Towards Skyrmion Stars: Large Baryon Configurations in
the Einstein-Skyrme Model
Bernard M. A. G. Piette∗ and Gavin I. Probert†
November 4, 2018
Department of Mathematical Sciences, University of Durham, Science Laboratories, South
Road, Durham. DH1 3LE. UK
Abstract
We investigate the large baryon number sector of the Einstein-Skyrmemodel as a possible
model for baryon stars. Gravitating hedgehog skyrmions have been investigated previously
and the existence of stable solitonic stars excluded due to energy considerations[1]. However,
in this paper we demonstrate that by generating gravitating skyrmions using rational maps,
we can achieve multi-baryon bound states whilst recovering spherical symmetry in the limit
where B becomes large.
∗email:b.m.a.g.piette@durham.ac.uk
†email:g.i.probert@durham.ac.uk
http://arxiv.org/abs/0704.0527v1
1 Introduction
The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in
a series of papers as a non-linear field theory of pions [2], [3]. Skyrme’s initial idea was to
think of baryons (in particular the nucleons) as secondary structures arising from a more
fundamental mesonic fluid. The key property of the model was that the baryons arose as
solitons in a topological manner and thus possessed a conserved topological charge identified
with the baryon number.
The lowest energy stable solutions of the model are termed Skyrmions and can be thought
of as baryonic solitons. The Skyrme model has been very successful in modelling the struc-
tures of various nuclei and has been shown by Witten et al. [4] to possess the general
features of a low energy effective field theory for QCD.
Some studies of the Skyrme model coupled to gravity have previously been undertaken
[1], [5], [6], mainly with the motivation of a comparison of its features with those of other non-
linear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory,
in which gravitationally bound configurations of non-abelian gauge fields are produced.
Other reasons for studying the Einstein-Skyrme model are cosmological and astrophysi-
cal ones. Various authors have studied black hole formation in the model, with the conclu-
sion that the so-called no-hair conjecture may not hold [7], [8].
The purpose of this paper is to study large baryon number Skyrmions or configurations
of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable
solitonic stars could exist within the model and to compare their properties to those of
neutron stars.
Preliminary studies of Skyrmion stars have predicted instability to single particle decay
[1]. However this was done using the hedgehog ansatz for baryon number larger than 1
which is known to lead to unstable solutions even for the usual Skyrme model. Since then,
it has been shown that the Skyrme model has stable shell-like solutions[9] which can be well
approximated by the so called rational map ansatz [10].
In this paper we use the rational map ansatz and its extension to multiple shells to
construct configurations in the Gravitating Skyrme model that have a very large number
of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound
even for very large baryon numbers.
To construct configurations that have a baryon number comparable to that of neutron
star, we have to introduce a further approximation, which we call the ramp ansatz. We show
that this anstaz introduces further errors of only a few percent and we use it to compute
very large Skyrmion configurations.
The paper is organised as follow: first we outline the Einstein-Skyrme model and discuss
the main features of the results on static gravitating SU(2) hedgehogs obtained by Bizon
and Chmaj [1]. We then use the rational map ansatz to construct shell like gravitating
multibaryon configurations and show that for a fixed value of the coupling constant, the
configurations exist only when the baryon number is below a certain critical value. Finally
we introduce a ramp profile approximation to construct solutions with extremely high baryon
numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.
2 The Einstein-Skyrme Model
The action for gravitating Skyrmions is formed from the standard Skyrme action for the
matter field and the Einstein-Hilbert action for the gravitational field.
LSk −
x. (1)
Here LSk is the Lagrangian density for the Skyrme model defined on the manifold M :
LSk =
Tr(∇µU∇µU−1) +
Tr[(∇µU)U−1, (∇νU)U−1]2, (2)
where U belongs to SU(2). As we eventually wish to study baryon stars, we take a spheri-
cally symmetric metric, such as associated with the line element
= −A2(r)
1− 2m(r)
1− 2m(r)
+ sin
), (3)
where A(r) and m(r) are two profile functions that must be determined by solving the
Einstein equations for the model. Our choice of ansatz is motivated by the fact that although
in some cases we will be studying non-spherical Skyrmion configurations, the regime we are
primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to
admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational
interaction is small compared to the Skyrme interaction and thus the use of a spherical
metric even with non-spherical configurations, is not a great problem.
