# Fractal dimension and topological invariants as methods to quantify
complexity in Yayoi Kusama’s paintings

Elsa de la Calleja  Instituto de Investigaciones en Materiales, Facultad de
Ingeniería, Universidad Nacional Autónoma de México, Av. Universidad 3000,
Ciudad de México, 04510, México  Roberto Zenit<EMAIL_ADDRESS>Instituto de
Investigaciones en Materiales, Facultad de Ingeniería, Universidad Nacional
Autónoma de México, Av. Universidad 3000, Ciudad de México, 04510, México
School of Engineering, Brown University, 184 Hope St, Providence, RI 02912,
USA

###### Abstract

Intricate patterns in abstract art many times can be wrongly characterized as
being complex. Complexity can be an indicator of the internal dynamic of the
whole system, regardless of the type of system in question, including art
creation. In this investigation, we use two different techniques to
objectively quantify complexity in abstract images: the fractal dimension and
the value of the Betti numbers. We first validate our technique by considering
synthetic images with a random distribution of dots, to then apply it to a
series of ‘Net obsession’ paintings by Yayoi Kusama. Surprisingly, we found
that although the fractal dimension of her works in this series is comparable
to those by Jackson Pollock in his dripping period, which could indicate a
high level of complexity, the value of the Betti numbers do show
disconnectedness and not high complexity. This is intuitively in agreement
with the visual assessment of such works.

##  I Introduction

The symmetry, complexity or periodicity of irregular spatial-temporal patterns
could be characterized as beautiful. For many years, numerous investigations
have tried to characterize the dynamical processes that lead to the creation
of such patterns. Structural properties in many complex systems are the final
manifestations of the internal dynamics. Through measures such as
connectivity, porosity, texture, roughness, symmetry or temporal correlations
to name a few, the complexity of systems can be characterized Ball (2016);
Barnsley (1993); Mandelbrot (1982); Etzenhouser et al. (1998); Daccord et al.
(1986); Goodchild and Mark (1987). Nature produces many examples of complex
patterns Caballero et al. (2012); Barnsley (1993); Andrienko (2018). Some
investigations have tried to described the dynamics behind natural
complexities, many of which are visual: for example Turing patterns Caballero
et al. (2012), symmetries, textures and patterns of marine invertebrates Ball
(2016), etc.

Art, of course, is full of complexity but its objective quantification is not
trivial. Some studies have quantified particular characteristics of abstract
paintings. For instance, Naumis _et al._ Aragón et al. (2008) evaluated the
_turbulent_ luminance of van Gogh paintings, Mekler _et al._ Martínez-Mekler
et al. (2009) measured size-ordered distributions in Vasili Kandinsky. The
physical process leading to the emergence of complex patterns in the David A.
Siqueiros accidental painting technique Palacios et al. (2019) and in Jackson
Pollock’s drip paintings Zetina et al. (2015) have been studied by our group.
The fractal dimension of artworks has been widely used to assess the degree of
complexity Mureika and Taylor (2013); Taylor et al. (1999); de la Calleja et
al. (2016); Redies et al. (2007); Palacios et al. (2019); Zetina et al.
(2015). This attempt has created a renewed interest in the subject and opens a
new area of study Abbot (2006); Coddington et al. (2008), even if their
validity and extend of applicability have been questioned Jones-Smith and
Mathur (2006); Jones-Smith et al. (2009). Despite these efforts, a general
consensus on the appropriate mathematical tools to quantify the complexity of
abstract art has not been reached.

In this report, the complexity of an image is evaluated using those two
different mathematical approaches: fractal dimension and Betti numbers. Both
measures are used to characterize structural properties as geometry, symmetry,
non-regularity, dimensionality and connectivity of a set. Felix Hausdorff
developed the mathematical description for Mandelbrot’s sets better known as
fractals Mandelbrot (1982). It is well accepted that the Hausdorff-Besicovitch
fractal dimension is the usual parameter to characterize irregular spatial-
temporal patterns as a result of temporal or spatial evolution of the elements
of the system Mecke (1996); Ball (2016); Forsythe et al. (2011). On the other
hand, the modern formulation of the theory of algebraic topology Mac Lane
(1986) is also a powerful tool to analyze complexity in many systems by using
the value of topological invariants, the so-called Betti numbers Pranav et al.
(2017); Gerarld (1990); de la Calleja and Zenit (2017); de la Calleja et al.
(2016). Despite being widely used in many other areas to quantify structural
properties Kosterlitz and Thouless (1973); Gottsche (1990); Spånslätt et al.
(2005); Kondic et al. (2012); Xia and Wei (2014), the use of topological
invariants as a measure of complexity has not been used much for the case of
art. Recently, the Betti numbers of abstract expressionist artworks were
computed and used as a measure of complexity de la Calleja and Zenit (2017),
with promising results to be applied in other cases.

To validate our proposed methodology to quantify complexity we first apply it
to sets of black and white synthetic abstract images formed by randomly adding
adding dots (or lines) in a two-dimensional empty area. The advantage of using
black-white two-dimensional images allows applying the technique without any
previous image processing. Then, the same methodology was used to measure the
complexity in Yayoi Kusama’s artworks. Her work is characterized by the use of
regular dots patterns. Some critics have cataloged such periodic distribution
of dots and nets as ‘simple’. The analysis of Kussama’s works is of particular
importance because we found that their fractal dimension hold similar values
as other abstract paintings, such as Pollock’s; however, the value of the
Betti numbers indicate a lack of connectivity and not necessarily complexity.
Hence, we show that the fractality of an image cannot be used alone to
quantify complexity. Furthermore, our results indicate that complexity is not
related to the visual appreciation of the image.

##  II Mathematical parameters to quantify complexity

To investigate the complexity in abstract images, two methods were considered
to assess the structural properties of images: the fractal dimension and the
value of topological invariants. Both procedures have been used in many cases
from spatial distribution of galaxies to music Pranav et al. (2017); Barnsley
(1993); Bigerelle and Iost (2000). In this study these mathematical parameters
were calculated for a set of synthetic abstract images composed with dots or
lines. As a result, two different measures were obtain for a given image. This
allows us to compare and contrast the structural properties of the images.
Evidently, in the art world the painting composition contains multiple
components, such as dense lines, layers of paint overlay, cluster nodes,
voids, or shapes and shadows. And many of those objects live in a three
dimensional space.

###  II.1 The Betti numbers, $\beta_{i}$

To sustain the characterization of the connectivity of abstract images using
topological invariants, a brief introduction is presented here. Algebraic
topology is the branch of mathematics that studies topological spaces Milnor
(1963); Munkres (1984); Edelsbrunner and Harer (2010). The formalism allows
the formulation of statements about topological spaces into the language of
group theory, offering substantial flexibility and a deeper understanding of
spatial structure and connectivities Pranav et al. (2017). A topological space
is characterized by its invariants, which are numbers that remain unchanged
under a homeomorphic transformation Gerarld (1990); Spanier (1966). All
homology groups are vector spaces, their dimension is the number of
independent $i-$dimensional cycles in a topological space. This is the formal
definition for the Betti numbers $\beta_{i}$, where $i=0,1,2,...,d$ Betti
(1871); Edelsbrunner and Harer (2010). The Betti numbers provide the general
description of structural topology in systems or sets where the mass
distribution play a relevant role.

The description of the boundaries of holes Munkres (1984) allow the
characterization of the connectivity of set in a space. For a two dimensional
space, only the two Betti numbers can be defined: $\beta_{0}$ measures the
number of simply-connected objects, path-connected or isolated components of
_X_ ; and $\beta_{1}$ counts the number of loops enclosing independent
tunnels, it means the number of holes within the simply-connected spaces or
the $\mu$-dimensional holes in _X_. By taking into consideration this reduced
definition, we used the values of these two numbers to assess to the
topological properties on abstract images and quantify their complexity.

###  II.2 The fractal dimension

A fractal object can be considered as an irregular set taking by reference the
irregularity in classical geometry. No matter how much the set is magnified,
smaller and smaller irregularities become visible Mandelbrot (1982); Gerarld
(1990). Roughly speaking, there are similar features on fractal objects: _(i)_
self-similarity; _(ii)_ recursive procedure for its construction; _(iii_ its
size is not quantified by the usual measures such as length; _(iv)_ the object
has a fine structure, which contains details at arbitrarily small scales; and
some others Falconer (2014). Taking into account the above characteristics the
fractal dimension is not a topological invariant property, since fractal
objects are not homeomorphic Mandelbrot (1982); Gerarld (1990); Harewicz and
Wallman (1941); Taylor and Tricot (1985); Taylor (1986); Falconer (1987);
Hilton (1988). Mandelbrot Mandelbrot (1982) defined a fractal as a set in
which the Hausdorff-Besicovitch dimension exceed the topological dimension.

