RUP-22-25
KEK-TH-2365
KEK-Cosmo-0280
KEK-QUP-2022-0011
YITP-22-143
CTPU-PTC-22-27

Threshold of Primordial Black Hole Formation against
Velocity Dispersion in Matter-Dominated Era

Tomohiro Haradaa, Kazunori Kohrib,c,d,e, Misao Sasakie,f,g,

Takahiro Teradah, and Chul-Moon Yooi

a Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
b Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba, Ibaraki
305-0801, Japan
c International Center for Quantum-field Measurement Systems for Studies of
the Universe and Particles (QUP, WPI), KEK, 1-1 Oho, Tsukuba, Ibaraki
305-0801, Japan
d The Graduate University for Advanced Studies (SOKENDAI),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
e Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa 277-8583, Japan
f Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
g Leung Center for Cosmology and Particle Astrophysics,
National Taiwan University, Taipei 10617, Taiwan
h Center for Theoretical Physics of the Universe,
Institute for Basic Science (IBS), Daejeon, 34126, Korea
i Division of Particle and Astrophysical Science, Graduate School of Science,
Nagoya University, Nagoya 464-8602, Japan

We study the effects of velocity dispersion on the formation of primordial
black holes (PBHs) in a matter-dominated era. The velocity dispersion is
generated through the nonlinear growth of perturbations and has the potential
to impede the gravitational collapse and thereby the formation of PBHs. To
make discussions clear, we consider two distinct length scales. The larger one
is where gravitational collapse occurs which could lead to PBH formation, and
the smaller one is where the velocity dispersion develops due to nonlinear
interactions. We estimate the effect of the velocity dispersion on the PBH
formation by comparing the free-fall timescale and the timescale for a
particle to cross the collapsing region. As a demonstration, we consider a
log-normal power spectrum for the initial density perturbation with the peak
value $\sigma_{0}^{2}$ at a scale that corresponds to the larger scale. We
find that the threshold value of the density perturbation $\tilde{\delta}_{\rm
th}$ at the horizon entry for the PBH formation scales as $\tilde{\delta}_{\rm
th}\propto\sigma_{0}^{2/5}$ for $\sigma_{0}\ll 1$.

##  1 Introduction

Black holes are widely accepted to exist in the Universe through various
observations such as the motion of surrounding celestial objects [1, 2, 3, 4],
gravitational-wave signals from binary-black-hole mergers [5, 6, 7], and the
black hole shadow [8]. Some of the black holes might have a primordial origin
[9, 10, 11]. Depending on their mass, primordial black holes (PBHs) may
explain dark matter abundance [12, 13, 14, 15], the binary black hole
abundance observed by the gravitational-wave events [16, 17, 18, 19], an
excess of microlensing events [20], a potential gravitational-wave signal at a
pulsar timing array [21, 22, 23, 24, 25, 26, 27], the seeds of supermassive
black holes and cosmic structures [28, 29, 30, 31, 32, 33, 34]. PBHs whose
lifetime is shorter than the age of the Universe may also play important roles
in the early Universe such as baryogenesis [35, 36, 37] and the production of
stochastic gravitational-wave background [38, 39, 40, 41, 42, 43, 44, 45].111
The stochastic background of the gravitational waves induced by the primordial
curvature perturbations is also associated with PBHs with longer lifetimes and
they are interesting observational targets. See Refs. [46, 47, 48, 49, 50, 51,
52, 53] and recent reviews [54, 55].

For any applications of PBHs, their abundance is a basic quantity that
determines their cosmological significance. For the estimation of the
abundance in a radiation-dominated (RD) era, three tools conventionally
adopted are: the criterion of PBH formation [56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68], the peaks theory for the statistical treatment of the
initial profiles [69, 70, 71, 72], and the critical behavior of the PBH
formation [73, 74, 75, 76] for sufficiently spherical collapses. The latest
theoretical estimation of the PBH abundance for the monochromatic curvature
power spectrum can be seen in Ref. [77] for Gaussian and local-type non-
Gaussian curvature perturbations (see also Refs. [78, 79, 80, 77, 81] for
effects of the non-Gaussianity), where the PBH mass function is calculated
based on an approximately universal criterion [65] and the peaks theory [72].

In a matter-dominated (MD) era, since the pressure of the fluid, which is the
most effective factor to impede PBH formation in an RD era, is absent and the
criterion of PBH formation obtained with the fluid approximation does not
apply. The absence of the pressure would make the gravitational collapse
easier, and lower peaks of the density perturbation may contribute to PBH
abundance [82, 83]. However, the high-peak approximation in peaks theory
becomes worse and the spherical symmetry assumption, which is relevant to rare
high peaks [69], may not be accurate. Furthermore, it is well known that the
deviation from the spherical symmetry can significantly grow in the
gravitational collapse in the case of pressureless matter [84, 85]. Thus, non-
spherical initial profiles are inevitable, and they may lead to the
complicated nonlinear dynamics associated with deformation and rotation. In
addition, the pressureless fluid approximation breaks down as soon as
nonlinearity becomes important due to the occurrence of shell-crossing
singularities. In reality, the pressureless matter obeys the collisionless
Boltzmann equation, and velocity dispersion will develop in the nonlinear
regime. Thus, careful studies that take account of various effects are
required in order to understand the PBH formation in an MD era. There are
several analytical studies in the literature, in which the effects of
asphericity [86], angular momentum [87], and inhomogeneity [82, 83, 88], were
estimated under some assumptions and approximations. For an analysis in
numerical relativity, see Ref. [89].

In this paper, we consider the effect of the velocity dispersion inside the
collapsing region and discuss the conditions for the velocity dispersion to
impede the PBH formation in the MD era.222 The effect of the velocity
dispersion was briefly mentioned in Ref. [90], in which the radius of the
virialized structure was compared to the Schwarzschild radius. We introduce a
more comprehensive discussion.  To make discussions simple and clear, we
consider two distinct length scales. The larger one is where gravitational
collapse occurs which could lead to PBH formation, and the smaller one is
where the velocity dispersion develops due to nonlinear interactions. The
paper is organized as follows. In Sec. 2, we overview the situation we
consider by describing our setup and assumptions. In Sec. 3, we derive the
conditions for the velocity dispersion to impede the PBH formation. In doing
so, we find there are essentially two different scenarios, depending on when
the velocity dispersion emerges, details of which are discussed in Appendix A.
In Sec. 4, the conditions to impede the PBH formation are rephrased in terms
of the density threshold for the PBH formation for a given density power
spectrum. In Sec. 5, using the obtained results, we compute the PBH production
probability as a function of the peak amplitude of the power spectrum of the
density perturbation. Our conclusions are given in Sec. 6. Throughout this
paper, we use the units in which both the speed of light and Newton’s
gravitational constant are unity, $G=c=1$.

##  2 Overview

To avoid any confusion in understanding how the velocity dispersion is
generated and works as a preventing factor against PBH formation, before
discussing the details, let us clarify our setup and assumptions.

First, the rough basic picture is as follows. We consider a high-density
region of the comoving scale $k_{\mathrm{PBH}}$ that undergoes gravitationally
collapse and may lead to PBH formation, which may be impeded by the velocity
dispersion generated by the dynamics on a scale $k$ where
$k>k_{\mathrm{PBH}}$. Here we suppose that a finite power of perturbation also
exists on these small scales. When the perturbations on the small scales enter
the nonlinear regime, that is, when the density perturbation becomes high
enough to render the fluid approximation invalid, the velocity dispersion is
assumed to be generated, so that the random motion of the constituent
particles can act against gravity. In this phase, if the repulsive effect due
to this generated random motion dominates the dynamics of the whole system
over the length scale $1/k_{\text{PBH}}$, the collapsing region will be
eventually virialized, so that the PBH formation will be impeded. Here, we
note that the velocity dispersion may get effective on the scale
$k_{\mathrm{PBH}}$ not immediately after it is generated on the scale $k$ but
later, depending on its magnitude. We describe more specific scenarios in the
following setup and assumptions.

Now let us describe the setup and assumptions.

  * •

We consider only the two separated comoving scales characterized by $k$ and
$k_{\rm PBH}$. We call $k_{\rm PBH}$ the large or PBH scale, while $k$ the
small scale. Throughout this paper, the tilded quantities are those for the
PBH scale $k_{\rm PBH}$, while those for the small scale $k$ are not tilded.
Thus in particular $\tilde{k}\equiv k_{\rm PBH}$. Two exceptions are the
velocity dispersion $\sigma_{v}(t)$ as a function of time, and the root-mean-
square amplitude of the density perturbation at the horizon entry
$\sigma_{\delta,{\rm ent}}(k)$ as a function of the comoving wavenumber.

  * •

Since we discuss a cosmological patch that is going to form a PBH (or to be
impeded by the velocity dispersion), the density perturbation on this patch
$\tilde{\delta}_{\rm ent}$ is exceptionally higher than the typical amplitude,
$\tilde{\delta}_{\rm ent}\gg\sigma_{\delta,{\rm ent}}(\tilde{k})$, at the time
the scale enters the Hubble horizon.

  * •

On the other hand, we suppose the amplitude of the smaller-scale perturbations
is given by a typical value, that is, the amplitude at the horizon entry
$\delta_{\rm ent}(k)$ is typically given by the standard deviation of the
density perturbation, $|\delta_{\rm ent}(k)|=\mathcal{O}(\sigma_{\delta,{\rm
ent}}(k))$. This magnitude is also assumed to be so small that the PBH
formation _on the small scale_ is negligible. Then, except for Sec. 5, we
assume for simplicity that the only impeding factor against PBH formation on
the scale $\tilde{k}$ is the velocity dispersion. In addition to the velocity
dispersion, other potential impeding factors against the PBH formation will be
briefly discussed in Sec. 5.

  * •

We assume that the fluid approximation (i.e., coherent flow of particles with
negligible velocity dispersion) is valid on the scale $k$ until caustics
appear, that is, until the perturbation enters the nonlinear regime (provided
that larger scales have not yet collapsed). The collapsing overdense regions
may be locally described by the closed Friedmann-Lemaître-Robertson-Walker
(FLRW) universe models (see, e.g., Ref. [91]). Then the collapsing time on the
PBH scale $\tilde{k}$ and that on the scale $k$ may be estimated by using the
closed FLRW collapsing model, which we denote by $\tilde{t}_{\mathrm{coll}}$
and $t_{\mathrm{coll}}$, respectively. It should be noted that, if
$t_{\mathrm{coll}}\gtrsim\tilde{t}_{\mathrm{coll}}$, the smaller-scale
perturbations cannot be independent of the surrounding over-density associated
with the PBH scale. Namely, the actual collapsing time will be earlier than
$t_{\rm coll}$ in that case.

It should be noted that the velocity dispersion generated on the scale $k$ may
not be immediately shared with the whole region on the scale $\tilde{k}$. If
the virialization is completed on the smaller scale before the collapse of the
larger scale, the matter is confined in the virialized halos. In this case,
the halos behave as _macroscopic self-gravitating comoving particles_ , hence
no impeding effect due to the velocity dispersion is expected against the
gravitational collapse on the scale $k_{\mathrm{PBH}}$. Therefore, we consider
the two possible scenarios: Case I and Case II as follows.