From (3), it can be shown that the Ricci scalar is
−A′′r2 − 2A′r + 2A′′rm+ A′m+ 3A′rm′ + Arm′′ + 2Am′
which, after integrating various terms by parts and noting that asymptotic flatness requires
both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational
part of the action to
Sgr =
−m′(r)
. (5)
For what follows, it will be convenient to scale to dimensionless variables by defining x =
eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the
model, α = πF 2πG. We note that taking Fπ = 186Mev and G = 6.72 × 10−45Mev−2, then
the physical value of the coupling is α = 7.3× 10−40.
As the Skyrme field is an SU(2) valued scalar field, at any given time one can think of it
as a map from R3 to the SU(2) manifold. Finite energy considerations impose that the field
at spatial infinity should map to the same point on SU(2), say the identity. Thus, one can
simply think of the Skyrme field as a map between three-spheres. All such maps fall into
disjoint homotopy classes characterised by their winding number. This winding number is
a conserved topological charge because no continuous deformation of the field and thus no
time evolution, can allow transitions between homotopy classes. It is this topological charge
that is interpreted as the baryon number.
3 Gravitating Hedgehog Skyrmions
Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties
of static spherically symmetric gravitating SU(2) skyrmions. Taking the Hedgehog Ansatz
for the Skyrme field
U = exp(i−→σ .r̂F (r)) (6)
subject to the boundary conditions
F (r = 0) = Bπ (7)
F (r = ∞) = 0 (8)
where B is the Baryon number associated with the Skyrmion configuration, they derived
the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the
model admit two branches of global solitonic solutions at each given baryon number, which
annihilate at a critical value of the coupling parameter. Above αcrit no further solutions
were found. In particular the value of the critical coupling decreased quite considerably
with increasing baryon number as αcrit ≈ 0.040378/B2 . It appears that the existence of a
critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact
the metric factor S(x) = (1 − 2µ(x)
) is non-zero at αcrit; there simply ceases to be any
stationary points of the action above the critical coupling.
The major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any
given value of α, MADM (B = N) > NMADM (B = 1). This is actually the case for the pure
Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the
lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1
are known not to be spherically symmetric[11] but are stable i.e. E(B = N) < N ∗E(B = 1).
It was actually shown by Houghton et al [10], [12] that the multi-baryon solutions of
the pure Skyrme model can be well approximated by the so called rational maps ansatz
which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz
separates its radial and angular dependence through a profile function and a rational map
respectively.
In the following sections we will generalise the construction of Houghton et al to approx-
imate the solution of the Einstein-Skyrme model.
4 The Rational Map Ansatz
The rational map ansatz introduced by Houghton et al.[10] works by decomposing the
field into angular and radial parts. Using the polar coordinates in R3 and defining the
stereographic coordinates z = tan(θ/2) expiφ the ansatz reads [10]
U = exp (i~σ · n̂RF (r, t)) (9)
where
n̂R =
1 + |R|2
2ℜ(R), 2ℑ(R), 1− |R|2
is a unit vector where R is a rational function of z.
It can be shown that the baryon number for Skyrmions constructed in this way, is equal
to the degree of the rational map providing we take the boundary conditions
F (r = 0) = π
F (r = ∞) = 0. (11)
Substituting the ansatz (9) into the action for the model and scaling to dimensionless
variables as earlier, we obtain the following reduced Hamiltonian
16πFπ
S(x)F (x)
+ Bsin2F (x)(1 + S(x)F (x)′2)
Isin4F (x)
where
S(x) = 1− 2µ(x)
From which one obtains the following field equations
S(x)x
F (x)
+ B sin2 F (x) + S(x)BF (x)′2 sin2 F (x) + I sin
4 F (x)
F (x)
S(x)V (x)
sin 2F (x)
B + S(x)BF (x)′2 + I sin
2 F (x)
− αS(x)F (x)
′3V (x)2
− S(x)′F (x)′V (x)− S(x)F (x)′V (x)′
= αA(x)F (x)
x+ 2B sin
2 F (x)
where, for convenience, we have defined V (x) as
V (x) = x
+ 2B sin2 F (x). (17)
B is the baryon number and
I = 1
1 + |z|2
1 + |R|2
2idzdz
(1 + |z|2)2
Its value depends on the chosen rational map R. To compute low energy configurations for
a given baryon charge B one must find the rational map R or degree B that minimize I.