Briefly, the formalism used consist of given a non-empty space of
n-dimensional Euclidean space, the box counting fractal dimension can be
calculated by the general equation $N(S)=(\epsilon/\epsilon_{0})^{-f(\alpha)}$
for small $\epsilon$ where $N(S)$ is the number of boxes used to cover the set
$S$, $\epsilon$ is the size of the box and $\epsilon_{0}$ as the minimum size
of box. The measure of fractal dimension is compute with a digital binarized
image which is divided into pieces of size $\epsilon$. Halsey et al. Halsey et
al. (1986) defined $P_{i,Q}^{Q}\sim\epsilon_{i}/\epsilon_{0}^{\alpha Q}$ as
the probability to visit a site in different scaling indices $Q$,
specifically, the number of times that $\alpha$ in $P_{i,q}$ takes a value
between $\alpha^{\prime}$ and $d\alpha^{\prime}$ defined as
$d\alpha^{\prime}\rho(\alpha^{\prime})\epsilon^{-f(\alpha^{\prime})}$ where
$f(\alpha^{\prime})$ is a continuous function. The fractal dimension for all
the scaling indices can be evaluated

$D_{Q}=\frac{1}{1-Q}\lim_{\epsilon\rightarrow
0}\frac{ln\sum_{i=1}^{N(\epsilon)}[P_{(i,Q)}]^{Q}}{ln(\epsilon/\epsilon_{0})}$
(1)

As _Q_ represents different scaling indices, we can define

$\sum_{i=1}^{N(\epsilon)}[P_{(i,Q)}]^{Q}=\int
d\alpha^{\prime}\rho(\alpha^{\prime})\epsilon^{-f(\alpha^{\prime})+Q\alpha^{\prime}}$
(2)

where the Lipschitz-Hölder exponent $\alpha_{i}$, characterizes the
singularity strength in the _ith_ box and quantifies the distribution of
complexity in an spatial location. There are similar definitions of fractal
dimension, and several techniques to measure it. To mention a few: the box-
counting method Barnsley (1993), Hausdorff measure Falconer (2014), Packing
dimension Gerarld (1990); Xiao (1997), fractal spectrum Chhabra and Jensen
(1989) on some others. Nevertheless, the methods are based on equivalent
definitions of fractal dimension. In this report the fractal spectrum with the
box-counting method is used to quantify the fractality on the sets Chhabra and
Jensen (1989).

##  III Methods and Methodology

In this work we evaluate the complexity of synthetic abstract images
considering the values of the fractal dimension and the Betti numbers. The
procedure to calculate complexity was the following: series of synthetic
abstract images were produced using a script written with a
Matlab$\copyright$. Two kind of images were tested: dots and lines. In both
cases a predetermined number of black objects (dots or lines) were placed
randomly within a rectangular blank space. To ensure repeatability, at least
three images were produced for each condition. By progressively adding dots or
lines on the given area, the black area increases as dots or lines overlap.
The area size was kept fixed for all cases. For the case of dots, three series
were constructed with dots of three mean diameters: $D_{1}=0.01$, $D_{2}=0.02$
and $D_{3}=0.03$. The dot size distribution has a standard deviation of 10% of
the mean diameter. For the case of lines objects of fixed length, $L=0.1$,
were placed randomly in the space with a random orientation. Three series were
generated by changing the thickness of the lines, obtained series of
$T_{1}=0.1,T_{2}=0.3$, and $T_{3}=0.5$. The mass distribution of dots or lines
on the blank space did not have any distribution preference, hierarchical
aggregation or imposed self-similarity. Typical images generated with dots are
shown in Fig.1.

Figure 1: Typical synthetic abstract images generated with random distribution
of dots.

###  III.1 Measuring complexity

The synthetic abstract images were binarized with the use of the free software
ImageJ Schneider et al. (2012). The fractal dimension of each image was
calculated with the FracLac plug-in Chhabra and Jensen (1989); Chhabra et al.
(1989). The software calculates the spectrum $f(\alpha)$ of an ordinary
uniform set which is a single point on the $f(\alpha)-\alpha$ plane. If the
set has fractal characteristics, it shows a line of consecutive points for
$Q\geq 0$ that starts on the left side of the spectrum climbing up to the
maximum value Halsey et al. (1986); Hentschel and Procaccia (1983); Schneider
et al. (2012). The maximum value for the generalized dimension corresponds to
$Q=0$. The maximum value was taken as the fractal dimension. The Betti numbers
were measured with the software Chomp cho (2016, accessed 10/27/2020). The
algorithm of this software calculates the local elementary reductions and
collapses to compute the homology of the images. The software calculates the
first two Betti numbers, $\beta_{0}$ number, and $\beta_{1}$ number. The black
objects over a blank background can be broadly related to the connectivity
among dots or lines and $\beta_{1}$ number indicates empty spaces
corresponding to the number of holes in the image.

##  IV Results for synthetic images

As mentioned previously, the aim of this report is to analyze the complexity
of synthetic abstract images by measuring their fractal dimension and the
value of their two Betti numbers. The case of synthetic images composed by
dots is presented and analysed here. The case for lines shows qualitatively
similar results; hence, it is presented in Appendix A. By observing the
changes in fractal dimension and Betti numbers as the number of dots increases
we can evaluate how the complexity evolves. We use ‘simple’ distributions of
dots motivated by our interest to evaluate the complexity of paintings by
Japanese avant-garde artist Yayoi Kusama. In 1959, she created _Infinity Nets_
, a 10 meters paint with meticulously inscribed patterns suggestive of a white
net, over a slightly darker grayish-white background. Her work has been
compared to Jackson Pollock’s Nakajima (2006); Sydney (1958). Visually, it is
evident that the two artists’ abstract expressionists works are significantly
different. However, as shown below, our measurements of their fractal
properties suggest that their paintings have similar structural properties.

###  IV.1 Fractal dimension of synthetic images

Figure 2(a) shows the fractal dimension $f(\alpha)$ of the three series of
abstract images computed with different sizes $D_{1},D_{2},D_{3}$ (black
squares, red dots and blue triangles, respectively) as a function of the
number of dots, $N_{d}$. The fractal dimension was measured for images
containing from from $2$ to $2500$ dots. Since the area of the image is
constant, an increase of the number of dots increases the density. Note also
that in these images, the dots are placed randomly and can overlap. The
fractal dimension gradually increases from $0.65$ as the number of dots
increases. A maximum fractal dimension of almost $2.0$ is reached at an
certain number of dots, which depends on the dot size. These maximum and
minimum fractal dimension values are expected Lesmoir-Gordon et al. (2000).
After the fractal dimension reaches a maximum, its value decreases as the
number of dots continues to increase. This is a consequence of the image
becoming entirely populated with dots. Typical examples of images for the
largest fractal dimension (and below and above) are shown in Figs. 2 (b), (c)
and (d)

(a) $f(\alpha)\approx 1.5,N_{d}=100$

(b) $f(\alpha)\approx 1.7,N_{d}=500$

(c) $f(\alpha)\approx 1.5,N_{d}=2000$

Figure 2: (a) The fractal dimension $f(\alpha)$ as a function of number of
dots, $N_{d}$, for a fixed surface area, for three different drop diameters,
$D_{1},D_{2},D_{3}$ (black squares, red dots, and blue triangles
respectively). (b,c,d) Typical images showing different values $f(\alpha)$,
below, at and above the maximum value for $D_{2}$=0.02.

###  IV.2 The Betti numbers of synthetic images

(a) $\beta_{0}/\beta_{1}>1,N_{d}=200$

(b) $\beta_{0}/\beta_{1}\approx 1,N_{d}=500$

(c) $\beta_{0}/\beta_{1}<1,N_{d}=1500$

Figure 3: (a) The value of $\beta_{0}$ (fill symbols) and $\beta_{1}$ (empty
symbols) as a function of $N_{d}$; (b) ratio $\beta_{0}/\beta_{1}$ as a
function of $N_{d}$ for the three different diameters $D_{1}$, $D_{2}$,
$D_{3}$ (black squares, red circles and blue triangles respectively); (c,d,e)
typical images showing different values of $\beta_{0}/\beta_{1}$ for
$D_{2}$=0.02.

The complexity characterized by the value of the Betti numbers is displayed in
Fig. 3(a). The value of the $\beta_{0}$, shown as filled symbols in the
figure, first increases with the number of dots since it measures the
topological connectivity. As more dots are added, each new dot represents a
simple-connected region, which indicates the linear increase of $\beta_{0}$.
As $N_{d}$ increases overlap begins to occur, and the new dots do not create
new simply-connected regions; hence, the rate if increased of $\beta_{0}$
slows down until a maximum value is observed. The maximum depends both on the
number of drops and the the drop diameter. As the area of the image is
progressively covered by drops, the value of $\beta_{0}$ decreases: the new
dots begin to fill space that connects regions leading to a decrease of
$\beta_{0}$. When the dots fill almost completely the area of interest, the
number of connections is again $1$ that is counted as one continuous set. For
larger drops, the area is filled with fewer dots and the maximum is reached
quickly. Conversely, the number of holes, quantified by the value if the
$\beta_{1}$ number (empty symbols in the figure) starts from zero. When the
dots are sparse and no overlaps occur, the simply-connected regions do not
have any voids leading to a small value of $\beta_{1}$. But as overlaps become
more frequent as the number of dots increases, some holes appear resulting in
higher $\beta_{1}$ values. As even more dots are added, the number of holes
reaches a maximum value because too many overlaps will lead to the new dots
filling up the holes. As with $\beta_{0}$ number, when the dots fill the area
of interest, $\beta_{1}$ number again approaches $0$.

De la Calleja _et. al_ de la Calleja and Zenit (2017) argued that images for
which $\beta_{0}\approx\beta_{1}$ had the maximum complexity. In other words,
the images had as many connected regions as holes. To evaluate this criteria
for the present images, Fig. 3 (b) shows the ratio $\beta_{0}/\beta_{1}$ as a
function of $N_{d}$. In the figure, the ratio $\beta_{0}/\beta_{1}$ decreases
as $N_{d}$ increases crossing a value of 1 at a certain number of dots, that
depends on the size of dot diameter. As $N_{d}$ increases, the ratio contonues
to decrease; for much large $N_{d}$ values the ratio appears to increase
again.. Typical images for $\beta_{0}/\beta_{1}$ smaller, equal and larger
than unity are shown in Figs. Fig. 3 (c), (d) and (e), respectively. At first
sight, the visual appearance of the images and their apparent complexity seems
to be in agreement with the images shown in Figs. 2 (b), (c) and (d), as the
fractal dimension crosses through a maximum value. However, the number of dots
at which the maximum fractal dimension occurs and at which
$\beta_{0}/\beta_{1}\approx 1$ is not the same. Also, the ratio
$\beta_{0}/\beta_{1}$ evolves from a high to small value, crossing through the
point of maximum complexity, while maximum complexity from the fractal
dimension is located at the maxium of $f(\alpha)$.