  * •

Case I: We assume the virialization of the perturbations on the smaller scale
is completed before the collapse of the larger scale, i.e.,
$t_{\mathrm{vir}}<\tilde{t}_{\mathrm{coll}}$. In this case, the velocity
dispersion is released when the mean density on the scale $\tilde{k}$ becomes
comparable to the density of a halo.333 Note that the mean density first
decreases as the Universe expands, but eventually increases as the region on
the scale $\tilde{k}$ contracts. The release of the velocity dispersion
happens while the mean density is increasing. At this point, the halos
_dissolve_ into ambient space. We identify the time when the halos dissolve
with the time ($\equiv t_{*}$) at which the velocity dispersion becomes
effective on the PBH scale $\tilde{k}$.

  * •

Case II: We assume $t_{\rm vir}>\tilde{t}_{\rm coll}$, i.e., there is no time
for the small scale to collapse and form virialized halos. In this case, the
velocity dispersion is directly released soon after the smaller-scale
perturbations grow into the nonlinear regime. Since the evolution of the
small-scale perturbations should be necessarily promoted (especially in case
II) by the evolution of the surrounding high-density region on the scale
$1/\tilde{k}$, we will consider the linear density perturbation $\delta_{\rm
com}$ in a FLRW universe in the synchronous comoving gauge for the evolution
of the small-scale perturbations. Then, the time $t_{*}$ when the velocity
dispersion becomes effective on the scale $\tilde{k}$ is identified with the
time when $\delta_{\rm com}\sim 1$, i.e., when caustics appear.

The detailed derivation of the criterion that distinguishes the two cases is
given in Appendix. Here let us briefly present an order-of-magnitude estimate.
Consider a region where a positive density perturbation $\tilde{\delta}$
exists. In a linear regime, it grows in proportion to the scale factor $a$ at
a matter-dominated stage, where $a\propto t^{2/3}$. Thus
$\tilde{\delta}(t)=(t/\tilde{t}_{\rm ent})^{2/3}\tilde{\delta}_{\rm ent}$,
where $\tilde{\delta}_{\rm ent}$ is the amplitude of the density perturbation
at horizon entry. The time at which the region collapses is sometime after the
amplitude reached $\mathcal{O}(1)$. Ignoring order-of-unity numerical factors,
we may identify the time when $\tilde{\delta}=1$ as the collapse time,
$\tilde{t}_{\rm col}$. When the region collapses but doesn’t lead to the
formation of a black hole, the system virializes approximately at the same
time, $\tilde{t}_{\rm vir}\sim\tilde{t}_{\rm col}$. Since the same argument
applies to the small scale, the region of the small scale will be virialized,
as $\delta_{\rm ent}$ is assumed to be very small. Thus we have

$\displaystyle{t}_{\rm vir}\sim{t}_{\rm ent}{\delta}_{\rm ent}^{-3/2}\,.$ (1)

On the other hand, on the PBH scale $\tilde{k}$, the region may form a black
hole if there is no impeding factors. So we estimate the collapse time as

$\displaystyle\tilde{t}_{\rm col}\sim\tilde{t}_{\rm ent}\tilde{\delta}_{\rm
ent}^{-3/2}\,.$ (2)

Now recalling that the comoving scale $k$ may be expressed in terms of its
horizon entry time as $k=(aH)_{\rm ent}\propto t_{\rm ent}^{-1/3}$, one finds

$\displaystyle\left(\frac{{t}_{\rm vir}}{\tilde{t}_{\rm
col}}\right)^{2/3}\sim\left(\frac{{t}_{\rm ent}}{\tilde{t}_{\rm
ent}}\right)^{2/3}\frac{\tilde{\delta}_{\rm ent}}{\delta_{\rm
ent}}=\frac{\tilde{k}^{2}}{{k}^{2}}\frac{\tilde{\delta}_{\rm ent}}{\delta_{\rm
ent}}\,.$ (3)

Thus the criterion is

$\displaystyle{\rm Case~{}I}$
$\displaystyle:~{}{t}_{\mathrm{vir}}<\tilde{t}_{\mathrm{coll}}\quad\Leftrightarrow\quad{k}^{2}\delta_{\rm
ent}>\tilde{k}^{2}\tilde{\delta}_{\rm ent}\,,$ $\displaystyle{\rm Case~{}II}$
$\displaystyle:~{}{t}_{\mathrm{vir}}>\tilde{t}_{\mathrm{coll}}\quad\Leftrightarrow\quad{k}^{2}\delta_{\rm
ent}<\tilde{k}^{2}\tilde{\delta}_{\rm ent}\,.$ (4)

In both cases, once the velocity dispersion is shared over the whole patch of
the PBH scale at $t=t_{*}$, the criterion for the PBH formation can be roughly
evaluated by comparing the gravitational free-fall time $\Delta\tilde{t}_{\rm
ff}\propto\tilde{\rho}^{-1/2}\propto\tilde{R}^{3/2}$ with the sound-wave
crossing time $\Delta\tilde{t}_{\rm cross}\sim\tilde{R}/\sigma_{v}$, where
$\tilde{\rho}$, $\tilde{R}$, and $\sigma_{v}$ are the density, the physical
size of the scale $\tilde{k}$, and the velocity dispersion, respectively. As
the velocity dispersion increases, the crossing time becomes shorter, and the
PBH formation is eventually impeded. The evaluation of the PBH formation
condition is given in Sec. 3.

In Sec. 4, assuming that the scale $k$ can be treated independently of the
scale $\tilde{k}$, we consider a log-normal density spectrum
$\ln\sigma(k)=\ln\sigma_{0}-\mu\ln(k/\tilde{k})^{2}$, where $\sigma_{0}$ and
$\mu$ are dimensionless parameters, and evaluate the threshold amplitude of
the density perturbation at horizon entry $\tilde{\delta}_{\rm ent}$. In Sec.
5, the PBH production rate is roughly evaluated following the procedure
provided in Ref. [87], and the effect of the velocity dispersion is compared
with the spin effects reported in Ref. [87].

##  3 Effect of the Velocity Dispersion against Collapse

The velocity dispersion may impede the gravitational collapse after the
velocity dispersion is shared in the whole region of the PBH scale
$\tilde{k}$. First, let us evaluate the impeding effect of the uniformly
distributed velocity dispersion. We consider a spherical ball on the PBH
scale. Once the velocity dispersion is shared over the collapsing ball, the
velocity dispersion simply increases as $\sigma_{v}\propto\tilde{\rho}^{1/3}$
as the system collapses adiabatically, according to Liouville’s theorem. More
explicitly, we have

$\displaystyle\sigma_{v}(t)=$
$\displaystyle\sigma_{v}(t_{*})\left(\frac{\tilde{\rho}(t)}{\tilde{\rho}(t_{*})}\right)^{1/3}=\sigma_{v}(t_{*})\frac{\tilde{R}(t_{*})}{\tilde{R}(t)},$
(5)

where $t_{*}<t$ and $t_{*}$ is the time when the velocity dispersion is shared
over the whole collapsing region.

The sound-wave crossing time across the region of the size $R(t)$ is estimated
as

$\displaystyle\Delta\tilde{t}_{\text{cross}}\simeq$
$\displaystyle\frac{\tilde{R}(t)}{\sigma_{v}(t)}=\frac{3}{2\sigma_{v}(t_{*})}\frac{\tilde{R}(t)^{2}}{\tilde{R}(t_{*})\tilde{R}_{\rm
ent}}\tilde{t}_{\rm ent},$ (6)

where $\tilde{R}_{\rm ent}$ is the radius at the horizon entry time
$\tilde{t}_{\rm ent}$ satisfying $\tilde{R}_{\rm ent}=1/\tilde{H}_{\rm
ent}=3\tilde{t}_{\rm ent}/2$ in the Einstein-de Sitter background with
$\tilde{H}_{\rm ent}$ being the Hubble parameter at the horizon entry. On the
other hand, the free-fall time is estimated as

$\displaystyle\Delta\tilde{t}_{\text{ff}}\simeq\left(\frac{\tilde{R}(t)}{\tilde{R}_{\text{ent}}}\right)^{3/2}\tilde{t}_{\text{ent}}.$
(7)

Noting that $\Delta\tilde{t}_{\rm cross}\propto\tilde{R}^{2}$ and
$\Delta\tilde{t}_{\rm ff}\propto\tilde{R}^{3/2}$, and $\Delta\tilde{t}_{\rm
cross}>\Delta\tilde{t}_{\rm ff}$ at the time of horizon entry, the collapse
halts when $\Delta\tilde{t}_{\rm ff}=\Delta\tilde{t}_{\rm cross}$ if the
radius is larger than the Schwarzschild radius at that time. Let us denote the
value of $\tilde{R}(t)$ at $\Delta\tilde{t}_{\rm ff}(t)=\Delta\tilde{t}_{\rm
cross}(t)$ by $\tilde{R}_{\rm halt}$. We find

$\tilde{R}_{\rm
halt}=\frac{4}{9}\sigma_{v}^{2}(t_{*})\frac{\tilde{R}^{2}(t_{*})}{\tilde{R}_{\rm
ent}^{2}}\tilde{R}_{\rm ent}\,.$ (8)

If this is larger than $2\tilde{M}$, i.e., $\tilde{R}_{\rm
halt}>\tilde{R}_{\rm ent}=1/\tilde{H}_{\rm ent}=2\tilde{M}$, the collapse
halts before the PBH formation. Conversely, a PBH forms if $\tilde{R}_{\rm
halt}<2\tilde{M}$. Thus, the condition for the PBH _non_ -formation can be
rewritten in the following simple form.

$\sigma_{v}(t_{*})\tilde{R}(t_{*})>\frac{3}{2}\tilde{R}_{\rm ent}.$ (9)

To understand how the velocity dispersion is shared over the whole region of
the size $1/\tilde{k}$, we introduce a closed FLRW collapsing model and
consider its perturbations as given in Appendix A. Deferring detailed
explanations of the complicated manipulations, here we just refer to the
results. According to the discussion in Appendix A, depending on whether the
halo formation on the scale $k$ can be completed before the collapse of the
scale $\tilde{k}$, we can rewrite Eq. (9), i.e., PBH non-formation condition,
as

$\displaystyle\delta_{\rm ent}<\frac{\tilde{k}^{4}}{k^{4}}$
$\displaystyle\text{for }~{}\text{Case I~{}: }\quad k^{2}\delta_{\rm
ent}>\tilde{k}^{2}\tilde{\delta}_{\rm ent}\,,$ (10) $\displaystyle\delta_{\rm
ent}>\frac{k}{\tilde{k}}\tilde{\delta}_{\rm ent}^{5/2}$
$\displaystyle\text{for }~{}\text{Case II: }\quad k^{2}\delta_{\rm
ent}<\tilde{k}^{2}\tilde{\delta}_{\rm ent}\,,$ (11)

where we have ignored factors of order unity. The qualitative features of the
above conditions can be understood as follows.

