginning of the universe, the energy
density of the excited Bose gas, $\rho_{g}$, identified as DM is assumed to
dominate the energy density of the condensation of the bosons, $\rho_{c}$
employed as DE. Next, by the expansion of the universe, the amount of
$\rho_{g}$ is decreased while $\rho_{c}$ is unchanged (This idea is
thermodynamically well-explained in [28]). Therefore, $\rho_{c}$ eventually
dominates $\rho_{g}$. This results in the accelerated expansion of the
universe at the large scale and the collapse of the condensation at the
smallest scale, yielding the localized compact objects as DM. The new energy
density $\rho_{l}$ is then introduced for the localized energy density due to
the collapse of the condensation at small scales.

According to the $\Lambda$CDM, the evolution of all these different forms of
energy densities are given by the following equations:

$\dot{\rho}_{c}=\Gamma\rho_{g},\ \ \
\dot{\rho}_{g}=-3H\rho_{g}-\Gamma\rho_{g},\ \ \ \dot{\rho}_{l}=-3H\rho_{l},$
(4.34)

where $\rho=\rho_{c}+\rho_{g}+\rho_{l}$ and
$H\equiv\frac{\dot{a}}{a}=\sqrt{8\pi G\rho/3}$. These equations are valid when
the energy density of the condensate has the condition
$\rho_{c}<\rho_{g}+\rho_{l}$. After the time $\Gamma^{-1}$, it becomes
$\rho_{c}>\rho_{g}+\rho_{l}$ and the inhomogeneous part of the condensate
would collapse and contributes to $\rho_{l}$. Then the condition
$\rho_{c}<\rho_{g}+\rho_{l}$ is realized again. In each time scale
$\Gamma^{-1}$, this situation, i.e., ’chase and collapse’, is repeated until
$\rho_{c}\approx\rho_{g}+\rho_{l}$, that is, until the ratio of the energy
densities of DE and DM becomes approximately order of one as predicted by
$\Lambda\text{CDM}$. This kind of dynamics is known as Self Organized
Criticality (SOC) as discussed in [28, 140].

##  Chapter 5 A Mechanism for the Formation of Bose-Einstein Condensation in
Cosmology

###  5.1 An Effective Minkowski Space Formulation

In this mechanism, based on the paper [35], we aim to realize BEC scalar field
in cosmology with particular emphasis on its microscopic description in
particle physics. To this end, by taking the FRW metric

$ds^{2}=-dt^{2}+a^{2}(t)[dr^{2}+r^{2}(d\theta^{2}+\text{sin}^{2}\theta
d\phi^{2})],$ (5.1)

we consider the following action

$S=\int
d^{4}x\sqrt{-g}\frac{1}{2}\\{-g_{\mu\nu}[\partial_{\mu}\phi\partial_{\nu}\phi+\partial_{\mu}\chi\partial_{\nu}\chi]-m_{\phi}^{2}\phi^{2}-m_{\chi}^{2}\chi^{2}-\mu\phi^{2}\chi\\}$
(5.2)

with its effectively Minkowskian form [72]

$S=\int d^{3}xd\eta\frac{1}{2}\\{\tilde{\phi}^{\prime\
2}-(\vec{\nabla}\tilde{\phi})^{2}+\tilde{\chi}^{\prime\
2}-(\vec{\nabla}\tilde{\chi})^{2}-\tilde{m}_{\phi}^{2}\tilde{\phi}^{2}-\tilde{m}_{\chi}^{2}\tilde{\chi}^{2}-\tilde{\mu}\tilde{\phi}^{2}\tilde{\chi}\\},$
(5.3)

which is obtained by use of the following transformations and derivatives:

$\displaystyle d\eta$ $\displaystyle=\frac{dt}{a(t)},\ \ \
\tilde{\phi}=a\phi,\ \ \ \tilde{\chi}=a\chi,\ \ \
a^{\prime}=\frac{da}{d\eta},\ \ \ \dot{a}=\frac{da}{dt},\ \ \
a^{\prime\prime}=\frac{d^{2}a}{d\eta^{2}},\ \ \
\ddot{a}=\frac{d^{2}a}{dt^{2}},$ (5.4) $\displaystyle\tilde{\mu}=a\mu,\ \ \
\tilde{m}_{i}^{2}=m_{i}^{2}a^{2}-\frac{a^{\prime\prime}}{a}=a^{2}\bigg{(}m_{i}^{2}-\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\bigg{)},$
(5.5)

where prime denotes the derivative with respect to conformal time $\eta$ and a
dot denotes the derivative with respect to ordinary time $t$ and the subscript
$i$ takes the values, $i=\phi,\chi$.

It is evident from Eq.(5.3) that locally (i.e., at each time $\eta$) we may
consider the spacetime effectively Minkowskian. This, in turn, implies that
for sufficiently small intervals of $\eta$, $\Delta\eta$, spacetime may be
approximately considered to be effectively Minkowskian. In this section, we
will clarify how small $\Delta\eta$ should be taken so that an effective
Minkowski formulation of spacetime is reliable, and we will show that the
effective Minkowski formulation is cosmologically relevant. Then, we consider
the problem of formation of scalar field condensation in the framework of
Minkowski space formulation in the following section. In this context, we use
the interaction term $\phi^{2}\chi$ in the Lagrangian, which shall induce the
evolution and the formation of the condensation via the particle physics
processes $\chi\chi\rightarrow\phi\phi$, during the time $\triangle
t=\frac{1}{n_{\chi}\beta\sigma v}$ for each process. Here, we assume that
initially, there are only relativistic $\chi$ particles causing the production
of non-relativistic $\phi$ particles, i.e.,
$\tilde{m}_{\chi}\ll\tilde{m}_{\phi}$. However, being in an effectively
Minkowskian space in $\triangle t$ is not enough to proceed with our model to
realize the condensation. We should also be able to use the tools of the usual
perturbative quantum field theory [73, 74, 75] for the calculation of the
rates and the cross sections in an effective Minkowski space for each process
given in Figure 5.1, where the leading order contributions to the production
of $\phi$ particles are given. To this end, we need the following two
conditions:

  * •

The first condition is the assumption that the rate of each process
$\chi\chi\rightarrow\phi\phi$ is much larger than the Hubble parameter,
$H\equiv\frac{\dot{a}}{a}$, during the so small time interval $\triangle t$.
Then one may take the masses $\tilde{m}_{\chi}$ and $\tilde{m}_{\phi}$
constant during each process; that is to say, there is no time dependence of
the effective masses in $\triangle t$, i.e, in microscopic scales, where each
particle process occurs while the time dependence is observed only at
cosmological scales. A concrete expression of this condition is

$\Bigg{\lvert}\frac{\triangle\tilde{m}^{2}}{\tilde{m}^{2}}\Bigg{\rvert}=\Bigg{\lvert}\frac{\triangle
t\bigg{(}\frac{da^{2}(m^{2}-\dot{H}-2H^{2})}{dt}\bigg{)}}{a^{2}(m^{2}-\dot{H}-2H^{2})}\Bigg{\rvert}\ll
1.$ (5.6)

  * •

The next condition requires the constancy of the effective coupling constant
$\tilde{\mu}$ in each time interval $\triangle t$ which is imposed by the
expression

$\bigg{\lvert}\frac{\triangle\tilde{\mu}}{\tilde{\mu}}\bigg{\rvert}=|H\triangle
t|\ll 1.$ (5.7)

In Appendix A, it is shown that by letting $H=\xi a^{-s}$, which includes all
simple interesting cases e.g. radiation, matter, stiff matter, cosmological
constant dominated universes, these two conditions (5.6) and (5.7) are
satisfied for a considerable range of parameters.

Figure 5.1: The leading order Feynman diagrams contributing to the production
of $\phi$ particles. Here
$\tilde{q}_{t}=\tilde{p}_{1}-\tilde{p}_{3}=\tilde{p}_{4}-\tilde{p}_{2},\
\tilde{q}_{u}=\tilde{p}_{1}-\tilde{p}_{4}=\tilde{p}_{3}-\tilde{p}_{2}$ are the
4-momenta carried in the internal lines. Note that the s-channel is forbidden
in this case by kinematics.

The derivation in Appendix B shows that the quantized expression of any field
$\chi$ (satisfying the conditions in $\triangle t$ mentioned above) is given
by

$\tilde{\chi}^{(i)}(\vec{r},\eta)\simeq\int\frac{d^{3}\tilde{p}}{(2\pi)^{\frac{3}{2}}\sqrt{2w_{p}^{(i)}}}\bigg{[}a_{p}^{(i)-}e^{i\big{(}\vec{\tilde{p}}.\vec{r}-w_{p}^{(i)}(\eta-\eta_{i})\big{)}}+a_{p}^{(i)+}e^{i\big{(}-\vec{\tilde{p}}.\vec{r}+w_{p}^{(i)}(\eta-\eta_{i})\big{)}}\bigg{]},$
(5.8)

where $\eta_{i}<\eta<\eta_{i+1}$, and the superscript $(i)$ refers to the
$i$th time interval between the $i$th and $(i+1)$th processes. In fact,
Eq.(5.8) is a consequence of the previous conditions.

Thus, by considering the conditions (5.6) and (5.7) together with the form of
Eq.(5.8) we have

$d\tilde{s}^{2}=-d\eta^{2}+d\tilde{x}_{1}^{2}+d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2}$
(5.9)

in each interval $\eta_{i}<\eta<\eta_{i+1}$, for $\chi$, where the masses and
the coupling constant of the particles are constant and they alter as one
passes from one interval to the other, and $\tilde{x}_{i}$ are related to
Eq.(5.1) by
$d\tilde{x}_{1}^{2}+d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2}=dr^{2}+r^{2}(d\theta^{2}+\text{sin}^{2}\theta
d\phi^{2})$. Therefore, one may employ the usual perturbative quantum field
theory tools in each interval in Eq.(5.9).

