# Saturon Dark Matter

Gia Dvali  Arnold Sommerfeld Center, Ludwig Maximilians University,
Theresienstraße 37, 80333 Munich, Germany  Max Planck Institute for Physics,
Föhringer Ring 6, 80805 Munich, Germany

###### Abstract

Saturons are macroscopic objects with maximal microstate entropy. Due to this
property, they can be produced via quantum transitions from a homogeneous
thermal bath, bypassing the standard exponential suppression characteristic of
ordinary extended objects. In this sense, saturons carry an advantage with
respect to other macroscopic objects such as black holes and ordinary
solitons. Due to unsuppressed thermal production, saturons can have
interesting cosmological implications. In particular they can serve as viable
dark matter candidates with some unique features. Unlike ordinary particle
dark matter, the superheavy saturons can freeze-in at very low temperatures. A
nucleation of a saturon can be described in terms of a saturated instanton.
This has implications for various phase transitions.

##  .1 Introduction

The origin of dark matter in the Universe is still a mystery. The especially
valuable candidates are the ones with calculable production mechanisms and
observational signatures. The known dark matter candidates, can be divided in
two general categories: elementary particles and macroscopic composite
objects. Each category has certain universal characteristic properties.

The elementary particle dark matter is bounded in mass. First, no quantum
particle can be heavier that the Planck scale, $M_{P}$. Secondly, the mass $m$
of a thermally-produced particle at temperature $T$ is bounded from above by
the temperature. The production of a heavier particle is exponentially
suppressed by the Boltzmann factor, $\exp(-m/T)$.

The second category of dark matter includes macroscopic objects. The examples
are provided by black holes and various solitons. Obviously, such dark matter
“particles” can be much heavier than the Planck mass. The challenge however is
the production mechanism.

In previous considerations, the macroscopic super-heavy dark matter is never
produced via a direct quantum transition from a radiation thermal bath. The
reason is an exponential suppression characteristic of ordinary macroscopic
objects. We shall show that the origin of this suppression is entropic. To be
more precise, it originates from an insufficient entropy of ordinary extended
objects as compared to the entropy of thermal radiation.

Due to this, the standard mechanisms for the production of macroscopic objects
do not rely on their quantum materialization. For example, topological
solitons can be produced during phase transitions via the Kibble mechanism
Kibble:1976sj . Its variations for non-thermal phase-transitions can also be
considered. On the other hand, a creation of gravitationally bound objects,
such as black holes, requires the presence of large inhomogeneities.

In the present article we shall propose a new class of dark matter candidates,
the so-called “saturons” Dvali:2020wqi . In certain sense, these formations
inherit the beneficiary features of both categories: the classical composite
objects and the quantum particles. On one hand, a saturon represents a
macroscopic object of an arbitrarily large mass/size. On the other hand, like
an elementary quantum, it can be produces thermally without an exponential
suppression.

The concept of saturons was introduced in Dvali:2020wqi . However, several
explicit examples have been constructed in various field theories in
Dvali:2019jjw ; Dvali:2019ulr ; Dvali:2021jto ; Dvali:2021rlf ; Dvali:2021tez
; Dvali:2021ofp . The defining property of saturons is that they saturate a
certain upper bound on the entropy. This bound is universally imposed by the
unitarity. That is, within the validity of a quantum field theoretic
description, the saturons are the objects of maximal microstate degeneracy.

As shown in Dvali:2019ulr , the phenomenon of saturation can take place for
objects with Lorentzian as well as the Euclidean signatures. In particular, it
was demonstrated that saturation is fully applicable to instantons. This has
clear physical consequences: upon a saturation, the transition process
mediated by the instanton saturates unitarity. We shall refer to saturated
instantons as “i-saturons”.

Due to the above special feature, saturons, despite being the macroscopic
entities, can be produced via a quantum scattering from a thermal bath. We
shall explain that a formation of a saturon can be described as a quantum
transition generated by a saturated instanton. That is, an i-saturon creates a
saturon. In fact, the saturated instantons can assist the phase transitions
even if no stable Lorentzian saturon exists at zero temperatures.

Due to their maximal entropy, the saturons share a number of striking
similarities with black holes. To be more precise, black holes represent a
special case of saturons. However, unlike non-gravitational saturons, the
black holes cannot be produced via a quantum process from a homogeneous
radiation thermal bath. We shall explain this fundamental difference in
detail.

Due to the above feature, saturons can offer a qualitatively new type of dark
matter. In particular, they can provide a superheavy macroscopic dark matter
produced at unusually low temperatures via a quantum transition process.

In the present paper we shall discuss the essence of the above phenomenon and
outline various aspects of saturon cosmology. In particular, we discuss its
implications for thermal transitions and dark matter.

##  .2 Saturons

The term “saturon” was introduced in Dvali:2020wqi  for describing an object
exhibiting the maximal microstate degeneracy permitted by unitarity. Various
examples of such objects and their connection with saturation of unitarity
were given in series of papers Dvali:2019jjw ; Dvali:2019ulr ; Dvali:2021jto ;
Dvali:2021rlf ; Dvali:2021tez ; Dvali:2021ofp . For the detailed discussions,
we refer the reader to the above articles. Here, we shall briefly recount the
key features of saturons that will be used in our analysis. First, we focus on
saturons in space-times with a Lorentzian signature.

We consider an effective field theory described by a set of degrees of freedom
interacting via a dimensionless quantum coupling $\alpha$. The validity of
their field theoretic description implies that the coupling is weak,
$\alpha\ll 1$.

We assume that these degrees of freedom form a macroscopic composite object of
mass/energy $E$ and a size $R$. For example, the role of such object can be
played by a soliton or any other bound state.

Being composite, the object can be described by a mean occupation number of
its constituents, which we denote by $n$. In the simplest cases the object has
a single characteristic scale, $R$, which sets its size. In such cases, the
dominant contribution to the energy of the bound state comes from the
constituent quanta of wavelengths $\sim R$. Each constituent contributes $\sim
1/R$ into the energy of the bound state. Correspondingly, the total energy is,
$E\sim n/R$.

For such objects, the exact classical limit is well-defined and reads,

$n\rightarrow\infty,~{}n\alpha\,=\,{\rm fixed},~{}R={\rm fixed}\,.$ (1)

It is easy to see that the self-sustainability condition implies, $\alpha\sim
1/n$. In this case, the bound state can be classically stable. However, it can
decay due to quantum effects that are controlled by $1/n$ corrections. Below,
we shall work at finite but large $n$.

As explained in Dvali:2020wqi , an important universal characteristic of any
bound state is the Goldstone scale. The reason is that macroscopic objects
break spontaneously a set of symmetries, both space-time as well as internal.

Of course, the concept of spontaneous breaking of symmetry becomes exact in
the limit $n\rightarrow\infty$. Only in this limit, the Hilbert spaces built
on different degenerated vacua are strictly orthogonal. However, the
description of objects in terms of spontaneous breaking of symmetry,
represents an excellent approximation for large $n$. That is, the would-be
Nambu-Goldstone modes are gapless at least to leading order in
$1/n$-corrections. All the bound states at weak coupling $\alpha$, satisfy
this condition.

Among various Goldstone modes, the Goldstone modes of Poincare symmetry are
especially useful. This is due to their universal nature. Obviously, any
localized state breaks spontaneously the Poincare symmetry. In particular, it
breaks both space and time translations as well as the Lorentz boosts.
Correspondingly, there exist a Goldstone boson with a well-defined scale,
which we denote by $f_{G}$. It is easy to express this scale through $R$ and
$n$, as $f_{G}^{2}\,=\,n/R^{2}$.

In conclusion Dvali:2020wqi , a generic bound state with a characteristic
single scale $R$, admits an universal description in terms of parameters $n$,
$E$ and $f_{G}$. They are uniquely expressed though $\alpha$ and $R$,

$\displaystyle{\rm Number}:~{}n\,=\,\frac{1}{\alpha},$ (2) $\displaystyle{\rm
Energy}:~{}E\,=\,\frac{n}{R}\,=\,\frac{1}{\alpha}\frac{1}{R},$ (3)
$\displaystyle{\rm
Goldstone~{}scale}:~{}f_{G}^{2}\,=\,\frac{n}{R^{2}}=\frac{E}{R}$
$\displaystyle\,.$ (4)

It is important to understand that the above are the universal relations for
bound states formed by an interaction with a scale $R$ and a weak coupling
$\alpha$. In particular, they are exhibited by all ordinary solitons. A new
feature that singles out a sub-class of saturons, is the maximal microstate
entropy.

This feature has been widely considered for black holes. However, the
microstate degeneracy is a property of a much wider class of field theoretic
objects. This degeneracy can originate from various sources. For example, it
can result from internal symmetries under which the object transforms.

Due to this, the composite states, such as solitons, can be realised in a
large number, $n_{st}$, of degenerate microstates. The notion of a
“microstate” that we use, is standard. It amount to distinct quantum states
that realize the same macroscopic (classical) properties of the object, such
as the energy, size and a total angular momentum. The states we refer to, are
pure. For such objects, a notion of the microstate entropy, defined as,

$S\,\equiv\,\ln(n_{st})\,,$ (5)

is fully applicable.

An important question, investigated in papers Dvali:2019jjw ; Dvali:2019ulr ;
Dvali:2020wqi  (for the summary, see Dvali:2021jto ), is a maximal microstate
degeneracy of a bound state. The answer is that an absolute upper bound on the
microstate entropy of a bound state is given by the following three
expressions Dvali:2020wqi ,

$\displaystyle{\rm Area-form}:~{}S_{sat}\,\sim\,$ $\displaystyle
R^{2}f_{G}^{2}$ (6) $\displaystyle{\rm Number-form}:~{}S_{sat}\,\sim\,$
$\displaystyle n$ (7) $\displaystyle{\rm Coupling-form}:~{}S_{sat}\,\sim\,$
$\displaystyle\frac{1}{\alpha}\,.$ (8)

The connection between these three formulations can be derived from the
relations (2), (3), (4). The saturons are defined as the objects exhibiting
the above maximal degeneracy.

Thus, from (3) and (7), (8), the saturon energy can be expressed through its
entropy, occupation number, or coupling, as,

$E_{\rm sat}\,=\,\frac{S_{\rm
sat}}{R}\,=\,\frac{n}{R}\,=\,\frac{1}{\alpha}\frac{1}{R}\,.$ (9)

It has been established that attaining a higher degeneracy is not possible
within the validity of any weakly-coupled set of degrees of freedom. That is,
an attempt of obtaining a bound state with a microstate entropy exceeding
$S_{\rm sat}$, puts its constituent degrees of freedom out of the validity
domain of the theory. Several markers of this effect have been identified in
form of the breakdown of the loop-expansion in coupling $\alpha$ Dvali:2019jjw
; Dvali:2019ulr ; Dvali:2020wqi , as well as, in saturation of the unitarity
bound by a set of scattering processes Dvali:2020wqi .

