# Ethical AI-Powered Regression Test Selection

Per Erik Strandberg  Mälardalen University, Sweden
<EMAIL_ADDRESS>Mirgita Frasheri  Aarhus University, Denmark
<EMAIL_ADDRESS>Eduard Paul Enoiu  Mälardalen University, Sweden
<EMAIL_ADDRESS>
###### Abstract

Test automation is common in software development; often one tests repeatedly
to identify regressions. If the amount of test cases is large, one may select
a subset and only use the most important test cases. The regression test
selection (RTS) could be automated and enhanced with Artificial Intelligence
(AI-RTS). This however could introduce ethical challenges. While such
challenges in AI are in general well studied, there is a gap with respect to
ethical AI-RTS. By exploring the literature and learning from our experiences
of developing an industry AI-RTS tool, we contribute to the literature by
identifying three challenges (assigning responsibility, bias in decision-
making and lack of participation) and three approaches (explicability,
supervision and diversity). Additionally, we provide a checklist for ethical
AI-RTS to help guide the decision-making of the stakeholders involved in the
process.

Figure 1: Overview of a development and test cycle (thick arrows). (1)
Developers update the system under test. (2) An artificial intelligence
powered regression test selection tool creates a test suite. (3) Because of
resource constraints, only some test cases are executed in the system under
test (4). It is possible, that a test case not selected for testing
(highlighted with dashed border), could have detected a flaw that would
otherwise lead to a bug in released software. Ethical challenges (bias,
responsibility and participation) are highlighted with bold red text and
question marks. (The figure is inspired by previous work on regression test
selection and the flow of information in software testing [13, 12].)

##  I Introduction

A well studied problem in the software engineering domain is regression test
selection (RTS), where one desires to select a good subset of tests from a
larger pool of test cases, in order to ensure that a system under test (SUT)
functions as expected. Consequently, the automation of test selection
represents a relevant research topic, for which several techniques have
already been proposed, including artificial intelligence (AI) based methods.
An AI system can be defined as a decision-maker, capable of some autonomous
actions, that operates in an environment, and tries to achieve a set of goals
[6]. The nature of these systems raises ethical concerns. While these are well
studied [5] in general, no prior work has been written that considers the
ethical aspects of RTS systems.

In this paper we aim to start filling the gap in the existing body of
knowledge on ethical AI-RTS by reasoning about a fictional AI-RTS tool,
inspired by an actual tool implemented in 2014, [13]. As opposed to an AI-RTS
with just poor quality, an unethical AI-RTS may select poorly because of
biased data, inscrutable decision-making, a poor understanding of what a just
selection ought to be, or an implementation not anchored with stakeholder
needs. We use a generic software development process with AI-RTS (as shown in
Figure 1) to illustrate different ethical challenges.

##  II Ethical Challenges in RTS

In early work on computer ethics, Moor reports of the invisibility factor: the
problem that one may not understand what the system is doing, possibly leading
to abuse or bias (intentional or not) [10]. If an AI-RTS was non-transparent
and hid reasons for decisions it would be difficult for developers or testers
to take responsibility. Even if the RTS system would explain why some test
cases are more important than others, if the operator cannot keep up with the
many decisions made, the system as a whole can still be perceived as opaque.

Cognitive biases are misconceptions that influence both humans and systems.
One example is automation bias – the tendency to have too much trust in
automated suggestions or solutions. If a system is very reliable, then
performance improves with automation. However, if a system has shortcomings,
these tend to go unnoticed because humans disregard contradicting information
[2]. Imagine if an AI-RTS had a programming error where, instead of increasing
priority of excluded tests after a week, it would only react after a month.
One might argue that this would be an innocent programming bug and not bias.
However, a consequence might be poorer overall performance of the testing
process.

Suppose that an AI-RTS would be used at a company, but that there would be
many parallel projects competing over the same resources for testing (this is
already the case at companies such as Google [9]). The common and shared goal
of “selecting a good set of test cases for testing” might expand to also
include the task of distributing test resources between departments. On this
larger scale there may be conflicting values and interests among stakeholders.
In this case, we may have to think of a society-in-the-loop and not just a
human-in-the-loop perspective. It may be motivated to oversee that a system
respects fundamental rights, follows ethical values, by ensuring participation
of various stakeholders, and by having humans watching the algorithms [4, 11]

The most prevalent ethical principle in AI guidelines is transparency, and
explicability is part of it [5]. Explicability means being able to explain as
to why a certain decision has been made, which might be required for assigning
responsibility. In order to understand decisions, a human-on-the-loop may need
logs showing various internal states of the AI-RTS. Deciding about the _level
of detail_ of such logs, as well as _what to log_ is not trivial. For the
Therac-25 accidents, logging error messages was insufficient as too much
output led to “error message blindness,” in particular because the operator
could not trace errors to documentation [8].

As seen in Figure 1, the subset of selected tests in the Test Suite is a
result of the joint contribution of different sources, including the data used
to train the decision-making AI, technical knowledge of developers, both AI-
RTS and SW, that might not necessarily overlap, as well as empirical know-how
gathered from experience on what usually works or not.

One of several approaches for avoiding bias and improving justice and fairness
of an AI [5], is to conduct supervision. However, most AI practitioners are
unsure how to [3].

Striving for diversity in AI is one way to improve participation, and a wide
range of guidelines promotes it, e.g. AI HLEG [4] and Amnesty [1]. According
to Jobin et al. [5] diversity can also target challenges with fairness and
responsibility.

##  III Ethical Questions around AI-RTS

In this paper, we discuss the challenges of responsibility, bias and
participation. We argue that many approaches could tackle these ethical
challenges and highlight the importance of explicability, supervision and
diversity. In practice, these approaches are somewhat connected. E.g.,
supervision can help tackle bias, but so could diversity. According to Jobin
et al. [5], there are challenges when applying ethical guidelines for AI,
therefore, we suggest AI-RTS stakeholders consider the following questions:

  1. 1.

How are humans in control of AI-RTS?

  2. 2.

Can the system explain its test selection to a human?

  3. 3.

Are decisions and malfunctions suitably logged?

  4. 4.

Who is responsible if the system under-performs?

  5. 5.

Have potential risks with the AI-RTS been considered?

  6. 6.

What would be considered a just test selection?

  7. 7.

Is the impact of decisions (e.g. test coverage) monitored?

  8. 8.

Have all relevant stakeholders had the opportunity to participate in the
design of the AI-RTS system?

  9. 9.

How are stakeholders made aware of RTS decisions?

  10. 10.

What is the system’s impact on human performance?

Jobin et al. [5] have processed a large number of ethical guidelines to find
common themes for AI in general. In particular, they identified that four
challenges when applying ethical guidelines for AI are: how they ought to be
interpreted, why they are important, where they are applicable, and how they
should be implemented. Of importance is also Eitel-Porter [3], which mentions
pitfalls in deploying ethical AI systems. We would also like to point to
Leveson, that reflects on safety-critical systems in a paper 30 years after
her very influential original Therac-25 paper [7, 8]. Challenges she
identified that are still relevant and might be applicable for AI-RTS include:
overconfidence in software, unrealistic risk assessments, as well as
insufficient user and government oversight and standards. Future work could
include studies of already deployed AI-RTS tools, and explore, in retrospect,
if the identified challenges are relevant. Alternatively, future work could
support the development of new AI-RTS tools.

## Acknowledgments

This research was partly funded by the 20150277 ITS ESS-H grant, Innovation
Foundation Denmark (AgroRobottiFleet), the Poul Due Jensen Foundation (Centre
for Digital Twin Technology at Aarhus University), and H2020 (957212).

## References

  * [1] Amnesty International and Access Now.  The Toronto Declaration….  https://www.torontodeclaration.org/, 2018.  Accessed 2021-06-30.
  * [2] M. L. Cummings.  Automation bias in intelligent time critical decision support systems.  In Decision Making in Aviation, pages 289–294. Routledge, 2017\.
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  * [4] Independent High-Level Expert Group on Artificial Intelligence set up by the European Commission (AI HLEG).  Ethics Guidelines for Trustworthy AI.  https://digital-strategy.ec.europa.eu/en/library/ethics-guidelines-trustworthy-ai, 2019\.  Accessed 2021-06-30.
  * [5] A. Jobin, M. Ienca, and E. Vayena.  The global landscape of ai ethics guidelines.  Nature Machine Intelligence, 1(9):389–399, 2019.
  * [6] S. Legg and M. Hutter.  A collection of definitions of intelligence.  Frontiers in Artificial Intelligence and applications, 157:17, 2007\.
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  * [9] A. Memon, Z. Gao, B. Nguyen, S. Dhanda, E. Nickell, R. Siemborski, and J. Micco.  Taming google-scale continuous testing.  In ICSE-SEIP’17. IEEE, 2017.
  * [10] J. H. Moor.  What is computer ethics?  Metaphilosophy, 16(4), 1985.
  * [11] I. Rahwan.  Society-in-the-loop: programming the algorithmic social contract.  Ethics and Information Technology, 20(1):5–14, 2018.
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# One stabilization is not enough
for closed knotted surfaces

Kyle Hayden  Department of Mathematics and Computer Science, Rutgers
University, Newark, NJ 07102, USA<EMAIL_ADDRESS>Sungkyung Kang
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang
37673, Korea<EMAIL_ADDRESS>Anubhav Mukherjee  Deapartment of
Mathematics, Princeton University, Princeton, NJ 08540, USA
<EMAIL_ADDRESS>
###### Abstract.

