xx 2018 xxx xx March 2018

# Capture of dark matter particles by a galaxy in the case of a bimodal
distribution of their velocities

Ruth Durrer1  Serge Parnovsky2,3 and Aleksei Parnowski4  1Département de
Physique Théorique and Centre for Astroparticle Physics, Université de Genève
24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland  2Taras Shevchenko
National University of Kyiv, Astronomical observatory, Observatorna str. 3,
Kyiv 04053, Ukraine  3CERN 1211 Geneva 23. Switzerland  4Space Research
Institute of National Academy of Sciences of Ukraine and State Space Agency of
Ukraine, prospekt Akademika Glushkova 40, build. 4/1, 03187 Kyiv, Ukraine

(xx August 2022)

###### Abstract

We have analyzed the rate of capture of dark matter (DM) particles by the
galaxy in the case of the existence of two different types of DM or a bimodal
velocity distribution function for DM. It is shown that, in addition to the
scenario considered in our previous work which is based on the assumption of
an unimodal distribution, more complex scenarios are possible in which the
transition to the state of intense capture and/or exit from it can occur in
two stages. A detailed description is given of the change in the curve
describing the rate of capture of dark matter particles as a function of the
rate of increase in the baryon mass of the galaxy for various values of the
rate of decrease of the DM density.

###### keywords:

Dark mattergalaxy formation 10.2018/LHEP000001

††journal: LHEP

##  1 Introduction

From the analysis of galaxy rotation curves, it is known that dark matter (DM)
is responsible for 85% of the mass of a typical galaxy. It forms a dark halo
that surrounds the galaxy and extends far beyond the limits of its visible
part, consisting of baryonic matter (BM). But galaxies were formed after
recombination as a result of the growth of density fluctuations. Therefore, a
natural question arises as to where this dark matter came from.

We have proposed our own answer to this question in article DP . The main
mechanism of the increase in the DM mass is the capture of DM particles flying
into the halo from intergalactic space. However, without BM accretion by the
galaxy, they would fly through the galaxy without being captured. If the
baryonic mass of the galaxy increases due to accretion of BM or the merger of
galaxies (see the details of galaxy formation in b ), then some of the passing
DM particles cannot escape due to a lack of kinetic energy. In this case, the
depth of the potential well has time to increase during the time the particle
passes through the galaxy. Correspondingly, the height of the potential
barrier, caused by the gravitational interaction of the particle with the
galaxy, also increases.

This is determined by the total mass of the galaxy $M$, which can increase due
to the accretion of BM and the capture of DM. Accordingly, the rate of its
increase $\dot{M}$ is equal to the sum of the rates of increase in DM,
designated as $\dot{M}_{DM}$, and the rate of increase in BM, designated as
$\dot{M}_{b}$:

$\dot{M}=\dot{M}_{DM}+\dot{M}_{b}\,.$ (1)

We consider $\dot{M}_{b}$ as an external quantity that can change with time,
and we try to calculate $\dot{M}$ as function of $\dot{M}_{b}$. We assume that
the mass of the galaxy $M$ is known at the moment of time under consideration
(assuming that it does not change significantly during the passage of the DM
particle, which cannot be captured due to its rather high initial velocity
$v_{0}$ far away from the galaxy). We also know the halo radius $R$, which we
consider to be constant, and the parameters of dark matter particles in
intergalactic space.

They include the particle mass $m$, particle density $N$, and the particle
velocity distribution. The last two quantities are determined by the
distribution function $f(v_{0})$. Let the number of particles with velocities
in the range from $v_{0}$ to $v_{0}+dv_{0}$ in a unit volume in extragalactic
space is equal to $dN=f(v_{0})dv_{0}$. So, the total cdensity of DM particles
is $N=\int_{0}^{\infty}f(v_{0})dv_{0}$.

We consider DM particles to be non-relativistic, and we use the formulas of
classical mechanics. Actually, nothing prevents us from taking into account
effects of Special Relativity, but this will not lead to a qualitative change
in the results of our model. We assume that the rest mass of the DM particle
is large enough not to take into account its wave properties, but otherwise
arbitrary.

We take into account the Hubble expansion to account for changes in the DM
density in intergalactic space. $N$ decreases rapidly with time because of
both Hubble expansion and the capture of a part of the DM particles by
galaxies. The first of these effects is stronger and ensures a dependence
according to the law $N\propto(1+z)^{3}$. The second makes the decrease in $N$
even faster. Other parameters of the galaxy like its mass, $M$ or of the
velocity distribution of DM, $f(v_{0})/N$ change much more slowly with time.

