# The Big Bang††thanks:  For reasons made clear later, the phrase ”Big Bang”
is used here (as in its original meaning) to mark the very beginning of the
Universe. The later decelerating, matter-creating priod (here) retains its
original name ”reheat” instead of Big Bang.
Origins and initial conditions from Self-Regulating Cosmology (SRC) model

Manasse R. Mbonye
Department of Physics, College of Science and Technology, University of
Rwanda, Kigali, Rwanda Contact<EMAIL_ADDRESS>
###### Abstract

Generating appropriate initial conditions for the Universe is key to
discussing cosmic evolution constructively. In standard cosmology the
traditional approach assumes an early Universe that emerges from an infinite
density, spacetime singularity. It then undergoes inflationary expansion,
followed by a matter-creating ”reheat” period. This approach produces results
generally in agreement with observations. However, to date it is not known how
(or even whether) a true past-directed spacetime singularity can generate a
regular spacetime that becomes the observed Universe. It has been suggested
that an appropriate approach should involve initial conditions that emerge
naturally from existing physics. In this paper we generate initial conditions
predicated on the Self-Regulating Cosmology (SRC) model recently presented
[1]. In SRC, the dynamics leads to a universe that also self-regenerates from
one evolutionary cosmic phase or (hereafter) kalpa111Wiki: Kalpa (Sanskrit
lit): a long period of time related to the lifetime of the universe
(creation). to another. Within each such kalpa cosmic dynamics evolves between
two different scales of de Sitter-like horizons. The end of a kalpa and the
beginning of the next interface through a phase transition whose features
naturally set all initial conditions for the new phase, including sourcing
entropy. The SRC Universe satisfies Poincare Recurrence Theorem, with
specified recurrence time. This facilitates a consistent co-application of
Boltzmann Anthropic Hypothesis with the Past Hypothesis (hence with the Second
Law of Thermodynamics). The issue of whether or not the early Universe
undergoes inflation becomes naturally self-manifest in this scenario. The
framework discusses two standing issues in cosmology: the old Cosmological
Constant Problem, and the new issue of over-mature structures observed by JWST
at high redshifts.

##  1 Introduction

Modern cosmology traces the origins of the Universe back to a singular event
of infinite density and pressure, named the Big Bang by Hoyle and considered
to be a space-like singularity [2]. Spacetime singularities, including this
initial one in cosmology, arise as solutions of General Relativity to problems
involving strong gravitational fields. In the sixties research in this area
led to statements now known as the Hawking-Penrose singularity theorems which
described features of singularities in black holes (future-directed) [2] and
in cosmology (past-directed) [3]. While theories of gravity predict spacetime
singularities, including the initial singularity [4], it is also the case that
no observational evidence for physical singularities exits yet, and there have
been inquiries on whether, or not, gravitational collapse can avoid them
[5].In line with such inquiries non-singular spacetimes have been proposed in
which at high densities regular matter turns into fluids of appropriate
equations of state in black holes (see for example [6] [7] [8] [9]) and in
cosmology (see for example [10]). Other efforts to smooth out the initial
singularity using quantum approaches have included the famous Hartle-Hawking
no-boundary approach [11] and the tunneling approach by Vilenkin.[12]. There
have also been discussions [13] questioning whether such quantum approaches do
indeed provide a satisfactory smooth beginning for cosmic spacetime. A
comprehensive review of quantum applied methods to smooth out singularities
can be found in [14].

In the 1980s the inflationary paradigm was introduced in cosmology to address
contemporary issues that included the horizon problem, the flatness problem
and the monopole problem [15]. It was even hoped then that a similar approach
could also possibly smooth out the initial singularity [16]. While the
inflationary paradigm was able to address several of these problems it does
not eliminate the singularity problem; though it merely pushes it back to a
time before inflation. The current most accepted model of modern cosmology,
$\Lambda CDM$, which has been quite successful interpreting observations [17],
incorporates the early era of inflation. As a result $\Lambda CDM$ as a model
still faces, among some of its challenges [18], the problem of singularity-
based cosmic origin with no known causal mechanism to generate inflation.
Agreeably, understanding the initial conditions for the Universe is certainly
key to discussing cosmic evolution accurately and predictively, and could
potentially aid in addressing better some of the other issues such as, the
initial entropy problem (IEP), and the origin of primordial fluctuations, to
name a few. The challenge here is how to rid spacetime of the pre-inflation
initial singularity and thereby infer what led to inflation if, indeed, there
was one.

In this paper we present different Big Bang initial conditions in the context
of a recently presented model of Self-Regulating Cosmology (SRC) [1]. In SRC,
the self-regulating cosmic dynamics leads the Universe into evolutionary
phases of self-regeneration, here called kalpas. As we shall find, the end of
a given kalpa and the beginning of the next are linked by an interface in form
of a phase transition, between two (de Sitter-like vacuum) states. We describe
evolutionary features from the preceding cosmic phase leading to this
transition through a simple language that makes use of known classical
processes as analogues, for intuition. The transition process sets initial
conditions for this, latter, new phase. The issue of whether or not there is
an inflationary era associated with the Big Bang finds a natural explanation
in this scenario. We close with a highlight on the framework’s potential to
discuss several standing issues in cosmology, including the cosmic initial
entropy problem IEP $\left[19\right]$, the cosmological constant problem
$\left[20\right]$ and the issue of mature structures observed by JWST at high
redshifts $\left[21\right]$. The rest of this paper is set up as follows. In
Section 2 we present a brief overview of the SRC model. In Section 3 we make a
case for the Big Bang based on SRC, and use it to present model motivated
cosmic initial conditions in section 4. In section 5 using the model framework
we highlight on some unreconciled cosmological observations. Section 6 offers
closing remarks.

##  2  SRC Model: Overview

The model of Self-Regulating Cosmology or SRC arises out of consideration that
cosmic dynamics is sourced from two related ingredients, namely: spacetime
(here referred to as basic space) whose structure provides the background
cosmic dynamics, and free matter fields which contribute perturbations to this
background dynamics222As shown [1] this means that even in the idealistic
situation of absent free matter-energy, the Universe would still expand.. The
two ingredients which are in persistent tension continuously interact to
produce a universe whose dynamics always seeks, as an attractor, the time-
linear or coasting path that basic space induces on to the Universe. The
process proceeds through basic space engaging in creation or annihilation of
free matter energy density with appropriate equations of state, in order to
offset the source of the perturbations. This is the basis for the self-
regulation character of the Universe, in the model. SRC model also leads to a
self-regenerating Universe with a cosmic phase interval or kalpa estimated at
$\sim 20Gyrs$. The model differs significantly from several cyclic [22],
ekryptic [23, 24], or periodic [25] models that have previously been
suggested. A full derivation of this framework can be found in [1]. In the
current work, we discuss how the SRC model sets conditions for (and predicts)
the Big Bang naturally. This, in turn, leads to the observed evolution that
follows thereafter. The underpinnings of the SRC model are summarized in a
statement referred to as the Dynamic Equilibrium Protection Proposal (DEPP)
made up of 3 propositions:

###### Proposition 1

An ideal universe, constituted purely by basic-space, remains in a state of
constant (coasting) expansion unless perturbed by free matter-energy
fields333The phrase ”free matter-energy fields” includes all but the fabric
(of Eqs. 2.1 to 2.2) making basic-space structure (of Eq. 2.3)..

###### Proposition 2

The Universe creates stability against perturbations that tend to shift it
from its coasting, dynamic equilibrium state. (Such can include density
perturbations growing from quantum fluctuations)

###### Proposition 3

When perturbed, the Universe will suppress the perturbations through creation
of free matter-energy fields with a net gravitational charge opposite that of
the perturbations.

DEPP implies that the classical features of elasticity of basic space causing
its expansion, in response to free matter-energy fields, are emergent features
that betray an underlying reality of its structure. It is argued [1] that (in
the absence of perturbing free matter-energy fields) basic space classically
behaves as a perfect fluid whose structure has a density $\sigma_{s},$ and
pressure $\hat{\pi}_{s}$, representable as a diagonal stress-energy tensor,
$T_{\nu}^{\mu}=diag(\sigma_{s},-\hat{\pi}_{s},-\hat{\pi}_{s},-\hat{\pi}_{s})$
[1],[26]. In such absence of free matter-energy fields it is shown the system
resides in dynamic equilibrium so that $\ddot{a}_{s}=0$. and satisfies.
Further, in satisfaction of the gravitational neutrality as mentioned before
we set $k=0$, which is also consistent with persisting observations $[27$]
that the physical Universe appears virtually spatially flat. Then, the
Einstein equations for the density and pressure evolutions read, respectively,

$\frac{\dot{a}_{s}^{2}}{a_{s}^{2}}=\frac{8\pi G}{3}\sigma_{s},$ (2.1)

and

$\frac{\dot{a}_{s}^{2}}{a_{s}^{2}}=-8\pi G\hat{\pi}_{s},$ (2.2)

leading to an equation of state $\hat{\pi}_{s}=-\frac{1}{3}\sigma_{s}$. As a
result, consistent with General Relativity, with this non-vanishing energy-
density the spacetime induces a metric

$ds^{2}=c^{2}dt^{2}-a_{s}\left(t\right)^{2}\left(dr^{2}+r^{2}d\Omega^{2}\right),$
(2.3)

where the scale factor is linear in time $a_{s}\left(t\right)\propto t$. Thus
the background basic space will induce a coasting or linearly expanding
universe (see also [27]), consistent with Proposition 1 in DEPP. Further, the
model posits that, to lowest scale, the dynamics of the Universe that takes
consideration of free matter fields as a perturbation will then take the form
$a\left(t\right)\sim a_{s}\left(t\right)\ +\delta a\left(t\right),$where
$\delta a\left(t\right)$ is the (yet to be determined) perturbation
contribution to $a\left(t\right)$ due to the free matter fields.

