Constraints on cosmic strings from ultracompact minihalos
Madeleine Anthonisen,
1
Robert Brandenberger,
1
and Pat Scott
2
1
Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada
2
Department of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road,
London SW7 2AZ, United Kingdom
(Received 12 April 2015; published 15 July 2015)
Cosmic strings are expected to form loops. These can act as seeds for accretion of dark matter, leading to
the formation of ultracompact minihalos (UCMHs). We perform a detailed study of the accretion of dark
matter onto cosmic string loops and compute the resulting mass distribution of UCM Hs. We then apply
observational limits on the present-day abundance of UCMHs to derive corresponding limits on the cosmic
string tension Gμ. The bounds are strongly dependent upon the assumed distribution of loop velocities and
their impacts on UCMH formation. Under the assumption that a loop can move up to a thousand times its
own radius and still form a UCMH, we find a limit of Gμ ≤ 1 × 10
−7
. We show, in opposition to previous
results, that strong limits on the cosmic string tension are not obtainable from UCMHs when more stringent
(and realistic) requirements are placed on loop velocities.
DOI: 10.1103/PhysRevD.92.023521 PACS numbers: 98.80.Cq
I. INTRODUCTION
Cosmic strings (see e.g. [1–4] for reviews) are topologi-
cal defects present in many theories of particle physics
beyond the standard model. They are lines of confined
energy density, analogous to defects such as vortex lines in
condensed matter systems like superconductors and super-
fluids. In all particle theories that permit cosmic strings, a
network of strings forms during a phase transition in the
very early Universe. Causality arguments [5,6] show that
this network persists to the present time. Using cosmo-
logical observations to hunt for gravitational effects of the
energy trapped in cosmic strings is therefore a powerful
way to probe particle physics beyond the standard
model [7].
Observable signatures of cosmic strings are typically
proportional to the mass per unit length μ of the string
[1–4], which is in turn related to the energy scale η at which
the strings form (μ ≃ η
2
[8]). Searching for cosmological
signatures of strings thus probes particle physics in a “top
down” manner, excluding higher energy scales most easily.
This makes searches for cosmic strings highly comple-
mentary to terrestrial accelerator experiments, which search
for new physics via a “bottom up” strategy.
As topological defects, by definition cosmic strings
cannot have ends. They must either exist as part of a
network of infinite strings, or as closed loops. Because they
are relativistic, a segment of infinite string will typically
have a translational velocity of the order of the speed of
light. The network of infinite cosmic strings follows a
“scaling solution” wherein the correlation length, which
describes the mean curvature and separation of string
segments, grows linearly with time. This causes the
contribution of the network to the energy density of the
Universe to remain constant. The analytical arguments for
the existence of this solution (see e.g. [4] for a review) have
been confirmed by numerical simulations [9–15]. The
scaling of the long string network is maintained by the
formation of loops when segments intersect, removing
energy from the network. Studying the distribution of string
loops numerically is much more demanding than following
the network, because a much larger hierarchy of scales
needs to be followed. However, current results indicate that
loops also follow a scaling solution (see recent references in
[9–15] for numerical evidence, and [16] for some more
recent analytical work).
String loops act as seeds for the growth of density
perturbations in the matter surrounding them. If they are
present early enough, and persist for long enough, they
can lead [17] to the formation of so-called ultracompact
minihalos of dark matter (UCMHs; [18–26]). UCMHs are
distinguished from regular dark matter (DM) halos by the
epoch at which they undergo gravitational collapse.
Regular halos do not transition to the nonlinear regime
of structure formation until z ≲ 30, whereas UCMHs
collapse shortly after matter-radiation equality, in iso-
lation. At this time, the background density field is still
cold, smooth and essentially featureless. This means that
UCMHs form by almost pure radial infall [22], giving
them far steeper central density profiles than regular
cold dark matter halos: ρ ∝ r
−9=4
[27–29] rather than ρ ∝
r
−1
[30].
If DM can self-annihilate, the rate goes as the square of
the particle density. The steep density profile of UCMHs
therefore makes them excellent candidates for indirect
detection of DM [23,31–35]. Searches for gamma-ray
sources with the LAT instrument aboard the Fermi
gamma-ray space telescope lead to the strongest limits
on the cosmological density of UCMHs [24]. Gravitational
lensing [22,36,37], neutrinos [38,39], reionization [40,41]
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and diffuse photon fluxes at various wavelengths
[24,42,43] provide supporting limits. With these limits
and a proper understanding of how to calculate the UCMH
yield from a particular scenario, it becomes possible to use
UCMHs to place limits on the spectrum of primordial
perturbations [24,32,38,44], non-Gaussianities [45] and
cosmic strings [17]. Here we provide an improved treat-
ment of UCMH formation around cosmic string loops, and
the resulting limits on the string tension Gμ.
