# Memory Burden Effect in Black Holes and Solitons:
Implications for PBH

Gia Dvali  Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany  Max-Planck-Institut für Physik,
Boltzmannstrasse 8, 85748 Garching, Germany  Juan Sebastián Valbuena-Bermúdez
ID<EMAIL_ADDRESS>Institut de Física d’Altes Energies (IFAE) and The
Barcelona Institute of Science and Technology (BIST), Campus UAB, 08193
Bellaterra (Barcelona), Spain  Michael Zantedeschi ID
<EMAIL_ADDRESS>Tsung-Dao Lee Institute and School of Physics and
Astronomy, Shanghai Jiao Tong University, Shengrong Road 520, 201210 Shanghai,
China

###### Abstract

The essence of the memory burden effect is that a load of information carried
by a system stabilizes it. This universal effect is especially prominent in
systems with a high capacity of information storage, such as black holes and
other objects with maximal microstate degeneracy, the entities universally
referred to as saturons. The phenomenon has several implications. The memory
burden effect suppresses a further decay of a black hole, the latest, after it
has emitted about half of its initial mass. As a consequence, the light
primordial black holes (PBHs), that previously were assumed to be fully
evaporated, are expected to be present as viable dark matter candidates. In
the present paper, we deepen the understanding of the memory burden effect. We
first identify various memory burden regimes in generic Hamiltonian systems
and then establish a precise correspondence in solitons and in black holes. We
make transparent, at a microscopic level, the fundamental differences between
the stabilization by a quantum memory burden versus the stabilization by a
long-range classical hair due to a spin or an electric charge. We identify
certain new features of potential observational interest, such as the model-
independent spread of the stabilized masses of initially degenerate PBHs.

##  I Introduction

The phenomenon of memory burden, originally described in Dvali (2018a), is
summarized in the following statement:

Information loaded in a system resists its decay.

Naturally, the effect is especially sound in systems with enhanced information
storage capacity. This capacity can be quantified by the number and degeneracy
of microstates that the system possesses for the given values of macroscopic
parameters, such as, e.g., the radius $\displaystyle R$ and the total energy.

Black holes are the prominent representatives of this category. This is
obvious from Bekenstein-Hawking entropy Bekenstein (1973),

$S_{\rm BH}\,=\,\pi R^{2}M_{\rm P}^{2}\,,$ (1)

where $\displaystyle M_{\rm P}$ is the Planck mass. Correspondingly, it was
suggested in Dvali (2018a); Dvali  _et al._ (2020) that the phenomenon of
memory burden must be applicable to black holes.

This effect explains, in terms of an explicit microscopic mechanism, why, at
the early stages of Hawking’s decay, the information stored in a black hole
cannot be released together with radiation. This matches the semi-classical
expectation.

However, an important new feature emerges. The internally maintained
information back-reacts and creates resistance against the decay of a black
hole. This is the effect of the memory burden phenomenon in a black hole.

Furthermore, by performing a detailed analysis of the prototype systems, it
was concluded in Dvali  _et al._ (2020) that not only do black holes undergo
the memory burden effect, but they likely are stabilized by it. That is, in
the process of a black hole decay, the memory burden grows, and after a
certain characteristic time, $\displaystyle t_{\rm M}$, reaches a critical
value. $\displaystyle t_{\rm M}$ is bounded from above by the age of a black
hole that lost about half of its initial mass.

At this point, the black hole has evolved into a “remnant” that cannot
continue an ordinary quantum decay. A remarkable feature is that the remnant
is macroscopic, with a mass comparable to the initial black hole. The fate of
this object cannot be determined by applying a standard semi-classical
analysis that would be valid for a young black hole.

As pointed out in Dvali  _et al._ (2020), at the current level of
understanding, it is not excluded that some new collective (classical)
instability sets in, leading to a disintegration of the memory-burdened
remnant.

Putting this possibility aside, after a black hole enters the memory burden
phase, its evaporation must slow down dramatically. From the analyticity
considerations, it was suggested in Dvali  _et al._ (2020) that the remaining
lifetime of a black hole scales as,

$\tau\,\sim\,R\,S_{\rm BH}^{1+k}\,.$ (2)

where $\displaystyle S_{\rm BH}$ is the entropy of initial black hole and
$\displaystyle k>0$ is an integer.

This has a number of implications which will be discussed later, after we
introduce the second part of the story.

On the other hand, it was shown recently Dvali (2021a, b, c) that black holes
are not the only objects with maximal information storage capacity. Rather, in
various consistent quantum field theories (QFTs) there exist a whole class of
objects, named “saturons” Dvali (2021a), that exhibit the identical
properties.

An important aspect is that saturons can emerge in the form of solitons and
other bound states in renormalizable QFTs at weak coupling where their
properties are fully under control and calculable Dvali (2021b, c, a, d);
Dvali and Venugopalan (2022); Dvali and Sakhelashvili (2022); Dvali  _et al._
(2022a, b, 2024a).

In order to fix the definition: a saturon represents an object that saturates
the QFT upper bound on the microstate degeneracy. The bound has been
formulated in Dvali (2021a) and can be given in two equivalent forms as the
bound on the microstate entropy, $\displaystyle S\equiv\ln(n_{\rm st})$, with
$\displaystyle n_{\rm st}$ number of degenerate microstates.

First, for a bound state of radius $\displaystyle R$ formed by QFT degrees of
freedom interacting via a running coupling $\displaystyle\alpha$, the upper
bound on the microstate entropy is,

$S\leqslant\frac{1}{\alpha}\,,$ (3)

where $\displaystyle\alpha$ must be evaluated at the scale $\displaystyle
1/R$.

Equivalently, the bound can be written in terms of the Goldstone scale,
$\displaystyle f$, of the spontaneously broken Poincare symmetry:

$S\leqslant\pi R^{2}f^{2}\,.$ (4)

As shown in Dvali (2021a), the above bounds set the maximal degeneracy
reachable within the validity of the QFT description. In particular, their
saturation is correlated with the saturation of unitarity by the scattering
amplitudes.

As already discussed in Dvali (2021a), Bekenstein-Hawking entropy of a black
hole (1) represents a particular case of saturation of both bounds. First,
applied to a black hole, the formula (4) is identical to (1), since for a
black hole of arbitrary mass, the scale of Poincare Goldstone is given by the
Planck mass, $\displaystyle f=M_{\rm P}$.

Simultaneously, the black hole entropy (1) is also equal to (3), since the
gravitational coupling, evaluated at energy scale $\displaystyle 1/R$, is
$\displaystyle\alpha_{\rm gr}=1/(\pi R^{2}M_{\rm P}^{2})$.

It has been observed that striking similarities between black holes and
saturons of renormalizable QFTs extend to their other key properties:

  * •

Impossibility of the information-retrieval classically Dvali (2021b, c, a, d);
Dvali and Sakhelashvili (2022); Dvali  _et al._ (2022a);

  * •

The minimal time-scale of the start of the information-retrieval,
$\displaystyle t\sim S\,R$, which is identical to the Page’s time in black
holes Dvali (2021a); Dvali and Sakhelashvili (2022); Dvali  _et al._ (2022a);

  * •

The existence of information horizon in the semi-classical theory Dvali  _et
al._ (2022a);

  * •

Thermal-like evaporation at initial stages of the decay Dvali and
Sakhelashvili (2022); Dvali  _et al._ (2022a);

  * •

The relation between the maximal spin and the entropy Dvali  _et al._ (2022b,
2024a).

The above correspondence makes the study of saturons important due to the
following reasons. First, it shows that the black hole properties are not
specific to gravity and can be understood within calculability domains of
renormalizable QFTs.

Secondly, saturons can serve as laboratories for understanding the microscopic
nature of known black hole properties and for discovering new features.

In the present paper, we shall apply this strategy to the memory burden effect
in two directions. First, following Dvali (2018a); Dvali  _et al._ (2020), we
outline the generic properties of the memory burden phenomenon within
Hamiltonian models of enhanced information capacity. We then demonstrate the
concrete manifestations of these properties within solitons, expanding the
earlier analysis of Dvali  _et al._ (2022a); Valbuena-Bermúdez (2023). Next,
we establish the link with analogous properties in black holes.

In studying the memory burden effect in black holes, we shall use two
strategies. On one hand, we shall extract the key features of the effect
relying solely on the universality of the phenomenon and the requirement of
QFT consistency of the system.

On the other hand, we shall cross-check our conclusions with a microscopic
theory of black hole’s quantum $\displaystyle N$-portrait Dvali and Gomez
(2013a, 2014a, 2014b). In this picture a black hole is described as a
saturated coherent state of gravitons, at criticality, which makes the origin
of its enhanced information storage capacity very explicit.

This approach allows us to predict certain new features of memory burden in
black holes, moving forward from the previous studies.

The memory burden effect has a wide range of applications. As already put
forward in Dvali  _et al._ (2020), one immediate application is the opening up
of a new window of the primordial black hole dark matter with masses below
$\displaystyle 10^{14}\text{g}$. In the standard treatment, this interval of
masses was ignored based on the assumption of the validity of the semi-
classical picture during the black hole’s entire lifetime. The memory burden
effect invalidates this assumption.

Some viable examples of PBH dark matter in the new mass window were given
already in Dvali  _et al._ (2020). Further studies, focusing on the formation
mechanisms Dvali  _et al._ (2021) and various constraints on memory burdened
PBH dark matter Alexandre  _et al._ (2024); Thoss  _et al._ (2024); Balaji
_et al._ (2024); Haque  _et al._ (2024), have also been performed.

In the present paper, we shall predict a new feature of potential
observational significance: the model-independent spread in PBH masses induced
by the memory burden effect.

As suggested in Dvali  _et al._ (2019a), another entity, likely subjected to
the memory burden effect, is a de Sitter Hubble patch of radius $\displaystyle
R$. Similarly to a black hole, de Sitter carries a maximal microstate entropy
of Gibbons-Hawking Gibbons and Hawking (1977) given by the expression (1). Due
to this, it falls in the category of systems with enhanced capacity for
information storage. This creates an avenue for studying the imprints of the
memory burden effect in inflationary cosmology, with the first steps taken in
Dvali  _et al._ (2019a).

However, it must be stressed that the situation in de Sitter is very different
from the black hole case since there exists no sensible notion of a
“stabilized” de Sitter. Instead, as discussed in Dvali  _et al._ (2019a);
Dvali (2021e), the memory burden gives a consistency upper bound on the
duration of classical de Sitter in terms of $\displaystyle t_{\rm M}$.
Originally, the bound on the duration of the de Sitter state was derived in
Dvali and Gomez (2014b, 2016); Dvali  _et al._ (2017) from the self-
entanglement of the de Sitter state caused by the back reaction from Gibbons-
Hawking radiation.

In this respect, we must notice that the memory burden effect completes a
bigger picture of previously suggested mechanisms leading to a breakdown of
the semi-classical description for a macroscopic system after a certain
critical time, so-called quantum break-time Dvali  _et al._ (2013).

In particular, it has been suggested Dvali and Gomez (2013b, 2014b) that a
black hole experiences quantum breaking while still being macroscopic. This
happens the latest after the loss of about half of the initial mass. It has
also been argued that at this point, the black hole acquires a significant
quantum hair, which could potentially lead to its stabilization Dvali and
Gomez (2013b).

The concept of quantum break-time has also been applied to de Sitter Dvali and
Gomez (2014b, 2016); Dvali  _et al._ (2017). However, unlike a black hole,
which can happily continue the existence beyond this point, the quantum break-
time represents a consistency upper bound on the duration of any classical de
Sitter state. In particular, in any consistent inflationary theory, the
inflation must end before the corresponding quantum break-time elapses 111The
quantum inconsistency of an eternally inflating Universe also follows from its
incompatibility with the $\displaystyle S$-matrix formulation of quantum
gravity Dvali (2020)..

The memory burden effect strengthens these earlier quantum break-time
proposals for black holes and for de Sitter, as it provides an additional
engine for quantum-breaking Dvali (2021e). It also makes the microscopic
dynamics of quantum breaking very explicit.

In general, due to its universal nature, the memory burden phenomenon strongly
affects the systems with high capacity of information storage, leading to a
number of physical consequences.

The rest of this paper is organized as follows. Next section describes the
memory burden effect in general systems of enhanced memory storage capacity,
which is then specialized to the case of solitons and black holes in Sec. III
and IV respectively. Sec. V focuses on the analogies and differences between
memory burden and classical extremality, while Sec. VI is dedicated to
numerical results showing examples of dynamical stabilization of solitons by
their memory. Finally, Sec. VII contains remarks on the phenomenological
consequences of our findings as well as our outlook. Visuals of our numerical
simulations can be found at the following URL.

##  II Essence of Memory Burden

The memory burden effect was introduced in Dvali (2018a), where it was
proposed that the loaded information pattern tends to stabilize the system
carrying this information. A more detailed analysis of prototype systems was
performed in two follow-up papers, Dvali  _et al._ (2019a) and Dvali  _et al._
(2020), with the purpose of applying the memory burden effect to cosmology and
black holes, respectively. The last work concluded that the slowdown of the
black hole decay due to the effect is imminent, the latest, by its half-decay.

In this chapter, following Dvali (2018a); Dvali  _et al._ (2019a, 2020), we
shall explain the essence of the memory burden phenomenon and give some
helpful classification of its regimes. In order to achieve this, we first go
through the main universal characteristics of the systems with the enhanced
capacity of information storage. This concept refers to the energetic
efficiency of quantum information storage and was introduced in Dvali (2017,
2018b, 2018c) at the level of basic Hamiltonians. The reader can find the
summary of the setup in Dvali  _et al._ (2019b); Dvali (2018d, 2021e).

The main characteristics of effective Hamiltonians describing systems of
enhanced information capacity can be determined by categorizing quantum
degrees of freedom according to the tasks they perform. The degrees of freedom
shall be introduced as quantum oscillators in number representations. These
shall later be identified with different modes of the quantum fields.

###  II.1 Memory modes

The first category of the degrees of freedom is the memory modes. We shall
introduce them as quantum fields denoted by $\displaystyle\theta^{j}$ where
the index $\displaystyle j=1,2,....,M$ labels their “flavor”. The total number
of species is $\displaystyle M$. The corresponding creation-annihilation
operators $\displaystyle\hat{a}_{j}^{\dagger},\,\hat{a}_{j}$ can satisfy
either fermionic or bosonic oscillator algebras. For definiteness, we shall
focus on the bosonic one:

$[\hat{a}_{i},\hat{a}_{j}^{\dagger}]\,=\,\delta_{ij}\,,~{}~{}\,[\hat{a}_{i},\hat{a}_{j}]=0\,.$
(5)

The role of the memory modes is to store information. The information is
stored in the patterns of their occupation numbers. These patterns are
represented via ket-vectors in the Fock space,

$|p\rangle=|n_{1},n_{2},...,n_{M}\rangle\,,$ (6)

where numbers $\displaystyle n_{j}\equiv\langle p|\hat{n}_{j}|p\rangle$
represents the eigenvalues of the corresponding number operators,
$\displaystyle\hat{n}_{j}\equiv\hat{a}_{j}^{\dagger}\hat{a}_{j}$.

The Hilbert sub-space formed by the vectors $\displaystyle|p\rangle$ shall be
referred to as the memory space. The dimensionality of it is defined by the
range of the occupation numbers of the memory modes $\displaystyle n_{j}$.
These can be subject to constraints.

An example of a simple constraint is to limit each occupation number to two
possible values, $\displaystyle n_{j}=0,1$. In this case, the free part of the
memory system is described by the Hamiltonian of $\displaystyle M$ independent
qubits,

$\hat{H}_{\rm free}\,=\,\sum_{j=1}^{M}\,m_{j}\,\hat{n}_{j}$ (7)

where $\displaystyle m_{j}$ are the energy gaps of the memory modes. The
corresponding dimensionality of the memory space is $\displaystyle n_{\rm
st}\,=\,2^{M}$. In general, the dimensionality of memory space grows
exponentially with the number of memory species $\displaystyle M$. Thus, the
existence of a large number of memory species is an essential condition for
the efficiency of information storage. However, it is not sufficient.