For Case I, the virialization of the scale $k$ occurs before the collapse of
the would-be PBH scale. The condition (10) is a bit counter-intuitive. So let
us see why it is the case. Since the gravitational potential $\Psi$ is
constant in time in the matter-dominated universe, and $\delta_{\rm
ent}\sim\Psi$ at horizon crossing, the virial velocity dispersion is estimated
as $\sigma_{v}^{2}\sim\Psi\sim\delta_{\rm ent}$ (see Eq. (43)). The virial
density is $\rho_{\rm vir}=\rho_{\rm ent}(a_{\rm ent}/a_{\rm
vir})^{3}=\rho_{\rm ent}(\delta_{\rm ent}/\delta_{\rm vir})^{3}\sim\rho_{\rm
ent}\delta_{\rm ent}^{3}$ since $\delta\propto a$ and $\delta_{\rm vir}=O(1)$.
As the collapse of a ball of the scale $\tilde{k}$ proceeds, the density
$\tilde{\rho}$ reaches $\rho_{\rm vir}$ at $t=t_{*}$, and the velocity
dispersion is released to the entire ball, i.e.,
$\tilde{\rho}(t_{*})=\rho_{\rm vir}$. Hence we obtain the relation,

$\displaystyle\delta_{\rm ent}$ $\displaystyle\sim$
$\displaystyle\left(\dfrac{\rho_{\rm vir}}{\rho_{\rm
ent}}\right)^{1/3}=\left(\dfrac{\rho_{\rm vir}}{\tilde{\rho}_{\rm
ent}}\right)^{1/3}\left(\dfrac{\tilde{\rho}_{\rm ent}}{\rho_{\rm
ent}}\right)^{1/3}=\left(\dfrac{\tilde{\rho}(t_{*})}{\tilde{\rho}_{\rm
ent}}\right)^{1/3}\left(\dfrac{a_{\rm ent}}{\tilde{a}_{\rm ent}}\right)$ (12)
$\displaystyle=$ $\displaystyle\dfrac{\tilde{R}_{\rm
ent}}{\tilde{R}(t_{*})}\dfrac{\tilde{k}^{2}}{k^{2}}\,.$

Recalling that $\delta_{\rm ent}\sim\sigma_{v}^{2}$ (Eq. (43)), this implies

$\displaystyle\sigma_{v}^{2}\dfrac{\tilde{R}(t_{*})^{2}}{\tilde{R}_{\rm
ent}^{2}}\sim\delta_{\rm
ent}\times\dfrac{\tilde{k}^{4}}{k^{4}}\frac{1}{\delta_{\rm
ent}^{2}}=\dfrac{\tilde{k}^{4}}{k^{4}}\frac{1}{\delta_{\rm ent}}\,.$ (13)

With Eq. (9), this gives the condition quoted in Eq. (10). Here we have
ignored all the coefficients of $\mathcal{O}(1)$. As one can see from the
above estimate, the reason why the condition gives an upper bound on
$\delta_{\rm ent}$ is because it would make the small-scale virialized halos
so small and dense that the large-scale region would collapse to a black hole
before the velocity dispersion is shared over the entire region of the PBH
scale $\tilde{k}$. More details are given in Appendix A.3.

Let us now turn to Case II where the velocity dispersion is shared by the
whole PBH region before the perturbations on the small scale $k$ are
virialized. The analysis in this case is a bit more complicated as one has to
consider the growth of the small-scale perturbation in the collapsing region
of the scale $\tilde{k}$. The detailed derivation of the condition (11) is
defered to Appendix A.4. Here we only mention that the reason why it depends
on $\tilde{\delta}_{\mathrm{ent}}$ is precisely because of this fact that the
small-scale perturbation grows within the collapsing ball of the large scale.
The reason why the condition gives a lower bound on $\delta_{\mathrm{ent}}$ is
simply because a larger $\delta_{\mathrm{ent}}$ gives a larger velocity
dispersion, and hence it is easier to impede the PBH formation.

##  4 Threshold of the Density Perturbation

As the effect of the perturbation on the small scale $k$ is statistical,
hereafter, we replace $\delta_{\rm ent}$ by its root-mean-square value
$\sigma_{\delta,{\rm ent}}(k)$. Then, the condition for impeding the PBH
formation due to velocity dispersion can be summarized as follows:

$\displaystyle\sigma_{\delta,\rm
ent}(k)<\left(\frac{k}{\tilde{k}}\right)^{-4}$ $\displaystyle\text{for
}~{}\text{Case I~{}:}\quad\sigma_{\delta,{\rm ent}}(k)>\tilde{\delta}_{\rm
ent}\left(\frac{k}{\tilde{k}}\right)^{-2},$ (14)
$\displaystyle\sigma_{\delta,{\rm ent}}(k)>\tilde{\delta}_{\rm
ent}^{5/2}\left(\frac{k}{\tilde{k}}\right)$ $\displaystyle\text{for
}~{}\text{Case II:}\quad\sigma_{\delta,{\rm ent}}(k)<\tilde{\delta}_{\rm
ent}\left(\frac{k}{\tilde{k}}\right)^{-2},$ (15)

where we remind that the tilded quantities are those associated with the PBH
scale. So only the range $k>\tilde{k}$ is meaningful.

Before proceeding to the threshold analysis, we mention a possible caveat
about when one considers a continuous perturbation spectrum. Although we
consider the case $k\to\tilde{k}$ in our analysis, since we treat the two
scales $k$ and $\tilde{k}$ as separate and independent scales, the validity of
our argument becomes questionable in the limit $k\to\tilde{k}$. To be more
precise, since we consider the effect of the small-scale perturbation on the
dynamics of the PBH scale only through the velocity dispersion, which is a
statistical quantity by definition, $k$ must be sufficiently larger than
$\tilde{k}$ to allow statistical interpretations of the small-scale velocity
effect. Therefore, to avoid complications, we introduce the minimum applicable
value for $k$, $k\geq k_{\rm min}$ where $k_{\rm min}>\tilde{k}$. Although we
never assign an explicit value to $k_{\rm min}$, we expect $k_{\rm
min}/\tilde{k}=\mathcal{O}(10)$. We also mention that the introduction of
$k_{\rm min}$ renders the condition derived below a _sufficient condition for
PBH non-formation_ or a _necessary condition for PBH formation_. This should
be kept in mind below.

With this minimum value $k_{\rm min}$ for $k$ in mind, we schematically depict
the region of PBH non-formation in Fig. 1, where light- and dark-shaded
regions correspond to Case I and Case II, respectively.

Figure 1: The parameter space of PBH _non_ -formation in the
$(k,\sigma_{\delta,\rm ent})$ plane. The shaded regions (Case I: light shaded,
Case II: dark shaded) show the parameter region of PBH non-formation. The
colored lines represent examples of density perturbation spectra peaked at the
PBH scale. From top to bottom, the velocity dispersion does not impede the PBH
formation (top red), does do so (bottom red and top blue), and does not do so
(bottom blue).

Once the density power spectrum $\sigma^{2}_{\delta,{\rm ent}}(k)$ is given as
a function of $k$, and if the line describing the density power spectrum
passes across the shaded region in Fig. 1, there exist small-scale
perturbations that prevent a density perturbation of an amplitude
$\tilde{\delta}_{\rm ent}$ on the scale $\tilde{k}$ to form a PBH. An
important point in this diagram is the place where the lines representing the
two conditions in (14) and (15) merge. It occurs at $(k,\sigma_{\delta,\rm
ent})=(\tilde{\delta}_{\rm ent}^{-1/2}\tilde{k},\tilde{\delta}_{\rm
ent}^{2})$.

Let us apply the above result to a couple of simple examples. When we consider
the PBH formation, it is often assumed that there is a narrow peak in the
power spectrum and abundant PBHs are produced on scales around the peak. In
particular, as a comoving wavenumber $k$ and its horizon crossing time $t$
during inflation is related as $k\sim{\mathrm{e}}^{Ht}$, a log-normal
distribution is naturally realized in many models of inflation. Then, around
the peak of the power spectrum, the power spectrum may be well approximated by
a log-normal spectrum:

$\sigma^{2}_{\delta,{\rm
ent}}(k)=\sigma^{2}_{0}\exp\left\\{-2\mu\left[\ln\left(\frac{k}{\tilde{k}}\right)\right]^{2}\right\\},$
(16)

or equivalently,

$\ln\sigma_{\delta,{\rm
ent}}(k)=-\mu\left[\ln\left(\frac{k}{\tilde{k}}\right)\right]^{2}+\ln\sigma_{0},$
(17)

where $\mu$ is the non-dimensional parameter characterizing the width of the
log-normal distribution.

First, let us consider the case when $\sigma_{\delta,{\rm ent}}(k_{\rm
min})<(k_{\rm min}/\tilde{k})^{-4}$ (blue lines in Fig. 1), namely,

$\displaystyle\ln\sigma_{0}<\ln\sigma_{\rm cr}(\mu,k_{\rm
min})\equiv\mu\left[\ln\left(\frac{k_{\rm
min}}{\tilde{k}}\right)\right]^{2}-4\ln\left(\frac{k_{\rm
min}}{\tilde{k}}\right).$ (18)

In this case, since the spectrum must lie below the shaded region in Fig. 1 to
form PBHs, we obtain the following inequality as the PBH formation condition
(see the first panel in Fig. 2):

$\displaystyle\ln\left(\frac{k_{\rm
min}}{\tilde{k}}\right)+\frac{5}{2}\ln\tilde{\delta}_{\rm
ent}>\ln\sigma_{\delta,{\rm ent}}(k_{\rm min})$ (19)
$\displaystyle\Leftrightarrow$ $\displaystyle\tilde{\delta}_{\rm
ent}>\exp\left[-\frac{2}{5}\mu\left[\ln\left(\frac{k_{\rm
min}}{\tilde{k}}\right)\right]^{2}\right]\left(\sigma_{0}\frac{\tilde{k}}{k_{\rm
min}}\right)^{2/5}.$ (20)

Figure 2:  The region for PBH non-formation and the spectral line (red curve)
for $\ln\sigma_{0}=-5$ (top panel) and $\ln\sigma_{0}=-2.5$ (bottom panel)
with $\ln(k_{\rm min}/\tilde{k})=1$, $\mu=1$ are shown. The top (bottom) panel
shows the case for $\sigma_{0}<\sigma_{\rm cr}(\mu,k_{\rm min})$
($\sigma_{0}>\sigma_{\rm cr}(\mu,k_{\rm min})$), which corresponds to the blue
(red) curves in Fig. 1. For $\tilde{\delta}_{\rm ent}<\tilde{\delta}_{\rm th}$
with $\tilde{\delta}_{\rm th}$ being defined in Eq. (24), the lines for
$\ln\sigma_{\delta,{\rm ent}}=\ln\tilde{\delta}_{\rm ent}-2\ln(k/\tilde{k})$
and $\ln\sigma_{\delta,{\rm ent}}=5/2\ln\tilde{\delta}_{\rm
ent}+\ln(k/\tilde{k})$ are shown by the blue and orange dashed lines,
respectively. The green-shaded region is the region for PBH non-formation.
Since the spectral line (red curve) passes through the green-shaded region, in
this case, there exist small-scale perturbations that prevent the PBH
formation on the scale $\tilde{k}$. If $\tilde{\delta}_{\rm ent}$ is increased
to the threshold value, i.e., $\tilde{\delta}_{\rm ent}=\tilde{\delta}_{\rm
th}$, the blue and orange dashed lines get shifted up to the corresponding
solid lines, so that the shaded triangle region of non-formation becomes
narrower to the hatched triangle and touches the spectral line at a vertex.