In Sec. II we review the important aspects of the cosmic
string loop scaling solution, before following the accretion
of DM by loops in Sec. III, and the subsequent formation of
UCMHs. In Sec. IV we apply those calculations to
determine the fraction of DM in loop-induced UCMHs,
including the effects of loop velocities. We derive limits on
the cosmic string tension in Sec. V, then summarize in
Sec. VI. For the most part we use natural units, with c set
to 1.
II. STRING LOOP SCALING
An important ingredient in investigating the formation
and evolution of dense structures seeded by cosmic string
loops is the number density of loops per unit loop radius,
nðR; tÞ. The quantity nðR; tÞdR is the number density of
loops in the cosmic string network at a time t with radii
between R and dR.
In this paper we use a one-scale model for the distribu-
tion of string loops [46,47], according to which all loops of
initial radius R
i
form at the same time t
i
ðR
i
Þ. Here R
i
and
t
i
ðR
i
Þ are linearly related by a constant α:
R
i
t
i
ðR
i
Þ
¼
α
β
: ð1Þ
The numerical value of α must be determined from
simulations; we adopt α ¼ 0.05 [15]. String loops warp
and twist as they evolve, so the probability that any given
loop is exactly circular at any point in time is virtually nil. It
is common to introduce a parameter β ≡ l=R that relates
the radius of a loop to its length l. Deviations from
circularity can then be accounted for by allowing β to
differ slightly from 2π (although for our final limits we
simply set β ¼ 2π).
The scaling solution implies that a constant number N of
loops are formed per expansion time per Hubble volume,
meaning that the number density of loops at the time of
their formation is
nðR
i
;t
i
Þ¼Nt
−4
i
¼ Nα
4
β
−4
R
−4
i
: ð2Þ
N is another constant that must be determined by simu-
lations; we take N ¼ 40 [15].
Neglecting, for the moment, slow decay of loops by
emission of gravitational radiation, the physical radius R of
a string loop remains constant as the Universe expands. The
number density redshifts, so that for t>t
i
ðRÞ the number
density of loops of radius R is
nðR; tÞ¼
zðtÞþ1
z
i
ðRÞþ1
3
nðR; t
i
Þ; ð3Þ
where zðtÞ is the cosmological redshift, and z
i
ðRÞ ≡
z½t
i
ðRÞ is the redshift at the time that loops of radius R
were created. Making use of the fact that t ∝ ð1 þ zÞ
−3=2
during matter domination and t ∝ ð1 þ zÞ
−2
during radia-
tion domination, the number density of loops is
nðR; tÞ¼Nα
2
β
−2
t
−2
R
−2
; ð4Þ
for loops formed during matter domination, and
nðR; tÞ¼Nα
5=2
β
−5=2
t
1=2
eq
t
−2
R
−5=2
ð5Þ
for loops formed during radiation domination (as evaluated
at some time t>t
eq
, where t
eq
is the time of equal matter
and radiation).
Emission of gravitational radiation by loops leads to a
reduction of R with time, at an approximately constant rate
proportional to Gμ. The actual rate has some distribution
over a population of loops, according to the loops’
individual geometries and the corresponding rate at which
they can each emit gravitational radiation—but there is a
typical value associated with the typical loop oscillation
frequency. Combined with the fact that smaller loops are
formed earlier, this means that there exists a radius R
dec
below which loops will have typically decayed by a given
time t,
R
dec
ðtÞ¼
Gμγ
β
t; ð6Þ
where γ is another numerical constant determined from
simulations [48]. We will use γ ¼ 10π.
From this point onward we make the “fast decay
approximation,” where we assume that loops decay essen-
tially instantaneously. In this approximation RðtÞ¼R
i
from the time of loop creation right up to decay, so R
and R
i
become essentially interchangeable in all the
equations that we have written so far. Under the fast decay
approximation, the impact of gravitational radiation can be
neglected for all radii R>R
dec
. Below this value, due to the
fact that there is always a tail in the distribution of
oscillation frequencies, and therefore some loops that
shrink in radius at every rate less than the typical one,
the physical number density of loops remaining will be
proportional to R. The density per unit loop radius therefore
becomes independent of R [1,2],
nðR; tÞ¼nðR
dec
;tÞ for R<R
dec
; ð7Þ
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023521-2
regardless of whether the loop was formed during matter or
radiation domination.