A similar analysis was conducted for the case of straight lines, extending the
study first used by De la Calleja _et. al_ de la Calleja and Zenit (2017). The
results for the case of lines hold the same qualitative behavior but the
values at which the maximum complexity is reached depend on the length and
thickness of the lines. These results are presented in detail and discussed in
Appendix A.

##  V Complexity of Yayoi Kusama’s infinity net

Yayoi Kusama is an international avant-garde Japanese artist. Her ‘net
painting’ style has been deemed as her most prototypical obsessional art
Pollock (2006). Some critics have interpreted her style as ‘culturally
specific’, ‘historically feminine’ and ‘subjective painting without image’
Pollock (2006). The art of the so-called queen of _Polka Dots_ is
characterized by constructions of by consecutive dots. The world-wide success
of her recent _Infinity Rooms_ , has made Yayoi Kusama one of the most
recognizable and admired contemporary artists.

Kusama’s paintings have evolved significantly from simple stylistic
observations in the 1950s to pure and personal expression of her mental
illness in the 1990s Nakajima (2006). Technically, Kusama’s brushstrokes are
very thick and do not seem to finish.

Considering the technique describe above, we analyse Kusama’s artworks in a
more objective manner. In order to quantify the complexity of Kusama’s
_infinity net_ style, $30$ paintings were selected. A complete list of these
works is included in Appendix B. Most of the artworks used here are from the
following collections: _Dot’s obsession_ Kusama (2019, accessed 10/27/2020a),
_Infinity Nets_ Kusama (2013), _My eternal soul_ Shimbun et al. (2017) and
_Silkscreens from 2009_ Kusama (2019, accessed 10/27/2020b). The specific
images were selected by considering the appearance of the characteristic
repeated dot pattern and the absence, as much as possible, of figurative
expressions or objects. The analysis of complexity in pieces of art is
possible because of the two dimensional nature of the paintings. Although it
is, in principle, possible to extend our proposal to 3-dimensional spaces it
was not attempted.

Figure 4 show four of the most iconic Kusama’s artworks: $(a)$ No. Red B
(1960), $(b)$ Pumpkin Yellow T (1992), $(c)$ Infinity Nets Yellow (1960) and,
$(d)$ Untitled (1967) Abrams (2018, accessed 10/27/2020). The pieces of art of
Kusama are full of the symmetric and irregular distribution of dots which
apparently are a result of simple compositions. However, our results provide
compelling evidence that this fact is questionable.

Figure 4: Four of the most representative Kusama’s paintings. $(a)$ No. Red B
(1960), $(b)$ Pumpkin Yellow T (1992), $(c)$ Infinity Nets Yellow (1960) and,
$(d)$ Untitled (1967). Images used with permission from the Yayoi Kusama
Foundation.

The fractal dimension $f(\alpha)$ for the selected set of paintings is shown
in Figure 5. As can be observed, the fractality of the Kusama’s artworks is
between $1.7<f(\alpha)<1.9$, with a mean value of $f(\alpha)=1.8285$ (dashed
black line). These range of fractal dimension is in good correspondence with
other abstract images with a high density of dots with the smallest diameter
as we observed in Figure 2. Also, Kusama’s artworks fractal dimensions can be
compared with those for other abstract expressionisms de la Calleja et al.
(2016). Interenstingly. the fractality of the art pieces of Yayoi Kusama
evaluated in this report have values similar to those calculated for Jackson
Pollock’s paintings Taylor et al. (1999); de la Calleja et al. (2016). This
result suggests that the despite the obvious differences in style and
composition the complexity of Kusama’s paintings, using this metric, is
similar to that of Pollock’s.

Figure 5: The fractal dimension of Kusama’s paintings. The paintings
considered are shown in Tables 1 and 2 in Appendix B. The dashed line shows
the mean value of $f(\alpha)$

Now, to complete the complexity characterization of Kusama’s paintings we
proceed to calculate the values of the Betti numbers. To do this, the images
are binarized choosing a threshold level following ordinary image processing
techniquesSchneider et al. (2012). The value of $\beta_{0}$ number quantifies
the connectivity of a set and, while the value of $\beta_{1}$ number counts
the holes. Figure 6 (a) show the Betti numbers for Kusama’s paintings listed
in Appendix B. Note that both $\beta_{0}$ and $\beta_{1}$ are normalized with
the area of each painting (A), to allow comparison between works of different
sizes. The mean value of $\beta_{0}$ (brown dashed line) and the mean value of
$\beta_{1}$ number (green dashed line) are also shown. In general, we find
that $\beta_{0}>\beta_{1}$ There is expected since the Kusama’s dots are
mostly disconnected, each one of them represents a single connected domain
with would lead to a high $\beta_{0}$ value. Since there are few overlaps,
only a few holes appear, therefore one would expect the images to have low
$\beta_{1}$ values. This, of course, is not always the case. In many instances
the high density of dots are in balance with the number of holes.

We have conjectured that a high image complexity corresponds to cases when the
number of connected regions is roughly the same as the number of holes,
$\beta_{0}\sim\beta_{1}$ de la Calleja et al. (2016). Fig.6$(b)$ shows the
ratio $\beta_{0}/\beta_{1}$ of all the works analyzed here. Overall, it is
clearly observed that $\beta_{0}>\beta_{1}$. This is good agreement with can
be deduced from the visual inspection of the paitings: dispersed non-
overlapping points do not lead to the formation of many holes. Of all the
works, only 1/3 of them a value of $\beta_{0}/\beta_{1}$ close to unity.
Hence, from this metric the complexity of Kusama’s paintings seems to be
smaller. This observation is in sharp contrast with what was concluded from
the fractal dimension analysis. It is also interesting to note that for the
case of Pollock’s paintings, the ratio $\beta_{0}/\beta_{1}$ was closed to
unity, at least for works close to the dripping period.

Figure 6: (a) Betti numbers per unit area of Kusama’s paintings, list in
Appendix B. Black circles: $\beta_{0}$; red circles, $\beta_{1}$. The black
dashed line shows the mean value of $\beta_{0}/A$ and the red dashed line show
the mean value of $\beta_{1}/A$. $(b)$ the ratio $\beta_{0}/\beta_{1}$ for the
same paintings. The solid line shows the mean value.

##  VI Discussion and Conclusion

The fractal dimension and Betti numbers can both be used to characterize
complexity with a non-predictable structure. Fractal properties are present in
a wide and extend the number of objects. While the fractal dimension measures
certain spectral properties of an image, the Betti numbers quantify the
topological shape. As demonstrated above, they can both provide a metric for
complexity of abstract images. One interesting aspect conclusion drawn from
this study was that, although Kusama’s art works have fractal dimensions
comparable to other abstract paintings, the value of the Betti numbers are
strikingly different.

The analysis of image complexity is a useful non-invasive technique to
characterize artworks. With the analysis presented here we demonstrate that it
is possible to capture essential information about painting styles. Yayoi
Kusama has painted many pieces with dots, and her influence and impact have
been significant. Kusama’s paintings are created with consecutive and order
distribution of dots. From our analysis, the fractality of Kusama’s art
indicates high complexity while the value of the topological invariants show
more homogeneous textures with a high degree of dispersiveness. Hence, we can
conclude the complexity of Kusama’s art work has a richer depth than that
observed for other abstract painters, including Jackson Pollock.

###  VI.1 Acknowledgments

The support of DGAPA-PAPIIT-UNAM (grant number IN108016) and ACT-FONCA (grant
number 04S.04.IN.ACT.038.18) are greatly acknowledged. We thank the Yayoi
Kusama foundation for their kind permission to use the images of the art
works.

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##  Appendix A Synthetic images with lines

De la Calleja _et. al_ de la Calleja and Zenit [2017] reported the complexity
of abstract images composed of rigid lines varying the length and the
thickness. In the present case, we are interested in the effect of changing
the line thickness. To evaluate the complexity abstract images were generated
using the same script written in Matlab. The procedure to generate images with
rigid lines in the three different thickness was the same as the one used with
dots. Black rigid lines were generated to fill a fixed black space. Three
different thickness were computed $T_{1}=0.1$, $T_{2}=0.3$ and $T_{3}=0.5$.
The lines were randomly distributed on the space. We follow the same
methodology to evaluate the complexity of images generated with dots, but
applied to images with lines. The fractal dimension ($f(\alpha)$) and Betti
numbers were calculated.

Figure 7: Typical synthetic abstract images generated with random distribution
of lines.

The fractal dimension $f(\alpha)$ as a function of the number of lines
($N_{L}$) is presented in Fig. 8. The fractal dimension start from a value of
unity for a single line, as expected. As more lines are added, the dimension
decreases slightly to soon after increase again. As more lines are added, the
space is filled rapidly which leading to a maximum fractal dimension of about
1.8. This is observed for the the three thicknesses, but the maxium is reached
at different $N_{L}$. After the maximum value of fractal dimension is reached,
which means that the empty space is fill with lines the fractal dimension
decreases agian. The Betti numbers obtained for the same images are shown in
Fig. 9. The results of $\beta_{0}$ of images with lines are presented in
$(a)$. The connectivity of lines is reached rapidly with a low number of lines
and this behavior is observed on images computed with the three thicknesses.
$\beta_{0}$ collapse to almost $1$ from images with $2000$ lines. This is
because as the number of lines increases, they connected and overlapped
between them and form one set. In Fig. 9 (b) the behavior of $\beta_{1}$ is
shown, which measures the number of empty spaces of a set. Images generated
with thin lines ($T_{1}$) reached a high number of holes. Contrary to the
cases when the lines are thick. In those cases, the empty space is filled
rapidly with a low number of lines. We have argued that a high level of
complexity is obtained when $\beta_{0}\sim\beta_{1}$. In Fig. 9 (c) the ratio
$\beta_{0}/\beta_{1}$ is shown as a fucntion of $N_{L}$. It shows that the
maximum complexity obtained from images with three different thicknesses
occurs at approximately the same number of lines $N_{L}\approx 100$
independently of the line thickness.