Notice Dvali:2020wqi  that the first relation in (9) coincides with the well
known Bekenstein bound on entropy BekBound . It emerges as a consequence of
more stringent bounds (6) - (8).

Due to this reason, using this bound as a sole guideline for our purposes is
not possible, as it carries no information about the strength of the coupling
(neither fundamental $\alpha$, nor collective $n\alpha$) or about the
occupation number of modes. Correspondingly, it is not sensitive to the
quantum field theoretic validity of the description and can formally be
satisfied beyond its domain. Therefore, we impose the necessary consistency
condition that the system respects the three bounds, (6), (7) and (8),
simultaneously. This gives the relation (9) as a byproduct. The saturon states
saturate all three and thereby satisfy (9).

The concept of saturation generlizes to objects with an Euclidean signature,
such as instantons Dvali:2019ulr . For such entities all three bounds (6) -
(8) are well-defined 111In contrast, the formulation of the Bekenstein bound
for instantons is not possible, due to absence of the notion of energy.. For
example, the instanton action exhibits the degeneracy with respect to broken
space and internal symmetries and the scale $f_{G}$ is unambiguously defined.
As explained in Dvali:2019ulr , an useful intuitive way for picturing the
instanton entropy is by using an auxiliary construct; mapping an instanton on
a soliton in Lorentzian theory with one additional time-like dimension.

In general, upon saturation, the microstate entropy of an instanton becomes
equal to is action. That is, the instanton becomes an i-saturon. This signals
that the corresponding transition process is close to saturating unitarity. We
shall come back to this point in the discussion of the role of saturons in
thermal transitions.

As shown in series of previous papers, Dvali:2019jjw ; Dvali:2019ulr ;
Dvali:2020wqi ; Dvali:2021jto ; Dvali:2021rlf ; Dvali:2021tez ; Dvali:2021ofp
, saturons can come in variety of forms. For example, they can be represented
by topological or non-topological solitons, baryons and other composite
objects. What unifies them is the saturation of the entropy bound.

Due to this, the saturons share some striking similarities with black holes.
This is already apparent from the area-law form of their entropy given by (6).
This expression is identical to the Bekenstein-Hawking entropy BekE  of a
black hole of radius $R$,

$S_{\rm BH}\,\sim\,R^{2}M_{P}^{2}\,,$ (10)

with the role of the scale of the Poincare Goldstone $f_{G}$ played by the
Planck mass, $M_{P}$. This is not surprising, since a black hole, just like
any other localized object, breaks the Poincare symmetry spontaneously. The
order parameter of this breaking is $M_{P}$.

As explained by the microscopic theory of “black hole’s quantum $N$-portrait”
Dvali:2011aa , in terms of the occupation number of gravitons and their
coupling, the black hole entropy is given by the equations (7) and (8)
respectively.

It has been shown that saturons exhibit other striking similarities with black
holes. These include a thermal like decay at initial stages as well as the
scaling of the information-retrieval time in close similarity with the Page’s
time of a black hole Dvali:2020wqi ; Dvali:2021jto ; Dvali:2021rlf ;
Dvali:2021tez ; Dvali:2021ofp . The relation between the maximal spin and the
mass (or entropy) is also identical to black holes Dvali:2021ofp .

From the above perspective, a black hole represents a saturon formed by a
gravitational interaction. As we shall see, surprisingly, this fact puts black
holes at some disadvantage, as compared to saturons produced by other
interactions (e.g., the gauge forces). Namely, a black hole cannot be produced
from the homogeneous thermal bath in the early Universe by means of a quantum
transition. In contrast, the non-gravitational saturons, can.

##  .3 Saturon lifetime

Before moving on, we comment on the saturon lifetime. First, saturons can be
stable due to conserved charges, both topological and non-topological. Of
course, in such cases the saturons with opposite charges must be produced in
equal numbers from the thermal bath.

However, even the saturons carrying no conserved charges are macroscopically
long-lived. This is one of the many properties that saturons share with black
holes. Similarly to a black hole, at the initial stages an unstable saturon
decays via the emission of the soft quanta: a quantum of energy $\sim 1/R$ is
emitted per time $R$. Correspondingly, the timescale during which an unstable
saturon looses about half of its mass, is given by $t_{\rm half}\sim S_{\rm
sat}R$.

This timescale is very important as it represents a so-called “quantum break-
time” of the system. In particular, beyond this point the semi-classical
approximation breaks down completely. There are several effects contributing
into this break-down.

The first one is an “inner entanglement”, originally pointed out for black
holes in Dvali:2013eja ; Dvali:2021bsy  and generalized to arbitrary high
entropy states in Dvali:2018xpy ; Dvali:2020wqi  (see also, Dvali:2022vzz ).
This entaglement is developed due to inner rescattering of constituents and it
reaches its maximum around $t_{\rm half}$. We shall explain it in more details
later on explicit examples of saturons.

Secondly, There are indications Dvali:2020wft  that after $t_{\rm half}$, the
decay slows down due to so-called “memory burden” effect Dvali:2018xpy . The
essence of the phenomenon is the following.

In saturons, the degenerate microstates are achieved by excitations of gapless
degrees of freedom, the so-called “memory modes”. Their presence is universal
in all saturons, while the emergence mechanisms can differ from case to case.
In a class of saturons (see examples below) the memory modes emerge as
Goldstone modes of spontaneously broken symmetries Dvali:2019jjw ;
Dvali:2019ulr ; Dvali:2020wqi ; Dvali:2021jto ; Dvali:2021rlf ; Dvali:2021tez
; Dvali:2021ofp .

In other cases they emerge as angular momentum modes that become gapless at
the critical point Dvali:2017nis . As a result, the saturon state is endowed
with $\sim n$ distinct “flavors” of gapless modes. Each distinct microstate
represents a specific pattern of their excitations. Their number is
exponentially large, resulting in the entropy $S\sim n$.

Due to the gaplessness of the memory modes, the quantum information encoded in
their excitation pattern, is energetically much cheaper than the analogous
information carried by the asymptotic free quanta. Correspondingly, the
saturon cannot decay without “offloading” this information pattern. Since
there exist no gapless carriers among the free quanta, the offloading of
information is a highly suppressed process. This effect tends to stabilize the
saturon.

The full quantum evolution has only been studied numerically Dvali:2020wft .
These studies confirm the analytic estimates by showing that, latest by half
decay, the process slows down.

Due to the limitations of numerical analysis, the estimate of the remaining
lifespan requires some guesswork. On one hand, it is unlikely that a saturon
with zero net charge gets fully stable, as no conservation law forbids its
decay. On the other hand, it is evident that the memory burden effect makes
the lifetime much longer than $t_{\rm half}$.

It is important to stress that the memory burden effect is an exclusive
property of objects with enhanced capacity of information storage, such as
saturons. This feature is not shared by composite objects of low microstate
entropy, such as ordinary solitons. This sets a difference between the decay
processes of these two classes of objects. However, the difference due to
memory burden becomes significant after the mass of the object is reduced by
order half.

The corpuscular resolution Dvali:2011aa  of classical objects, such as black
holes and solitons (for a corpuscular description of topological charges, see
Dvali:2015jxa ) as composites of high occupation number of quanta, reveals
close similarities in their quantum decays at initial stages. This decay can
be understood as a result of quantum depletion due to the re-scatterings of
the constituent quanta. This mechanism is equally operative in black holes,
solitons and other field configurations Dvali:2011aa ; Dvali:2017eba ;
Dvali:2017ruz ; Dvali:2020wqi ; Dvali:2022vzz ; Dvali:2021rlf ; Dvali:2021tez
. In particular, in Olle:2020qqy  this picture has been applied to specific
breathing scalar field configurations, called “oscillons” Copeland:1995fq .
This study was motivated by oscillons as dark matter Olle:2019kbo .

However, the similarities hold only up until the half-decay time. Beyond this
point, exclusively for saturons, the inner-entanglement and memory burden
effects set in. No analogous effect is apparent for ordinary low entropy
solitons 222Of course, endowing the ordinary solitons, such as, e.g.,
oscillons, with maximal microstate entropy, promotes them into the category of
saturons, with all the consequences, such as memory burden and inner
entanglement..

In summary, the feature of a maximal entropy gives an advantage to saturons,
as compared to ordinary solitons. First, even in the absence of a conserved
charge, saturons tend to get an extended lifespan via the memory burden
effect. This longevity shall be assumed in what follows. Secondly, the maximal
entropy allows for an unsuppressed quantum production of saturons from a
homogeneous thermal bath. We now give a detailed discussion of this effect.

##  .4 Thermal production of saturons

Saturons can certainly be produced by the conventional mechanisms
characteristic of objects from the same universality class. For example,
saturons of the type of topological defects Dvali:2019jjw ; Dvali:2019ulr ;
Dvali:2020wqi , can be produced by the Kibble mechanism Kibble:1976sj .
Analogously, those saturons Dvali:2021tez ; Dvali:2021ofp  that can be placed
in the category of $Q$-balls Lee:1991ax ; Coleman:1985ki , can be produced by
one of the mechanisms characteristic for ordinary $Q$-balls Kusenko:1997si ;
Kusenko:1997vp ; Dvali:1997qv  (for a recent discussion, see, e.g.,
Bai:2022kxq ).

In the case of production by such conventional mechanisms, the novelty, as
compared to ordinary solitons, is that saturons can be produced in
exponentially large number of microstates and this affects their subsequent
evolution. In particular, it suppresses the rate of annihilation.
Correspondingly, a much higher number of saturons can survive as compared to
ordinary solitons produced by the same mechanism. This comparisons require a
separate investigation which is beyond the subject of the present paper.

Perhaps, the most interesting novelty, which we shall focus on, is the
possibility of a thermal production. Ordinarily, the thermal production of
extended objects, such as solitons, is exponentially suppressed. A classic
example illustrating this suppression is the computation by Linde Linde:1981zj
, who estimated the rate of thermal production of the domain wall bubble at
temperature $T$. We start with the regime of a “thin-wall” approximation in
which the bubble radius $R$ exceeds its thickness as well as the inverse
temperature. In this regime Linde’s result amounts to,

$\Gamma(T)\sim T^{4}\left(\frac{A_{3}}{T}\right)^{\frac{3}{2}}\,{\rm
e}^{-\frac{A_{3}}{T}}\,,$ (11)

where $A_{3}$ is a three-dimensional Euclidean action which essentially is
equal to the energy of the bubble, $E_{\rm bub}$ 333We changed notation of
Linde:1981zj  in order to use different symbols for the action ($A$) and the
entropy ($S$)..