Even with large $\displaystyle M$, the Hamiltonian (7) does not necessarily
represent a system of enhanced information capacity. This is because the
energy span of the memory space can be very large if the gaps $\displaystyle
m_{j}$ are high.

Therefore, the second important characteristic is the energy cost of an
information pattern,

$E_{p}=\sum_{j=1}^{M}m_{j}\,n_{j}\,,$ (8)

as well as the gaps between different memory patterns,
$\displaystyle|p\rangle$ and $\displaystyle|p^{\prime}\rangle$,

$E_{p}-E_{p^{\prime}}\,.$ (9)

In short, the efficiency of the information storage by a system is determined
by the density of states: the number $\displaystyle n_{\rm st}$, of states
$\displaystyle|p\rangle$, that can fit within a physically meaningful smallest
energy gap $\displaystyle\Delta E$. Usually, this is set by a typical
uncertainty in the system’s energy, such as the level-width. Naturally, many
flavors $\displaystyle M$ and smaller gaps $\displaystyle m_{j}$ achieve more
efficiency.

The resulting information storage capacity is quantified by the microstate
entropy of the system defined as,

$S\equiv\ln(n_{\rm st})\,.$ (10)

In quantum field theory (QFT), the limit to memory capacity is set by saturons
Dvali (2021a): the objects with microstate entropy that saturates the upper
bounds (3) and (4) imposed by the validity of a given QFT description.

###  II.2 Master modes: assisted gaplessness

The second category of the degrees of freedom are so-called master modes. We
shall denote them by $\displaystyle\phi_{\alpha}$, with creation-annihilation
operators,
$\displaystyle\hat{a}_{\phi_{\alpha}},\hat{a}^{\dagger}_{\phi_{\alpha}}$,
where $\displaystyle\alpha=1,2,...,$ is their flavor index.

The role of the master modes is to assist the memory modes in becoming
gapless. As opposed to the memory modes, which must come in a large number of
flavors $\displaystyle M$, the number of master mode species can be much less.

To illustrate the mechanism of the assisted gaplessness, and the resulting
effect of memory burden, a single flavor of the master mode is sufficient. The
corresponding number operator shall be denoted by
$\displaystyle\hat{n}_{\phi}\equiv\hat{a}^{\dagger}_{\phi}\hat{a}_{\phi}$.

With the above conventions, the effect of assisted gaplessness Dvali (2017,
2018b, 2018c); Dvali  _et al._ (2019b) can be illustrated by using the
following simple prototype Hamiltonian Dvali (2018a); Dvali  _et al._ (2019a,
2020); Dvali (2021e),

$\displaystyle\displaystyle\hat{H}\,$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\,\hat{H}_{\rm ms}\,+\,\hat{H}_{\rm mem}\,,$ (11)
$\displaystyle\displaystyle{\rm with}:~{}\hat{H}_{\rm ms}$
$\displaystyle\displaystyle\equiv$ $\displaystyle\displaystyle
m_{\phi}\hat{n}_{\phi}\,,$ $\displaystyle\displaystyle\hat{H}_{\rm mem}$
$\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\left(1-\frac{\hat{n}_{\phi}}{N_{\phi}}\right)^{q}\sum_{j}m_{j}\hat{n}_{j}\,,$

where, for physical clarity, we have split the Hamiltonian into the master
$\displaystyle\hat{H}_{\rm ms}$ and memory $\displaystyle\hat{H}_{\rm mem}$
parts, respectively.

The parameters $\displaystyle m_{\phi}$ and $\displaystyle m_{j}$ represent
the intrinsic energy (or mass) gaps of the master and memory modes, i.e., the
gaps around the Fock vacuum state $\displaystyle n_{\phi}=n_{j}=0$. The number
$\displaystyle N_{\phi}\gg 1$ is a large number that sets the coupling between
master and memory modes as $\displaystyle 1/N_{\phi}$. Notice that in QFT
systems, for a given $\displaystyle N_{\phi}$, $\displaystyle M$ is bounded
as,

$M\leqslant N_{\phi}\,.$ (12)

This bound can be understood as a manifestation of a general bound

$({\rm coupling})\times({\rm number~{}of~{}species})\leqslant 1\,,$ (13)

violation of which invalidates the QFT description Dvali (2021a). Thus, within
the validity of QFT, the maximal number of memory patterns is achieved for
$\displaystyle M\simeq N_{\phi}$.

The parameter $\displaystyle q$, which we take as some positive even number,
requires clarification. This quantity parameterizes the functional dependence
of the effective energy gaps of the memory modes on $\displaystyle n_{\phi}$.
Of course, in general, this function can be more complicated. However, near
the point of the assisted gaplessness, it is well-described by a monomial with
a power $\displaystyle q$. In this form, the assisted gaplessness takes place
for the critical occupation number of the master mode,

$n_{\phi}=N_{\phi}\,.$ (14)

Indeed, if we keep the master mode in the Fock vacuum state, $\displaystyle
n_{\phi}=0$, and form a memory pattern via the excitations of the memory
modes, $\displaystyle|p\rangle=|n_{1},...,n_{M}\rangle$, the cost of energy is

$E_{p}=\langle p|\hat{H}_{\rm mem}|p\rangle\,=\,\sum_{j}m_{j}n_{j}\,.$ (15)

This can be extremely high if $\displaystyle m_{j}$-s are large.

However, the system possesses another state (14), with the exact same memory
pattern as $\displaystyle|p\rangle$, but with a critically excited memory
mode. We shall denote it by $\displaystyle|\tilde{p}\rangle$, where tilde
indicates the difference in $\displaystyle n_{\phi}$.

On this state, the contribution to the energy from the memory mode-dependent
part of the Hamiltonian is zero:

$E_{\rm mem}\,=\,\langle\tilde{p}|\hat{H}_{\rm mem}|\tilde{p}\rangle=0\,.$
(16)

This is because the effective gaps for the memory modes are no longer given by
$\displaystyle m_{j}$, but rather by the quantities,

$\omega_{j}\,\equiv\,\left(1-\frac{n_{\phi}}{N_{\phi}}\right)^{q}m_{j}\,,$
(17)

which vanish on a state with $\displaystyle n_{\phi}\,=\,N_{\phi}$.

Thus, for the information pattern it is energetically favorable that the
master mode, instead of being in the vacuum $\displaystyle n_{\phi}=0$, is in
the critical state $\displaystyle n_{\phi}=N_{\phi}$. This is the essence of
the mechanism of the assisted gaplessness Dvali (2017, 2018b, 2018c); Dvali
_et al._ (2019b) (the term was coined in Dvali  _et al._ (2019b)).

However, the price to pay is the energy of the master mode:

$E_{\rm ms}\,=\,\langle\tilde{p}|\hat{H}_{\rm
ms}|\tilde{p}\rangle=m_{\phi}N_{\phi}\,.$ (18)

Due to this, for a given information pattern $\displaystyle p$, the
energetically optimal state is determined by the balance between the two
entries $\displaystyle E_{\rm mem}$ and $\displaystyle E_{\rm ms}$. This fixes
the occupation number of the master mode, $\displaystyle n_{\phi}$, to a
certain optimal value. The energy difference between this optimal state and
master mode’s Fock vacuum ($\displaystyle n_{\phi}=0$), both evaluated for the
same memory pattern $\displaystyle p$, determines the energy efficiency of the
information storage.

It is useful to quantify this efficiency by defining the memory-efficiency
coefficient as the ratio of the actual cost of an information pattern
$\displaystyle E_{\rm mem}$ to its cost in the master mode Fock vacuum:

$\epsilon\,\equiv\,\frac{E_{\rm mem}}{E_{p}}\,.$ (19)

Moving away from the above optimal state, requires climbing an energetic
barrier. This creates a resistance against abandoning the state of the
enhanced memory capacity. This is the essence of the memory burden effect
Dvali (2018a); Dvali  _et al._ (2020).

###  II.3 Memory burden effect

We now wish to give certain universal characteristics of the memory burden
effect Dvali (2018a); Dvali  _et al._ (2019a, 2020).

####  II.3.1 Generalities

In order to understand the energetic balance leading to the memory burden
effect, let us minimize the Hamiltonian (37) with respect to $\displaystyle
n_{\phi}$ in a state with a memory pattern
$\displaystyle|p\rangle=|n_{1},...,n_{M}\rangle$. The intrinsic (vacuum)
energy cost of the pattern, $\displaystyle E_{p}$, is given by (15).

Notice that for minimization we can use Bogoliubov approximation in which we
treat the operator $\displaystyle\hat{n}_{\phi}$ as a $\displaystyle c$-number
$\displaystyle n_{\phi}$. This is justified, since the occupation number
around the states of interest is macroscopic and the $\displaystyle c$-number
approximation of the operator works up to corrections of $\displaystyle
1/n_{\phi}$ (for more detailed discussion of $\displaystyle c$-number method
see Dvali  _et al._ (2019b)).

We can distinguish the two regimes depending on whether $\displaystyle E_{p}$
is above or below the following critical value,

$E_{*}\,\equiv\,\frac{1}{q}m_{\phi}N_{\phi}\,.$ (20)

For

$E_{p}\,\geqslant\,E_{*}\,,$ (21)

the minimum of the energy of the system is achieved for

$n_{\phi}\,=\,N_{\phi}\left(1\,-\,\left(\frac{E_{*}}{E_{p}}\right)^{\frac{1}{q-1}}\right)\,,$
(22)

and it is equal to,

$E\,=\,m_{\phi}N_{\phi}\left(1\,-\,\frac{q-1}{q}\left(\frac{E_{*}}{E_{p}}\right)^{\frac{1}{q-1}}\right)\,.$
(23)

Taking into account (20) and (21), we can easily see that this energy is less
than the would-be vacuum energy cost of the pattern $\displaystyle E_{p}$.
That is, in this case, it is energetically favourable to stabilize the system
in the state with non-zero occupation number of the master mode $\displaystyle
n_{\phi}$.

On the other hand, for

$E_{p}\leqslant E_{*}\,,$ (24)

the minimum energy state is,

$n_{\phi}=0\,,$ (25)

with the energy,

$E\,=\,E_{p}\,.$ (26)

Of course, at the critical value $\displaystyle E=E_{*}$, the two regimes give
the same minimal energy

$E\,=\,E_{p}\,=\frac{1}{q}m_{\phi}N_{\phi}\,.$ (27)

That is, if the vacuum energy cost of the pattern is below the critical value
(20), it is not worth energetically to keep the system in $\displaystyle
n_{\phi}>0$ state. Otherwise, the system is stabilized in a state with
$\displaystyle n_{\phi}>0$ by the memory pattern. This is the key ingredient
of the memory burden effect.

Notice, the Hamiltonian (11) conserves the number of the master mode.
Correspondingly, the states with arbitrary $\displaystyle n_{\phi}$ represent
its eigenstates and do not evolve in time. Therefore, in order to observe the
stabilizing effect of the memory burden dynamically, we must add interactions
that do not conserve $\displaystyle n_{\phi}$ and allow for quantum
transitions of $\displaystyle\phi$-quanta into some “external” degrees of
freedom $\displaystyle\hat{b}^{\dagger},\hat{b}$, with energy gaps
$\displaystyle m_{b}$. This can be achieved by the inclusion of the
interaction terms of the form,

$(\hat{b}^{\dagger})^{\beta}(\hat{a}_{\phi})^{\alpha}\,+\,{\rm h.c.}\,,$ (28)

where $\displaystyle\alpha,\beta$ are integers.

The dynamical memory burden effect is clearly illustrated already in the
simplest case $\displaystyle\alpha=\beta=1$, with $\displaystyle
m_{\phi}=m_{b}$, which can be solved analytically Dvali (2018a). The
Hamiltonian (11) is supplemented by

$\hat{H}_{\rm
int}\,=\,\frac{\tilde{m}}{\sqrt{N_{\phi}}}\,\hat{b}^{\dagger}\hat{a}_{\phi}\,+\,\frac{\tilde{m}^{*}}{\sqrt{N_{\phi}}}\,\hat{a}^{\dagger}_{\phi}\hat{b}\,\,+\,m_{\phi}\hat{n}_{b}\,,$
(29)

where $\displaystyle\hat{n}_{b}\equiv\hat{b}^{\dagger}\hat{b}$ is the
$\displaystyle b$-mode number operator and $\displaystyle\tilde{m}$ is a
complex parameter of the dimensionality of energy. Notice that $\displaystyle
1/\sqrt{N_{\phi}}$ in the mixing terms makes the parametrization consistent
with the normalization of the coupling of the master mode in (11).

The above system can be solved in two ways. We can first find the minimum of
energy for a fixed total occupation number

$n_{\phi}\,+\,n_{b}\,=\,N_{\phi}\,,$ (30)

using the Bogoliubov approximation for $\displaystyle a_{\phi}$ and
$\displaystyle b$ modes. Given the constraint (30), without any loss of
generality, we can parameterize these modes replacing the operators by the
$\displaystyle c$-number functions,

$\hat{a}_{\phi}\,=\,\cos(\theta)\,e^{i\alpha_{\phi}}\sqrt{N_{\phi}}\,,~{}\hat{b}\,=\,\sin(\theta)\,e^{i\alpha_{b}}\sqrt{N_{\phi}}\,,$
(31)

where $\displaystyle\alpha_{\phi}$ and $\displaystyle\alpha_{b}$ are phases
and $\displaystyle\theta$ is an angle that parameterizes the distribution of
the occupation number among $\displaystyle a_{\phi}$ and $\displaystyle
b$-modes. Notice that the phases $\displaystyle\alpha_{\phi}$ and
$\displaystyle\alpha_{b}$ drop out of the minimization procedure, since they
align with the phase of $\displaystyle\tilde{m}$ and give the negative over-
all sign of the mixing term.

The total effective Hamiltonian becomes a function of a single variable
$\displaystyle\theta$,

$\hat{H}\,=\,\sin(\theta)^{2q}\,E_{p}\,-\,|\tilde{m}|\,\sin(2\theta)\,.$ (32)

It is clear that the larger is $\displaystyle E_{p}$, the closer is the
minimum of the energy to $\displaystyle\theta=0$, which implies $\displaystyle
n_{\phi}=N_{\phi}$ and $\displaystyle n_{b}=0$.

For $\displaystyle qE_{p}/|\tilde{m}|\gg 1$, the system gets stabilized in the
state of enhanced memory capacity with the depleted number of master modes,
$\displaystyle\Delta n_{\phi}\,=\,N_{\phi}\,-\,n_{\phi}$, given by,

$\frac{\Delta
n_{\phi}}{N_{\phi}}\,=\,\sin^{2}{\theta}\simeq\left(\frac{|\tilde{m}|}{qE_{p}}\right)^{\frac{2}{2q-1}}\,.$
(33)

Correspondingly, this number gets transferred to the $\displaystyle b$-mode.

On the other hand, for $\displaystyle E_{p}\,=\,0$, the minimum is achieved
for $\displaystyle\theta=\pi/4$ implying that the average occupation numbers
satisfy,

$n_{\phi}\,=\,n_{b}\,.$ (34)

These results are fully confirmed by the explicit quantum evolution of the
system.

Indeed, following Dvali (2018a); Dvali  _et al._ (2020), let us time-evolve
the system from the initial state $\displaystyle n_{\phi}=N_{\phi}$ and
$\displaystyle n_{b}=0$.

For $\displaystyle E_{p}=0$, the occupation numbers evolve in time as,

$\frac{n_{\phi}(t)}{N_{\phi}}\,=\,\cos^{2}\left(\frac{|\tilde{m}|}{\sqrt{N_{\phi}}}t\right)\,,~{}~{}\,\frac{n_{b}(t)}{N_{\phi}}\,=\,\sin^{2}\left(\frac{|\tilde{m}|}{\sqrt{N_{\phi}}}t\right)\,,$
(35)

which, for the averaged values, reproduce (34).