Figure 8: The Fractal dimension $f(\alpha)$ reaches rapidly almost a value of
$2.0$. It happens for the three thickness $T_{1},T_{2},T_{3}$ (black squares,
red dots, and blue triangles respectively). Figure 9: In $(a)$ is presented
that high connectivity measured with $\beta_{0}$ on abstract images generated
with lines is reached with a low number of lines on the three different
thickness: $T_{1}$, $T_{2}$ and $T_{3}$ (black squares, red dots, and blue
triangles respectively). $(b)$ exhibits the complexity measured by the number
of holes obtained from the overlapped lines on the images. Images constructed
with thin lines (black squares) exhibit the highest value of $\beta_{1}$. In
(c) the ratio $\beta_{0}/\beta_{1}$ of abstract images generated with lines of
three different thickness shows a decrease of complexity when the images
contain a high number of lines.

##  Appendix B Kusama’s paintings

The body of work by Yayoi Kusama is vast. Many of her paintings have been
exposed in important galleries around the world, including performances and
pieces were the public can interact with the artwork. Among them ‘Net
painting’ has been considered to be her most prototypical example of
‘obsessional art’.

In Yayoi Kusama’s own words: ‘My life, a dot, namely, one among millions of
particles. It was in 1959 that I gave my manifesto that [my art] obliterates
myself and others with the void for a net woven with an astronomical
accumulation of dots’. Since the end of the 1950s, Kusama has produced a many
artworks under different titles, different sizes, color schemes but with an
apparently repetitive pattern: infinity nets.

The ‘self-obliteration’ of Kusama’s words appear as the extreme opposite of
the ‘action painting’. During the painting process, her brush does not leave
the canvas, in sharp contrast to Pollock’s free-hand action Palacios et al.
[2019].

At first sight, Kusama’s paintings may appear to be simple monotonous patterns
of a net. However, the monotonous look may prompt various associations with
the obsessions of the artist. The ideological, social, and physiological
aspects of Kusama’s art, have been discussed in diverse forums. Kaniichi
Kaniichi [1988] discusses the negation of action painting and posses the
question: can the practice of ‘Net’ production be a feminine alternative to
the masculine staging of ‘action’ as she appeared to produce the ‘opposite’ of
Pollock type paint?

In _Net painting_ , the act of painting is austerely restrained to minimal
action. If action painting is a trace of a progressive ‘cutting off’ of the
body of the painting, net painting might suggest a ‘touching’ without harming.
Pollock’s action is oriented to detach the artist form the body of the
painting; Kusama’s continuous painting aspires to produce contiguity with her
object Nakajima [2006].

our results show that despite having contrastingly different styles and
approach to painting, Kusama and Pollock have similar values of complexity
evaluated with fractal dimension. However, if are the Betti number the
parameter of characterization, Kusama’s work shows low level of complexity.

Paint | Year | Name | Size | Source
---|---|---|---|---
1 | 1960 | White No. 28 | 147.6x111.1 |  | https://www.christies.com/
---
lotfinder/Lot/yayoi-kusama-b-1929-
white-no-28-5846063-details.aspx
2 | 1998 | Sunlight | 59.7x47.8 |  | http://www.artnet.com/artists/
---
yayoi-kusama/sunlight-EzrsqLuca
Tdm45F84jNNgA2
3 | 1960 | Infinity nets | 73x91.1 |  | https://www.artsy.net/artwork/
---
yayoi-kusama-infinity-nets-1960-1
4 | 1992 | Pumpkin (Yellow T) | 72.3x60.4 |  | https://www.artsy.net/artwork/
---
yayoi-kusama-pumpkin-yellow-t
5 | 1988 | Petals | 53x45 |  | https://www.phillips.com/detail/
---
yayoi-kusama/UK010612/143
6 | 1988 |  | Waves on the
---
Hudson River
45.5x38 |  | https://www.mutualart.com/Artwork/
---
Waves-on-the-Hudson-River/
3D6A54B3E798DCD0
7 | 2009 |  | Night Ripples
---
[TOWSS]
130.3x162 |  | https://www.roslynoxley9.com.au/
---
artwork/yayoi-kusama-
night-ripples-towss-0/33-14041
8 | 2006 | A Flowing River | 162x130.3 |  | https://www.roslynoxley9.com.au/
---
exhibition/selected-works-
silkscreens/vyft4
9 | 1953 | Waves | 26.4x33 |  | http://www.artnet.com/artists/
---
yayoi-kusama/waves-
W2Lf-D1xCrZ_Wy1IEBlFrg2
10 | 1960 |  | Yellow net
---
(Infinity)
240x294.6 |  | https://www.nga.gov/collection
---
/art-object-page.124183.html
11 | 2011 | Infinity nets LNXA | 97x130.3 |  | https://www.phillips.com/detail/
---
YAYOI-KUSAMA/UK010612/139
12 | 1986 | Mountain country | 38x45.7 |  | http://www.barbaramathesgallery.
---
com/exhibition/yayoi-kusama-
from-here-to-infinity/selected-works/3
13 | 1992 |  | The sky in the
---
evening glow
(Red rain)
161.9x227.3 |  | https://guyhepner.com/product/
---
rain-evening-glow-yayoi-kusama/
14 | 1988 | Passage of the wind | 53.3x45.7 |  | https://www.artsy.net/artwork/
---
yayoi-kusama-passage-of-the-wind
15 | 2011 | Red dots | 100x100 |  | https://www.christies.com/
---
lotfinder/Lot/yayoi-kusama-b-1929-
red-dots-6191623-details.aspx

Table 1: List of Yayoi Kusama’s art works used in this investigation, part I.

Paint | Year | Name | Size | Source
---|---|---|---|---
16 | 2017 |  | I want to live
---
forever
194x194 |  | https://www.cobosocial.com/dossiers/
---
yayoi-kusama-a-dot-in-the-universe/
17 | 2009 |  | Standing on the
---
riverbank of my
hometown
145.5x112 |  | https://www.victoria-miro.com/
---
exhibitions/427/works/f05e6aa9496d58
18 | 2007 |  | Birth, Ageing,
---
Sickness and Death
[QXPAT]
130.3x162 |  | https://ocula.com/art-galleries/
---
victoria-miro-gallery/artworks/
yayoi-kusama/birth-ageing-
sickness-and-death-qxpat//
19 | 2014 |  | Infinity dots
---
(EFY)
130.3x97 |  | https://ocula.com/art-galleries/
---
victoria-miro-gallery/artworks/
yayoi-kusama/infinity-dots-(efy)
20 | 2014 |  | Infinity-Dots
---
[HOFS]
130.3x97 |  | https://ocula.com/art-galleries/
---
victoria-miro-gallery/artworks/
yayoi-kusama/infinity-dots-hofs/
21 | 1994 | Yellow threes | 162.1x390 |  | https://www.fronterad.com/
---
yayoi-kusama-una-delicada-locura/
22 | 2008 |  | Infinity nets
---
[OWTWQB]
162x130.3 |  | https://ocula.com/art-galleries/
---
roslyn-oxley9/artworks/yayoi-kusama/
infinity-nets- owtwqb/
23 | 1991 | The Galaxy | 61x91.4 |  | https://www.phillips.com/detail/
---
yayoi-kusama/NY010117/20
24 | 2009 |  | Late night Chat
---
is Filled with
Dreams
162x162 |  | http://amorartinfo.blogspot.com/2012/08/
---
yayoi-kusama-un-mundo-de-pesadillas.html
25 | 2001 |  | When the soul
---
bursts into flames
82x62.5x21 |  | https://www.roslynoxley9.com.au/
---
artists/49/ $Yayoi_{K}usama$/516/39817/
26 | 1967 | Untitled | 101.6x127 |  | https://www.artsy.net/article/
---
artsy-editorial-6-works-explain-yayoi-
kusamas-rise-art-stardom
27 | 2013 | Infinity Nets PEAA | 100X100 |  | https://www.artsy.net/artwork/
---
yayoi-kusama-infinity-nets-peaa
28 | 1967 | Untitled | 40.6x45.7 |  | https://ackland.org/files/2020/04/
---
Kusama-Exhibition-Text-and-Images.pdf
29 | 2014 | All the eternal love | 194x194 |  | https://www.briefltd.com/yayoi-kusama/
---
30 | 2010 | Endless life of people | 40.6x40.6 |  | https://www.artsy.net/artwork/
---
yayoi-kusama-endless-life-of-people-6

Table 2: List of Yayoi Kusama’s art works used in this investigation, part II.


# Free Streaming Length of Axion-Like Particle After Oscillon/I-ball Decays

Kaname Imagawa  Masahiro Kawasaki  Kai Murai  Hiromasa Nakatsuka  and Eisuke
Sonomoto

###### Abstract

Axion-like particles (ALPs) are pseudoscalar bosons predicted by string
theory. The ALPs have a shallower potential than a quadratic one, which
induces the instability and can form the solitonic object called
oscillon/I-ball. Although the lifetime of oscillons can be very long for some
type of potentials, they finally decay until the present. We perform the
numerical lattice simulations to investigate the decay process of oscillons
and evaluate the averaged momentum of ALPs emitted from the oscillon decay. It
is found that, if oscillons decay in the early universe, the free-streaming
length of ALPs becomes too long to explain the small-scale observations of the
matter power spectrum. We show that oscillons with long lifetimes can change
the density fluctuations on small scales, which leads to stringent constraints
on the ALP mass and the oscillon lifetime.