In this brief note, we show that there exist smooth 4-manifolds (with nonempty
boundary) containing pairs of exotically knotted 2-spheres that remain exotic
after one (either external or internal) stabilization. It follows that the
“one is enough” theorem of Auckly-Kim-Melvin-Ruberman-Schwartz does not hold
for closed surfaces whose homology classes are characteristic.

Kyle Hayden, Sungkyung Kang, Anubhav Mukherjee

##  1\. Introduction

Exotic phenomena in dimension four is fundamentally unstable. For example,
results of Wall [Wal64] and Gompf [Gom84] show that exotic compact 4-manifolds
become diffeomorphic after _stabilizing_ by taking repeated connected sums
with $S^{2}\\!\times\\!S^{2}$ (in the orientable case) or
$S^{2}\tilde{\times}S^{2}$ (in the non-orientable case). Analogous results
hold for pairs of surfaces $\Sigma$ and $\Sigma^{\prime}$ in a 4-manifold $X$
that are _exotically knotted_ , i.e., isotopic through ambient homeomorphisms
of $X$ but not ambient diffeomorphisms of $X$. In particular, results of
Perron [Per86] and Quinn [Qui86] imply that such surfaces become smoothly
isotopic after sufficiently many _external stabilizations_ , where the ambient
4-manifold $X$ itself is stabilized away from $\Sigma$ and $\Sigma^{\prime}$.
More recently, Baykur-Sunukjian [BS15] showed that exotic surfaces also become
smoothly isotopic after repeated _internal stabilization_ , i.e., repeatedly
summing each surface with a standard $T^{2}\subset S^{4}$.

It remains a fundamental problem to understand the number of stabilizations
required to dissolve exotic phenomena, and it is unknown whether or not a
single stabilization suffices in the original setting of closed, simply-
connected exotic 4-manifolds. One stabilization is enough for most known
examples of exotic phenomena (c.f., [Auc03, Akb02, BS13, AKMR15, Akb15,
Kim19]), as well as in multiple more general settings [AKM+19, AS21, RS22].
However, a spate of recent results has revealed contexts in which a single
stabilization is not enough [Lin20, LM21, Gut22, Kan22, GHKP, KMT22, Hay23].

To date, all examples of exotic 4-manifolds or exotic surfaces that remain
exotic after one stabilization have had nonempty boundary. With an eye towards
the closed case, this note considers the intermediate setting of closed
surfaces in 4-manifolds with boundary.

######  Theorem 1.1.

There exist smooth 4-manifolds with boundary containing pairs of exotically
knotted 2-spheres that remain exotic after one (external or internal)
stabilization.

To the authors’ knowledge, these are also the first examples of exotic pairs
of surfaces that survive both internal and external stabilization. The key
input for Theorem 1.1 is the second author’s recent construction of exotic
contractible 4-manifolds that survive one stabilization with
$S^{2}\\!\times\\!S^{2}$ [Kan22]. We prove these exotic 4-manifolds become
diffeomorphic after a single _twisted_ stabilization, and this enlarged
4-manifold contains the desired pair of exotically knotted 2-spheres. Along
the way, we use the results of [AKM+19] to see that a single (possibly
twisted) stabilization suffices to dissolve a large class of corks — namely,
those obtained as surgeries along slice disks in $B^{4}$; see §2.

######  Remark 1.2.

(a) In [AKM+19], Auckly–Kim–Melvin–Ruberman–Schwartz use Gabai’s light bulb
theorem [Gab20] to establish a “one is enough”-type result: If $X$ is a
smooth, simply-connected 4-manifold and $\Sigma,\Sigma^{\prime}\subset X$ are
smoothly embedded surfaces of the same genus and homology class such that

  * (i)

$\Sigma$ and $\Sigma^{\prime}$ are each connected surfaces,

  * (ii)

$[\Sigma]=[\Sigma^{\prime}]$ is an _ordinary_ homology class, i.e., not dual
to $w_{2}(X)$, and

  * (iii)

$X\setminus\Sigma$ and $X\setminus\Sigma^{\prime}$ are simply-connected,

then $\Sigma$ and $\Sigma^{\prime}$ become smoothly isotopic after one
external stabilization. In [LM21], Lin and the third author showed that
hypothesis (i) is necessary. In this note, our ambient 4-manifold $X$ is
simply-connected and the surfaces are connected and have simply-connected
complements, so our examples show that hypothesis (ii) is necessary, i.e., the
surfaces must be ordinary. (b) In forthcoming work [Auc], Auckly produces
exotic surfaces in _closed_ 4-manifolds that remain distinct after internal
stabilization.

### Acknowledgements

The authors thank Dave Auckly for helpful comments. KH is supported by NSF
grant DMS-2243128. SK is supported by the Institute for Basic Science
(IBS-R003-D1).

##  2\. Some topological constructions

###  2.1. Surgery along a slice disk

We begin by recalling a construction of contractible 4-manifolds based on
slice disks (c.f., [Gor75]). Given a knot $K\subset S^{3}$ bounding a smooth
slice disk $D\subset B^{4}$, fix a compact tubular neighborhood $N(D)\cong
D\times D^{2}$ and a choice of meridian $\\{p\\}\times\partial D^{2}\subset
S^{3}$ for some point $p\in K=\partial D$. By taking the disk exterior
$B^{4}\setminus\mathring{N}(D)$ and attaching a $(-k)$-framed 2-handle along
the meridian $\mu$, we obtain a contractible 4-manifold whose boundary is
$\smash{S^{3}_{\nicefrac{{1}}{{k}}}(K)}$. We refer to this 4-manifold as
$1/k$-surgery along the slice disk $D\subset B^{4}$, and we denote it by
$B^{4}_{\nicefrac{{{1}}}{{k}}}(D)$.

We can also describe this construction in terms of Kirby diagrams. Given a
handle diagram of $B^{4}$ in which $K$ is unknotted and $D$ is its standard
slice disk, we replace $K$ with a dotted 1-handle curve and add a
$(-k)$-framed 2-handle along a meridian to $K$. An example is illustrated in
Figure 1, where $K$ is the stevedore knot.

$D$
---
(a)
---
$D$
---
(b)
---
$0$
---
$-k$
---
(c)
---
$0$
---
Figure 1. (a) A slice disk $D$ bounded by the stevedore knot. (b) A handle
diagram for $B^{4}$ in which $D$ is standard. (c) The 4-manifold
$B^{4}_{\nicefrac{{{1}}}{{k}}}(D)$.

###  2.2. Log transformation

Consider a smooth 4-manifold $X$ containing a smoothly embedded torus $T$ with
square zero, and fix a framed tubular neighborhood $N(T)\cong T^{2}\times
D^{2}$. A _torus surgery_ along $T$ consists of removing $\mathring{N}(T)$ and
regluing $N(T)$ using a non-trivial self-diffeomorphism of its boundary
$\partial N(T)\cong T^{2}\times\partial D^{2}=T^{3}$. Our arguments will
involve a special case of this operation known as a _logarithmic transform of
multiplicity zero_ , where the self-diffeomorphism of $T^{3}\cong
T^{2}\times\partial D^{2}$ exchanges $\partial D^{2}$ with an essential simple
closed curve in $T^{2}$. Note that the result is independent of the choice of
curve in $T^{2}$, as any two such curves can be exchanged by a diffeomorphism
$f$ of $T^{2}$ and the gluing diffeomorphism $f\times\operatorname{{id}}$ of
$T^{2}\times\partial D^{2}$ extends to $T^{2}\times D^{2}$. For a handle-
theoretic description of this operation, see Figure 2 (c.f., [GS99, Figure
8.25]).

$\mu$
---
$\beta$
---
$\alpha$
---
$0$
---
$0$
---
$\alpha$
---
$\mu$
---
$\beta$
---
Figure 2. A log transform of multiplicity zero.

##  3\. Proof of the main theorem

We begin by observing that a single (possibly twisted) stabilization suffices
for any pair of exotic 4-manifolds obtained as surgeries along a pair of slice
disks in $B^{4}$ with the same boundary.

######  Lemma 3.1.

Let $D$ and $D^{\prime}$ be slice disks in $B^{4}$ bounded by a knot $K\subset
S^{3}$.

  1. (a)

If $k$ is even, then
$B^{4}_{\nicefrac{{1}}{{k}}}(D)\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\\!\times\\!S^{2})\cong
B^{4}_{\nicefrac{{1}}{{k}}}(D^{\prime})\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\\!\times\\!S^{2})$
rel boundary.

  2. (b)

If $k$ is odd, then
$B^{4}_{\nicefrac{{1}}{{k}}}(D)\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\tilde{\times}S^{2})\cong
B^{4}_{\nicefrac{{1}}{{k}}}(D^{\prime})\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\tilde{\times}S^{2})$
rel boundary.