III. ACCRETION OF DARK MATTER
BY STRING LOOPS
Accretion of cold dark matter by a cosmic string loop
leads to a spiky DM distribution [49,50] (see also [27,28]).
If the resulting structure undergoes gravitational collapse
sufficiently early, a UCMH will result. To determine the
total fraction of DM contained in loop-induced UCMHs at
the present day, we must study the accretion of DM by
string loops in some detail.
There are several pitfalls to navigate in doing this. It is
not valid to simply naively take all loops at t
eq
, and apply
the growth factor from linear perturbation theory to the
initial mass of each loop, as some loops may decay before
such a period of accretion could become effective. It would
be equally incorrect to just eliminate loops that decay
before a certain time (e.g. the present day, or t
eq
), as such
loops may have accreted enough mass before their decay to
have already created nonlinear structures—possibly even
during radiation domination. Even if such loops do not
induce nonlinear structures by the time of their decay, the
velocity perturbation that they induce will persist, and
continue to grow in time after t
eq
even if the loop has
already decayed. That growth may eventually lead to
gravitational collapse in time to create a UCMH, even
though the loop decayed much earlier. Thus, to study
formation of UCMHs from string loops we must perform a
careful study of the accretion of DM onto string loops both
before and after t
eq
, and before and after their time of decay
t
d
¼
α
γGμ
t
i
: ð8Þ
To compute the accretion of mass by loops of radius R
produced at time t
i
ðRÞ, we use the Zel’dovich approxima-
tion [51]; specifically, the spherical collapse model. We
consider fluctuations that are initially isothermal, where the
initial fluctuation is exclusively determined by the cosmic
string loop. The loop is the source of gravitational
attraction, and over time will lead to an inhomogeneous
DM distribution.
We focus on a spherical shell of matter of physical radius
r about the center of the loop. We are interested in the time
evolution of this radius, as the shell moves toward the
center of the loop. It is convenient to parametrize this radius
in the form
rðxÞ¼aðxÞbðxÞζ
i
; ð9Þ
where aðxÞis the scale factor, ζ
i
is the comoving coordinate
associated with the spherical shell at the initial time t
i
, and
bðxÞ measures the difference between the motion of the
spherical region in the presence of the string loop compared
to how it would evolve under simple cosmological expan-
sion. Here a convenient parameter for time is
x ≡
aðtÞ
aðt
eq
Þ
¼
z
eq
þ 1
zðtÞþ1
: ð10Þ
In the case of cold dark matter we can neglect thermal
motion of matter, and in this approximation each shell will
be characterized by the relative mass fluctuation parameter
ΦðrÞ ≡ δM=MðrÞ, where MðrÞ is the total mass within the
shell of initial radius r, and δM is the mass fluctuation due
to the string loop (which is independent of r).
As derived in [52], in the Zel’dovich approximation we
have the following equation of motion for bðxÞ:
xðx þ 1Þ
d
2
b
dx
2
þ
1 þ
3
2
x
db
dx
þ
1
2
1 þ Φ
b
2
− b
¼ 0;
ð11Þ
where the information about which shell we are considering
is hidden in the value of Φ for that shell.
The solutions of this equation depend on whether we are
in the radiation-dominated phase t<t
eq
or in the matter-
dominated phase t>t
eq
, and whether the loop has decayed
or not. In the radiation-dominated phase the DM is a
subdominant component of matter, so on the sub-Hubble
scales that we are considering when we study accretion by
string loops, we expect only logarithmic growth of the
fluctuations after loop decay. In the matter-dominated
phase, we expect linear growth in x. Hence, we must
study the solutions of Eq. (11) in various cases. We will in
fact find that the nonlinear mass grows linearly in x even in
the radiation phase, as long as the seed loop is still present.
The size at time x of a compact object formed via
accretion about the string loop is given by the radius of the
shell which is “turning around” at the time x. The turn-
around time t
TA
for a fixed shell is the time when
_
r ¼ 0.At
that time, the shell of matter disconnects from the Hubble
flow to collapse, forming a virialized clump. Adopting the
coordinates of Eq. (9), the turnaround condition becomes
b þ x
db
dx
¼ 0: ð12Þ
There is a critical turnaround time (with an associated
critical redshift, z
c
) after which a collapsing overdensity
will not contain sufficiently pristine material to form by
radial infall, and so cannot form a UCMH. The precise time
at which the radial infall approximation breaks down and a
collapsing halo can no longer be said to form a UCMH is
still rather uncertain [24,36,45]. Certainly a “latest collapse
redshift” of z
c
∼ 1000 is a conservative and very safe choice
[22], but UCMH formation down to redshifts as low as
z
c
∼ Oð100Þ is not inconceivable [24,45]. Here we will
assume z
c
¼ 1000 for our final limits on the cosmic string
CONSTRAINTS ON COSMIC STRINGS FROM … PHYSICAL REVIEW D 92, 023521 (2015)
023521-3
tension Gμ. Adopting a smaller yet still plausible redshift
would lead to improved constraints, because later collapse
redshifts allow progressively smaller perturbations time to
collapse and form UCMHs.