For a saturated bubble, the thin-wall approximation is applicable only order
of magnitude wise. We shall therefore estimate the rate by a more general
method, which does not rely on such approximation. As a consistency check, we
show that our estimate agrees with Linde’s result.

Since the goal of our analysis is to point out a qualitatively new mechanism,
we shall be limited by an order of magnitude estimates. The unimportant
numerical factors will be set equal to one. Of course, in applications to
realistic models, more precise estimates must be performed.

We assume that a theory admits a solitonic object of some characteristic size
$R$ and energy $E_{\rm bub}$. For simplicity, we shall assume a spherical
symmetry, although the arguments extent beyond this approximation. The role of
the object in particular can be played by a bubble separating the two vacua,
as in Linde’s analysis. However, our estimate applies to a larger class of
objects.

Let us consider a thermal bath at temperature $T$. For the beginning, we limit
ourselves with a single type of scalar species in thermal equilibrium. Then,
the energy density of the thermal bath is $\sim T^{4}$, whereas the entropy
density is given by $\sim T^{3}$. Correspondingly, the energy ($E_{\rm rad}$)
and the entropy ($S_{\rm rad}$) of a spherical region of radius $R$ are given
by,

$S_{\rm rad}(T)=(RT)^{3}\,,~{}\,E_{\rm rad}(T)=T^{4}R^{3}\,.$ (12)

Let us estimate the probability for quantum transition of this spherical
region into a state of the soliton (e.g., a vacuum bubble) of the same radius
$R$.

We shall assume that the soliton is created out of a thermal region of the
equal radius. Although different regimes are also possible, as we shall
explain, they do not change the estimate substantially. Then, in order for the
transition to take place, the soliton energy must not exceed the energy of the
thermal bath within the same sphere:

$E_{\rm bub}\leqslant E_{\rm rad}(T)\,.$ (13)

Taking into account (12), the “on-threshold” production of a bubble gives a
condition,

$\frac{E_{\rm bub}(T)}{T}=S_{\rm rad}(T)\,.$ (14)

Let us first assume that the bubble carries a zero (or a very small) entropy.
This is an implicit assumption in conventional discussions. In particular,
this is the case in Linde’s analysis.

Then, creation of such a bubble represents a transition from a high entropy
state (radiation) into a low-entropy one (soliton). The corresponding rate is
suppressed by the entropy factor,

$\Gamma(T)\,\sim\,{\rm e}^{-S_{\rm rad}(T)}\,.$ (15)

Taking into account (14), it is clear that the expression (16) recovers the
exponential factor given in (11). The difference is that we did not use any
thin-wall approximation.

The above result explains in clear physical terms why the thermal production
of ordinary solitons is exponentially suppressed. It is highly improbable for
the thermally-distributed particles to get organized into a coherent state of
a very low entropy, such as the soliton.

The same picture explains why the situation with saturons is fundamentally
different. The reason is that the saturon entropy is maximal for a given
quantum field theoretic system. In particular, it can exceed the entropy of
the radiation bubble. Correspondingly, the suppression factor can be
compensated by the entropy of the saturon,

$\Gamma(T)\,\sim\,{\rm e}^{-S_{\rm rad}(T)+S_{\rm sat}(T)}\,.$ (16)

Of course, the formation probability cannot exceed one. In the extreme case,
the saturon formation will saturate the unitarity of the time-evolution.

Let us discuss this phenomenon in more details. For generality, we consider a
thermal bath with $N_{*}$ particle species in thermal equilibrium. Then, the
entropy and the energy of a radiation bubble of radius $R$ are given by,

$S_{\rm rad}(T)=N_{*}(RT)^{3}\,,~{}\,E_{\rm rad}(T)=N_{*}T^{4}R^{3}\,.$ (17)

In order for a saturon to form, the following two conditions must be
satisfied,

$S_{\rm rad}(T)\leqslant S_{\rm sat}\,,~{}~{}~{}\,E_{\rm sat}\leqslant E_{\rm
rad}(T)\,.$ (18)

Taking into account (17) and (3), we obtain the following window of
temperatures for the formation of a saturon,

$\left(\frac{S_{\rm sat}}{N_{*}}\right)^{\frac{1}{3}}\gtrsim
TR\gtrsim\left(\frac{S_{\rm sat}}{N_{*}}\right)^{\frac{1}{4}}\,.$ (19)

Expressing the saturon entropy through the occupation number of its
constituents, $n$, via the relation (7), we can rewrite the above expression
as,

$\left(\frac{n}{N_{*}}\right)^{\frac{1}{3}}\gtrsim
TR\gtrsim\left(\frac{n}{N_{*}}\right)^{\frac{1}{4}}\,.$ (20)

Notice that the window exists only for $\frac{S_{\rm sat}}{N_{*}}\,\gtrsim\,1$
(equivalently, for $\frac{n}{N_{*}}\gtrsim 1$), which shall be assumed.

In fact, as we shall explain later, the saturon formation probability is
peaked at the lower bound of this inequality. That is, the transition
probability is maximized when the saturon mass exactly matches the energy of
the radiation within the same sphere. In other words, the optimal transition
happens for,

$E_{\rm sat}\,=\,E_{\rm rad}(T)\,=\,N_{*}T_{s}^{4}R^{3}\,$ (21)

Taking this into account, we obtain that the optimal temperature $T$ for the
formation of the saturon of radius $R$ is,

$T\sim\frac{1}{R}\left(\frac{S_{\rm
sat}}{N_{*}}\right)^{\frac{1}{4}}\sim\frac{1}{R}\left(\frac{n}{N_{*}}\right)^{\frac{1}{4}}\,.$
(22)

Correspondingly, using the expressions (3) and (9), the saturon energy can be
expressed as the function of the optimal temperature and the saturon entropy
(or the occupation number),

$E_{\rm sat}\,\sim\,T\,S_{\rm
sat}^{\frac{3}{4}}N_{*}^{\frac{1}{4}}\sim\,T\,n^{\frac{3}{4}}N_{*}^{\frac{1}{4}}.$
(23)

We must note that while at the optimal temperature saturons can be produced
out of thermal bath without an exponential suppression, the process cannot
equilibrate easily. First, if the saturon entropy is higher than the entropy
of radiation, the inverse process is exponentially suppressed. The second
factor is the expansion rate of the Universe which suppresses the probability
of equilibration.

##  .5 Abundance of saturon dark matter

###  .5.1 Bottom-up: estimate of required suppression

Saturons may compose a part or an entirety of the dark matter in the Universe.
In the latter case, we can put a bottom-up constraint on the probability of
their production.

Let $p$ denote a total probability of the transition of a thermal sphere of
radius $R$, into a saturon that occupies the same sphere. The transition rate
is maximal around the optimal temperature and quickly falls off beyond it.
Thus, below $T$, we shall end up on average with one saturon per volume
$V\,=\,R^{3}/p$. Taking into account the condition (21), we get the following
energy density of saturons at optimal formation temperature $T$,

$\rho(T)\,=\,pN_{*}T^{4}\,.$ (24)

After this moment, the saturon energy density red-shifts as matter, and for
todays temperature $T_{0}\simeq 2.4$K, reaches the value,

$\rho_{\rm sat}(T_{0})\,=\,pN_{*}TT_{0}^{3}\,.$ (25)

Equating this to the current dark matter density, $\rho_{\rm DM}$, we get the
following condition on $p$,

$p\,=\,\frac{\rho_{\rm DM}}{N_{*}TT_{0}^{3}}\,.$ (26)

In terms of the present day radiation energy density, $\rho_{\rm
rad}(T_{0})\,\sim\,T^{4}_{0}$, we write,

$p\,\sim\,\frac{\rho_{\rm DM}}{\rho_{\rm
rad}(T_{0})}\frac{T_{0}}{T}\frac{1}{N_{*}}\,\sim\,\frac{\rm eV}{N_{*}T}\,.$
(27)

For example, saturons produced around the temperatures $T\sim$ MeV, can serve
as dark matter candidates provided they satisfy $p\sim 10^{-7}$.

###  .5.2 Top-down: probability of production

Let us now estimate the probability of production, $p$, in top-down approach.
We consider a thermal bath at temperature $T$ in an expanding Universe. We are
interested in the probability of production of a saturon of radius $R$ from
the spherical domain of the same radius filled with radiation. The saturon
formation process can be viewed as the transition of thermal quanta of
occupation number $n_{\rm rad}=N_{*}(TR)^{3}=S_{\rm rad}$ into a coherent
state of saturon of occupation number $n\,=\,S_{\rm sat}$. Since $S_{\rm
rad}\leqslant S_{\rm sat}$, we have $n_{\rm rad}\leqslant n$.

That is, the transition process represents a conversion of $n_{\rm rad}$
thermally distributed particles into $n$ coherently organized quanta of
softness $1/R$. Mentally, this can be viewed as a scattering process

$n_{\rm rad}\rightarrow n\,,$ (28)

obtained by summing over different transition amplitudes. The momentum
transfer per particle is $\sim 1/R$. Correspondingly, the characteristic time-
scale of the process is $t_{tr}\sim R$. We shall call it a “transition-time”.

The transition takes place for the optimal temperature (21), for which the
energy of radiation $E_{\rm rad}(T)$ matches the saturon mass, $E_{\rm sat}$.
Let us denote this optimal temperature by $T_{s}$.

For $T\neq T_{s}$, the saturon production is suppressed. Indeed, for
$T<T_{s}$, one needs to produce a saturon out of a thermal sphere larger than
its size. This causes an extra exponential suppression. For, $T>T_{s}$, the
system produces a state that is not a saturon ground state and correspondingly
carries a small fraction of the available entropy.

This physical intuition is supported by the analysis of the scattering
amplitudes Dvali:2020wqi , which shows that the probability for the saturon
production in a scattering process at center of mass energy $E$, quickly
diminishes outside of the following window,

$\Delta E\,\equiv\,|E-E_{\rm sat}|\lesssim\frac{E_{\rm sat}}{S_{\rm sat}}\,.$
(29)

In the present case, $\Delta E$ must be taken as the allowed spread of the
thermal energy within the sphere of radius $R$ that is transiting into a
saturon state.