###### Proof.

Consider the 0-framed knot trace $X_{0}(K)$ and let $S$ and $S^{\prime}$
denote the 2-spheres obtained from $D$ and $D^{\prime}$, respectively, by
taking their union with the core disk in the 2-handle. Let $Z$ denote the
4-manifold obtained from $X_{0}(K)$ by attaching a $(-k)$-framed 2-handle
along the meridian to $K$. Then $B^{4}_{\nicefrac{{1}}{{k}}}(D)$ and
$B^{4}_{\nicefrac{{1}}{{k}}}(D^{\prime})$ are obtained from $Z$ by surgering
along the 2-spheres $S$ and $S^{\prime}$, respectively. Observe that $S$ and
$S^{\prime}$ have simply-connected complements (because their meridians bound
core disks of a 2-handle) and that these represent a characteristic homology
class if and only if $k$ is odd. By [AKM+19], $S$ and $S^{\prime}$ are
isotopic in $Z$ (rel $\partial Z$) after one stabilization with either
$S^{2}\\!\times\\!S^{2}$ if $k$ is even or $S^{2}\tilde{\times}S^{2}$ if $k$
is odd. This ambient isotopy (rel boundary) induces a diffeomorphism of the
surgered 4-manifolds, namely the once-stabilized copies of
$B^{4}_{\nicefrac{{1}}{{k}}}(D)$ and
$B^{4}_{\nicefrac{{1}}{{k}}}(D^{\prime})$. ∎

For the narrower purposes of proving Theorem 1.1, we can get away with the
weaker conclusion that $B^{4}_{\nicefrac{{1}}{{k}}}(D)$ and
$B^{4}_{\nicefrac{{1}}{{k}}}(D^{\prime})$ become diffeomorphic after one
(possibly twisted) stabilization where, a priori, the diffeomorphism may be
nontrivial along the boundary. (In explicit cases, as in the proof of Theorem
1.1, one may often promote this to the stronger rel boundary conclusion.)

This weaker alternative admits an elementary, handle-theoretic proof: Begin by
fixing a handle structure on $B^{4}$ relative to the slice disk $D$, so that
$D$ appears as an unknotted, boundary-parallel disk in the 0-handle and $K$
appears as an unknot, with additional handles attached to the complement of
$D$. This is depicted schematically in part (a) of Figure 3. Part (b) of the
figure depicts $B^{4}_{\nicefrac{{1}}{{k}}}(D)$. In part (c), we perform a
stabilization, which is twisted if and only if $k$ is odd. After a handle
slide, we obtain the diagram in (d). By sliding the $(-k)$-framed 2-handle
over the 0-framed meridian in the top right, we obtain the diagram in (e).
Sliding all the original 2-handles off of the 1-handle yields the diagram in
(f), and a 1-/2-handle cancellation then yields the diagram in (g). Observe
that this final diagram is simply the union of $B^{4}$ with a 0-framed
2-handle attached along $K$ and a $k$-framed 2-handle attached along a
meridian $\mu$ of $K$; this 4-manifold only depends on $K$ and $k$, and not on
$D$ or $D^{\prime}$.

(a)
---
$K$
---
$D$
---
$\mu$
---
$-k$
---
$-k$
---
$k$
---
$0$
---
$0$
---
$0$
---
$-k$
---
$0$
---
$0$
---
$-k$
---
$0$
---
$-k$
---
$0$
---
$0$
---
$-k$
---
(b)
---
(c)
---
(d)
---
(e)
---
(f)
---
(g)
---
Figure 3. Surgery along a slice disk after one (possibly twisted)
stabilization. $0$
---
$0$
---
(f)
---
(d)
---
$0$
---
$0$
---
$0$
---
(c)
---
$0$
---
$0$
---
$0$
---
(b)
---
$S\\#T^{2}$
---
$0$
---
(e)
---
$0$
---
$0$
---
(a)
---
$S$
---
$0$
---
Figure 4. Relating internal and external stabilization via logarithmic
transform.

We also make note of a relationship between internal and external
stabilization. The following is well-known to experts, but we sketch its proof
for completeness.

######  Lemma 3.2.

Let $S$ be a smoothly embedded 2-sphere of square zero in a smooth 4-manifold
$X$, and let $X_{0}$ denote the result of surgering $X$ along $S$. Then
$X_{0}\raisebox{1.0pt}{\text{\footnotesize$\\#$}}S^{2}\\!\times\\!S^{2}$ is
obtained from $X$ by a logarithmic transformation of multiplicity zero along
$S\raisebox{1.0pt}{\text{\footnotesize$\\#$}}T^{2}\subset X$.

###### Proof.

Fix a handle decomposition of $X$ relative to a neighborhood $N(S)\cong
S^{2}\times D^{2}$. This is depicted schematically in Figure 4(a), where the
grey arcs depict 2-handles that may intersect $S$. Since the diagram starts
with $S^{2}\times D^{2}$ (which is shown in Figure 4(a) as the 0-trace of an
unknotted component) which contains the given sphere $S$ as
$S^{2}\times\\{0\\}$, it is clear that while 1-handles and 3-handles can
exist, we may assume that they are attached away from $S$. As illustrated in
parts (b) and (c) of the figure, we may locally stabilize $S$ to produce the
torus $S\raisebox{1.0pt}{\text{\footnotesize$\\#$}}T^{2}$ and introduce
canceling 1-/2-handle pairs to adapt the diagram to
$S\raisebox{1.0pt}{\text{\footnotesize$\\#$}}T^{2}$. The diagram in (d)
depicts the result of a multiplicity-zero log transform. After canceling the
topmost 1-/2-handle pair and performing minor isotopy, we obtain (f), which is
a diagram for
$X_{0}\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\\!\times\\!S^{2})$.
(We also note that it does not matter which $S^{1}$-factor in $T^{2}$ was
exchanged with the meridian to
$S\raisebox{1.0pt}{\text{\footnotesize$\\#$}}T^{2}$; either transformation
yields the same result.) ∎

###  3.1.  Proof of the main theorem

Let $W$ and $W^{\prime}$ denote the pair of compact, contractible 4-manifolds
constructed by the second author in [Kan22]. These 4-manifolds are
homeomorphic but not diffeomorphic, and they remain nondiffeomorphic after a
single stabilization with $S^{2}\\!\times\\!S^{2}$. The construction begins
with a pair of $(+1)$-surgeries along disks $D,D^{\prime}\subset B^{4}$ with
the same boundary $K$. While these 4-manifolds are only exotic rel boundary,
results of Akbulut-Ruberman [AR16] can be used to extend $B^{4}_{+1}(D)$ and
$B^{4}_{+1}(D^{\prime})$ by an invertible homology cobordism $C$ from
$S^{3}_{+1}(K)$ to another integer homology sphere $Y$ with the property that
every self-diffeomorphism of $Y$ has an extension to $C$ that is the identity
on $S^{3}_{+1}(K)$. The extended 4-manifolds $W=B^{4}_{+1}(D)\cup C$ and
$W^{\prime}=B^{4}_{+1}(D^{\prime})\cup C$ are then _absolutely exotic_ and
remain so after one stabilization with $S^{2}\\!\times\\!S^{2}$ [Kan22].

By Lemma 3.1, the 4-manifolds $B^{4}_{+1}(D)$ and $B^{4}_{+1}(D^{\prime})$
become diffeomorphic rel boundary after one twisted stabilization with
$S^{2}\tilde{\times}S^{2}$. Since $W^{\prime}$ can be obtained from $W$ by
cutting out $B^{4}_{+1}(D)$ and gluing in $B^{4}_{+1}(D^{\prime})$ via the
identity, we obtain the following:

######  Proposition 3.3.

$W\\#(S^{2}\tilde{\times}S^{2})$ and $W^{\prime}\\#(S^{2}\tilde{\times}S^{2})$
are diffeomorphic rel boundary. $\square$

We are now in position to prove the main theorem.

###### Proof of Theorem 1.1.

Let the knot $K$, the slice disks $D$ and $D^{\prime}$, and the 4-manifolds
$W$ and $W^{\prime}$ be as described above. We also let $Z$ denote the
enlarged 4-manifold $Z=W\\#(S^{2}\tilde{\times}S^{2})\cong
W^{\prime}\\#(S^{2}\tilde{\times}S^{2})$. As in the proof of Lemma 3.1, $Z$
can also be obtained from $B^{4}$ by attaching a 0-framed 2-handle along $K$
and a $(-1)$-framed 2-handle along a meridian of $K$.

Consider the 2-spheres $S$ and $S^{\prime}$ in $Z$ obtained by capping off the
slice disks $D$ and $D^{\prime}$ with the core of the 0-framed 2-handle
attached along $K$. As in the proof of Lemma 3.1, these spheres have simply-
connected complements, hence are topologically isotopic by results of Perron
[Per86] and Quinn [Qui86] (building on work of Freedman [Fre82]). On the other
hand, surgering $Z$ along $S$ or $S^{\prime}$ yields $W$ or $W^{\prime}$,
respectively, so these spheres are not smoothly isotopic. Moreover, since $W$
and $W^{\prime}$ remain nondiffeomorphic after one stabilization with
$S^{2}\\!\times\\!S^{2}$, we see that $S$ and $S^{\prime}$ remain smoothly
nonisotopic after one external stabilization.

