Stellar evolution constrains primordial black holes as dark matter candidates

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PoS(ICRC2015)1186
Stellar evolution constrains primordial black holes
as dark matter candidates
F. Capela
DAMPT, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road,
Cambridge CB3 0WA, U.K.
M.S. Pshirkov∗
Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetsky prospekt
13, 119992, Moscow, Russia
Institute for Nuclear Research of the Russian Academy of Sciences, 117312, Moscow, Russia
Pushchino Radio Astronomy Observatory, 142290 Pushchino, Russia
E-mail: pshirkov@sai.msu.ru
P. G. Tinyakov
Université Libre de Bruxelles, Service de Physique Théorique, CP225, 1050, Brussels, Belgium
By considering adiabatic contraction of the dark matter (DM) during star formation, we estimate
the amount of DM trapped in stars at their birth. If the DM consists partly of primordial black
holes (PBHs), they will be trapped together with the rest of the DM and will be finally inherited
by a star compact remnant – a white dwarf (WD) or a neutron star (NS), which they will destroy
in a short time. Observations of WDs and NSs thus impose constraints on the abundance of PBH.
We show that the best constraints come from WDs and NSs in globular clusters which exclude
the DM consisting entirely of PBH in the mass range 1016 −1022 g, with the strongest constraint
on the fraction ΩPBH/ΩDM ≤10−4being around PBH masses 1021 g.
The 34th International Cosmic Ray Conference,
30 July- 6 August, 2015
The Hague, The Netherlands
∗Speaker.
c
Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/

PoS(ICRC2015)1186
Stellar evolution constrains PBHs M.S. Pshirkov
1. Introduction
There are compelling evidences that almost 30% of the energy density of the Universe is
contained in the form of a non-relativistic non-baryonic dark matter (DM) [1]. The DM nature
remains essentially unconstrained, leaving room for a host of candidates. It is frequently assumed
that the DM is composed of some kind of new particles beyond the Standard Model of particle
physics. However, other candidates such as primordial black holes (PBH) may provide a viable
alternative: the obvious advantage is that no new particles beyond the extemely well tested Standard
Model are required.
Stars, especially their remnants, could be used as a very sensitive probe to this kind of dark
matter candidates: in higher density environment of compact stars stars, the accretion on even
smlall black holes is sufficiently efficient to destroy the star in a short time (see Ref. [2]. Even if a
single PBH is captured by a compact star, the latter gets destroyed. Requiring that the probability
of such an event is small leads to the constraints on the PBH abundance [3, 4].
The strength of the constraints is determined by the amount of captured DM (firstly we well
speak about DM in general, discussing the particular PBH case later). There are two different
capture mechanisms. A star can capture DM from its surrounding environment, such as the Galactic
halo, during its lifetime, including a compact remnant stage. The DM particles passing through the
star may interact with the nucleons, losing enough energy to become gravitationally bound [5, 6].
Then each subsequent orbit will also pass through the star, so that eventually, after many collisions,
the DM particle will sink to the center of the star. Such capture process can lead to the accumulation
of a considerable amount of DM inside the compact star throughout its lifetime [7].
Also, the DM could also be captured during the star formation. In the course of gravitational
collapse of a prestellar core in a giant molecular cloud, the DM that was initially gravitationally
bound to the core undergoes adiabatic contraction, forming a cuspy profile centered at the star,
with the density
ρ
(r)behaving like
ρ
(r)∝r−3/2. We are interested, however in a amount of DM
captured by a star as a result of its formation. The usually considered DM density profile after the
contraction is not sufficient for that purpose, and one needs to know in more detail the distribution
of DM in the phase space.
To get captured, the DM particles have to lose their energy by interactions with the star mate-
rial. Therefore, one needs to calculate the number of particles whose orbits cross the star after the
adiabatic contraction and, what is crucial, to account correctly for particles that spend only a small
fraction of time inside the star, because they consitute the ovewhelming majority of all particles.
2. Adiabatic contraction of DM during star formation
2.1 Star formation and adiabatic contraction of DM
Stars are formed in giant molecular clouds (GMCs) with a typical mass of 105−106M⊙and
density
ρ
B∼500 GeV/cm3. Eventually, GMCs fragment into smaller clumps, each forming a
protostar after the contraction of the gas. In these regions baryonic densities are quite high, so for
most environments the gravitational effect of the DM on the formation of stars is negligible and the
behavior of DM is determined by the gravitational potential of the baryons.
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PoS(ICRC2015)1186
Stellar evolution constrains PBHs M.S. Pshirkov
The baryons contracting into a protostar create a deepening gravitational potential which drags
the DM particles along – the DM distribution develops a density profile that is peaked at the core
of the prestellar cloud.
2.2 Simulation of DM orbits
The core parameters were taken from the Ref. [4], the typical values being the baryonic density
ρ
∼5×106GeV/cm3and the size ∼5000 AU.
In the adiabatic approximation only the initial and final states matter. The gravitational po-
tential of the contracting baryons was modeled as a two-component mass distribution: the uniform
spherical cloud and a point mass in its center. The total mass of the cloud and the value of the cen-
tral mass are chosen in such a way that the latter linearly grows from zero at t=0 to the maximum
value at t=T, while the former decreases from the maximum value to zero, the total mass of the
system being constant. The change of mass was performed slowly over a period of time Tthat is
at least several times longer than free fall time tff in order for adiabatic condiions to be satisfied.
DM was initially distributed uniformly in space (characterized by some background DM den-
sity ¯
ρ
DM), and had Maxwellian distribution in velocities with the dispersion ¯v,
dn =¯nDM (3
2
π
¯v2)3/2
exp{−3v2
2 ¯v2}d3v,(2.1)
where ¯nDM =
ρ
DM/mDM is the mean DM number density. An estimate of the initial density of DM
that is gravitationally bound inside the pre-stellar core has been computed for several star masses
in [4]. From eq. (2.1), at zero velocity vthe ambient phase space density is (cf. [8])
Qmax =(3
2
π
)3/2¯
ρ
DM
m4
DM ¯v3.(2.2)
The density of DM gravitationally bound to the pre-stellar core is smaller as compared to the
ambient DM density by the factor (v/¯v)3≪1, vbeing the escape velocity from the pre-stellar core.
The simulation proceeds as follows: at t=0 we inject a DM particle with a random uniformly
distributed initial position and velocity. If the particle is gravitationally bound at t=0, it is evolved
through the equations of motion, otherwise it is rejected; 3 ×107trajectories were simulated in this
way.
The density profile can be obtained by recording the positions of particles at a randomly cho-
sen time t>Tafter the end of the contraction. The distribution in rcan be then converted into
particle density. An accurate estimate of the number of captured particles, however, needs addi-
tional information, because many trajectories pass through the star, but spend most of their time
outside. We calculate and record the periastron rmin and apastron rmax of each particle orbit at
the end of star formation to get full information about particle trajectory. The distribution of rmin
determines the number of particles that ever get within a given distance from the center and could
be eventually captured.
Fig. 1 shows the resulting distributions as a function of r/¯
R, where ¯
Ris the initial radius of the
cloud. The actual star radius is also indicated on the plot. The lower curve shows the fraction n(r)of
particles that are, in a given moment of time, within the radius rfrom the center. Points with errors
3