###  5.2 Realizing the Conditions for Condensation

Having introduced the effective flat space formulation with the constancies of
the particle’s effective masses and effective coupling constant by Eq.(5.6)
and Eq.(5.7) for each process $\chi\chi\rightarrow\phi\phi$ during $\triangle
t$ , we are now in a position to observe the tendency of the $\phi$’s towards
Bose-Einstein condensation via the curved space effects in cosmology [36]. To
this end, we need to realize the coherence and correlation for the de Broglie
wavelengths of the particles, and achieve large, finite number density of the
particles.

####  5.2.1 Achieving Coherence and Correlation

To begin with, for each process occurring in the effective flat space, we
consider the center of mass frame where the conservation of energy amounts to
$\vec{\tilde{p}}^{2}+\tilde{m}_{\chi}^{2}=\vec{\tilde{k}}^{2}+\tilde{m}_{\phi}^{2}$.
Here $\vec{\tilde{p}}=\vec{\tilde{p}}_{1}$ and
$\vec{\tilde{k}}=\vec{\tilde{p}}_{3}$ (see Figure 5.1). Then, by the
definition of the effective masses given in Eq.(5.5), we get

$\vec{\tilde{p}}^{2}-\vec{\tilde{k}}^{2}=a^{2}(m_{\phi}^{2}-m_{\chi}^{2}).$
(5.10)

To proceed with our study on condensation using Eq.(5.10), one should see
whether the effective momenta are time dependent or not. To this end, we
should compare the effective momentum (corresponding to the metric (5.9)) with
the physical momentum (corresponding to the metric (5.1)) as follows:

$|\vec{\tilde{p}}|\,=\,\tilde{m}_{\chi}\sqrt{\tilde{g}_{ij}\frac{d\tilde{x}^{i}}{d\tilde{\tau}}\frac{d\tilde{x}^{j}}{d\tilde{\tau}}}\,=\,a\,m_{\chi}\sqrt{g_{ij}\frac{dx^{i}}{d\tau}\frac{dx^{j}}{d\tau}}\,=\,a|\vec{p}|\;.$
(5.11)

In the above, tilde $``\ \tilde{}\ "$ over a quantity refers to its form for
the metric (5.9) while the quantities without a tilde refer to its form for
the metric (5.1), and $d\tau^{2}=-ds^{2}$,
$d\tilde{\tau}^{2}=-d\tilde{s}^{2}$. Eq.(5.11) is obtained by use of
$g_{ij}=a^{2}\tilde{g}_{ij}$ and
$\frac{d\tilde{x}^{i}}{d\tilde{\tau}}=a^{2}\left(\frac{m}{\tilde{m}}\right)\frac{dx^{i}}{d\tau}$
which, in turn, follows from the geodesic equations for the actions
$\int\,\tilde{m}_{\chi}\,\sqrt{-\tilde{g}_{\mu\nu}\frac{d\tilde{x}^{\mu}}{d\tilde{\tau}}\frac{d\tilde{x}^{\nu}}{d\tilde{\tau}}}\,d\tilde{\tau}$
and
$\int\,m_{\chi}\,\sqrt{-g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}\,d\tau$
for the metrics (5.9) and (5.1), respectively, after requiring that
$|\vec{\tilde{p}}|$ and $|\vec{p}|$ coincide for $a=\mbox{constant}=1$. We
then observe that $\lvert\vec{\tilde{p}}\rvert$ and $\lvert\vec{p}\rvert$ are
related to each other by $\lvert\vec{\tilde{p}}\rvert=a\lvert\vec{p}\rvert$.
This relation implies that $\lvert\vec{\tilde{p}}\rvert=\text{constant}$
because the redshift dependence of the physical momentum is given by
$\lvert\vec{p}\rvert\propto\frac{1}{a}$. Thus the constancies of
$\lvert\vec{\tilde{p}}\rvert$ and $\lvert\vec{\tilde{k}}\rvert$ imply that if
Eq.(5.10) is satisfied for a value of
$\vec{\tilde{p}}^{2}-\vec{\tilde{k}}^{2}$ (at the moment of some transition
$\chi\chi\rightarrow\phi\phi$ ) then, in general, it is not satisfied at a
later time by the same $\vec{\tilde{p}}^{2}-\vec{\tilde{k}}^{2}$. This means
that during a process $\chi\chi\rightarrow\phi\phi$, the scale factor $a$ is
unchanged. However, during another process (being a later time), $a$ is again
unchanged except that it has a new increased constant value with respect to
the previous process. Then for each next process, $a$ increases while it
behaves as a constant value at the moment of these processes. In this regard,
for a given value of $\lvert\vec{\tilde{p}}\rvert$ in Eq.(5.10), the
corresponding $\lvert\vec{\tilde{k}}\rvert$ will get smaller and smaller over
time via the scale factor $a$. Here the curved space effect is then provided
by the scale factor $a$. Therefore, this observation is one of the key points
for deriving the tendency of the $\phi$’s towards Bose-Einstein condensation
in the level of cosmology.

Another approach to the tendency for the condensation of $\phi$ particles can
also be given by considering the phase space evolution of the particles by the
evolution of the distribution function for one of the final state particles
[118] through the equation

$\displaystyle\frac{d\tilde{f}(\vec{\tilde{p_{4}}},\eta)}{d\eta}=$
$\displaystyle\frac{1}{32(2\pi)^{5}\tilde{E}_{4}}\int\int\int\delta^{(4)}(\tilde{p_{1}}+\tilde{p_{2}}-\tilde{p_{3}}-\tilde{p_{4}})|\tilde{M}|^{2}$
$\displaystyle\times\\{\tilde{f}_{1}\tilde{f}_{2}(1+\tilde{f}_{3})(1+\tilde{f}_{4})-\tilde{f}_{3}\tilde{f}_{4}(1+\tilde{f}_{1})(1+\tilde{f}_{2})\\}\frac{d^{3}\vec{\tilde{p_{1}}}}{\tilde{E}_{1}}\frac{d^{3}\vec{\tilde{p_{2}}}}{\tilde{E}_{2}}\frac{d^{3}\vec{\tilde{p_{3}}}}{\tilde{E}_{3}},$
(5.12)

which is written for the effective Minkowski space as clarified by $``\
\tilde{}\ "$ over the quantities. In Eq.(5.12), $\tilde{M}$ denotes the
transition matrix element for $\chi\chi\rightarrow\phi\phi$ dominated by the
tree-level diagrams in Figure 5.1 and
$\tilde{f}_{i}=\tilde{f}(\vec{\tilde{p_{i}}},\eta)$ is the number density in
phase space. For the distribution of $\phi$ particles in phase space at early
times after the start of the process $\chi\chi\rightarrow\phi\phi$, we
consider the initial times for the sake of simplicity. Then we may write

$\displaystyle\tilde{f}^{(f)}(\vec{\tilde{p_{j}}},\eta)\simeq 0\ \ \
\text{with}\ \ \ j=3,4$ (5.13)
$\displaystyle\tilde{f}^{(i)}(\vec{\tilde{p_{j}}},\eta)\simeq$
$\displaystyle\tilde{n}_{\chi}\frac{[\Theta(|\vec{\tilde{p}}|_{\text{max}}-|\vec{\tilde{p}}|_{\text{min}})-\Theta(|\vec{\tilde{p}}|_{\text{min}}-|\vec{\tilde{p}}|_{\text{max}})]}{4\pi(|\vec{\tilde{p}}|_{\text{max}}-|\vec{\tilde{p}}|_{\text{min}})|\vec{\tilde{p}}_{j}|^{2}}$
$\displaystyle\times[\Theta(|\vec{\tilde{p}}_{\text{j}}|-|\vec{\tilde{p}}|_{\text{min}})-\Theta(|\vec{\tilde{p}}_{\text{j}}|-|\vec{\tilde{p}}|_{\text{max}})]\
\ \ \ \ \ \ \ \ \ \ \ \text{with}\ \ \ j=1,2,$ (5.14)

where $\Theta$ denotes the Heaviside function (i.e., unit step function). For
obtaining Eq.(5.14), we get use of the expression $\tilde{n}_{\chi}(\eta)=\int
d^{3}\tilde{p}\tilde{f}^{(i)}(\vec{\tilde{p}},\eta)$ (where $\tilde{n}_{\chi}$
is the number density of $\chi$ in the co-moving frame defined by Eq.(5.9)).
Here we also assume that the momentum of the $\chi$ particles (that have
enough energies to induce $\chi\chi\rightarrow\phi\phi$ processes) satisfies
$0\leq|\vec{\tilde{p}}|_{\text{min}}=\sqrt{\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}}<|\vec{\tilde{p}}|<|\vec{\tilde{p}}|_{\text{max}}=\sqrt{\vec{\tilde{k}}^{2}_{\text{max}}+\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}}$,
and the spatial distributions of the $\chi$ particles are homogeneous and
isotropic. Then, after using the expressions (5.13) and (5.14) in Eq.(5.12),
we get

