# Equilibrium states from gravitational collapse of minimally coupled scalar
field with non-zero potential

Dipanjan Dey<EMAIL_ADDRESS>Department of Mathematics and Statistics,
Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5  Koushiki
<EMAIL_ADDRESS>International Centre for Space and Cosmology, School
of Arts and Sciences, Ahmedabad University, Ahmedabad, GUJ 380009, India
Pankaj S. Joshi<EMAIL_ADDRESS>International Centre for Space and
Cosmology, School of Arts and Sciences, Ahmedabad University, Ahmedabad, GUJ
380009, India

###### Abstract

We study the model of spherically symmetric and spatially homogeneous
gravitational collapse of a minimally coupled scalar field. Our study focuses
on obtaining the scalar field potential that leads to a final equilibrium
state in the gravitational collapse. We demonstrate the existence of a class
of scalar field solutions that can indeed result in such an end equilibrium
state.

key words: Gravitational collapse, Scalar field collapse, equilibrium
condition.

##  I Introduction

As predicted by the well-known singularity theorems Penrose , the
gravitational collapse of a massive star, when it runs out of it’s internal
fuel, would result in a spacetime singularity where physical quantities
diverge and the spacetime breaks down. Such singularities would be hidden
within an event horizon of gravity, forming a blackhole Oppenheimer_1939 ;
Datt_1938 , or they could be cosmic singularities visible to faraway observers
in the universe PSJ1 ; PSJ2 ; PSJ3 ; Mosani ; Mosani2 ; Mosani3 ; Mosani4 ;
Mosani5 ; DeyK1 . This is a major issue of the contemporary debate in black
hole physics and cosmology today.

All the same, gravitational collapse is a fundamental process in nature.
Cosmological structures such as galaxies, galactic clusters, and various
structures in the universe form due gravitational collapse of matter clouds at
various scales. In order to help the structure formation process, the
gravitational collapse of a massive matter cloud that was initiated at various
epochs and scales in the universe must equilibrate or slow down considerably.
There are different models of gravitational collapse where the stabilization
of cosmic structures is explained. One of the frequently used models is the
top-hat collapse model Gunn . In the top-hat collapse model, the initial over-
densities in our Universe are considered to be isolated from the external
universe, and the dynamics of the over-dense regions are described by the
closed Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime which is in this
model seeded by dustlike matter. In that model, the pressureless matter first
expands with the background albeit at a slower rate, and at a certain time,
due to its own gravitational pull, the expansion halts and the over-dense
region starts collapsing. Now, since the matter is pressureless in the top-hat
model, there is no way one can show any sign of stabilization of matter using
the general relativistic Einstein equations. Therefore, in that model, the
Newtonian virialization technique is used to stabilize the collapsing matter.
Though in the top-hat collapse model, the Newtonian virialization technique is
used in an ad-hoc way, this model produces theoretical results which closely
resemble the observations. Since there exists no general relativistic
definition of the Newtonian virialization process (at least as per our
knowledge), this model is frequently used to describe cosmic structures.
However, there are some gravitational collapse models where the collapsing
matter asymptotically reaches an equilibrium state due to the effective non-
zero pressure JMN11 ; Joshi:2013dva ; Bhattacharya:2017chr ; Dey:2019fja .
Recently, in Dey:2023qdt , it is shown that the equilibrium condition can be
obtained from the covariant counterpart of the virialization equation.
However, the authors in that paper do not claim that the equilibrium condition
is the general relativistic virialization condition. More research in this
direction is necessary. It should be noted that in JMN11 ; Joshi:2013dva ;
Bhattacharya:2017chr ; Dey:2019fja ; Dey:2023qdt , the entire dynamics of the
gravitational collapse leading to the end equilibrium state are done within
the framework of general relativity. Moreover, in Dey:2023qdt , the authors
demonstrate how this gravitational collapse model can be made cosmologically
relevant. They also illustrate that this model can yield similar outcomes to
those obtained from the top-hat collapse model.

In the present paper, we consider the matter constituent of the gravitational
collapse model as a minimally coupled spatially homogeneous scalar field. We
restrict ourselves to such fields because spatial inhomogeneity here would
result in a type-II matter field, not observed physically hawking . Our
objective is to investigate the specific type of scalar field potential that
could possibly give rise to the final equilibrium state. We demonstrate that a
specific class of scalar field potentials enables the gravitational collapse
of the scalar field to attain the end equilibrium state. In the type of
collapsing dynamics described here, the non-zero negative pressure plays an
important role. Since dark matter is generally considered to be pressureless,
this type of equilibrium configuration can exist in a certain cosmological
length scale where the effect of dark energy cannot be ignored. In Maor:2005hq
; Lahav:1991wc ; Shapiro ; Horellou:2005qc ; Basilakos:2003bi ;
Basilakos:2006us ; Basilakos:2009mz , the authors extensively examine
cosmological scenarios where the presence of dark energy influences the
formation of dark matter structures. These studies reveal that considering the
non-zero effect of dark energy can lead to variations in the virialization
radius. In those studies, the authors utilized the Newtonian virialization
technique. However, in the paper Dey:2023qdt , the authors investigated
similar scenarios by considering a gravitational collapse model leading to a
final equilibrium state. In this paper, as mentioned above, we derive a class
of scalar field potentials that exhibit similar collapsing dynamics as shown
in Dey:2023qdt . It is important to note that we do not propose the scalar
field as dark matter or dark energy. Our focus is solely on obtaining the
potential of the scalar field leading to an equilibrium, where the scalar
field serves as the constituent of the collapsing matter. Therefore, the
present work is primarily motivated by mathematical considerations, rather
than cosmological ones.