This has been done in [10] and [11] for several values of B. Moreover when b is large, one
can use the approximation[11] I ≈ 1.28B2 . The value of I so obtained is then used as a
parameter and one can solve equations (14) - (16) for the radial profiles F (x), A(x) and
µ(x).
We should point out here that for the pure Skyrme model the rational map ansatz
produce very good approximation to the multi skyrmion solutions [10]: the energies are only
3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by
very little. All the solutions computed by Battye and Sutcliffe[11], when B is not too small,
have somehow the shape of a hollow shell. The baryon density is very small everywhere
outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.
5 Hollow Skyrmion Shells
Using the rational map ansatz, we will now solves the field equations (14) - (16) to compute
some low action configurations. These solutions will correspond, initially, to a hollow shell
of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure
Skyrme model. In the following sections we will show how our ansatz can be generalised to
allow for more realistic configuration made out of embedded shells.
The first thing to note about our solutions is that we again obtain two branches of
solutions at each baryon number (Fig. 1). Obtaining this same qualitative behaviour is not
surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual
B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered
for the rational map generated configurations. Namely, we observe that it decreases as
approximately 0.040378/B
2 (Fig. 2). In particular this means that for a given value of the
coupling, the rational map generated skyrmions can possess a much higher topological charge
than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively
if Bhedgehog is the maximum baryon number for which hedgehog solutions can be found at
a given value of the coupling, then the highest baryon number rational map solution found
at the same value of α will be approximately B4hedgehog . Again we observe that the metric
function S(x) is non-zero at the critical coupling for all the solutions we have found and as
such a horizon has not formed.
In Table 1 we present the radius, ADM mass per baryon and minimum value of the
metric function, S(x), for configurations up to the maximum baryon number allowed at
α = 1× 10−6. These values were obtained by direct numerical solution of equations (14) -
(16), where we have used the boundary data as specified in (11).
We didn’t didn’t use the physical value of α (7.3×10−40) because for this value, the ratio
between the width of the shell and its radius is so small when we reach the maximum value
of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10−6
is small enough to allow for a shell with a large baryon number to exist but large enough
to make it possible to compute these solution nears the critical value of B for a single shell
configuration.
 0  0.005  0.01  0.015  0.02  0.025  0.03  0.035
Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the
rational map ansatz.
The major difference between these configuration and the solutions of Bizon and Chmadj
is that the rational map ansatz configurations become more bound when the baryon number
increases This suggests the possibility that giant gravitating Skyrmions can be bound and
consequently, that the Skyrme model can be used to study baryon stars.
Another interesting feature of the data is the observed change in the radius of the solu-
tions with increasing baryon number. We note that the radius grows as approximately B
However there are two main deviations from this. Firstly, the constant of proportionality
relating the radius to the square root of the baryon number decreases slightly but persis-
tently as we increase the baryon number, indicating the gravitational interaction becoming
more important as the number of baryons increases.
As we approach the maximum baryon charge that can exist at α = 1 × 10−6, we also
notice that the radius of the skyrmion actually decreases as we add more baryons. This
shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.
This is a tantalising property when one considers that generally a neutron star’s radius must
decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
 0.035
 0.04
 0.045
 0.05
 2  4  6  8  10  12  14  16  18  20
Figure 2: Plot of the decrease in αcrit with increasing baryon number, for configurations generated
with the rational map ansatz. +: αcr for the minimum value of I; curve: αcr = 0.0404B
−1/2.
to support the star.
To motivate the further approximation that we will introduce in the next section, we
now look at the profiles of the configuration that we have computed. First of all, we observe
that the profile function F (x) stays approximately at its boundary value, π, for a finite
radial distance before decreasing monotonically over some small region and finally attaining
its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x)
and the metric field A(x) (see Fig. 3). Furthermore, as we increase the baryon number
the structure becomes more pronounced, with the distance before the fields change (shell
radius) increasing significantly, whilst the distance over which the fields change (shell width)
settles to a constant size. We conclude that at large baryon numbers, those configurations
correspond to hollow shells where the baryons are distributed on a tight lattice over the
shell. As such the, structures are nearly spherical, validating our choice of radial metric.