$\frac{d\tilde{f}^{(f)}(\vec{\tilde{p}}_{4},\eta)}{d\eta}\simeq\tilde{n}^{2}_{\chi}\frac{\delta(\sqrt{\frac{1}{4}\vec{\tilde{p}}^{2}_{4}+\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}}-\sqrt{\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}})}{(\sqrt{\frac{1}{4}\vec{\tilde{p}}^{2}_{4}+\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}}-\sqrt{\tilde{m}_{\phi}^{2}-\tilde{m}_{\chi}^{2}})}\mathcal{B}(|\vec{\tilde{p}}_{4}|),$
(5.15)

where

$\displaystyle{\cal B}(|\vec{\tilde{p}}_{4}|)$ $\displaystyle=$
$\displaystyle\frac{1}{128(2\pi)^{7}\tilde{E}_{4}}\int\int\int\,\frac{d^{3}\vec{\tilde{p}}_{1}}{\tilde{E}_{1}}\frac{d^{3}\vec{\tilde{p}}_{2}}{\tilde{E}_{2}}\frac{d^{3}\vec{\tilde{p}}_{3}}{\tilde{E}_{3}}\,\delta^{(4)}(\tilde{p}_{1}+\tilde{p}_{2}-\tilde{p}_{3}-\tilde{p}_{4})\,|\tilde{M}|^{2}$
(5.16)
$\displaystyle\times\Bigg{\\{}\,\frac{\left[\Theta\left(|\vec{\tilde{p}}_{1}|-|\vec{\tilde{p}}|_{min}\right)-\Theta\left(|\vec{\tilde{p}}_{1}|-|\vec{\tilde{p}}|_{max}\right)\right]}{\vec{\tilde{p}}_{1}^{2}\,\vec{\tilde{p}}_{2}^{2}}$
$\displaystyle\,\,\,\,\,\,\,\,\,\times\,\frac{\left[\Theta\left(|\vec{\tilde{p}}_{2}|-|\vec{\tilde{p}}|_{min}\right)-\Theta\left(|\vec{\tilde{p}}_{2}|-|\vec{\tilde{p}}|_{max}\right)\right]}{\vec{\tilde{p}}_{1}^{2}\,\vec{\tilde{p}}_{2}^{2}}\,\Bigg{\\}}.$

and $\delta$ denotes Dirac delta function for which we have used that
$\text{lim}_{x\rightarrow y}\frac{\Theta(y)-\Theta(x)}{y-x}=\delta(x)$.

(Note that the $|\vec{\tilde{k}}|_{max}$ dependence in Eq.(5.16) may be
eliminated (in favor of $|\vec{\tilde{p}}_{4}|$ and a constant
$|\alpha|_{max}$) by making use of

$\displaystyle\vec{\tilde{p}}_{3}=\tilde{m}_{3}\frac{\vec{\tilde{P}}}{\tilde{m}_{3}+\tilde{m}_{4}}+\vec{\tilde{k}}=\frac{\vec{\tilde{P}}}{2}+\vec{\tilde{k}}~{},~{}~{}\vec{\tilde{p}}_{4}=\tilde{m}_{4}\frac{\vec{\tilde{P}}}{\tilde{m}_{3}+\tilde{m}_{4}}-\vec{\tilde{k}}=\frac{\vec{\tilde{P}}}{2}-\vec{\tilde{k}}$
(5.17)
$\displaystyle\vec{\tilde{P}}=\vec{\tilde{p}}_{3}+\vec{\tilde{p}}_{4}~{},~{}~{}\vec{\tilde{k}}=\frac{(\tilde{m}_{2}\vec{\tilde{p}}_{3}-\tilde{m}_{1}\vec{\tilde{p}}_{4})}{\tilde{m}_{3}+\tilde{m}_{4}}=\frac{\vec{\tilde{p}}_{3}-\vec{\tilde{p}}_{4}}{2}\;.$
(5.18)

Eq.(5.18) implies that, for a fixed $\vec{\tilde{p}}_{4}$, there exist the
largest number $|\alpha|_{max}$ that maximizes $|\vec{\tilde{k}}|$ with
$\vec{\tilde{p}}_{3}=-|\alpha|_{max}\vec{\tilde{p}}_{4}$, i.e.,
$|\vec{\tilde{k}}|_{max}=\frac{1}{2}\left(1+|\alpha|_{max}\right)|\vec{\tilde{p}}_{4}|$.)

The final form Eq.(5.15) of the phase space evolution of $\phi$ particles
indicates that the evolution of their momenta are towards
$|\vec{\tilde{p}}_{4}|=0$. This is the main indication of the Bose-Eeinstein
condensation because this gives rise to coherence (necessary for the $\phi$
particles being at the same de Broglie wavelengths) and correlation by
$\lambda\gg n_{\phi}^{-1/3}$ (necessary for the overlap of the de Broglie
wavelengths of the particles) of the system.

One should here notice that while Eq.(5.15) is an important equation to
indicate the formation of BEC, it does not show its complete formation. For
the derivation of Eq.(5.15) we have used Eq.(5.14), which is given for the
initial time when $\tilde{f}^{(f)}(\vec{\tilde{p}}_{j},\eta)\ll\,1$ (when the
formation of BEC has not been realised yet); on the other hand, Eq.(5.15) is
given for much later times when $|\vec{\tilde{k}}|_{max}\,\rightarrow\,0$.
This implies that we have not given proof but a hint towards the formation of
BEC at later times. Additionally, even while evolution towards
$|\vec{\tilde{p}}_{4}|=0$ is a crucial point towards forming of BEC, it is
insufficient [141]. In fact, essentially, Eq.(5.10) also indicates
$|\vec{\tilde{p}}_{4}|\,\rightarrow\,0$ by time. Eq.(5.15) reaffirms and
supports this finding and encourages a more thorough investigation. Therefore,
we have only shown the tendency of the $\phi$’s towards Bose-Einstein
condensation instead of proving its formation. For a thorough examination of
condensation formation, the same procedures must be followed at all times,
including the times when Bose statistics’ impact cannot be disregarded (i.e.,
considering the times when $\tilde{f}^{(f)}(\vec{\tilde{p}}_{j},\eta)$ cannot
be disregarded on the right hand side of Eq.(5.12)) and then solve the
equation for $\tilde{f}^{(f)}(\vec{\tilde{p}}_{4},\eta)$ and show that it has
a delta function of the form of Eq.(5.15). This requires separate work in
future. It is also important to note that there are also other processes
$\phi\phi\,\rightarrow\,\chi\chi$ and $\chi\phi\,\rightarrow\,\chi\phi$.
However, we assume that we have only $\chi$ particles at initial times, i.e.,
$n_{\phi}$ in phase space is small. Therefore, the rates of these processes
can be ignored. Additionally, the process $\chi\phi\,\rightarrow\,\chi\phi$
does not change the number density of the particles, so the condensation is
not affected by this process. While it seems that the process
$\phi\phi\,\rightarrow\,\chi\chi$ changes the number density of $\phi$, there
is a general rise in $n_{\phi}$ in case of considering both of
$\chi\chi\,\rightarrow\,\phi\phi$ and $\phi\phi\,\rightarrow\,\chi\chi$. As a
result, the general evolution, at least until chemical equilibrium, is towards
forming BEC. All these factors must be thoroughly considered in future
research to obtain a more complete and accurate picture of the evolution of
condensation.

####  5.2.2 Achieving Finite Number Density of $\phi$ Particles

The other condition for BEC is to realize the macroscopic nature of BEC. To
this end, the number density of $\phi$ particles, i.e., $n_{\phi}$, should
achieve a remarkable finite value in the presence of cosmological expansion.
Let us clarify this point as follows: By use of the relation
$\tilde{n}=\int\tilde{f}d^{3}\tilde{p}$ by using Eq.(5.12) for the initial
times of the transition $\chi\chi\rightarrow\phi\phi$, where $\tilde{f}_{3}$
and $\tilde{f}_{4}$ are negligible, we obtain

$\frac{d\tilde{\eta}_{4}(\eta)}{d\eta}=\tilde{\beta}\tilde{n}_{1}\tilde{n}_{2}\tilde{\sigma}\tilde{v},$
(5.19)

where $\tilde{\beta}$ is a constant that corresponds to average effective
depth of the collisions, $\tilde{\sigma}$ is the total cross-section of the
process, $\tilde{v}$ is the average velocity of two inital particles in the
spaced defined by (5.9). In Eq.(5.19), the total effective cross-section for
the transitions with $\tilde{m}_{\chi}\ll\tilde{m}_{\phi}$ is given by

$\tilde{\sigma}=\frac{(2\pi)^{4}}{4\sqrt{(\tilde{p}_{1}.\tilde{p}_{2})^{2}-\tilde{m}_{\chi}^{4}}}\int\int\delta^{(4)}(\tilde{p_{1}}+\tilde{p_{2}}-\tilde{p_{3}}-\tilde{p_{4}})|\tilde{M}|^{2}\frac{d^{3}\vec{\tilde{p_{3}}}}{\tilde{E}_{3}}\frac{d^{3}\vec{\tilde{p_{4}}}}{\tilde{E}_{4}}\simeq\bigg{(}\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\bigg{)}^{4}\frac{|\vec{\tilde{k}}|}{64\pi|\vec{\tilde{p}}|^{2}\tilde{m}_{\phi}},$
(5.20)

where we have used the definition of the transition matrix to find

$\tilde{M}=\tilde{\mu}^{2}\bigg{[}\frac{1}{(\tilde{p}_{1}-\tilde{p}_{3})^{2}+\tilde{m}_{\phi}^{2}}+\frac{1}{(\tilde{p}_{1}-\tilde{p}_{4})^{2}+\tilde{m}_{\phi}^{2}}\bigg{]}\simeq\bigg{(}\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\bigg{)}^{2},$
(5.21)

which is obtained by considering the center of mass frame with the assumptions
$\vec{\tilde{p}}^{2}\gg\tilde{m}_{\chi}^{2}$ and
$\vec{\tilde{k}}^{2}\ll\tilde{m}_{\phi}^{2}$.