For $\displaystyle qE_{p}/|\tilde{m}|\gg 1$, the story is very different. The
oscillation amplitude is now suppressed, so that the average value of
$\displaystyle\Delta n_{\phi}$ is given exactly by (33). For a detailed
numerical analysis, see Dvali  _et al._ (2020).

Of course, the above example of the system’s decay represents an
oversimplified toy model, with the main purpose of illustrating the system
with a swift memory burden effect. The time evolution must be taken with a
grain of salt, especially in the later oscillatory period.

For QFT systems, such as a black hole or a soliton, submerged in infinite
space, the master mode can decay into a continuum of the external
$\displaystyle b$-modes. These correspond to different momentum (or spin)
eigenstates of outgoing radiation. In such cases, the time-evolution of the
system is dissipative rather than oscillatory. Regardless, as long as the
system maintains the information pattern with $\displaystyle E_{p}\gg
m_{\phi}$, it is subjected to memory burden, latest for $\displaystyle\Delta
n_{\phi}/N_{\phi}\sim 1$. That is, the memory burden phase starts latest by
the time the system gets rid of an order-one fraction of its initial energy.

The way of avoiding the memory burden would be for the system to get rid of
the information pattern very fast Dvali (2018a). However, in systems of
enhanced information capacity, due to the assisted gaplessness, this is an
extremely suppressed process Dvali  _et al._ (2020). For a detailed numerical
analysis of complex prototype systems, demonstrating this outcome, the reader
is referred to the above paper.

We now move to discussions of different parameter regimes and distinguish the
two extreme realizations of the memory burden effect.

####  II.3.2 Type-$\displaystyle I$ regime

The first regime, which we call Type-$\displaystyle I$, takes place when the
intrinsic frequencies of the master and memory modes are of the same order. We
shall take them to be given by an universal mass gap $\displaystyle m$:

$m=m_{\phi}=m_{j}\,.$ (36)

The Hamiltonian (11) becomes,

$\hat{H}\,=\,m\,\hat{n}_{\phi}\,+\,m\,\left(1\,-\,\frac{\hat{n}_{\phi}}{N_{\phi}}\right)^{q}\sum_{j=1}^{M}\hat{n}_{j}\,.$
(37)

Notice that it is invariant under an arbitrary $\displaystyle U(M)$
transformation of the memory modes: $\displaystyle\hat{a}_{j}\rightarrow
U_{jk}\hat{a}_{k}$.

Let us evaluate the Hamiltonian on a memory pattern, with total occupation
number of all species given by,

$\sum_{j=1}^{M}n_{j}\,=\,N_{\rm G}\,.$ (38)

Due to $\displaystyle U(M)$ symmetry this number can be arbitrarily
redistributed among $\displaystyle M$ memory modes, which creates the
following number of degenerate microstates Dvali (2021a),

$n_{\rm states}\simeq\left(1+\frac{N_{\rm
G}}{M}\right)^{M}\left(1+\frac{M}{N_{\rm G}}\right)^{N_{\rm G}}\,.$ (39)

The above expression represents a binomial coefficient
$\displaystyle\binom{N_{\rm G}}{M}$ evaluated using Stirling approximation for
large $\displaystyle N_{\rm G}$ and $\displaystyle M$.

This degeneracy can be understood in terms of the Goldstone phenomenon, since
the occupation number of modes $\displaystyle N_{\rm G}$ breaks the
$\displaystyle U(M)$ symmetry spontaneously down to $\displaystyle U(M-1)$.

The memory pattern back reacts on the master mode. In order to understand this
back reaction, following Dvali (2018c), we minimize the Hamiltonian (37) with
respect to $\displaystyle n_{\phi}$. This gives,

$\left(1\,-\,\frac{n_{\phi}}{N_{\phi}}\right)^{q-1}=\frac{N_{\phi}}{qN_{\rm
G}}\,.$ (40)

We see that for $\displaystyle N_{\rm G}\geqslant N_{\phi}$, $\displaystyle
n_{\phi}\simeq N_{\phi}$. Plugging (40) in the generic expression (17) for the
effective gaps of the memory modes, we get,

$\omega\,=\,m\left(\frac{N_{\phi}}{qN_{\rm G}}\right)^{\frac{q}{q-1}}\,.$ (41)

In particular, for $\displaystyle q\gg 1$, we obtain,

$\frac{\omega}{m}\,\simeq\,\frac{N_{\phi}}{qN_{\rm G}}\,.$ (42)

Notice also that the contribution to the energy from the master and memory
modes

$E_{\rm ms}\,=\,m\,N_{\phi}\,,~{}~{}E_{\rm mem}\,=\,\omega\,N_{\rm G}\,,$ (43)

relate as

$E_{\rm mem}\,\simeq\,\frac{1}{q}E_{\rm ms}\,.$ (44)

From (43) and (42), it is clear that in type-$\displaystyle I$, the memory-
efficiency coefficient is:

$\epsilon_{I}\,\simeq\,\frac{\omega}{m}\simeq\,\frac{N_{\phi}}{qN_{\rm G}}\,.$
(45)

In realistic systems, due to interactions with external degrees of freedom,
the occupation number $\displaystyle n_{\phi}$ can change in time. In the
previous section we reproduced an example of Dvali (2018c) in which the
transition takes place into a single $\displaystyle b$-mode. In this case, the
behaviour is oscillatory. However, when the number of external decay channels
is large, the inverse transitions practically never happen and $\displaystyle
n_{\phi}$ decreases irreversibly.

For example, this is what happens in black holes Dvali  _et al._ (2020), where
the master mode gets depleted into outgoing Hawking particles Dvali and Gomez
(2013a). In such cases, the system dynamically evolves until it gets
stabilized (or semi-stabilized) by the memory burden effect.

In other words, the information drives the system towards the state in which
the energy cost of the information pattern is optimal. Correspondingly, the
system resists against the attempts of driving it away from such a state. We
shall later discuss this dynamics in solitons and in black holes and compare
them.

####  II.3.3 Type-$\displaystyle II$ regime

We shall now consider the regime in which the intrinsic energy gaps of the
memory modes are much higher than that of the master mode:

$m_{j}\gg m_{\phi}\,.$ (46)

We shall refer to it as type-$\displaystyle II$. In this case the minimization
of the Hamiltonian with respect to $\displaystyle n_{\phi}$ gives,

$\left(1\,-\,\frac{n_{\phi}}{N_{\phi}}\right)^{q-1}=m_{\phi}\frac{N_{\phi}}{qE_{p}}\,.$
(47)

Taking into account (13) and (46), we notice that for a memory pattern with
$\displaystyle N_{\rm G}\gtrsim M$, the r.h.s., of the above equation is much
less than one. Thus, similarly to type-$\displaystyle I$ case, we have
$\displaystyle n_{\phi}\simeq N_{\phi}$. Correspondingly, the energy cost of
the master mode is

$E_{\rm ms}\,\simeq\,m_{\phi}N_{\phi}\,.$ (48)

At the same time, the actual energy cost of the memory pattern is,

$E_{\rm
mem}=\left(m_{\phi}\frac{N_{\phi}}{qE_{p}}\right)^{\frac{q}{q-1}}E_{p}\simeq\frac{1}{q}m_{\phi}N_{\phi}\,.$
(49)

Thus, we arrive to the same relation (44) as in type-$\displaystyle I$ case.

However, the efficiency coefficient $\displaystyle\epsilon$ is now extra-
suppressed. For example, taking all $\displaystyle m_{j}$ to be set by a
single scale, $\displaystyle m_{j}=m\gg m_{\phi}$, we get

$\epsilon_{II}\,\equiv\,\frac{N_{\phi}}{qN_{\rm G}}\frac{m_{\phi}}{m}\,.$ (50)

This is suppressed with respect to type-$\displaystyle I$ case (45) by an
additional factor $\displaystyle m_{\phi}/m$.

The basic lesson is the following:

Increasing the vacuum gaps of the memory modes with respect to the master
mode, the system becomes more and more efficient in information cost.
Correspondingly, the memory burden effect is stronger.

##  III Memory Burden in Solitonic Saturon

We shall now study the memory burden effect in solitonic saturons. Following
Dvali (2021a), we first introduce an example of a saturon in form of a vacuum
bubble. Next, we discuss their stabilization via memory burden effect closely
following Dvali  _et al._ (2022a).

The bubble is a solution in a renormalizable $\displaystyle SU(N)$-invariant
theory of a scalar field $\displaystyle\Phi$ in the adjoint representation
with the following Lagrangian density,

$\mathcal{L}[\Phi]=\frac{1}{2}{\rm{Tr}}\,(\partial_{\mu}\Phi)(\partial^{\mu}\Phi)-V(\Phi)\,,$
(51)

where the potential is chosen as,

$V(\Phi)\,=\,\frac{\alpha}{2}{\rm{Tr}}\left(f\Phi-\Phi^{2}+\frac{I}{N}{\rm{Tr}}\,\Phi^{2}\right)^{2}\,.$
(52)

Here $\displaystyle\alpha$ is a coupling constant and $\displaystyle f$ is a
scale. $\displaystyle I$ is an unit matrix in $\displaystyle SU(N)$-space.

The validity of QFT description imposes the following bound on the parameters
of the model,

$\alpha N\lesssim 1\,.$ (53)

Basically, the quantity $\displaystyle\alpha N$, which is analogous to ’t
Hooft coupling, must not exceed the critical value. In the opposite case
($\displaystyle\alpha N\gg 1$), $\displaystyle\Phi$ no longer represents a
good QFT degree of freedom. This is unambiguously signalled by the breakdown
of the loop-expansion, as well as, by the saturation of unitarity in multi-
particle scattering amplitudes Dvali (2021a). Notice that after we map the
above theory on a generic Hamiltonian of enhanced memory capacity (11), it
will become clear that the bound (53) represents a particular case of the
bound (13).

The system (51) has multiple degenerate vacua satisfying the condition

$f\,\Phi_{a}^{b}-\left(\Phi^{2}\right)^{b}_{a}+\frac{\delta^{b}_{a}}{N}\,{\rm{Tr}}\,\Phi^{2}\,=\,0\,.$
(54)

We shall focus on a pair of neighbouring vacua.

In the first one, the vacuum expectation value (VEV) is $\displaystyle\Phi=0$
and the global $\displaystyle SU(N)$-symmetry is unbroken. This vacuum
exhibits a mass gap

$m=\sqrt{\alpha}f\,,$ (55)

which sets the minimal energy cost for all particle excitations.

In the second vacuum of our interest, the VEV is,

$\Phi_{0}\,=\,f\frac{1}{\sqrt{N(N-1)}}{\rm{diag}}\left(N-1,-1,...,-1\right)\,,$
(56)

and $\displaystyle SU(N)$-symmetry is spontaneously broken down to
$\displaystyle SU(N-1)\times U(1)$. Due to spontaneous breaking of symmetry,
in this vacuum we have the set of massless Goldstone bosons

$\theta^{j}(x)\,,\quad j=1,2,\dotsc,M\,,$ (57)

corresponding to broken generators $\displaystyle T^{j}$. Their total number
is

$M\,=\,2(N-1)\,.$ (58)

These Goldstones form a fundamental representation of the $\displaystyle
SU(N-1)$ group of complex dimensionality $\displaystyle N-1$ with non-zero
charge under $\displaystyle U(1)$.

The effective Lagrangian of the Goldstone bosons has the following form,

$\mathcal{L}_{\text{eff}}\,=\,f^{2}\left(\partial_{\mu}{\mathcal{U}}(x)\right)\left(\partial^{\mu}{\mathcal{U}}(x)\right)\,,$
(59)

with

${\mathcal{U}}(x)\equiv f^{-1}U(x)^{\dagger}\Phi_{0}U(x)\,,$ (60)

where $\displaystyle U(x)$ is the space-time dependent $\displaystyle
SU(N)$-transformation matrix,

$\quad U(x)=\exp\left[-i\theta^{j}(x)T^{j}\right]\,.$ (61)

Up to second order in $\displaystyle\theta^{j}(x)$-s and leading order in
$\displaystyle 1/N$, the Goldstone Lagrangian can be written as,

$\mathcal{L}_{\text{eff}}\simeq\frac{1}{4}f^{2}\sum_{j}\left(\partial_{\mu}\theta^{j}\right)\left(\partial^{\mu}\theta^{j}\right)\,.$
(62)

###  III.1 Vacuum Bubbles

Due to the degeneracy of vacua, there also exist the domain wall
configurations separating them,

$\Phi(x)=\phi(x)\frac{1}{\sqrt{N(N-1)}}{\rm{diag}}\left(N-1,-1,...,-1\right)\,,$
(63)

where the function $\displaystyle\phi(x)$ interpolates between $\displaystyle
0$ and $\displaystyle f$. We shall be interested in a spherically symmetric
bubble (closed wall) of $\displaystyle\phi=f$ vacuum embedded in
$\displaystyle\phi=0$ one.

Let us first consider the following configuration,

$\Phi=\frac{\phi(r,t)}{f}\Phi_{0}\,.$ (64)

The field $\displaystyle\phi(r,t)$ satisfies the following equation of motion,

$\partial_{t}^{2}\phi-\partial_{r}^{2}\phi-\frac{2}{r}\partial_{r}\phi+\frac{\partial
V}{\partial\phi}\,=\,0\,,$ (65)

where,

$V(\phi)\,=\,\dfrac{\alpha}{2}\phi^{2}\left(\phi-f\right)^{2}\,.$ (66)

The boundary conditions are such that for $\displaystyle t=0$,
$\displaystyle\phi(r=\infty)=0$ and
$\displaystyle\partial_{t}\phi|_{t=0}=\partial_{r}\phi|_{r=0}=0$.

If the initial radius of the bubble satisfies $\displaystyle R\gg m^{-1}$,
this configuration can be approximated by

$\phi(r)=\dfrac{f}{2}\left[1+\tanh\left(\dfrac{m(R-r)}{2}\right)\right]\,.$
(67)

The bubble wall has a thickness $\displaystyle\delta_{w}\simeq m^{-1}$ and a
tension, $\displaystyle\sigma\simeq m^{3}/(6\alpha)$. Order of magnitude wise,
this remains true also for thick-walls bubbles for which $\displaystyle
R\sim\delta_{w}$.

In the interior of the bubble, the $\displaystyle SU(N)$ symmetry is broken
spontaneously down to $\displaystyle SU(N-1)\times U(1)$. Due to this, there
are $\displaystyle M=2(N-1)$ gapless Goldstone species localized within the
bubble. These species do not exist outside of the bubble, since $\displaystyle
SU(N)$-symmetry is restored there.

As a consequence, the bubble has a large microstate degeneracy. This
degeneracy follows from spontaneously broken $\displaystyle SU(N)$-symmetry,
since a bubble obtained from (63) by an arbitrary $\displaystyle
SU(N)$-transformation $\displaystyle U$,

$\Phi(x)\rightarrow U^{\dagger}\,\Phi(x)\,U,$ (68)

represents a classical solution of the same energy.

In quantum theory, the number of degenerate microstates is not infinite, since
only the orthogonal states must be counted. These states are obtained by
redistribution of the constituent quanta of the bubble among different
$\displaystyle SU(N)$-flavors which (in large-$\displaystyle N$) amounts to
the following number of degenerate microstates Dvali (2021a),

$n_{\rm
states}\simeq\left(1+\frac{s(R)}{2N}\right)^{2N}\left(1+\frac{2N}{s(R)}\right)^{s(R)},$
(69)

where the quantity $\displaystyle s(R)=4\pi(R\,m)^{3}/\alpha$ is the time-
averaged space integral of $\displaystyle\phi^{2}(r)$. This quantity
effectively measures the mean occupation number of constituent quanta of the
bubble soliton viewed as a coherent state Dvali  _et al._ (2015a).