In the case $\sigma_{\delta,{\rm ent}}(k_{\rm min})>(k_{\rm
min}/\tilde{k})^{-4}$ (red lines in Fig. 1), namely,

$\ln\sigma_{0}>\ln\sigma_{\rm cr}(\mu,k_{\rm min}),$ (21)

the spectrum must lie above the shaded region to form PBHs. This condition is
satisfied if the point $(k,\sigma_{\delta,\rm ent})=(\tilde{\delta}_{\rm
ent}^{-1/2}\tilde{k},\tilde{\delta}_{\rm ent}^{2})$ is below the spectral
curve. This leads to the condition,

$\displaystyle 2\ln\tilde{\delta}_{\rm ent}$ $\displaystyle<$
$\displaystyle\ln\sigma_{\delta,{\rm ent}}\left(k=\tilde{\delta}_{\rm
ent}^{-1/2}\tilde{k}\right)=-\mu\left(\ln\tilde{\delta}_{\rm
ent}^{-1/2}\right)^{2}+\ln\sigma_{0}\,.$ (22)

Noting that $\tilde{\delta}_{\rm ent}<1$, we find that the inequality (22)
implies the condition (see the second panel in Fig. 2),

$\tilde{\delta}_{\rm
ent}>\exp\left[-\frac{2}{\mu}\left(2+\sqrt{4+\mu\ln\sigma_{0}}\right)\right].$
(23)

To summarize, the _necessary_ condition for the PBH formation, (20) and (23),
is

$\tilde{\delta}_{\rm ent}>\tilde{\delta}_{\rm th}(\sigma_{0},\mu,k_{\rm
min})\equiv\left\\{\begin{array}[]{ccl}\exp\left[-\dfrac{2}{5}\mu\ln^{2}\left({k_{\rm
min}}/{\tilde{k}}\right)\right]\left(\sigma_{0}({\tilde{k}}/{k_{\rm
min}})\right)^{2/5}&;&\sigma_{0}<\sigma_{\rm cr}(\mu,k_{\rm min}),\\\
\exp\left[-\dfrac{2}{\mu}\left(2+\sqrt{4+\mu\ln\sigma_{0}}\right)\right]&;&\sigma_{0}>\sigma_{\rm
cr}(\mu,k_{\rm min}).\end{array}\right.$ (24)

We note that, as is explicitly shown in Fig. 3, $\tilde{\delta}_{\rm th}$ is
discontinuous at $\sigma_{0}=\sigma_{\rm cr}(\mu,k_{\rm min})$ as a function
of $\sigma_{0}$ if

$\left.\frac{d\ln\sigma_{\delta,{\rm ent}}(k)}{d\ln k}\right|_{k=k_{\rm
min}}>-4\quad\Leftrightarrow\quad\mu\ln\left(\frac{k_{\rm
min}}{\tilde{k}}\right)<2\,.$ (25)

Figure 3: The density threshold $\tilde{\delta}_{\rm th}(\sigma_{0},\mu,k_{\rm
min})$ for PBH formation as a function of $\sigma_{0}$ for $(\mu,\ln(k_{\rm
min}/\tilde{k}))=(1,0)$, $(1,1)$, $(1,2)$, $(1/2,1)$ and $(2,1)$.

Some comments on the presence of the discontinuity in $\tilde{\delta}_{\rm
th}$ are in order. If the root-mean-square value of perturbations on the small
scale $\sigma_{\delta,\rm ent}(k)$ is large enough, the resulting virialized
halos in a PBH scale region become so compact that they behave like particles
at rest and never impede the PBH formation. On the other hand, if
$\sigma_{\delta,\rm ent}(k)$ is very small, the velocity dispersion becomes
too small to affect the PBH formation. Thus the presence of the discontinuity
reflects the fact that the velocity dispersion becomes an effective impeding
factor only for a finite range of $\sigma_{\delta,\rm ent}(k)$. This results
in the two types of spectra that may lead to the PBH formation, as shown by
the top red and the bottom blue curves in Fig. 1. The transition from one type
to the other corresponds to the discontinuity. On general grounds, however,
one may expect that such a discontinuity would be smoothened when we improve
the analyses. For example, we required
$\Delta\tilde{t}_{\text{cross}}<\Delta\tilde{t}_{\text{ff}}$ for the non-
formation of PBHs, but the fate of would-be PBH over-dense regions may be
probabilistic when
$\Delta\tilde{t}_{\text{cross}}\simeq\Delta\tilde{t}_{\text{ff}}$. In
addition, once we take into account the other impeding factors which we
ignored in this paper, namely, asphericity [86], angular momentum [87], and
inhomogeneity [82, 83, 88], the discontinuity will not play any role in
determining the PBH formation threshold for most of the parameter space of our
interest (compare Figs. 3 and 4).

For cosmological applications, one may expect that the width of the peak $\mu$
of the log-normal spectrum (17) would be of order unity, whereas the peak
height $\sigma_{0}$ would not be very close to unity since otherwise PBHs
would be overproduced. Thus we expect that $\sigma_{0}<\sigma_{\rm cr}$ and
the threshold in the first line of Eq. (24), $\tilde{\delta}_{\rm
th}\propto\sigma_{0}^{2/5}$, or the condition from Case II, is likely to be
relevant in such cases. We mention that if the peak is very sharp so that the
width of the spectrum is much less than $k_{\rm min}$, namely,
$\mu\ln^{2}(k_{\rm min}/\tilde{k})\gg 1$, the velocity dispersion becomes
completely ineffective. In such a case, the other impeding factors we
mentioned above will play a decisive role.

##  5 PBH Formation Probability

In this section, we consider the PBH formation probability by taking into
account not only the effect of velocity dispersion but also all the other
impeding factors discussed previously in the literature. First, let us recall
the effect of angular momentum discussed in Ref. [87]. The threshold values
due to the angular momentum estimated at first- and second-order perturbations
are given in Eq. (4.12) in Ref. [87] as

$\displaystyle\tilde{\delta}_{\rm th(1)}$ $\displaystyle=$
$\displaystyle\frac{3\cdot 2^{2}}{5^{3}}q^{2},$ (26)
$\displaystyle\tilde{\delta}_{\rm th(2)}$ $\displaystyle=$
$\displaystyle\left(\frac{2}{5}\mathcal{I}\sigma_{0}\right)^{2/3},$ (27)

where $q$ and $\mathcal{I}$ are parameters of order unity describing the
initial reduced quadrupole moment and the second-order tidal contribution to
the angular momentum, respectively, and $\sigma_{0}$ is the standard deviation
of the density perturbation at the PBH scale. Assuming $q$ is independent of
$\sigma_{0}$, the different dependence of the two thresholds on $\sigma_{0}$
leads to different shapes of the PBH production probabilities as shown in Fig.
4.

Second, let us turn to the effect of anisotropy. According to Ref. [86], in an
MD era, taking the anisotropy of the system into account, the PBH production
probability $\beta_{0}$ is estimated as

$\displaystyle\beta_{0}\simeq\int_{0}^{\infty}\mathrm{d}\alpha\int_{-\infty}^{\alpha}\mathrm{d}\beta\int_{-\infty}^{\beta}\mathrm{d}\gamma\,\Theta\left(\tilde{\delta}(\alpha,\beta,\gamma)-\delta_{\text{th}}\right)\Theta(1-h(\alpha,\beta,\gamma))w(\alpha,\beta,\gamma),$
(28)

where $\alpha$, $\beta$, and $\gamma$ are the variables representing the
anisotropy, $\Theta(\cdot)$ is the Heaviside step function,
$\tilde{\delta}(\alpha,\beta,\gamma)=\alpha+\beta+\gamma$ is the density
perturbation in terms of the anisotropy parameters, $h(\alpha,\beta,\gamma)$
is a function that determines the threshold based on the hoop conjecture [92],
and
$w(\alpha,\beta,\gamma)\propto\exp\left[-3\left(2(\alpha^{2}+\beta^{2}+\gamma^{2})-(\alpha\beta+\beta\gamma+\gamma\alpha)\right)/(2\sigma_{0}^{2})\right]$
is the Doroshkevich probability distribution function [93]. The result is that
$\beta_{0}\propto\sigma_{0}^{5}$ with $\tilde{\delta}_{\rm th}=0$. See Refs.
[86, 87] for more details.

Finally, let us consider the effect of inhomogeneities discussed in [88].
Their result indicates that $\beta_{0}\propto\sigma^{3/2}$ if only the effect
of inhomogeneities is taken into account. If we assume this impeding effect to
work independently of the effect of anisotropy, we would obtain
$\beta_{0}\propto\sigma^{5}\times\sigma^{3/2}=\sigma^{13/2}$. However, as
noted in [88], the threshold obtained there is a sufficient condition for the
PBH formation. In other words, it is not a necessary condition for PBH
formation. Therefore, to be conservative, we do not take its effect into
account in this paper.

Figure 4: The dependence of the PBH production probability $\beta_{0}$ on the
standard deviation $\sigma_{0}$ of the density perturbations at the PBH scale.
The solid curves are our results including the effects of velocity dispersion
for $(\mu,\ln(k_{\rm min}/\tilde{k}))=(1,0)$ (top panel; purple), $(1,1)$
(both panels; blue), $(1,2)$ (top panel; magenta), $(1/2,1)$ (bottom panel;
purple), and $(2,1)$ (bottom panel; magenta). The vertical solid lines
indicate the corresponding values of $\sigma_{\rm cr}$. For comparison, the
first-order and second-order results in Ref. [87] are plotted as the orange
(right) and dark orange (left) dashed lines, respectively. For these,
parameters are chosen as $\mathcal{I}=1$ and $q=\sqrt{2}$ [87]. The common
slope $\beta_{0}\simeq 0.05556\,\sigma_{0}^{5}$ represented by the green
dashed line is the effect of anisotropy [86]. For comparison, the prediction
of Carr’s formula
$\mathrm{Erfc}[\tilde{\delta}_{\text{th}}/(\sqrt{2}\sigma_{0})]/2$ for the PBH
production in the RD era is also shown by the dashed gray line with
$\tilde{\delta}_{\text{th}}=0.42$.