##  1 Introduction

The $\Lambda$CDM model, which contains cold dark matter and cosmological
constant, explains the cosmological observations very well for scales above
$\mathcal{O}$(Mpc) [1, 2, 3]. However, below $\mathcal{O}$(Mpc), some
observations seem inconsistent with the $\Lambda$CDM model, which is called
the small-scale crisis [4, 5, 6, 7]. As a candidate to solve this crisis,
fuzzy dark matter (FDM), dark matter with ultralight mass, has been attracting
much attention [8, 9]. In fact, FDM with a mass of about $10^{-22}$ eV, whose
de Broglie wavelength is $\mathcal{O}$(kpc), smears out the fluctuations on
small scales and provides good agreement with the observations on small
scales.

Axion-like particle (ALP) is one of the candidates of FDM. ALP is predicted by
string theory and its mass is naturally small because of its shift symmetry
[10]. The ALP is often considered to have a cosine potential on the analogy of
the QCD axion. However, the ALP can have a different potential in general [11,
12, 13]. If the ALP potential is shallower than a quadratic one, the ALP field
can be spatially destabilized and non-topological solitons called oscillons or
I-balls are formed [14, 15, 16, 17, 18, 19, 20, 21].

Oscillons are quasi-stable and have long lifetimes due to the adiabatic
invariance which is approximately conserved. Oscillons slowly lose their
energy emitting scalar radiation. In this paper, we call this process
“evaporation” of oscillons. The evaporation rate is calculated perturbatively
[22, 23]. In this quasi-stable phase, oscillons have a configuration in which
the field oscillates periodically and the oscillation frequency slowly
increases with time as the evaporation proceeds. When the oscillation
frequency reaches a certain value, the oscillon decays at once and the
spatially localized energy is released in a very short time compared to its
lifetime. We call this process non-perturbative decay or simply “decay” of
oscillons, which is mainly investigated in this paper. When the oscillons emit
ALPs with large velocities through the decay, their free streaming washes out
small-scale density fluctuations, which could lead to a stringent constraint
on the ALPs by observations of the matter power spectrum. If ALPs account for
the dark matter of the universe, a sizable ratio of ALPs forms oscillons, and
hence it is important to investigate the evolution of the ALPs taking into
account the effects of oscillons.

In this paper, we investigate the effects of the free streaming of ALPs
emitted from the oscillon decay on the matter power spectrum. To this end, we
evaluate the momentum distribution of ALPs at the time of oscillon decays and
free-streaming length (FSL) using lattice simulations. Even when ALPs
constitute 100% of the dark matter, not all the ALPs form oscillons and some
ALPs remain behaving as cold dark matter [24]. Therefore, just after the
oscillon decays, dark matter is composed of warm (or hot) ALPs emitted from
oscillons, cold ALPs, and the other cold dark matter (if any). This situation
is similar to a two-component system containing cold and warm dark matter,
which may smooth out the small-scale structures depending on the FSL and
abundance of the warm component. Based on this similarity, we study the matter
power spectrum of the universe, including ALPs from oscillons and cold dark
matter. As a result, when the oscillons decay at a temperature below
$\mathcal{O}$(1) eV, we obtain stringent constraints on the ALP mass and the
oscillon lifetime from the observations of CMB, large-scale structure of the
universe, and Lyman-$\alpha$ forest.

This paper is organized as follows. In Sec. 2, we derive the initial profile
of an oscillon from its basic properties and formulate the ALP emission from
the oscillon. In Sec. 3, we perform numerical simulations on the time
evolution of the oscillon and estimate the ALP velocity from the oscillon
decay. In Sec. 4, we limit the lifetime and mass of the oscillon from the
results obtained in Sec. 3. Sec. 5 is devoted to conclusion and discussion.
All cosmological parameters in this paper are extracted from the results of
Planck 2018 [3].

##  2 Oscillons of Axion-like Particle

In this section, we review the properties of oscillons. In Sec. 2.1, we derive
the equations that determine the stationary solution of the oscillon profile,
which is used for the initial condition in the lattice simulation in Sec. 3,
based on Refs. [25, 26]. In Sec. 2.2, we explain the evaporation of oscillons
which is described perturbatively. In Sec. 2.3, we introduce the decay of
oscillons. In this stage, the spatially localized energy is released in a very
short time compared to its lifetime. We also describe how to calculate the
physical quantities such as the momentum distribution of the radiation from
the oscillons calculated in the lattice simulation.

###  2.1 Oscillon Profile

We consider the ALP field $\phi$ whose action in the Minkowski space-time is
written as

$S=\int\mathrm{d}^{4}x\,\left[\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right].$
(2.1)

Here we do not consider the cosmic expansion and the validity of this
assumption is discussed in Sec. 4.1. We decompose the ALP potential $V(\phi)$
as

$V(\phi)=\frac{1}{2}m^{2}\phi^{2}+V_{\mathrm{nl}}(\phi),$ (2.2)

where $m$ is the ALP mass and $V_{\mathrm{nl}}(\phi)$ represents the higher
order terms. In this paper, we adopt the pure natural type potential derived
from the pure Yang-Mills gauge theory [27, 28, 29]

$V(\phi)=\frac{m^{2}M^{2}}{2p}\left[1-\left(1+\frac{\phi^{2}}{M^{2}}\right)^{-p}\right],\hskip
28.45274pt(p>-1),$ (2.3)

where $M$ is the effective decay constant of the ALP. This potential is
shallower than a quadratic one for $p>-1$, leading to the formation of
oscillons [30, 21].

We briefly derive the differential equations that the oscillon profile should
satisfy following Refs. [25, 26]. Since oscillons show an oscillatory
behavior, we write the oscillon configuration as

$\phi_{\mathrm{osc}}(t,\bm{x})=2\psi(\bm{x})\cos{(\omega_{\mathrm{osc}}t)},$
(2.4)

where $\omega_{\mathrm{osc}}$ is the internal frequency of oscillons.
Oscillons have an approximately conserved quantity called an adiabatic
invariant which is written as

$I=\frac{1}{\omega_{\mathrm{osc}}}\int\mathrm{d}^{3}x\,\overline{\dot{\phi}^{2}_{\mathrm{osc}}},$
(2.5)

where $\overline{\dot{\phi}^{2}_{\mathrm{osc}}}$ represents the time average
of $\dot{\phi}^{2}_{\mathrm{osc}}$ over the oscillation period
$2\pi/\omega_{\mathrm{osc}}$. The equation of motion for $\phi$ is

$\frac{\partial^{2}\phi}{\partial
t^{2}}-\nabla^{2}\phi+m^{2}\phi+\frac{\partial
V_{\mathrm{nl}}}{\partial\phi}(\phi)=0.$ (2.6)

Substituting Eq. (2.4) into Eq. (2.6), we obtain

$-\omega_{\mathrm{osc}}^{2}\psi\cos(\omega_{\mathrm{osc}}t)-\nabla^{2}\psi\cos(\omega_{\mathrm{osc}}t)+m^{2}\psi\cos(\omega_{\mathrm{osc}}t)+\frac{1}{2}\frac{\partial
V_{\mathrm{nl}}}{\partial\phi}(2\psi\cos\left[\omega_{\mathrm{osc}}t\right])=0.$
(2.7)

Multiplying Eq. (2.7) by $\cos(\omega_{\mathrm{osc}}t)$ and taking the average
with respect to time, we obtain

$\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial}{\partial
r}\right)\psi=(m^{2}-\omega_{\mathrm{osc}}^{2})\psi+\frac{1}{2}\frac{\partial
V_{\mathrm{eff}}}{\partial\psi}(\psi),$ (2.8)

where
$V_{\mathrm{eff}}(\psi)=\overline{V_{\mathrm{nl}}(\phi=2\psi\cos[\omega_{\mathrm{osc}}t])}$.
Here we assume a spherically symmetric configuration $\psi(\bm{x})=\psi(r)$
and impose the boundary conditions

$\lim_{r\rightarrow 0}\frac{\partial\psi(r)}{\partial
r}=\lim_{r\rightarrow\infty}\psi(r)=0.$ (2.9)

In order for Eq. (2.8) to have a nonzero solution under these boundary
conditions, $\omega_{\mathrm{osc}}$ should be smaller than a critical value
$\omega_{c}$. The solution of Eq. (2.8) with the boundary conditions (2.9) is
used as the initial condition for the lattice simulation. Considering the
evaporation, $\omega_{\mathrm{osc}}$ gradually increases, and the oscillons
suddenly decay when $\omega_{\mathrm{osc}}$ reaches $\omega_{\mathrm{c}}$.

###  2.2 Evaporation of Oscillons

Oscillons continuously radiate ALPs in the quasi-stable phase [31, 32, 23]. We
call this phenomenon evaporation of oscillons and also call the emitted
radiation perturbative radiation. The evaporation rate is calculated as
follows [22, 23]. The total energy of an oscillon is given by

$\displaystyle E_{\mathrm{tot}}(t)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\bm{x}\left[\frac{1}{2}\dot{\phi}^{2}+(\nabla\phi)^{2}+V(\phi)\right],$
(2.10)

where the size of the integration region is much larger than the oscillon
radius. Then the evaporation rate is defined as [22]

$\Gamma=\frac{1}{\bar{E}_{\rm{tot}}(t)}\left|\frac{\mathrm{d}}{\mathrm{d}t}\bar{E}_{\rm{tot}}(t)\right|.$
(2.11)

The ALP field is decomposed as

$\phi(t,\bm{x})=\phi_{\mathrm{osc}}(t,r)+\xi(t,r),$ (2.12)

where $\xi$ is the perturbative radiation of the oscillon. Here and hereafter,
we assume the spherical symmetry. In the linear order, the equation of motion
for $\xi(t,r)$ is given by