We must verify that t
TA
for the innermost shell of DM
around a loop occurs before this critical collapse time, if the
loop is to be said to have seeded a UCMH. The turnaround
time is affected by the loop decay time; the earlier a loop
evaporates, the longer it will take for the turnaround
to occur.
A. Formation and accretion before t
eq
(x
i
< 1, x < 1)
In this section we study the accretion of DM by a string
loop during radiation domination. The analysis of this
section is applicable to all loops formed before t
eq
,but
different parts of the analysis apply depending on whether
the loop decays before or after t
eq
. We carry out the
calculation in two different regimes. The first regime covers
the period from loop formation (x
i
) up to either loop decay
(x
d
) or equality (x
eq
¼ 1), whichever is earlier. This treat-
ment is all that is required during radiation domination for
loops that decay after equality, as in that case it is valid right
up to t
eq
. For loops that decay before equality (x
d
< 1), the
solution for the first regime must be matched on to the
solution for the second at x ¼ x
d
. The second regime
extends from loop decay to equality ðx ¼ 1Þ. In the fast
decay approximation, Φ ¼ 0 in this regime, as the loop is
absent.
1. Regime I (x < x
d
)
It is well known from the study of accretion of cold dark
matter onto point seeds [49,50] that the innermost shells are
the first to decouple from the Hubble flow, turn around and
collapse back onto the seed mass. To see whether any mass
shell turns around by a particular time for a string loop with
initial radius R, we must hence focus on the shell of initial
radius R. We first express the matter overdensity inside this
shell in terms of cosmic string parameters. As long as the
loop decay can be neglected, the value of the overdensity
parameter ΦðRÞ is constant. Starting from
ΦðRÞ¼
δM
MðRÞ
; ð13Þ
where δM ≡ M
loop
¼ μβR is the mass of the loop and
MðRÞ¼
4
3
πR
3
ρ
DM
ðt
i
Þð14Þ
is the total mass of DM contained within the region of
initial radius R (the loop radius) at the time of loop
formation, we find
ΦðRÞ ≡ Φðx
i
Þ¼4β
3
α
−2
ðGμÞf
−1
χ
κ
−1
x
−1
i
: ð15Þ
Here f
χ
≡ Ω
DM
=Ω
m
and we have used the value of x at the
time of loop formation to label the innermost shell instead
of the loop radius R. Also, we have made use of the
background Friedmann equation of motion (after rescaling
the DM density to the time t
eq
and using the fact that the
total density at that time is twice the matter density), and
defined the constant κ by
H
2
eq
≡ κt
−2
eq
; ð16Þ
i.e.
κ ≡
2
4
πGρ
DM
ðt
eq
Þt
2
eq
3f
χ
; ð17Þ
which falls in the range 1=4 < κ < 4=9.
During radiation domination x is small, so the second
derivative term in Eq. (11) becomes negligible. With initial
conditions bðx
i
Þ¼1 for x
i
≪ 1, the approximate solution
of (11) is
bðxÞ
3
¼ 1 −
3
2
xΦðRÞ; ð18Þ
for x ≫ x
i
. Inserting the result from Eq. (15) into this, we
obtain
bðxÞ
3
¼ 1 − 6β
3
α
−2
f
−1
χ
κ
−1
ðGμÞ
x
x
i
: ð19Þ
Inserting this result in the turnaround equation Eq. (12)
yields the following expression for the turnaround time
x
TA
¼
1
2ΦðRÞ
¼ 2
−3
β
−3
α
2
f
χ
κx
i
ðGμÞ
−1
ð20Þ
for the innermost shell.
Consider now shells outside the innermost one, with
initial physical radius r>R. For those shells the value of Φ
is reduced to
ΦðrÞ¼
R
r
3
ΦðRÞ: ð21Þ
The mass that has turned around by “time” x is
MðxÞ¼
4π
3
ρ
DM
ðt
i
Þr
TA
ðxÞ
3
; ð22Þ
where r
TA
ðxÞ is the initial radius of the shell turning around
at time x. From the first line of (20) it follows that
x ¼
1
2Φðr
TA
Þ
; ð23Þ
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so that
r
TA
ðxÞ
3
¼ 2xR
3
ΦðRÞ: ð24Þ
From (21) the nonlinear mass that has accreted around the
loop at time x is
MðxÞ¼
x
x
TA
MðRÞ¼2xM
loop
; ð25Þ
where x
TA
still refers exclusively to the turnaround of the
innermost shell.