In Fig. 4, we plot $\beta_{0}$ with $\tilde{\delta}_{\text{th}}$ determined by
the consideration of the velocity dispersion using Eq. (24). For comparison,
we also plot $\beta_{0}$ with $\tilde{\delta}_{\text{th}}$ determined by the
consideration of the spin effects [87] using Eqs. (26) and (27). The common
slope can be approximated as $\beta_{0}\simeq 0.05556\,\sigma_{0}^{5}$ [86]
(green dashed line). This is the effect of anisotropy. Different criteria on
the density threshold result in different exponential falloffs. They can be
understood by Carr’s formula [94], according to which the PBH formation
probability is roughly proportional to $\exp[-\tilde{\delta}_{\rm
th}^{2}/(2\sigma_{0}^{2})]$. Applying this to our result (24) in the case
$\sigma_{0}<\sigma_{\rm cr}$, $\tilde{\delta}_{\rm
th}^{2}\propto\sigma_{0}^{4/5}$, the exponent is found to be proportional to
$\sigma_{0}^{-6/5}$, as shown by the solid curves in Fig. 4. On the other
hand, the exponents are proportional to $\sigma_{0}^{-2}$ and
$\sigma_{0}^{-2/3}$, respectively, for the first-order and second-order spin
effects, respectively, as shown by the orange dashed (first-order) and dark
orange dashed (second-order) curves.

From Fig. 4, we see that the effect of the velocity dispersion can be the
dominant factor against the PBH formation. Although the value of $\sigma_{0}$
where the probability exponentially drops significantly depends on
$k_{\mathrm{min}}$, if the first-order spin effect is small, which is the case
if $q\ll 1$, or if $k_{\mathrm{min}}\simeq\tilde{k}$, the effect of the
velocity dispersion is most significant. Here it should be noted that, in
reality, all the impeding factors should be taken into account at the same
time although we treated the effects of the spin and the velocity dispersion
independently in this paper. 444 From Fig. 4, we find that the abundance
produced in the RD epoch can be larger than the one in the MD at least for
$\sigma_{0}\gtrsim\mathcal{O}(0.1)$.

##  6 Conclusions

In this paper, we discussed the effect of velocity dispersion on PBH
formation. For this purpose, we considered two distinct scales: One is a large
scale that would form a PBH, called the PBH scale, if there were no impeeding
factors. The other is a small scale on which the velocity dispersion is
generated. Then we derived the condition for the velocity dispersion on the
small scale to impede the PBH formation. Once the velocity dispersion is
shared in the whole collapsing region, the diffusion associated with the
velocity dispersion competes with the gravitational contraction. We obtained
the simple criterion $\sigma_{v*}\tilde{R}_{*}<3\tilde{M}$ for the formation
of a black hole of mass $\tilde{M}$ in terms of the size $\tilde{R}_{*}$ and
the velocity dispersion $\sigma_{v*}$ at the time when the velocity dispersion
is shared in the whole region (see Eq. (9)). To apply this formula to the PBH
formation, we considered two scenarios of the process.

If the density perturbation on the small scale is sufficiently large, the
virialization takes place before the collapsing time of the PBH scale. Then,
the velocity dispersion in the virialized halos is released to the whole
would-be PBH region when the mean density of that region becomes comparable to
the virial density of halos, and it may impede PBH formation. However, if the
density perturbation on the small scale is extremely large, the virialized
halos become too compact and dense, and the dissolution of halos takes place
too late to impede the PBH formation. Thus we obtain an upper bound for the
amplitude of the small-scale density perturbation below which the PBH
formation is impeded.

On the other hand, if the density perturbation on the small scale is extremely
small, the virialization can never be achieved before a PBH is formed.
However, even in this case, if the small-scale density perturbation is not too
small, it grows and may eventually enter the nonlinear regime. This growth of
the small-scale perturbation is enhanced by the fact that it behaves like a
perturbation in a closed collapsing background universe which the would-be PBH
region mimics. Then assuming that the velocity dispersion is generated and
shared over the PBH scale when the amplitude of the small-scale density
perturbation exceeds unity, the PBH formation may be impeded. Thus, this
scenario renders the perturbation with a somewhat smaller amplitude to impede
the PBH formation.

The two conditions are summarized in the inequalities (14) and (15). These
conditions can, in principle, apply to any functional form of the density
power spectrum. Based on this result, we computed the PBH production rate,
assuming that the power spectrum is of a log-normal form, with the peak at the
PBH scale. The result is shown in Fig. 4, where comparisons with the other
impeding factors, namely anisotropy and angular momentum, are also made. We
find that the effect of velocity dispersion can be a dominant factor to
prevent PBH formation, although all these effects are subjected to various
uncertainties, in addition to the spectral shape dependence, that affects the
threshold value of the density perturbation amplitude. It should be also noted
that, if the spectrum has a sufficiently sharp peak around the PBH scale, the
velocity dispersion is likely to be ineffective. When we translate the
observational constraints on the PBH abundance into those on model parameters
in PBH formation scenarios, since the effect of the velocity dispersion may
act as an additional factor to prevent PBH formation, the allowed regions for
the model parameters will get wider (see Figs. 10 and 11 in Ref. [13] for the
observational constraints on the PBH abundance).

One of the future directions is to properly include the effect of deviations
from the idealized dust picture during the collapse. Besides the apparent fact
that the collapsing matter is not dust but collisionless cold particles in the
simplest situation, which naturally gives rise to the dispersion in the
momentum space as well as in the real space, if it interacts with each other
or with other fields non-gravitationally, an additional pressure arises during
virialization. If the matter is a massive scalar field, the quantum pressure
due to the uncertainty principle may play an essential role. Recently, the PBH
formation right after inflation when the inflaton is oscillating, which is
effectively an MD era, has been discussed in Refs. [95, 96, 97, 98, 99, 90,
100, 101]. It will be interesting to extend our general relativistic
formalisms to these setups.

## Acknowledgment

The authors thank Yukawa Institute for Theoretical Physics at Kyoto
University, where this work was initiated during the YITP-X-21-02 “Brain-
storming workshop on Primordial Black Holes and Gravitational Waves”. This
work was supported in part by IBS under the project code, IBS-R018-D1 (TT),
and in part by JSPS KAKENHI Grant Numbers JP19H01895 (CY, TH, MS), JP20H05850
(CY), JP20H05853 (CY, TH, MS), JP19K03876 (TH), JP17H01131 (KK), JP20H04750
(KK), JP22H05270 (KK), and JP20H04727 (MS).

##  Appendix A Generation of Velocity Dispersion

###  A.1 Closed FLRW Collapsing Model

To describe the evolution of over-dense regions, we introduce closed FLRW
universes as models for the over-dense regions. We assume that the closed FLRW
model can be applied to all over-dense regions on any scale, that is, they can
apply to both scales of $\tilde{k}$ and $k$. Therefore, the equations given in
this section can be applied to the scale $\tilde{k}$ replacing all the bare
quantities ($a_{\rm ent}$, $\Omega_{\rm ent}$, $t_{\rm ent}$, $t_{\rm coll}$,
$H_{\rm ent}$, $\delta_{\rm ent}$, $R_{\rm ent}$, $E$, $\mathcal{K}$ and $M$)
by the corresponding tilded quantities ($\tilde{a}_{\rm ent}$,
$\tilde{\Omega}_{\rm ent}$, $\tilde{t}_{\rm ent}$, $\tilde{t}_{\rm coll}$,
$\tilde{H}_{\rm ent}$, $\tilde{\delta}_{\rm ent}$, $\tilde{R}_{\rm ent}$,
$\tilde{E}$, $\tilde{\mathcal{K}}$ and $\tilde{M}$).

The Friedmann equation for a closed universe can be written in the following
form:

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=$ $\displaystyle
H_{\text{ent}}^{2}\left[\Omega_{\text{ent}}\left(\frac{a_{\text{ent}}}{a}\right)^{3}+\left(1-\Omega_{\text{ent}}\right)\left(\frac{a_{\text{ent}}}{a}\right)^{2}\right],$
(29)

where $a$, $H_{\rm ent}$ and $\Omega_{\rm ent}$ are the scale factor, Hubble
parameter, and cosmological density parameter at the horizon entry $t=t_{\rm
ent}=2/(3H_{\text{ent}})$, respectively. Note that, at this stage, the horizon
entry time is just the time for reference because we have not introduced the
scale of the over-dense region. The spatial curvature $\mathcal{K}$ is given
by

$\mathcal{K}:=(\Omega_{\rm ent}-1)H_{\rm ent}^{2}a_{\rm ent}^{2}.$ (30)

The parametric solution for the scale factor $a$ is

$\displaystyle\frac{a}{a_{\text{ent}}}=$
$\displaystyle\frac{\Omega_{\text{ent}}}{2(\Omega_{\text{ent}}-1)}(1-\cos\theta),$
(31) $\displaystyle t=$
$\displaystyle\frac{\Omega_{\text{ent}}}{2H_{\text{ent}}(\Omega_{\text{ent}}-1)^{3/2}}(\theta-\sin\theta),$
(32)

where the parameter $\theta$ is related to the conformal time $\eta$ by
$\theta=\sqrt{\mathcal{K}}\eta$.

Taking the Einstein-de Sitter (EdS) universe as the background universe and
considering the uniform Hubble slice, we can obtain the value of the density
perturbation at the horizon entry as

$\delta_{\rm ent}=\Omega_{\rm ent}-1,$ (33)

where we have used the fact that the background density is given by the
critical density $3H^{2}/(8\pi)$ for the uniform Hubble slice. Hereafter we
assume $\delta_{\text{ent}}\ll 1$, which can be satisfied even for a rarely
high-density region leading to PBH formation in an MD era.

For $\theta=\sqrt{\mathcal{K}}\eta\ll 1$, we find $a\propto t^{2/3}$ at the
leading order, and the over-dense region can be approximated by an EdS
universe. More specifically, we find

$\displaystyle\frac{a}{a_{\text{ent}}}=$ $\displaystyle\frac{\Omega_{\rm
ent}}{4(\Omega_{\rm ent}-1)}\mathcal{K}\eta^{2}+\frac{1}{\Omega_{\rm
ent}-1}\mathcal{O}\left[(\mathcal{K}\eta^{2})^{2}\right],$ (34) $\displaystyle
t=$ $\displaystyle\frac{\Omega_{\rm ent}a_{\rm ent}\eta}{12(\Omega_{\rm
ent}-1)}\mathcal{K}\eta^{2}+\frac{a_{\rm ent}\eta}{\Omega_{\rm
ent}-1}\mathcal{O}\left[(\mathcal{K}\eta^{2})^{2}\right].$ (35)

Since the value of $\eta$ at the horizon entry is given by $\eta_{\rm ent}\sim
1/(H_{\rm ent}a_{\rm ent})\sim\delta_{\rm ent}/\sqrt{\mathcal{K}}$, we find
$\eta<\eta_{\rm ent}\ll 1/\sqrt{\mathcal{K}}$ before the horizon entry, and
then the EdS approximation is valid before the horizon entry. The time at the
maximum expansion ($\theta=\pi$), $t_{\text{max}}$, and the collapsing time
($\theta=2\pi$), $t_{\text{coll}}$, are given by

$\displaystyle t_{\text{coll}}=2t_{\rm max}=\frac{\pi\Omega_{\rm
ent}}{H_{\text{ent}}(\Omega_{\rm ent}-1)^{3/2}}\simeq$
$\displaystyle\frac{\pi}{H_{\text{ent}}\delta_{\text{ent}}^{3/2}}.$ (36)

At the maximum expansion, the radius of the over-dense region,
$R_{\text{max}}$, becomes

$R_{\text{max}}\simeq\frac{1}{H_{\text{ent}}\delta_{\text{ent}}}=\frac{R_{\text{ent}}}{\delta_{\text{ent}}}.$
(37)

At this moment, the energy of the over-dense region can be parametrized as

$E=-h\frac{M^{2}}{R_{\text{max}}}$ (38)

with $M=1/(2H_{\text{ent}})$ being the horizon mass at the horizon entry and
$h$ is an $\mathcal{O}(1)$ dimensionless parameter. For a uniform ball,
$h=3/5$.