Now, due to the expansion of the Universe, the temperature drops and the
energy of radiation within the radius $R$ gets red-shifted. Correspondingly,
the window of opportunity during which $\Delta E$ satisfies (29), is open only
for a finite time $\Delta t_{s}$. If this time is shorter than the transition
time $t_{tr}\sim R$, the probability gets suppressed.

The time $\Delta t_{s}$ can be easily estimated. First, the energy within the
sphere is related with the change of temperature, $T\,=\,T_{s}+\Delta T$, as,

$\frac{\Delta E}{E_{\rm sat}}\sim\frac{\Delta T}{T_{s}}\,.$ (30)

The time interval required for this change is,

$\Delta t_{s}\sim\frac{\Delta T}{T_{s}H}\,.$ (31)

Using (29), we get the optimal time interval,

$\Delta t_{s}\sim\frac{1}{S_{\rm sat}H}\,.$ (32)

For convenience, we define a parameter that controls the suppression,

$\beta\equiv\frac{\Delta t_{s}}{t_{tr}}\sim\frac{1}{S_{\rm sat}HR}\,.$ (33)

When $\beta\gtrsim 1$, in absence of additional suppression factors, the
transition probability is close to one, $p\sim 1$. In the opposite case,
$\beta\ll 1$, the probability is suppressed. To leading order, we can estimate
$p\sim\beta$.

Using the relation (22) and the formula for the Hubble (38), we can express
the parameter $\beta$ as,

$\beta\,\sim\,\frac{M_{P}}{T_{s}}\frac{1}{(N_{*}S_{\rm
sat}^{5})^{\frac{1}{4}}}\,.$ (34)

Finally, remembering the relation (8) between the entropy of the saturon and
the coupling, we can write,

$\beta\,\sim\,\frac{M_{P}}{T_{s}}\frac{\alpha^{\frac{5}{4}}}{N_{*}^{\frac{1}{4}}}\,.$
(35)

We wish to notice the following curios thing. In scenarios with $\beta\ll 1$,
we can fix the required entropy by equating the expressions (27) and (34).
Then, the temperature drops out of the equation and the proper entropy is
fixed as,

$S_{\rm sat}\,\sim\,\left(\frac{M_{P}}{\rm
eV}\right)^{\frac{4}{5}}N_{*}^{\frac{3}{5}}\sim 10^{22}N_{*}^{\frac{3}{5}}\,.$
(36)

We shall now apply the above knowledge to cosmological production of saturons.
We distinguish two cases, to which we refer to as “freeze-out” and “freeze-in”
scenarios. The names are borrowed from the terminology of ordinary particle
dark matter. In particular, the term “freeze-in” was introduced in Hall:2009bx
.

However, these names require some clarifications. Unlike the case with a
particle dark matter, saturons are never in thermal equilibrium. Rather, the
terms “freeze-out” and “freeze-in” refer to the state of the constituent
quanta at the time of the saturon production. In the “freeze-out” scenario the
constituents are in thermal equilibrium, whereas in case of “freeze-in”, they
are not. This is the reason for the name choices. However, the physics is
fundamentally different. This is due to the fact that saturons are collective
phenomena, and their production mechanisms are not reducible to the ones of
ordinary particle dark matter.

###  .5.3 Freeze-out

In a freeze-out scenario, the saturon constituent species are in thermal
equilibrium at the moment of the saturon production. This puts certain
conditions on the parameters.

First, the temperature must satisfy $T\gtrsim 1/R$, where $R$ is the size of a
saturon. Since the energy of each constituent is $\sim 1/R$, for a lower
temperature their numbers would be Boltzmann-suppressed.

Secondly, the re-scattering rate of the saturon constituents,

$\Gamma_{2\to 2}\,\sim\,\alpha^{2}T\,,$ (37)

where $\alpha$ is the coupling, must be larger than the Hubble expansion rate,

$H\,\sim\,N_{*}^{1/2}\,\frac{T^{2}}{M_{P}}\,.$ (38)

This condition puts the following constraint on the coupling,

$\alpha\,\gtrsim\,N_{*}^{1/4}\,\left(\frac{T}{M_{P}}\right)^{1/2}\,.$ (39)

Using (8), this equation can be rewritten as the bound on the saturon entropy:

$S_{\rm
sat}\,\lesssim\,\frac{1}{N_{*}^{\frac{1}{4}}}\left(\frac{M_{P}}{T}\right)^{\frac{1}{2}}\,.$
(40)

Correspondingly, through (3) and (22), the above translates into the following
bound on the saturon mass,

$E_{\rm
sat}\,\lesssim\,N_{*}^{\frac{1}{16}}T^{\frac{5}{8}}M_{P}^{\frac{3}{8}}\,.$
(41)

Taking into account the relation (39), in the expressions (34) and (35), we
get the following lower bound on the parameter $\beta$,

$\beta\,\gtrsim\,\left(\frac{M_{P}}{T}\right)^{\frac{3}{8}}N_{*}^{\frac{1}{16}}\,.$
(42)

Thus, in a freeze-out scenario, the probability of saturon formation at the
optimal temperature can be $p\sim 1$.

This finding should not be understood as the rejection of the freeze-out
scenario for a saturon dark matter. There exist additional factors that can
easily reduce the saturon density to an acceptable level.

First, the saturon entropy can be slightly below the unitarity upper bound.
Since the sensitivity of $p$ with respect to the saturon entropy is
exponential, this situation is rather natural.

Secondly, the saturon density can be reduced by the additional stages of
reheating. For example, due to a relatively late decay of heavy particle
species. Such a decay injects an additional energy into radiation and
correspondingly dilutes the density of the saturon matter.

###  .5.4 Freeze-in

In a freeze-in scenario, the saturon constituents are not in thermal
equilibrium. At the time of saturon production, their abundance is negligible.
This requires that their re-scattering rate (37) is lower than the Hubble rate
(38). Correspondingly, we obtain,

$\alpha\,\lesssim\,N_{*}^{1/4}\,\left(\frac{T}{M_{P}}\right)^{1/2}\,.$ (43)

Using (8), this imposes the following lower bound on the saturon entropy,

$S_{\rm
sat}\,\gtrsim\,\frac{1}{N_{*}^{\frac{1}{4}}}\left(\frac{M_{P}}{T}\right)^{\frac{1}{2}}\,.$
(44)

As discussed, the optimal temperature for saturon production is (22). We can
estimate $\beta$ under the assumption that the number freezes around this
epoch.

Taking into account the inequality (43) in (35), we get the following lower
bound on the parameter $\beta$ in freeze-in scenario,

$\beta\,\lesssim\,\left(\frac{M_{P}}{T}\right)^{\frac{3}{8}}N_{*}^{\frac{1}{16}}\,.$
(45)

Since the right hand side of the equation is larger than one, this bound is
less informative than in the freeze-out scenario. Thus, in freeze-in scenario,
the probability suppression has to be calculated on a case by case basis.

In a freeze-in scenario, the mass of a generic saturon is limited essentially
only by the energy of the entire Hubble patch. For example, solar mass
saturons can freeze-in below the QCD temperatures.

##  .6 $SO(N)$ examples

For purely illustrative purposes, we shall give two examples of saturon states
realized in theories with large symmetries such as $SO(N)$ (or $SU(N)$). They
shall emerge as the bound states of solitonic and non-solitonic nature
respectively. Many variations of the models are possible.

###  .6.1 Vacuum bubble saturon

As the first example, we construct a saturon as a vacuum bubble in $SO(N)$
theory. For this, we slightly modify the model of Dvali:2020wqi ;
Dvali:2021tez , using the same basic mechanism for achieving the entropy
saturation of the bubble. The key idea Dvali:2019jjw ; Dvali:2019ulr ;
Dvali:2020wqi  is to have solitons with spontaneously broken large global
symmetry in their interior. The general conclusions are not limited to a
particular model. In Dvali:2020wqi ; Dvali:2021tez  these solitons were chosen
in form of the vacuum bubbles, interpolating between the unbroken and broken
phases.

We shall construct a similar model with a slightly simplified representation
content. The model consists of two real scalar fields: $\phi_{j}$,
transforming as $N$-dimensional representation under $SO(N)$ symmetry (with
$j=1,2,...,N$, the $SO(N)$ index) and a singlet $\chi$. The Lagrangian has the
following form,

$\displaystyle
L\,=\,\frac{1}{2}\left(\partial_{\mu}\phi_{j}\right)^{2}\,+\,\frac{1}{2}\left(\partial_{\mu}\chi\right)^{2}\,-$
$\displaystyle-\,\frac{\alpha}{4}\left(f_{G}^{2}-\phi_{j}^{2}-\chi^{2}\right)^{2}-\frac{\alpha}{2}\phi_{j}^{2}\chi^{2}\,.$
(46)

where $\alpha$ is a coupling and $f_{G}$ is a scale. In addition to $SO(N)$,
the theory also possesses a discrete $z_{2}$ symmetry under which
$\chi\rightarrow-\chi$. This symmetry is not essential for our considerations
and can be explicitly broken by additional terms.

Notice that unitarity forces the following bound on the parameters of the
theory Dvali:2019jjw ; Dvali:2019ulr ; Dvali:2020wqi ,

$\alpha N\lesssim 1\,.$ (47)

The product $(N\alpha)$ plays the role analogous to ’t Hooft coupling planar .
For values exceeding the bound (47), the $\phi_{j}$ a $\chi$ fields are no
longer the valid degrees of freedom. In particular, this is signalled by the
breakdown of the loop expansion in power series in $(N\alpha)$. Any discussion
of objects obtained as trustable solutions in terms of $\phi_{j}$ and $\chi$
fields, is bounded by (47). This fact plays a fundamental role in limiting the
entropy of such objects Dvali:2019jjw ; Dvali:2019ulr ; Dvali:2020wqi .

The theory has the two sets of degenerate vacua. One is the vacuum with
unbroken $SO(N)$-symmetry, $\phi_{j}=0,\chi=f_{G}$. This vacuum is doubly
degenerate with respect to the $z_{2}$ symmetry. In this vacuum the theory
exhibits a mass gap of order $m\equiv\sqrt{\alpha}f_{G}$ for all the fields.

In addition, there exists a vacuum $\phi_{i}^{2}=f_{G}^{2},\chi=0$, in which
$SO(N)$ is spontaneously broken down to $SO(N-1)$. Correspondingly, the
spectrum includes $N-1$ massless Goldstone bosons with the decay constants
given by $f_{G}$. In addition, the vacuum houses two fields with masses $m$:
the $\chi$-boson and a radial mode of the $\phi_{j}$-field.