If we instead consider internal stabilization, we obtain tori
$T=S\raisebox{1.0pt}{\text{\footnotesize$\\#$}}T^{2}$ and
$T^{\prime}=S^{\prime}\raisebox{1.0pt}{\text{\footnotesize$\\#$}}T^{2}$.
Towards a contradiction, suppose there is a smooth isotopy carrying $T$ to
$T^{\prime}$. This carries a tubular neighborhood $N(T)\cong T^{2}\times
D^{2}$ to a tubular neighborhood $N(T^{\prime})$. This diffeomorphism
preserves the class of the meridian $\partial D^{2}$ in $\partial N(T)$ and
$\partial N(T^{\prime})$, so the 4-manifolds obtained from $Z$ by log
transforms of multiplicity zero along either $T$ or $T^{\prime}$ are
diffeomorphic. (Note that, as discussed in §2, the result of the 0-log
transform is independent of the choice of framing for these neighborhoods.)
However, Lemma 3.2 shows that the resulting 4-manifolds can also be obtained
by surgering $Z$ along $S$ or $S^{\prime}$ and then stabilizing with
$S^{2}\\!\times\\!S^{2}$, yielding
$W\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\\!\times\\!S^{2})$ and
$W^{\prime}\raisebox{1.0pt}{\text{\footnotesize$\\#$}}(S^{2}\\!\times\\!S^{2})$.
This is a contradiction, as these latter 4-manifolds are not diffeomorphic. ∎

## References

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# Structure Formation after Reheating: Supermassive Primordial Black Holes and
Fermi Ball Dark Matter

Marcos M. Flores  Department of Physics and Astronomy, University of
California, Los Angeles
Los Angeles, California, 90095-1547, USA  Yifan Lu  Department of Physics and
Astronomy, University of California, Los Angeles
Los Angeles, California, 90095-1547, USA  Alexander Kusenko  Department of
Physics and Astronomy, University of California, Los Angeles
Los Angeles, California, 90095-1547, USA  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), UTIAS
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

In the presence of (relatively) long-range forces, structures can form even
during the radiation dominated era, leading to compact objects, such as Fermi
balls or primordial black holes (PBHs), which can account for all or part of
dark matter. We present a detailed analysis of a model in which fermions are
produced from the inflaton decay developing some particle-antiparticle
asymmetry. These fermions undergo clustering and structure formation driven by
a Yukawa interaction. The same interaction provides a cooling channel for the
dark halos via scalar radiation, leading to rapid collapse and the formation
of a compact object. We discuss the criteria for the formation of either PBHs
and Fermi balls. In the PBH formation regime, supermassive PBHs can seed the
active galactic nuclei or quasars found at high redshift. Alternatively, Fermi
balls can account for all of the cold dark matter, while evading microlensing
constraints.

††preprint: IPMU23-0027

##  I Introduction

The structures in some species can grow even during the radiation dominated
era in the presence of relatively long-range forces Gradwohl and Frieman
(1992); Amendola  _et al._ (2018); Savastano  _et al._ (2019); Domènech and
Sasaki (2021); Flores and Kusenko (2021a, 2023, b); Flores  _et al._ (2022a,
b); Domènech  _et al._ (2023); Flores  _et al._ (2023). One example is Yukawa
interactions which can form halos of interacting fermions Flores and Kusenko
(2021a); Domènech  _et al._ (2023). The fate of such halos depends on their
cooling. In the absense of dissipation, these halos would remain intact until
the present day, or until their constituent particles decay. However, the same
Yukawa interaction also introduces a dissipation channel in the form of scalar
radiation. These scalar waves allow for the rapid removal of energy and
angular momentum from the halo Flores and Kusenko (2021a, b). Consequently,
the halo can collapse to highly condensed fermionic matter, leading to either
a compact Fermi ball or a primordial black hole (PBH).

PBHs are fascinating objects that could form in the early Universe. They were
initially conjectured to be formed from the collapse of large overdensities
Hawking (1971); Carr and Hawking (1974); Carr (1975) in the primordial power
spectrum. Over the years, other formation mechanisms have been proposed. For
example, PBHs could arise from the unstable scalar field condensate formed
after inflation, such as Q balls Cotner and Kusenko (2017a, b); Cotner  _et
al._ (2019) or oscillons Cotner  _et al._ (2018). Phase transitions in the
dark sector can lead to the collapse of vacuum bubbles and produce PBHs even
after photon decoupling Lu  _et al._ (2023). Unlike the formation scenario
from critical collapse Khlopov (2010); Choudhury  _et al._ (2023a, b), the PBH
(and Fermi ball) mass function in our model does not depend on the details of
the underlying inflation theory.

Early structure formation does allow for the formation of halos composed of
both fermions and antifermions. In fact, there are some cases in which the
lack of an asymmetry might be useful Flores  _et al._ (2022a, 2023).
Generically however, the formation of compact objects requires a particle-
antiparticle asymmetry to avoid pair annihilation. Generation of a dark matter
asymmetry has been explored in many contexts, with a particular focus on
addressing the coincidence problem, i.e., the fact that the dark matter
abundance is remarkably close to the abundance of ordinary matter Petraki and
Volkas (2013); Zurek (2014).

In this work, we will focus on the early structure formation with a dark
sector asymmetry generated through inflaton decay. As we will describe,
depending on the temperature at the onset of early structure formation,
compact objects can be formed in two classes of time; either instantaneously
or noninstantaneously. The resulting compact object can span a wide range of
masses, including sub-Planck mass Fermi balls up to supermassive black holes
(SMBH). Finally, we will discuss possible consequences in cosmology – the
produced supermassive PBHs could seed the high redshift active galactic nuclei
(AGN) observed by JWST Bunker  _et al._ (2023), while all of cold dark matter
(CDM) can exist in the form of Fermi balls.

##  II Asymmetry in the Dark Sector

Let us consider a dark sector with three generations of fermions $\psi_{i}$,
$\bar{\psi}_{i}$ ($i=1,2,3$) and a light scalar $\chi$. We work in the basis
where the fermion mass matrix is diagonalized and, for simplicity, we assume
the fermions have a common mass $m_{\psi}$. In addition, we consider three
generations of Majorana fermions $N_{i}$ with a common mass $M$. If we take
these Majorana fermions to be right-handed neutrinos, the decay of $N_{i}$’s
can also generate a net lepton number Fukugita and Yanagida (1986). Inflation
is driven by a complex scalar field $\varphi$. We impose a global $U(1)$
symmetry on $\varphi$ and the dark sector. The charge assignment is as
following:

|  $\varphi$ |  $\psi$ |  $\bar{\psi}$
---|---|---|---
$U(1)_{D}$ |  -1 |  1 |  -1

The Lagrangian of the dark sector and inflaton is given by

$\displaystyle\mathcal{L}_{\rm ds}=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\chi)^{2}-\frac{1}{2}m_{\chi}^{2}\chi^{2}-m_{\psi}\bar{\psi}_{i}\psi_{i}-y_{ij}\chi\bar{\psi}_{i}\psi_{j}+h.c.$
(1) $\displaystyle\mathcal{L}_{\rm inf}=$
$\displaystyle\partial_{\mu}\varphi^{\dagger}\partial^{\mu}\varphi-
m_{\varphi}^{2}\lvert\varphi\rvert^{2}-V(\varphi)-\frac{1}{2}MN^{c}_{i}N^{c}_{i}$
$\displaystyle+\kappa_{ij}\varphi
N^{c}_{i}\psi_{j}+\bar{\kappa}_{ij}\varphi^{\dagger}N^{c}_{i}\bar{\psi}_{j}+h.c.$

Here, $V(\varphi)$ is the potential that drives inflation and all the fermions
in this model are two component spinors. We note that structure formation
under the Yukawa potential in the dark sector is independent of the
predictions from the inflation theory. Specifically, our model do not rely on
a particular form of the inflation potential or the primodial power spectrum.

Reheating is driven by the inflaton decay to the dark sector via
$\varphi\rightarrow N_{i}+\bar{\psi}_{j}$ and the subsequent decay of
$N_{i}$’s which also generates the Standard Model (SM) plasma. Since the mass
matrices are all diagonalized, the couplings $\kappa_{ij}$ and
$\bar{\kappa}_{ij}$ are in general complex. This leads to a different decay
rate for the charge conjugated decay mode. Due to the $\psi$ number violating
interactions, an asymmetry in the dark sector will be generated after
reheating. The asymmetry generated per decay can be calculated from the
interference of the diagrams in Fig. 1 (and the corresponding diagrams for the
decay of $\varphi$). We note that this calculation is reminiscent of similar
scenarios encountered in leptogenesis Fukugita and Yanagida (1986); Davidson
_et al._ (2008).