###  A.2 Virialization

After the collapse, if a black hole does not form, the collapsed region is
expected to be virialized in a time scale comparable to the collapsing time,

$\displaystyle t_{\text{vir}}=st_{\text{coll}},$ (39)

with $s=\mathcal{O}(1)>1$ being another dimensionless parameter. Let us
consider the generation process of the velocity dispersion and how it can be
shared in the whole region of the scale $\tilde{k}$. As is stated in Sec. 2,
here we consider two possible scenarios depending on whether the virialization
of the scale $k$ can be realized before $t=\tilde{t}_{\rm coll}$ or not. The
condition for the realization of the virialization is given by

$t_{\rm vir}=st_{\rm coll}<\tilde{t}_{\rm coll}.$ (40)

By using Eq. (36), the inequality can be rewritten as

$\displaystyle\qquad sH_{\rm ent}^{-1}\delta_{\rm ent}^{-3/2}<\tilde{H}_{\rm
ent}^{-1}\tilde{\delta}_{\rm ent}^{-3/2}$ $\displaystyle\Leftrightarrow$
$\displaystyle\qquad sk^{-3}\delta_{\rm
ent}^{-3/2}<\tilde{k}^{-3}\tilde{\delta}_{\rm ent}^{-3/2}$
$\displaystyle\Leftrightarrow$
$\displaystyle\qquad\frac{k^{2}}{\tilde{k}^{2}}\frac{\delta_{\rm
ent}}{\tilde{\delta}_{\rm ent}}>s^{2/3}=\mathcal{O}(1),$ (41)

where we have used the relation $\tilde{H}_{\rm ent}/H_{\rm
ent}=\tilde{k}^{3}/k^{3}$. Hereafter we individually discuss the two cases
$t_{\rm vir}<\tilde{t}_{\rm coll}$ (case I) and $t_{\rm vir}>\tilde{t}_{\rm
coll}$ (case II) in subsections A.3 and A.4, respectively.

###  A.3 Case I: Generation through the Virialization

Here, we consider the case $t_{\rm vir}=st_{\rm coll}<\tilde{t}_{\rm coll}$,
namely, the condition (41) is satisfied. The virial theorem states that the
kinetic energy $K$ and the potential energy $U$ satisfies
$K=-U/2=h{M^{2}}/{R_{\rm vir}}$, where $R_{\rm vir}$ is the radius of the
virialized halo. Since the total energy is conserved and is given by
$-h{M^{2}}/{R_{\rm max}}$ at the maximum expansion, we obtain

$E=K+U=-K=-\frac{1}{2}M\sigma_{v}^{2}=-h\frac{M^{2}}{R_{\rm max}}.$ (42)

From this equation, we find

$\sigma_{v}^{2}=2h\frac{M}{R_{\rm max}}=h\frac{2M}{R_{\rm ent}}\frac{R_{\rm
ent}}{R_{\rm max}}\simeq h\delta_{\rm ent},$ (43)

where we used $2M=R_{\rm ent}$ and Eq. (37). In addition, from $E=U/2=U_{\rm
max}$, we find $R_{\rm vir}=R_{\rm max}/2$, or equivalently,

$\rho_{\rm vir}=8\rho_{\rm max}\simeq 8\rho_{\rm ent}\delta_{\rm ent}^{3}.$
(44)

The virialized halos behave as macroscopic particles until the mean density of
the scale $\tilde{k}$ becomes comparable to the virial density $\rho_{\rm
vir}$. Thus, we identify the time $t=t_{*}$ by $\tilde{\rho}(t_{*})=\rho_{\rm
vir}=8\rho_{\rm ent}\delta_{\rm ent}^{3}$. This equation can be rewritten as

$2\delta_{\rm ent}=\left(\frac{\tilde{\rho}(t_{*})}{\rho_{\rm
ent}}\right)^{1/3}=\frac{\tilde{R}(t_{\rm
ent})}{\tilde{R}(t_{*})}=\frac{\tilde{R}_{\rm
ent}}{\tilde{R}(t_{*})}\frac{a_{\rm ent}}{\tilde{a}_{\rm
ent}}=\frac{\tilde{R}_{\rm
ent}}{\tilde{R}(t_{*})}\frac{\tilde{k}^{2}}{k^{2}}.$ (45)

From this equation and $\sigma_{v}^{2}(t_{*})\simeq h\delta_{\rm ent}$, we
obtain

$\sigma_{v}^{2}(t_{*})\tilde{R}^{2}(t_{*})\simeq\frac{h}{4\delta_{\rm
ent}}\frac{\tilde{k}^{4}}{k^{4}}\tilde{R}_{\rm ent}^{2}.$ (46)

Then, the condition (9) can be reduced to

$\delta_{\rm ent}<\frac{h}{9}\frac{\tilde{k}^{4}}{k^{4}}.$ (47)

At first glance, the last inequality seems counterintuitive because we need a
smaller value of $\delta_{\rm ent}$ to obtain the significant effect of the
velocity dispersion originating from the smaller scale perturbations. However,
this inequality indeed makes sense as follows. Since the virial radius is
given by $R_{\rm vir}=R_{\rm ent}/({2\delta_{\rm ent}})$, the larger
$\delta_{\rm ent}$ is, the smaller the size of the virialized halos becomes.
Therefore, under the condition that the virialization is realized before
$t=\tilde{t}_{\rm coll}$, if the value of $\delta_{\rm ent}$ is larger than
the critical value, the virialized halos are too small and the halo
dissolution is too late to prevent PBH formation.

###  A.4 Case II: Direct Generation from nonlinear Perturbations

Here we consider the case $t_{\rm vir}=st_{\rm coll}>\tilde{t}_{\rm coll}$,
namely, the following inequality is satisfied:

$\frac{k^{2}}{\tilde{k}^{2}}\frac{\delta_{\rm ent}}{\tilde{\delta}_{\rm
ent}}>s^{2/3}=\mathcal{O}(1).$ (48)

Even in this case, the growth of the smaller-scale perturbations is promoted
as the contraction of the scale $\tilde{k}$ proceeds. Then, the smaller-scale
perturbations may get into the nonlinear regime and produce caustics, and
velocity dispersion is generated before the collapse of the scale $\tilde{k}$.
We assume that the velocity dispersion is shared in the whole region of the
scale $\tilde{k}$ at the same time $t_{*}$ of its generation. To
quantitatively evaluate the value of the velocity dispersion, let us consider
the linear-perturbation equation in the background closed universe.

####  A.4.1 Density perturbations in the closed FLRW universe

In the comoving gauge, the equation for the density perturbations of the scale
$\tilde{k}$ in the closed universe is given by (see, e.g., Ref. [102])

$(1-\cos\theta)\frac{\mathrm{d}^{2}}{\mathrm{d}\theta^{2}}\delta_{\rm
co}+\sin\theta\frac{\mathrm{d}}{\mathrm{d}\theta}\delta_{\rm co}-3\delta_{\rm
co}=0.$ (49)

The growing-mode solution is given as follows (see, e.g., Ref. [102]):

$\delta_{\rm co}=\frac{\delta_{\rm
co,max}}{2}\left(\frac{5+\cos\theta}{1-\cos\theta}-\frac{3\theta\sin\theta}{(1-\cos\theta)^{2}}\right)=\delta_{\rm
co,max}\left(\frac{\theta^{2}}{20}+\mathcal{O}(\theta^{4})\right),$ (50)

where $\delta_{\rm co,max}=\delta_{\rm co}(\pi)$.

In order to relate the evolution of the perturbation $\delta_{\rm co}$ to the
density perturbation $\delta$ defined in the flat FLRW model with the uniform
Hubble slice, we need to clarify the relation between the values of
$\delta_{\rm ent}$ and $\delta_{\rm co,max}$. For this purpose, let us
consider the value of $\delta_{\rm co}$ at the horizon entry. The horizon
entry of the scale $k$ is earlier than that of the scale $\tilde{k}$, so the
EdS approximation is valid at the horizon entry time as is discussed in Sec.
A.1. Then, we obtain

$\displaystyle a_{\rm ent}H_{\rm ent}=k,$ (51) $\displaystyle H_{\rm
ent}=\tilde{H}_{\rm ent}\frac{k^{3}}{\tilde{k}^{3}}=\frac{2}{3t_{\rm ent}}.$
(52)

From the second equation, we find

$\displaystyle t_{\rm ent}$ $\displaystyle=$
$\displaystyle\frac{2}{3\tilde{H}_{\rm ent}}\frac{\tilde{k}^{3}}{k^{3}}.$ (53)

Let us express $t_{\rm ent}$ by using the conformal time from Eq. (35) as
follows:

$t_{\rm ent}\simeq\frac{1}{12\tilde{H}_{\rm ent}}\left(\tilde{H}_{\rm
ent}\tilde{a}_{\rm ent}\eta_{\rm ent}\right)^{3},$ (54)

where we have used the approximation $\tilde{\Omega}_{\rm ent}\simeq 1$. Then
we obtain

$\eta_{\rm ent}=2\frac{\tilde{k}}{k}\frac{1}{\tilde{a}_{\rm ent}\tilde{H}_{\rm
ent}}.$ (55)

From Eq. (50), we can find

$\delta_{\rm co,ent}\simeq\tilde{\mathcal{K}}\eta_{\rm
ent}^{2}\frac{\delta_{\rm co,max}}{20}=\frac{1}{5}(\tilde{\Omega}_{\rm
ent}-1)\frac{\tilde{k}^{2}}{k^{2}}\delta_{\rm
co,max}=\frac{1}{5}\frac{\tilde{k}^{2}}{k^{2}}\tilde{\delta}_{\rm
ent}\delta_{\rm co,max}.$ (56)