Other extrema of the potential are: the maximum at $\phi_{j}=\chi=0$ and the
saddle points at $\phi_{i}^{2}=\chi^{2}=f_{G}^{2}/3$ which connect the vacua
with $SO(N)$ and $SO(N-1)\times z_{2}$ symmetries.

Following Dvali:2020wqi ; Dvali:2021tez , the saturon solution can be obtained
from the bubble of an $SO(N-1)$-invariant vacuum embedded in $SO(N)$-invariant
one. In order to understand the structure, we first consider a a spherical
bubble in a thin-wall regime. This is the regime when the radius of the
bubble, $R$, is much larger than its thickness $\sim 1/m$. We remark
preemptively that for the saturated bubbles one has $R\sim 1/m$.
Correspondingly, the thin-wall approximation only works as an order of
magnitude estimate. Nevertheless, extrapolation from this regime is useful for
understanding the saturon properties.

For a thin-wall bubble the absolute values are represented by the functions

$\sqrt{\phi_{i}^{2}}=\rho(r,t),~{}|\chi|=\rho_{\chi}(r,t)\,,$ (48)

where $r$ is a radial coordinate. The function $\rho(r)$ interpolates from
$\rho(0)\simeq f_{G}$ to $\rho(\infty)=0$ over an interval $R$, which
determines the size of the bubble. In fact, for the thin-wall bubble, $R\gg
1/m$, the transition mostly happens within the bubble-wall which has a
thickness $\sim 1/m$.

The function $\rho_{\chi}(r,t)$ has the opposite asymptotic behaviour,
$\rho_{\chi}(0)\simeq 0,~{}\rho_{\chi}(\infty)=f_{G}$, interpolating from
almost a zero value at the bubble center to one of the $z_{2}$-degenerate
vacua at $r=\infty$. Also here, the transition takes place within the bubble
wall.

Depending whether it carries an overall $SO(N)$ charge, the bubble can be
either stationary or oscillatory. In both cases the lifetime is macroscopic in
$N$. In both cases, the bubble exhibits a large degeneracy, due to the
spontaneous breaking of the $SO(N)$-symmetry in its interior. Obviously, due
to $SO(N)$-symmetry of the theory, any bubble obtained by the
$SO(N)$-transformation of the solution, is also a solution. This results in a
microstate degeneracy. The bubbles that saturate the entropy bound exist in
the regime $\alpha N\sim 1$, which according to (47) is also an unitarity
bound on the description in terms of degrees of freedom $\phi_{j}$ and $\chi$.

From the analysis of Dvali:2020wqi ; Dvali:2021tez , it is easy to see that
for a saturon bubble, which is the main focus of our interest, $m\sim 1/R$.
The occupation number $n$ of its constituents is

$n\simeq N\simeq\frac{1}{\alpha}\,.$ (49)

The microstate degeneracy of a bubble is given by a binomial coefficient,

$n_{st}\simeq\frac{(n+N)!}{n!}{N!}\sim\left(1+\frac{N}{n}\right)^{N}\left(1+\frac{n}{N}\right)^{n}\,,$
(50)

where we have used Stirling approximation for large $n$ and $N$. At the
saturation point (49) the entropy is

$S_{\rm sat}\simeq N\simeq\frac{1}{\alpha}\,.$ (51)

The saturated bubble represents a bound state of $n\sim N$ quanta of
wavelengths $R\sim m^{-1}$. It is clear that the bubble satisfies all the
properties (2), (3), (4), as well as the expressions for the saturon entropy,
(6), (7), (8).

As shown in Dvali:2021tez , there exist bubbles for which the functions
$\rho(r),\rho_{\chi}(r)$ are time-independent.

In the present model, for a stationary bubble, the field $\phi$ has the
following form,

$\phi\,=\,\hat{O}^{\dagger}\Phi\,,$ (52)

where

$\Phi\,=\,\rho(r)\big{[}1,0,\dotsc,0\big{]}\,,$ (53)

and

$\hat{O}\,=\,\exp\mspace{-3.0mu}\big{[}i\,\omega t\,\hat{T}\big{]}\,.$ (54)

Here $\omega$ is the frequency and $\hat{T}$ is one of the broken generators
of $SO(N)$. Thus, the bubble spins, but only in an internal $SO(N)$-space.

The functions $\rho(r)$ and $\rho_{\chi}(r)$ again have the opposite
asymptotics, $\rho(0)\sim f_{G}\,~{}\,\rho(\infty)\,=\,0$ and
$\rho_{\chi}(0)\,<\,f_{G}\,~{}\,\rho_{\chi}(\infty)\,=\,f_{G}$. Notice that
$\rho(0)\,=\,f_{G}$ and $\rho_{\chi}(0)=0$ are realized only for $R=\infty$
limit.

The classical stability of the solution is due to the fact that it is
energetically favourable to store the $SO(N)$ charge in form of a bubble
rather than in form of free particles. In this sense, the bubble is similar to
a $Q$-ball Lee:1991ax ; Coleman:1985ki . However, it carries a maximal
microstate entropy which makes all the difference in the present context.

From the analysis of Dvali:2020wqi ; Dvali:2021tez , it is easy to see that
for an internally-spinning saturon bubble (52), $\omega\sim m\sim 1/R$ and the
occupation number $n$ of its constituents satisfies (49). Correspondingly, its
entropy is given by (51).

###  .6.2 Example of scalar force bound states

In our second example, the spontaneous breaking of global $SO(N)$-symmetry is
not required. Instead, the bound state is formed due to an attractive force
mediated by a scalar $\pi$. The Lagrangian is,

$\displaystyle
L=\frac{1}{2}\left(\partial_{\mu}\phi_{j}\right)^{2}\,+\,\frac{1}{2}\left(\partial_{\mu}\pi\right)^{2}-\frac{1}{2}m_{\pi}^{2}\pi^{2}\,-$
$\displaystyle-\frac{1}{2}m^{2}\phi_{j}^{2}\,-\,\sqrt{\alpha}\,m\,\pi\phi_{j}^{2}\,+\,...\,,$
(55)

where again $\alpha$ is the coupling and $m,m_{\pi}$ are masses. We assume
that the energy is bounded from below due to the higher order non-linear
terms, not shown explicitly. As in the previous example, the validity domain
of the description is bounded by (47).

The scalar $\pi$ mediates an attractive Yukawa type force with strength
$\alpha$. The attractive potential energy between two non-relativistic
$\phi_{j}$ particles is

$V_{\pi}(r)=-\alpha\frac{{\rm e}^{-m_{\pi}r}}{r}\,.$ (56)

For $m_{\pi}\lesssim m$, this force can produce a bound state of many non-
relativistic $\phi_{j}$ quanta. The size of the bound state is given by $R\sim
1/m$, whereas the occupation number of quanta in the bound state is

$n\simeq 1/\alpha\,.$ (57)

This can be seen by balancing, at distance $r\sim 1/m$, the kinetic energy of
each quantum of de Broglie wavelength $\simeq 1/m$, with the attractive
potential energy from the rest,

$\displaystyle(E_{kin}\simeq m)\,\sim\,(E_{pot}\simeq(n\alpha)m)\,.$ (58)

This gives (57). The mass of the bound state is,

$\displaystyle E_{\rm sat}\sim mn\,\sim\frac{m}{\alpha}\,.$ (59)

Notice that such a bound state exists even for a single $j$-flavor of
$\phi_{j}$, i.e., for $N=1$.

However, the existence of many species creates an exponentially large
microstate degeneracy. Due to $SO(N)$-symmetry, the bound state has the same
mass for arbitrary set of flavors. For fixed $n$ and $\alpha$, the degeneracy
increases exponentially with $N$. For example, a symmetrized state of $n$
quanta, has a degeneracy given by (50).

As in the previous example, due to unitarity bound (47) and relation between
the occupation number and coupling (57), the maximal entropy of the bound
state is reached for (49) and is given by (51). For this choice of parameters,
the bound state becomes a saturon. In a sense, the bound state is similar to
the atomic nuclei in a theory of large number of nucleon flavors.

As in the previous example, the degeneracy can be understood in terms of
Nambu-Goldstone phenomenon of spontaneously broken $SO(N)$ symmetry by the
saturon bound state.

As already discussed, in general, the saturon bound states are macroscopically
long lived. They can also be exactly stable. For the bound states transforming
non-trivially under $SO(N)$, the stability can be understood by the
conservation of $SO(N)$-charge, which costs less energy in the bound state as
compared to the state of free quanta.

However, even $SO(N)$ invariant bound states, can be very long-lasting. The
long timescale of quantum decay is a general feature of the saturated bound
states of $N$ quanta. The first reason is a highly suppressed rate of re-
scattering (see, Dvali:2020wqi ; Dvali:2021tez ). In addition, the decay is
slowed down by the memory burden effect Dvali:2018xpy ; Dvali:2020wft . As a
result, even saturons carrying no conserved charges can be sufficiently long-
lived for being dark matter. We shall comment more on this later.

##  .7 General bounds

We wish to study the above examples of $SO(N)$-saturons in cosmological
context. However, since they include a large number $N$ of particle species,
we must take into account some general constraints.

###  .7.1 Species bound

First, we must take into account that the gravitational cutoff is lowered to a
so-called “species” scale Dvali:2007hz ,

$M_{\rm sp}\equiv\frac{M_{P}}{\sqrt{N}}\,.$ (60)

This constraint is imposed by black hole physics and thereby is fully non-
perturbative in nature. In particular, it cannot be avoided by any sort of a
fine-tuning of radiative corrections or by a resummation of perturbative
series. Notice that due to low cutoff, the theories with large $N$ are
motivated by the Hierarchy Problem Dvali:2007hz .

For the present purposes, we must keep in mind that the species scale $M_{\rm
sp}$ limits from above the temperature of the Universe.

It is important to notice that although the scale (60) represents an upper
bound on the masses of elementary particles, it does not restrict the masses
of composite objects such as saturons or black holes. Correspondingly, even
the superheavy saturons can be produced at very low temperatures.

This is one of the features that makes the present case very different from
the particle dark matter scenarios with many species such as discussed, e.g.,
in Dvali:2009fw  or in Cohen:2018cnq .

###  .7.2 Bounds from BBN

Other important bounds on the parameters of the $SO(N)$ sector come from the
big bang nucleosynthesis (BBN).

First, the abundance of new species during BBN must be negligible. Secondly,
the BBN epoch must be within a so-called “normalcy” temperature. This concept
was introduced in Arkani-Hamed:1998sfv  in the contexts of cosmological
production of Kaluza-Klein graviton species in the theory of large extra
dimensions Arkani-Hamed:1998jmv .