Taking the maximal value of $\displaystyle N$ given by (53), for thick-wall
bubbles $\displaystyle R\sim 1/m$, we get the following expression for
entropy,

$S\sim N\sim\frac{1}{\alpha}\,.$ (70)

It is clear that this expression represents the area ($\displaystyle\sim
1/m^{2}$) in units of the Goldstone scale $\displaystyle f$. Thus, for thick
wall bubbles the entropy saturates the area-law bound (4).

For $\displaystyle R\gg 1/m$, the entropy is,

$S\,\sim\,N\ln(mR)\,.$ (71)

Due to the above degeneracy, the bubble represents a system of enhanced
capacity of the memory storage. The memory modes are Goldstone modes
$\displaystyle\theta^{j}$. The master mode is the radial mode,
$\displaystyle\phi(x)$.

Setting other components to zero, the effective Lagrangian of Goldstone modes
and of the radial mode $\displaystyle\phi$ is given by

$\mathcal{L}_{\text{eff}}\,=\,\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}\phi^{2}\left(\partial_{\mu}{\mathcal{U}}(x)\right)\left(\partial^{\mu}{\mathcal{U}}(x)\right)-V(\phi)\,.$
(72)

The Goldstone modes are well-defined exceptionally for $\displaystyle\phi\neq
0$.

###  III.2 Stabilization by memory burden

We can now observe the stabilization by the memory burden effect. In this, we
shall closely follow Dvali  _et al._ (2022a), where this effect was studied.
This effect requires that a non-zero information pattern is stored in non-zero
frequency excitations of Goldstone modes. That is, some of the Goldstone modes
(i.e., the memory modes) of non-zero frequencies are occupied to numbers
$\displaystyle n_{a}$.

On such a state, $\displaystyle|p\rangle\equiv|n_{1},n_{2},...,n_{M}\rangle$,
the measure of the memory burden effect is the expectation value,

$\langle
p|\left(\partial_{\mu}{\mathcal{U}}(x)\right)\left(\partial^{\mu}{\mathcal{U}}(x)\right)|p\rangle\,.$
(73)

In order to understand the effect, let us first consider a classical
configuration (60) with,

$U=e^{it\omega_{j}T^{j}}\,.$ (74)

It is easy to see that this gives the equation,

$\partial_{t}^{2}\phi\,-\,\partial_{r}^{2}\phi-\frac{2}{r}\partial_{r}\phi-\omega^{2}\phi+\frac{\partial
V}{\partial\phi}\,=\,0\,,$ (75)

with

$\omega^{2}\,\equiv\,\sum_{j=1}^{M}\omega_{j}^{2}\,.$ (76)

Different patterns $\displaystyle(\omega_{1},\omega_{2},...)$ can be obtained
from one another by $\displaystyle SU(N-1)$ transformation, which leaves the
$\displaystyle\Phi_{0}$ invariant.

The above equation always has a stationary bubble solution with time-
independent $\displaystyle\phi(r)$. The asymptotic values are
$\displaystyle\phi(\infty)=0$ and $\displaystyle\phi(0)\neq 0$. Since various
patterns $\displaystyle(\omega_{1},\omega_{2},...,\omega_{M})$ are related by
a symmetry transformation, it is sufficient to discuss the case
$\displaystyle\omega^{j}=\delta^{j1}\omega$ and later generalize to different
patterns.

In the thin-wall limit, $\displaystyle R\gg 1/m$, the profile of the bubble is
given by (67), where the radius $\displaystyle R$ can be determined by
extremizing the energy of the bubble as function of $\displaystyle R$,

$E=\dfrac{2\pi}{3\alpha}m^{3}R^{2}(1-\dot{R}^{2})^{-1/2}+\dfrac{2\pi}{3\alpha}m^{2}\omega^{2}R^{3}\,,$
(77)

subject to the condition that the quantity

$Q=\dfrac{2\pi}{3}f^{2}\omega R^{3}=\dfrac{2\pi}{3\alpha}m^{2}\omega R^{3}\,,$
(78)

represents a conserved charge. This determines the radius of the stationary
bubble as,

$R_{0}=\dfrac{2}{3}\frac{m}{\omega^{2}}\,.$ (79)

Correspondingly, the energy of the bubble is,

$E_{0}=\dfrac{\omega}{\alpha}\,\frac{m^{5}}{\omega^{5}}\,\left(\frac{40\pi}{81}\right)\,.$
(80)

As already discussed in Dvali  _et al._ (2022a), the stationary bubble
solution can be mapped on a $\displaystyle U(1)$ non-topological soliton or a
$\displaystyle Q$-ball Lee and Wick (1974); Friedberg  _et al._ (1976);
Coleman (1985) formed by a complex scalar with modulus $\displaystyle\phi(r)$
and the $\displaystyle U(1)$-charge $\displaystyle Q$ given by (78) (for some
implications of $\displaystyle Q$-balls, see, e.g., Safian  _et al._ (1988);
Lee and Pang (1992); Kusenko (1997); Dvali  _et al._ (1998); Kusenko and
Shaposhnikov (1998); Kusenko  _et al._ (1998); Kim  _et al._ (1993); Volkov
and Wohnert (2002)). However, despite the similarity of classical solutions,
such a soliton will not exhibit a significant microstate degeneracy and will
not represent a system of enhanced capacity of information storage.
Correspondingly, the charge $\displaystyle Q$ cannot be associated with any
information pattern. For a significant microstate degeneracy, having
$\displaystyle SU(N)$-symmetry is crucial. Only in such a case the total
charge $\displaystyle Q$ can be distributed among an exponentially large
number of memory patterns.

The stationary bubbles carrying arbitrary patterns are obtained by
$\displaystyle SU(N-1)$ transformations and have identical macroscopic
features such as the energy and the radius. However, they have different
characteristics with respect to an asymptotic probe with fixed $\displaystyle
SU(N-1)$ quantum numbers relative to which the rotation is performed.

In other words, the information is stored in $\displaystyle SU(N-1)$-rotations
of a pattern relative to a fixed observer. Obviously, all rotated bubbles have
identical energies but they are distinguished by the observer who can perform
a scattering experiment using a fixed probe.

The stationary bubble represents a bubble stabilized by the memory burden
effect. In terms of occupation numbers of Goldstones, $\displaystyle n_{j}$,
the quantities $\displaystyle\omega_{j}$ read,

$\omega_{j}^{2}\,=\,\omega\,n_{j}\,\frac{3}{2\pi f^{2}R^{3}_{0}}\,.$ (81)

From the quantum perspective, it is illuminating to express the energy of the
bubble in terms of the occupation numbers of the radial (master) mode,
$\displaystyle N_{\phi}$, and of the Goldstone modes,

$N_{\rm G}\equiv\sum_{j=1}^{M}n_{j}\,.$ (82)

Splitting the energy of the stationary bubble (80) into two contributions, we
have:

$E_{0}\,=\,E_{\text{mem}}\,+\,E_{\text{ms}}\,,$ (83)

where

$E_{\text{mem}}=\omega N_{\rm G}\,,~{}~{}\text{with},\,~{}~{}N_{\rm
G}\equiv\dfrac{1}{\alpha}\dfrac{m^{5}}{\omega^{5}}\left(\dfrac{16\pi}{81}\right)\,,$
(84)

is the energy of memory (Goldstone) modes, whereas

$E_{\text{ms}}=mN_{\phi}\,,~{}~{}\text{with},\,~{}~{}N_{\phi}\equiv\dfrac{1}{\alpha}\,\dfrac{m^{4}}{\omega^{4}}\,\left(\dfrac{8\pi}{27}\right)\,,$
(85)

is the energy of the master (radial) mode $\displaystyle\phi(r)$. At the
stationary point we have

$E_{\text{mem}}=\dfrac{2}{3}E_{\text{ms}}\,.$ (86)

Notice that this reproduces the generic equation (44) with $\displaystyle
q=3/2$.

From (78) and (84) it is easy to notice that we have,

$Q=N_{\rm G}\,.$ (87)

That is, the classical charge in quantum theory counts the total occupation
number of Goldstone modes. Various memory patterns are obtained by
distribution of $\displaystyle N_{\rm G}$ among $\displaystyle M=2(N-1)$
flavors of Goldstones.

Next, notice that the memory pattern with the same $\displaystyle N_{\rm G}$
in the $\displaystyle SU(N)$-invariant vacuum would cost

$E_{p}=mN_{\rm G}\,,$ (88)

since in this vacuum the minimal energy gap of any elementary excitation with
$\displaystyle SU(N)$ quantum number is $\displaystyle m$.

Taking this into account, we can evaluate the memory-efficiency coefficient
(19), which gives,

$\epsilon\,=\,\frac{\omega}{m}\simeq\,\frac{2N_{\phi}}{3N_{\rm G}}\,.$ (89)

In the last equality we have expressed $\displaystyle\omega/m$ through
occupation numbers via (84) and (85). This reproduces the equation (45) with
$\displaystyle q=3/2$. We thus see that the vacuum bubble realizes the
type-$\displaystyle I$ memory burden effect, with $\displaystyle q=3/2$.

The type-$\displaystyle I$ nature of the memory burden is not surprising,
since in the $\displaystyle SU(N)$ invariant vacuum, the energy gaps of all
excitations are equal to $\displaystyle m$.

The stationary bubble represents a bubble stabilized by the memory burden.
Notice that for such a bubble we have,

$Q=N_{\rm
G}=\dfrac{\pi}{\alpha}(mR_{0})^{5/2}\left(\dfrac{2}{3}\right)^{3/2}\,.$ (90)

A bubble with an insufficient memory burden will not be stationary. In
particular, such is a bubble with the radius larger than the critical value
$\displaystyle R\gg R_{0}$. Obviously, for such a bubble,

$N_{\rm
G}\ll\dfrac{\pi}{\alpha}(mR_{0})^{5/2}\left(\dfrac{2}{3}\right)^{3/2}\,,$ (91)

and

$E_{\text{mem}}\ll\dfrac{2}{3}E_{\text{ms}}\,.$ (92)

The memory burden is insufficient for stabilizing such a bubble. Instead, as
shown in Dvali  _et al._ (2022a), the bubble starts to collapse and oscillate
emitting energy. This decreases the occupation number of the master mode. This
process shall continue until the balance is restored to (86). At this point
the bubble gets stabilized by the memory burden effect.

In particular, the initial bubbles of equal energies but different values of
$\displaystyle N_{\rm G}$ will evolve into the stabilized bubbles of different
energies as we shall discuss next.

###  III.3 The energy splitting by memory burden

We wish to discuss the following important effect which is characteristic of
the phenomenon of memory burden stabilization. Namely, the initially-
degenerate energy states with different memory patterns $\displaystyle N_{\rm
G}\neq N_{\rm G}^{\prime}$, upon stabilization by the memory burden, split in
energies.

Let us start with two initial bubble states with different values of the
Goldstone occupation numbers, $\displaystyle N_{\rm G}$ and $\displaystyle
N_{\rm G}^{\prime}$ respectively. We assume that both numbers are subcritical
(91). Thus, initially the memory burdens are dynamically insignificant.
According to (93), the contribution to the energy from the memory modes is,

$E_{\text{mem}}=\omega N_{\rm G}\,\sim N_{\rm G}\sqrt{\frac{m}{R}}\,.$ (93)

Notice that order-of-magnitude wise this expression holds also for (91), since
$\displaystyle\omega\sim\sqrt{\frac{m}{R}}$.

Correspondingly, the energy splitting between the two bubble states with fixed
difference, $\displaystyle\Delta N_{\rm G}\equiv N_{\rm G}-N_{\rm
G}^{\prime}$,

$\Delta E\,\sim\,\Delta N_{\rm G}\,\sqrt{\frac{m}{R}}\,,$ (94)

vanishes for $\displaystyle R\rightarrow\infty$. Thus, for sufficiently large
and equal radii, the two bubble states are nearly degenerate.

Notice that the states with different $\displaystyle N_{\rm G}$ are fully
legitimate microstates of the same macrostate. Semi-classically, the detection
of the Goldstone charge is not possible. In quantum theory any attempted
measurement is suppressed by powers of $\displaystyle\Delta N_{\rm
G}/N_{\phi}$, which vanishes in the semi-classical limit. Thus, the
microstates with different $\displaystyle N_{\rm G}$ are equally
indistinguishable semi-classically as the microstates with equal
$\displaystyle N_{\rm G}$, but relatively rotated in $\displaystyle
SU(N)$-space.

Now, since the occupation numbers of Goldstones, $\displaystyle N_{\rm
G},N_{\rm G}^{\prime}$, are subcritical (91), the bubbles will evolve in time
by oscillating and emitting energy until they get stabilized by the memory
burden effects. Since during this process the radii diminish and
correspondingly the Goldstone frequencies ($\displaystyle\omega$) grow, so
does the energy splitting between the two bubble states.

At the stabilization point, the bubble energies can be highly split if the
relative difference of Goldstone numbers is large. For example, for
$\displaystyle\Delta N_{\rm G}/N_{\rm G}\sim 1$, the energy splitting between
the two bubble “remnants” is of order their masses. Of course, the energy of a
remnant plus radiation is the same in both cases.

To summarise: For a soliton that has not yet entered the memory burden
stabilization phase (i.e., an under-burdened soliton with (91)), we can
distinguish the two types of degenerate microstates: 1) the microstates that
have the same memory burden $\displaystyle N_{\rm G}$; and 2) the microstates
with different values of $\displaystyle N_{\rm G}$.

The first category belongs to the same orbit in $\displaystyle SU(N)$-space
and, therefore, remains degenerate throughout the bubble evolution. However,
the energies of the second category undergo the splittings according to (94).

The above situation is generic for the memory burden effect. In particular, as
we shall discuss, the same phenomenon takes place in black holes. There too,
after entering the memory burden domination phase, the masses of initially-
degenerate black holes with different information patterns can become vastly
different - see Fig. 1 for a schematic visual. We shall also demonstrate this
effect numerically in Sec. VI.

Figure 1: Schematic view of the splitting of energies of the microstates of
either a soliton or a black hole, time-evolved from initially (nearly)
degenerate microstates with different memory patterns. The solid lines
represent the energies of the remnant states, whereas the black dashed line
represents the total energy of the system (the remnant mass plus the energy of
emitted radiation). Notice that each solid line is not a single state but
accounts for all the states that remain degenerate due to identical memory
burdens. The degeneracy can be exact due to symmetry. For example, such are
the vacuum bubble microstates related by $\displaystyle SU(N)$-symmetry. The
line can also exhibit a finer-structure, in case the symmetry is only
approximate.

Finally, we wish to stress that while the memory burden stabilizes the
systems, the converse is not excluded: the stable systems with very little
information capacity can happily exist. An example of such a system would be
the $\displaystyle Q$-ball with the $\displaystyle U(1)$-charge. Although the
$\displaystyle U(1)$-charge carries very little information, it nevertheless
can stabilize the system dynamically.

The important thing about the systems with high information content (such as
saturons) is that they have no choice: the memory burden effect stabilizes
them necessarily.

##  IV Memory burden in black holes

Already in the original papers on memory burden Dvali (2018a); Dvali  _et al._
(2019a, 2020) it has been argued that black holes must be subjected to this
phenomenon. Furthermore, the analysis of Dvali  _et al._ (2020) revealed that
the stabilizing nature of the effect, with high likelihood, leads to the
prolongation of black hole’s life-time. Here, we shall further scrutinize this
question in light of our analysis.

Let us first structure the arguments that justify the presence of the memory
burden effect in black holes. This conclusion can be reached by three
independent lines of evidence.