To determine the contribution $\delta a\left(t\right)$ due to free matter
fields we start by considering a flux of a fluid of density $n$, satisfying
$n_{;\mu}u^{\mu}+n\Theta=n\Gamma$, with a generating source of rate $\Gamma$
and expansion $\Theta$. For an adiabatic case one finds
$\sigma_{s}+3H(\sigma_{s}+\pi_{s}+P_{c})=0$, where
$P_{c}=-\frac{1}{3H}\left(\sigma_{s}+\pi_{s}\right)\Gamma$ is identified as
the creation pressure [28]. Using thermodynamic arguments [28][29] and Eqs.
2.2 and 2.3 along with DEPP [1] one finds that $\delta a\left(t\right)$ takes
on a harmonic form $\delta a\left(t\right)=a_{m}\sin\left(\omega
t+\psi\right)$, with $a_{m}$ as the oscillation amplitude and $\psi$ as an
arbitrary phase angle.

As a result, the overall scale factor $a(t)$ for cosmic dynamics is found to
be given by

$a(t)=\upsilon^{\ast}\left[t+\tilde{\tau}\sin\left(\frac{t}{\tilde{\tau}}+\psi\right)\right],$
(2.4)

where $\upsilon^{\ast}$ is a normalizing factor with dimensions of inverse
time, $\tilde{\tau}=\frac{1}{\omega}$ is a (reduced) period (a new
cosmological parameter to be determined) and $\psi=\frac{2\pi
t_{f}}{\tilde{\tau}}$, the phase angle with $t_{f}$ (fundamental interval)
later fixed by boundary conditions as Planck time.$t_{f}=t_{p}$. One finds
$a\left(t\right)$ to be a solution to
$\tilde{\tau}^{2}\ddot{a}+a=\upsilon^{\ast}t$. From Eq. 2.4, the corresponding
Hubble parameter takes the form

$H(t\mathbf{)=}\frac{1+\cos\left(\frac{t}{\tilde{\tau}}+\psi\right)}{t+\tilde{\tau}\sin\left(\frac{t}{\tilde{\tau}}+\psi\right)}\mathbf{.}$
(2.5)

Eqs. 2.4, 2.5 provide information on cosmic evolution in the SRC model.

The SRC model has been shown to posses several advantages over several other
models, including $\Lambda CDM$. These advantages include the ability to
construct the correct values of the observed parameters, based on very
restricted data class. Thus armed only with accurate observational information
on when the current cosmic acceleration commenced, the model determines the
current values of the standard cosmological parameters accurately. For cosmic
acceleration estimated to begin at $t_{ac}=4.07Gyr$ ago, based on observations
[30], the model finds the for the Universe, the corresponding age to be
$t_{0}=13.76Gyr$, the current Hubble parameter to be $H_{0}=68.22km/s/mpc$ and
the corresponding matter and dark energy parameters to be $\Omega_{m}=30.1\%$
and $\Omega_{de}=69.9\%$, respectively. These results are consistent with the
most recent observations, [31], [32]. The model also reproduces the general
evolutionary profile, consistent with the early matter dominating phase and
the latter dark energy dominating phase. It, further, makes predictions about
the future evolution of the Universes, including a graceful exit from the
current acceleration and its future fate. In the model the Universe expands
monotonically, while oscillating about a coasting path. Figure 1 sketches the
time evolution of the scale factor $a\left(t\right)$ depicted in Eq. 2.4.

Figure 1: Figure 1 shows the general features of the time evolution of the
scale factor $a(t)$, as depicted by the SRC model. The net expansion (red)
which is a combination of two independent influences (basic-space and free
matter-energy) is shown as an oscillatory curve about the linear (equilibrium)
expansion rate of basic-space. Here we have used $\tau_{\frac{1}{2}}=9.69Gyr$
as the working cosmic half period. Note: $a_{0}\ =1$ at $t=13.76Gyr$.

One of the distinguishing features of the Universe as predicted by SRC (see
Figure 1) is self regeneration or self re-initiation at the end of each phase
$\tau$. In our discussion we shall consider this period $\tau$ to commence at
a Big Bang (such as that predicted to have occurred $13.76Gyrs$ ago). The
period $\tau$ then extends through the decelerated structure formation phase,
into the current accelerated phase and to the latter’s predicted end (as
according to SRC), at which point another Big Bang ensues. Whether this latter
(or for that matter any) Big Bang initiates a new universe or whether it
simply marks a new period of a continued universe, is essentially academic,
here, and likely just semantic generally. Therefore, in the rest of this work
we shall, for consistency, take the position that evolution of the Universe
continues on, punctuated by periods of equal size $\tau$, each commencing with
a Big Bang. As noted before, for purposes of identification we shall refer to
each such period as a kalpa. In the next section we focus attention on the
description of the period around the Big Bang, commencing a given cosmic
period or kalpa, described in terms of its initial conditions from end of a
previous cosmic period. We consider the beginning of the current period as a
particular example of a Big Bang in SRC, which we discuss. In using SRC for
interpretation of other known evolutionary characteristics, the discussion
will later include features like entropy evolution, issues like the
cosmological constant problem and mature structures currently observed in
early Universe.

##  3 Making a case for the Big Bang

In the SRC model the Universe is monotonically expanding while also self-
regulating, meaning it is always in search of a dynamic equilibrium state.
Further, the Universe is self-regenerating, with a specific self-regeneration
interval $\tau$ [1]. In this section we apply these features of SRC to the
observed Universe to study its initial conditions. As we find, such conditions
will turn out to describe features and events at a point of self-regeneration.
To this end, we shall treat the current cosmic phase (see Figure 1), whose
initial conditions we seek, as the $k^{th}$ kalpa. Here, the counting index
$k$ can take on any positive integral values $k=\left\\{1,2,....\right\\}$ and
its current value is arbitrary, and not any special for our present
considerations. We shall denote the time evolution in this particular kalpa by
$t_{k}$, with $0\leq t_{k}\leq\tau$. That cosmic phase is, in turn, bounded by
2 others (not shown in Figure 1), including $\left(k-1\right)^{th}$ the
immediate past kalpa whose time evolution is denoted by $0\leq
t_{k-1}\leq\tau$ and, including $\left(k+1\right)^{th}$ the immediate future
one,with a time evolution denoted by $0\leq t_{k+1}\leq\tau$. The aim is to
link the end of one cosmic kalpa with the commencement of the next one. For
our discussion, the consideration here focuses on the events identified with
the period interfacing the end of the $\left(k-1\right)^{th}$ and commencement
of the $k^{th}$ kalpas. This period will give rise to the very initial
conditions for the current phase, here referred to as the $k^{th}$ kalpa.

###  3.1 Evolutionary features at the phase interface

Recall cosmic dynamics, here, is represented (see Eq. 2.4, Figure 1) by the
evolution of a 2-term scale factor $a\left(t\right)=a_{s}\left(t\right)+\delta
a\left(t\right)$, with one background contribution $a_{s}\left(t\right)\ $from
basic space and the other $\delta a\left(t\right)$ arising as a perturbation
from free matter fields. As shown in Section 2, in the idealized situation in
which matter fields are absent the Universe still expands along a coasting
path $a_{s}\left(t\right)$. We first focus on the effect of matter fields
sector on cosmic dynamics. The periodic nature of $\delta a\left(t\right)$
(see Eq. 2.4) implies that $\delta a\left(t_{k-1}=\tau\right)=\delta
a\left(t_{k}=0\right)=0$. Thus (from the matter sector) the end of the
$\left(k-1\right)^{th}$ cosmic kalpa and the beginning of the next $k^{th}$
coincide. This would then suggest that, in general, taking into consideration
the entire dynamics. including the contribution due to basic space and that
due to matter fields, events at $t_{k-1}=\tau\ $and those at $t_{k}=0$ should
coincide, so that both $t_{k-1}=\tau\ $in the $\left(k-1\right)^{th}$ kalpa
and $t_{k}=0$ in the consequent $k^{th}$ one represent the same temporal
event. The question is, do they, in general? In the rest of this Section we
investigate the soundness of this assertion, and explore its consequences, if
valid.