The structure of this paper is organized as follows. In Section (II), we
discuss the conditions necessary for the formation of a general relativistic
equilibrium configuration, which arises as the final outcome of gravitational
collapse. In Sec. (III), we obtain a class of minimally coupled scalar field
potentials which can lead to an end equilibrium state of gravitational
collapse. In Sec. (IV), we take one example of such collapse dynamics which
lead to an end equilibrium state and show the potential of the scalar field
belongs to the class of potentials discussed in Sec. (III). We draw this
example from Dey:2023qdt . In Sec. (V), we conclude by discussing our results
and their possible implications. Throughout the paper, we use a system of
units in which the velocity of light and the universal gravitational constant
(multiplied by $8\pi$), are both set equal to unity.

##  II End equilibrium state in gravitational collapse

The metric of a general spherically symmetric collapsing system can be written
as

$\displaystyle ds^{2}=-e^{2\nu(r,t)}dt^{2}+{R^{\prime 2}\over
G(r,t)}dr^{2}+R^{2}(r,t)~{}d\Omega^{2}\,,$ (1)

where $\nu(r,t)$, $R(r,t)$ and $G(r,t)$ are functions of local coordinates $r$
and $t$. Since none of the functions depends upon angular coordinates, the
metric given above can be used to describe a spherical gravitational collapse.
Due to the presence of the undetermined functions (i.e., $\nu(r,t)$, $R(r,t)$
and $G(r,t)$) of $r$ and $t$, the above metric can describe a large class of
collapsing models of matter-fields. Here, a dot and a prime above any function
are used to specify a derivative of that function with respect to coordinate
time and radius, respectively. As we know, the singularity theorems state that
under a certain physically reasonable set of conditions such as retaining the
causal regularity, non-violation of an energy condition on matter fields, and
the existence of trapped surfaces, a spacetime singularity would be formed
necessarily due to the catastrophic collapse of a general matter distribution.
Due to the formation of trapped surfaces, the whole matter distribution
ultimately sinks into that singular point. However, there exist various
scenarios in gravitational collapse where trapped surfaces do not form due to
the violation of strong energy conditions, or due to different evolution of
geometries, and therefore the collapsing matter distribution may stop
collapsing further, and ultimately reach an equilibrium state.

In the realm of general relativity, a self-gravitating system can reach an
equilibrium state JMN11 ; Joshi:2013dva ; Bhattacharya:2017chr ; Dey:2019fja .
The conditions for the equilibrium state of a spherical over-dense region of
physical radius $R$ is

$\displaystyle\lim_{t\to\infty}\dot{R}=\lim_{t\to\infty}\ddot{R}=0\,\,,$ (2)

which implies the collapsing system can reach the end equilibrium state in the
asymptotic comoving time. There exists a scaling degree of freedom in the
metric mentioned above and using that freedom one can write JMN11 :

$\displaystyle R(r,t)=r~{}a(r,t)\,,$ (3)

which implies at the equilibrium state:

$\displaystyle\dot{a}_{e}(r)=\ddot{a}_{e}(r)=0\,,$ (4)

where we use subscript $e$ to denote the equilibrium value of any quantity as

$a_{e}(r)\equiv\lim_{t\to\infty}a(r,t)\,.$

The above equilibrium condition can be redefined in a coordinate-invariant way
using Cartan scalars. In an invariant frame, the frame derivatives $D\Psi_{2}$
and $\Delta\Psi_{2}$ are Cartan scalars, where the spin frame operators
$D\equiv\ell^{a}\nabla_{a}\equiv o^{A}o^{A^{\prime}}\nabla_{AA^{\prime}}$,
$\Delta\equiv n^{a}\nabla_{a}\equiv i^{A}i^{A^{\prime}}\nabla_{AA^{\prime}}$,
where $(o^{A},i^{A})$ and $(o^{A^{\prime}},i^{A^{\prime}})$ are the spin frame
and its conjugate spin frame bases, respectively and
$\Psi_{2}=C_{abcd}l^{a}m^{b}\bar{m}^{c}n^{d}=C_{1342}$. Here, $C_{abcd}$ is
the Weyl curvature tensor and $(l^{a},n^{a},m^{a},\bar{m}^{a})$ are the
corresponding null frame bases. The Cartan scalars $D\Psi_{2}$ and
$\Delta\Psi_{2}$ can be combined to construct a new scalar. It can be shown
that one can redefine the above equilibrium conditions by using a scalar
$(D-\Delta)^{2}\Psi_{2}$ which can be constructed from $D\Psi_{2}$ and
$\Delta\Psi_{2}$ Dey:2023qdt . The redefined equilibrium condition is:

$\displaystyle\lim_{t\to\infty}(D-\Delta)^{2}\Psi_{2}=0~{}\forall r.$ (5)

The above definition of the equilibrium condition is coordinate-independent.