$\left[\\!\mathop{}\mathbin{\Box}+V^{{}^{\prime\prime}}(\phi_{\mathrm{osc}}(t,r))\right]\xi=-V^{{}^{\prime}}(\phi_{\mathrm{osc}}(r,t))+2\overline{V^{{}^{\prime}}(\phi_{\mathrm{osc}}(t^{\prime},r))\cos(\omega_{\mathrm{osc}}t^{\prime})}\cos(\omega_{\mathrm{osc}}t),$
(2.13)

where the primes represent the derivative with respect to $\phi$. We decompose
the radiation as $\xi(t,r)=\sum_{j=3}\xi_{j}(r)\cos(j\omega_{\mathrm{osc}}t)$
and then the equation of motion for $\xi_{j}$ is given by

$\left[\nabla^{2}+\kappa_{j}^{2}\right]\xi_{j}=-\mathcal{S}_{j},$ (2.14)

where $\kappa_{j}=\sqrt{j^{2}\omega^{2}-m^{2}}$ and $\mathcal{S}_{j}(r)$ is
the source term including the combination of $\xi_{k}$’s [23]. The radiative
solution of $\xi$ for $r\to\infty$ is given as

$\displaystyle\xi_{j}(t,r)$ $\displaystyle\simeq$ $\displaystyle\frac{1}{4\pi
r}\sum_{j=3}\tilde{\mathcal{S}}_{j}(\kappa_{j})\cos(\kappa_{j}r-j\omega_{\mathrm{osc}}t),$
(2.15) $\displaystyle\tilde{\mathcal{S}_{j}}(k)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\bm{x}^{\prime}\mathcal{S}_{j}(r^{\prime})e^{-i\bm{k}\cdot\bm{x}^{\prime}}.$
(2.16)

Thus the time-averaged classical energy loss rate is obtained by

$\frac{\mathrm{d}\bar{E}_{\mathrm{tot}}(t)}{\mathrm{d}t}=4\pi
r^{2}\overline{\partial_{t}\xi(t,r)\partial_{r}\xi(t,r)}=-\frac{1}{8\pi}\sum_{j=3}j\omega_{\mathrm{osc}}\kappa_{j}\tilde{\mathcal{S}}_{j}(\kappa_{j})^{2}.$
(2.17)

###  2.3 Decay of Oscillons

In this subsection, we explain the decay of oscillons and introduce some
physical quantities to investigate particle emission from oscillons. Oscillons
emit a large amount of radiation at the end of their life, releasing all of
their energy promptly. We call this phenomenon oscillon decay and also call
the particle emission non-perturbative particle emission or simply particle
emission.

We first describe the ALP emission to find the energy distribution for each
momentum mode and the Poynting vector that represents the ALP energy flow. The
classical ALP field in momentum space is written as

$\xi(t,\bm{x})=\int\frac{\mathrm{d}^{3}\bm{k}}{\sqrt{2\omega_{k}(2\pi)^{3}}}\,\left[\xi_{\bm{k}}e^{i\omega_{k}t-i\bm{k}\cdot\bm{x}}+\xi^{*}_{\bm{k}}e^{-i\omega_{k}t+i\bm{k}\cdot\bm{x}}\right],$
(2.18)

where $\xi_{\bm{k}}$ is a complex coefficient and
$\omega_{k}=\sqrt{|\bm{k}|^{2}+m^{2}}$. On the other hand, in the lattice
simulation, we obtain the Fourier transform of the field value
$\xi(t,\bm{x}_{0})$ with respect to the time as

$\xi(t,\bm{x}_{0})=\int_{-\infty}^{\infty}\mathrm{d}\omega\,\xi_{\omega}(\bm{x}_{0})e^{i\omega
t},$ (2.19)

where $\bm{x}_{0}$ is some fixed point and we take this point to be far enough
away from an oscillon. To compare the two formulas, we rewrite the equation as
follows

$\displaystyle\xi(t,\bm{x}_{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\omega\left[\xi_{\omega}(\bm{x}_{0})e^{i\omega
t}+\xi^{*}_{\omega}(\bm{x}_{0})e^{-i\omega t}\right]$ (2.20)
$\displaystyle\simeq$
$\displaystyle\int_{0}^{\infty}\left(\mathrm{d}k\frac{k}{\omega_{k}}\right)\left[\xi_{\omega}(\bm{x}_{0})e^{i\omega_{k}t}+\xi^{*}_{\omega}(\bm{x}_{0})e^{-i\omega_{k}t}\right]$
(2.21) $\displaystyle=$
$\displaystyle\int\frac{d^{3}\bm{k}}{\sqrt{2\omega_{k}(2\pi)^{3}}}\left[\sqrt{\frac{\pi}{k^{2}\omega_{k}}}\xi_{\omega}(\bm{x}_{0})e^{i\omega_{k}t}+\sqrt{\frac{\pi}{k^{2}\omega_{k}}}\xi^{*}_{\omega}(\bm{x}_{0})e^{-i\omega_{k}t}\right],$
(2.22)

where, in Eq. (2.20), we use $\xi^{*}_{\omega}=\xi_{-\omega}$, and, in Eq.
(2.21), we use the on-shell condition $\omega_{k}=\sqrt{|\bm{k}|^{2}+m^{2}}$
to rewrite the measure and neglect the range of integration from $0$ to $m$.
Comparing Eq. (2.22) with Eq. (2.18), we get

$|\xi_{k}|=\sqrt{\frac{\pi}{k^{2}\omega_{k}}}|\xi_{\omega_{k}}|,$ (2.23)

which connects the analytical field values and numerical values obtained by
lattice simulations. We use the spherical symmetry to drop the
$\bm{k}$-direction dependence of $\xi_{\bm{k}}$. Here and hereafter, we omit
the $\bm{x}$-dependence of $\xi_{\omega}$.

Let us consider the non-perturbatively emitted energy from oscillons for each
momentum mode after the decay of oscillons. In this stage, the solitonic
solution (2.4) no longer holds, and the energy is promptly released (see also
Sec. 3.2). Therefore, after the oscillon decay, we can ignore the field value
of oscillons and only consider radiation. The total energy of the radiation is

$\displaystyle E_{\mathrm{rad}}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\bm{x}\,\left[\frac{1}{2}\dot{\xi}^{2}+\frac{1}{2}(\nabla\xi)^{2}+\frac{1}{2}m^{2}\xi^{2}\right]$
(2.24) $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\bm{k}\,\omega_{k}|\xi_{k}|^{2},$ (2.25)

where $k=|\bm{k}|$. Here, we assume $|\phi/M|$ is small enough to neglect the
non-linear terms of $V(\phi)$ in estimating $E_{\text{rad}}$. We numerically
confirmed that the condition $|\phi/M|\ll 1$ is satisfied even immediately
after the oscillon decay. From Eqs. (2.25) and (2.23), we have

$\frac{\mathrm{d}E_{\mathrm{rad}}}{\mathrm{d}\ln{k}}=4\pi
k^{3}\omega_{k}|\xi_{k}|^{2}=4\pi^{2}k|\xi_{\omega}|^{2}.$ (2.26)

This formula (2.26) is valid for both perturbative radiation and non-
perturbative particle emission.

Now we can calculate the average momentum when the ALPs are emitted from
oscillons. The average momentum is given by

$\bar{k}=\frac{\int\mathrm{d}^{3}k\,n_{k}k}{\int\mathrm{d}^{3}k\,n_{k}}=\frac{\int_{0}^{\infty}\mathrm{d}k\,4\pi^{2}k|\xi_{\omega}|^{2}/\omega_{k}}{\int_{0}^{\infty}\mathrm{d}k\,4\pi^{2}|\xi_{\omega}|^{2}/\omega_{k}},$
(2.27)

where $n_{k}=E_{k}/\omega_{k}$ and $E_{k}$ is the Fourier mode of the
radiation energy as $E_{\mathrm{rad}}=\int\mathrm{d}^{3}\bm{k}\,E_{k}$.

We evaluate the Poynting vector near the boundary to see the evolution of the
energy of the outgoing ALPs. Since, at the position far from the origin, the
oscillon profile $\phi_{\mathrm{osc}}$ is negligibly small, the Poynting
vector is written as

$\frac{1}{4\pi
r^{2}}\frac{\mathrm{d}E_{\mathrm{tot}}(t)}{\mathrm{d}t}=T_{0r}(t)\simeq\frac{\partial\xi(r,t)}{\partial
r}\frac{\partial\xi(r,t)}{\partial t}.$ (2.28)

##  3 Simulation

We study the free streaming of the ALPs released from the non-perturbative
decay of oscillons and its effect on the density fluctuations. For this
purpose, we evaluate the momenta of the ALPs emitted at the time of the non-
perturbative decay of oscillons using lattice simulations.

###  3.1 Simulation Setup

Imposing the spherical symmetry, the equation of motion for the ALP field
$\phi$ is given by

$\left(\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial
r^{2}}-\frac{2}{r}\frac{\partial}{\partial r}+m^{2}\right)\phi=-\frac{\partial
V_{\mathrm{nl}}(\phi)}{\partial\phi},$ (3.1)

which we solve numerically. In the lattice simulation, we normalize the field
and coordinates by $m$ and $m^{-1}$ as

$\phi\rightarrow m\phi,\quad t\rightarrow\frac{t}{m},\quad
x\rightarrow\frac{x}{m}.$ (3.2)

The simulation parameters are summarized in Table 1. The box size is chosen to
be sufficiently large compared to the radius of the oscillon ($\sim
10m^{-1}$).

$\omega_{\mathrm{osc}}|_{t=0}$ | varying
---|---
Box size $L$ | 64 $m^{-1}$
Grid size $N$ | 1024
Initial time | 0
Final time | $5.0\times 10^{5}~{}m^{-1}$
Time step | $1.0\times 10^{-2}~{}m^{-1}$
Table 1: Simulation parameter.