The model described in the article DP  assumes that the halo is spherically
symmetrical and homogeneous, i.e. has no inhomogeneities, on which the DM
particles could perform a gravitational maneuver. Therefore, particles moving
radially can either be captured at $v_{0}<v_{pc}$ or fly through the galaxy at
$v_{0}>v_{pc}$ (the subscript $pc$ indicates passage through the center). The
estimate detailed in DP  gives

$v_{pc}\simeq a\sqrt{\dot{M}},\quad a=\left(\frac{2GR}{M}\right)^{1/4}\,.$ (2)

A non-linear equation

$\dot{M}_{DM}\simeq\pi
R^{2}\int_{0}^{v_{pc}}\left(1-\frac{v_{0}^{4}}{v_{pc}^{4}}\right)mf(v_{0})v_{0}dv_{0}\,$
(3)

was also obtained in the article DP . From the analysis of eqs. (1-3) we
obtained a qualitative picture of the process of capture of DM particles and
of the accumulation of DM mass in the galaxy. In this article, we will
continue the analysis, considering additionally some more exotic variants.

These options arise due to the fact that the equation (3) includes an unknown
distribution function $f(v_{0})$. We have no observational data to establish
its type or convincing theories to predict it. We do not know if DM particles
are in a state of thermal equilibrium. It is only known that dark matter has
characteristic velocities much smaller than the speed of light and belongs to
the so-called cold or warm dark matter.

Nevertheless, one can make a number of reasonable assumptions about the form
of $f(v_{0})$. We suppose that the function $f(v_{0})\geq 0$ is continuous,
vanishes at $v_{0}=0$, reaches a maximum at some value $v_{0}=v_{\max}$, and
quickly decreases to zero at high-speed end, most likely exponentially in
$v_{0}^{2}$. At small $v_{0}$ we can put

$f(v_{0})\approx a_{1}v_{0}^{2}$ (4)

due to the three independent Cartesian velocity components. The coefficient
$a_{1}\propto N$.

This is enough to analyze the solution of the system (1-3) and derive a number
of qualitative conclusions. They are based primarily on the shape of the curve
$\dot{M}(\dot{M}_{b})$. We immediately note that this form is constantly
changing due to the time dependence of the parameters included in equations
(2) and (3), primarily $N$.

In this case, as we have found in DP , a qualitative change in the shape of
the curve occurs. When $N$ is less than some critical value $N_{1}$, the curve
increases monotonically. In this case, the capture of dark matter occurs
weakly and only at $\dot{M}_{b}>0$. At $N>N_{1}$ the curve acquires a
characteristic s-shape. In this case it consists of three branches. Two of
them have a positive derivative and correspond to stable solutions. They are
connected by a section with a negative derivative corresponding to unstable
solutions. The behavior of the solution in this case is well studied in
catastrophe theory. It is well described in many books including ar  and s .
This shape is called the fold catastrophe.

The solution $\dot{M}_{b}=\dot{M}=0$ is stable, as is any point on the stable
branches. Therefore, in the absence of accretion, there is no capture of DM
particles on the lower branch. But for $\dot{M}_{b}>0$ we have
$\dot{M}_{DM}>0$. So capture in this case is a consequence of synergy, i.e.
combined action of BM accretion and passage of DM particles. If the system is
on the lower stable branch and an external influence, i.e. the value of
$\dot{M}_{b}$ increases so much that it reaches the right boundary of the
lower branch (we will refer to it as point B) and continues to increase, then
the state of the system jumps to the upper stable branch and remains on it
even after decreasing $\dot{M}_{b}$.

The upper branch describes the galaxy in a state of intense capture of DM
particles. In this case, the ratio between the growth rates of the dark and
baryon masses can be very large. This explains the high abundance of dark
matter in galaxies. In our opinion, the galaxies captured almost all of their
DM in the state of intense capture. In DP  the mass of dark matter that could
be captured by the Galaxy was estimated if the process of intense capture
began at the redshift parameter $z_{0}$. It turned out that the proposed
mechanism can ensure the capture of all dark matter in the Galaxy at
$z_{0}=15...25$. The process of intense capture could have begun even before
the appearance of the first stars and reionization.

The upper stable branch has a left boundary, i.e. the minimum value
$\dot{M}_{b}$. We will refer to it as the point C. As $\dot{M}_{b}$ decreases,
the system located on the upper branch can reach point C and, as $\dot{M}_{b}$
decreases further, jump to the lower branch. In this case, the galaxy leaves
the stage of intense capture. Note that this is possible only if
$\dot{M}_{b}>0$ in point C. As $N$ decreases, the curve changes so that point
C shifts to the right, crossing at some moment of time the y-axis. Only after
this crossing is it possible to end the stage of intensive capture with a
decrease in $\dot{M}_{b}$.