To see the curved space effect on the number density, it is needed to convert
the form of Eq.(5.19) into the form defined by the space (5.1) as follows.
Because we have $\tilde{f}_{i}=f_{i}$ (where i=$\phi,\chi$) followed by

$\tilde{f}_{i}\,=\,\tilde{f}(\vec{\tilde{p}}_{i},\eta)\,=\,\frac{d\tilde{N}(\eta)}{d^{3}\tilde{p}_{(i)}d^{3}\tilde{x}_{(i)}}\,=\,\frac{d\tilde{N}(\eta)}{d^{3}p_{(i)}\,d^{3}x_{(i)}}\,=\,\frac{dN(t)}{d^{3}p_{(i)}\,d^{3}x_{(i)}}\,=\,f(\vec{p}_{i},t)\,=\,f_{i}\;,$
(5.22)

we get $\tilde{n}_{i}=a^{3}n_{i}$ by the definition of number density just
given above (here the sub-index $(i)$ refers to the i’th particle, and we have
used in Eq.(5.22) $\vec{p}=\frac{1}{a}\vec{\tilde{p}}$,
$\vec{x}=a\vec{\tilde{x}}$ where $x$ is the physical length scale).

At this point, we should notice that there are three different possible cases
for $m_{\chi}^{2}$ in Eq.(5.5):
i) $\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\,\leq\,m_{\chi}^{2}$ and
$\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\,\geq\,0$,
ii) $\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\,\leq\,m_{\chi}^{2}$ and
$\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\,<\,0$,
iii) $\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\,>\,m_{\chi}^{2}$
The case iii) above should be excluded to get rid of troublesome tachyons.
Therefore the cases i) and ii) remain as the only safe choices. In the case of
Eq.(A.1), i) implies that
$\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\,=\,\xi^{2}(2-s)\,a^{-2s}\,\geq\,0$,
i.e., $s\,\leq\,2$ while ii) implies that $\xi^{2}(2-s)\,a^{-2s)}\,<\,0$,
i.e., $s\,>\,2$. Most of the straightforward cosmic periods that are
physically significant - namely, the radiation, matter, and cosmological
constant eras - correspond to $s\,\leq\,2$ while the only physically
interesting era for the case $s\,>\,2$ is a possible stiff matter dominated
era where $s=3$. Therefore we consider the case i) here while we study the
extreme case
$|\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}|\,\gg\,m_{\phi}^{2}$ of
$s\,>\,2$ in the Appendix C to see the basic implications of ii).

For the case i) above we have
$|\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}|\,\ll\,m_{\phi}^{2}$ since
$m_{\chi}^{2}\,\ll\,m_{\phi}^{2}$. This, in turn, implies that
$\sigma\,=\,a^{2}\tilde{\sigma}\simeq\,a^{2}\left(\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\right)^{4}\frac{|\vec{\tilde{k}}|}{64\pi\,\vec{\tilde{p}}^{2}\tilde{m}_{\phi}}\,\propto\,a$
since $\tilde{m}_{\phi}\,\simeq\,a\,m_{\phi}$, $\tilde{\mu}\,\simeq\,a\,\mu$,
and $|\vec{\tilde{p}}|$, $|\vec{\tilde{k}}|$ are independent of redshift by
Eq.(5.11). For the relative velocity of the particles, $\vec{\tilde{v}}$, we
have
$|\vec{\tilde{v}}|=\frac{|d\vec{\tilde{r}}|}{d\eta}=\frac{|d\tilde{r}|/a}{dt/a}=|\vec{v}|=v_{0}$.
Thus, with these conversions by using $n_{i}(t)=\frac{C_{i}(t)}{a^{3}(t)}$,
Eq.(5.19) results in two equations

$\frac{\dot{C}_{\chi}}{a^{3}}=-\beta\bigg{(}\frac{C_{\chi}}{a^{3}}\bigg{)}\sigma_{0}v_{0}a\
\ \ \ \ \
\frac{\dot{C}_{\phi}}{a^{3}}=\beta\bigg{(}\frac{C_{\chi}}{a^{3}}\bigg{)}\sigma_{0}v_{0}a.$
(5.23)

Integrating out the first equation for $C_{\chi}$ and using it in the second
equation to obtain $C_{\phi}$ with considering cosmologically relevant cases
defined by $H=\xi a^{-s}$, we get

$C_{\phi}=C_{1}-\frac{C_{1}}{\frac{C_{1}\beta\sigma_{0}v_{0}}{(s-2)\xi}(a^{s-2}-a_{1}^{s-2})+1},$
(5.24)

where $C_{1}=C_{\chi}(t_{1})$ is the value of $C_{\chi}$ at the start of the
conversion of $\chi$s to $\phi$s. Eq.(5.24) can be examined for two different
cases:

  * •

At initial times $t\simeq t_{1}$, it may be approximated as

$C_{\phi}\simeq\frac{C_{1}\beta\sigma_{0}v_{0}}{(|s-2|)\xi}a^{|s-2|}\bigg{[}1-\bigg{(}\frac{a_{1}}{a}\bigg{)}^{|s-2|}\bigg{]}\
\ \ \text{for}\ \ \ s-2>0,$ (5.25)
$C_{\phi}\simeq\frac{C_{1}\beta\sigma_{0}v_{0}}{(|s-2|)\xi}a_{1}^{-|s-2|}\bigg{[}1-\bigg{(}\frac{a_{1}}{a}\bigg{)}^{|s-2|}\bigg{]}\
\ \ \text{for}\ \ \ s-2<0.$ (5.26)

Eq.(5.25) and Eq.(5.26) imply that $C_{\phi}$ initially reaches higher values
by the increase of the scale factor $a$ for $s<2$ while $C_{\phi}$ grows
faster for $s>2$.

  * •

For the late times $t\gg t_{1}$, Eq.(5.24) is approximated as

$C_{\phi}\sim C_{1}-\frac{(|s-2|)\xi}{\beta\sigma_{0}v_{0}}a^{-|s-2|}\ \ \
\text{for}\ \ \ s-2>0\ \ \ \text{and}\ \ \
\frac{C_{1}\beta\sigma_{0}v_{0}}{(|s-2|)\xi}a^{|s-2|}\gg 1$ (5.27)
$C_{\phi}\sim C_{1}-\frac{(|s-2|)\xi}{\beta\sigma_{0}v_{0}}a^{|s-2|}\ \ \
\text{for}\ \ \ s-2<0\ \ \ \text{and}\ \ \
\frac{C_{1}\beta\sigma_{0}v_{0}}{(|s-2|)\xi}a_{1}^{-|s-2|}\gg 1.$ (5.28)

The last equations imply that if the process $\chi\chi\rightarrow\phi\phi$
continues till very late times then $C_{\phi}$ reaches its maximum value of
$C_{1}$ for $s>2$ while it may be smaller than the value in the case $s<2$.
However, the study of $C_{\phi}$ for late times is not very reliable in that
the effects of the processes $\phi\phi\rightarrow\chi\chi$ and their
statistics are neglected in these times while they can be ignored at initial
times. These equations are reliable only if $n_{\phi}$ has not reached a large
value at late times yet. To better understand the problem at late times, more
thorough research will be required in future.

Finally, it is crucial to examine the scope of applicability of Eq.(5.6) by
considering Eq.(5.20) in such perturbative calculations (where
$\left(\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\right)\,<\,1$) because $\sigma$
becomes smaller for smaller $\frac{\tilde{\mu}}{\tilde{m}_{\phi}}$. However,
$|\vec{\tilde{k}}|$, $|\vec{\tilde{p}}|$, and $\tilde{m}_{\phi}$ itself also
affect the quantity $\tilde{\sigma}$, as well as
$\frac{\tilde{\mu}}{\tilde{m}_{\phi}}$. Further, instead of $\sigma$ alone,
the more relevant quantity in Eq.(5.6) is $n\sigma$ (given in $\bigtriangleup
t$). In this regard, even in the regime of perturbation, there is a sizeable
parameter space where such an effective Minkowski space formulation is valid.
For example, one may identify $\phi$ by dark matter and let
$\frac{\tilde{\mu}}{\tilde{m}_{\phi}}=0.1$,
$\frac{|\vec{\tilde{k}}|}{|\vec{\tilde{p}}|}=0.01$,
$|\vec{\tilde{p}}|\sim\,\tilde{m}_{\phi}\,c$; then
$n_{0}\,\sim\,\frac{10^{-3}\,eV\,cm^{-3}}{\tilde{m}_{\phi}\,c^{2}}$, so by
Eq.(5.20),
$n_{0}\tilde{\sigma}_{0}\simeq\,\frac{10^{-3}\,eV\,cm^{-3}}{\tilde{m}_{\phi}\,c^{2}}\left(\frac{\tilde{\mu}}{\tilde{m}_{\phi}}\right)^{4}\left(\frac{|\vec{\tilde{k}}|}{|\vec{\tilde{p}}|}\right)\left(\frac{\hbar\,c}{\tilde{m}_{\phi}\,c^{2}}\right)^{2}$
$\sim$ $10^{-19}\times\left(\frac{eV}{\tilde{m}_{\phi}}\right)^{3}\,cm^{-1}$
which satisfies
$\frac{H_{0}}{n_{0}\sigma_{0}\,v}\simeq\,\frac{10^{-28}cm^{-1}}{n_{0}\sigma_{0}(v/c)}\,\ll\,1$
if $\frac{v}{c}\sim\,1$ and $\tilde{m}_{\phi}\,\ll\,10^{3}\,eV$ (including the
phenomenologically interesting case of ultra light dark matter). We should
here notice that in the calculations, we cannot use
$|\vec{\tilde{k}}|\,\simeq\,0$ because doing so would mean a complete BEC,
where the entire system acts as a single quantity, and in this case the
standard description of scattering regarding single particles in quantum field
theory cannot be considered. In fact, our primary goal in this research is to
indicate how curved space effects promote the formation of BEC in a model that
incorporates $\phi^{2}\chi$ kind of interaction terms instead of showing that
an effective Minkowski space formulation holds true in every scenario in
cosmology. We expect that this formulation and this research may give
additional insight to understand the formation of BEC in cosmology.