PoS(ICRC2015)1186
Stellar evolution constrains PBHs M.S. Pshirkov
R/¯
R10−510−410−310−210−11
r/ ¯
R
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
fraction of particles
n(r)ν(r)
n(r)∝r1.5
ν(r)∝r
radial distribution
periastron distribution
Figure 1: Lower curve: The fraction of particles n(r)that are found within radius rat the end of the adiabatic
contraction. ¯
Ris the initial radius of the prestellar core. Upper curve: The fraction of particles
ν
(r)whose
orbits have the periastron smaller than r. Lines show the power laws n(r)∝r1.5and
ν
(r)∝r. The errorbars
represent statistical errors (adopted from [9]).
represent the results of the simulation. The straight line is a fit by the power law n(r)∝r1.5. As
n(r)is proportional to the density integrated up to the radius r, we can conclude that
ρ
(r)∝r−1.5,
in agreement with the results of Refs. [4] and [10]
The upper curve shows the fraction
ν
(r)of particles whose orbits have periastra within the ra-
dius r. The fraction of such orbits is larger than n(r): after the adiabatic contraction, a considerable
amount of DM particles have very elongated orbits and spend most of their time at radii larger than
r. Indeeed,
ν
(r)scales differently,
ν
(r)∝r. The ratio of the two curves gives the enhancement
factor as a function of the radius. At the star radius r=R⊙this factor is
ν
(R⊙)/n(R⊙) = 1.84×103.
To summarize, as a result of the formation of a star, the DM bound to the prestellar core
experiences the adiabatic contraction and develops a cuspy profile with the density
ρ
∝r−1.5. The
final distribution of the DM particles in orbits becomes such that there are much more particles
that ever come within given radius rthan there are within it at any given moment. Whether all
DM particles crossing the star surface (rmin <R⊙can be finally captured by the star, depends on
whether there is enough time for them to lose energy via the DM-nucleon interactions.
3. Constraints on PBH
3.1 Constraints on PBH from star formation
From now on we would concentrate on particular DM candidate – PBH. Since only the PBHs
that can lose energy can be captured, only the trajectories that pass through or in the very near
vicinity of the star are relevant for our analysis. To calculate the number of captured PBHs, for
each PBH trajectory of this type one has to determine whether there is enough time for the PBH to
lose energy and end up in the compact remnant of the star — WD or NS.
There are two stages of the energy loss process. First, the PBH spends part of the time outside
the target star at orbits that can be approximately considered as radial, except a small number of
cases when the apastron is of order R∗. The capture time in this case has been estimated in [3] to
4