Since for a given comoving radius $r$, the scale factor $a(r,t)$ is a smooth
monotonically decreasing function of comoving time $t$, the condition
$\dot{a}=\ddot{a}=0$ may be satisfied in a finite comoving time in some
collapsing scenarios, however, that not as an equilibrium state. The condition
$\dot{a}=\ddot{a}=0$ at a finite comoving time ($t_{e}$), in general, does not
imply an equilibrium state. An equilibrium state at the finite comoving time
$t_{e}$ requires:

$\displaystyle a(r,t)$ $\displaystyle>$ $\displaystyle a_{e}(r)~{}\forall
t\in[0,t_{e})\implies\dot{a}<0~{}\forall t\in[0,t_{e})\,\,,$
$\displaystyle\text{and}~{}a(r,t)$ $\displaystyle=$ $\displaystyle
a_{e}(r)~{}\forall t\geq t_{e}\implies\dot{a}=\ddot{a}=\dddot{a}=...$ (6)
$\displaystyle=$ $\displaystyle a^{(n)}=0~{}\forall t\geq t_{e}\,\,.$

Therefore, if the collapsing system reaches the equilibrium state at a finite
comoving time ($t_{e}$) then the first derivative of the scale factor $a(r,t)$
at that time becomes discontinuous (i.e., $\mathcal{C}^{0}$ function) which is
not possible if we consider $g_{\mu\nu}(t,r)$ is at least $\mathcal{C}^{2}$.
Therefore, for all possible scenarios of gravitational collapse, the
equilibrium conditions can only be satisfied at the asymptotic comoving time.

##  III Homogeneous scalar field collapse

Here, we consider a homogeneous scalar field $\phi(t)$ that seeds a closed
FLRW spacetime. An FLRW spacetime is the appropriate choice as the metric
given in Eq.(1) will become such in a homogeneous case. The line element of
FLRW spacetime is:

$ds^{2}=-dt^{2}+\frac{a^{2}(t)}{1-k~{}r^{2}}dr^{2}+r^{2}~{}a^{2}(t)~{}d\Omega^{2}.$
(7)

Here, $k=1$ since we consider closed FLRW spacetime. Using the Einstein
equations the energy density and pressure can be written as:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}+V(\phi)=3\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right)\,\,,$
(8) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}-V(\phi)=-2\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}-\frac{k}{a^{2}}\,\,,$
(9)

where $V(\phi)$ is the potential of scalar field. At the equilibrium state,
the energy density ($\rho_{eq}$) and pressure ($p_{eq}$) become,

$\displaystyle\rho_{e}$ $\displaystyle=$
$\displaystyle\frac{3k}{a_{e}^{2}}\,\,,$ (10) $\displaystyle p_{e}$
$\displaystyle=$ $\displaystyle-\frac{k}{a_{e}^{2}},$ (11)

which implies that at the equilibrium state, the equation of state
($\omega_{\phi}$) of the scalar field should be: $\omega_{\phi}=-\frac{1}{3}$.
One can also verify that at the equilibrium state:

$\displaystyle\dot{\phi}^{2}_{e}=V_{e}(\phi)=\frac{2k}{a_{e}^{2}}\,\,,$ (12)

which implies:

$\displaystyle(T_{\phi})_{e}=\frac{1}{2}\dot{\phi}^{2}_{e}=\frac{1}{2}V_{e}(\phi)\,\,,$
(13)

where $T_{\phi}$ is the kinetic energy of the scalar field. The above relation
between the potential energy and the kinetic energy of the scalar field at
equilibrium is surprisingly similar to the virialization condition in
Newtonian mechanics. It can be easily verified that the equilibrium conditions
cannot be achieved by the spatially flat FLRW spacetime, since for that we get
trivial solutions: $\dot{\phi}_{e}=0$ and $V_{e}(\phi)=0$ which is
meaningless.