The first line developed in Dvali (2018a); Dvali  _et al._ (2020) is based on
the universality of the phenomenon. The review of the effect given in Chapter
II, makes it evident that it is impossible to construct a hermitian
Hamiltonian with efficient information storage that avoids the memory burden
phenomenon. The fact that the black holes are efficient (in fact, the most
efficient) storing devices, indicates that the effect must be present there.

Further justification is provided by the analysis of Dvali  _et al._ (2022a)
and by its continuation in the present work. This analysis shows the presence
of the memory burden effect in solitonic saturons. These objects share with
black holes all the known key features such as information horizon, area-law
entropy, thermal decay and Page-like time of information retrieval. This gives
a strong indication that memory burden must belong to this list.

Finally, the third line of reasoning is based on hints from a microscopic
theory, a so-called black hole’s quantum $\displaystyle N$-portrait Dvali and
Gomez (2013a, 2014a, 2014b). According to this theory, a black hole represents
a saturated coherent state (or a condensate) of gravitons, at criticality. The
term criticality implies that the mean occupation number is inverse of the
gravitational coupling. In the present language, these coherent gravitons play
the role of a master mode. This theory offers a microscopic explanation of
black hole properties within the domain of calculability.

In particular, the entropy of a black hole is explained by the emergence of
$\displaystyle M$ “flavors” of the gapless modes in the above critical state
of gravitons. These can be described as Bogoliubov/Goldstone modes of the
critical graviton condensate Dvali and Gomez (2014a); Dvali  _et al._ (2015b);
Dvali and Panchenko (2015); Averin  _et al._ (2016a).

In the present language, they represent the memory modes. In our reasoning of
establishing the nature of the memory burden in black holes, we shall borrow
only very general features of $\displaystyle N$-portrait, as this was done in
Dvali (2018a); Dvali  _et al._ (2020). We shall stay strictly within the
calculability domain of the QFT framework.

Namely, we shall rely on the following rather conservative (and natural)
starting points:

1) Both memory and master modes of a black hole are describable as modes of
the graviton field. Although additional field species can certainly exist,
e.g., in form of string excitations, they do not change the essence of the
effect (although, can lead to certain modifications Dvali (2021e)). We come to
the role of the extra species later, in Subsec. IV.3.

2) The memory modes can be classified according to the symmetries that are
left unbroken by the black hole metric. This is analogous to how in our bubble
example the Goldstone memory modes have been characterized by their quantum
numbers under the $\displaystyle SU(N-1)\times U(1)$ unbroken symmetry.

Equipped with the above guidelines, we shall discuss the black hole memory
burden effect. Let us first identify the master and memory modes.

The identification of the master mode is easy, since at the initial times,
this mode has macroscopic occupation number and is therefore well-described
classically. Moreover, since the Bogoliubov approximation works for such
modes, as a control parameter we can use the suitable characteristics of the
classical metric. Basically, the master modes are Fourier harmonics of the
classical metric. For a black hole of radius $\displaystyle R$, the dominant
contribution comes from the modes of energy gap,

$m_{\phi}\sim 1/R\,.$ (95)

Of course, the black hole metric viewed as the coherent state of gravitons is
a distribution peaked around the above value Dvali and Gomez (2013a). For our
purposes, it suffices to consider a single mode rather than a sharply peaked
distribution.

Let us now identify the memory modes. As already said, we shall only use very
general points outlined above. We know that the memory modes must come from
graviton modes and can be classified according to symmetries of the classical
metric. Moreover, the number of their flavors, $\displaystyle M$, must be

$M\sim S_{\rm BH}\,,$ (96)

where $\displaystyle S_{\rm BH}$ is the Bekenstein-Hawking entropy which for a
black hole of mass $\displaystyle M_{\rm BH}$ and radius $\displaystyle R$ is
Bekenstein (1973),

$S_{\rm BH}=\pi(RM_{\rm P})^{2}=4\pi\frac{M_{\rm BH}^{2}}{M_{\rm P}^{2}}\,.$
(97)

There exist unique candidates fulfilling the above requirements that can
emerge as the assisted gapless modes Dvali (2018b, a). These are the modes of
the graviton corresponding to various spherical harmonics, $\displaystyle
Y_{lm}$. Only the harmonics with momenta up to cutoff of the theory must be
included, since only such modes are the legitimate (weakly-coupled) QFT
degrees of freedom. In Einstein gravity, with no additional light species
besides graviton, the cutoff is $\displaystyle M_{\rm P}$. We thus need to
include all possible spherical harmonics of graviton up to the Planck mass.

For a black hole of radius $\displaystyle R$, this counting gives the
multiplicity of memory flavors that scales as the area of a black hole in
units of $\displaystyle M_{\rm P}$,

$M\sim(RM_{\rm P})^{2}\,,$ (98)

and therefore precisely matches the entropy demand (97) Dvali (2018b, a).

As an additional supporting evidence, the above fully matches the counting of
gapless modes from the black hole symmetries derived in Averin  _et al._
(2016b, a).

Of course, the majority of modes comes from the highest spherical harmonics.
We thus identify the gapless memory modes of a black hole as the modes of
angular momenta $\displaystyle\sim M_{\rm P}R$. Their counterparts in the
asymptotic vacuum are the same angular harmonics $\displaystyle Y_{lm}$ of a
free graviton. However, these carry the energy gaps $\displaystyle m_{j}\sim
M_{\rm P}$.

Notice that the above explains why a black hole cannot emit information
efficiently Dvali (2018a): in order to “escape” from a black hole via a
quantum process, a memory mode has to climb an extremely high energetic
barrier.

The existence of modes with the same $\displaystyle Y_{lm}$ but with largely
split energy gaps inside and outside of a black hole is not surprising. The
black hole breaks the Poincare symmetry at the scale $\displaystyle M_{\rm P}$
Dvali (2021a). Due to this, the memory modes of a black hole, despite having
the high orbital momenta, are gapless, whereas the asymptotic modes with
similar momenta are gapped by $\displaystyle\sim M_{\rm P}$.

The above knowledge suffices for adapting the generic Hamiltonian (11) to a
black hole situation. The effective Hamiltonian of black hole memory and
master modes is described by (11) with the understanding that index
$\displaystyle j$ labels spherical harmonics $\displaystyle Y_{lm}$.
Correspondingly, the intrinsic gaps of master and memory modes are:

$m_{\phi}\sim\frac{1}{R}\,~{}{\rm and}~{}\,m_{j}\sim M_{\rm P}\,,$ (99)

respectively. The coupling of the master mode to memory modes is set by the
standard gravitational coupling $\displaystyle\alpha_{\rm gr}(R)$ evaluated at
scale $\displaystyle R$,

$\frac{1}{N_{\phi}}\sim\alpha_{\rm gr}(R)\sim\frac{1}{(RM_{\rm P})^{2}}\,.$
(100)

As pointed out in Dvali and Gomez (2013a), this coupling is equal to the
inverse of the Bekenstein-Hawking entropy $\displaystyle S_{\rm BH}$. This is
not an accident. As explained in Dvali (2021a), this equality puts
$\displaystyle S_{\rm BH}$ in accordance with the generic bound on entropy
(3).

Now, in this description, the initial state of a classical black hole
corresponds to the critical state $\displaystyle n_{\phi}=N_{\phi}$ in which
the memory modes are essentially gapless, $\displaystyle\omega_{j}\,=\,0$.

As already discussed in Dvali (2018a), this explains why at the initial stages
of evaporation, the black hole information cannot come out. Indeed, due to the
conservation of the angular momentum, translating the information stored in
the memory modes into the asymptotic quanta, would require radiating the
quanta of energies $\displaystyle\sim M_{\rm P}$, which is not possible. In
addition, a pair-wise annihilation of memory modes of frequencies
$\displaystyle\omega$ into the soft (low angular momentum) external quanta is
suppressed as $\displaystyle\sim\omega_{j}^{5}/M_{\rm P}^{4}$ and is
negligible (see below). Thus, neither the emission of the memory modes nor
their conversion into the external quanta is an option.

At the same time, black hole can emit the master mode via quantum scattering,
reproducing the ordinary Hawking radiation Hawking (1975). Basically, the
process is a conversion of a master mode into asymptotic quanta of similar
frequencies. This process is unsuppressed, since the occupation number of the
master mode is equal to its inverse coupling. As a result, the suppression by
powers of coupling is compensated by the occupation number, and the rate of
emission is Dvali and Gomez (2013a),

$\Gamma_{\rm
ms}\sim\frac{1}{R}\frac{n_{\phi}^{2}}{N_{\phi}^{2}}\sim\frac{1}{R}\,,$ (101)

which reproduces the Hawking rate. This simple quantum explanation of the
Hawking effect is one of the successes of $\displaystyle N$-portrait Dvali and
Gomez (2013a). However, due to generic nature of the effect, it holds for
arbitrary saturons Dvali and Sakhelashvili (2022); Dvali  _et al._ (2022a).

Since we start in the critical state $\displaystyle n_{\phi}\,=\,N_{\phi}$, at
the initial stages the memory burden effect is weak. At this stage, the
equation (101) is a good approximation. It tells us that, on average,
$\displaystyle n_{\phi}$ decreases by $\displaystyle\Delta n_{\phi}\sim 1$
over time $\displaystyle\Delta t\sim R$.

As it is obvious from (11), in general, decreasing the occupation number of
the master mode by $\displaystyle\Delta n_{\phi}$, affects its gap by,

$\Delta m_{\phi}\sim\frac{q}{N_{\phi}}\left(\frac{\Delta
n_{\phi}}{N_{\phi}}\right)^{q-1}E_{p}\,.$ (102)

Taking into account (99), (100) and $\displaystyle N_{\rm G}\sim M\sim S_{\rm
BH}$, this is of order,

$\Delta m_{\phi}\sim q\left(\frac{\Delta
n_{\phi}}{N_{\phi}}\right)^{q-1}M_{\rm P}\,.$ (103)

The memory burden effect becomes dominant by the time when
$\displaystyle\Delta m_{\phi}\sim 1/R$, which gives,

$\frac{\Delta n_{\phi}}{N_{\phi}}=(qM_{\rm P}R)^{-\frac{1}{q-1}}\,.$ (104)

The same expression is obtained from (47) after taking into account (99).

The effective gaps of the memory modes also grow with $\displaystyle\Delta
n_{\phi}$ as

$\omega_{j}\,=\,\left(\frac{\Delta n_{\phi}}{N_{\phi}}\right)^{q}M_{\rm P}\,,$
(105)

and by the time of validity of (104), they become,

$\omega_{j}\,=\,\frac{1}{qR}\frac{1}{(qM_{\rm P}R)^{\frac{1}{q-1}}}\,.$ (106)

In the above expressions, the quantities $\displaystyle R$ and $\displaystyle
S_{\rm BH}$ must be understood as the parameters of the initial black hole.

We must remember that we track the evolution of an initial state with a given
memory pattern. The microstate entropy $\displaystyle S_{\rm BH}$ accounts for
the degeneracy of various patterns. However, as the evolution goes on and
memory modes gain energy gaps, the initially degenerate memory patterns get
split in energy. Of course, the total energies of the systems, i.e., black
hole plus radiation, obtained by evolving different initial patterns, are
degenerate. However, the fractions of energy that remain stored in the black
hole relative to radiated portion are different for states evolved from
different initial patterns (see Fig. 1).

The equations (104) and (106) determine the onset of the memory burden using
as a clock the change in occupation number of the master mode
$\displaystyle\Delta n_{\phi}$. Since before the onset of the memory burden
effect, this number changes by usual Hawking emission, essentially it measures
the time in units of $\displaystyle R$: $\displaystyle t=\Delta n_{\phi}R$.

It also measures the onset of memory burden in terms of the emitted fraction
of the initial mass of a black hole as $\displaystyle\Delta M_{\rm BH}/M_{\rm
BH}=\Delta n_{\phi}/N_{\phi}$.

Equations (104) and (106) depend on an unknown parameter $\displaystyle q$.
However, independently of the precise value of this parameter, the tendency is
very clear: the domination of the memory burden takes place latest by the time
the black hole radiates away about half of its mass. That is, the expected
upper bound on memory burden time of a black hole is

$t_{\rm M}\,\sim\,S_{\rm BH}R\,.$ (107)

This is the case for relatively large $\displaystyle q$. For smaller values of
$\displaystyle q$, the memory burden can dominate sooner. For example, for
$\displaystyle q=2$, this takes place for $\displaystyle\Delta
n_{\phi}\sim\sqrt{S_{\rm BH}}$, i.e., after the time $\displaystyle t_{\rm
M}\sim\sqrt{S_{\rm BH}}R$.

Since the intrinsic gaps of the memory and master modes are very different -
$\displaystyle m_{\phi}/m_{j}\sim 1/(M_{\rm P}R)$ \- the memory burden effect
in black holes is type-$\displaystyle II$.

The coefficient of memory efficiency for a black hole can be estimated
following the reasoning of Dvali (2018a). In a black hole of mass
$\displaystyle M_{\rm BH}$, the energy difference between the most distant
information patterns is $\displaystyle\sim 1/R$. If we would store the same
pattern by a non-gravitational device of size $\displaystyle R$ on top of a
flat space vacuum, the energy cost would be of order $\displaystyle E_{p}\sim
M_{\rm p}S_{\rm BH}$. The memory-efficiency coefficient at the beginning of
evaporation, when black hole is still classical, is therefore

$\epsilon_{\rm initial}\sim\frac{1}{(M_{\rm P}R)S_{\rm BH}}\sim\frac{1}{S_{\rm
BH}^{\frac{3}{2}}}\,.$ (108)

By the onset of the memory burden effect this becomes,

$\epsilon_{\rm MB}\sim\frac{1}{RM_{\rm P}}\sim\frac{1}{S_{\rm
BH}^{\frac{1}{2}}}\,.$ (109)

In the next section we discuss the mass-splitting among the stabilized black
hole states.

###  IV.1 Spread of black hole masses

Since the mass of a stabilized black hole is a function of the memory burden
$\displaystyle E_{p}$, the prediction is that initially-degenerate black holes
over time become spread in masses, as schematically given in Fig. 1. This is
similar to the evolution of equal mass vacuum bubbles with different values of
$\displaystyle N_{\rm G}$.

In order to share with the reader a general sense of scaling, let us analyse
an oversimplified toy model in which the memory modes of a black hole are
treated as independent qubits with the Hamiltonian (11) with equal intrinsic
gaps $\displaystyle m_{j}=M_{\rm P}$. In this case, $\displaystyle
E_{p}=M_{\rm P}N_{\rm G}$. Correspondingly, the memory burden effect can be
measured by the total occupation number of the memory modes $\displaystyle
N_{\rm G}$. For $\displaystyle M$ qubits, this number is bounded by
$\displaystyle N_{\rm G}\leqslant M$ and the total number of patterns is given
by $\displaystyle n_{\rm st}=2^{M}$. This gives the entropy $\displaystyle
S_{\rm BH}\sim M$ (more precisely, $\displaystyle S_{\rm BH}=M\ln 2$).

For estimating the spread in this toy model we can use the equation (103) and
take into account that the memory burden effect becomes significant for
$\displaystyle\Delta m_{\phi}\sim 1/R$. We also take into account that for a
black hole $\displaystyle m_{\phi}\sim 1/R,~{}m_{j}\sim M_{\rm P}\sim M$ and
$\displaystyle N_{\phi}\sim M\sim S_{\rm BH}$. However, we keep $\displaystyle
N_{\rm G}$ as a free parameter.