The initial field values are set by solving Eqs. (2.8) and (2.9). At the
initial time $t=0$, we set

$\dot{\phi}(t=0,r)=0,$ (3.3)

and we always impose

$\left.\frac{1}{r}\frac{\partial\phi(t,r)}{\partial r}\right|_{r=0}=0.$ (3.4)

At the boundary of the simulation box $r=L$, we impose the absorbing boundary
condition, which leads to the absorption of the ALPs emitted from the oscillon
[33]. The time evolution and spatial derivative are described by the fourth-
order symplectic integration scheme and fourth-order central difference
scheme, respectively. We confirmed that the simulation results do not
significantly depend on the box size $L$, grid size $N$, and time step.

###  3.2 Simulation Results

We numerically calculate the total energy in the simulation box and Poynting
vector using Eqs. (2.10) and (2.28), respectively.

Figure 1:  Time evolution of the energy and Poynting vector. The left panel
shows the time evolution of the total energy in the simulation box. Note that,
since we are using the absorbing boundary condition, the total energy is not
conserved. The right panel shows the time evolution of the Poynting vector at
$r=60m^{-1}$. The Poynting vector is averaged over a time interval
$10^{3}m^{-1}$. The gray vertical dashed lines separate the three regions:
before, immediately after, and after the oscillon decays. The gray lines of
the left and right represent the times when the oscillon decays and when 95%
of the energy is lost respectively. Eq. (2.3) is used for the potential with
$p=2$.

The left panel of Fig. 1 shows the time evolution of the energy inside the
simulation box, and the right panel of Fig. 1 shows the time-averaged Poynting
vector near the boundary. From the left panel of Fig. 1, we can see that there
are three stages for the time evolution of the system. During the early stage,
$t\lesssim 2.6\times 10^{5}m^{-1}$, the oscillon is almost stable and
gradually releases its energy as radiation. In the transient stage $2.6\times
10^{5}m^{-1}\lesssim t\lesssim 2.7\times 10^{5}m^{-1}$, the oscillon decays
non-perturbatively and releases most of its energy. After the oscillon decays
$t\gtrsim 2.7\times 10^{5}m^{-1}$, the value of the Poynting vector in the
right panel of Fig. 1 decreases rapidly. From Fig. 1, we can see

$\frac{t_{\text{end}}-t_{\rm{decay}}}{t_{\rm{decay}}}=\frac{\Delta
t_{\rm{decay}}}{t_{\rm{decay}}}\simeq\frac{10^{4}m^{-1}}{3\times
10^{5}m^{-1}}\simeq 0.03,$ (3.5)

where $t_{\mathrm{decay}}$ is the decay time, $t_{\mathrm{end}}$ is the time
when 95 % of the energy is lost and $\Delta
t_{\text{decay}}=t_{\text{end}}-t_{\text{decay}}$. Thus, an oscillon releases
most of its energy in a very short time scale compared to its lifetime.

In order to obtain the energy distribution of the emitted ALPs immediately
after the oscillon decay, we perform the Fourier transform of $\xi(t,r\sim L)$
with respect to time and obtain $\xi_{\omega}$ [see Eq. (2.19)]. Since we use
the field values near the boundary of the simulation box, we can safely
neglect the effect of the oscillon profile. The result is shown in Fig. 2. The
energy distribution of the ALP radiation before the oscillon decay is shown by
the blue line in the figure, where the mode with $\omega\sim
3\omega_{\mathrm{osc}}$ is dominant. This is consistent with the analytic
calculation of evaporation [23, 22]. On the other hand, immediately after the
oscillon decays, the mode with $\omega\sim 1.1m$ is dominant as shown by the
orange line. After that, all modes reduce the energy as shown by the green
line.

Figure 2:  Fourier transform of $\xi(t,r=60m^{-1})$. The vertical axis is
proportional to the energy of the particle with each frequency (see Eq.
(2.26)). Blue, green, and orange lines correspond to the results calculated in
different time intervals of $[2.50,\,2.51]$, $[2.66,\,2.67]$, and
$[3.50,\,3.51]$ (in units of $10^{5}m^{-1}$), respectively. Gray dashed lines
represent $\omega\simeq\omega_{\mathrm{osc}},3\omega_{\mathrm{osc}}$, and
$5\omega_{\mathrm{osc}}$, respectively with $\omega_{\mathrm{osc}}$ at
$t=2.50\times 10^{5}m^{-1}$.

Next, we evaluate the momentum distribution of the ALP emission using the on-
shell condition $\omega=\sqrt{k^{2}+m^{2}}$. We can safely neglect the mode
with $\omega\lesssim m$ since the value of $|\xi_{\omega}|^{2}$ with
$\omega\lesssim m$ is much smaller than the peak at $\omega\sim 1.1m$. Then we
rewrite the horizontal axis of Fig. 2 in terms of the momentum $k$ as shown in
Fig. 3. We can say from Fig. 3 that the dominant contribution of the emitted
energy is not one with $k/m\sim 3$, which is the radiation emitted in the
quasi-stable stage of oscillon, but one with $k/m\sim\mathcal{O}(0.1)$. From
the result in Fig. 3 and Eq. (2.27), we obtain

$\displaystyle\bar{k}$ $\displaystyle\simeq$ $\displaystyle 0.195m,$ (3.6)
$\displaystyle\bar{v}$ $\displaystyle=$
$\displaystyle\bar{k}/\sqrt{\bar{k}^{2}+m^{2}}\simeq 0.191,$ (3.7)

where $\bar{v}$ is the averaged velocity. From Eq. (3.7), we can see that the
velocity of the ALPs emitted in the non-perturbative decay is
$\mathcal{O}(10)\%$ of the speed of light. As we will see later in Sec. 4.2,
the release of the ALPs with non-negligible momenta can suppress the small-
scale density perturbations.

Figure 3:  Fourier transform of $\xi(t,r=60m^{-1})$. The dotted, solid and
dashed lines represent the Fourier transform with $p=1$, $2$, and $3$
respectively. This time interval with $p=2$ corresponds to that of the orange
line in Fig. 2. We use the on-shell condition $\omega=\sqrt{k^{2}+m^{2}}$ to
rewrite the orange line of Fig. 2 to Fig. 3 and also use Eq. (2.26). The
spikes at $k/m\sim 3$ are the perturbative radiations from the oscillon with
$\omega\simeq 3\omega_{\mathrm{osc}}$ , which are emitted when oscillons are
in the quasi-stable phase.

##  4 Cosmological Effect of Oscillon Decay

###  4.1 Free-Streaming Length of Axion-like particles

In this subsection, we estimate the free-streaming length (FSL) of the ALPs
emitted from the non-perturbative decay of oscillons. We focus on the time
when oscillons release large amounts of their energy in a short period of
time. As shown in the right panel of Fig. 1, the Poynting vector during the
quasi-stable phase of the oscillon is much smaller than that at the time of
the non-perturbative decay. Therefore, we ignore the free-streaming due to the
evaporation.

We assume that the ALPs are instantaneously emitted at $t_{\mathrm{decay}}$
and that the cosmic expansion is negligible in calculating the value of
$\bar{k}$, which is justified if

$\displaystyle\Delta t_{\mathrm{decay}}\ll
H(t_{\mathrm{decay}})^{-1}~{}~{}\Leftrightarrow~{}~{}\Delta
t_{\rm{decay}}/t_{\rm{decay}}\ll 1,$ (4.1)

where $H(t)^{-1}$ is the Hubble time, and
$H(t_{\mathrm{decay}})^{-1}=2t_{\mathrm{decay}}$ in the radiation dominated
era. Since, from our numerical simulation, we obtained $\Delta
t_{\text{decay}}/t_{\text{decay}}\simeq 0.03$ as shown in Eq. (3.5), the
condition (4.1) is satisfied.

The comoving FSL is calculated as

$\displaystyle\ell_{\mathrm{FSL}}(t_{\mathrm{decay}})$
$\displaystyle=\int_{t_{\mathrm{decay}}}^{t_{0}}\mathrm{d}t\,\frac{v(t)}{a(t)}$
$\displaystyle=H_{0}^{-1}\int_{a(t_{\mathrm{decay}})}^{a(t_{0})}\,\frac{\mathrm{d}\tilde{a}}{\tilde{a}^{2}\sqrt{\Omega(\tilde{a})}}\frac{k(\tilde{a})}{\sqrt{k(\tilde{a})^{2}+m^{2}}},$
(4.2)

where $a=a(t)$ is the scale factor, $H_{0}$ is the present Hubble parameter,
$\Omega(a)=(\Omega_{\mathrm{r,0}}/a^{4}+\Omega_{\mathrm{m,0}}/a^{3}+\Omega_{\rm{\Lambda}})^{1/2}$,
$\Omega_{i,0}$ (i=r,m,$\Lambda$) is the present density parameters of
radiation, matter, and cosmological constant, $v(t)$ is the ALP velocity, and
$k(a)=\bar{k}\,a(t_{\mathrm{decay}})/a$ is the average physical momentum with
$\bar{k}$ given by Eq. (3.6).

###  4.2 Matter Power Spectrum of Axion-like particles

We consider the matter power spectrum in the scenario where oscillons decay at
the cosmic temperature $T=T_{\mathrm{decay}}$. For simplicity, we assume that
the ALPs account for all dark matter of the universe. When oscillons are
formed, 50% to 70% of ALPs are confined in the oscillons [24] and the rest of
the ALPs do not form oscillons but behave as cold dark matter. Therefore, at
the time of the oscillon decay, dark matter consists of two components, warm
ALPs emitted from oscillons and cold ALPs that have not formed oscillons. We
also assume that all oscillons decay at the same time as an approximation.
Actually, the produced oscillons have some mass distribution. However, since
the evaporation rate is larger for larger oscillons, oscillons quickly evolve
to ones with a smaller size and spend most of their lifetimes. Thus, the
decays of oscillons occur in the similar redshift.