Something like a phase transition occurs at $N=N_{1}$ with a change in the
regime of particle capture. But by this time, the intensive capture mode will
most likely have had time to end.

We can illustrate all of the above with the example of the Maxwell-Boltzmann
distribution

$f(v)=\frac{4Nv^{2}}{\sqrt{\pi}v_{max}^{3}}\exp{\left[-\left(\frac{v}{v_{max}}\right)^{2}\right]},\,v_{max}=\sqrt{\frac{2k\Theta}{m}}$
(5)

with temperature $\Theta$ and maximum at $v=v_{max}$. Fig. 1 shows the
dependence of quantities proportional to $\dot{M}$ and $\dot{M}_{b}$.

$x=\gamma\dot{M}_{b},\quad
y=\gamma\dot{M},\quad\gamma=\sqrt{\frac{2GR}{M}}v_{\max}^{-2}.$ (6)

All parameters included in (1-5) are reduced to a single parameter

$\mu=\frac{2R^{2}mN}{v_{\max}}\sqrt{\frac{2\pi GR}{M}},$ (7)

which decreases due to the decrease of $N$. In Fig. 1 one can see both the
transition from the s-shaped curve to a monotone one and the intersection of
point C with the y-axis.

Figure 1: Dependence of $x$ and $y$ defined in (6) proportional to the rates
of increase in the total and baryonic masses of the galaxy for the case of the
Maxwell distribution of DM particle velocities. The curves from left to right
correspond to the different values of the parameter $\mu$ from (7) equal to
10, 7, 5 and 4.

We can describe the process of accumulation of dark matter inside the halo of
the galaxy. As a result of the growth of small density fluctuations, regions
of increased density emerge, in which galaxies can form. The surrounding
matter, both baryonic and dark, begins to fall into them. In the absence of
accretion of baryonic matter, DM particles fly through protogalaxies and are
not captured.

But the situation changes with an increase in the mass of the baryonic
component of the galaxy. A fraction of the DM particles passing into the
galaxy at sufficiently low speed is being captured. The mass of dark matter
inside the galaxy begins to grow, but the rate of its increase is smaller than
the rate of increase in the mass of BM.

If the rate of increase in the mass of the BM exceeds a certain threshold
value, corresponding to point B, a jump on the upper branch occurs. The galaxy
goes into the regime of intense capture of DM particles. The system remains on
the upper branch until either the particle density drops below $N_{1}$, or
until the point C crosses the y-axis, and the baryon mass growth rate becomes
less than the abscissa of the point C. The galaxy comes out of the regime of
intense capture of DM particles.

##  2 Exotic examples of DM capture

Figure 2: Unimodal (a) and bimodal (b) functions $H$ associated with the
velocity distribution of dark matter particles.

Let us try to add something to this picture and show that sometimes the
process of transition to the stage of intense capture or exit from it occurs
in a more complicated way. This possibility arises in some cases when the
distribution function $f(v)$ is bimodal. Can such a function be realized? As
already mentioned, we do not know the form of $f(v)$. It is possible that some
processes in the early Universe could lead to a such distribution of DM
particle velocities.

But there is also another possibility. There may be two (or more) different
types of dark matter with different particles and different velocity
distributions, then we can include both in equation (3). For this, instead of
the particle density distribution, we must consider the distribution of their
mass density $mf(v)$ over velocities. It is obtained by summing the
distribution for each particle type. If the properties of different DM
particles are very different, then the velocities corresponding to the modes
of the two distributions are also likely to be different. By summing two
distributions with different modes, we can get a total bimodal distribution.
Therefore, it is worth analyzing the behavior of the curves describing the
dependence of $\dot{M}$ on $\dot{M}_{b}$ for the case of an arbitrary
distribution $f(v)$ that satisfies the formulated conditions.

Note that all points on the curve that are of most interest to us, like points
B and C, have vertical tangents. The same applies to the moment of the phase
transition at $N=N_{1}$, when a point with a vertical tangent appears on the
curve. At these points we have the condition $d\dot{M}/d\dot{M}_{b}=\infty$ or
$d\dot{M}_{b}/d\dot{M}=0$. From (1) we then get
$d\dot{M}_{DM}/d\dot{M}=d\dot{M}_{DM}/dv_{pc}\cdot dv_{pc}/d\dot{M}=1$. From
(3) and (2) we then obtain

$H(v_{pc})=\frac{1}{2\pi R^{2}mN}\left(\frac{M}{2GR}\right)^{1/2}\,,$ (8)

where we have introduced the function $H$ defined by

$H(v_{pc})\equiv N^{-1}v_{pc}^{-6}\int_{0}^{v_{pc}}v_{0}^{5}f(v_{0})dv_{0}.$
(9)

The factor $N^{-1}$ is introduced such that this function does not directly
depend on $N$. It compensates for the fact that $f\propto N$. As a result, the
function $H$ changes with time, but slowly, not as fast as the particle
density $N$.