The essence of this constraint is that in the BBN epoch, the thermal bath must
cool-down predominantly due to the expansion of the Universe. In the presence
of the additional light species, a new cooling channel opens up. The bath can
cool-down via re-scatterings of the ordinary particles into the new species.
In order not to affect the successful predictions of BBN, the rate of such a
process must be less than the Hubble expansion rate.

If we denote the coupling between the Standard Model and $SO(N)$ species by
$\alpha_{N}$, the normalcy constraint on the temperature reads,

$T\alpha_{N}^{2}N\,\lesssim\,\sqrt{N_{*}}\frac{T^{2}}{M_{P}}\,.$ (61)

Notice that unitarity puts a constraint similar to (47) on the coupling,
$\alpha_{N}N<1$. For the coupling at its upper bound, the normalcy temperature
must satisfy,

$T\,\gtrsim\,\frac{M_{P}}{N\sqrt{N_{*}}}\,.$ (62)

Taking $T$ of order the BBN temperature, $\sim$ MeV, we get the following
bound on $N$ for species with $m<T$,

$N\,\gtrsim\,10^{22}\,.$ (63)

Of course, for $m\gg$ MeV, there is no BBN constraint on $N$, since the
production of species is exponentially suppressed.

###  .7.3 Star cooling bounds

Another potential constraint on the light particle species comes from the
processes of star cooling. Of course, this constraint only applies if the
masses of new species are lower than the temperature in a star. For a star of
core temperature $T$, the cooling rate due to emission of species is given by
the left hand side of (61). In each particular case we must make sure that
this rate is much less than the rate of ordinary cooling induced by the
emission of the Standard Model quanta such as the neutrino.

##  .8 Thermal production of $SO(N)$-saturons

We now estimate the rate of a thermal production of $SO(N)$ saturons. We start
with a freeze-out scenario.

###  .8.1 Freeze-out

We consider a thermal bath of particles formed over the vacuum with unbroken
$SO(N)$-symmetry. This applies to both examples of saturons. We assume that
the temperature is $T\sim m$, and that the particles are still in approximate
thermal equilibrium. In theory (.6.1) the vacuum bubbles can form by the
Kibble mechanism, provided the Universe start out at temperatures $T\gg m$
above the symmetry restoration point. We are not interested in such a
scenario. Instead, we shall study the saturon formation directly from the
thermal bath as a quantum transition process.

Since we are working at large $N$, for definiteness, let us assume that the
total number of species is dominated by the saturon constituents, i.e.,

$N_{*}\simeq N\,.$ (64)

Then, the entropy and the energy of the thermal bubble of radius $R$ are given
by (12) with $N_{*}=N$,

$S_{\rm rad}(T)=N(RT)^{3}\,,~{}~{}~{}\,E_{\rm rad}(T)=NT^{4}R^{3}\,.$ (65)

Since we also have $T\sim 1/R\sim m$, the condition (14) of the equality
between the energies of the saturon bubble and the corresponding thermal
sphere, automatically implies the equality of their entropies,

$S_{\rm sat}=S_{\rm rad}=N\,.$ (66)

Taking into account (66) and the general formula (40), we get that $N$ and $T$
must satisfy,

$N\,\lesssim\,\left(\frac{M_{P}}{T}\right)^{\frac{2}{5}}\,.$ (67)

Correspondingly, we get the following bound on the saturon energy

$~{}E_{\rm sat}\,\lesssim\,T^{\frac{3}{5}}M_{P}^{\frac{2}{5}}\,.$ (68)

The thermal production of saturons stops below the temperature $T\sim 1/R\sim
m$ and the saturon number quickly freezes out.

###  .8.2 Freeze-in

We now discuss a freeze-in scenario. In this case we assume that saturon
constituents are not in thermal equilibrium. Thereby, their re-scattering rate
must be lower than the Hubble rate. According to (44), this implies,

$N\,\gtrsim\,\frac{1}{N_{*}^{\frac{1}{4}}}\left(\frac{M_{P}}{T}\right)^{\frac{1}{2}}\,.$
(69)

Expressing via (66) the saturon entropy through $N$, from general formula (23)
we get the following expression for saturon energy,

$E_{\rm sat}\,\sim\,TN^{\frac{3}{4}}N_{*}^{\frac{1}{4}}\,.$ (70)

The bound (69) translates as the lower bound on this energy,

$E_{\rm
sat}\,\gtrsim\,N_{*}^{\frac{1}{16}}T^{\frac{5}{8}}M_{P}^{\frac{3}{8}}\,.$ (71)

Assuming that to the leading order the probability suppression is
$p\sim\beta$, and applying the equation (36) to the present case, we fix $N$
as,

$N\,\sim\,10^{22}N_{*}^{\frac{3}{5}}\,.$ (72)

###  .8.3 Examples

Example 1

Let us take the optimal temperature $T\sim$ TeV and assume that only the
Standard Model species are in equilibrium. This implies $N_{*}\sim 100$. Then,
from (72) we get, $N\sim 10^{23}$, the saturon size, $R\sim 10^{-12}$cm and
energy $E_{\rm sat}\sim 10^{20}$ GeV.

Notice that the population of free species in $SO(N)$ sector is negligible.
Their production rate remains subdominant to the Hubble rate throughout the
cosmological history. The $SO(N)$-sector contributes into the energy density
of the Universe in form of the saturon bound states that constitute dark
matter.

Example 2

Let us find out a phenomenologically acceptable maximal value of $N$. This is
achieved by maximizing the number of species $N_{*}$ that are in thermal
equilibrium at the freeze-in temperature. Of course, these species cannot
belong to $SO(N)$ sector. They can remain in thermal equilibrium latest by BBN
and must decouple before this epoch. We shall therefore assume that their
masses are $m_{*}\sim$MeV. Due to the unitarity constraint analogous to (47),
their coupling is bounded from above by $\alpha_{*}\sim 1/N_{*}$. Since, by
assumption, these species are in thermal equilibrium, their scattering rate
must exceed the Hubble rate prior to BBN, which puts a constraint,

$N_{*}\,\lesssim\,(M_{P}/T_{BBN})^{2/5}\sim 10^{9}\,.$ (73)

Taking $N_{*}\sim 10^{9}$, from (72) we obtain $N\sim 10^{28}$. The
corresponding mass of the saturon (70) is,

$E_{\rm sat}\,\sim\,10^{20}{\rm GeV}\,.$ (74)

Notice that this saturon is a macroscopic object of the size

$R\,\sim\,10^{-6}{\rm cm}\,,$ (75)

which exceeds its own Compton wavelength by $28$ orders of magnitude. This is
due to the fact that it represents a bound state of $N\sim 10^{28}$ particles.

This scenario barely satisfies the star-cooling constraint. However, the
question about extra $N_{*}$ species must be addressed. They drop out of
thermal equilibrium before BBN. If stable, they will contribute into dark
matter density together with saturons.

Notice that the above scenario is fully consistent with the species bound
(60). For $N\sim 10^{28}$, the gravitational cutoff is about $M_{\rm sp}\sim
100$TeV. Correspondingly, the freeze-in temperature somewhere above or around
BBN, is safely below it. As already discussed Dvali:2007hz , due to a low
cutoff, the theories with many species have an independent motivation from the
point of view of the Hierarchy Problem. In this class of theories, the
possibility of saturon dark matter comes as an extra bonus.

##  .9 Comment on inner entanglement

As we have already discussed, in both $SO(N)$ examples the stability of
certain saturons is due to $SO(N)$ charge. A saturon, which represents a bound
state of $\phi$-quanta of occupation number $n\sim N$, forms a representation
of the $SO(N)$ group. Due to the binding potential, the saturon mass is less
than the energy of free $\phi$-quanta that match the $SO(N)$ quantum numbers
of the saturon. Basically, a saturon is stabilized by the $SO(N)$-charge in a
way similar to a stabilization of ordinary structures, such as nuclei or
planets, by a conserved baryon number. The new feature is that the microstate
degeneracy of the object saturates the bounds (6) - (8).

Due to this property, even a system of saturons with opposite quantum numbers
is macroscopically long lived. That is, a system consisting of a saturon and
an anti-saturon placed on top of each other, decays very slowly. This is due
to $1/N$-suppression of the annihilation process.

The energy of a saturon-anti-saturon pair is by $\Delta E\sim nm$ lower than
the energy of $2n$ free quanta. Correspondingly, the decay products must
contain by $\sim n$ less free $\phi$-particles than the initial number of
constituents, which is equal to $2n$. That is, during the decay process, the
number of quanta must be reduced by $\sim n$.

However, an explosive annihilation of large number of constituents into a
small number of quanta, a so-called “quantumization” process Dvali:2022vzz ,
is exponentially suppressed. Notice that the suppression of quantum transition
matrix elements between the states of large and small numbers of particles can
be derived from a very general argument Dvali:2020wqi . Correspondingly, the
dominant process is a gradual decay due to annihilation/scattering of small
number of constituent quanta into the asymptotic ones Dvali:2011aa ;
Dvali:2013eja ; Dvali:2017eba ; Dvali:2017ruz ; Dvali:2021rlf ; Dvali:2021tez
; Dvali:2022vzz .

During the decay process, the saturons are subject to generation of an inner
entanglement. Originally this effect was pointed out within the quantum
$N$-portrait resolution of black holes and of de Sitter state as composites of
$N$ gravitons Dvali:2013eja ; Dvali:2017eba . However, the inner entanglement
takes place in generic objects with enhanced capacity of information storage
(enhanced microstate entropy) and it goes hand in hand with the memory burden
effect Dvali:2018xpy ; Dvali:2020wft ; Dvali:2021bsy ; Dvali:2022vzz .

In the present case, this is due to the internal rescattering of
$\phi$-quanta, without leaving the bound state. For example, a pair of quanta
of a flavor $j$ scatters into a two-particle state entangled with respect to
the flavor index,

$\left|\,\phi_{j},\phi_{j}^{\prime}\right>\,\rightarrow\,\frac{1}{N}\sum_{l}\left|\,\phi_{l},\phi_{l}^{\prime}\right>\,,$
(76)

where prime indicates the distinction with respect to other quantum numbers
such as the angular momentum. This process contributes into the generation of
an inner entanglement which reaches its maximum approximately after time $\sim
NR$. Beyond this point, the system represents a maximally entangled
$n$-particle state. Not even approximately, such a state can be treated semi-
classically.