After these results, we can now conclude that via the construction of the
effective Minkowski space formulation at the time scale oparticle physics
process $\chi\chi\rightarrow\phi\phi$ (induced by $\phi^{2}\chi$ type of
interaction), we have obtained all necessary conditions, i.e., coherence,
correlation, and finite number density, for the condensation of $\phi$
particles at initial times in cosmology, where the curved space effect is
provided by the scale factor $a$.

##  Chapter 6 Curved Space and Particle Physics Effects on the Formation of
Bose-Einstein Condensation around a Reissner - Nordstrøm Black Hole

In the previous chapter, we have shown the evolution of $\phi$ particles is
towards condensation in a curved space, particularly in cosmology described by
FRW metric. A similar approach can also be applied to other curved spaces. To
this end, this chapter [50] aims to observe if a condensate of the scalar
field is realized around a black hole.

In particular, we consider an RN black hole [51] (a non-rotating black hole
with a non-vanishing electric charge) instead of the simpler case of a
Schwarzschild black hole. This is because charged particles of charge $q$ (of
the same charge as the black hole) with low enough energies $\omega$ with
$0\,<\,\omega\,<q\frac{Q}{r_{+}}$ (where $r_{+}=M+\sqrt{M^{2}-Q^{2}}$ is the
radius of the event horizon) can be scattered by RN black holes [51]. On the
other hand, in the case of Schwarzschild black holes, there is no scattering
from the black hole. Hence, the particle physics processes essential for the
formation of condensation cannot be efficiently realized around a
Schwarzschild black hole. Therefore, we consider the problem of scalar field
condensation around RN black holes as a simple and efficient framework to
study the condensation of scalar fields around black holes.

###  6.1 Framework

The RN metric is defined as

$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2}),$
(6.1)

where

$f=\bigg{(}1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\bigg{)}.$ (6.2)

Here $M$ and $Q$ are the black hole’s mass and charge.

We assume that initially, there are a homogeneous distribution of a diluted,
relativistic $\chi$ fields in this background, and they are converted into the
non-relativistic $\phi$ fields by the particle physics processes
$\chi\chi\,\rightarrow\,\phi\phi$. Here we consider an interaction term
$\chi^{*}\chi\phi^{*}\phi$, which will be introduced in Section 6.1.2.

Most of the collisions of the $\chi$ particles occur between the ones
scattered from the black hole and the others coming from infinity. The
relativistic particles extract energy from the black hole after being
scattered if they obey the condition $m\,<\,\omega\,<\,q\frac{Q}{r_{+}}$. This
type of extracting energy from a black hole is known as superradiance.

In this study, we focus on the radial head-on collisions of the $\chi$
particles and examine the behaviour of the produced $\phi$ particles. In this
respect, we first study the motion of the scalar particles in the radial
direction at the level of test particles as given in the following subsection.

####  6.1.1 Motion in Radial Direction

To proceed, we begin with examining the radial behaviours of the fields. To
this end, we consider the 1+1 dimensional subspace of the 3+1 dimensional
space (6.1)

$ds^{2}\,=\,-f\,dt^{2}\,+\,f^{-1}dr^{2}.$ (6.3)

The Lagrangian for a test particle of mass $m$ in Eq.(6.3) is

$L\,=\,m\frac{ds}{d\tau}\,=\,m\sqrt{-f\dot{t}^{2}\,+\,f^{-1}\dot{r}^{2}},$
(6.4)

where $\dot{t}=\frac{dt}{d\tau}$, $\dot{r}=\frac{dr}{d\tau}$ with
$d\tau=\sqrt{-ds^{2}}$. The Lagrange equation for the coordinate $t$ results
in the conservation of $h$, i.e., the total energy (including the potential
energy) per unit mass of a test particle, namely

$h\,=\,f\,\dot{t}\,=\,\frac{1}{m}\frac{\partial\,L}{\partial\,\dot{t}}\,=\,C\,=\,\mbox{constant},$
(6.5)

where we have used

$\left(-f\dot{t}^{2}+f^{-1}\dot{r}^{2}\right)\,=\,-1$ (6.6)

for massive particles. Eq.(6.6) in combination with Eq.(6.5) results in

$\dot{r}^{2}\,=\,C^{2}\,-\,\left(1\,-\,\frac{2M}{r}\,+\,\frac{Q^{2}}{r^{2}}\right).$
(6.7)

As $r\,\rightarrow\,\infty$, Eq.(7.13) becomes

$\dot{r}_{\infty}^{2}\,=\,C^{2}\,-\,1.$ (6.8)

Here we let $\chi$ particles that are initially at rest fall towards the black
hole from a significant distance that may be approximated by infinity, which
implies $C_{\chi}=1$ by Eq.(7.14). Moreover, at the center of mass frame, the
conservation of energy for a given process $\chi\chi\,\rightarrow\,\phi\phi$
implies $m_{\chi}C_{\chi}=m_{\phi}C_{\phi}$ or
$C_{\phi}=C_{\chi}\frac{m_{\chi}}{m_{\phi}}$. Because $m_{\chi}<m_{\phi}$, we
then get $C_{\phi}<C_{\chi}$. This result means that the produced $\phi$
particles scattered by the black hole cannot reach infinity. Therefore, they
can reach only a finite distance from the black hole, which is the greater
root of $(1-C_{\chi}^{2})r_{0}^{2}-2Mr_{0}+Q^{2}=0$ for $Q\,<\,M$, which may
be found by equating $\dot{r}$ in Eq.(7.13) to zero. In this respect, there
will be a belt of $\phi$ particles with zero or almost zero momenta around the
black hole. This provides a suitable condition for the formation of Bose-
Einstein condensation. This result is obtained at the level of test particles.
We now search for the same result at the level of field theory in the
following sections.

####  6.1.2 The Field Equations for the Scalars

We consider the following action with the background of Eq.(6.1)

$S=\int\;d^{4}x\,\sqrt{-g}\,\\{-g^{\mu\nu}\left[D_{\mu}\phi\,\left(D_{\nu}\phi\right)^{*}\,+\,D_{\mu}\chi\,\left(D_{\nu}\chi\right)^{*}\right]\,-\,m_{\phi}^{2}\left|\phi\right|^{2}\,-\,m_{\chi}^{2}\left|\chi\right|^{2}\,-\,\lambda\,\phi^{*}\phi\chi^{*}\chi\\},$
(6.9)

where $D_{\mu}\,=\,\partial_{\mu}\,+\,iq\,A_{\mu}$ with $q$ being the electric
charge of the scalar field and $A_{\mu}\,=\,\left(\frac{Q}{r},0,0,0\right)$
denoting the electric field of the black hole. Here we let both $\chi$ and
$\phi$ have the same charge q. In Eq.(6.9), due to the small coupling constant
of electromagnetic interactions and we have assumed the low scalar particle
density, we have ignored their impact on scalar particle interactions.

If we take $\lambda$ small in comparison with the other terms in Eq.(6.9), the
corresponding field equation for $\phi$ is then given by

$D_{\mu}\,D^{\mu}\,\phi\,-\,m_{\phi}^{2}\,\phi\,=\,0.$ (6.10)

The corresponding field equation for $\chi$ may be obtained by replacing
$\phi$ in Eq.(6.10) with $\chi$. Although we have neglected the term with the
coefficient $\lambda$ in the field equations to obtain approximate solutions,
the effect of the interaction is still imposed by the previously obtained
expression

$\displaystyle C_{\phi}=C_{\chi}\frac{m_{\chi}}{m_{\phi}}.$ (6.11)

Using the ansatz [119]

$\phi_{\omega}\left(t,r,\theta,\varphi\right)\,=\,\sum_{l,m}e^{-i\omega\,t}Y_{l}^{m}\left(\theta,\varphi\right)\frac{\psi_{\omega}\left(r\right)}{r},$
(6.12)

the field equation (6.10) results in

$f^{2}\frac{d^{2}}{dr^{2}}\psi_{\omega}+ff^{\prime}\frac{d}{dr}\psi_{\omega}+\left[\left(\omega-\frac{qQ}{r}\right)^{2}-V\right]\psi_{\omega}\,=\,0,$
(6.13)

where $\prime$ denotes derivative with respect to r, and

$V=f\left(\frac{l(l+1)}{r^{2}}+\frac{f^{\prime}}{r}+m_{\phi}^{2}\right).$
(6.14)

In the case of the radial motion for which $l=0$ and by considering
$\frac{f^{\prime}}{r}\,\ll\,m_{\chi}^{2}\,<\,m_{\phi}^{2}$ (which is satisfied
for a wide range of $m_{\chi}^{2}$ if $r$ is not close to $r_{+}$, as will be
mentioned in Section 6.2.2), Eq.(6.13) reduces to

$\frac{d^{2}\psi_{\omega}}{dr_{*}^{2}}\,+\,\left[\left(\omega-\frac{qQ}{r}\right)^{2}\,-\,\tilde{m}_{\phi}^{2}\right]\,\psi_{\omega}\,=\,0,$
(6.15)

where $dr_{*}\,=\,f^{-1}dr$, $\tilde{m}_{\phi}^{2}\,=\,f\,m_{\phi}^{2}$.
Eq.(6.15) is the same expression as the field equation obtained from the
corresponding 1+1 dimensional form of the action (6.9) (see Appendix D).