It is interesting to note the linear growth in x, the same
growth obtained in linear perturbation theory after matter-
radiation equality. It is also interesting to note that if the
loop survives until x ¼ 1, the nonlinear mass which has
collapsed at that point is exactly twice the loop mass.
Now we consider under which conditions the resulting
halo will actually collapse in this regime, as all halos that
collapse already during radiation domination are sure to
lead to UCMHs. Collapse of the innermost shell in Regime
I requires that x
TA
< minðx
d
; 1Þ. First of all, we see that
only loops formed at t
i
< αγGμt
eq
decay before equality,
which corresponds to radii
R
t
eq
< α
2
β
−1
γGμ: ð26Þ
First we deal with the case where decay occurs before
equality (x
d
< 1). Noting that during radiation domination
t
d
=t
i
¼ðx
d
=x
i
Þ
2
, from Eq. (8) we obtain
x
d
¼ α
1=2
ðγGμÞ
−1=2
x
i
: ð27Þ
Comparing the expressions (20) and (27), we see that if
decay occurs before equality, the condition for turnaround
in Regime I is independent of the value of x
i
(and therefore
also R). For such loops, collapse in Regime I can only
occur if
Gμ > 2
−6
β
−6
α
3
γκ
2
f
2
χ
: ð28Þ
For loops that decay after equality (x
d
> 1), the con-
dition for collapse in Regime I instead becomes
Gμ > 2
−3
β
−5=2
α
3=2
κf
χ
R
t
eq
1=2
: ð29Þ
For values of Gμ larger than these critical values,
nonlinear dark matter clumps will have formed before
matter-radiation equality about cosmic string loops, and
have therefore formed UCMHs. For R corresponding to
decay before equality and values of Gμ smaller than
Eq. (28), we must also study how DM accretion continues
between loop decay and equality to determine if UCMHs
might still be created during radiation domination, but in
Regime II instead of Regime I.
2. Regime II (x > x
d
)
In cases where the loop decays before equality, we must
continue to evaluate the evolution of the clump in the
regime x
d
<x<1. Once again, the small x approximation
is valid and we can neglect the second derivative term in
Eq. (11). The difference here with the calculation in
Regime I is that we set Φ ¼ 0, as the loop is absent,
and use bðxÞ from Regime I [Eq. (18)] evaluated at x ¼ x
d
as the initial condition. This leads to the solution
b
3
ðxÞ¼
2b
3
ðx
d
Þ − 3x þ 3b
3
ðx
d
Þx þ 3x
d
2 þ 3x
d
¼ 1 − ð3=2Þx
d
ΦðRÞð2 þ 3xÞð2 þ 3x
d
Þ
−1
: ð30Þ
Inserting this into the turnaround condition [Eq. (12)]
gives the turnaround time
x
TA
¼
1
2ΦðRÞ
þ
1
3x
d
ΦðRÞ
−
1
2
ð31Þ
for the innermost shell. Making the correction
1
2
→
2
3
to the
final constant in order to match solutions exactly between
Regimes I and II at x
d
,
1
we see that only for Gμ larger than
the critical value Eq. (28) do the second two terms give a
positive correction to the result for Regime I [Eq. (20)].
This indicates (as expected) that there is always some
growth in Regime II, but the correction is small, so we can
see that the additional growth is minimal. Indeed, the
critical relationship between Gμ and R that results from
demanding that collapse happens within Regime II [i.e.
Eq. (31) < 1],
R=t
eq
< 2
2
· 3
−2
β
−1
γðGμÞ
− 2
4
β
2
α
−3=2
γ
1=2
f
−1
χ
κ
−1
ðGμÞ
3=2
þ 2
10
· 3
−2
β
5
α
−3
f
−2
χ
κ
−2
ðGμÞ
2
; ð32Þ
is not even in the correct regime when taken to lowest order
in Gμ [i.e. when only the first line of this expression is
compared to Eq. (26)]. R in this expression can only
actually be large enough for decay to happen before
equality if the contributions of the higher-order terms are
included and Gμ is sufficiently large.
Shells outside the innermost one also exhibit the reduced
growth. Following the same treatment as for Regime I,
these will turn around at
1
This is needed in order to account for the terms we neglected
in Regime I when we took the approximation x
i
≪ 1.
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