##  .10 Comparison with black holes

It is illuminating to compare the non-gravitational saturons to black holes.
The key difference is that a quantum production of a black hole from a
homogeneous thermal bath is strongly suppressed. Instead, for a cosmological
black hole production, it is necessary that some other mechanism prepares
large density inhomogeneities which then collapse into black holes
Escriva:2022duf .

This difference may sound somewhat surprising, since a black hole represents a
particular case of a saturon. The specifics is that it is a bound state
produces by the gravitational interaction. Interestingly, this property makes
a black hole production from a radiation thermal bath highly improbable.

The “secret” is that, unlike other saturons, in case of a black hole one and
the same interaction (gravity) governs the two important time-scales:

1) a would-be quantum transition-time, $t_{tr}$;

and

2) the Hubble expansion time of the Universe.

In order to see this, we apply the previous reasoning to a quantum creation of
a black hole of radius $R$, from the thermal sphere of the same radius, in a
homogeneous radiation dominated Universe. We first assume that $R$ is much
smaller than the Hubble radius $R_{H}$. In this regime, the effects of the
Hubble curvature can be ignored.

For such a black hole, the mass, $M_{\rm BH}$, and the radius, $R$, satisfy
the standard flat-space relation:

$R\,\sim\,\frac{M_{\rm BH}}{M_{P}^{2}}\,.$ (77)

We equate this mass to the initial energy of radiation within the equal-size
sphere,

$M_{\rm BH}\,\sim\,N_{*}T^{4}R^{3}\,.$ (78)

Next, dividing both sides of the equation by $M_{P}^{2}$ and taking into
account the relation (77), we get,

$R^{2}\,\sim\,\frac{M_{P}^{2}}{N_{*}T^{4}}\,\sim\,R_{H}^{2}\,.$ (79)

The scale $R_{H}$ is the Hubble radius.

Thus, the minimal radius of a thermal sphere that matches the energy of the
same size black hole, is the Hubble radius. The ratio of black hole to
radiation entropy is,

$\frac{S_{\rm BH}}{S_{\rm rad}}\,\sim\,\frac{M_{P}}{\sqrt{N_{*}}T}\,.$ (80)

Since, the temperature is always less than the species scale (60), this number
is larger than one. Thus, the entropy of a Hubble size black hole is always
higher than the entropy of radiation.

However, for a Hubble size black hole the flat space analysis breaks down,
since the Hubble expansion must be taken into account. This expansion prevents
the black hole formation. This is due to the following two factors.

First, a would-be quantum transition time, $t_{tr}\sim R$, is equal to the
Hubble time. Thereby, during this time the expansion of the Universe dilutes
the energy within the initial Hubble sphere.

Secondly, the Hubble patch has the Gibbons-Hawking entropy Gibbons:1977mu ,

$S_{\rm GH}\,\sim\,(R_{H}M_{P})^{2}\,.$ (81)

Notice that this entropy is equal to the Bekenstein-Hawking entropy (10) of a
flat space black hole of radius $R=R_{H}$. However, we are not in a flat space
and a would-be black hole is substituted by the Hubble sphere. That is, in an
expanding Universe, the most entropic state of the size of the Hubble horizon
is the Hubble patch itself.

The system, therefore, has no incentive for creating any other state, since
the saturon in form of the Hubble patch is already present. In other words,
the dominant fraction of the Hilbert space is taken up by the microstates of
the Hubble patch. As a result, no black hole formation takes place.

Let us now consider an alternative possibility. Since black holes come in all
possible sizes, one can think of a quantum process of spontaneous collapse of
a radiation sphere into a sub-horizon black hole. Such processes are
exponentially suppressed. In order to see this, let us consider a quantum
collapse of a spherical region of radius $R\ll R_{H}$ into a black hole of the
same mass. For $R\ll R_{H}$ the radius of an equal mass black hole satisfies
$R_{BH}\ll R$. Its Bekenstein-Hawking entropy is,

$S_{\rm BH}\,\sim\,N_{*}T^{4}R^{3}R_{BH}=S_{\rm rad}\,(R_{BH}T)\,.$ (82)

For $R_{BH}\gg T^{-1}$, this exceeds the initial entropy of the radiation
sphere. Nevertheless, this entropy is not sufficient for compensating the
exponential suppression of the quantum collapse due to the size difference.

The Euclidean action over a most symmetric trajectory leading to such a
collapse can be estimated as $S_{E}\sim N_{*}(TR)^{4}$. This exceeds the black
hole entropy by a factor $\sim R/R_{BH}$. Correspondingly, the excess of the
black hole entropy over the entropy of radiation is insufficient for
compensating the exponential suppression of the quantum collapse. The over-all
suppression factor,

${\rm e}^{-S_{E}+S_{\rm BH}-S_{\rm rad}}\,=\,{\rm e}^{-(\sim S_{\rm
BH}R/R_{BH})}\,,$ (83)

renders the process highly improbable.

Thus, in a homogeneous Universe a radiation thermal bath cannot collapse into
a black hole spontaneously. This goes in contrast with generic saturons. As we
have seen, they can materialize via quantum transitions from a radiation
thermal bath. This difference is due to the following two factors: 1) the
universal nature of the gravitational interaction, and 2) the fact that a non-
gravitational binding force can be much stronger than gravity.
Correspondingly, a saturon that is formed by such a force, can be much lighter
than a same-size black hole. Due to this, it can easily match the size and the
energy of the thermal bath within a sub-horizon sphere. In this respect,
saturons are similar to other extended objects, such as solitons.

At the same time, the saturons share with black holes the property of maximal
entropy, which the ordinary objects do not posses. Therefore, while matching
the energy and the size of a thermal sphere, a saturon dominates the entropy
count.

In other words, saturons inherit those features from both classes of objects
that allow for unsuppressed quantum formation in a radiation dominated
universe.

##  .11 Saturated instantons and enhanced bubble nucleation

So far, we were discussing the saturons defined as objects in space-times with
Lorentzian signature, either at zero or at finite temperatures. However, as
shown in Dvali:2019ulr , this concept fully generalizes to Euclidean field
configurations, such as instantons. Of course, for such entities the notion of
energy is not defined. However, the bounds (6) - (7) do not rely on energy.
Correspondingly, these bounds are well defined for instantons.

The microstate entropy of an instanton is defined according to the degeneracy
of its Euclidean action. As shown Dvali:2019ulr , the saturation of the bounds
(6) - (7) by this entropy, has a trasparent physical meaning.

In particular, the existence of a saturated instanton, an i-saturon, signals
the saturation of unitarity by the corresponding transition process. A
violation of the bounds (6) - (7) by an instanton is not possible as this
would invalidate the respective quantum field theoretic description. For
example, it has been demonstrated that in large-$N$ gauge theories, the
instantons saturate the entropy bound when the theory approaches the regime of
confinement Dvali:2019ulr .

A saturated instanton describes a creation of a saturated object. In short:

a transition generated by an i-saturon creates a saturon.

This effect can have interesting cosmological implications, regardless dark
matter. In particular, it can affects the dynamics of the high temperature
phase transitions. For example, the rate of the bubble nucleation at finite
$T$ can be dramatically enhanced by Euclidean i-saturons, even if no stable
Lorentzian saturon exists in the theory at $T=0$. Instead, it suffices that a
critical bubble at finite $T$ is a saturated state. The existence of such a
bubble goes hand in hand with the existence of a saturated instanton that
creates it.

We illustrate this on explicit example of a thermal nucleation of a saturated
bubble. We consider a saturated vacuum bubble introduced in Dvali:2020wqi .
The model consists of a scalar field $\hat{\phi}$ transforming in adjoint
representation of $SU(N)$-group. The Lagrangian density is given by,

$L=\frac{1}{2}{\rm
Tr}(\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi})-V(\hat{\phi})\,,$ (84)

where the zero temperature scalar potential has the following form,

$V(\phi)\,=\,\frac{\alpha}{2}{\rm
Tr}\left(f\hat{\phi}-(\hat{\phi}^{2}-\frac{\hat{I}}{N}{\rm
Tr}\hat{\phi}^{2})\right)^{2}\,.$ (85)

Here, $\hat{I}$ is the unit $N\times N$ matrix and $\hat{\phi(x)}$ is an
$N\times N$ traceless hermitian matrix. The parameter $\alpha$ is the coupling
and $f$ is the scale. As in the previous large-$N$ examples, the validity of
the description imposes the bound (47) on the ’t Hooft coupling $\alpha N$.

At zero temperature, the theory has a set of degenerate vacua. They consists
of a $SU(N)$-symmetric vacuum, as well as the vacua in which the $SU(N)$
symmetry is spontaneously broken down to various maximal subgroups,
$SU(N-K)\times SU(K)\times U(1)$ with $0<K<N$.

At non-zero temperature, the vacua are no longer degenerate. In particular at
$T\rightarrow\infty$, only the $\hat{\phi}=0$ vacuum is present. We shall
assume that the degeneracy is lifted by the additional terms in a way that,
below some critical temperature, the broken symmetry vacuum is lower than the
unbroken one. In such a case the system can tunnel via a bubble nucleation.

For definiteness, we assume that the order parameter of the transition is
oriented along the following component,

$\hat{\phi}\,=\,\phi(x)\,{\rm
diag}((N-1),-1,....,-1)\frac{1}{\sqrt{N(N-1)}}\,.$ (86)

That is, the transition takes place from $SU(N)$-symmetric vacuum to
$SU(N-1)\times U(1)$-symmetric one.

Then, the temperature dependent potential of the order parameter $\phi$ can be
arranged in the following form,

$V(T,\phi)\,=\,\frac{\alpha(T)}{2}\phi^{2}\left(f(T)-\phi\right)^{2}\,\,-\,\mu^{2}(T)\phi^{2}\,.$
(87)

Of course, the effective parameters $\alpha(T),f(T)$ and $\mu(T)$ depend on
$T$.

The parameter $\mu^{2}>0$ introduces a bias, which lowers the free energy of
the broken symmetry vacuum $\phi\neq 0$ below the symmetric one $\phi=0$. The
thin-wall approximation corresponds to $\mu^{2}\ll m^{2}$, where
$m^{2}\,\equiv\,\alpha\,f^{2}$ is the effective mass of the $\phi$ excitation
at temperature $T$.

The saturated bubble and the corresponding saturated instanton exists for
$\mu\sim m$. Correspondingly, the thin-wall approximation is valid order of
magnitude wise. It therefore makes sense to use it as a starting point and
approach the parameter regime of saturation from this side.