PoS(ICRC2015)1186
Stellar evolution constrains PBHs M.S. Pshirkov
1015 1020 1025 1030 1035 1040
BH mass, g
10−8
10−6
10−4
10−2
100
ΩPBH/ΩDM
Femto EROS+MACHO
Hawking+γ-rays
FIRAS
WMAP3
Star Formation
Superradiance
104GeV cm−3
103GeV cm−3
102GeV cm−3
Figure 2: Constraints on the fraction of PBHs as DM. Shaded regions are excluded. The blue
shaded regions correspond to the revised constraints derived in this paper assuming the DM densities of
(104,103,102) GeV/cm3and the velocity dispersion of 7 km/s (adopted from [9]).
be of order
tcapt ≃2
τ
√
ξ
0∼2×108yrs(1022 g
mBH ),(3.1)
where
τ
=
π
R5/2
∗v2
4GmBH√GM ln Λ,
with
ξ
0=rmax/R∗,vthe escape velocity of the star, lnΛthe Coulomb logarithm that takes a value
close to lnΛ≃30 for a main-sequence star, and R∗and Mthe radius and the mass of the star,
respectively. For the numerical value in eq. (3.1) we have taken rmax/¯
R≃0.1 corresponding to a
value of
ξ
0≃4.4×104. Thus, the fraction of simulated PBH orbits that would be captured by the
star during its lifetime, could be calculated for each PBH mass.
The second stage starts when PBH is fully inside the star. It then continues to lose energy
through the dynamical friction [11] and sinks towards the star center until the moment when the
star turns into a compact remnant. If the radius to which the PBH has been able to sink to during
the lifetime of the star is smaller than the radius of the compact remnant, the latter inherits a PBH.
The efficiency of the dynamical friction grows with the PBH mass. At large masses all the PBH
captured by a star have time to sink to within the radius of the future remnant in a lifetime of the
star. At small masses only a fraction of captured PBH can make it. At the second stage there is
no difference with the calculations of Ref. [4]. Combining the two stages gives the fraction of the
PBHs that ends up inside the compact remnant.
Finally, the number of simulated trajectories should be related with the mean density of DM
in a given environment. This can be done as described in Ref. [4] by calculating the fraction of the
DM particles which are gravitationally bound to the prestellar core before the adiabatic contraction.
This fraction is proportional to the total DM density and inversely proportional to the cube of the
DM velocity dispersion, use of this relation allows to rescale the results for one particular DM
density and velocity dispersion to other values of these parameters.
Naturally, most stringent constraints come from observations of compact stars in regions
with a high DM density and a low DM velocity dispersion. We consider the values
ρ
DM =
5

PoS(ICRC2015)1186
Stellar evolution constrains PBHs M.S. Pshirkov
(104,103,102)GeV cm−3and ¯v=7 km s−1, as a benchmark. The resulting constraints are shown
in Fig. 2 together with other existing constraints. The strongest constraints shown in light blue
correspond to the highest value of DM density considered, i.e
ρ
DM ∼104GeV cm−3and decrease
linearly for lower values of DM density. These (rather extreme) conditions could have been present
in the cores of metal-poor globular clusters at the epoch of star formation, if they are proved to be
of a primordial origin [12] (see detailed discussions in [4, 3]). Another place where favourable con-
ditions could exist are dwarf spheroidal galaxies that are considered to be DM dominated [13, 14]
and have very low velocity dispersions [13]. However, at present compact objects such as NS or
WD have not been observed in dwarf spheroidal galaxiesn although, surveys for pulsars and X-ray
binaries have already revealed some hints on NSs existence in dSph galaxies [15, 16].
4. Conclusions
The adiabatic contraction of DM during the formation of a star was investigated by simulating
the behavior of ∼30 million particles. In particular, the number of particles n(r)within a given
radius rwas found to be proportional to r1.5, which corresponds to the DM density profile
ρ
(r)∝
r−1.5, in agreement with previous calculations and the Liouville theorem.
At the same time, the adiabatic contraction creates a rather special distribution of particle
orbits: if one considers the particles that contribute to n(r)for a small r, a substantial (O(1))
fraction of them have very elongated orbits with periastra smaller than r. In fact, the number of
particles
ν
(r)that have periastra smaller than rscales as
ν
(r)∝r. Such particles spend only a
small fraction of time close to the center, so their individual contributions to the density at small r
are suppressed. At r=R⊙, there are about 1.8×103more particles that have periastra smaller than
rthan there are particles within r.
This has direct implications for the possible DM capture by stars after their formation. A large
number of particles that constitute the DM cusp around the newly-formed star have orbits that cross
the star, and that can potentially lead to their capture.
The most stringent constraints are obtained from observations of compact stars in the regions
with high DM density and small velocity dispersion. The examples corresponding to the densities
ρ
= (104,103,102)GeV cm−3and a low velocity dispersion ¯v=7 km s−1are shown in Fig. 2.
Such conditions could have been present at the cores of metal-poor globular clusters at the epoch
of star formation if they are of a primordial origin, or — with a DM density somewhat smaller than
103GeV/cm3— in dwarf spheroidal galaxies
Acknowledgements
The work of F.C. and P.T. is supported in part by the IISN and the Belgian Science Policy
Belgian Science Policy under IUAP VII/37. The work of M.P. is supported by RSF grant No. 14-
12-00146. M.P. acknowledges the fellowship of the Dynasty foundation.
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PoS(ICRC2015)1186
Stellar evolution constrains PBHs M.S. Pshirkov
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