During the $\left(k-1\right)^{th}$ kalpa one has, from Eq. 2.5, that at
$t_{k-1}=\tau$ and the Hubble parameter here takes the value:
$H\left(\tau\right)=\frac{1+\cos\psi}{\tau+\tilde{\tau}\sin\psi}$. Recalling
$\psi=\frac{2\pi t_{f}}{\tilde{\tau}}$, with $t_{f}\lll\tau$ (later we can
justifiably take $t_{f}\sim t_{pl}$ and also often use observational estimate
in [1] that $\tau=19.38Gyr$) we can then legitimately apply $\cos\psi\simeq 1$
and $\sin\psi\simeq\psi$ so that at $t_{k-1}=\tau$ we effectively have

$H\left(\tau\right)=\frac{2}{\tau+t_{f}}\bumpeq\frac{2}{\tau}.$ (3.1)

On the other hand, from Eq. 2.5 during the $k^{th}$ kalpa, at $t_{k}=0$ one
finds $H(t_{k}=0\mathbf{)=}\frac{1+\cos\psi}{\tilde{\tau}\sin\psi}$, so that
effectively

$H\left(0\right)=\frac{2}{t_{f}}$ (3.2)

Clearly, results of the Hubble parameters in Eqs. 3.1 and 3.2 do not match,
particularly given that $\left|\tau\right|\gg\left|t_{f}\right|$. In
particular, the two results represent disparate magnitudes of energy density,
$\sim H^{2}$. However, we note that metrics
$g_{\mu\nu}^{\left(k-1\right)^{th}}$ and $g_{\mu\nu}^{k^{th}}$ at these points
can be conformaly similar. Thus setting
$g_{\mu\nu}\left(\eta\right)=\exp-\left[2\left(H\left(t_{k}=0\right)\right)t\right]g_{\mu\nu}^{k^{th}}\left(t_{k}=0\right)=\exp-\left[2\left(H\left(t_{k-1}=\tau\right)\right)t\right]g_{\mu\nu}^{\left(k-1\right)^{th}}\left(t_{k-1}=\tau\right)$,
where $\eta$ is the conformal time, we have that

$g_{\mu\nu}^{k^{th}}\mid_{t_{k}=0}=\varphi^{2}g_{\mu\nu}^{\left(k-1\right)^{th}}\mid_{t_{k-1}=\tau}$
(3.3)

where now $\varphi=\exp\left(\frac{1}{t_{f}}-\frac{1}{\tau}\right)t$ is a
conformal factor.

Given the similarities implied in Eq. 3.3 and the differences implied between
Eqs..3.1 and 3.2, what useful inference can one draw? To proceed we shift
slightly backwards in time and note that at the instant $t_{k-1}=\tau-t_{f}$,
just before (but in the neighborhood of) $t_{k-1}=\tau$, the dark–energy-
induced acceleration vanishes, $\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\left(\rho+3p\right)=0\ $(use Eq. 2.4 to verify). Here then the fields
satisfy an equation of state $w=-\frac{1}{3}$. Thus this point $t_{k-1}=\tau-
t_{f}$ marks the end of a long streak$\ \left(\frac{\tau}{2}Gyrs\right)\ $of
cosmic acceleration and the Universe is here in a state of dynamic
equilibrium, as demanded by DEPP. One consequence of this is that, barring
remaining scanty clumpy structures with significant spatial intervals in
between, the Universe has by now become quite dilute indeed, leaving mostly
massless fields moving on null geodesics. As has previously been pointed out
by Penerose [33], based on the (Einstein-Planck) energy relations
$mc^{2}=E=\hbar\nu$, time only flows for massive bodies which makes them
clocks $m\sim\frac{1}{t}$. Massless particles moving on null geodesics, on the
other hand, are time-flow-insensitive. The latter reside on a conformably flat
space and one could therefore infer the metrics $g_{\mu\nu}\mid_{t_{k}=0}$and
$g_{\mu\nu}\mid_{t_{k-1}=\tau}$ in Eq. 3.3 to reasonably represent the same
spacetime $g_{\mu\nu}\left(\eta\right)$.

On the other hand, unlike the case in Conformal Cyclic Cosmology [22] in which
a given cosmic expansion phase goes on infinitely in proper time, in SRC
cosmic acceleration (above) is only proper-time finite [1] and a complete
spatial dilution (and complete black hole evaporation) into massless fields
are not features of the model. In short, as previously pointed out, there will
still be massive structures, albeit at significant intervals. One may then
want to ask what a physical observer might infer from such a scenario. Our
working interpretation, from here on, is a temporal region dominated by
spacetime voids and with scanty structure. As we find going forward (next
section), this picture will turn out to have consequences, some of which can
bear influence on the next cosmic phase. Thus, in this model, complete
disappearance everywhere of fields moving on time-like geodesics is not a
necessary condition for a new cosmic phase to commence. Instead, as we find,
the existence of space voids will play an important role.

We now take a closer look at features of space in these voids, as we seek to
answer the original question. In cosmology space stretches as originally
described by Lamaitre and observed by Hubble. Moreover, in SRC basic space is
shown to expand monotonically, even in the absence of free matter fields,
leading to the conclusion that space has a structure [1]. Putting these
concepts together allows one to make use of an analogy in which basic space in
SRC manifests dynamical characteristics akin to those of a classical material
medium, specifically an elastic medium. This behavior is actually already
apparent in SRC. Explicitly, the demonstrated harmonic cosmic expansion
response, $\ddot{a}\propto a\left(t\right)$, of basic space to stresses due to
free matter fields in [1], (see also Eq. 2.4), is reminiscent of an elastic
medium in a classical sense. Taken in this context then, basic space here
behaves as an elastic medium which, under stress, obeys a simple Hooke’s law-
type relation

$\ddot{a}_{m}+\kappa a_{m}=0,$ (3.4)

with an elastic constant
$\kappa=\left(\frac{2\pi}{\tau}\right)^{2}=\frac{1}{3}\left|\Gamma\right|H_{m}$.
The notion that spacetime behaves as an elastic medium is not new. Besides our
simple result in Eq. 3.4, there have been several works alluding to space as
an elastic medium (see e.g. example [34][35] [36]). Thus, for example, by
considering continuum space as an elastic medium Padmanabhan [34] has derived
the Einstein equations as a consistency condition, and without the usual need
to vary the metric. Here, Padmanabhan makes an important observation that the
metric $g_{\mu\nu}$ and the energy stress tensor $T_{\mu\nu}$ are not
fundamental variables.

One notes, once more in this treatment, that by the time under consideration
$t_{k-1}=\tau$, the Universe has already been stretched under the dark-energy-
driven monotonous acceleration, for as long as $\frac{\tau}{2}\approx
9.68Gyrs$. The dark-energy induced acceleration ends at $\tau-t_{f}$, just
before the end of the kalpa, and after this the Universe is driven by the
energy associated with the phase constant $\psi$. Therefore, taking the space
as an elastic medium analogy further, our interpretation is that here space
attains (the analogue of) its elastic limit and reaches a yielding point, to
undergo a phase transition thereafter during which the spacetime strain no
longer follows a simple linear Hook’s law-type relation. A general classical
consideration of an elastic medium undergoing a phase transition into a
plastic state (in this case) was considered by [37] as a spontaneous breaking
of symmetry. Analogously, at a classical level, we shall consider the above
changes (Eq. 3.2 to Eq. 3.3) in the Hubble parameter $H$ at the cosmic kalpa
interface $t_{k-1}/t_{k}$ to signal some (1st order) phase transition. The
phase transition is initiated through localized spacetime defects driven by
the energy $\sim\frac{1}{t_{f}^{2}}$, in the phase constant
$\psi=2\pi\frac{t_{f}}{\tau}$ the only operative feature at the
$t_{k-1}/t_{k}$ interface (or at $t_{k-1}=\tau$, $t_{k}=0$). Thus, at the
phase change boundary, the Hubble parameter is a step function, whose time
rate of change here can be described by

$\dot{H}_{\left(t=0\right)}\sim
2\left[\frac{1}{t_{f}^{2}}-\frac{1}{\tau^{2}}\right]\delta\left(t_{k-1}-t_{k}\right)$
(3.5)

and an associated energy density change, $\Delta\varepsilon=\frac{3}{8\pi
G}\Delta H^{2}\sim\frac{3}{2\pi
G}\left[\left(\frac{1}{t_{f}}\right)^{2}-\left(\frac{1}{\tau}\right)^{2}\mathit{.}\right]$.

We can now provide an answer to the original question. The original space of
$H\left(\tau\right)=\frac{2}{\tau}\ $underwent a phase transition at
$t_{k-1}=\tau$ creating within it defect(s) in form of fast expanding
$H\left(0\right)=\frac{2}{t_{f}}$ patch(es) or bubble(s) Evidently, here and
only here, within this patch created at the phase transition, events at
$t_{k-1}=\tau\ $and those at $t_{k}=0$ coincide, so that both $t_{k-1}=\tau\
$in the $\left(k-1\right)^{th}$ kalpa and $t_{k}=0$ in the consequent $k^{th}$
kalpa now represent the same temporal event. And from the point of view of an
observer in the spacetime defect patch’s future a new spacetime is now
created, whose horizon is made initially small by the associated extreme
expansion (Hubble) rate, $H_{\left(t_{f}\right)}\sim\frac{1}{t_{f}}$, of the
patch or bubble. From the point of view of space as an elastic medium that
reached its elastic limit considered above, a consequence of the phase
transition is the expansion law for the new patch is no longer Hooke’s
(linear) law-type. Instead, as we show below (see Eq. 3.4) it is de Sitter-
like inflationary. To this end we discuss, in what follows below, the
implications of this highlighted phase transition process and the resulting
features of the new patch/bubble, in context of the Big Bang and initial
conditions for the new cosmic kalpa (the currently observable Universe).