Correspondingly, for a black hole of the initial mass $\displaystyle M_{\rm
BH}$ and the memory burden $\displaystyle N_{\rm G}$, the relative change in
mass by the time of stabilization is,

$\frac{\Delta M_{\rm BH}}{M_{\rm BH}}\,\sim\,\left(\frac{S_{\rm BH}}{q(M_{\rm
P}R)N_{\rm G}}\right)^{\frac{1}{q-1}}\,\sim\,\left(\frac{\sqrt{S_{\rm
BH}}}{qN_{\rm G}}\right)^{\frac{1}{q-1}}\,.$ (110)

One must remember that the above expression is valid as long as

$N_{\rm G}\geqslant\frac{1}{q}\sqrt{S_{\rm BH}}\,.$ (111)

This inequality represents a manifestation of the general formula (21) for a
black hole. In the opposite case the stabilization is not efficient. Thus, for
$\displaystyle N_{\rm G}$ at its lower bound (111), the stabilization takes
place at $\displaystyle\Delta M_{\rm BH}/M_{\rm BH}\sim 1$. On the other hand,
for $\displaystyle N_{\rm G}\sim S_{\rm BH}$, we recover (104).

The equation (110) tells us that the spread in masses of stabilized remnants
is determined by the statistical distribution of $\displaystyle N_{\rm G}$
among the initial black holes. For simplicity of estimate, let us assume that,
at the time of formation, the occupation number of the black hole master mode
is critical, $\displaystyle n_{\phi}=N_{\phi}$. That is, the memory patterns
are strictly degenerate. In practice, this is expected to be an extremely good
approximation, since the initial black hole is well-described classically up
to corrections $\displaystyle\sim 1/N_{\phi}$.

In such a case, we can assume that the probability distribution of patterns is
flat, with no energy bias. Correspondingly, the probability of a pattern with
given $\displaystyle N_{\rm G}$ is,

${\mathcal{P}}_{N_{\rm G}}\,=\,2^{-M}\frac{M!}{(M-N_{\rm G})!N_{\rm G}!}\,,$
(112)

which is maximal for $\displaystyle N_{\rm G}=M/2$ with

${\mathcal{P}}_{M/2}\,\sim\frac{1}{\sqrt{M}}\,,$ (113)

and the width of $\displaystyle\sqrt{M}/2$. On the other hand, for
$\displaystyle N_{\rm G}\ll M$, the probability is exponentially suppressed
as,

${\mathcal{P}}_{N_{\rm G}\ll M}\,\sim 2^{-M}\left(\frac{M{\rm e}}{N_{\rm
G}}\right)^{N_{\rm G}}\,,$ (114)

where we dropped polynomial factors for simplicity. Taking into account that
the black hole entropy is $\displaystyle S_{\rm BH}\sim M$, the above implies
that the probability of forming a black hole with memory burden $\displaystyle
N_{\rm G}\ll S_{\rm BH}$ is exponentially small. Thus, it is expected that the
most probable black holes are the ones with $\displaystyle N_{\rm G}\sim
S_{\rm BH}$.

To summarize, as it is clear from the equation (110), the change of the black
hole mass prior to its stabilization is set by $\displaystyle N_{\rm G}$ as,

$\Delta M_{\rm BH}\,\propto\,N_{\rm G}^{-\frac{1}{q-1}}\,,$ (115)

where the coefficient of proportionality, $\displaystyle M_{\rm
BH}\,\left(\sqrt{S_{\rm BH}}/q\right)^{\frac{1}{q-1}}$, is fully determined by
the initial mass of a black hole. At the same time, the memory burden
$\displaystyle N_{\rm G}$ is determined by the mass only statistically, via
(112).

This fact has potentially important observational implications since it
predicts a statistical spread of $\displaystyle\Delta M_{\rm BH}$ in addition
to the initial mass distribution determined by particularities of a
cosmological scenario (see Sec. VII).

###  IV.2 Fate of a black hole burdened by memory

What happens after a black hole enters the memory burden phase requires a more
detailed understanding of the picture. The two possible outcomes were
discussed in Dvali  _et al._ (2020).

The first option is that a new classical (collective) instability sets in and
the (former) black hole evolves through it. From our current understanding, it
cannot be excluded that due to this instability the remnant can disintegrate
via some non-linear process.

The second (more conservative) option assumes no immediate classical
instability. In such a case, the black hole continues to decay via a quantum
process. However, due to the memory burden, the process is extremely slow. In
Dvali  _et al._ (2020), the remaining life-time of a black hole was given as,

$\tau\sim R\,S_{\rm BH}^{1+k}\,,$ (116)

where $\displaystyle k>0$ is an integer. This form follows from the fact that
the prolonged life-time is an analytic function of $\displaystyle S_{\rm BH}$.
The analyticity in $\displaystyle S_{\rm BH}$ is enforced by the requirement
that the decay rate must be analytic in occupation numbers as well as in
gravitational couplings, all of which are set by $\displaystyle S_{\rm BH}$.
The case $\displaystyle k=0$ (zero memory burden) would correspond to a
standard extrapolation of Hawking’s decay rate.

The value $\displaystyle k>0$, can be understood from the following argument.
In order to continue its decay, the black hole must get rid of the memory
burden. That is, the excited memory modes must get de-excited. This can only
be done by the scattering processes that involve at least a pair of the memory
modes. The memory modes must annihilate into the modes of lower angular
momenta in order to match their energies. That is, each mode $\displaystyle
Y_{lm}$ must find a partner $\displaystyle Y_{l^{\prime}m^{\prime}}$ with very
close values of $\displaystyle l,m$. Such pairs are extremely rare and their
annihilation rate is

$\Gamma\sim\frac{\omega_{j}^{5}}{M_{\rm P}^{4}}\sim\frac{1}{R^{5}M_{\rm
P}^{4}}\,.$ (117)

In terms of the initial entropy, the life-time translates to

$\tau\gtrsim R\,S_{\rm BH}^{2}\,.$ (118)

This reinforces equation (2), giving $\displaystyle k=1$ as the most
conservative estimate.

###  IV.3 The Effect of Species

It is known that black holes provide a link between the number of QFT species
$\displaystyle N_{\rm sp}$ and the upper bound on the scale of strong quantum
gravity Dvali (2010a); Dvali and Redi (2008); Dvali (2010b); Dvali and Gomez
(2009); Dvali and Lust (2010); Dvali and Gomez (2010),

$M_{\rm sp}\,=\,\frac{M_{\rm P}}{\sqrt{N_{\rm sp}}}\,.$ (119)

The expression is fully non-perturbative and cannot be removed by any re-
summation of perturbative series. The physical meaning of the scale
$\displaystyle M_{\rm sp}$, called “species scale”, is that a black hole of
size $\displaystyle<\,1/M_{\rm sp}$, cannot be treated semi-classically. This
can be seen from a number of arguments.

For example, if the standard semi-classical evaporation rate would apply to
such a black hole, one would conclude that the black hole would live shorter
than its radius, which is absurd.

Similarly, the violation of the bound (119) is excluded by quantum information
arguments Dvali and Gomez (2009): a black hole of size $\displaystyle<1/M_{\rm
sp}$, would have the microstate degeneracy, and correspondingly the
information storage capacity, exceeding the one of Bekenstein-Hawking entropy
(97). In other words, the “species entropy” would exceed the entropy of
Bekenstein-Hawking Dvali (2021a).

In general, the increased number of species shortens the standard stage of
black hole evaporation by a factor $\displaystyle 1/N_{\rm sp}$. This has
implication for the memory burden, since the time for reaching this phase is
correspondingly shortened Dvali (2021e):

$t_{\rm M}\,\sim\,S_{\rm BH}R\frac{1}{N_{\rm sp}}\,.$ (120)

The effect of the species beyond this point is less certain, since due to
quantum hair, no universality in decay rates of different species is
guaranteed. In other words, the black hole can develop a “species hair” Dvali
(2010b). Therefore, the further decay of the remnant can be biased towards
some of the species.

As discussed in Alexandre  _et al._ (2024), already the formula (120) can have
important implications for PBH dark matter, as it can shift the masses of
memory burdened black holes towards higher values. Furthermore, assuming that
decay of the remnant remains approximately democratic in species, the life-
time in memory burden phase will subsequently be shortened by the quantity
$\displaystyle\sim 1/N_{\rm sp}$.

##  V Quantum memory burden versus classical extremality

It is well known that black holes, as well as solitons, can be stabilized
against the quantum decay by classical charges, either Noether or topological.
The term “classical” implies that the charge is classically-detectable. In
other words, the object must carry a classical “hair” with respect to this
charge.

In case of a black hole, in the light of “no-hair” theorems Ruffini and
Wheeler (1971); Bekenstein (1972); Teitelboim (1972), such are the charges
associated with massless gauge fields that can be measured via Gaussian fluxes
at infinity 222 The seeming exceptions, such as a classical skyrmion hair
Luckock and Moss (1986); Bizon and Chmaj (1992); Droz  _et al._ (1991), have
been shown Dvali and Gußmann (2017, 2016) to be equivalent to Aharonov-Bohm
type hair under a discrete gauge symmetry Krauss and Wilczek (1989); Preskill
and Krauss (1990).. Usually, the amount of charge capable of stabilization is
comparable to the mass of a black hole (or a soliton).

We now wish to confront a mechanism of stabilization of an object by a
classical hair with the one of a quantum memory burden. For achieving this, we
first need to understand the stabilization of a black hole (or a soliton) by a
classical charge in a fully quantum language of a microscopic theory.

The first steps in this direction were undertaken in Dvali and Gomez (2013a).
This microscopic description was further generalized to topological solitons
in Dvali  _et al._ (2015a).

In black hole’s quantum $\displaystyle N$-portrait Dvali and Gomez (2013a), a
purely gravitational black hole represents a critical coherent state or a
condensate of master mode gravitons. The typical energy gaps of these modes
are $\displaystyle m_{\phi}\sim 1/R$ and their mean-occupation number,
$\displaystyle n_{\phi}$, is critical, $\displaystyle n_{\phi}\sim S_{\rm
BH}$.

Correspondingly, as in any other system with enhanced information capacity,
prior to stabilization, the master modes are the main contributors into the
mass of an initially-classical black hole,

$M_{\rm BH}\sim n_{\phi}\,m\sim S_{\rm BH}\frac{1}{R}\,.$ (121)

In this quantum picture, the existence of a classical charge of a black hole
means that, on top of the master mode, some other mode carrying this
particular charge becomes macroscopically occupied. In other words, a second
master mode appears.

For example, the electrically charged Reissner-Nordström black hole contains a
macroscopic occupation number of photons. These photons compose a coherent
state describing the classical electric field of the black hole. That is, when
a black hole carries a classical charge, the “status” of a master mode is
shared between the master graviton and some other mode.

Now, as argued in Dvali and Gomez (2013a), the occupation number of the master
graviton sets the upper bound on other occupation numbers. That is, for a
black hole the occupation numbers of master modes satisfy the bound,

$n_{\rm any~{}mode}\leqslant n_{\phi}\,.$ (122)

In this way, the theory Dvali and Gomez (2013a) provides a quantum explanation
of why a black hole charge can never exceed its mass. Notice that this is true
for an arbitrary saturon. In particular, (122) is satisfied by saturated
vacuum bubbles discussed in section III.

According to Dvali and Gomez (2013a), a black hole reaches extremality when
the occupation number of a certain mode, other than the master graviton,
saturates the above bound. In this description, the stability of an extremal
Reissner-Nordström black hole can be viewed as the saturation of the bound
(122) by the photon constituents of a black hole.

It is intuitively clear Dvali and Gomez (2013a) as well as evident from
computations Dvali and Eisemann (2022) that in this case the black hole
evaporation must stop. The reason is that for maintaining the saturation of
the bound (122), a further lowering of the occupation number of the graviton
master mode, $\displaystyle n_{\phi}$, must be accompanied by the depletion of
the electric field. This is not possible without emission of the charged
quanta. However, the fact that the black hole of size $\displaystyle R$ has an
unscreened electric charge to start with, implies that in the theory the mass
of the lightest charged particle, $\displaystyle m_{\rm e}$, satisfies:
$\displaystyle m_{\rm e}\gg 1/R$. In the opposite case, the charge would be
screened by the Schwinger effect.

Correspondingly, the quantum emission of electric charge is exponentially
suppressed, since such a process requires a rescattering of many soft
constituent quanta (of energies $\displaystyle\sim 1/R$) into a quantum of
energy $\displaystyle m_{\rm e}\gg 1/R$. The transition rate of such processes
is bounded from above by $\displaystyle e^{-m_{\rm e}R}$, as indicated by a
general argument given in Dvali (2021a). This is confirmed by explicit
computations of corresponding multiparticle processes Dvali and Eisemann
(2022). The suppression of the transitions many$\displaystyle\rightarrow$few
can also be extracted from explicit computations of graviton scattering
processes Dvali  _et al._ (2015c); Addazi  _et al._ (2017). The analogous
suppression is exhibited by scalar theories Brown (1992); Argyres  _et al._
(1993); Voloshin (1992); Gorsky and Voloshin (1993); Libanov  _et al._ (1994,
1995); Son (1996); Monin (2018).

Thus, the characteristics of black hole stabilization by a classical charge is
the existence of a long-range classical hair with respect to that charge. At
the same time, the quantum information pattern carried by the black hole plays
no role. For example, the extremal black holes of the same electric charge and
mass can carry information patterns of very different content.

In contrast, when a black hole is stabilized by a quantum memory burden, no
classical charge associated with a long-range gauge field is required. Of
course, a macroscopic “hair” emerges in form of a memory burden parameter,
$\displaystyle N_{\rm G}$. This parameter is in principle measurable by a
scattering experiment, but there is no inconsistency with the no-hair
properties: since the memory-burdened black hole is a quantum object, it has
no reason to respect classical no-hair theorems. Therefore, unlike classical
extremal case, there is no specific gauge charge or a long-range Gaussian flux
associated with the quantity $\displaystyle N_{\rm G}$.

The difference between the two mechanisms is also clear from the fact that the
decay of a memory-burdened black hole is not suppressed exponentially but only
slows down as a power-law (2).

Interestingly, the solitons stabilized by the memory burden unify both
features. The main reason is the existence of two types of microstates
discussed above.

The first category of microstates have equal $\displaystyle N_{\rm G}$ but are
distinguished by the relative $\displaystyle SU(N)$-transformations. These
remain exactly degenerate even after the memory burden sets in. The
microstates belonging to the second category are distinguished by values of
$\displaystyle N_{\rm G}$. The energies of such microstates get split after
stabilization.

Both sets of microstates have counterparts in a black hole. As discussed, the
microstates of a black hole correspond to degeneracy of patterns obtained by
occupation numbers of different memory modes $\displaystyle Y_{lm}$. Among
these states, there exist both categories: with equal and distinct memory
burdens. As time elapses, the level splitting develops according to the memory
burdens they carry, as given in Fig. 1.

Another example of a classical “charge” that can stabilize a black hole
against quantum decay is the angular momentum. The similar case in solitons
reveals a certain interesting peculiarity.

The stationary $\displaystyle SU(N)$-bubble with non-zero spin is obtained via
giving the winding number $\displaystyle n$ to the Goldstone mode Dvali  _et
al._ (2022b). Basically, the matrix $\displaystyle U$ in (74) is replaced by

$U=e^{i\left(t\omega+n\varphi/\sqrt{2}\right)\frac{\omega_{j}}{\omega}T^{j}}\,,$
(123)

where $\displaystyle\varphi$ is the polar angle and the factor
$\displaystyle\sqrt{2}$ is added to ensure the phase continuity.

The winding of the Goldstone phase produces the vorticity and, simultaneously,
the angular momentum,

$J=\int{\rm d}^{3}x\,T_{0\varphi}=n\,N_{G}=n\,Q\,,$ (124)

where $\displaystyle T_{\mu\nu}$ is the energy momentum tensor.