###  6.2 A Special Solution

####  6.2.1 Derivation

In the hope of obtaining a wave-like solution to Eq.(6.15) we consider the
following type of solution

$\displaystyle\psi_{\omega}=e^{isr_{*}}g(r_{*}),$ (6.16)

where $s^{2}=\omega^{2}-m^{2}$. Eq.(6.15), after using Eq.(6.16), becomes

$\displaystyle\frac{g^{\prime\prime}}{g}+2is\frac{g^{\prime}}{g}-\frac{2(qQ\omega-m^{2}M)}{r}+\frac{(q^{2}-m^{2})Q^{2}}{r^{2}}=0,$
(6.17)

where $\prime$ denotes the derivative with respect to $r_{*}$. We try the
following choice

$\frac{g^{\prime}}{g}=\frac{\alpha_{1}}{r}+\frac{\beta_{1}}{r^{2}}+\frac{\gamma_{1}}{r^{3}},$
(6.18)
$\frac{g^{\prime\prime}}{g}=\frac{\alpha_{2}}{r}+\frac{\beta_{2}}{r^{2}}+\frac{\gamma_{2}}{r^{3}},$
(6.19)

where $\alpha_{1}$, $\beta_{1}$, $\gamma_{1}$, $\alpha_{2}$, $\beta_{2}$,
$\gamma_{2}$ are some constants whose explicit forms will be derived below.
Hence, if such a solution exists, then Eq.(6.16) becomes

$\displaystyle\psi_{\omega}=e^{isr_{*}}\,\exp{\left[\int_{r_{0}}^{r}{\frac{\alpha_{1}\,r}{r^{2}-2Mr+Q^{2}}dr}+\int_{r_{0}}^{r}{\frac{\beta_{1}}{r^{2}-2Mr+Q^{2}}dr}+\int_{r_{0}}^{r}{\frac{\gamma_{1}}{r^{3}-2Mr^{2}+Q^{2}r}dr}\right]}.$
(6.20)

The explicit form of the solution $\psi_{\omega}$ for a given $M$ and $Q$ may
be obtained as follows. Eq.(6.18) and Eq.(6.19) solve Eq.(6.17) if (see
Appendix E)

$\alpha_{1}=\frac{9}{5}-\frac{9}{20}\bigg{(}\frac{M}{Q}\bigg{)}^{2}\,,\ \ \ \
\ \ \beta_{1}=-\frac{9}{2}M\,,\ \ \ \ \ \ \gamma_{1}=3Q^{2}\,,$ (6.21)
$\displaystyle\alpha_{2}=0\,,\ \ \ \ \ \
\beta_{2}=\bigg{[}\frac{9}{5}-\frac{9}{20}\bigg{(}\frac{M}{Q}\bigg{)}^{2}\bigg{]}\bigg{[}\frac{4}{5}-\frac{9}{20}\bigg{(}\frac{M}{Q}\bigg{)}^{2}\bigg{]}\,,\
\ \ \ \ \
\gamma_{2}=2\bigg{[}\frac{-9}{5}+\frac{63}{40}\bigg{(}\frac{M}{Q}\bigg{)}^{2}\bigg{]}M.$
(6.22)

Inserting Eq.(6.21) and Eq.(6.22) into Eq.(6.17) and using
$s^{2}=\omega^{2}-m^{2}$, we get three equations for three unknown quantities
$\omega$, $q$, $m$ for a given $M$ and $Q$:

$\displaystyle(m^{2}-\omega^{2})-\frac{M^{2}}{9Q^{4}}\bigg{(}\frac{9}{5}-\frac{63}{40}\frac{M^{2}}{Q^{2}}\bigg{)}^{2}=0,$
(6.23)
$\displaystyle-\frac{3M^{2}}{Q^{2}}\bigg{(}\frac{9}{5}-\frac{63}{40}\frac{M^{2}}{Q^{2}}\bigg{)}+\bigg{(}\frac{9}{5}-\frac{9}{20}\frac{M^{2}}{Q^{2}}\bigg{)}\bigg{(}\frac{4}{5}-\frac{9}{20}\frac{M^{2}}{Q^{2}}\bigg{)}+(q^{2}-m^{2})Q^{2}=0,$
(6.24)
$\displaystyle\frac{2M}{3Q^{2}}\bigg{(}\frac{9}{5}-\frac{63}{40}\frac{M^{2}}{Q^{2}}\bigg{)}\bigg{(}\frac{9}{5}-\frac{9}{20}\frac{M^{2}}{Q^{2}}\bigg{)}-2(qQ\omega-m^{2}M)=0.$
(6.25)

By plugging Eq.(6.23) and Eq.(6.21) into Eq.(6.20), we obtain the following
solution

$\displaystyle\psi_{\omega}$ $\displaystyle=$
$\displaystyle\exp\Bigg{\\{}\left(\frac{\pm\,3M\left(7M^{2}-8Q^{2}\right)}{40Q^{4}\sqrt{Q^{2}-M^{2}}}\right)$
(6.26)
$\displaystyle\times\,\Bigg{[}\left(2M^{2}-Q^{2}\right)\Bigg{(}\tan^{-1}\bigg{(}\frac{M-r}{\sqrt{Q^{2}-M^{2}}}\bigg{)}-\tan^{-1}\bigg{(}\frac{M-\text{r}_{0}}{\sqrt{Q^{2}-M^{2}}}\bigg{)}\Bigg{)}$
$\displaystyle-\sqrt{Q^{2}-M^{2}}\Big{(}M\log\left(-2Mr+Q^{2}+r^{2}\right)-M\log\left(-2M\text{r}_{0}+Q^{2}+\text{r}_{0}^{2}\right)+r-\text{r}_{0}\Big{)}\Bigg{]}\
$ $\displaystyle+\
\left(\frac{3(6M^{3}-4MQ^{2})}{40Q^{2}\sqrt{Q^{2}-M^{2}}}\right)\Bigg{(}\tan^{-1}\bigg{(}\frac{M-r}{\sqrt{Q^{2}-M^{2}}}\bigg{)}-\tan^{-1}\bigg{(}\frac{M-\text{r}_{0}}{\sqrt{Q^{2}-M^{2}}}\bigg{)}\Bigg{)}$
$\displaystyle-\frac{3\left(3M^{2}+8Q^{2}\right)}{40Q^{2}}\Big{(}\log\left(-2Mr+Q^{2}+r^{2}\right)-\log\left(-2M\text{r}_{0}+Q^{2}+\text{r}_{0}^{2}\right)\Big{)}$
$\displaystyle+3\log(r)-3\log(\text{r}_{0})\Bigg{\\}},$

where the exponential function can be either an increasing or decreasing real
exponential in $r$ depending on the values of $M$ and $Q$. The increasing one
corresponds to the unphysical case. Therefore, we consider the decreasing one
for the implication of the solution $(\ref{H1b})$, as will be examined in the
next section. Further, if we solve Eq.(6.23), Eq.(6.24), and Eq.(6.25) for
$m$, $\omega$, and $q$ by Mathematica [120], we get the following results:

$\displaystyle m^{2}_{1}=-\frac{9\left(7M^{4}+2M^{2}Q^{2}\right)}{40Q^{6}},\ \
\ \ \
m^{2}_{2}=\frac{9\left(7M^{2}-8Q^{2}\right)^{2}\left(4Q^{2}-3M^{2}\right)}{1600Q^{6}\left(Q^{2}-M^{2}\right)},$
(6.27)
$\displaystyle\omega_{1,2}=\mp\frac{3iM\left(7M^{2}+12Q^{2}\right)}{40Q^{4}},\
\ \ \ \
\omega_{3,4}=\mp\frac{3i\left(7M^{4}-22M^{2}Q^{2}+16Q^{4}\right)}{40\sqrt{Q^{8}\left(M^{2}-Q^{2}\right)}},$
(6.28) $\displaystyle
q_{1,2}=\mp\frac{3i\left(17M^{2}-8Q^{2}\right)}{20Q^{3}},\ \ \ \ \
q_{3,4}=\pm\frac{3iMQ\left(27M^{2}-28Q^{2}\right)}{40\sqrt{Q^{8}\left(M^{2}-Q^{2}\right)}},$
(6.29)

where $\\{m^{2}_{1},\ \omega_{1,2},\ q_{1,2}\\}$ and $\\{m^{2}_{2},\
\omega_{3,4},\ q_{3,4}\\}$ are two different set of solutions.