###  3.2 Horizons

The preceding discussion describes the phase transition process in the SRC
model which takes place at (or rather constitutes) the interface of two cosmic
kalpas, an outgoing kalpa and a commencing one. This phase transition occurs
at the end of a long streak of dark-energy-driven cosmic acceleration (in the
model [1] estimated as $\frac{\tau}{2}=9.68Gyrs$). During the phase
transition, the process packs an incredibly large energy density change from
the original state of

$\varepsilon_{\min}\sim\frac{3}{2\pi G\tau^{2}}$ (3.6)

to nucleate a patch or bubble in an energy density state of

$\varepsilon_{\max}\sim\frac{3}{2\pi
Gt_{pl}^{2}}=\frac{24\pi}{G}\frac{1}{\left(\tau\psi\right)^{2}}$ (3.7)

This latter energy state of the formed patch/bubble is associated with the
phase constant $\psi$, which dominates the patch/bubble dynamics at this
instant (see Eqs.2.4 and 2.5). Note that in SRC, $\psi$, which initially
appears as an integration constant used to fix boundary conditions in the
model, does play an important role here analogous to that played by the
cosmological constant in $\Lambda CDM$ (and more as we see below).

Associated with these states are corresponding horizons that define the
observable Universe at the time. Thus, just before the phase transition the
only horizon is (what is now the outer horizon) $R_{\left(\tau\right)}$
corresponding to $H_{\left(\tau\right)}=H(t_{k-1}=\tau)$ and to the vacuum
energy density $\varepsilon_{\min}$ and given by

$R_{\left(\tau\right)}=\frac{c}{H_{\left(\tau\right)}}\sim\frac{1}{2}c\tau,$
(3.8)

where $c$ is the vacuum speed of light. In SRC $R_{\left(\tau\right)}$ is a de
Sitter-like horizon that bounds the observable Universe, corresponding to end
of a cosmic phase or kalpa at the latter’s maximum age estimated [1] at
$\tau=19.38Gyrs$. On the other hand, the extreme space expansion associated
with the phase transition implies a nucleated patch, with an internal bounding
or (inner) horizon $R_{\left(t_{f}\right)}$ corresponding to
$H_{\left(t_{f}\right)}=H(t_{k}\approx t_{f})$ and to the vacuum energy
density $\varepsilon_{\max}$ (see also Eq.3.6) and given by

$R_{\left(t_{f}\right)}=\frac{c}{H_{\left(t_{f}\right)}}\sim\frac{1}{2}ct_{f},$
(3.9)

For the (would-be) observers inside the bubble/patch at the end of the phase
transition, this smaller de Sitter-like horizon $R_{\left(t_{f}\right)}$ will
now at its initial age $t_{k}=0$, form the boundary for a new observable
Universe.

To sum up, the preceding analysis suggests existence of two de Sitter-like
horizons: one from an ending cosmic phase and one from a commencing phase both
joined by a phase transition described by Eq. 3.5. Further, noting that by
implication, the occurrence of such phase transition processes can be
generalized in the SRC model to occur at other kalpa interfaces, one concludes
that the smaller horizon marking the beginning of the $k^{th}$ kalpa in Eq.
3.9 is no different from the one that marked the beginning of the
$\left(k-1\right)^{th}$ before it. Hence the $\left(k-1\right)^{th}$ kalpa,
and any other kalpa in the model (for that matter) is bounded by two de
Sitter-like horizons, namely $R_{\left(t_{f}\right)}$ associated with early-
universe dark-energy, $\varepsilon_{ede}=\varepsilon_{\max}\sim\frac{3}{2\pi
G}\left(\frac{2}{t_{f}}\right)^{2}$that initiates the cosmic kalpa, and
$R_{\left(\tau\right)}$ associated with late-universe dark energy
$\varepsilon_{lde}=\varepsilon_{\min}\sim\frac{3}{2\pi
G}\left(\frac{2}{\tau}\right)^{2}$at the end of a cosmic kalpa. These results
will now be used to discuss initial conditions for the Universe.

##  4 Cosmic initial conditions from SRC

Based on the preceding discussion, one can now sum up the conditions for the
Big Bang implied by the SRC model. In this section we shall present these
conditions in two groups, namely dynamical initial conditions and entropy
initial conditions.

###  4.1 Initial conditions I: Dynamical conditions

One way to present these conditions is to ask whether and how (if at all) they
predict the currently accepted early line of events of the Universe as
interpreted through observations such as CMB. Below we only highlight on each
of these events qualitatively, and leave a detailed quantitative description
for elsewhere.

####  4.1.1 Big Bang (Phase Transition)

Just before the transition the observable Universe has been dominated by dark
energy and has been accelerating for half a kalpa, $\frac{\tau}{2}\sim
9.69Gyrs$. However, both because most black holes have not radiated away and
also because the finite time acceleration has not yet generated infinite
spacetime dilution, there still exists massive structure available, albeit
scanty. The space constituting the pre-transition Hubble sphere is not at the
time just pure space, with massless fields, devoid of structure. Thus in
applying the previous metaphor of space here as elastic medium, the resulting
phase transition may be limited to the void or massless regions. It is in one
or some of such regions that the resulting extreme spacetime stretch may lead
to a local ”elastic limit” to nucleate a defect in form of a de Sitter patch
or bubble. Such a phase transition description at the 2-kalpa interface
constitutes the Big Bang in SRC. Initially, this transition begins with a
Universe at the end of its dark-energy driven acceleration and with a Hubble
horizon size $R_{\left(\tau\right)}\sim\frac{c}{H_{\left(\tau\right)}}=c\tau$.
During the phase transition the dynamics is dominated by the phase constant
$\psi=\frac{2\pi t_{f}}{\tilde{\tau}}$ at the defect nucleation site that ends
up as an inflationary patch or bubble,
$H(t_{k}=0)=\frac{1+\cos\psi}{\tilde{\tau}\sin\psi}\rightarrow
H\left(0\right)=\frac{2}{t_{f}}$, bounded by a de Sitter-like horizon
$R_{\left(t_{f}\right)}=\frac{c}{H_{\left(t_{f}\right)}}=\frac{1}{2}ct_{pl},$
embedded in the larger old horizon $R_{\left(\tau\right)}$.

Building on experience of thermodynamic phase transitions, it is reasonable to
expect that the above phase transition process may initially involve one
patch, or even several patches/bubbles nucleated at the sane time. It is also
possible that the transition could instantaneously involve the whole Hubble
sphere $R_{\left(\tau\right)}$ (possibly through less usual communication
processes such as spacetime entanglement). Indeed, one notes that from the
perspective of an observer that ends up inside one of these horizons, it does
not matter how many bubbles are initiated or how large the space involved by
the phase transition is initially since their observation will only be
restricted to a Universe that initiates with a size of the horizon
$R_{\left(t_{f}\right)}\sim ct_{f}$. In general, however, the nucleation
process of a small single patch or probably few such patches is likely much
more probable (see discussions below on constraints from: (i) initial entropy
and (ii) from current observations). This tiny patch or bubble of size
$R_{\left(t_{f}\right)}\sim t_{pl},$ bounded by a de Sitter horizon now
constitutes the new ($k^{th}$ phase) Universe at time $t=t_{k}=0$.

####  4.1.2 Inflation

The patch that becomes the new Universe evolves with a scale factor that can
be written down as $a\left(t\right)=a_{0}\exp\left(\int Hdt\right)$. At $t=0$,
$H\left(0\right)=\frac{4\pi}{\psi\tau}=\frac{2}{t_{f}}=cons\tan t$ and$\
a\left(t\right)=\exp\left[H_{\left(t_{f}\right)}t\right]$ to give

$a\left(t\right)=\exp\left(\frac{4\pi}{\psi\tau}\right)t=\exp\left(\frac{2}{t_{f}}t\right).$
(3.10)

Thus, SRC predicts the end of the Big Bang phase transition to give rise to a
new Universe that is inflationary, right from the start, and with a de Sitter-
like horizon (Eq. 3.9) of size
$R_{\left(t_{f}\right)}=\frac{c}{H_{\left(t_{f}\right)}}\sim\frac{1}{2}ct_{pl}$.
Therefore early Universe inflation occurs naturally as a consequence of the
several evolutionary trends already described that precede it. In particular,
the inflationary process in SRC is predicted by the model, as a direct and
immediate consequence of the described Big Bang phase transition and does not
originate from a spacetime singularity. Moreover this means inflation’s
observed effects of stretching a previously causally related spacetime and
flattening the curvature out of it, while amplifying any primordial
perturbations, can all be explained as predictable events having their roots
in the big bang phase transition and the evolutionary events that preceded it.
Below we point out how, as naturally as inflation results from the Big Bang
phase transition in SRC it also naturally finds a graceful exit shortly after,
which is predictively demanded by the SRC model.