The fact that the winding number $\displaystyle n$ induces the spin given by
(124) has been known for $\displaystyle Q$-balls with $\displaystyle
U(1)$-charge Kim  _et al._ (1993); Volkov and Wohnert (2002); Kleihaus  _et
al._ (2005). The remarkable thing about the saturated $\displaystyle SU(N)$
vacuum bubbles is that the equation (124), describing the relation between the
maximal spin and the entropy, is strikingly similar to the one satisfied by a
black hole Dvali  _et al._ (2022b),

$J_{\rm max}=S\,.$ (125)

A saturon vacuum bubble of $\displaystyle SU(N)$, which has entropy

$S=N_{\rm G}\,,$ (126)

reproduces the relation (125) due to the limited vorticity, $\displaystyle
n\sim 1$, that it can sustain. Attaining a higher spin would require a larger
vorticity, $\displaystyle n\gg 1$, which would increase the mass and the size
of the bubble, thereby making it undersaturated.

Note that the relation (125) is shared by other saturons. For example, it is
automatic in saturated baryons in QCD with large number of colors and flavors
Dvali (2021b).

The same relation is shown Dvali  _et al._ (2024b) to be shared by a saturated
version of a spinning cosmic string loop ( a so-called vorton Davis and
Shellard (1989)).

To summarize, the saturon bubbles of maximal spin exhibit the exact same
relation between spin and entropy as the extremal spinning black holes. At the
same time, in extremal saturon bubbles this relation reveals an explicit
microscopic meaning in terms of vorticity.

Notice also that vorticity gives a topological meaning to the quantum
stability of the extremal bubble Dvali  _et al._ (2022b). Indeed, the decay of
a bubble via the particle emission is blocked because of conservation of the
topological winding number. The decay is only possible if the entire vortex is
ejected via a quantum tunnelling which is an exponentially suppressed process.

In this sense, the extremal spin represents another form of a “classical”
memory burden. As in the case of the burden induced by a pure $\displaystyle
SU(N)$-charge, the burden induced by the spin is due to a critical occupation
number $\displaystyle N_{\rm G}$ of a Goldstone/memory mode. For a saturon
bubble, $\displaystyle\omega\sim m$, this memory mode becomes an additional
master mode. The difference from non-spinning case is that the
macroscopically-occupied memory mode has a non-zero angular momentum.

This analogy is fully extended to black holes if we conjecture that a spinning
black hole corresponds to the one in which a mode of a particular angular
harmonic $\displaystyle Y_{lm}$ is macroscopically occupied. In this light, it
is interesting to discuss a conjecture of Dvali  _et al._ (2022b) stating that
the extremal spin in a black hole is accompanied by vorticity, similarly to
the case of a spinning saturon bubble. Again, as in the case of a bubble, the
vorticity gives a topological explanation to the quantum stability of the
extremal black hole. In the case of the spin, similarly to other classical
charges, the extremal black hole does carry a classical hair in form of the
angular momentum.

To summarize, there exist fundamental differences between the black holes
stabilized via classical charges and the ones stabilized by a quantum memory
burden effect. One obvious difference is that the classical extremality is
necessarily accompanied by a long-range classical gauge hair.

This is not the case for a black hole stabilized by the memory burden effect.
Instead, for a memory-burdened black hole, the hair is short-range and carries
no Gaussian flux at infinity.

More importantly, a memory-burdened black hole carries a significant quantum
hair. The significance is measured by the fact that the energy of the quantum
information pattern is comparable to the mass of the remnant.

Such a remnant is neither a black hole nor classical, and, therefore, is not
constrained by classical no-hair theorems. Although macroscopic, it
nevertheless is a quantum entity. This “macro-quantumness” is a feature of
black holes Dvali and Gomez (2014b) as well as of other saturons Dvali
(2021a).

The above important difference between the quantum memory-burdened and (quasi)
extremal black holes must be taken into account when applying such mechanisms
of stabilization to light PBH dark matter. In particular, if the naive semi-
classical time-scale required for approaching extremality is longer than the
memory burden time $\displaystyle t_{\rm M}$, the latter effect becomes the
dominant one and must be taken into account.

In addition to differences at a fundamental level, this will have
observational consequences. For example, the memory-burdened PBH dark matter
Dvali  _et al._ (2020, 2021); Alexandre  _et al._ (2024); Thoss  _et al._
(2024); Balaji  _et al._ (2024); Haque  _et al._ (2024) will be subjected to
the mass-spread discussed in the present paper. This spread is absent for PBHs
stabilized by extremality within the validity of semi-classical regimes, such
as discussed in de Freitas Pacheco  _et al._ (2023) (and references therein).

The fundamental difference between the prolongations of a black hole life-time
due to a classical hair versus the quantum memory burden effect, extends
beyond the ordinary black holes. The example is provided by a black hole
smaller than the compactification radius in the framework of large extra
dimensions Arkani-Hamed  _et al._ (1998).

Such a black hole is a solution of Einstein’s equations in high-dimensional
gravity and its radius and temperature are defined by the fundamental Planck
mass, $\displaystyle M_{\rm F}$, which is suppressed relative to the four-
dimensional one by the volume of extra space Arkani-Hamed  _et al._ (1998):

$M_{\rm F}\,=\,\frac{M_{\rm P}}{\sqrt{V_{\rm extra}}}\,.$ (127)

Correspondingly, the high-dimensional black holes are colder relative to
would-be four-dimensional black holes of the same mass. It was proposed in
Anchordoqui  _et al._ (2024, 2022) that such black holes can serve as dark
matter.

Let us confront the Kaluza-Klein (KK) hair of young high-dimensional black
holes with the memory burden effect. Of course, the question arises solely
from the point of view of a four-dimensional observer, since from high-
dimensional perspective these are usual classical black holes.

From four-dimensional perspective, the decay of high-dimensional black holes
can be understood in terms of species. First, as discussed in Dvali (2010a),
the equation (127) can be viewed as a particular case of (119), with the role
of particle species played by KK gravitons and the role of the four-
dimensional species scale assumed by the fundamental Planck mass
$\displaystyle M_{\rm sp}=M_{\rm F}$.

When evaluating the evaporation rate, we must take into account that the black
hole is smaller than the compactification radius, correspondingly, it develops
the species hair Dvali (2010b). That is, the black hole sources a tower of
massive KK gravitons to which it also evaporates. The evaporation into many KK
species is equivalent to the emission of a single high-dimensional graviton
Dvali and Lust (2010).

Therefore, the evaporation of a young black hole is well-described semi-
classically through a high-dimensional Hawking process. From the point of view
of a four-dimensional observer it is an object that caries a classical species
hair Dvali (2010b) under KK gravitons. Such a classical hair is fundamentally
different from the quantum memory burden that sets in at a later stage.

This is already evident from the fact that quantum memory burden will be
experienced by a high dimensional black hole after it gets sufficiently “old”.
Extrapolating our results, we can say that this will happen the latest by
half-decay. Therefore, for a high-dimensional PBH, the quantum memory burden
effect must be taken into account in the same way as for ordinary Einsteinian
black holes.

##  VI Numerical Analysis: stabilization of solitons

We summarize here our numerical findings. We focus on the dynamics of
underburdened solitons, whose evolution, as discussed in Sec. V, offers
analogy with evaporating black holes that are stabilised by their quantum
memory.

The idea is to start with a vacuum bubble, with the value of the charge
$\displaystyle N_{\rm G}$ given by (92). As discussed in Subsec. III.2, this
amount of charge is insufficient for an immediate stabilization. In
particular, for such a bubble the energy of the memory pattern is negligible
as compared to the energy of the master mode (92), which constitutes the main
source of the energy.

Such a bubble oscillates, emitting energy and correspondingly shrinking in
size. We map this stage of evolution on the one of a black hole before
entering the memory burden phase.

The decay process of the bubble continues up until the Goldstone charge
fulfils the condition (90). Simultaneously, the energy of the memory pattern
catches up with the energy of the master mode (86). Correspondingly, the
bubble is stabilized by the burden of memory. At this point, the bubble
becomes a stationary $\displaystyle Q$-ball.

Let us denote the reference stationary $\displaystyle Q$-ball charge by
$\displaystyle Q_{\rm s}=N_{\rm G}$ and its frequency by
$\displaystyle\omega_{\rm s}$. We then vary the initial charge by changing the
frequency of constituents, while keeping the profile fixed, to the following
values

  * •

$\displaystyle\omega=3\omega_{\rm s}/4$, $\displaystyle Q=3Q_{\rm s}/4$

  * •

$\displaystyle\omega=\omega_{\rm s}/2$, $\displaystyle Q=Q_{\rm s}/2$

  * •

$\displaystyle\omega=\omega_{\rm s}/4$, $\displaystyle Q=Q_{\rm s}/4$

  * •

$\displaystyle\omega=0$, $\displaystyle Q=0$.

resulting in underburdened configurations of different $\displaystyle N_{\rm
G}$ (therefore, not related by a $\displaystyle SU(N)$ transformation) in the
regime (92) and (91).

Our numerical analysis supports the analytical discussion of the previous
Sections. Namely, the bubbles of different initial charges - but of similar
energies - while they have analogous dynamics at initial times, are stabilised
towards $\displaystyle Q$-ball configurations of different masses.

The statement holds true for all regimes analysed in our study - regardless of
other details such as the winding number, the inclusion of new interactions
and the thickness of the bubble wall. This shows that the mechanism of
stabilization by memory is a general consequence of the high-memory storage
capacity of the object and is only secondarily affected by other features,
such as, for example, saturation or vorticity. Consequently, this further
supports the prediction of Subsec. III.3 that an analogous mass splitting is
expected in the asymptotic distribution of black holes of equal initial masses
but different memory burdens.

The rest of this Section is dedicated to showing how the above statement comes
about. For numerical simplicity, we focus on the $\displaystyle SU(4)$
symmetric case. Unless otherwise stated, we fix $\displaystyle\alpha=m=1$ in
units of $\displaystyle f$ and choose $\displaystyle\omega=0.3\,m$. Since we
are interested in the winding $\displaystyle n=0$ and $\displaystyle n=1$
cases, a $\displaystyle 2+1$ dimensional analysis is sufficient. Finally, some
of the simulation visuals can be found at the following URL.

### Winding $\displaystyle n=0$ case

Figure 2: Energy density evolution for different initial charges and winding
$\displaystyle n=0$. The third column shows the integrated charge and energy
as functions of time.

The energy density evolution is shown in Fig. 2. For $\displaystyle Q=3Q_{\rm
s}/4$ (first panel), the system is close to the stationary case, and therefore
has a significant burden. In fact, the bubble starts collapsing, but the
information (charge) stored within immediately backreacts on the dynamics.
This, in turn, results in an oscillatory behaviour analogous to what was
already observed in Dvali  _et al._ (2022a). The reason the bubble does not
relax to its stationary configuration is energetic. Simply, the excitations
are smaller than the mass gap outside of the configuration. As a consequence,
(almost) no charge is emitted and the information is retained within the
bubble. This is a manifestation of the existence of the information horizon
Dvali  _et al._ (2022a). The notion becomes exact in the semi-classical limit
which corresponds to large-$\displaystyle N$.

This can be seen explicitly in the third column of Fig. 2 where the integrated
energy and charge densities are shown as functions of time (the integrated
charge, corresponding to the blue line, is obscured by the cyan curve of the
case $\displaystyle Q=Q_{\rm s}/2$).

An analogous situation is observed in the second panel of Fig. 2. Since in
this case the initial charge is smaller - $\displaystyle Q=Q_{\rm s}/2$ \- the
resulting oscillations are more pronounced. Still, the burden forbids the
collapse, and the energy gap ensures that almost no charge is emitted. The
oscillatory frequency of the profile in both cases is roughly given by
$\displaystyle R^{-1}$ as it can be qualitatively seen from the plot. A
perturbative analysis of the system energy would inform us of this fact, as
already shown in Dvali  _et al._ (2022a).

In the first panel of the second row, for $\displaystyle Q=Q_{\rm s}/4$, the
energy of the collapse is sufficient for overcoming the mass gap outside of
the $\displaystyle Q$-ball. As a consequence, charge is emitted and part of
the information is lost in the first oscillation as it can be seen in Fig. 2.
At later stages, the pulsation energy becomes insufficient for exciting new
quanta, resulting in a stationary configuration with smaller charge and
radius. For analysis of such excitations in $\displaystyle U(1)$ symmetric
$\displaystyle Q$-balls, see e.g., Saffin  _et al._ (2023).

Finally, in the zero charge no burden is present to stop the collapse. The
system is effectively a bubble interpolating between two degenerate vacua.
Therefore, the collapse due to the surface tension takes place within one
oscillation and is rather violent. However, for smaller values of coupling,
the oscillation can last longer.

### Winding $\displaystyle n=1$ case

Figure 3: Energy density evolution for different initial charges for winding
number $\displaystyle n=1$. The third column shows the integrated charge and
energy and angular momentum as functions of time

Memory burden has an analogous stabilizing impact on underburdened bubbles
endowed with vorticity characterized by winding number $\displaystyle n=1$. We
refer the reader to Zantedeschi (2022) for the numerical construction of the
stationary solution.

As discussed in Sec. V, vorticity is realized in terms of a macroscopic
occupation number of Goldstones under the ansatz (123), leading to non-
vanishing angular momentum according to (124).

Since saturated bubbles with vorticity and black holes obey the same maximal-
spin bound - c.f., (125) - one might be tempted to map underburdened bubbles
with vorticity on evaporating black holes dynamically approaching extremality.
Of course, within the semi-classical picture, whether or not a black hole
evolves towards such point, is determined by the particle spectrum
participating in the emission Page (1976). However, the semi-classical
analysis does not take into account the memory burden effect which invalidates
it. This must be kept in mind when mapping the quantum evolution of a black
hole on the dynamics of underburdened bubble in our numerical analysis.

The first panel in Fig. 3 shows the evolution of energy as a function of time
for the case $\displaystyle Q=3Q_{\rm s}/4$. Although the simulation is
performed in two spatial dimension, for illustrative purposes, we slice here
along the $\displaystyle x$ coordinate, fixing $\displaystyle y$ along the
center of the vortex. The situation is similar to the $\displaystyle n=0$ case
with similar initial charge. Notice that in this case the energy of the vortex
in the central region oscillates together with the pulsating radius of the
configuration. The period of pulsation also here is of order $\displaystyle
R^{-1}$ \- $\displaystyle R$ being the radius of the stationary configuration
(and therefore, also the initial radius at the simulation time). As expected,
the bubble is stabilized by the conserved charge and the angular momentum in
its interior - see third column of Fig. 3.

A similar behaviour is observed in the second panel for $\displaystyle
Q=Q_{\rm s}/2$. Indeed here the pulsations are more pronounced due to the
larger initial off-balance between memory and master mode energy.
Concomitantly, a mild emission of energy and charge is observed through the
initial collapse. Notice that at the end of the simulation, at large time, the
axial symmetry is broken. Consequently, instability mode grows in time,
eventually leading to the emission of the vortex. The numerical duration of
the simulation is insufficient to capture the impact of this phenomenon on the
integrated quantities in Fig. 3. However, we have already characterized it in
Dvali  _et al._ (2024a) when studying soliton mergers. Therein, we noted that
a vortex localized in the merged configuration is unstable, therefore
resulting in its eventual ejection. Examples of such dynamics can be found at
the following URL1 and URL2.

A different behaviour is observed in the first panel of the second row for
$\displaystyle Q=Q_{\rm s}/4$. In this case, the first collapse is extremely
violent, and one might naively expect the features analogous to the case
$\displaystyle n=0$. Instead, only a negligible charge is emitted by the
configuration, which is localized in a large ring around its center. The ring
is, effectively, a slowly rotating (quasi-)oscillon. Eventually, due to the
broken axial symmetry, for $\displaystyle t\gtrsim 120m^{-1}$, the system
fragments into smaller $\displaystyle Q$-balls (a phenomenon known in the case
of $\displaystyle U(1)$-symmetric $\displaystyle Q$-balls, see, e.g., Battye
and Sutcliffe (2000); Kinach and Choptuik (2023)), which cannot be visualized
due to the $\displaystyle y$ cut of the plane performed in the figure. The
phase keeps rotating between the fragmented $\displaystyle Q$-balls due to the
angular momentum of the configuration. Correspondingly, significant amount of
energy and angular momentum are expelled from the configuration, as can be
visualized at the following URL and Fig. 3 show.