###  III.1 Scalar field potential responsible for the end equilibrium state of
the gravitational collapse

Along with the two field equations of the scalar field (Eqs. (8, 9)), one can
also write down the following conservation equation or Klein-Gordon for the
scalar field:

$\displaystyle\dot{\rho}_{\phi}+3\frac{\dot{a}}{a}\left(\rho_{\phi}+p_{\phi}\right)=0\implies\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}+V_{,\phi}=0.$
(14)

However, the above equation is not independent since it can be derived from
Eqs. (8, 9). Therefore, we have two equations and three unknowns:
$\phi(a),V(\phi)$, and $\dot{a}(a)$. Therefore, we have the freedom to choose
one free function. Here, we choose the function $\dot{a}(a)$ in such a way
that the equilibrium conditions are satisfied. We consider following
functional form of $\dot{a}(a)$:

$\displaystyle\dot{a}(a)=\beta~{}\left(f(a)-f(a_{e})\right)^{\alpha}~{}~{}\forall
a\in[a_{e},a_{0}]\,,$ (15)

where the constant parameters $\beta<0$, and $a_{0}$ is the initial value of
the scale factor where $a_{0}>a_{e}$. Here, we consider another assumption
that $f(a)>f(a_{e}),~{}\forall a>a_{e}$ and $f(a)$ should be
$\mathcal{C}^{\infty}~{}~{}\forall a\in[a_{e},a_{0}]$ . We can relax the last
assumption by considering $\dot{a}(a)=\beta~{}\lvert
f(a)-f(a_{e})\rvert^{\alpha}$. Since it is a collapsing scenario, we consider
$\beta<0$. The above expression of $\dot{a}(a)$ ensures $\dot{a}\to 0$ as
$a\to a_{e}$. Using the above expression of $\dot{a}(a)$, we get the following
expression of $\ddot{a}(a)$:

$\displaystyle\ddot{a}(a)=\alpha\beta^{2}f^{\prime}(a)~{}\left(f(a)-f(a_{e})\right)^{2\alpha-1}~{}~{}\forall
a\in[a_{e},a_{0}]\,,$ (16)

which implies $\alpha>\frac{1}{2}$, since we need $\ddot{a}(a_{e})=0$. As it
was discussed previously, the condition $\dot{a}=\ddot{a}=0$ at a finite
comoving time does not imply the equilibrium state considering $a(t)$ as a
smooth monotonically decreasing function of comoving time. When a collapsing
system approaches the equilibrium state asymptotically, not only $\dot{a}$,
$\ddot{a}$ tend to zero but all the higher order derivatives of $a$ with
respect to the comoving time should show similar behavior. Therefore,
considering the smooth behavior of $a(t,r)$ for a given value of $r$, we can
define the following modified version of the general relativistic equilibrium
state:

$\displaystyle\lim_{t\to\infty}a^{(n,~{}0)}(t,r)=0,~{}~{}\forall
n\in\mathbb{Z}^{+},$ (17)

where $a^{(n,~{}0)}(t,r)$ implies $n$th order partial derivative of $a(t,r)$
with respect to comoving time $t$ and zeroth order partial derivative of the
same with respect to comoving radius $r$. Using the Cartan scalars, the above
equation can be written as:

$\displaystyle\lim_{t\to\infty}(D-\Delta)^{n}\Psi_{2}=0~{}\forall
r,~{}~{}\forall n\in\mathbb{Z}^{+}.$ (18)

Since here we express the time derivatives of the scale factor as a function
of the scale factor (i.e., $a^{(n)}(a)$), the above definition (i.e., Eq.
(17)) of the general relativistic equilibrium state is very much crucial to
differentiate the scenario where the $\dot{a}$ and $\ddot{a}$ become zero in
finite comoving time from the scenario where the collapsing system
asymptotically reaches an equilibrium state. Now, we can write down the
following equilibrium condition for the above-mentioned homogeneous collapse:

$\displaystyle a^{(n)}(a_{e})=0,~{}~{}\forall n\in\mathbb{Z}^{+},$ (19)

Using the expression of $\dot{a}(a)$ we can write:

$\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle h(a)\,\,,$
$\displaystyle\ddot{a}$ $\displaystyle=$ $\displaystyle
h(a)~{}\partial_{a}h(a)\,\,,$ $\displaystyle\dddot{a}$ $\displaystyle=$
$\displaystyle
h^{2}(a)~{}\partial^{2}_{a}h(a)+h(a)~{}(\partial_{a}h(a))^{2}\,\,,$
$\displaystyle\ddddot{a}$ $\displaystyle=$ $\displaystyle
h^{3}(a)~{}\partial^{3}_{a}h(a)+4h^{2}(a)\partial_{a}h(a)~{}\partial_{a}^{2}h(a)+h(a)~{}(\partial_{a}h(a))^{3}\,\,,$
$\displaystyle\vdots$ $\displaystyle a^{(n+1)}$ $\displaystyle=$
$\displaystyle\partial_{t}^{n}h(a)=\sum~{}\frac{n!}{k_{n1}!~{}k_{n2}!...k_{nn}!}\partial_{a}^{k_{n}}h(a)\left(\frac{\partial_{t}a}{1!}\right)^{k_{n1}}\left(\frac{\partial^{2}_{t}a}{2!}\right)^{k_{n2}}...\left(\frac{\partial^{m}_{t}a}{m!}\right)^{k_{nm}}...\left(\frac{\partial^{n}_{t}a}{n!}\right)^{k_{nn}}\,$
(20)