####  4.1.3 Reheat

Recall that in SRC model [1] the dynamical evolution of the Universe is
governed by a scale factor given in Eq. 2.4. In the very early Universe, in
the neighborhood of $t\sim 0$, the dynamics is dominated by energy associated
(Eq. 3.7) with the phase angle constant $\psi$, and as a consequence the early
Universe is naturally inflationary, as previously demonstrated. Now,
dynamically inflation is a non-equilibrium process which tends to shift cosmic
dynamics away from its preferred equilibrium path [1] of $R=ct$. Thus,
according to DEPP (see Section 2), during inflation the Universe must
naturally seek to offset this non-equilibrium perturbation through creation of
appropriate matter-energy fields that increase its inertia to slow it down. To
this end, the Universe will produce matter-energy fields with a positive or
attractive gravitational signature. This is manifest from recalling that in
cosmology, the source of inertia is $\frac{\ddot{a}}{a}+\frac{4\pi
G}{3}\left(\rho+3p\right)=0$, and in SRC this scales as
$\frac{\ddot{a}}{a}\propto-\sin\left(\frac{t}{\tilde{\tau}}+\psi\right)$,
which leads to $\ddot{a}<0$ for $\psi<\frac{t}{\tilde{\tau}}<\pi$. Clearly the
result is a matter-creating deceleration that slows down the Universe with
dissipation and increased temperature. It follows that in SRC, inflation ends
and dissipative matter production proceeds immediately when
$\frac{t}{\tilde{\tau}}\gtrsim\psi$ or $t\gtrsim t_{f}$, since at this point
the phase angle constant $\psi=\frac{2\pi t_{f}}{\tilde{\tau}}$ no longer
dominates the dynamics (as in in Eq. 3.10). It is in this way that the
Universe exits inflation naturally, shortly after it enters it, producing
radiation in the process. As noted, the matter production process is also
dissipative, which adds friction to the gravitational inertia to produce heat,
and raise cosmic temperature.

Now, in traditional cosmology, the end of inflation and beginning of matter
production happen to occur together. However traditionally, the search was
always one of (fitting the observations by) finding a graceful exist from
inflation. Thus, besides being just a by-product to this sought-for exit
process, matter production in general has no traditionally established purpose
in evolution of cosmic dynamics. In SRC, on the other hand, the need to
regulate cosmic dynamics demanded implies a bid to exit inflation which drives
the Universe into production of positively gravitating matter-energy (through
DEPP [1]) to increase cosmic inertia and dissipation through dynamic friction.
In this way, cosmic inflation which in SRC emerges naturally also finds a
natural graceful exit, through matter creation according to DEPP. Moreover,
inflation and its graceful exit are achieved here with no need to invoke a
singularity-sourced scalar field with a fine-tuned potential and other
characteristics.

One can sum all this up by observing that the events of the previous cosmic
phase, which led to a phase transition, ended up producing a new cosmic phase
Universe, with a natural initial inflationary period that stretches the
Universe out of any curvature (generated during the phase transition) before
naturally regulating its dynamics through a SRC-predicted dissipative matter-
creating process, referred to here as the reheat process. Thereafter, the
Universe in SRC evolves according to observations [31].

###  4.2 SRC initial conditions II: Initial Entropy

We now touch on the issue of initial cosmic entropy as a boundary condition,
in the context of SRC model. We start with a brief overview of the current
state of the topic and thereafter inquire on any potentially useful
contributions suggested by the SRC model.

####  4.2.1 Background: The Initial Entropy Problem (IEP)

The cosmic microwave background CMB radiation produced from the early Universe
is known to be virtually thermal, and hence with a maximal entropy. But the
Universe has evolved since and continues to, suggesting increasing entropy.
This apparent contradiction is part of the Initial Entropy Problem (IEP)
[38][39][40][41], which part is now addressed by adding gravitational entropy
to the thermodynamic (radiation) entropy, as first suggested by Penrose [42].
But the other part of the IEP calls for a need to explain how it is that the
initial entropy was low in the first place, given that there are many more
ways to choose a Universe under equilibrium conditions of high entropy [43].
The idea, which has a significant following in cosmology (see for example
[45][46]), is that equilibrium conditions (including highest entropy) do form
boundary conditions for an evolving system and so the Universe should have
initially evolved from such conditions. Thus, initial cosmic conditions here
would involve a dynamical mechanism by which a low entropy fluctuation evolves
from a background high entropy state. The interpretation [41] then is the
initial conditions should not be assumed a priori, but should rather be
generically set up by existing physics. Such thinking is consistent with the
Boltzmann Entropy Hypothesis[47][48], in which Boltzman first proposed that
the low entropy of the Universe should be a random fluctuation from a maximal
entropy state. The fluctuation theorem [44] implies that small fluctuations
from highest equilibrium entropy are exponentially more likely in this
scenario than large fluctuations.

On the other hand, proponents of inflation for cosmic initial conditions point
out that a low entropy initial condition (false vacuum from scalar field that
does not clump) is favored over a high entropy one by virtue of the Second Law
of Thermodynamics and the Past Hypothesis [41] [49]. Indeed both the absence
of such a background high entropy pool and that of a dynamical mechanism by
which a low entropy fluctuation could evolve from such a pool, are at the base
of the (other remaining part of) the initial entropy problem (IEP).

More generally, at issue appears to be a subtle but deeper and potentially
intricate argument of boundary conditions. Supporters of the Boltzman
Hypothesis, who argue that low initial entropy should be explained from a low
entropy fluctuation from a high entropy equilibrium background, point to the
logic for upholding the concept of time-symmetry, implied in physical
laws[46]. Proponents of inflation for initial conditions, on the other hand
[38] [41] [49] point out that the Past Hypothesis and the Second Law of
Thermodynamics which are both well grounded-concepts in physics, suggest time-
asymmetric cosmic evolution. What makes this even much more challenging is the
fact that as physicists we are all trained to subscribe to both the logic of
time-symmetry as implied in physical laws, and to the logic of time asymmetry
as implied by the arrow of time in the Second Law of Thermodynamics and in the
Past Hypothesis. Framed this way, this dichotomy in time symmetry makes the
IEP probably one of the most outstanding unresolved issues in physics,
probably side-by-side with the Cosmological Constant Problem.

####  4.2.2 Addressing the IEP from SRC

As has been discussed, according to SRC model, in one kalpa the Universe
evolves between two de Sitter-like boundaries, from the inner initial horizon
size $R_{\left(t_{f}\right)}$ to outer final horizon size
$R_{\left(\tau\right)}$. We have found that this final state, by then
dominated by extended massless voids, due to excessive expansion, later gives
rise to a new phase by nucleation of de Sitter-like bubble of size
$R_{\left(t_{f}\right)}=ct_{f}\ $generated by the then dominating phase
constant $\psi=2\pi\frac{t_{f}}{\tau}$ with an associated energy density
$\varepsilon_{\max}$. We now highlight on the likelihood of the process from
statistical estimates.

Following Gibbons and Hawking [50] one can write the entropy enclosed by the
horizon $R_{\left(\tau\right)}$ (Eq. 3.8) at the end of the cosmic phase as

$S_{\left(\tau\right)}=k_{B}\pi\frac{R_{\left(\tau\right)}^{2}}{l_{pl}^{2}}$
(4.1)

We have noted that in this final state, while very dilute, the Universe still
contains some structures in the form of unradiated black holes, galaxies and
others. One can still estimate the combined entropy. In fact Gibbons and
Hawking [50] gave a prescription on calculation of combined entropy, here
$S_{c}$ of such configuration which contains extra matter. This gives

$S_{c}=\left[S_{\left(\tau\right)}-\sqrt{S_{b}S_{\left(\tau\right)}}+S_{b}\right]<S_{\left(\tau\right)},$
(4.2)

where $S_{b}$ is the entropy of the black hole with equivalent ADM mass to the
localized matter. This means that introduction of localized structure or even
radiation inside such horizon depresses the horizon entropy. It follows that
de Sitter horizon entropy is maximal [45]. We will take Eq. 4.2 to represent
the net entropy at the end of the cosmic phase. For purposes of the discussion
here, though, we shall with no loss of generality, still apply the maximal
entropy $S_{\left(\tau\right)}$ of Eq. 4.1 as the background equilibrium
entropy.

One can, further, write the horizon entropy of the nucleated post-transition
de Sitter patch or bubble (Eq. 3.9) as

$S_{\left(t_{f}\right)}=k_{B}\pi\frac{R_{\left(t_{f}\right)}^{2}}{l_{pl}^{2}}.$
(4.3)

That the entropy $S_{\left(t_{f}\right)}$ of the patch is induced on the
background equilibrium entropy $S_{\left(\tau\right)}$ during the transition,
implies the resulting combined entropy $S_{\left(rc\right)}$ becomes,

$S_{\left(rc\right)}=\left[S_{\left(\tau\right)}-\sqrt{S_{\left(t_{f}\right)}S_{\left(\tau\right)}}\right],$
(4.4)

where (compared with Eq. 4.2) we neglect the last term in
$S_{\left(t_{f}\right)}$ (recall
$\frac{R_{\left(t_{f}\right)}}{R_{\left(\tau\right)}}\lll 1$). Thus the patch
entropy $S_{\left(t_{f}\right)}$ in Eq. 4.3 emerges as a fluctuation of this
background entropy, $S_{\left(\tau\right)}$ in Eq. 4.1. Following [45] one can
set the probability for such a fluctuation $P_{\left(t_{f}\right)}$ as

$P_{\left(t_{f}\right)}=\exp\left(-\sqrt{S_{\left(t_{f}\right)}S_{\left(\tau\right)}}\right),$
(4.5)

where $S_{\left(\tau\right)}$ and $S_{\left(t_{f}\right)}$ are given by Eqs.
4.1 and 4.3, respectively.