$\varphi^{\dagger}$$N_{i}$$\psi_{j}$$\varphi^{\dagger}$$N_{i}$$\psi_{j}$$\psi_{k}$$\varphi$$N_{l}$

Fig. 1: Diagrams of inflaton decay at tree and one loop levels.

We find the result to be

$\varepsilon=\frac{2\ln
2-1}{8\pi}\left(\frac{M}{m_{\varphi}}\right)^{2}\frac{\operatorname{Im}(\xi-\bar{\xi})}{\lvert\kappa_{ij}\rvert^{2}+\lvert\bar{\kappa}_{ij}\rvert^{2}},$
(2)

where

$\xi=\kappa_{ij}^{\ast}\kappa_{ik}^{\ast}\kappa_{lj}\kappa_{lk},$ (3)

and $\bar{\xi}$ is obtained by replacing $\kappa$ with $\bar{\kappa}$. In
general, the couplings $\kappa_{ij}$ and $\bar{\kappa}_{ij}$ are independent
of each other, so the asymmetry produced by the decay of $\varphi$ will not be
canceled by the decay of $\varphi^{\dagger}$. Observe that in Eq. (2), the
fermion mass suppression factor $(M/m_{\varphi})^{2}$ will lead to final
asymmetry which is expected to be very small.

The evolution of the $\psi$ number density $n=n_{\psi}-n_{\bar{\psi}}$ during
reheating follows the Boltzmann equation

$\dot{n}+3Hn=\varepsilon\Gamma\frac{\rho_{\varphi}}{m_{\varphi}}-\frac{3KT^{5}}{4\pi^{3}m_{\varphi}^{4}}n,$
(4)

with

$K=\lvert\kappa_{ij}\bar{\kappa}_{kl}\rvert^{2}+\lvert\bar{\kappa}_{il}\kappa_{jk}\rvert^{2}.$
(5)

In Eq. (4), the first term on the right-hand side corresponds to the
production of asymmetry from the inflaton decay. Washout of the generated
asymmetry will also occur due to the $\psi\psi\rightarrow NN$ and $\psi
N\rightarrow\bar{\psi}N$ interactions which are mediated by the inflaton. This
leads to an overall asymmetry which will be suppressed. The effect of washout
is encapsulated in the second term on the right-hand side of Eq. (4). The
temperature $T$ is the plasma temperature during reheating, which we take to
be the same between the dark sector and SM sector. The radiation density and
the inflaton energy density follow the equations:

$\displaystyle\dot{\rho}_{\varphi}+3H\rho_{\varphi}$
$\displaystyle=-\Gamma\rho_{\varphi},$ (6)
$\displaystyle\dot{\rho}_{R}+4H\rho_{R}$ $\displaystyle=\Gamma\rho_{\varphi},$

where $\rho_{R}=g_{\star}\pi^{2}T^{4}/30$ and $g_{\star}=123.5$ is the degrees
of freedom of the SM and the dark sector including three generations of
Majorana fermions. The inflaton decay rate is given by

$\Gamma=\frac{1}{8\pi}(\lvert\kappa_{ij}\rvert^{2}+\lvert\bar{\kappa}_{ij}\rvert^{2})m_{\varphi}.$
(7)

The $\psi$ asymmetry produced is commonly parametrized by $\eta=n/s$, where
$s$ is the entropy density of the ambient plasma. At the beginning of
reheating, both of asymmetry production and washout interactions are active
due to the high plasma temperature. As the temperature decreases during
reheating, the washout interactions gradually freeze out and a constant
asymmetry $\eta$ is produced near the end of inflaton decay. If we neglect the
washout terms, then the final asymmetry produced will be $\eta\sim\epsilon$.
In order to compute the final asymmetry including the washout effects, we
numerically solve Eqs. (4) and (6) by taking
$\kappa_{ij}\sim\bar{\kappa}_{ij}\equiv\kappa$ for all $i,j$. Due to the mass
suppression $M/m_{\varphi}$, we can choose the value of the parameter
$\epsilon$ independently from $\kappa$ and require
$\epsilon\ll\lvert\kappa\rvert^{2}$. In addition, the coupling $\kappa$ should
also be sufficiently suppressed such that the flatness of the inflaton
potential is preserved at the quantum level. An example of the time evolution
of the energy densities and the asymmetry is shown in Fig. 2. We see that
indeed a constant asymmetry $\eta$ is produced. As expected, washout effects
reduce the value $\eta$ to be much smaller than the parameter $\epsilon$.

Since $\varphi$ carries a dark quantum number, fluctuations of the complex
inflaton could generate isocurvature perturbations in the dark sector. As we
will show later on, the dark sector fermions will collapse into Fermi balls
which provide all of the CDM that we observe today. This translates to CDM
isocurvature fluctuations that are constrained by Planck Akrami  _et al._
(2020). However, this process depends on the inflation trajectory and possible
additional $U(1)_{D}$ violating operators in the inflaton potential. So the
analysis of this effect is tangential to the main purpose of this paper. A
detailed calculation is done in the case of baryogenesis in Ref. Cline  _et
al._ (2020) and a two-field inflation model is allowed for a suitable choice
of the inflaton potential $V(\varphi)$.

Fig. 2: Inflaton, radiation energy density and asymmetry as functions of time.
The parameters are chosen to be $\kappa=2\times 10^{-4}$, $m_{\varphi}=1\times
10^{16}$ GeV, $\varepsilon=1\times 10^{-20}$ and the initial inflaton energy
is taken to be $1\times 10^{16}$ GeV.

##  III Structure formation mechanism

The inclusion of light scalars $\chi$ in Eq. (1) introduces an instability in
the $\psi$ fluid, analogous to the gravitational instability in conventional
structure formation. Gravity alone does not permit for the growth of structure
during the earliest epochs of the Universe. However, the inclusion of Yukawa
forces, which are generally stronger than gravity, can generate structure on
timescales significantly faster than the Hubble time – even during radiation
domination. This principle of early structure formation has been examined in
many contexts Gradwohl and Frieman (1992); Amendola  _et al._ (2018);
Savastano  _et al._ (2019); Domènech and Sasaki (2021); Flores and Kusenko
(2021a, 2023, b); Flores  _et al._ (2022a, b); Domènech  _et al._ (2023);
Flores  _et al._ (2023).

In its simplest form, early structure formation involves a heavy fermion
interacting via a Yukawa interaction. The growth of $\psi$ overdensities
$\Delta_{k}\equiv(n_{\psi}-\bar{n}_{\psi})/\bar{n}_{\psi}$ is described by the
differential equation Domènech  _et al._ (2023),

$\Delta_{k}^{\prime\prime}+\frac{2+3x}{2x(1+x)}\Delta_{k}^{\prime}=\frac{3}{2x(1+x)}\Delta_{k}f_{\psi}\left[1+\frac{2\beta^{2}}{1+(k\ell)^{-2}}\right]$
(8)

where $\prime\equiv d/dx$ with $x=a/a_{\rm eq}$ being the scale factor
normalized to the time where $\rho_{R}=\rho_{m}+\rho_{\psi}$ and lastly,
$f_{\psi}=\rho_{\psi}/(\rho_{m}+\rho_{\psi})$. Here, $\beta$ characterizes the
strength of the force relative to gravity and $\ell$ is the distance over
which the Yukawa interaction is relevant. For simplicity, in our analysis we
assume $\beta$ is a constant given by $\beta=yM_{\rm Pl}/m_{\psi}$ and
$\ell=m_{\chi}^{-1}$, though in general, additional terms in the $\chi$
potential can induce a density-dependent $\beta$ and $\ell$ Domènech  _et al._
(2023).

In the limit of a massless mediator, i.e. $\ell\to\infty$, the $k$ dependence
from Eq. (8) is eliminated. In massless mediator limit, the growing mode from
Eq. (8) is given by

$\Delta_{k}(t)=\Delta_{k}(0)I_{0}\left(4\sqrt{p}\left(t/t_{\mathrm{eq}}\right)^{1/4}\right),$
(9)

where

$p=\frac{3}{8}\left(1+\beta^{2}\right),$ (10)

and $I_{0}$ is the modified Bessel function of the first kind. This solution
exhibits exponential growth behavior as long as $\sqrt{p}(t/t_{\rm
eq})^{1/4}\gg 1$. This constrains the temperature for rapid structure
formation in the radiation dominated era to be

$T\ll\left(\frac{g_{\star s,\rm eq}}{g_{\star s}}\right)^{1/3}\beta^{2}T_{\rm
eq}\equiv T_{\rm form}.$ (11)

For simplicity, we only require $T<T_{\rm form}$ in the following analysis. In
addition to this restriction, we also require that
$\bar{\psi}\psi\leftrightarrow\chi\chi$ interactions to have frozen out, so
that halos may be generated from a stable population of particles. Lastly, we
will also require that early structure formation occurs once $T\lesssim
m_{\psi}/3$. These three conditions allow us to define the “growth
temperature” Flores and Kusenko (2021a),

$T_{g}=\min\left(\frac{m_{\psi}}{3},\frac{m_{\psi}}{\ln{(y^{4}m_{p}/m_{\psi})}},T_{\rm
form}\right),$ (12)

at which early structure formation begins. For the remainder of this work, we
focus on structure formation after reheating so our model is valid only for
$T_{g}<T_{\rm RH}$. The possibility of structure formation during the inflaton
decay is also interesting and will be left for future work.