Figure 3: Characteristic curves of dependence $\dot{M}$ on $\dot{M}_{b}$ at
different values of particle density $N$. a) $N<N_{1}$, line 1 on Fig. 2. b)
$N=N_{1}$, line 2 on Fig. 2. c) S-shaped curve for lines 3 and 7. d) The same
as c) but with a vertical tangent point outside BC segment for line 4. e)
Double s-shaped curve for line 5. f) The same as c) but with a vertical
tangent point inside BC segment for line 6.

Let us consider its properties. The function $H(v_{pc})\geq 0$ and tends to
zero as $v_{pc}\to 0$ because of asymptotic behavior (4). It also tends to
zero as $v_{pc}\to\infty$ because the integral in (9) tends to a finite value,
but the factor $v_{pc}^{-6}$ ensures that it falls according to the law
$H(v_{pc})\xrightarrow[v_{pc}\to\infty]{}O(v_{pc}^{-6})$. This means that it
has a maximum at a certain $v_{pc}=v_{1}$.

Can it have several local maxima with minima in between? At each extremum, the
condition

$\frac{dH(v_{pc})}{dv_{pc}}=N^{-1}v_{pc}^{-7}\left(v_{pc}^{6}f(v_{pc})-6\int_{0}^{v_{pc}}v_{0}^{5}f(v_{0})dv_{0}\right)=0$
(10)

must be satisfied. After integration by parts, it reduces to

$\int_{0}^{v_{pc}}v_{0}^{6}{f^{\prime}(v_{0})}dv_{0}=0,$ (11)

where the prime means the derivative with respect to the argument. If the
function $f(v_{0})$ is bimodal and has two maxima at the points $v_{max1}$ and
$v_{max2}>v_{max1}$ and a local minimum at $v_{min1}$ between them, then
${f^{\prime}}$ is positive on the segments $(0,v_{max1})$,
$(v_{min1},v_{max2})$ and negative on the segments $(v_{max1},v_{min1})$,
$(v_{max2},\infty)$. Condition (11) can be met at some point between
$v_{max1}$ and $v_{min1}$ only if

$\int_{0}^{v_{max1}}v_{0}^{6}{f^{\prime}(v_{0})}dv_{0}<\int_{v_{max1}}^{v_{min1}}v_{0}^{6}{|f^{\prime}(v_{0})|}dv_{0},.$
(12)

If this condition is not satisfied, then there are no extrema of the function
$H$ also on the segment $(v_{min1},v_{max2})$. A simple analysis shows that
$H$ can be bimodal only if $f$ is bimodal or multimodal, and the maxima of $f$
are at a sufficiently large distance from each other. But this variant is
possible in principle. However, the bimodality of $f$ does not guarantee that
$H$ is bimodal.

Figure 4: Different options of the curve in Fig. 3e provide a transition to
the upper and lower branches with a different number of jumps.

Consider the function $H$ at some moment of time. It can be unimodal as in
Fig. 2a or bimodal. On Fig. 2b shows an example of such a function. Of course,
any of the two maxima can be the global maximum at $v_{pc}=v_{1}$. In Fig. 2
it is the left one, which has a smaller value of $v_{pc}$ and hence $\dot{M}$.
But it is easy to construct curves for the case when the right maximum is
higher.

According to (8), the points with vertical tangents on the dependence of
$\dot{M}$ on $\dot{M}_{b}$ correspond to the intersections of the curve
$H(v_{pc})$ with the horizontal line of height given by the right hand side of
the equation (8) which decreases with increasing $N$. We denote

$N_{1}=\frac{1}{2\pi R^{2}mH(v_{1})}\left(\frac{M}{2GR}\right)^{1/2}.$ (13)

At $N<N_{1}$ we have no solution and the points with vertical tangents do not
exist. On Fig. 2 this case corresponds to the horizontal line 1 passing above
the curve. The characteristic form of dependence $\dot{M}$ on $\dot{M}_{b}$
for this case is shown in Fig. 3a.