where $k_{n1}+k_{n2}+..+k_{nm}+..+k_{nn}=k_{n}$,
$k_{n1}+2~{}k_{n2}+..+m~{}k_{nm}+..+n~{}k_{nn}=n$, and the summation is over
all the partitions of $n$. To get the above expression of $a^{(n+1)}$, we use
Faá di Bruno’s rule for the derivatives of composite functions. One can remove
all the expressions of $\partial_{t}$ in the above expression of $a^{(n+1)}$
and can express $a^{(n+1)}$ as a function of $h(a)$ and its derivatives only,
i.e., $a^{(n+1)}\equiv
a^{(n+1)}(h(a),\partial_{a}h(a),...,\partial_{a}^{n}h(a))$. This can be done
by using the expression of $\partial_{t}^{i}a$ recursively until $i=1$ is
reached. The expression of
$a^{(n+1)}(h(a),\partial_{a}h(a),...,\partial_{a}^{n}h(a))$ is very important
since in Eq. (15), $\dot{a}$ is written as a function of $a$. Now, since
$h(a)=\beta~{}\left(f(a)-f(a_{e})\right)^{\alpha}~{}~{}\forall
a\in[a_{e},a_{0}]$, we can write:

$\displaystyle\partial_{a}h(a)$ $\displaystyle=$
$\displaystyle\beta\alpha\left(f(a)-f(a_{e})\right)^{\alpha-1}f^{\prime}(a)\,\,,$
$\displaystyle\partial_{a}^{2}h(a)$ $\displaystyle=$
$\displaystyle\beta\alpha(\alpha-1)\left(f(a)-f(a_{e})\right)^{\alpha-2}(f^{\prime}(a))^{2}+\beta\alpha\left(f(a)-f(a_{e})\right)^{\alpha-1}f^{\prime\prime}(a)\,\,,$
$\displaystyle\vdots$ $\displaystyle\partial_{a}^{p}h(a)$ $\displaystyle=$
$\displaystyle\beta\sum~{}\frac{p!}{q_{1}!~{}q_{2}!~{}...q_{p}!}~{}\frac{\alpha!}{(\alpha-q)!}\left(f(a)-f(a_{e})\right)^{\alpha-q}\left(\frac{\partial_{a}f}{1!}\right)^{q_{1}}\left(\frac{\partial^{2}_{a}f}{2!}\right)^{q_{2}}...\left(\frac{\partial^{m}_{a}f}{m!}\right)^{q_{m}}...\left(\frac{\partial^{p}_{a}f}{p!}\right)^{q_{p}}\,,$
(21)

where as previously done $q_{1}+q_{2}+..+q_{m}+..+q_{p}=q$,
$q_{1}+2~{}q_{2}+..+m~{}q_{m}+..+p~{}q_{p}=p$, and the summation is over all
the partitions of $p$. Now, in order to get the expression of $a^{(i)}$ as a
function of $f(a)$ and its higher order derivatives we need to put the above
expression of $\partial_{a}^{i}h(a)$ in Eq. (20). Now, we are ready to
investigate the constrain on $\alpha$ required for an asymptotic equilibrium
state. In order to get the lower-bound of $\alpha$ we need to find out the
terms having the lowest power of $\left(f(a)-f(a_{e})\right)$ in the
expression of $a^{(n+1)}$ written in Eq. (20). Followings are the lowest power
of $\left(f(a)-f(a_{e})\right)$ for a given value of $n$:

$\displaystyle\dot{a}~{}~{}~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}~{}~{}\left(f(a)-f(a_{e})\right)^{\alpha}\,\,,$
$\displaystyle\ddot{a}~{}~{}~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}~{}~{}\left(f(a)-f(a_{e})\right)^{2\alpha-1}\,\,,$
$\displaystyle\dddot{a}~{}~{}~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}~{}~{}\left(f(a)-f(a_{e})\right)^{3\alpha-2}\,\,,$
$\displaystyle\ddddot{a}~{}~{}~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}~{}~{}\left(f(a)-f(a_{e})\right)^{4\alpha-3}\,\,,$
$\displaystyle\vdots$ $\displaystyle a^{(n+1)}~{}~{}~{}$
$\displaystyle\rightarrow$
$\displaystyle~{}~{}~{}\left(f(a)-f(a_{e})\right)^{(n+1)\alpha-n}\,\,,$ (22)

where one can get the lowest power of $\left(f(a)-f(a_{e})\right)$ for
$a^{(n+1)}$ by using the expression of $\partial_{a}^{p}h(a)$ in the above
expression of $\partial_{t}^{n}h(a)$. Therefore, at equilibrium state (i.e.,
$a=a_{e}$)

$\displaystyle\lim_{a\to
a_{e}}a^{(n+1)}=0~{}~{}\text{iff}~{}~{}\alpha>\frac{n}{n+1}$
$\displaystyle\implies\lim_{\begin{subarray}{c}a\to a_{e}\\\
n\to\infty\end{subarray}}a^{(n+1)}=0~{}~{}\text{iff}~{}~{}\alpha\geq 1.$ (23)