The effective potential (87) has two minima,

$\phi=0\,~{}{\rm
and}~{}\,\phi\,\simeq\,f\left(1+\,\frac{2\mu^{2}}{m^{2}}\right)\,.$ (88)

The energy split between them is

$\epsilon\simeq f^{2}\mu^{2}\,.$ (89)

The transition between the vacua takes place via a bubble nucleation. The
nucleation rate is exponentially suppressed by the Euclidean action of the
instanton $A_{inst}$ that materializes a critical bubble,

$\Gamma(T)\sim{\rm e}^{-A_{inst}}\,.$ (90)

We now estimate this action. We first find the radius $R$ of the critical
bubble, via extremizing $A_{inst}$.

For $T\gg R^{-1}$, the rate of the bubble nucleation is given by the equation
(11). In this regime, the action effectively factorizes Linde:1981zj ,

$A_{inst}\,\rightarrow\,\frac{A_{3}}{T}\,,$ (91)

where the three-dimensional part, $A_{3}$, coincides with the energy of the
static bubble.

In the thin-wall approximation the energy of a static bubble of radius $r$ is
given by,

$A_{3}=E_{bub}=-\frac{4}{3}\pi r^{3}\epsilon\,+\,4\pi r^{2}A_{1}\,,$ (92)

where $A_{1}$ is the tension of the bubble wall. For a thin-wall bubble this
quantity is,

$A_{1}\,=\,\frac{1}{6}f^{3}\sqrt{\alpha}\,=\,\frac{1}{6}\frac{m^{3}}{\alpha}.$
(93)

The radius of the critical bubble, $r=R$, is found by extremizing (92) and it
is given by,

$R\,=\,\frac{2A_{1}}{\epsilon}\,=\,\frac{m}{3\mu^{2}}.$ (94)

Substituting this into (92), we get the following value for the energy of the
critical bubble,

$E_{bub}\,=\,\frac{16\pi
A_{1}^{3}}{3\epsilon^{2}}\,=\,\frac{2\pi}{81}\frac{m^{5}}{\mu^{4}}\frac{1}{\alpha}\,.$
(95)

The action of the corresponding instanton is,

$A_{inst}\,=\,\frac{16\pi
A_{1}^{3}}{3\epsilon^{2}}\,=\,\frac{2\pi}{81}\frac{m^{5}}{\mu^{4}T}\frac{1}{\alpha}$
(96)

So far, we gave the standard analysis of the bubble nucleation along the lines
of Linde:1981zj . However, an important novelty in the present case is that
both the critical bubble as well as the corresponding Euclidean instanton are
exponentially degenerate. This is due to the presence of $\sim N$ Goldstone
modes of spontaneously broken $SU(N)$ symmetry. This symmetry is spontaneously
broken both by the bubble as well as by the related instanton.

Now, the critical bubble becomes a saturon for $N\sim 1/\alpha$ and $R\sim
1/m$. Since species are in equilibrium, this also implies $T\sim m\sim\mu$.
Extrapolating the thin-wall expressions (95) and (96) to this regime, we see
that, simultaneously with the critical bubble, the entropy bounds (6) - (8)
are saturated by the Euclidean instanton.

At the saturation point the action of the instanton becomes equal to its
microstate entropy.

$A_{inst}=S_{inst}\sim\frac{1}{\alpha}$ (97)

The instanton entropy provides an additional exponential enhancement factor in
the transition rate,

$\Gamma(T)\sim{\rm e}^{-A_{inst}+S_{inst}}\,.$ (98)

At the saturation point, this factor compensates the exponential suppression
due to the action $A_{inst}$. At this point the transition probability becomes
unsuppressed. The result of the transition is the creation of the saturon
bubble. That is, the Euclidean instanton and the critical bubble that it
materializes, become saturated simultaneously.

Finally, we note that the bubble nucleation process preserves the translation
invariance of the thermal bath. This is regardless of saturation of the
bubbles and of corresponding instantons. The bubbles are produced in a
translation-invariant quantum superposition. The state is classicalized by
inhomogeneities.

##  .12 Summary and outlook

The purpose of the present paper is to create awareness about certain
potentially-interesting cosmological implications of saturons Dvali:2020wqi .
These macroscopic objects can come in various forms of bound states such as
solitons, baryons and other creatures Dvali:2020wqi ; Dvali:2019jjw ;
Dvali:2019ulr ; Dvali:2021jto ; Dvali:2021rlf ; Dvali:2021tez ; Dvali:2021ofp
. The defining property is that saturons carry a maximal microstate entropy
within the validity domain of the quantum field theoretic description. This
feature is quantified by saturation of the bounds (6) -(8).

Due to this property, saturons can be produced in the early Universe via a
quantum transition process from the homogeneous thermal bath. This production
mechanism is not operative for other macroscopic objects, such as black holes
or ordinary solitons, albeit due to distinct reasons.

The thermal production of ordinary solitons is exponentially suppressed. An
example is provided by the suppressed bubble nucleation at finite temperature
Linde:1981zj . We showed that this effect can be understood in terms of an
universal phenomenon of the entropy suppression which applies to generic
macroscopic objects such as solitons. The entropy of an ordinary soliton
(e.g., the vacuum bubble of the previous paper) is typically very low as
compared to the entropy of a thermal bath of equal volume and energy. It is
therefore insufficient for overcoming the exponential suppression of the
process.

The story with black holes is different. The Bekenstein-Hawking entropy of a
black hole (10) BekE  does saturate the bounds (6) - (8). Due to this and
other fundamental similarities, it is clear that black holes represent a
special category of saturons Dvali:2020wqi . In fact, a black hole is a
saturon that is predominantly composed by $N$ gravitons that are held together
by gravity Dvali:2011aa .

Despite this similarity, the thermal formation of black holes in a homogeneous
radiation dominated Universe is highly suppressed. This is because, the very
same gravitational interaction that keeps black hole in tact, is at the same
time responsible for the cosmological evolution of the Universe. Due to this,
a would-be black hole matches the energy and size of the thermal radiation
only if it covers the entire Hubble patch. However, at this scale the entropy
is fully dominated by the Gibbons-Hawking entropy (81) Gibbons:1977mu  of the
Hubble patch. Correspondingly, no black hole forms.

In contrast, the non-gravitational saturons can bypass both difficulties. On
one hand, just like black holes, they carry the maximal entropy. On the other
hand, they can match the energy and entropy of the radiation sphere at sub-
Horizon scales. Thereby, saturons can be created as a result of a quantum
transition directly from a homogeneous thermal bath.

Due to an efficient production in the radiation epoch, saturons can serve as
viable dark matter candidates. We have considered two versions of their
cosmological production, referred to as “freeze-out” and “freeze-in”
scenarios. The terminology, which we borrowed from ordinary particle dark
matter discussions, refers to the initial level of thermal equilibration of
the species that after transition compose a saturon. The rest of physics is
fundamentally different, since saturons are collective phenomena that never
come to thermal equilibrium. We gave qualitative discussions of both scenarios
within explicit models of saturons.

The maximal entropy endows saturons with some out of ordinary properties. For
example, the superheavy macroscopic saturons can be created at very law
temperatures. Their interactions and mergers can create the potentially
observable signatures such as gravitational waves.

It has been shown Dvali:2019ulr  that the concept of entropy saturation
generalizes to Euclidean configurations such as instantons. The entropy bounds
(6)-(8) are well-defined for such entities. The instanton entropy emerges from
a microstate degeneracy of its action. For the saturated instanton, i-saturon,
the entropy and action are equal. The instanton entropy provides an additional
enhancement factor in the transition rate which compensates the suppression
due to the action. Thereby, the saturation marks the regime in which the rate
becomes unsuppressed. Its further increase is not possible, due to
invalidation of the quantum field theoretic description.

We have shown that an unsuppressed thermal nucleation of a saturon bubble can
be understood in terms of a saturated instanton. That is, the saturon vacuum
bubble is created by a saturated thermal instanton. In other words, existence
of a saturon bubble is correlated with the existence of an Euclidean
i-saturon. This is the case even if no stable saturon exists at $T=0$. This
effect can have important implications for various phase transitions. In
particular, it can affect the thermal history of the Universe. It also applies
to creation of defect during inflation, discussed by Basu, Guth and Vilenkin
inflation . The effect of saturation, must enhance the rate of such
transitions.

It is known that for certain parameter regimes the spontaneously broken
symmetries are never restored at high temperatures Weinberg:1974hy ;
Mohapatra:1979qt ; Mohapatra:1979vr . Instead of vanishing above a certain
critical point, the order parameter grows with increasing temperature. In such
cosmologies, due to the absence of phase transitions, the topological defects
cannot form via the Kibble mechanism. Hence, their quantum production from the
thermal bath remains the only channel, which for ordinary defects is
exponentially suppressed. Based on this, in Dvali:1995cc ; Dvali:1996zr  and
Dvali:1995cj  it was argued that symmetry non-restoration eliminates the
cosmological problems related with domain walls Zeldovich:1974uw  and
monopoles Mon1 ; Mon2 .

As we discussed, saturons avoid the suppression of thermal production, thanks
to their entropy. Due to this, the saturated defects, such as, e.g., magnetic
monopoles or the vacuum bubbles, can be produced thermally even in the absence
of the Kibble mechanism. This can reintroduce the monopole problem in
cosmological scenarios with symmetry non-restoration.

A related interesting question is whether saturated cosmic strings of
cosmological importance can be produced via the thermal quantum transition
described in this paper.

It has been argued recently Dvali:2021ooc  that QCD contains a saturon state
in form of the Color Glass Condensate Gelis:2010nm . The present analysis
indicates that such a state must have been formed in the early Universe.
Cosmological implications of it remain to be understood.

Other potentially observable signatures of saturons concern the collider
experiments. Due to their maximal microstate entropy, saturons with masses up
to few TeV can be produced in high energy collisions at particle accelerators
such as LHC. Again, this distinguishes them from ordinary solitons with low
entropy which are subject to an exponential suppression. This distinction also
enables for direct collider searches for certain categories of saturon dark
matter. Some discussions of the generic properties of saturon production in
particle collisions can be found in Dvali:2020wqi .

The present paper gives some glimpses of saturon cosmology. The discussed
examples are indicative and the estimates are qualitative, often relying on
intuitive arguments. Nevertheless, the picture appears to be physically sound
and interesting from the cosmological perspective.

Acknowledgments — We thank Akaki Rusetsky for useful discussions on aspects of
large-$N$ and Goran Senjanović for discussions on symmetry non-restoration.
This work was supported in part by the Humboldt Foundation under Humboldt
Professorship Award, by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2111 -
390814868, and Germany’s Excellence Strategy under Excellence Cluster Origins.

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