Analogous dynamics, eventually leading to fragmentation is observed also for
vanishing charge (red line). Energy is discretely emitted from the
configuration as the outer wall of the bubble squeezes the central vortex.
These correspond to the three observed pulses in the fourth panel of Fig. 3.
Characterization of both phenomena of fragmentation, as well as vortex
ejection, are beyond the scope of this work.

### Derivative interaction case

Figure 4: Energy density evolution for different initial charges for winding
number $\displaystyle n=0$ (top row) and $\displaystyle n=1$ (bottom row) for
non-vanishing interaction with derivatively coupled singlet
$\displaystyle\chi$. Third column shows the integrated charge and energy as
functions of time

In order to make the analogy between bubbles and black holes maximally
transparent, and to allow for a significant relaxation of the configuration
pulsations, we further extend the system by derivatively coupling the soliton
field to a $\displaystyle SU(N)-$singlet scalar $\displaystyle\chi$ via the
Lagrangian,

$\mathcal{L}\supset\frac{1}{2}\partial_{\mu}\chi\,\partial^{\mu}\chi+\frac{g}{2}\chi\,{\rm
Tr}(\partial_{\mu}\Phi)(\partial^{\mu}\Phi)-\frac{\lambda}{4}\chi^{4}\,.$
(128)

The above interaction term is sensitive only to the master mode of the
saturon, while leaving the charge unaffected. The mechanism of stabilization
by memory proceeds similarly to the previous cases.

In units of $\displaystyle m$, the couplings are $\displaystyle g\simeq 0.5$
and $\displaystyle\lambda\simeq 0.15$ and we further initialize
$\displaystyle\chi$ to be vanishing. The resulting energy density evolution,
adopting analogous initial conditions to the cases discussed in the previous
Subsection, are displayed in Fig. 4. The top row shows the dynamics for the
case of winding number $\displaystyle n=0$, while the bottom corresponds to
the case with vorticity. The cases with charge $\displaystyle Q=3Q_{\rm s}/4$
(first column) and $\displaystyle Q=Q_{\rm s}/2$ (second column) are
sufficient for our discussion.

Although $\displaystyle\chi$ is massless, a full relaxation is not possible.
In fact, the derivative interaction sources a tadpole for $\displaystyle\chi$,
which therefore localizes on the bubble support. A quartic coupling has been
added in (128), in order to tame the growth of $\displaystyle\chi$. This setup
effectively generates a local mass gap for $\displaystyle\chi$ of magnitude
$\displaystyle\sim(\sqrt{g}\lambda\,\omega^{2}\,f^{2})^{1/3}$. This backreacts
on the capacity of the system to fully relax to a stationary configuration,
and constitutes the main reason we observe oscillations - although much less
pronounced - at late times. This is in stark contrast to the case without
singlet $\displaystyle\chi$ of the previous Subsection cf., Figs. 2 and 3, in
which the bubble simply pulsates with almost constant frequency and amplitude.

The integrated energy confirms this behaviour as shown in the third column of
Fig. 4. Clearly, significantly more energy is emitted in the case of initial
charge $\displaystyle Q=Q_{\rm s}/2$ (red lines) as this is further away from
being stationary and, therefore, has more energy available for the emission.
In general, for $\displaystyle n=1$ (dashed lines), the energy damping is both
faster and more efficient, due to the extra gradient energy of the bubble
background around the vortex region.

Figure 5: Energy density evolution for different initial charges for winding
number $\displaystyle n=0$ (top row) and $\displaystyle n=1$ (bottom row) for
vanishing (left column) and non-vanishing (central-column) interaction with
derivatively coupled singlet $\displaystyle\chi$. The third column shows
energy and charge evolution as functions of time for $\displaystyle n=0$
winding.

Nevertheless, charge is well-conserved - within $\displaystyle 1\%$ \-
throughout the simulation time. This is expected given the nature of the
coupling, not sensitive to the flavour of the bubble.

Compared to the case without $\displaystyle\chi$, the charge of the bubble is
seen to be significantly more oscillatory. Notice that the derivative
interaction leads to a redefinition of charge by a factor $\displaystyle
1+g\chi$. In fact, the oscillatory behaviour is seen in correspondence of
$\displaystyle\chi$ oscillations on the bubble support. The effect is larger
at the beginning of the simulation since the $\displaystyle\chi$ field is very
far from equilibrium and this leads to violent oscillations, modestly
challenging the capabilities of our numerical implementation (as expected,
given that we are dealing with a derivative interaction term)333 In order to
be exactly charge and energy conserving, an indirect inversion method should
be used when computing the time-evolution involving a matrix of both
$\displaystyle\phi$ and $\displaystyle\chi$ at each lattice point. This is not
very efficient, as a huge matrix should be inverted at once, leading to a
significant restriction on the lattice dimension. Instead, we adopt a direct
method, based on finite difference, and compute each $\displaystyle t+1$
component of the field restricting to neighbouring points from time slices at
$\displaystyle t$ and $\displaystyle t-1$ \- $\displaystyle t$ being the index
characterizing the step number. In practice, an exact method for the
$\displaystyle\phi$ field evolution is used, evaluating all time derivatives
with central scheme at the time-slice $\displaystyle t$ and using it to obtain
the $\displaystyle t+1$ component. However, then to update the
$\displaystyle\chi$ field it becomes necessary to use a backward derivative
for source term proportional to $\displaystyle g$. This induces an error in
our algorithm dependent on the time-step size which, we verified, reflects -
partly - the amplitudes of the peaks in the first panel of the third column of
Fig. 4..

Finally, analogously to the case of the previous Subsection, in the last panel
we can observe the onset of the vortex ejection phenomenon towards the final
stage of the evolution.

### Thick Wall regime

So far, our numerical analysis focused on bubbles in the thin-wall regime.
Although such bubbles are undersaturated in terms of the bound (4), they
nevertheless represent the systems of enhanced capacity of information
storage. This is fully sufficient for understanding the key features of the
memory burden effect and mapping it on black holes. In fact, the thin-wall
bubbles are very well suited for capturing the type-$\displaystyle II$ nature
of the black hole memory burden effect.

On the other hand, in the thick-wall regime the bubble/$\displaystyle Q$-ball
represents a saturon (see discussion around Eq. (69)). This has advantage of
reproducing the feature of a black hole in terms of the area-law scaling of
entropy (4). Therefore, we shall also study this case.

As we will see, at a qualitative level, statements analogous to the ones of
the previous Subsection hold also in this regime, providing further evidence
for the generality of the stabilization by the burden of memory. However,
there are some quantitative differences, since the memory burden effect in
thick-wall bubbles is of type-$\displaystyle I$.

The two panels in the third column of Fig. 5 show the energy and the charge of
the configuration as functions of time for the four cases of underburdened
bubbles, corresponding to the four different colors and compare the cases
without (continuous) and with (dashed line) the derivatively-coupled singlet
$\displaystyle\chi$. For simplicity, we focus on the case of zero winding.
Moreover, we chose $\displaystyle\omega_{\rm s}=0.8\,m$, while keeping other
parameters unchanged.

One notable feature in the thick wall regime is that the constituents within
the bubble have larger frequency. Therefore, it is easier to excite modes
above the asymptotic mass gap, as compared to thin-wall case. This, in turn,
leads to a progressively larger relative emission of charged particles as the
initial charge of the configuration decreases. This is due to the fact that
the system is in the type-$\displaystyle I$ regime, which, for larger
$\displaystyle\omega$, has a worse storing efficiency - see Eq. (89).

Remarkably, charge emission seems alleviated in presence of
$\displaystyle\chi$-field, as another channel becomes available for
relaxation, allowing the system to reach equilibrium while retaining
information. On the other hand, a significant amount of pulsation energy is
depleted due to the presence of the singlet.

A further comment related to the behaviour of the system at small and
vanishing charge (green and red line respectively) is due. In this case, the
system does not collapse, as opposed to the thin wall scenario. Instead, only
a small fraction of the initial energy is emitted; the reason being that the
“oscillon” configuration is not sufficiently excited to produce asymptotic
quanta. This basically results in stationary configurations. Equally stated,
the oscillation frequency of the oscillon is smaller than the asymptotic mass
gap.

For completeness, we report in Fig. 5 also the energy density evolution of
underbudened bubbles in the thick-wall regime for the winding numbers
$\displaystyle n=0$ (first row) and $\displaystyle n=1$ (bottom row). In the
left column the singlet coupling is $\displaystyle g=0$, while in the central
column $\displaystyle g=0.6$. In the $\displaystyle n=1$ case, we can observe
a marginal energetic emission due to the presence of the singlet
$\displaystyle\chi$ channel of about $\displaystyle 5\%$-$\displaystyle 10\%$
of its total energy.

Noticeably, in the third panel, a vortex ejection takes place at
$\displaystyle t\simeq 150m^{-1}$. Correspondingly, the configuration looses
about $\displaystyle 40\%$ ($\displaystyle 30\%$) of its initial energy
(charge). Moreover, since the vortex is responsible for the spin, its ejection
causes a drop of about $\displaystyle 90\%$ of the total angular momentum (we,
once gain, refer the reader to Dvali  _et al._ (2024a) for a detailed
discussion of this phenomenon). In the case of derivative interaction (central
panel) the ejection takes place later, towards the end of the simulation time,
indicating that the derivative coupling can stabilize the vortex within the
$\displaystyle Q$-ball support.

##  VII Implications and Outlook

The goal of the present paper was to further investigate the physical nature
and the extent of universality of the memory burden phenomenon Dvali (2018a);
Dvali  _et al._ (2020). After introducing the essence and generic features of
the phenomenon, we have studied its manifestations in solitons building up on
the previous work Dvali  _et al._ (2022a).

We have established a close correspondence with the expected features of the
memory burden phenomenon in black holes. In this analysis, we took a double
approach. On one hand, following Dvali (2018a); Dvali  _et al._ (2020), we
have modelled the memory burden effect in black holes relying exclusively on
their well-established features. These features unambiguously place black
holes in the category of objects of enhanced capacity of information storage
susceptible to the memory burden effect.

On the other hand, we used a microscopic theory of black hole’s quantum
$\displaystyle N$-portrait Dvali and Gomez (2013a, 2014a) as the reference
point for the consistency checks of our results.

The emerging picture speaks in favor of close similarities between black holes
and other objects of enhanced capacity of information storage, such as the
solitonic saturons Dvali (2021b, c, a, d); Dvali and Venugopalan (2022); Dvali
and Sakhelashvili (2022); Dvali  _et al._ (2022a, b, 2024a).

We have confronted the stabilization via memory burden with the stabilization
by means of the classical hair and outlined the fundamental differences
between the two mechanisms.

We have predicted that the memory burden effect induces the mass splitting
between initially-degenerate black holes according to differences in their
information patterns.

We have performed numerical analysis for various regimes of the solitonic
memory burden, with or without the topological winding numbers. The results of
these numerical simulations fully confirm our analytic conclusions.

Let us now briefly go over some implications of the memory burden effect and
the future prospects.

### New dark matter window for PBH

As put forward in Dvali  _et al._ (2020), one immediate implication for black
hole stabilization by the memory burden effect is the opening of a new window
for PBH dark matter in the range of masses below $\displaystyle\sim 10^{14}$g.
The idea of PBH goes back to Zel’dovich and Novikov (1967); Hawking (1971);
Carr and Hawking (1974), and the proposal of PBH composition of dark matter is
also old Chapline (1975) (for a review, see Carr  _et al._ (2016)). However,
in the standard treatment, in which one extrapolates the Hawking regime till
the very end of black hole existence, the PBH of masses below
$\displaystyle\sim 10^{14}$g were assumed to be excluded from dark matter,
since they were expected not to survive till the current epoch.

The understanding that black holes undergo the memory burden effect Dvali
(2018a) and can get stabilized by it Dvali  _et al._ (2020), drastically
changes the standard view. PBHs in the wide range of masses below
$\displaystyle 10^{14}$g can now account for the entirety of dark matter. The
proof-of-concept type examples of light PBHs (e.g., with masses
$\displaystyle\sim 10^{8}$g) which satisfy all the known constraints, were
originally given in Dvali  _et al._ (2020).

Furthermore, the possibility of dark matter in the form of PBHs stabilized by
memory burden in the mass range above $\displaystyle\sim 10^{4}\rm g$
(corresponding to most conservative value $\displaystyle k=1$ in (116)) was
discussed in Dvali  _et al._ (2021). This paper proposes an explicit
cosmological mechanism for the formation of such PBHs in the right abundance.

A further analysis of various constraints on light ($\displaystyle\lesssim
10^{4}\rm g$) PBH dark matter stabilized by memory burden was offered in two
recent papers Alexandre  _et al._ (2024); Thoss  _et al._ (2024). In
particular, it was shown that PBHs that enter the memory burden phase prior to
BBN epoch, are unconstrained from BBN and CMB. For the most conservative
estimate of the post memory-burden lifetime, corresponding to $\displaystyle
k=1$ in (2), these are PBHs with masses below $\displaystyle 10^{9}$g. Such
PBHs can easily compose the entirety of dark matter. Further implications of
the memory burden effect for PBHs can be found in Balaji  _et al._ (2024);
Haque  _et al._ (2024) (see Riotto and Silk (2024) for some future prospects).

The analysis of the present paper predicts a spread in PBH dark matter masses,
regardless of their production mechanism. This is due to the mass-splitting of
the remnants evolving from initially-degenerate black hole states, as shown in
equation (110) and schematically described by Fig. 1.

Let us consider a black hole, formed at time $\displaystyle t_{\rm f}$ with
initial mass $\displaystyle M_{\rm BH}(t_{\rm f})$, which entered the memory
burden phase around some time $\displaystyle t_{\rm M}$, prior to today’s
Hubble time. For definiteness, let us assume that the subsequent change of the
mass is negligible. For instance, this will be the case for $\displaystyle
k>0$ and $\displaystyle M_{\rm BH}(t_{\rm f})\gtrsim 10^{4}$g.

However, the mass of PBH at stabilization is determined by the memory burden
that it carries (115). To the leading order, the burden is controlled by the
occupation number of the memory modes, $\displaystyle N_{\rm G}$. As we have
discussed, statistically, the most probable value of this number is of the
order of the PBH initial entropy $\displaystyle S_{\rm BH}$. The probability
of forming a black hole with memory pattern $\displaystyle N_{\rm G}\ll S_{\rm
BH}$ is exponentially small. This implies that most of the PBHs that we
observe today as dark matter carry the memory burdens of order the initial
entropy $\displaystyle N_{\rm G}\sim S_{\rm BH}$, with the statistical spread
given by (112). This translates as the corresponding spread in PBH masses via
(115).

In particular, PBHs of masses $\displaystyle M_{\rm BH}\lesssim 10^{14}{\rm
g}$ are subjected to this spread. Notice that in case of existence of a large
number of hidden particle species, the time $\displaystyle t_{\rm M}$ gets
shortened as (120) Dvali (2021e). Due to this, the memory burden can affect
much heavier PBH dark matter Alexandre  _et al._ (2024). Correspondingly, the
prediction of the mass-spread will extend to such black holes.

Notice that the spread is predicted even if the production of PBHs is sharply
peaked at a particular mass.

Figure 6: $\displaystyle f_{\rm PBH}$ as a function of $\displaystyle M_{\rm
BH}$. Shaded areas represent existing constraints - we refer the reader to
Carr  _et al._ (2010); Auffinger (2023); Thoss  _et al._ (2024) for an
accurate description. The ones on the left follows from Hawking evaporation
and are, therefore, only mildly affected by the most conservative estimates of
memory burden effect, $\displaystyle k=1$ [see (2)]. Dotted lines represent
the monochromatic distribution at formation time $\displaystyle t_{\rm f}$.