The subsequent evolution of the halos depends on the range of the Yukawa
interaction $1/m_{\chi}$ and the horizon size $H_{g}^{-1}$ at $T_{g}$. If
$m_{\chi}\gg H_{g}$, then structure formation occurs rapidly at $T_{g}$ and
the subhorizon sized halos quickly undergo scalar radiation cooling. As we
will discuss shortly, these subhorizon halos can radiatively collapse, and
depending on the parameters under consideration, result in either PBHs or
Fermi balls. Since the timescale for halo formation is much smaller than the
Hubble time, we refer to this scenario as “instantaneous formation.” If
$m_{\chi}\ll H_{g}$, then subhorizon sized halos form initially. However, as
the horizon expands, Yukawa force can also drive halo mergers, which we expect
to be very efficient compared to gravitational mergers. The maximum mass of
the halo is determined by the time when the length scale $1/m_{\chi}$ enters
the horizon. This happens at a temperature lower than $T_{g}$ and we refer to
this process as “noninstantaneous formation.”

The overdensities described by Eq. (8) quickly grow to exceed unity, thus
entering the nonlinear regime. At this stage, the overdenisites evolve into
virialized $\psi$ halos. In the absence of dissipation, these virialized halos
would simply remain as such, unless the constituent particles themselves can
decay. However, the same Yukawa interaction which generated overdensities in
the $\psi$ fluid also enables the emission of scalar radiation from
accelerating $\psi$ particles. Initially, when the halo is sufficiently
dilute, energy is primarily dissipated through pair interactions, i.e.,
through free-free or bremmstrahlung emission. As a given $\psi$ halo
contracts, its density will increase until $\chi$ radiation becomes trapped.
In this regime, radiation is only emitted from the surface of the halo. For
radiative collapse to lead to the formation of a compact object, the
characteristic timescale associated with cooling,

$\tau_{\rm cool}\equiv\frac{E}{dE/dt}=\frac{E}{P_{\rm brem}+P_{\rm
surf}+\cdots}$ (13)

must be shorter that the Hubble time $H^{-1}$. Since all halos must traverse
the inefficient surface cooling regime before reaching their final form, the
surface cooling timescale dictates whether a halo can collapse quickly enough.
Using the Stefan-Boltzmann law, we find that $\tau_{\rm cool}\sim\tau_{\rm
surf}\sim R_{h}$. For $m_{\chi}\gg H$, the maximum halo radius $R_{h}\sim
m_{\chi}^{-1}\ll H^{-1}$ implying that subhorizon halos will cool rapidly
compared to the Hubble time. This process corresponds to the “instantaneous
formation” regime discussed above.

Alternatively, when the mediator mass is smaller than the Hubble parameter,
subhorizon halos will form, but these smaller halos will quickly merge until
the combined halos produce a larger halo of radius $R_{h}\sim H^{-1}$. This
process continues until $m_{\chi}\sim H$. Once halos become small relative to
the horizon size, surface cooling will be efficient enough to drive them
toward collapse. This process corresponds to the “noninstantaneous formation”
regime discussed previously.

These different scenarios will each lead to different constraints on parameter
space. Therefore, we will tackle the each situation individually.

###  III.1 Instantaneous formation

The instantaneous formation scenario relies on the mediator mass, $m_{\chi}$,
being larger than the Hubble parameter as structures begin to grow. This sets
a strict lower bound on the mediator mass,

$m_{\chi}>\sqrt{\frac{4\pi^{3}g_{\star}}{45}}\frac{T_{g}^{2}}{m_{p}}.$ (14)

Under this restriction, the halo mass is well approximated by the total $\psi$
mass within the $1/m_{\chi}$ radius:

$M_{h}=\frac{4\pi}{3}R^{3}\rho_{\psi}=\frac{8\pi^{3}g_{\star
s}}{135}\frac{\eta m_{\psi}T_{g}^{3}}{m_{\chi}^{3}}.$ (15)

At the final stage of the halo collapse, the halo is reheated. Subsequently,
the washout effect of the dark sector asymmetry, caused by the $\psi$ number
violating scattering, is reactivated. This reactivation has the potential to
annihilate the existing asymmetry within the collapsing halo. The timescale of
this process is determined by the washout terms in Eq. (4):
$\tau_{\mathrm{ann}}^{-1}\sim\kappa^{4}T^{5}_{h}/m^{4}_{\varphi}$. For
successful PBH or Fermi ball formation, the $\psi$ particles inside the halo
cannot be totally annihilated before the end of collapse. This requires the
collapse timescale $\tau_{c}<\tau_{\mathrm{ann}}$. We estimate $\tau_{c}$ as
the following: For simplicity, we assume that $y_{ij}=y$ among all
generations. By the virial theorem,

$\frac{4\pi^{4}}{90}R^{3}T_{h}^{4}=\frac{y^{2}N^{2}}{R},$ (16)

which implies the temperature of the halo increases as $1/R$. In the Yukawa
energy term, we neglected possible order 1 factors that comes from the radial
density profile or non-spherical distribution of matter in the halo.
Annihilation is only efficient at high halo temperatures, and radiation
cooling at this stage is dominated by the surface radiation channel:

$\frac{dE}{dt}=4\pi R^{2}T_{h}^{4}\sim\frac{y^{2}N^{2}}{R^{2}}\frac{dR}{dt}.$
(17)

From this, we can calculate the collapse timescale $\tau_{c}=R/(dR/dt)$ by

$\tau_{c}=\frac{y^{2}N^{2}}{R^{3}T_{h}^{4}}\sim\frac{\sqrt{yN}}{T_{h}}\sim\frac{\sqrt{y\eta}}{T_{h}}\left(\frac{T_{g}}{m_{\chi}}\right)^{3/2},$
(18)

where we again used the virial theorem. Comparing this with $\tau_{\rm ann}$,
we obtain the following constraint:

$\left(\frac{m_{\chi}}{T_{g}}\right)^{3/2}>\kappa^{4}\sqrt{y\eta}\left(\frac{T_{h}}{m_{\varphi}}\right)^{4},$
(19)

which must be satisfied for all achievable halo temperature $T_{h}$. The most
stringent case occurs at the radius $R_{f}$ of the final compact object. The
maximum temperature is estimated to be

$T_{\rm max}=\frac{T_{g}}{m_{\chi}R_{f}}.$ (20)

Deducing the type of the compact object formed (PBH/Fermi ball) will require a
more involved analysis and we will discuss this in detail in Sec. III.3.
However, the qualitative behavior can be understood by comparing the scaling
of $R_{f}$. For PBHs, the final radius is simply $R_{f}=2GM_{h}$ so $R_{f}\sim
N$. Alternatively, the Fermi balls radius scales as $R_{f}\sim
N^{2/3}/m_{\psi}$ (see Eq. (30)). It follows then that the Schwarzschild
radius of the halo will exceed the Fermi ball radius at large $N$. Therefore,
our model generally predicts PBHs on the higher mass end while Fermi balls are
formed on the lower mass end.

Finally, the Compton wavelength of $\psi$’s must be larger than $R_{f}$ for
the halos to collapse. This is always satisfied for Fermi balls with $N\gg 1$
which can be seen from the scaling formula. We also explicitly check this
condition in the PBH formation regime.

###  III.2 Noninstantaneous formation

In the noninstantaneous formation scenario, subhorizon sized Fermi balls are
formed initially. As the Universe cools down, more Fermi balls enter the
horizon and the Yukawa interaction continues to drive the structure formation
through Fermi ball merges. There are two temperatures of importance in this
scenario. The first is the growth temperature $T_{g}$, at which density
perturbation grows and halos form, previously defined in Eq. (12). The second
relevant temperature, $T_{c}$, corresponds to the point in time at which
mergers stop and halos start to radiatively cool and collapse. Based on our
previous discussion, $T_{c}$ occurs at the temperature when the length scale
$1/m_{\chi}$ enters the horizon:

$T_{c}=\left(\frac{45}{4\pi^{3}g_{\star}}\right)^{1/4}\sqrt{m_{\chi}m_{p}}.$
(21)

Much like conventional structure formation, we expect a rich history of
mergers which steadily generate larger dark matter halos, albeit on a
significantly shorter timescale. The exact mass function of such objects
requires many-body simulation of the merger dynamics. Here, we simply
approximate the halo mass function as monochromatic and peaked at $M_{h}$,
where the halo mass $M_{h}$ is given by the total mass of all the available
halos within the horizon at $T_{c}$. Since the halo mergers are driven by
Yukawa interactions, which again, are significantly stronger than gravity, we
expect mergers to be very efficient. The approximation is justified by recent
$N$-body simulations which have illustrated the development of structure
during radiation domination Domènech  _et al._ (2023). Crucially, these
simulation show a marked absence of filimentary structure, indicating that
nearly all of the heavy fermions end up in halos, with the halo’s radius being
determined by the mediator’s Compton wavelength.