####  6.2.2 Viability of the Solution

To examine the viability of our solution, we rescale all the expressions given
in Eq.(6.15) in terms of the multiples of $M$ or $M_{\circ}$ (the mass of the
sun) as given below:

$\displaystyle\overline{r}_{*}=\frac{c^{2}}{GM_{\circ}}r_{*},~{}~{}\overline{r}=\frac{c^{2}}{GM_{\circ}}r,$
(6.30)
$\displaystyle\overline{\omega}=\omega\frac{GM_{\circ}}{c^{3}}=\bigg{(}\frac{\omega}{s^{-1}}\bigg{)}5\times
10^{-6},$ (6.31) $\displaystyle\overline{m}=m\frac{GM_{\circ}}{\hbar
c}=\bigg{(}\frac{m}{kg}\bigg{)}\,4.5\times 10^{45},$ (6.32)
$\displaystyle\overline{M}=\frac{M}{M_{\circ}}=\bigg{(}\frac{M}{kg}\bigg{)}2\times
10^{-30},$ (6.33)
$\displaystyle\overline{Q}=\frac{Q\sqrt{k}}{M_{\circ}\sqrt{G}}=\bigg{(}\frac{Q}{C}\bigg{)}5,7\times
10^{-21},$ (6.34) $\displaystyle\overline{q}=q\frac{M_{\circ}\sqrt{Gk}}{\hbar
c}=\bigg{(}\frac{q}{C}\bigg{)}4,8\times 10^{55},$ (6.35)

where $k$ is the Coulomb’s constant and we have explicitly written $G$, $c$,
$\hbar$, $k$ (that we had set to 1) to see the phenomenological contents of
these quantities. With these redefinitions, Eq.(6.15) is given by (see
Appendix F)

$\displaystyle\frac{d^{2}\psi_{\omega}}{d\overline{r}_{*}^{2}}+\bigg{[}\big{(}\overline{\omega}-\frac{\overline{q}\overline{Q}}{\overline{r}}\big{)}^{2}-\overline{m}_{\phi}^{2}\left(1-\frac{2\overline{M}}{\overline{r}}+\frac{\overline{Q}^{2}}{\overline{r}^{2}}\right)\bigg{]}\psi_{\omega}=0.$
(6.36)

We should here notice that the rescaled form of our field equation (6.36)
contains the barred forms of Eqs.(6.27)-(6.29). In this regard, we have
obtained the following results:

  * •

In Eq.(6.29) $\bar{q}_{1,2}$ takes the imaginary values. Then the solution set
$\\{\bar{m}^{2}_{1},\ \bar{\omega}_{1,2},\ \bar{q}_{1,2}\\}$ does not
correspond to the physical one. Therefore only the other solution set
$\\{\bar{m}^{2}_{2},\ \bar{\omega}_{3,4},\ \bar{q}_{3,4}\\}$ is the physical
one if $|\bar{Q}|>\bar{M}$.

  * •

We also see that there are no physical solutions for $\bar{q}=0$ (or $q=0$)
because $\bar{q}=0$ corresponds to the case of $\bar{M}>|\bar{Q}|$, which is
not in the physical range of our set of solution. Therefore, in agreement with
our model, the solution given in this study is relevant only for $q\neq 0$.

  * •

The allowed ranges are $1.44<m^{2}Q^{2}<\infty$, $1.2<\omega Q<\infty$,
$0<qQ<\infty$, where $m^{2}Q^{2}$ and $\omega Q$ are in the order of $1$ for
most of the values of $M$ and $Q$. $qQ$ is in the order of $10^{-1}$ for most
of the values of $M$ and $Q$.

By taking into account our set of solutions, as discussed above, we are now in
a position to check if our wave solution (6.26) results in the wave profiles
that are expected from a test particle treatment where we examine the tendency
of the condensation (the accumulation of the $\phi$ particles with zero
momenta) around the black hole. By considering our physically relevant set
$|\bar{Q}|>\bar{M}$, we have plotted $\psi_{\omega}$ plots for various values
of its parameters by using a Mathematica code [120]. We have found mainly two
types of behaviours for $|\psi_{\omega}|$, namely, exponentially increasing
with increasing $r$, exponentially decreasing with increasing $r$ with a local
peak. We discard the exponentially increasing ones because they are not
physical. Finally, as an example we obtain the desired Figure 6.1 (in
agreement with our test particle treatment), as given below, with $\bar{Q}=17$
for a given $\bar{M}=10$. It is also observed that for some $M$, $Q$ pairs the
absolute value of $\psi_{\omega}$ peaks at some $Q$ values, for example, as in
Figure 6.2. We have also checked whether these wave profiles correspond to the
$\phi$ particles or not. For a given value $\bar{Q}=17$ and $\bar{M}=10$ we
obtain $\frac{\omega}{m}\,<\,1$, which implies the behaviour of the wave
profile of the form of Figure 6.1 for the scalar field $\phi$. Therefore, we
conclude that the wave profile in Figure 6.1 is consistent with the
accumulation of the scalar particles at some distance from the black hole
suggested by the test particle behaviour as discussed in Section 6.1.1.

Figure 6.1: $|\psi_{\omega}|$ versus $\bar{r}$ graph for $\bar{M}=10$,
$\bar{Q}=17$, $\bar{r}_{0}=20$ Figure 6.2: $|\psi_{\omega}|$ versus $\bar{Q}$
graph for $\bar{M}=10$, $\bar{r}=20$, $\bar{r}_{0}=30$

We realize that the $\frac{f^{\prime}}{r}$ term can be ignored with respect to
$m_{\chi}^{2}$ for phenomenologically feasible values of the parameters as can
be seen below

$\frac{f^{\prime}}{\bar{r}}=\frac{2}{\bar{r}^{2}}\left(\frac{\bar{M}}{\bar{r}}-\frac{\bar{Q}^{2}}{\bar{r}^{2}}\right),$
(6.37)

where we have used $\bar{r}$, $\bar{M}$, $\bar{Q}$ in place of $r$, $M$, $Q$
by use of the Eqs.(6.30)-(6.35). We here consider the outside of the horizon,
i.e., $\bar{r}>\bar{r}_{+}$. In the case of small values of $\bar{r}$, where
$\bar{r}\simeq\bar{M}$, we have $\frac{\bar{M}}{\bar{r}}\simeq 1$. Then the
term $\left(\frac{\bar{M}}{\bar{r}}-\frac{\bar{Q}^{2}}{\bar{r}^{2}}\right)$
ensures that $\frac{f^{\prime}}{\bar{r}}\rightarrow 0$ with respect to
$m_{\chi}^{2}$ as $|\bar{Q}|\rightarrow\bar{M}$. In the case of large values
of $\bar{r}$, the term $\frac{2}{\bar{r}^{2}}$ ensures
$\frac{f^{\prime}}{\bar{r}}\sim\,0$, which is compatible with the wave profile
6.1.

For the sake of simplicity, to prevent the fields in this study from altering
the space’s geometry, we considered their low energy densities. As a result,
it is quite challenging to find these dark matter candidates. On the other
hand, as long as we are at a sufficiently large distance from the black hole
to satisfy $\frac{f^{\prime}}{r}\,\ll\,m_{\chi}^{2}\,<\,m_{\phi}^{2}$ and the
spherical shape of the wave profile is retained, i.e., $l=0$, we do not
anticipate a significant change in the form of the geometry even when the
energy density of the fields is raised. In that situation, the RN metric will
still characterize the compact object’s geometry. In this case, the
gravitational effect of these field(s), such as their impact on the rotation
curve(s) of their galaxies, can be used to detect the presence of the scalar
fields charged with a dark electric charge surrounding an RN black hole
(charged with the same dark electric charge) (while such an analysis will have
additional, non-trivial points to be addressed). Each of these topics requires
a distinct and detailed analysis. All of these aspects must be taken into
account in thorough, independent, and in-depth future analysis in order to
draw a firm and reliable conclusion regarding the impact of non-negligible
energy densities of the scalar fields. In fact, literature has addressed the
issue of galaxy rotation curves in the context of Bose-Einstein condensate
dark matter, which is made up of a self-gravitating scalar dark matter gas
cloud [41, 146]. There are also studies in the literature that consider the
source of gravitational as a point source [147] while they do not discuss the
predictions for galaxy rotation curves. Most of these studies are non-
relativistic while there are also studies in the context of general relativity
[41, 148]. [41] studies the problem through the postulation of a mass density
for the scalar field. On the other hand, [148] studies the problem in the
context of charged black holes including RN black holes while the scalar field
is taken to be neutral. This paper also finds a localized solution of the
scalar field while the explicit form of the solution is derived only at the
limits $r_{\star}\pm\infty$. In other words, some studies have the same topics
of interest as the current paper. The current work still includes some novel
aspects, such as a model of two interacting scalars that leads to
$C_{\phi}=C_{\chi}\frac{m_{\chi}}{m_{\phi}}$ (which may be considered as the
mechanism behind the Bose-Einstein condensation in this context) and the
solution given in Section 6.2 is a new analytical solution.