Rewriting $\delta_{\rm co,max}$ by using $\delta_{\rm co,ent}$, we obtain

$\delta_{\rm co}=\frac{5}{2}\frac{k^{2}}{\tilde{k}^{2}}\frac{\delta_{\rm
co,ent}}{\tilde{\delta}_{\rm
ent}}\left(\frac{5+\cos\theta}{1-\cos\theta}-\frac{3\theta\sin\theta}{(1-\cos\theta)^{2}}\right).$
(57)

It is known that the density perturbations $\delta_{\rm co}$ in the comoving
gauge and $\delta$ in the uniform Hubble gauge is related to each other as
$\delta_{\rm co}\simeq({3}/{5})\delta$ on super-horizon scales (see, e.g.,
Ref. [63]). Then, we extrapolate this relation to the horizon entry, that is,
we use the relation $\delta_{\rm co,ent}\simeq({3}/{5})\delta_{\rm ent}$. By
using this relation, we finally obtain the following expression:

$\delta_{\rm co}=\frac{3}{2}\frac{k^{2}}{\tilde{k}^{2}}\frac{\delta_{\rm
ent}}{\tilde{\delta}_{\rm
ent}}\left(\frac{5+\cos\theta}{1-\cos\theta}-\frac{3\theta\sin\theta}{(1-\cos\theta)^{2}}\right).$
(58)

####  A.4.2 Estimation of the velocity dispersion

To evaluate the velocity dispersion, we consider the velocity perturbation in
the comoving gauge. In the comoving gauge, the velocity potential $\phi_{v}$
is related to the density perturbation as [102]

$\phi_{v}=-(\triangle+3\tilde{\mathcal{K}})^{-1}\frac{\mathrm{d}}{\mathrm{d}\eta}\delta_{\rm
co},$ (59)

where $\triangle$ is the Laplacian. The spatial curvature contribution can be
estimated as $\tilde{\mathcal{K}}=\tilde{\delta}_{\rm ent}\tilde{k}^{2}\ll
k^{2}$, and the contribution is negligible compared to the Laplacian term.
Then, we have

$\phi_{v}\simeq k^{-2}\frac{\mathrm{d}}{\mathrm{d}\eta}{\delta}_{\rm co}.$
(60)

The absolute value $v$ of the velocity can be estimated as follows:

$v\simeq{k}\phi_{v}\simeq k^{-1}\frac{d}{d\eta}{\delta}_{\rm co}.$ (61)

From the following equations

$\displaystyle\tilde{R}(\theta)$ $\displaystyle=$ $\displaystyle\tilde{R}_{\rm
ent}\frac{\tilde{a}}{\tilde{a}_{\rm ent}}=\frac{\tilde{R}_{\rm
ent}}{2}\frac{\tilde{\Omega}_{\rm ent}}{\tilde{\Omega}_{\rm
ent}-1}(1-\cos\theta),$ (62) $\displaystyle v(\theta)$ $\displaystyle\simeq$
$\displaystyle\frac{\frac{\mathrm{d}}{\mathrm{d}\eta}{\delta}_{\rm
co}}{k}=\frac{9}{2}\frac{k}{\tilde{k}}\frac{1}{\sqrt{\tilde{\Omega}_{\rm
ent}-1}}\delta_{\rm
ent}\frac{2\theta+\theta\cos\theta-3\sin\theta}{(1-\cos\theta)^{2}},$ (63)

we obtain

$\frac{\tilde{R}(\theta)}{\tilde{R}_{\rm
ent}}v(\theta)\simeq\frac{9}{4}\frac{k}{\tilde{k}}\tilde{\delta}_{\rm
ent}^{-3/2}\delta_{\rm
ent}\frac{2\theta+\theta\cos\theta-3\sin\theta}{1-\cos\theta},$ (64)

where we have used $\tilde{\Omega}_{\rm ent}-1=\tilde{\delta}_{\rm ent}$.

Let us suppose that the linear perturbations continue to grow until they enter
the nonlinear regime, and caustics appear. Assuming this happens around when
$\delta_{\rm co}\sim 1$, we adopt the value of $v$ at the time $t=t_{*}$
defined by $\delta_{\rm co}=1$ for the value of $\sigma_{v}(t_{*})$. Letting
$\theta_{*}$ denote the value of $\theta$ at $t=t_{*}$, we find

$1=\delta_{\rm
co}(\theta_{*})=\frac{3}{2}\frac{k^{2}}{\tilde{k}^{2}}\frac{\delta_{\rm
ent}}{\tilde{\delta}_{\rm
ent}}\frac{5-4\cos\theta_{*}-\cos^{2}\theta_{*}-3\theta_{*}\sin\theta_{*}}{(1-\cos\theta_{*})^{2}}.$
(65)

From the inequality (48), we can find $\theta_{*}\gtrsim 2.2$. Arranging the
equation for $\theta_{*}$, we can derive the following equation:

$(1-\cos\theta_{*})^{-1}=2^{2/3}3^{-2/3}\tilde{\delta}_{\rm
ent}^{2/3}\left(\frac{k}{\tilde{k}}\right)^{-4/3}\delta_{\rm
ent}^{-2/3}\left(\frac{5-4\cos\theta_{*}-\cos^{2}\theta_{*}-3\theta_{*}\sin\theta_{*}}{\sqrt{1-\cos\theta_{*}}}\right)^{-2/3}.$
(66)

Substituting this equation into Eq. (64), we obtain

$\displaystyle\frac{\tilde{R}(\theta_{*})}{\tilde{R}_{\rm
ent}}\sigma_{v}(\theta_{*})\simeq\frac{\tilde{R}(\theta_{*})}{\tilde{R}_{\rm
ent}}v(\theta_{*})\simeq
3^{4/3}2^{-4/3}\left(\frac{k}{\tilde{k}}\right)^{-1/3}\tilde{\delta}_{\rm
ent}^{-5/6}\delta_{\rm ent}^{1/3}\Xi(\theta_{*}),$ (67)

where

$\displaystyle\Xi(\theta_{*}):=(2\theta_{*}+\theta_{*}\cos\theta_{*}-3\sin\theta_{*})\left(\frac{5-4\cos\theta_{*}-\cos^{2}\theta_{*}-3\theta_{*}\sin\theta_{*}}{\sqrt{1-\cos\theta_{*}}}\right)^{-2/3}.$
(68)

Because $\Xi(\theta_{*})$ is a monotonic function of $\theta_{*}$ satisfying
$\Xi(2.2)\simeq 0.56$ and $\Xi(2\pi)=(3\pi)^{1/3}\simeq 2.1$, we can treat
$\Xi(\theta_{*})$ as a factor of order 1. Then the PBH non-formation condition
(9) can be rewritten as

$b\frac{\tilde{k}}{k}\tilde{\delta}_{\rm ent}^{-5/2}\delta_{\rm ent}>1,$ (69)

where

$b:=\frac{3}{2}\Xi^{3}(\theta_{*})=\mathcal{O}(1).$ (70)

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# LOCV calculations for polarized liquid ${}^{3}\mathrm{He}$ with the spin-
dependent correlation

G.H. Bordbar111Corresponding author 222E-Mail<EMAIL_ADDRESS>and
M.J. Karimi  Department of Physics, Shiraz University, Shiraz 71454, Iran

###### Abstract

We have used the lowest order constrained variational (LOCV) method to
calculate some ground state properties of polarized liquid ${}^{3}He$ at zero
temperature with the spin-dependent correlation function employing the
Lennard-Jones and Aziz pair potentials. We have seen that the total energy of
polarized liquid ${}^{3}He$ increases by increasing polarization. For all
polarizations, it is shown that the total energy in the spin-dependent case is
lower than the spin-independent case. We have seen that the difference between
the energies of spin-dependent and spin-independent cases decreases by
increasing polarization. We have shown that the main contribution of the
potential energy comes from the spin-triplet state.

Keywords: Liquid ${}^{3}He$, Spin polarized, Correlation, Spin-dependent

##  I Introduction

Helium has two stable isotopes: that of mass 4 is readily available as helium
gas or liquid from the atmosphere or gas wells, while that of mass 3 is
extremely rare in nature and only became available commercially in the 1950s
Dob . Liquid ${}^{3}He$ is particularly suited to study correlation among the
strongly interacting many-body fermionic systems. Several approaches have been
used for investigating the properties of normal liquid ${}^{3}He$. These are
mainly based on the STLS scheme Nd , mott localization Gl , spin fluctuation
theory Ms , Green’s function Monte Carlo (GFMC) GFMCa , FN-DMC, DMC, VMC and
EMC simulations DVEF , CBPT formalism FPS , nonperturbative renormalization
group equation KW , nonlocal density functional formalism PT  , correlated
basis functions (CBF) CBFa  and Fermi hyper-netted chain (FHNC) FHNCa . The
spin polarized liquid ${}^{3}He$ as an interesting many-body system has been
investigated using different approaches such as FHNC FHNCb , GFMC GFMCb , CBF
CBFb  and transport theory Trans .

In recent years, we have studied both normal and polarized liquid ${}^{3}He$
at zero and finite temperature Gha1 ; Gha2 ; Gha3 ; Gha4 . In these
calculations, the lowest order constrained variational (LOCV) method based on
the cluster expansion of the energy functional has been used. This method is
fully self-consistent, since it does not introduce any free parameter to the
calculations. We have also used the LOCV method in many-body calculations of
dense matter Ghb . Recently, we have used this method to calculate some
properties of the polarized neutron matter and the polarized symmetrical and
asymmetrical nuclear matters Ghc . In these works, a comparison of our results
and those of other many-body techniques indicates that the LOCV method is a
powerful microscopic technique to calculate the properties of the polarized
matter.

In this work, we use the LOCV method to compute the ground state energy of the
polarized liquid ${}^{3}He$ at zero temperature by employing the spin-
dependent correlation function with the Lennard-Jones LJ  and Aziz Aziz1 ;
Aziz2  pair potentials.