Such structures immediately pose an interesting question. Can the gravitating Skyrmions
exist as shells with more than one layer? To investigate this we note that it is possible to
B R( 2
) MADM (
2etopconv
) Smin
1 0.8763 1.2315 1.0000
4 1.7728 1.1365 1.0000
8 2.5065 1.1180 1.0000
100 8.6829 1.0845 0.9999
500 19.3994 1.0827 0.9998
1× 103 27.4314 1.0825 0.9997
1× 104 86.7192 1.0821 0.9989
1× 105 274.0397 1.0814 0.9963
1× 106 864.6968 1.0792 0.9883
1× 107 2715.0729 1.0722 0.9628
1× 108 8377.4601 1.0500 0.88192
1× 109 23585.5315 0.9743 0.6107
1.5× 109 26860.2040 0.9463 0.5020
1.8× 109 27470.2449 0.9302 0.4256
1.81× 109 27456.5804 0.9296 0.4225
1.85× 109 27357.9201 0.9274 0.4090
1.9× 109 27078.6014 0.9246 0.3886
1.95× 109 26126.5508 0.9217 0.3517
1.951× 109 26050.7695 0.9217 0.3495
1.952× 109 25937.4210 0.9216 0.3463
Table 1: Properties of the one shell low energy configuration for α = 1× 10−6
 1210  1215  1220  1225  1230  1235
 1210  1215  1220  1225  1230  1235
 0.99
 0.995
 1.005
 1.01
 1210  1215  1220  1225  1230  1235
Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 106 and
α = 1× 10−6.
modify the boundary condition (11) to read
F (r = 0) = Nπ (19)
F (r = ∞) = 0 (20)
whilst still ensuring that the Skyrme field is well defined at the origin. This idea was first
used in[12] to construct two shell configurations for the pure Skyrme model.
The baryon charge is now N times the degree of the rational map. Fig. 4 shows the
structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion
now exists as a N-layered structure. This is exhibited in the form of the profile, mass and
metric functions which interpolate between the boundary values in N distinct steps of equal
 1200  1205  1210  1215  1220  1225  1230
 1200  1205  1210  1215  1220  1225  1230
 0.995
 1.005
 1.01
 1.015
 1.02
 1200  1205  1210  1215  1220  1225  1230
Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations
(F (0) = 2π) when B = 2× 106 and α = 1× 10−6.
size stacked next to each other.
We can therefore think of this as a naive way of constructing a gravitating Skyrmion.
Instead of using the boundary conditions as in (11) and a rational map of degree B we
consider constructing the B-Skyrmion using a rational map of degree B/N (with the asso-
ciated value of I) and the boundary condition (20). This is a crude construction as we are
effectively considering N adjacent shells of baryons, all with the same baryon number. We
might realistically expect that the baryon number per shell and distribution of shells may
vary significantly for the minimum energy configuration. Nevertheless we shall study the
properties of such structures. In fact, in the case where the baryon number is large and the
number of shells is small, we expect this crude construction to be quite valid. That is, we
do not expect the baryon number to change significantly over the few shells at large radius.
B R( 2
) MADM (
2etopconv
) Smin
4 1.2898 1.6179 1.0000
8 1.7858 1.4072 1.0000
100 6.1754 1.1363 0.9999
1× 103 19.4157 1.0913 0.9996
1× 104 61.3207 1.0833 0.9985
1× 105 193.7006 1.0812 0.9949
1× 106 610.6271 1.0779 0.9835
1× 107 1911.3704 1.0680 0.9475
1× 108 5825.2626 1.0362 0.8325
9.0× 108 13736.9982 0.9302 0.4258
9.7× 108 13263.0853 0.9224 0.3644
9.76× 108 12998.2817 0.9217 0.3480
9.764× 108 12931.5189 0.9216 0.3444
9.7647× 108 12895.6984 0.9216 0.3425
9.76472× 108 12891.4247 0.9216 0.3423
9.764724× 108 12889.8645 0.9216 0.3422
Table 2: Table of properties of double layer solutions obtained numerically at α = 1× 10−6
For the remainder of this section we will restrict ourselves to the case where N = 2.