In this study, it is also possible to consider $\phi$ and/or $\chi$ particles
as the only source or as one of the sources of DM in the universe. The idea
that this mechanism is the only source of DM cannot be completely ruled out
due to the special nature of both the source and DM. An example to a possible
scenario may be given that $\chi$ particles, which are non-relativistic at
considerable distances from the black hole, may operate as the component of
DM/DE at cosmic scales, whereas $\phi$ particles are thought to act in the
component of DM at galactic scales. That if such a situation is feasible or
not may be considered in future in a separate study. Another relevant non-
trivial point is that a constant pair of $M$, $Q$ gives a constant pair of
$m$, $q$ as seen in Eq.(6.27) and Eq.(6.29); that is to say, for constant
values of $m_{\phi}$, $q_{\phi}$ the number of $M$, $Q$ pairs is unchanged. At
first sight, one may infer that these fixed numbers of values lead to a
particular case of the solitonic solution for the scalar DM behaviour around
the black hole, so this study is insufficient to be regarded as an important
source of DM in cosmology. Given the following discussions, such a
justification may not be valid. The values of $M$ and $Q$ of RN black holes
are expected to be changed by accretion, i.e., by swallowing the particles
$\chi$ and $\phi$. Because the charge is an additive quantum number, the value
of $Q$ increases proportional to the number of swallowed particles. However,
this may not be the same for the value of $M$. The value of $M$ is expected to
be smaller than the number of swallowed particles times their masses since the
effect of the gravitational potential between the particles should also be
considered. At the initial times of the accretion, due to the small number
density of the particles, the effect of the gravitational potential between
them can be ignored. Therefore, it is expected that $\frac{\Delta Q}{\Delta
M}\simeq\frac{q}{m}$ at these times but $\frac{\Delta Q}{\Delta
M}>\frac{q}{m}$ at later times. Hence, by the time the values of $Q$ and $M$
increase and at some time the pair of $Q$ and $M$ reaches some values so that
the solitonic solution is formed. Then, $Q$ and $M$ proceed to evolve in order
for the formation of the localized wave form. However, at later times this
wave profile may be dispersed due to the ongoing accretion and at this time
the effect of the gravitational potential between the particles may be
significant because of the non-negligible number densities of $\phi$
particles. Thus its mass and charge will effectively change depending on the
distance from the black hole. This indicates that at any radial distance the
black hole’s mass and charge will always meet the criteria required for
forming the solitonic solution. This, as well as the gravitational attraction
between the particles, tends to prevent the wave profile’s dispersion.
However, this point requires a more detailed further study to ensure these
results. Further, if we let $C_{\chi}$ take values in an interval about 1,
i.e., if we take $\omega$ to be in an interval about $m_{\chi}$ rather than
taking a sharp value for $\omega$, there will be then more chance that a value
of $(m_{\phi}^{2}-\omega^{2})$ in the interval provides Eq.(6.23) for a given
set of $M$, $Q$. Given these discussions and the novelty of the mechanism and
the solution mentioned here, in our opinion, this new mechanism and the
solution need in-depth follow-up research that examines all of their
characteristics and potential in further studies.

##  Chapter 7 A Metric for Gravitational Collapse around a Schwarzschild Black
Hole

This chapter aims to be the first step towards a more realistic picture for
examining BEC around an accreting black hole - fluid system [52]. To this end,
we consider the problem of gravitational collapse of a fluid under the effect
of a small Schwarzschild black hole. We postpone the study of the behaviour of
BEC in such a background to future studies. In this chapter, we start with the
most general spherically symmetric metric and its corresponding non-zero
components of the Einstein tensor, where $G^{0}_{\;1}$ is responsible for the
accretion, i.e., radial inward power flux. To ensure a Schwarzschild and
Robertson-Walker metrics as limiting cases of our study, we first derive a
condition in the case of non-accretion, i.e., $G^{0}_{\;1}=0$. We next propose
the simplest, most plausible mass function that describes twice the total
energy in a spherically collapsing shell at some distance and time.
Considering the condition derived in limiting cases and the proposed mass
function, we derive a metric for gravitational collapse around a Schwarzschild
black hole. We ensure that this metric corresponds to the case of
$G^{0}_{\;1}\neq 0$. In the limiting cases of $r\,\rightarrow\infty$ or
$M\,\rightarrow 0$, we observe that $G^{0}_{\;1}\,\rightarrow 0$ and the
metric derived is reduced to the Robertson-Walker metric. Our discussion on
these limits shows that the accretion is due to the presence of the black hole
instead of the behaviour of the fluid itself. We also discuss the apparent
horizon and the singularities of our solution in a particular case of dust. We
see that all the singularities are hidden behind the apparent horizon.
Additionally, the collapse of dust allows negative pressures implying that the
collapsing dust may induce dark energy-like behaviour.

We organize this chapter as follows. We first introduce our framework, where
we derive a condition that ensures Robertson-Walker and Schwarzschild’s
metrics are particular solutions. In this section, we also propose a mass
function. With the condition and mass function, we derive a metric for
accreting Schwarzschild black hole-fluid system. In the next section, we
discuss some implications of the metric where we examine the general behaviour
of our solution with its apparent horizons and singularities in the case of
dust. In the final section, we comment on the energy density and the pressures
for dust collapse.

###  7.1 Framework

In this section, we first quickly review the background information that is
crucial to understanding this research. Then, by requiring consistency between
two distinct formulations of the Misner-Sharp mass function for a marginally
bound dust, we derive a requirement crucial to understanding the remaining
sections of the study.

####  7.1.1 Preliminaries

The most general spherically symmetric metric corresponds to [149, 150, 151]

$ds^{2}=-e^{2\lambda}dt^{2}+e^{2\psi}dr^{2}+R^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(7.1)

where $\lambda=\lambda(r,t)$, $\psi=\psi(r,t)$, $R=R(r,t)$. The non-zero
components of Einstein tensor for the metric (7.1) are given by [149, 151]

$\displaystyle
G^{0}_{\;0}\,=-\,\frac{F^{\prime}}{R^{2}R^{\prime}}\,+\,\frac{2\dot{R}\,e^{-2\lambda}}{R\,R^{\prime}}\left(\dot{R}^{\prime}-\dot{R}\lambda^{\prime}-\dot{\psi}R^{\prime}\right),$
(7.2) $\displaystyle
G^{1}_{\;1}\,=-\,\frac{\dot{F}}{R^{2}\dot{R}}\,-\,\frac{2R^{\prime}\,e^{-2\psi}}{R\,\dot{R}}\left(\dot{R}^{\prime}-\dot{R}\lambda^{\prime}-\dot{\psi}R^{\prime}\right),$
(7.3) $\displaystyle
G^{0}_{\;1}\,=-\,e^{2\psi-2\lambda}\,G^{1}_{\;0}\,=\,\frac{2e^{-2\lambda}}{R}\left(\dot{R}^{\prime}-\dot{R}\lambda^{\prime}-\dot{\psi}R^{\prime}\right),$
(7.4) $\displaystyle
G^{2}_{\;2}\,=\,G^{3}_{\;3}\,=\,\frac{e^{-2\psi}}{R}\,\left[\left(\lambda^{\prime\prime}+{\lambda^{\prime}}^{2}-\lambda^{\prime}\psi^{\prime}\right)R\,+\,R^{\prime\prime}+R^{\prime}\lambda^{\prime}-R^{\prime}\psi^{\prime}\right]$
$\displaystyle-\,\frac{e^{-2\lambda}}{R}\,\left[\left(\ddot{\psi}+\dot{\psi}^{2}-\dot{\lambda}\dot{\psi}\right)R\,+\,\ddot{R}+\dot{R}\dot{\psi}-\dot{R}\dot{\lambda}\right],$
(7.5)

where the indices $0,1,2,3$ corresponds to $t,r,\theta,\phi$, and

$F(r,t)=R(1-e^{-2\psi}R^{\prime 2}+e^{-2\lambda}\dot{R}^{2})$ (7.6)

is the Misner-Sharp mass function [152, 149].

####  7.1.2 Derivations of a Condition that Insures Robertson-Walker and
Schwarzschild Metrics being Particular Solutions

In the following, we first calculate the dust’s Misner-Sharp mass function.
Despite the fact that this mass function and its derivation are widely known
[152, 149], we shall repeat them in order to get one of our fundamental
requirements, namely, Eq.(7.17). Additionally, we shall also generalize the
mass function to various fluids.

In case of the non-accreting black hole, i.e., in case of $G^{0}_{\;1}\,=\,0$
(e.g. as in Lemaître-Tolman-Bondi metric), the energy momentum tensor
$T^{\mu}_{\nu}$ of the Einstein equations may be then considered to be of type
I in the Hawking-Ellis classification [154], i.e., of the form of a perfect
fluid

$T^{\mu}_{\nu}=\text{diag}(-\rho,P_{1},P_{2},P_{3})$ (7.7)

in comoving coordinates. $G^{0}_{\;0}$ component in Eq.(7.2) for a perfect
fluid is then written as [150, 151, 156]

$\frac{F^{{}^{\prime}}}{R^{2}R^{{}^{\prime}}}\,=\,8\pi\rho,$ (7.8)

which leads to the following mass function [151, 152]

$F=8\pi\int_{0}^{R}\rho\,\tilde{R}_{t}^{2}\,d\tilde{R}_{t}.$ (7.9)

Here $R_{t}$ corresponds to $R(r,t)$ with $t$ being fixed in the integration.
Eq.(7.9) suggests that, for a perfect fluid, $F(r,t)$ in general, describes
twice the total energy inside a sphere of radius $R(r,t)$ in the local
Minkowski space at a given $r$ and $t$. For this reason, we must define
$\frac{1}{2}F(r,t)$ as the entire energy contained within a collapsing shell
at $r$ and $t$ if no power flow occurs in the radial direction.

We assume that at first, a small black hole seed (of mass M) is embedded in a
dust cloud. This may be considered to model a small primordial black hole in
dust with initial approximate uniform distribution [157]. We demand that the
effect of the black hole disappears in the limit of vanishing black hole mass
and at infinitely great distances from the black hole. In other words, as
$r\,\rightarrow\,\infty$ or $M\,\rightarrow\,0$ we expect that the
corresponding metric should become Robertson Walker metric. Moreover, we
consider the spherically symmetric gravitational collapse which means that

$R(r,t)\,=\,f(t)\,r,$ (7.10)

where $f(t)$ is some function of time. The dust initially, almost everywhere,
may be well approximated by a homogenous isotropic energy density with
$\rho=m\frac{n_{0}}{f^{3}(t)}$ (i.e., we consider the dust as gravitationally
marginally bound [149]) where $m$ is the rest mass energy of each dust
particle, $n_{0}$ is the number density for $f(t)=1$. The mass function for
dust may be obtained from Eq.(7.9) as [158, 159, 160]

$F\,=\,m\,n_{0}\,\frac{8\pi}{3}r^{3},$ (7.11)