Therefore, $\alpha\geq 1$ ensures the null values of all the higher order
derivatives of $a(t)$ in the limit $a\to a_{e}$, where
$\dot{a}(a)=\beta~{}\left(f(a)-f(a_{e})\right)^{\alpha}~{}~{}\forall
a\in[a_{e},a_{0}]$. If $\alpha=\frac{n}{n+1}$ (i.e., $\alpha<1$) then all the
higher order derivatives of $a(t)$ become zero except $a^{(n+1)}$, which
implies that the collapsing dynamics is not settling down a stable equilibrium
state. A positive or negative value of $a^{(n+1)}(a_{e})$ implies that
$a^{(n)}$ is going to have positive or negative values, respectively as
comoving time progresses. If $\frac{n-1}{n}<\alpha<\frac{n}{n+1}$, all the
derivatives from $\dot{a}$ to $a^{(n)}$ would become zero at $a=a_{e}$,
however, remaining all the higher order derivatives of $a(t)$ (i.e.,
$a^{(n+1)},a^{(n+2)},\cdots a^{(\infty)}$) would blow up at $a=a_{e}$ which is
obviously not the equilibrium scenario. Therefore, $\alpha\geq 1$ isolates
equilibrium scenarios from the various other dynamics where $\dot{a},\ddot{a}$
may become zero at a certain comoving time but do not imply an equilibrium
state. Therefore, from now on, we are going to consider only $\alpha\geq 1$.

Using the expression of $\dot{a}$ and $\ddot{a}$ in Eq. (15) and Eq. (16),
respectively we get:

$\displaystyle(\partial_{a}\phi)^{2}$ $\displaystyle=$
$\displaystyle-\frac{2\alpha~{}f^{\prime}(a)}{a\left(f(a)-f(a_{e})\right)}+\frac{2}{\beta^{2}~{}a^{2}\left(f(a)-f(a_{e})\right)^{2\alpha}}+\frac{2}{a^{2}}\,\,,~{}~{}\forall
a\in[a_{e},a_{0}],$ (24) $\displaystyle V(a)$ $\displaystyle=$
$\displaystyle\frac{\alpha\beta^{2}f^{\prime}(a)}{a}\left(f(a)-f(a_{e})\right)^{2\alpha-1}+\frac{2\beta^{2}}{a^{2}}\left(f(a)-f(a_{e})\right)^{2\alpha}+\frac{2}{a^{2}}\,\,,~{}~{}\forall
a\in[a_{e},a_{0}].$ (25)

The above equations show the functional dependence of $\partial_{a}\phi$ and
the scalar field potential $V$ on $a$ for the collapsing dynamics where
collapsing system asymptotically reaches the equilibrium state. Since
$\alpha\geq 1$, asymptotically $V(a_{e})=\frac{2}{a_{e}^{2}}$ and
$\partial_{a}\phi\to\pm\infty$. Close to the equilibrium state i.e., when
$a\to a_{e}$ we can write:

$\displaystyle\partial_{a}\phi$ $\displaystyle\approx$
$\displaystyle\frac{\pm\sqrt{2}}{\beta~{}a\left(f(a)-f(a_{e})\right)^{\alpha}}\,\,,$
(26) $\displaystyle V(a)$ $\displaystyle\approx$
$\displaystyle\frac{2}{a^{2}}\,.$ (27)

Using the above approximate solutions of $\partial_{a}\phi$ and $V(a)$, we can
analytically show the expression of scalar field potential as a function of
$\phi$ i.e., $V(\phi)$ which would be applicable near equilibrium state only.
From the above approximate solution of $\partial_{a}\phi$, we can write:

$\displaystyle\phi(a)\approx\pm\frac{\sqrt{2}}{\beta}\int_{a}^{a_{0}}\frac{da}{a\left(f(a)-f(a_{e})\right)^{\alpha}}.$
(28)

For analytical expression of $\phi(a)$, lets consider $f(a)=a$. Now, for this
simple form of $f(a)$ we get the following solutions of $V(\phi)$ near the
equilibrium states:

$\displaystyle\text{when}~{}~{}\alpha=1:~{}~{}V(\phi)$ $\displaystyle\approx$
$\displaystyle\frac{2}{a_{e}^{2}}\left(1-2\eta~{}e^{\frac{\beta
a_{e}}{\sqrt{2}}\phi}\right)~{}~{}\textbf{for}~{}~{}\phi>0\,\,,$
$\displaystyle\approx$
$\displaystyle\frac{2}{a_{e}^{2}}\left(1-2\eta~{}e^{-\frac{\beta
a_{e}}{\sqrt{2}}\phi}\right)~{}~{}\textbf{for}~{}~{}\phi<0\,\,,$
$\displaystyle\text{when}~{}~{}\alpha>1:~{}~{}V(\phi)$ $\displaystyle\approx$
$\displaystyle\frac{2}{a_{e}^{2}}\left(1-\frac{2~{}\zeta}{\phi^{\frac{1}{\alpha-1}}}\right)~{}~{}\textbf{for}~{}~{}\phi>0\,\,,$
$\displaystyle\approx$
$\displaystyle\frac{2}{a_{e}^{2}}\left(1-\frac{2~{}\zeta}{(-\phi)^{\frac{1}{\alpha-1}}}\right)~{}~{}\textbf{for}~{}~{}\phi<0\,\,,$