Given that $S_{\left(t_{f}\right)}<<$ $S_{\left(\tau\right)}$, the information
embodied in Eq. 4.5 is consistent with the demands of the fluctuation theorem
[44] that small entropy fluctuation(s) about the background entropy are
exponentially more likely than large ones. Moreover, the result in Eq. 4.3
supports the idea advanced by Albrecht [45] that the small initial cosmic
entropy (here $S_{\left(t_{f}\right)}$) should, nevertheless, be de Sitter
maximal. In general, these results are consistent with the Boltzmann Anthropic
Hypothesis [47][48], that the low entropy of the Universe should be a random
fluctuation from a maximal entropy state. In general, these results are
consistent with the Boltzmann Anthropic Hypothesis [47][48], that the low
entropy of the Universe should be a random fluctuation from a maximal entropy
state. Lastly, the observation that the patch or bubble that initiates the
$k^{th}$ phase (and hence its entropy $S_{\left(t_{f}\right)}$ in Eq. 4.3) is
a consequence of the outgoing $\left(k-1\right)^{th}$ phase and a fluctuation
of the latter’s background entropy $S_{\left(\tau\right)}$, is consistent with
the Penrose view [43] [41] that such initial conditions should not be fixed, a
priori, but instead should generically emerge from existing physics.

In summary our results then suggest that SRC does provide a dynamical
mechanism by which, as part of initial cosmic conditions, a low entropy
fluctuation $S_{\left(t_{f}\right)}$ could have evolved from a background
state with high entropy $S_{\left(\tau\right)}$. One observes that upholding
these results indicating that the low initial cosmic entropy always emerges
from the large entropy of the background, also points to the logic for
upholding a related phenomenon, namely: the concept of Poincare-recurrency of
evolution of the associated dynamical system, here the Universe. This is
because the Boltzmann’s Anthropic Hypothesis is an example of the Poincare
Recurrence Theorem [51] which states that, under certain conditions, a system
will eventually return arbitrarily close to its initial state in phase space.
This feature of the Universe is already justifiably apparent as we point out
below.

On the other hand, observe that locally within a given kalpa in SRC, the
entropy evolution beginning as $S_{\left(t_{f}\right)}$ from the start of the
new cosmic phase and ending as $S_{\left(\tau\right)}$ at the phase end, is
expectedly a monotonically increasing quantity and hence consistent with both
the Past Hypothesis and the Second Law. Consequently, in this model, it is
also necessary to uphold both the Second Law of Thermodynamics and the Past
Hypothesis.along with the implied future-directed time-asymmetry.

The issue now is how (or even whether) these two seemingly diametrically
opposed interpretations can be reconciled. As it turns out, in SRC, the two
concepts can be reconciled, as we now briefly point out. One notes that the
above results imply that in SRC the Universe manifests two kinds of dynamical
processes which obey two different kinds of time-applications. First, there
are kalpa-to-kalpa or extended time processes, which appear self-repeating and
consistent with Poincare Recurrence Theorem. For these processes the Poincare
Recurrence Time is the SRC cosmic phase/kalpa duration $\tau$. Therefore, that
the Universe in the SRC model is itself an example of a Poincare recurrence
system is here already manifest. Secondly, within a given kalpa interval,
there are internal $\left(0<t_{k}<\tau\right)\ $cosmic phase processes which
uphold the Second Law and the Past Hypothesis. Such dual character of global
and local cosmic processes is already at the defining center of the SRC model.
Recall [1] that in SRC cosmic evolution is driven by a two-term scale factor
$a\left(t\right)=a_{s}\left(t\right)+\delta a\left(t\right)$. The first term
which depicts the dynamics of basic space, evolves the Universe on a global,
equilibrium, coasting expansion path $R=ct$. It is the expansion path the
Universe would follow in the idealized situation where it is massless. The
second term is the contribution from free matter-energy sector which
influences the details of cosmic evolution internally in a given cosmic phase.
It moves the Universe on an expansion trajectory that oscillates about the
above-mentioned equilibrium path in response to a contemporary domination of
matter or dark energy, respectively (see Figure 1). The interplay between the
two contributions, which is summarized in DEPP, constitutes the rationale for
regulated cosmic dynamics in SRC. In this way, SRC explains and reconciles the
Past Hypothesis, the Second Law of Thermodynamics and the implied future-
directed time asymmetry, on the one hand, with Boltzmann Anthropic Hypothesis,
the recurrence of the Universe seen in the kalpa-to-kalpa dynamics with a
fixed Poinare Recurrence Time $\tau$, on the other hand.

The outgoing discussion suggests SRC can contribute towards a resolution of
the Initial Entropy Problem. It provides appropriate initial entropy
conditions, requiring that the initial low entropy of the Universe should be a
random fluctuation from a maximal entropy state. As a dynamical system the
Universe is seen to obey the Poincare Recurrence Theorem [51], from kalpa to
kalpa, with a well-defined Poincare Recurrence Time $\tau$, as described by
its dynamical equations and by DEPP [1]. On the other hand, locally within a
given phase $\left(0<t_{k}<\tau\right)$ cosmic evolution upholds the Past
Hypothesis and the Second Law of Thermodynamics. Thus, locally, the concept of
a future-directed time-arrow is in force. These findings suggest that cosmic
dynamics is an interplay of two time scales a universal kalpa-to-kalpa
recurrent one, and a local and future-directed asymmetric one. the findings
also imply SRC provides a guiding cosmological scenario in which initial
conditions [43] such as the Past Hypothesis for example [52] are predicted by
the dynamics, and not merely assumed. A detailed discussion on this topic will
appear elsewhere.

###  4.3 Similarities and contrasts: Does SRC improve on $\Lambda$CDM initial
conditions?

In the following brief discussion we outline general similarities and
contrasts in the cosmic initial conditions from the SRC model and those from
the $\Lambda$CDM model.

####  4.3.1 Origin and Big Bang

(i) In $\Lambda$CDM the Universe originates from a past-directed spacelike
singularity, often referred to as the initial singularity, about $13.8Gyrs$
ago. The origin of the initial singularity is usually not addressed by the
model, neither is the process by which spacetime does evolve from such a
physical singularity. It is believed that during the trans-planckian interval
that follows, gravity breaks away from the other forces and spacetime
(gravitational) fluctuations develop soon after, which later evolve into
density perturbations that, in turn, seed structure formation.

(ii) In SRC the Universe evolves from one cosmic phase or kalpa to another
through a (first order) phase transition that involves changes in both energy,
volume and entropy (EVE changes), as discussed. The initial seed of the phase
transition is a spacetime defect induced when the Universe is dominated by
energy provided by the phase constant $\psi$ in the model (Eqs. 2.4 and 2.5
and 3.7) which operates like the cosmological constant in $\Lambda$CDM. During
the phase transition the Universe evolves from one de Sitter state with a
time/energy scale $\tau\simeq 19.38Gyrs$ to a de Sitter state with a planckian
time/energy scale $t_{f}\gtrsim t_{pl}$. This phase transition should also
generate spacetime fluctuations which will later grow into density
perturbations to seed structure formation.

####  4.3.2 Inflation

(i) In $\Lambda$CDM an inflationary mechanism must be invoked to address the
flatness, horizon and monopole problems and to amplify the fluctuations into
density perturbations. This is achieved through introduction, by hand, of a
scalar field with fine-tuned characteristics that ensure the desired goal,
including a potential that initially flattens to ensure exponential spacetime
expansion. No prior physics predicts inflation, the inflaton, its
characteristics or how it evolves from the initial spacetime singularity.

(ii) In SRC, after the phase transition, the nucleated patch forms the new
cosmic phase Universe which settles into a post-planckian de Sitter-like, and
hence inflationary state. This de Sitter state in SRC is possible because of
the characteristics of the phase $\psi=\frac{2\pi t_{f}}{\tau}$, where as
before $\tau\simeq 19.38Gyrs$ and $t_{f}\approx t_{pl}$. This phase constant
manifests characteristics reminiscent of the cosmological constant $\Lambda$,
in $\Lambda CDM$. In particular one can relate the two such that
$\Lambda\approx\left(\frac{2\pi}{\tau\psi}\right)^{2}$. Therefore inflation is
naturally generated in SRC. The need to address the flatness, the horizon and
the monopole problems does not exist here since these issues are naturally
addressed by the contemporary state of the evolving Universe, and so is the
amplification of the spacetime perturbations through the consequent inflation.

####  4.3.3 Inflation exit and Reheat Inflation must end and the Universe must
both create matter and reheat

(i) In $\Lambda$CDM this is achieved through fine-tuning the post-inflation
scalar field to role down a steep potential and thereafter oscillate at the
bottom. No physical principle here appears make the end of inflation
inevitable.