##  II Lowest Order Constrained Variational Method

We consider a system of $N$ interacting ${}^{3}He$ atoms with $N^{+}$ spin up
and $N^{-}$ spin down atoms. The total number density ($\rho$) and spin
asymmetry parameter ($\xi$) are defined as

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho^{+}+\rho^{-},$
$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\frac{N^{+}-N^{-}}{N}.$ (1)

$\xi$ shows the spin ordering of matter which can have a value in the range of
$\xi=0.0$ (unpolarized matter) to $\xi=1.0$ (fully polarized matter). For this
system, we consider the energy per particle up to the two-body term in the
cluster expansion,

$E=E_{1}+E_{2},$ (2)

where

$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\frac{3}{10}\frac{\hbar^{2}}{m}(3\pi^{2}\rho)^{\frac{2}{3}}[(1+\xi)^{\frac{5}{3}}+(1-\xi)^{\frac{5}{3}}],$
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{i,j}\langle ij\mid w(12)\mid ij-ji\rangle.$
(3)

In the above equation, $w(12)$ is the effective pair potential,

$w(12)=-\frac{\hbar^{2}}{2m}[F(12),[\nabla_{12}^{2},F(12)]]+F(12)V(12)F(12),$
(4)

where $F(12)$ is the two-body correlation operator and $V(12)$ is the pair
potential between the helium atoms. In our calculations, we use the Lennard-
Jones LJ  and Aziz Aziz1 ; Aziz2  pair potentials. The Lennard-Jones pair
potential is as follows LJ ,

$V(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right],$
(5)

where

$\epsilon=10.22K,\hskip 56.9055pt\sigma=2.556A\cdot$ (6)

The Aziz pair potential has the following form Aziz1 ; Aziz2 ,

$V(r)=\epsilon\left\\{Ae^{-\alpha
r/r_{m}}-\left[C_{6}\left(\frac{r_{m}}{r}\right)^{6}+C_{8}\left(\frac{r_{m}}{r}\right)^{8}+C_{10}\left(\frac{r_{m}}{r}\right)^{10}\right]F(r)\right\\},$
(7)

where

$\displaystyle F(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{tabular}[]{lll}$e^{-(\frac{Dr_{m}}{r}-1)^{2}}$&;&$\frac{r}{r_{m}}\leq
D$\\\ 1&;&$\frac{r}{r_{m}}>D$,\end{tabular}\right.$ (10)

and

(15)

Now, we consider a spin-dependent correlation function as follows

$F(12)=f_{0}(r_{12})P_{0}+f_{1}(r_{12})P_{1},$ (16)

where

$\displaystyle P_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\sigma_{1}.\sigma_{2}),$ $\displaystyle P_{1}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(3+\sigma_{1}.\sigma_{2}).$ (17)

$f_{0}$ and $f_{1}$ indicate the spin-singlet and spin-triplet two-body
correlation functions, respectively. With the above two-body correlation
function, we have derived the following relation for the effective pair
potential,

$w_{s}(r)=\frac{\hbar^{2}}{m}({f_{s}{{}^{\prime}}}(r))^{2}+f_{s}^{2}(r)V(r),$
(18)

and then the two-body energy $E_{2}$ is found by

$E_{2}=2\pi\rho\sum_{s=0,1}\int_{0}^{\infty}drr^{2}w_{s}(r)a_{s}.$ (19)

In Eq. (19)

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\xi^{2})[1+l(k_{F^{+}}r)l(k_{F^{-}}r)],$
$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[(1+\xi)^{2}(1-l^{2}(k_{F^{+}}r))+(1-\xi)^{2}(1-l^{2}(k_{F^{-}}r))$
(20) $\displaystyle+(1-\xi^{2})(1-l(k_{F^{+}}r)l(k_{F^{-}}r))].$

$k_{F^{\pm}}=(6\pi^{2}\rho^{\pm})^{\frac{1}{3}}$ is the Fermi momentum and
$l(x)$ is given by

$l(x)=\frac{3}{x^{3}}[\sin(x)-x\cos(x)].$ (21)

Now, we minimize the two-body energy Eq. (19) with respect to the variations
in the two-body correlation function subject to the normalization constraint
Clark ,

$\frac{1}{N}\sum_{i,j}\langle ij|h^{2}(12)-F^{2}(12)|ij-ji\rangle=1.$ (22)

The normalization constraint is conveniently re-written in the integral form
as

$4\pi\rho\sum_{s=0,1}\int_{0}^{\infty}drr^{2}[h^{2}(r)-f_{s}^{2}(r)]a_{s}=1,$
(23)

where the Pauli function $h(r)$ is

$h(r)=\\{1-\frac{1}{4}[(1+\xi)^{2}l^{2}(k_{F^{+}}r)+(1-\xi)^{2}l^{2}(k_{F^{-}}r)]\\}^{-\frac{1}{2}}.$
(24)

The minimization of the two body energy $E_{2}$ gives the following Euler-
Lagrange differential equation for the two-body correlation function
$f_{s}(r)$,

$f_{s}^{{}^{\prime\prime}}(r)+(\frac{2}{r}+\frac{a_{s}^{{}^{\prime}}}{a_{s}})f_{s}^{{}^{\prime}}(r)-\frac{m}{\hbar^{2}}\left(V(r)-2\lambda\right)=0.$
(25)

The Lagrange multiplier $\lambda$ imposes by normalization constraint. For
$s=0$ and $s=1$ states, the two-body correlation function $f_{s}(r)$ is
obtained by numerically integrating Eq. (25). Using this two-body correlation
function we can determine the effective pair potential $w_{s}(r)$ as a
function of interatomic distance from Eq. (18). Finally, the two-body energy
$E_{2}$ and the total energy of system can be calculated.

##  III Results and Discussion

We have calculated some ground state properties of the polarized liquid
${}^{3}He$ at zero temperature with the Lennard-Jones LJ  and Aziz Aziz1 ;
Aziz2  pair potentials using the spin-dependent correlation function. Our
results are as follows.

The spin-dependent two-body correlation functions at $s=0$ state and $s=1$
states for different values of spin asymmetry parameter ($\xi$) are shown in
Fig. 1. These figures show that the correlation function at $s=1$ state heals
to pauli function, $h(r)$, more rapidly than $s=0$ state. Therefore, the $s=1$
state has a shorter correlation length with respect to $s=0$ state. For large
values of $r$, $f_{0}(r)$ and $f_{1}(r)$ have the same values and therefore,
the spin-dependent part of correlation operator (Eq. 16) is vanished. In these
figures, the spin-independent two-body correlation function are also plotted
for comparison. It is seen that the spin-dependent correlation function
differs from spin-independent correlation function, except for the fully
polarized matter ($\xi=1.0$). This is due to the fact that for fully polarized
matter, there is only $s=1$ state. From Fig. 1, we can see that the
correlation functions with the Lennard-Jones and Aziz potentials are nearly
identical.

In Fig. 2, we have shown the total energy of polarized liquid ${}^{3}He$
versus number density calculated both with the spin-dependent correlation and
the spin-independent correlation at different values of spin asymmetry
parameter $\xi$. We can see that the total energy increases by increasing
$\xi$. Fig. 2 indicates that in the spin-dependent case, the total energy of
the liquid ${}^{3}He$ is lower than the spin-independent case. It is also seen
that for all values of $\xi$, the energy curve has a minimum which shows the
existence of a bound state for this system. It is shown that the difference
between the energies of spin-dependent case and spin-independent case
decreases by increasing $\xi$ and it becomes zero as $\xi$ approaches to one.
It is seen that for all values of the density and spin asymmetry parameter,
the total energy with the Aziz pair potential is grater than that of the
Lennard-Jones pair potential.

The potential energy of the polarized liquid ${}^{3}He$ for different values
of $\xi$ are presented in Fig. 3, for spin-dependent and spin-independent
cases. This figure indicates that the potential energy decreases by increasing
the polarization. According to the above results, we can conclude that the
increasing of kinetic energy dominates and this leads to the increasing of
total energy by increasing $\xi$. Fig. 3 shows that the potential energy in
the spin-dependent case has lower values with respect to the spin-independent
case. It is also seen that the difference between the potential energies of
spin-dependent and spin-independent cases decrease by increasing $\xi$. We see
that the potential energies with the Aziz and the Lennard-Jones pair
potentials are different. This difference increases by increasing the density.

In Fig 4, the potential energies of $s=0$ and $s=1$ states for different
values of $\xi$ are compared. We have seen that the potential energy at $s=1$
state is lower than at $s=0$ state. It can be concluded that the spin-triplet
state has the main contribution in the potential energy of polarized liquid
${}^{3}He$. It is also seen that the potential energy of $s=0$ ($s=1$) state
increases (decreases) by increasing $\xi$. For $s=0$ state, we can see that at
low densities, the potential energies with the Lennard-Jones and Aziz pair
potentials are nearly identical. However, for $s=1$ state and high densities,
the difference between these potential energies becomes appreciable.

The equation of state of polarized liquid ${}^{3}He$, $P(\rho,\xi$), can be
obtained using

$\displaystyle P(\rho,\xi)=\rho^{2}\frac{\partial E(\rho,\xi)}{\partial\rho}$
(26)

In Fig. 5, we have presented the pressure of liquid ${}^{3}He$ as a function
of the density ($\rho$) for fully polarized ($\xi=1.0$) and unpolarized
($\xi=0.0$) cases. This figure shows that for different values of the
polarization, the equations of state of liquid ${}^{3}He$ are nearly
identical. From Fig. 5, it is seen that for both $\xi=1.0$ and $\xi=0.0$, the
equation of state with the Aziz pair potential is stiffer than that of the
Lennard-Jones pair potential.

##  IV Summary and Conclusion

We have considered a system consisting of Helium atoms $(^{3}He)$ with
asymmetrical spin configuration and derived the two-body term in the cluster
expansion of the energy functional by employing spin-dependent correlation
function. Then, we have minimized the two-body energy term under the
normalization constraint and obtained the Euler-Lagrange differential
equation. By numerically solving this differential equation, we have computed
the correlation function and then calculated the other properties of this
system with the Lennard-Jones and Aziz pair potentials. It is shown that for
the two different spin-singlet and spin-triplet states, the correlation
functions are different from each other. Our results show that the
introduction of the spin-dependent term in the correlation operator reduces
the total energy of system by about $10\%$. It is also shown that the total
energy increases by increasing the polarization. The difference between the
energies of the spin-dependent and spin-independent cases decreases by
increasing the polarization. We have seen that, the potential energy of these
states have a remarkable difference. It is shown that the main contribution of
the potential energy comes from $s=1$ state. Our calculations show that there
is a difference between the results with the Lennard-Jones and Aziz pair
potentials, especially at high densities.

###### Acknowledgements.

Financial support from the Shiraz University research council is gratefully
acknowledged.

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Figure 1: The correlation function with the Aziz (dashed curves) and Lennard-
Jones (full curves) pair potentials in the case of spin-dependent at $s=0$ and
$s=1$ states for $\xi=0.0$ (a), $\xi=0.33$ (b), $\xi=0.66$ (c) and $\xi=1.0$
(d). Our results for the spin-independent correlation function are also
presented for comparison.

Figure 2: Our results for the total energy of the polarized liquid ${}^{3}He$
with the Aziz and Lennard-Jones (LJ) pair potentials in the case of spin-
dependent (full curve) and spin-independent (dotted curve) correlation
functions for $\xi=0.0$ (a), $\xi=0.33$ (b), $\xi=0.66$ (c) and $\xi=1.0$ (d).

Figure 3: As Fig. 2, but for the potential energy of the polarized liquid
${}^{3}He$.

Figure 4: Our results for the potential energy of the polarized liquid
${}^{3}He$ with the Aziz and Lennard-Jones (LJ) pair potentials at $s=0$
(dotted curve) and $s=1$ (full curve) states for $\xi=0.0$ (a), $\xi=0.33$
(b), $\xi=0.66$ (c) and $\xi=1.0$ (d).  Figure 5: The equation of state of the
fully polarized (dotted curve) and unpolarized (full curve) liquid ${}^{3}He$
with the Aziz and Lennard-Jones (LJ) pair potentials.