where $\eta$ is the integration constant,
$\zeta=\left[-\frac{\beta\left(\alpha-1\right)}{\sqrt{2}a_{e}^{\alpha}}\right]^{-\frac{1}{\alpha-1}}$,
and since $\beta<0,\text{and}~{}\alpha>1$, $\zeta>0$. It can be verified that
in the limit $a\to a_{e}$, as $\phi\to\pm\infty$, $V(\phi)$ approaches to
$\frac{2}{a_{e}^{2}}$ asymptotically. It should be noted that the above
approximate expressions of $V(\phi)$ are only valid close to the equilibrium
state, however, the expression of $V(a)$ given in Eq. (25) is valid throughout
the gravitational collapse, i.e., for $a\in[a_{e},a_{0}]$.

Figure 1: Figure depicts the scalar field potential near the equilibrium where
the dotted red curve corresponds to the approximate potential near the
equilibrium and the black solid line is for the exact solution of potential.

##  IV Scalar Field Potential in a Bonafide Model of Gravitational Collapse
Leading to an Equilibrium State

In this section, we derive the scalar field potential that is necessary for
achieving an equilibrium state at the end of the gravitational collapse of
homogeneous matter in a bonafide model. The model we refer to is extensively
discussed in Dey:2023qdt . Here, we employ the same dynamical model and obtain
the scalar field potential that is required, assuming that the collapsing
fluid is composed of the minimally coupled scalar field. In Dey:2023qdt , the
gravitational collapse of homogeneous matter is modeled by considering a
closed Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime described by the
following metric:

$\displaystyle
ds^{2}=-dt^{2}+\frac{a^{2}(t)}{1-r^{2}}dr^{2}+r^{2}a^{2}(t)~{}d\Omega^{2}\,\,.$

To capture a collapsing scenario leading to an equilibrium state, the collapse
model incorporates the following expression for $\dot{a}$:

$\displaystyle\dot{a}(t_{r})=-{A(t_{r}-1)\over(1+B(t_{r}-1)^{2})^{2}}~{},~{}t_{r}={t\over
t_{\rm max}}~{},$ (32)

Here, $A$ and $B$ represent real positive constants, while $t_{\rm max}$
denotes the time when the spherical homogeneous region begins to collapse
under its own gravitational force. At $t=t_{\rm max}$, the function $a(t)$
reaches its maximum value, denoted as $a_{\rm max}$. In Dey:2023qdt , the
bonafide model is employed to describe a cosmological scenario where
spherically symmetric over-dense regions of dark matter and dark energy can
achieve an equilibrium configuration at a specific cosmological length and
time scale. In this dynamics, the over-dense region initially expands with the
background, albeit at a slower rate. However, at a certain time $t=t_{\rm
max}$, the expansion comes to a halt, and the over-dense region starts
collapsing. Consequently, at $t=t_{\rm max}$, the value of $\dot{a}$ becomes
zero which can be verified from Eq. (32). Using the above expression of
$\dot{a}$, we can get the following expression of the scale factor $a(t_{r})$:

$\displaystyle a(t_{r})=a_{e}+\frac{A}{2B\left(1+B(t_{r}-1)^{2}\right)}\,,$
(33)

where $a_{e}$ is the value of the scale factor at equilibrium which has the
following expression considering zero pressure at time $t=t_{\rm max}$:

$\displaystyle a_{e}={B-A^{2}\over 2AB}\,.$ (34)

where we need $B>A^{2}$ to get a positive value of $a_{e}$. From Eqs. (32,
33), one can see that the equilibrium configuration can be achieved by the
collapsing system at asymptotic time. Using Eqs. (32, 33), we can write down
the $\dot{a}$ as a function of $a$:

$\dot{a}=-\frac{4B^{2}}{A}{\left[\frac{A}{2B^{2}}\frac{1}{(a(t)-a(t_{e}))}-\frac{1}{B}\right]}^{\frac{1}{2}}{(a(t)-a(t_{e}))}^{2}\,\,,$
(35)

which in the limit $a\to a_{e}$ becomes:

$\displaystyle\dot{a}(a)\approx\beta\left(a~{}-~{}a_{e}\right)^{\frac{3}{2}}\,\,,$
(36)

where $\beta=-\frac{2\sqrt{2}}{\sqrt{A}}B$. As discussed in the previous
section, in this case, $\alpha=\frac{3}{2}$ which is greater than one, and
therefore, this collapsing system has an end equilibrium state. It can be
verified for this present scenario,
$\lim_{t\to\infty}a^{(n,~{}0)}(t,r)=0,~{}~{}\forall n\in\mathbb{Z}^{+}$ which
implies the existence of end equilibrium state. Using Eqs. (26, 27), and the
expression of $V(\phi)$ near the equilibrium in Eq. (III.1), we get:

$\displaystyle\phi(a)$ $\displaystyle\approx$
$\displaystyle\pm\frac{2\sqrt{2}}{\beta
a_{e}}\frac{1}{\left(a~{}-~{}a_{e}\right)^{\frac{1}{2}}}\,\,,$ (37)
$\displaystyle V(\phi)$ $\displaystyle\approx$
$\displaystyle\frac{2}{a_{e}^{2}}\left(1-\frac{2\zeta}{\phi^{2}}\right)\,\,,$
(38)

where $\zeta=\frac{8B}{A^{2}}\frac{1}{\left(\frac{B}{A^{2}}-1\right)^{3}}$.
Since $B>A^{2}$, $\zeta>0$. In Fig. (1), we can see that the approximate
solution of the potential of the scalar field (depicted by dotted red curve)
almost coincides the exact solution (depicted by black solid line) near the
equilibrium configuration. Here, we consider $A=16.57,B=369.3$ which are
relevant to the cosmological scenario discussed in Dey:2023qdt . Therefore,
the system starts collapsing at $a_{\rm max}=0.0302$ and asymptotically
reaches the equilibrium state where $a_{e}=0.00774$.

##  V Conclusion

In this paper, we study the gravitational collapse of a minimally coupled
scalar field with non-zero potential, and we obtain a class of scalar field
potentials that lead to an equilibrium end state of gravitational collapse. In
order to do that we first express the equilibrium condition in the following
way

$\displaystyle a^{(n)}(a_{e})=0,~{}~{}\forall n\in\mathbb{Z}^{+},$ (39)

which is just a different form of the condition stated in Eq. (2). While
obtaining the potential of the scalar field, we express every dynamical
variable as a function of the scale factor. Now at equilibrium, the scale
factor becomes $a_{e}$. A scale factor can reach a specific value at a given
comoving time. However, it becomes challenging to ascertain whether that
particular value of the scale factor corresponds to the equilibrium state, as
all dynamical variables are expressed in terms of the scale factor rather than
the comoving time. To address this, one can verify the higher-order
derivatives of the scale factor. The null values of all higher-order
derivatives along with the first and second at a certain scale factor value
$a_{e}$ ensure the existence of an end equilibrium state of the collapsing
system. The null values of the first and second-order derivatives of the scale
factor at an asymptotic comoving time do imply the existence of an equilibrium
state. However, when one expresses all the dynamical variables in terms of
scale factor, null values of the first two derivatives at a certain value of
scale factor $a_{e}$ do not imply an equilibrium state, since there is no
information of comoving time (e.g. $a=a_{e}+(t_{e}-t)^{n},~{}\forall n>2$, the
first two derivatives become zero at $a=a_{e}$ but the collapsing system has
no equilibrium state at $a=a_{e}$). We have to show null values of all the
higher-order derivatives at a certain value of $a$. We show the lower limit of
a parameter $\alpha$ for which all the higher order derivatives of the scale
factor become zero at $a=a_{e}$. One can use the comoving time to express an
equilibrium dynamic. However, with that procedure, one cannot cover the large
class of collapsing dynamics that can reach the equilibrium state. In
Dey:2023qdt , the authors show one example of such a dynamics, where the scale
factor has a particular functional dependence on comoving time. By expressing
all variables as functions of the scale factor $a$, we eliminate the need to
determine the dependence of $a$ on the comoving time, except for one crucial
constraint: all higher-order derivatives of $a$ must become zero at $a=a_{e}$.
In this way, we cover a large class of collapsing dynamics leading to an end
equilibrium state. Additionally, if one expresses everything in terms of
comoving time then it may be very difficult to get an analytical form of
$V(\phi)$ and then we would have to show the potential by a numerical plot
which is not of our interest in this paper.

In Sec. (IV), we derive the scalar field potential required to achieve an
equilibrium state at the conclusion of the gravitational collapse of
homogeneous matter in a bonafide model discussed in Dey:2023qdt . We show that
the scalar field potential corresponding to the example belongs to the class
of potentials discussed in Sec. (III). As stated before, in this paper, we
refrain from presenting the scalar field as a candidate for dark matter or
dark energy. Instead, our primary objective lies in deriving the potential of
the scalar field, where it plays a crucial role as the constituent of the
collapsing matter. At a certain cosmological length and time scale, if the
gravitational collapse of a certain homogeneous matter distribution consisted
of a scalar field approach towards the general relativistic equilibrium state,
then after a large comoving time, the potential of the scalar field may have a
similar functional form given in Eqs. (III.1, III.1). Now, the next step would
be to find out some cosmological implications of the aforementioned scalar
field potential. However, we have deferred this aspect to future work,
acknowledging the need for further exploration and analysis in this direction.

##  VI Acknowledgement

DD would like to acknowledge the support of the Atlantic Association for
Research in the Mathematical Sciences (AARMS) for funding the work.

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