(ii) Inflation is non-equilibrium dynamics and in SRC the need to regulate
cosmic dynamics towards dynamic equilibrium, demanded by DEPP implies a
deceleration to exit inflation naturally and gracefully. The deceleration sets
in early [1] when $t>t_{f}$ so that $\psi$ no longer dominates the dynamics
and the Universe creates matter-energy dissipatively to increase its inertia
and heat. In this way, cosmic inflation which in SRC emerges naturally also
finds a natural graceful exit, through matter creation according to DEPP. Once
again, inflation and its graceful exit are both achieved here with no need to
invoke a singularity-sourced scalar field with a fine-tuned potential and
other characteristics.

####  4.3.4 Entropy

(i) In $\Lambda$CDM, invoking an inflaton for the very initial condition
implies that a low entropy initial condition is provided by the false vacuum
from a scalar field that does not clump [41]. According to $\Lambda$CDM, by
virtue of the Second Law of Thermodynamics and the Past Hypothesis, a low
entropy initial condition is favored over a high entropy one implied by the
Boltzman Hypothesis. Note that the inflaton and its properties do not emerge
from any existing physics, as would be recommended [42][52] but are instead
fixed, a priori.

(ii) In SRC the low entropy early Universe emerges as a fluctuation of the
large background entropy of the outgoing cosmic phase. The process is a
consequence of the SRC model which naturally reconciles the Boltzman’s
Anthropic Hypothesis with both the Past Hypothesis and the Second Law of
Thermodynamics. The prescription identifies internal evolution of a cosmic
phase as subject to local future-directed time evolution while, on the other
hand, recurrence exists between consecutive phases.

##  5 SRC on some unreconciled observations

As an application of SRC we present brief views on a couple of some currently
unreconciled observations in cosmology.

###  5.1 The Cosmological Constant Problem

The Cosmological Constant Problem [20][53] arises from a mismatch between the
observed vacuum energy density [54] and that theoretically calculated [55].
Calculations assume a Planck energy cut-off give $M_{pl}^{4}\sim
10^{76}\left(GeV\right)^{4}$ as the value of the vacuum energy density while
on the other hand $\rho_{eff}\sim 10^{-47}\left(GeV\right)^{4}$ is usually
quoted as the observed background vacuum energy density [54]. The result is a
catastrophic mismatch of $\sim 123$ orders of magnitude between theory and
observation, known as the Cosmological Constant Problem and considered the
largest mismatch ever in the history of physics. There have been several
attempts to address the Cosmological Constant Problem ever since Zeldovich
[56] defined it. For a recent review of the problem and contributed solutions
(see e.g. [57]). The SRC model suggests a natural approach to addressing the
Cosmological Constant Problem. Below we outline the argument (a detailed
version will appear elsewhere).

In the preceding discussion of the Big Bang, it is shown that the Universe
evolves between 2 de Sitter-like horizons. Thus, first there is the inner or
initial bubble horizon of size $R_{\left(t_{f}\right)}=\frac{2c}{H\max}\sim
2ct_{f}$. The parameter $t_{f}$ is related to the phase constant
$\psi=2\pi\frac{t_{f}}{\tau}$ in SRC. Recall, $\psi$ initially arises as an
integration constant which is fixed by boundary conditions as an appropriate
boundary [1] that renders the scale-factor $a\left(t\right)$ non-singular,
based on an augment that relates $t_{f}$ directly to zero-point energy density
at $t=0$. This horizon is associated with an energy density (see Eq.3.7)
$\varepsilon_{\max}\sim\frac{3}{2\pi G}\left(\frac{2}{t_{f}}\right)^{2}$. In
the model we have justifiably taken $t_{f}\sim t_{pl}$. Secondly, there is an
outer horizon of size
$R_{\left(\tau\right)}=\frac{2c}{H_{\left(\tau\right)}}\sim 2c\tau$, of a
fully evolved Universe appearing at the end of a cosmic phase, where $\tau$ is
the age of the Universe at the end of a cosmic phase. Recall the parameter
$\tau$ arises, in this model [1] as a period, in the derivation of the scale
factor $a\left(t\right)$ and is related to the frequency
$\omega=\frac{2\pi}{\tau}$ of the harmonic regulation of cosmic dynamics. The
parameter $\tau$ is also related to $\psi$ through
$\psi=2\pi\frac{t_{f}}{\tau}$. Its operating value of $\tau=19.38Gyrs$
generically emerges from existing physics, being calculated based solely on
the (observations) look back time $t_{ac}\ $when the current cosmic
acceleration began [30] taken in [1] as $t_{ac}\simeq 4.07Gyrs$. This horizon
is associated with an energy density (see Eq.3.8) of
$\varepsilon_{\min}\sim\frac{3}{2\pi G}\left(\frac{2}{\tau}\right)^{2}$

From this our estimate for the ratio between the vacuum energy densities at
the beginning of a cosmic phase and that at its end becomes

$\frac{\varepsilon_{\max}}{\varepsilon_{\min}}=\left(\frac{\psi}{2\pi}\right)^{2}=1.287\times
10^{122}$ (5.1)

Eq. 4.6 also emphasizes the important role played by the dimensionless phase
constant $\psi$ here, as in the model in general. The result indicates the
vacuum energy density at the Big Bang is some 122 orders of magnitude as large
as that at the end of the cosmic phase, $19.38Gyrs$ later. This is in
agreement with the result in [54], though actually a larger value than that of
121 orders of magnitude usually quoted in the literature for the cosmological
constant problem, and we now explain why. Our interpretation of the
implication of this extreme disparity is that, in between during the evolution
of a cosmic phase, the vacuum energy density of the Universe is time evolving
.$\varepsilon_{v}=\varepsilon_{v}\left(t,\psi\right)$. To determine the
functional form of $\varepsilon_{v}=\varepsilon_{v}\left(t,\psi\right)$ we
will impose two constraints. First, the evolution of the vacuum energy density
should be independent of the free matter-energy fields within the Universe.
Secondly, one should recover the energy density at the beginning, and at the
end, of the cosmic phase in the limit cosmic evolution approaches these times.

The Hubble function of Eq. 2.5 suggests that the most inclusive form of an
evolving vacuum-energy function $\varepsilon_{v}\sim H_{v}^{2}$ that is
independent of free matter fields should take the form

$\varepsilon_{v}=\frac{3}{8\pi
G}\left[\frac{1+\cos\psi}{t+\frac{\tau}{2\pi}\sin\psi}\right]^{2}\approx\frac{3}{2\pi
G}\left[\frac{1}{t+t_{f}}\right]^{2}.$ (5.2)

A simple check shows that Eq. 4.7 also reproduces the desired limits. Thus
$\underset{t\rightarrow t_{f}}{\lim}\varepsilon_{v}\left(t,\psi\right)=$
$\varepsilon_{\max}\sim\left(\frac{1}{t_{pl}}\right)^{2}$recovers Eq. 3.7 as
the upper energy density limit at the Big Bang $t=0$, while
$\underset{t\rightarrow\tau}{\lim}\varepsilon_{v}\left(t,\psi\right)=$
$\varepsilon_{\max}\sim\left(\frac{1}{\tau}\right)^{2}$recovers Eq. 3.6 as the
lower energy density limit at the Big Bang $t=\tau$. At the current cosmic age
of $t_{0}=13.76Gyrs$ the approach gives the current vacuum energy ratio as
$\frac{\varepsilon_{\max}}{\varepsilon_{v_{0}}}\sim 6.49\times 10^{121}$. This
result is consistent with estimates based on observations [54][55].

Therefore according to SRC, the vacuum energy density should evolve as
$\varepsilon_{v}=\varepsilon_{v}\left(t,\psi\right)$, between two boundary
$\varepsilon_{\max}$ and $\varepsilon_{\min}$ values (see Eqs 3.6-3.7) fixed
by the horizons (in Eqs. 3.8-3.9) and whose ratio is given by Eq. 4.6. This
suggested solution to the Cosmological Constant Problem is based on the key
feature of SRC that cosmic dynamics $R\left(t\right)$ evolves in phases or
kalpas, each bounded by two de Sitter-like horizons,
$R_{\left(t_{f}\right)}\leq R\left(t\right)\leq R_{\left(\tau\right)}$.

###  5.2 On mature structures observed (by JWST) at high redshits

In the preceding discussion we have shown that the Big Bang marks an interface
between one outgoing cosmic kalpa and an incoming one. The new phase initiates
as an inflating de Sitter patch-bubble, with a Hubble horizon that coincides
with the de Sitter horizon. We also established that the nucleated patch is
small, and why that is. In the background and hidden by the tiny horizon of
the patch, is the rest of the Universe that constituted the preceding phase.
This background too is under relatively fast expansion, having just ended a
period of dark energy-driven acceleration which, according to SRC estimates,
would have lasted for the past period of $\frac{\tau}{2}\sim 9.69Gyrs$.
Therefore, while it still contains some structure, this background space is a
lot more dilute, with structure a whole lot more sparse (than currently for
example). It is against such a background, in the first place, that the phase
transition nucleates new inflating patches (incidentally, only one such
patch/bubble, bounded by its horizon, is relevant for our purposes here). We
have shown that the formation of such de Sitter-like patch at $t=0$ (the
transition) results from the fact that cosmic evolution at the time (see Eqs.
2.4-2.5) is dominated by a large constant energy density
$\sim\frac{1}{t_{f}^{2}}$ associated with the very short time-scale phase
constant $\psi=2\pi\frac{t_{f}}{\tau}$. In our analogy this domination
stretches space beyond its elastic limit, hence the phase transition at the
nucleated de Sitter-like defects. Shortly after this, for $t>t_{f}$ (see Eqs.
2.4-2.5) the phase constant no longer dominates the dynamics and inflation
naturally ends. This process of cosmic self-regulation, demanded by DEPP in
SRC leads to deceleration of the fast expanding space, in search of its
natural dynamic equilibrium, as can be verified from Eq. 2.1. To achieve this
deceleration the system must becomes inertial and dissipative with
matter/radiation creation. This is the reheat process. And thereafter,
according to SRC [1] the Universe will continue to decelerate for the next
period of $\frac{\tau}{2}\sim 9.69Gyrs$ before dark energy dominates again to
commence the next acceleration [1].