Here, we compute the halo mass and derive the constraints again as in the
instantaneous formation scenario. The halo density before the mergers is given
by

$n_{h}=\frac{1}{H_{g}^{-3}}\frac{s_{c}}{s_{g}},$ (22)

where $s_{c}$ and $s_{g}$ are the entropy densities at $T_{c}$ and $T_{g}$,
respectively. Assuming that $g_{\star s}$ is almost a constant between $T_{g}$
and $T_{c}$, the total halo mass

$M_{h}=\eta m_{\psi}s_{g}H_{g}^{-3}\frac{4\pi
n_{h}}{3m_{\chi}^{3}}=\frac{8\pi^{3}g_{\star s}}{135}\frac{\eta
m_{\psi}T_{c}^{3}}{m_{\chi}^{3}}.$ (23)

The halo temperature at formation can also be estimated using the virial
theorem:

$T_{i}=\left(\frac{90}{4\pi^{4}}\right)^{1/4}\frac{\sqrt{yN}}{R}=\left(\frac{32\pi^{2}g_{\star
s}^{2}}{405}\right)^{1/4}T_{c}\sqrt{\frac{y\eta T_{c}}{m_{\chi}}}.$ (24)

The estimation of $\tau_{c}$ is slightly different in this scenario for the
halo mass Eq. (23), and we obtain

$\tau_{c}\sim\frac{\sqrt{y\eta}}{T_{h}}\left(\frac{T_{c}}{m_{\chi}}\right)^{3/2}.$
(25)

So the annihilation constraint becomes

$\left(\frac{m_{\chi}}{T_{c}}\right)^{3/2}>\kappa^{4}\sqrt{y\eta}\left(\frac{T_{h}}{m_{\varphi}}\right)^{4},$
(26)

with the maximum halo temperature given by

$T_{\rm max}=\left(\frac{32\pi^{2}g_{\star
s}^{2}}{405}\right)^{1/4}\frac{\sqrt{y\eta}}{R_{f}}\left(\frac{T_{c}}{m_{\chi}}\right)^{3/2}.$
(27)

The Compton wave length of $\psi$’s are always larger than the PBH/Fermi ball
radius, as in the case of instantaneous formation.

Scalar cooling occurs alongside the Yukawa mergers and could lead to the
formation of compact objects before mergers reach the $1/m_{\chi}$ threshold.
To estimate this effect, we note that all mergers occur at the scale $\ll
1/m_{\chi}$. The potential due to a single halo is $V(r)\sim y^{2}N/r$. The
timescale for the initial mergers can be estimated by the infall time of the
two initially static halos:

$t_{\rm infall}=\frac{\pi}{4}\frac{m_{\psi}d^{3/2}}{yM_{h}^{1/2}},$ (28)

where $d\sim H_{g}^{-1}$ is the average halo distance. For the cooling time, a
conservative lower bound is given by the initial halo radius $\sim
H_{g}^{-1}$. However, even if $t_{\rm infall}>\tau_{\rm cool}$, the initially
formed compact objects will be Fermi balls of small mass which interact via
the Yukawa interaction. Therefore, it is still possible to form objects with
the final mass given by (23).

In Fig. 3, we demonstrate an example of the available parameter space and the
PBH/Fermi ball mass distribution. The instantaneous formation scenario occurs
for larger values of $m_{\chi}$ (to the right of the white dashed line). For
heavy $\psi$ particles ($\gtrsim 10^{9}$ GeV), the growth temperature $T_{g}$
is determined by the condition $T_{g}<T_{\rm form}$ whereas for lower $\psi$
mass, the growth of structure occurs immediately after the decoupling of
$\psi$. This leads to the change in the boundary direction between
instantaneous/noninstantaneous formation.

Fig. 3: PBH/Fermi ball mass on the available parameter space. The purple grid
marks the region where Fermi balls form and the instantaneous/noninstantaneous
formation is separated by the white dashed line. The region covered by red and
orange mesh is excluded by the annihilation and total particle number
constraint (see Sec. III.3) and the gray mesh marks the region where cooling
could be faster than mergers. The contour for Planck mass Fermi balls is shown
by the solid cyan line. The parameters are chosen to be $m_{\varphi}=10^{16}$
GeV, $y=5\times 10^{-2}$ and $\kappa=2\times 10^{-4}$ and $\eta=10^{-25}$.

###  III.3 The halo destiny

In the final stages of scalar cooling, compact objects with high fermion
density are formed. The final state of these compact objects, whether it be as
PBHs or Fermi balls, depends on the nuclear physics in the dark sector. At the
final stage of the collapse, the dark fermions are reheated to become highly
relativistic. The effective fermion mass

$m_{\ast}=m_{\psi}+y\langle\chi\rangle$ (29)

are reduced by the nonzero vacuum expectation value (vev) of $\chi$ within the
halo Domènech and Sasaki (2021). This enhances the Fermi degeneracy pressure
and an equilibrium state can be reached if the Yukawa attraction is balanced
by this enhanced pressure. This is explored in Refs. Gresham  _et al._ (2017);
Gresham and Zurek (2019); Gresham  _et al._ (2018) using relativistic mean
field theory from the Walecka model Walecka (1974); Serot and Walecka (1986)
and also in Ref. Del Grosso  _et al._ (2023) as fermionic soliton solutions in
general relativity. The qualitative behavior of the system is summarized as
follows: At large $N$, when the radius of the Fermi ball is much larger than
$1/m_{\chi}$, the system is relativistic and reaches the saturation limit
where $R\sim N^{1/3}$. The stability of the resulting object is determined by
solving the relativistic equation of state together with the Tolman-
Oppenheimer-Volkof (TOV) equations in Ref. Gresham and Zurek (2019). At medium
$N$, the constituents remain relativistic but $R\ll 1/m_{\chi}$. One has to
take into account the nontrivial spatial variation of the scalar vev within
the Fermi ball. The correct scaling of the radius is found to be $R\sim
N^{2/3}$. This will be the relevant case in our scenario since the final
compact objects are formed by radiative cooling of the initial $1/m_{\chi}$
sized halos. In the limit $m_{\chi}\rightarrow 0$, the stable radius is given
by Gresham  _et al._ (2017)

$R_{\rm
FB}=\left(\frac{243}{2048\pi^{2}}\right)^{1/6}\frac{yN^{2/3}}{m_{\psi}}.$ (30)

For a finite $m_{\chi}$, the saturation limit is reached for

$N_{\rm sat}\sim\left(\frac{m_{\psi}}{ym_{\chi}}\right)^{3/2}.$ (31)

In our model,

$N=\frac{8\pi^{3}g_{\star s}}{135}\frac{\eta T^{3}}{m_{\chi}^{3}},$ (32)

where $T=T_{c}$ or $T_{g}$. Therefore, Eq. (30) is valid as long as

$T^{3}\ll\frac{1}{\eta}\left(\frac{m_{\psi}m_{\chi}}{y}\right)^{3/2}.$ (33)

Using Eqs. (14) or (21), one can show that it suffices to require
$\eta\ll(m_{\psi}/ym_{p})^{3/2}$ in both formation scenarios. This is easily
satisfied as typically, $\eta<10^{-20}$ and $m_{\psi}\gg 10^{5}$ GeV in our
model. In addition, we require that $N>10^{6}$ so that there are always enough
particles for halo formation.

In principle, the condition for PBH formation requires a TOV analysis of the
stability of the system to take into account general relativistic effects.
Here, we adopt a heuristic argument and declare that a PBH has formed when
$R_{\rm FB}<2GM_{f}$. We leave the detailed analysis combining TOV equations
as future work. The region of parameter space relevant for the formation of
Fermi balls is marked by the purple grid in Fig. 3.

##  IV Significance in Cosmology

In our scenario, either Fermi balls or PBHs are formed in the early radiation
dominated Universe. These compact objects can be produced across a large span
of masses. Of particular interest to this work will be (i) PBHs with masses
$\sim 10^{5}-10^{6}\ M_{\odot}$, i.e., SMBH and (ii) Fermi balls with masses
$<10^{-10}M_{\odot}$.

###  IV.1 Supermassive black holes

Fig. 4: SMBH mass (in $M_{\odot}$) predicted from our model (solid contours).
The black dashed line indicates the required abundance (with negligible
accretion) by quasar observations. The purple grid marks the region where
Fermi balls instead of PBHs form. The temperature at final halo formation (in
the noninstantaneous scenario) is labeled on the dash-dotted lines. The
parameters are chosen to be $m_{\varphi}=10^{16}$ GeV, $y=5\times 10^{-2}$ and
$\kappa=2\times 10^{-4}$ and $\eta=10^{-30}$

PBHs have the ability to grow through accretion and therefore may seed high
redshift AGNs Clesse and García-Bellido (2015); Bean and Magueijo (2002). This
possibility has the potential to explain the apparent formation of galaxies at
redshifts higher than predicted by $\Lambda\rm CDM$. This aspect is
particularly interesting given the recent JWST observations of AGN at $z\sim
11$ Bunker  _et al._ (2023). In addition, PBHs could also act as progenitors
of the SMBHs found in the quasars at $z\sim 6$ Volonteri and Stark (2011);
Willott  _et al._ (2010). However, we should note that unlike the ordinary
formation scenario where PBHs are produced from the primordial density
fluctuations, the abundance of PBHs can be produced in our model without
requiring a highly non-Gaussian tail of the fluctuation Nakama  _et al._
(2018, 2016); Garcia-Bellido  _et al._ (2017) (see Refs. Garriga  _et al._
(2016); Huang  _et al._ (2023) for other SMBH formation scenarios). Following
the discussion in Ref. Serpico  _et al._ (2020), we note that the PBHs could
either be SMBH at formation or intermediate mass black holes and undergo
significant growth before redshift $z\sim 6$. In the later scenario, accretion
near the Eddington limit is needed but a growth factor of $\sim 10^{5}$ is
still achievable for the accretion model discussed in Ref. Serpico  _et al._
(2020).