We adopt the following formula used in the context of warm dark matter to
calculate the matter power spectrum [34, 35]

$\frac{P_{\mathrm{ALP}}(k)}{P_{\Lambda\mathrm{CDM}}(k)}=(1-f_{\mathrm{osc}})+\frac{f_{\mathrm{osc}}}{(1+k/k^{\prime}_{\mathrm{d}})^{0.7441}},$
(4.3)

where $f_{\text{osc}}$ and $k_{\text{d}}^{\prime}$ are given by

$\displaystyle f_{\mathrm{osc}}(r_{\mathrm{osc}})$ $\displaystyle=$
$\displaystyle 1-\exp(-a\frac{r^{b}_{\mathrm{osc}}}{1-r^{c}_{\mathrm{osc}}}),$
(4.4) $\displaystyle k^{\prime}_{\rm{d}}(k_{\mathrm{d}},r_{\mathrm{osc}})$
$\displaystyle=$ $\displaystyle k_{\mathrm{d}}/r_{\mathrm{osc}}^{1/2},$ (4.5)
$\displaystyle k_{\mathrm{d}}(k_{\mathrm{fs}})$ $\displaystyle=$
$\displaystyle
2.206h\,\mathrm{Mpc}^{-1}\left(\frac{k_{\mathrm{fs}}}{h\mathrm{Mpc}}\right)^{1.703},$
(4.6)

with

$(a,b,c)=(1.551,0.5761,1.263).$ (4.7)

Here
$k_{\mathrm{fs}}(t_{\mathrm{decay}})=2\pi/\ell_{\mathrm{FSL}}(t_{\mathrm{decay}})$
and $r_{\mathrm{osc}}=\Omega_{\mathrm{osc}}/\Omega_{\mathrm{DM}}$ is the
density ratio of the oscillons to all dark matter at the time of the oscillon
decay. $P_{\Lambda\mathrm{CDM}}(k)$ in Eq. (4.3) is calculated by CAMB [36].
The formula of the matter power spectrum (4.3) is obtained by N-body
simulations for the two-component system of warm and cold dark matter and the
parameters (4.7) are determined by fitting the N-body result to Eq. (4.4).
Note that the $r_{\mathrm{osc}}$-dependence of the damping scale
$k^{\prime}_{\mathrm{d}}$ in Eq. (4.5) differs from that in Ref. [34], where
the authors assume the thermal warm dark matter. In Ref. [34], it is presumed
that the damping scale $k^{\prime}_{\mathrm{d}}$ has the same dependence on
$r_{\mathrm{osc}}$ as the Jeans scale which is written as

$k_{\mathrm{J}}=\left.a\sqrt{\frac{4\pi
G\rho_{\mathrm{M}}}{\sigma^{2}}}\right|_{t=t_{\mathrm{eq}}},$ (4.8)

where $G$ is the gravitational constant, $\rho_{\mathrm{M}}$ is the matter
energy density, and $\sigma^{2}$ is the mass-weighted mean squared velocity of
the whole dark matter. In the present case,
$\sigma^{2}=r_{\mathrm{osc}}\sigma_{\mathrm{osc}}^{2}$ with
$\sigma_{\mathrm{osc}}\sim\bar{k}/\sqrt{\bar{k}^{2}+m^{2}}=\bar{v}$. Since
$\bar{v}$ is independent of $t_{\text{osc}}$, we have $k_{\mathrm{J}}\propto
1/\sqrt{\sigma^{2}}\propto 1/r_{\mathrm{osc}}^{1/2}$.

Figs. 4 and 5 show the matter power spectra for $r_{\mathrm{osc}}=0.6$ [24]
and $T_{\text{decay}}=5$ eV, $1$ eV, $0.1$ eV. From these figures, we can see
that the matter power spectrum is inconsistent with the observed data
(especially Lyman-$\alpha$ forest) for $T_{\text{decay}}\lesssim 1$ eV. Fig. 6
shows the matter power spectrum for $T_{\mathrm{decay}}=1$ eV and
$r_{\mathrm{osc}}=$0.4, 0.6, and 0.8. From this figure, it can be seen that
the larger the value of $r_{\mathrm{osc}}$ is, the less it fits the
observations.

Figure 4:  Matter power spectrum with $r_{\mathrm{osc}}=0.6$. The blue, orange
and green solid lines represent the matter power spectra with the oscillon
decays at the temperature $T_{\mathrm{decay}}=$ 5 eV, 1 eV, and 0.1 eV,
respectively. The black dashed line represents the matter power spectrum in
the $\Lambda$CDM model, which is calculated by CAMB [36]. The blue, red, and
yellow data points with error bars are the observational data of Planck 2018
[3], DES Y1 cosmic shear [37], and eBOSS DR14 Lyman-$\alpha$ forest [38, 39],
respectively.  Figure 5:  Same as Fig. 4 but for the range
$0.1<k\,[h\mathrm{/Mpc]}<5$.  Figure 6:  Matter power spectrum with
$T_{\mathrm{decay}}=1$ eV. The blue, orange and green solid lines represent
the matter power spectra with $r_{\mathrm{osc}}=$ 0.4, 0.6 and 0.8
respectively. The other explanations are the same as Figs. 4, 5.

We comment on the values of $(a,b,c)$ [Eq. (4.7)], which is determined for the
thermally produced warm dark matter model [34]. These values might be changed
for our case since $k^{\prime}_{d}$ differently depends on the density ratio
of the warm component,
$k^{\prime}_{d}\propto(\Omega_{\mathrm{warm}}/\Omega_{\mathrm{DM}})^{-5/6}$
for thermally produced warm dark matter and $k^{\prime}_{d}\propto
r_{\mathrm{osc}}^{-1/2}$ for our case. We check the robustness of our results
over the different values of Eq. (4.7). We compute the matter power spectra
for the respective powers of $k^{\prime}_{d}\propto r_{\mathrm{osc}}^{-5/6}$
and $\propto r_{\mathrm{osc}}^{-1/2}$ using the same parameter of Eq. (4.7)
and the result is not changed, that is the oscillon decay temperature below
$\mathcal{O}$(1) eV is inconsistent with observation.

Fig. 7 shows the observational constraints on the lifetime of oscillons and
ALP mass from the matter power spectrum. The blue shaded region in Fig. 7 is
excluded because the free-streaming length of the radiated ALPs is too large
to form the small-scale structures. The ALP with mass smaller than about
$10^{-24}$ eV (green shaded region) is excluded from the CMB observations
[40]. The black dotted lines show the numerically calculated lifetimes of the
oscillon for the potential (2.2) [24]. From Fig. 7, the theoretical lifetimes
corresponding to various values of $p$ in the potential (2.2) are limited.

Figure 7:  Constraints on the lifetime of oscillons $m\tau$ and ALP mass with
$r_{\mathrm{osc}}=0.6$. The blue and orange dashed lines represent the
lifetimes of oscillons that decay at $z=0$, $6$ and $T_{\mathrm{decay}}=1$ eV,
respectively. The blue region is excluded because the free streaming of ALPs
is too large to explain the small-scale structures as shown in Figs. 4 and 6.
The green region is excluded because of the CMB observations [40], in which it
is assumed that ALPs constitute 100% of dark matter. The black dotted lines
are the typical lifetime of oscillons with the potential (2.3) at
$p=-0.75,0.5$ and $3$ respectively [24]. In the upper left region labeled
“Stable Oscillon”, oscillons have a long lifetime and remain in the current
universe. In the lower right region labeled “Little Effect on Small Scale”,
the lifetime of oscillons is too short and the FSL of the ALPs is too short to
have any effect on the small scale structures.

##  5 Conclusion

In this paper, we studied the decay of oscillons produced from ALPs and
estimated the FSL of ALPs emitted from the decay. Since the long FSL
suppresses small-scale density perturbations, we can obtain constraints on the
ALP mass and decay epoch of the oscillons from the small-scale observations.
We have found that the matter power spectrum contradicts the observational
data such as the Lyman-$\alpha$ forest if the ALP oscillons decay at the
cosmic temperature below $1$ eV, from which we obtained a limit on the mass of
the ALPs.

In this paper, we adopt the pure natural type (2.3) as the ALP potential,
which leads to long lifetimes $\tau\gg 10^{3}m^{-1}$ and satisfies Eq. (4.1).
On the other hand, the cosine potential that is often used for simplicity
predicts shorter lifetimes than $10^{3}m^{-1}$ [23]. In this case, as
suggested from Fig. 7, the oscillon formation does not affect the small-scale
structure of the universe.

The value of $\bar{k}$ is obtained for the potential (2.3). In addition, we
confirmed that the order of the value of $\bar{k}$ is almost the same
($\bar{k}\simeq 0.1m$) for the $\phi^{6}$ potential which is used in [22, 31].
For simplicity, we assume that all oscillons have the same lifetime, but in
reality, oscillons with various lifetimes are formed. The effect of ALPs on
the matter power spectrum in such a system will be investigated in the future.

Note that we changed the $r_{\mathrm{osc}}$-dependence of $k^{\prime}_{d}$ in
Eq. (4.5) from the original paper [34] due to physical requirements, and this
should also change the fitting parameters $(a,b,c)$ of Eq. (4.7). Although the
usage of the fitting parameters [34] is justified as in Sec. 4.2, in order to
determine these parameters accurately, N-body calculations are required.

We assume the only particles that affect the free-streaming are those emitted
from the oscillon decays. In fact, ALPs radiated before the decay of oscillon
are less likely to affect the FSL because they lose momentum more rapidly due
to the larger effect of cosmic expansion. However the effect of evaporation to
reduce $r_{\mathrm{osc}}$ is not negligible and the effect of free-streaming
particles emitted by evaporation should also be considered for a more detailed
analysis.

## Acknowledgments

This work is supported by the Grant-in-Aid for Scientific Research Fund of the
JSPS 20H05851(M. K.), 21K03567(M. K.), 20J20248 (K. M.), and 19J21974 (H. N.).
M. K. and K. M. are supported by World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan (M. K. and K. M.). K. M. is supported
by the Program of Excellence in Photon Science. H. N. is supported by Advanced
Leading Graduate Course for Photon Science.

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