One consequence of the inertia-induced deceleration phase that commences
immediately after inflation, is the disappearance of the de Sitter horizon
that causally separated the initial patch from the background, leaving a
regular Hubble horizon. The other is that the deceleration increases the size
of the Hubble horizon. Thus as the new phase Universe expands with a slowing
speed and hence with a growing Hubble horizon into its background, old
structures such as galaxies and black holes from the past phase (and outside
the horizon) can then cross the growing Hubble horizon, into the new cosmic
phase (Universe). The first of the structures to enter the horizon would be
those that previously were within the neighborhood of the patch/bubble site
that formed the new phase now containing the observer. Future observers could
distinguish such ”alien” structures (coming from the past phase) from those
other large-scale structures actually formed in the contemporary new phase. In
particular, those past phase structures that crossed the horizon early during
the Dark Ages (before the expected commencement of the large-scale structure
formation period of the new cosmic phase commenced) would appear distinctly
misplaced as galaxies or black holes too mature for that cosmic era [58].
Figure 2 is a diagrammatic sketch of cosmic evolution in comoving coordinates
(based on $\Lambda CDM$444For ease of communicating our point, we have chosen
to use the familiar $\Lambda CDM$ cosmic horizon evolution diagram with no
loss of generality. The SRC (not shown here) appears slightly different mostly
around the matter-dark energy equality line about 4.07 Gyrs ago.).We have
modified the original diagram (Courtesy of Physics Stack Exchange) to add
features identifiable with the past cosmic phase. The solid baseline
represents the extent of the horizon of the past phase at the instant of the
phase transition. The phase transition (Big Bang) takes place at the (past-
future cosmic phase-interface) intersection of that solid baseline with the
observer’s worldline (solid vertical line). The dotted lines parallel to the
solid vertical line show comoving worldlines of ”alien” structures
(galaxies/black holes) from the past phase. Each of the arrows indicates the
instant of entry of the associated structure into the Hubble sphere (oval
region) through the Hubble horizon of the contemporary (current) phase of the
Universe. The structures are observed at their crossing of the observer’s past
light cone.

Figure 2: A sketch of cosmic evolution in comoving coordinates in based on
$\Lambda CDM$.(Courtesy of Physics Stack Exchange) modified to describe the
process. The outline of the oval region shows the evolution of the Hubble
horizon. The vertical solid line is the observer’s worldline. The solid
baseline shows the Hubble sphere size of past cosmic phase at phase
transition. The dotted lines show comoving structures from the previous cosmic
phase entering (at the arrows) the Hubble horizon of the new cosmic phase.

The scenario just depicted in above paragraph is reminiscent of recent
observations by the James Webb Space Telescope (JWST). Lately the JWST has
consistently brought in pictures, interpreted to be those of unusually mature
structures at high redshifts, such as mature galaxies [58][59] and mature
black holes [60]. Based on the outgoing analysis, our projection, is that some
of these mature structures observed by JWST at high redshifts may be
structures that entered the cosmic horizon from the previous kalpa or phase of
the Universe. Identification of such unusually mature structures in the early
universe, as ”alien” structures would help validate the SRC.

##  6 Discussion

Understanding initial conditions for the Universe is central to discussing
cosmic evolution. In standard cosmology the traditional approach is to assume
initial conditions in which an early Universe emerges from an infinite density
spacetime physical singularity that somehow undergoes a period of inflationary
expansion followed by a matter creating ”reheat” period. This approach has
produced results which have generally been quite consistent with observations.
The challenge, though, is that to date it is not known how (or even whether) a
true past-directed spacetime singularity can give rise to a future regular
spacetime, leave alone an inflationary one. This questions the source of the
invoked initial fields like the inflaton, the source of its physical
characteristics such as those of its potential or its thermodynamic features
such as the initial entropy. In this paper we have presented a different set
of initial conditions based on a framework provided by the SRC model, the
evolution of which can justifiably lead to the observed Universe. In the SRC
model, cosmic dynamics is self-regulating allowing the Universe to evolve
through self-regenerating cosmic phases or kalpas. The end of a given kalpa
and the beginning of the next are linked by an interface that sets conditions
for this latter new phase. The approach is consistent with the observation
that has been expressed by Penrose [43] and by Carroll [46] that such cosmic
initial conditions should not be fixed, a priori, but instead should
generically emerge from existing physics.

In our approach we have, with justification, considered space to behave as a
material medium with an elastic constant $\kappa\sim\frac{1}{\tau^{2}}$, where
$\tau$ is the duration of a cosmic phase. Taking the analogy further, we noted
that at $\tau-t_{f}$, when cosmic acceleration ends just before the end of the
kalpa, this space elasticity reaches a yielding point to undergo a phase
transition thereafter. The phase transition is initiated through localized
spacetime sites with nucleated defects in which the spacetime strain no longer
follows a simple linear Hook’s law-type relation. Instead the defects generate
de Sitter-like patches so that space here becomes inflationary. From the point
of view of an observer inside the spacetime defect patch’s future a new
spacetime is created whose horizon is made initially small by the associated
extreme expansion (Hubble) rate, $H_{\left(t_{f}\right)}\sim\frac{1}{t_{f}}$,
of the the patch or bubble. This phase transition of space between two de
Sitter like horizons, from the old phase horizon $R_{\left(\tau\right)}=c\tau$
to a new phase with horizon $R_{\left(t_{f}\right)}=ct_{f}$ constitutes, the
Big Bang in SRC. The process also seeds all the initial conditions for the new
Universe phase. First, it is shown how the process leads to a new inflationary
Universe, with a short-lived de Sitter-like interval. We have discussed how
the inflationary phase here ends naturally and gracefully, evolving into a
matter-creating dissipative and reheating phase, consistent with predictions
of SRC in DEPP. Thereafter, observations show the Universe becomes radiation
dominated. As a consequence, in SRC there is no need to invoke some arbitrary
inflaton scalar field. The initial conditions of the new dynamical phase
Universe evolve from the previous phase.

The paper also discussed the issue of initial cosmic entropy as part of
initial conditions. We have shown how the evolution of the Universe in SRC is
based on self-regenerating cosmic phases or kalpas. It therefore obeys the
Poincare Recurrence Theorem with a Poincare Recurrence Time equal to the kalpa
interval $\tau$, taken in the model as $\tau\simeq 19.38Gyr$. We showed that
at the inter-kalpa phase transition the Boltzman’s Anthropic Hypothesis, which
is an example of Poincare Recurrence Theorem, holds and the new phase
Universe’s initial low entropy evolves as a fluctuation from the higher
entropy of the outgoing phase. Thereafter, within the new phase, the entropy
evolves consistent with the Second Law of Thermodynamics and the Past
Hypothesis. Consequently, while the Boltzman Hypothesis, under the umbrella of
Poincare Recurrence operates globally between kalpas, the Past Hypothesis and
Second Law operate locally within a kalpa. In our view this reconciles the
Boltzman’s Hypothesis with the Second Law of Thermodynamics, in the model.

The paper closes with SRC approaches to address some currently unreconciled
observations. First, we discuss the Cosmological Constant Problem in the
framework of SRC in which the Universe evolves in phases, each bounded by two
de Sitter-like horizons. We how the existence of these two bounding horizons
allows a logical consideration of an evolving vacuum-energy in an approach
leading to a solution of the cosmological constant problem. Lastly SRC
predicts astrophysical structures such as galaxies or black holes, from
previous phases entering the new phase Universe, through the latter’s Hubble
horizon. We have therefore conjectured that the JWST-observed mature
structures such as old spiral galaxies and supermassive black holes at high
red-shifts are likely those coming from the previous phase Universe,
consistent with SRC predictions.

Assuming the SRC model correctly interprets the observed Universe, there is
room to discuss the processes in more detail including those that need a
micro-physical level framework. For example, the postulated phase transition
linking adjacent cosmic phases takes place mostly in the Planck regime where
both quantum and gravitational effects are expectedly significantly in force.
It will require an appropriate quantum gravitational framework to
quantitatively describe processes in this regime, including how de Sitter-like
bubbles can initially be nucleated as defects in such a background. Other
features not discussed here, such as the issue of initial fluctuations, will
be addressed elsewhere.

Acknowledgement:

Acknowledgement: I would like to thank Edwin Gatwaza for diagram edits and
Albert Munyeshyaka for style editing. This work was supported, in part, by the
Swedish International Development Cooperation Agency (SIDA) through the
International Science Program (ISP) grant No̱ RWA:01.

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