The observed AGNs at $z=6$ determines the fraction of dark matter in PBHs
above $10^{6}M_{\odot}$ to be $f_{\rm PBH}\equiv\Omega_{\rm PBH}/\Omega_{\rm
DM}\sim 2.9\times 10^{-9}$ Serpico  _et al._ (2020); Volonteri and Stark
(2011); Willott  _et al._ (2010). Due to theoretical uncertainties involving
PBH accretion prior to $z=6$, we assume that the supermassive PBHs produced in
our mechanism have negligible accretion rates before $z=6$. Even in this
extreme case, the required PBH abundance is still consistent with the CMB
bound Serpico  _et al._ (2020); De Luca  _et al._ (2020), which constrains the
PBH with mass $\sim 10^{5}M_{\odot}$ to have $f_{\rm PBH}\lesssim 10^{-9}$.

The abundance of PBHs in our scenario will rely on the fact that essentially
all of the heavy fermions $\psi$ are captured into halos. This allows us to
use the fermion energy density to determine the fraction of PBHs present
today,

$\begin{split}f_{\rm PBH}&=\frac{\Omega_{\rm PBH}}{\rm
DM}\approx\frac{\Omega_{\psi}}{\Omega_{\rm DM}}\\\\[7.11317pt] &\approx
0.23\left(\frac{m_{\psi}}{10^{10}\rm
GeV}\right)\left(\frac{\eta}{10^{-20}}\right).\end{split}$ (34)

The observed abundance can be produced by taking $\eta=10^{-30}$. The
corresponding scalar mass falls in the range of ultralight scalars. In Fig. 4,
we labeled several contours that produce the SMBH mass of interest.

###  IV.2 Fermi balls as dark matter candidates

Fig. 5: Predicted Fermi ball mass. The green dashed line gives $f_{\rm FB}=1$.
The solid contour in cyan indicates Fermi balls with Planck mass. The blue
grids cover the region that is constrained by microlensing and gray region
marks where cooling could be faster than mergers. The parameters are chosen to
be $m_{\varphi}=10^{16}$ GeV, $y=5\times 10^{-2}$ and $\kappa=2\times 10^{-4}$
and $\eta=10^{-20}$. Fig. 6: Mass and radius for Fermi balls (or PBHs) with
$f_{\rm FB}=1$ in Fig. 5. The blue meshed region is constrained by
microlensing observations. For comparison, the typical Einstein radius in EROS
is also shown. The vertical dotted line indicates Planck-massed Fermi balls.

Besides PBHs, our model can also accommodate the formation of compact Fermi
balls with a wide range of mass that even extend down below the Planck mass.
We demonstrate such an example in Fig. 5. Since the Fermi balls only interact
with the SM matter through gravity, they also serve as candidate of dark
matter. We define $f_{\rm FB}$ as the fraction of DM in Fermi balls in a
similar fashion. In Fig. 6, we show the Fermi ball (or PBH) mass and the
radius that correspond to the $f_{\rm FB}=1$ contour in Fig. 5 (green dashed
line). In the Fermi balls mass range, their abundance can be constrained from
the microlensing observations Smyth  _et al._ (2020); Niikura  _et al._
(2019); Griest  _et al._ (2013, 2014); Tisserand  _et al._ (2007); Allsman
_et al._ (2001); Oguri  _et al._ (2018); Zumalacarregui and Seljak (2018);
Wilkinson  _et al._ (2001). For example, the EROS experiment Tisserand  _et
al._ (2007) monitors the microlensing events of stars in the Magellanic
Clouds. The typical Einstein radius of such lensing events for a
$10^{-5}M_{\odot}$ Fermi ball is estimated to be

$R_{E}=\sqrt{4GM\frac{D_{\rm LS}D_{\rm L}}{D_{\rm LS}+D_{\rm L}}}\sim
10^{6}\mathrm{km},$ (35)

where $D_{\rm LS}\sim 50$ kpc is the distance from the source to the lensing
object and $D_{\rm L}\sim 5$ kpc is the typical distance from the earth to the
lensing object. However, such a Fermi ball in Fig. 5 in the region with
$f_{\rm FB}\sim 1$ has radius less than $1$ meter. Therefore, the current
microlensing constraints also apply to Fermi balls. This rules out the
possibility of $f_{\rm FB}=1$ in the mass range $10^{-10}<M/M_{\odot}<10^{9}$
Carr  _et al._ (2021). The window for Fermi balls to account for all of dark
matter remains open below $10^{-10}M_{\odot}$ (however, see the disputed
result regarding femtolensing of $\gamma$-ray bursts Barnacka  _et al._
(2012); Katz  _et al._ (2018)). In particular, it includes the asteroid mass
range that is currently unconstrained. It is proposed in Ref. Das  _et al._
(2022) that stellar shocks produced by collision of a star with compact
objects like Fermi balls could produce limits in this mass range. On the lower
mass end, constraints from PBH evaporation do not apply to Fermi balls as they
do not emit particles via Hawking radiation. So the mass window for $f_{\rm
FB}=1$ could extend down to the lowest mass predicted in our model.

The Fermi balls behave as noninteracting CDM, even though the dark sector
Yukawa attraction is still present. To demonstrate this, we estimate the Fermi
ball density at the Galactic center using the generalized Navarro–Frenk–White
(NFW) profile

$\rho(r)=\frac{2^{3-\alpha}\rho_{0}}{(r/r_{s})^{\alpha}(1+r/r_{s})^{3-\alpha}},$
(36)

where the scale radius and density are taken to be $r_{s}=8$ kpc and
$\rho_{0}=1.08\times 10^{-2}M_{\odot}\ \rm pc^{-3}$ to match the local value
of the DM density. We average this density profile within 1 pc and find the
average distance between Fermi balls to be

$\langle d\rangle\sim\begin{cases}0.08(M_{\rm FB}/M_{\odot})^{1/3}\ \rm
pc\quad&\alpha=1\\\ 0.004(M_{\rm FB}/M_{\odot})^{1/3}\ \rm
pc\quad&\alpha=2\end{cases}.$ (37)

Then the ratio of the average distance to the Yukawa force range can be
parametrized by

$\langle d\rangle m_{\chi}\sim\zeta\left(\frac{m_{\psi}}{10^{11}\rm
GeV}\right)^{1/3}\left(\frac{\eta}{10^{-20}}\right)^{1/3}\left(\frac{T}{1\rm
GeV}\right),$ (38)

where $\zeta=0.7\times 10^{10}\ (\rm or\ 3.7\times 10^{8})\gg 1$ for
$\alpha=1\ (\rm or\ 2)$ is the numerical factor coming from parametrization.
Depending on the formation scenario, the temperature $T$ in Eq. (38) is either
$T_{g}$ or $T_{c}$. We check that in both cases, $T$ is always greater than 1
GeV. Therefore, the self-interaction between Fermi balls is negligible.

In summary, we described formation of primordial compact objects driven by
Yukawa interactions starting from the inflaton decay and followed by halo
formation in the dark sector. There is a wide range of possible masses, and
the outcome falls into two distinct formation scenarios. The case of
noninstantaneous formation admits a rich merger history and could be better
understood with the aid of future numerical simulations that take into account
dissipation effects. The end result of the growth of structure is a population
of either PBHs or Fermi balls. The massive PBHs could seed the high redshift
AGNs, while, at the lower mass end, the Fermi balls could account for all of
the dark matter. Since the constraints for evaporating PBHs do not apply to
compact Fermi balls with mass less than $10^{15}$ g, future work is needed to
constrain this possible mass range.

###### Acknowledgements.

We thank Michael S. Turner for helpful discussions. This work was supported by
the U.S. Department of Energy (DOE) Grant No. DE-SC0009937 and by the UC
Southern California Hub, with funding from the UC National Laboratories
division of the University of California Office of the President. The work of
A.K. was also supported by the World Premier International Research Center
Initiative (WPI), MEXT, Japan, and by Japan Society for the Promotion of
Science (JSPS) KAKENHI Grant No. JP20H05853. M.M.F was supported by the
University of California, Office of the President Dissertation Year Fellowship
and donors to the UCLA Department of Physics & Astronomy.

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