At $N=N_{1}$ the line 2 touches the curve at its maximum. This value
corresponds to the transition to an s-shaped curve from the monotonic one. The
characteristic form of the curve for this case is shown in Fig. 3b. At
$N>N_{1}$ we have several possibilities. For the line 3 below the maximum (but
above the second mode for the bimodal $H$) we find two solutions of the
equation (8). The solution with smaller $v_{pc}$ corresponds to point B, one
with larger $v_{pc}$ to point C. For small $v_{pc}$ we can use the
approximation (4) and obtain the estimate for coordinates of point B. The
characteristic form of curve for this case is shown in Fig. 3c.

The following lines are only of interest in the case of the bimodal
distribution shown in Fig. 2b. Line 4 touches the top of the lower mode. The
characteristic curve for this case is depicted in Fig. 3d. Line 5 has four
points of intersection with $H(v_{pc})$. The characteristic curve depicted in
Fig. 3e, its shape resembles two letters s one above the other. Line 6 passing
through the local minimum provides the characteristic curve depicted in Fig.
3f. The characteristic curve for line 7 below the local minimum in Fig. 2b
qualitatively the same as in Fig. 3c.

Recall that lines 1-7 correspond to an increase in the particle density $N$,
which corresponds to earlier stages of the evolution of the galaxy. In the
process of this evolution, the curve $\dot{M}$ on $\dot{M}_{b}$ changes its
form, successively acquiring the forms shown in Figures 3c, 3f, 3e, 3d, 3c,
3b, and 3a.

Of all these stages, the most exotic is by far the one whose graph is shown in
Fig. 3e. However, this conditional image corresponds to 4 possible variants of
the mutual arrangement of the abscissas of points with vertical tangents. All
of them are shown in Fig. 4. The curve has three stable branches AB, IH, CE
and two unstable ones BI, HC.

Let us consider the different possibilities one by one. The curve in Fig. 4a
corresponds to the case when the transition to and exit from the stage of
intense capture of DM particles occurs in two jumps BD + HJ and CK + IF. In
Figure 4b, both transitions occur in one jump. The states on the branch IH are
stable, but unattainable for the system. In Fig. 4c the upward jump BD occurs
in one stage, downward in two, Fig. 4d depicts the opposite case.

Let us discuss the question of which direction the dependence curve shifts
during galaxy formation, when the value of $N$ rapidly decays. Consider an
arbitrary point on this curve that has some fixed ordinate, i.e. constant
value $\dot{M}$. From (2) we find the dependence of $v_{pc}$. The value of $a$
does not depend on $N$. It slowly decreases due to the increase in the mass of
the galaxy $M$, but this can be neglected in comparison with the effect of the
rapidly falling particle density $N$. The value of $\dot{M}_{DM}\propto N$
also decreases rapidly. The abscissa of the point $\dot{M}_{b}$ shifts to the
right with decreasing $N$ according to (1). Therefore, in the process of
evolution, the curve $\dot{M}$ vs. $\dot{M}_{b}$ shifts to the right. Fig. 1
illustrates this.

This also applies to the abscissas of the points B, C and H, I, if the last
two are present. But their ordinate also changes. Fig. 2 shows that the value
of $v_{pc}$ and hence also $\dot{M}$ increases for points B and H and
decreases for points C and I as $N$ decreases.

Can the number of stages required to jump to the upper stable branch and to
descend to the lower stable branch change during the evolution? One can try to
answer this question, but it is not so essential for astrophysics.

##  3 Conclusions

We have analyzed the rate of capture of DM particles by the galaxy for two
different types of DM particles or a bimodal velocity distribution function of
DM particles. It is shown that, in addition to the scenario considered in our
previous work which was based on the assumption of unimodality of the velocity
distribution, more complex scenarios are possible in which the transition to
the state of intense capture and/or exit from it occurs in two stages.

A detailed description is given of the change in the curve describing the rate
of capture of dark matter particles as a function of the rate of increase in
the baryon mass of the galaxy for various values of the density of dark matter
which decreases with time, $N$. As $N$ decreases, the curve shifts to the
right, increasing the value of $\dot{M}_{b}$ at a practically constant
$\dot{M}$.

##  4 CONFLICTS OF INTEREST

The authors declare that there are no conflicts of interest regarding the
publication of this paper.

## References

  * [1] R. Durer and S. Parnovsky, arxiv:2208.08843 [astro-ph] (2022).
  * [2] J. Binney and S. Tremaine, Galactic dynamics [2nd ed.]. Princeton University Press (2008).
  * [3] V. I. Arnold, Catastrophe Theory, Springer Verlag, Heidelberg (2003).
  * [4] P. T. Saunders, An introduction to catastrophe theory, Cambridge University Press (1980).