Table 2 summarises the properties of double layered gravitating skyrmions up to the
maximum baryon charge allowed at α = 1 × 10−6. Briefly, we note the main features.
Firstly, for all baryon numbers, the radius of the double layered solutions is significantly
less than their single layered counterparts. This is not surprising as the baryon charge exists
over a thicker region and so the mean radius can decrease with the baryon density remaining
the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense,
the double layer solutions are energetically favourable when one compares the ADM mass
with the single layer solutions. Finally we note that the maximum baryon number allowed
(at the given coupling) is almost twice as much in the case of the single layer skyrmions.
Of course the results of this section are not really the main regime of interest. We clearly
need to study configurations of extremely high baryon number (of order 1058) relevant for
baryon stars.
We will now discuss this high baryon number regime.
6 The Ramp-profile Approximation
Unfortunately, at very high baryon numbers, eqns. ((14) - (16)) become difficult to handle
numerically. This is largely because the radius of the solutions becomes much larger than
the distance over which the fields change. That is, we need to integrate over a region which
is much less than 10−16radius, and so even double precision data types have insufficient
precision.
Moreover, single shell configurations are not physically relevant and multiple shells will
only yield configuration that looks like a star if the number of layers is very large, typically
well over 1017. With such a large number of layers we won’t be able to solve the equation nu-
merically as we will need at least 10 times as many sampling points for the profile functions.
We must thus resort to another level of approximation: approximate the profile functions
by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14]
except that our ansatz has to be piecewise linear to be able to generate configurations with
a huge number of layers. After defining the ansatz for an arbitrary number of layers, we
will show that for a single layer configuration the ansatz produces configurations that are
in good agreements with the rational map ansatz configuration. Then we will use the new
ansatz to construct configurations that are made out of a very large no of layers.
We have shown, in the previous section, that one can construct shell like structures
with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell
like structures. That is, the fields are constant nearly everywhere except in a small region
corresponding to the shell. In that region, the profile look like linear functions smoothly
linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate
the fields by the ramp-functions
F (x) =
− (x− x0)
, (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (21)
µ(x) =
+ (x− x0)
, (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (22)
A(x) =
(1 + A0)
+ (x− x0)
(1−A0)
, (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (23)
In the above there are four free parameters, namely the central radius x0 of the shell
over which the fields change, the width of the shell W , the mass field at spatial infinity M
and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number
of layers we wish to study and, as such, is treated as an input parameter.
The picture is of a gravitating skyrmion with very high baryon number existing as N
thin layers or shell of small thickness.
The above ansatz, allow us to find an approximation to the integrated energy. To do
this we use the fact that the shell width is much smaller than the radius at large baryon
numbers. In particular to evaluate the action integral we can approximate expressions of
the type
G(x) sinp F (x) for any function G(x) that varies very little over the width of the
shell by
G(x0) sin
p F (x). We then use the fact that
Z x0+NW/2
x0−NW/2
F (x) =
y dy. (24)
This leads to the following expression for the energy:
E = −16πFπ
1 +A0
1 + A0
0 −Mx0 +
1− A0
W 2x0
− MWx0
1 + A0
−W − π
− 3IW
16x20
1 + A0
To find the configurations which minimize this energy we first minimised it with respect to
A0 and M algebraically in order to find an expression for the energy as a function of the
width and radius only. Then we minimised this numerically using Mathematica. We will
now discuss the features of these configurations.
First of all, we must compare the results obtained with the ramp-profile when N = 1
and compare them to the result obtained with the full profile. Tables. 3 and 4 show
the properties of solutions we obtained using the ramp-profile approximation, again at
α = 1 × 10−6. All the general features of the full numerical solutions are reproduced. In
particular, the approximate B
2 scaling and then decrease of the radius, the decreasing ADM
mass and the differences between the double and single layer solutions are all exhibited by
the data obtained using the ramp-profile approximation.
Quantitatively though, there are some differences. The approximation allows a signif-
icant increase in the maximum allowed baryon charge. Also, the radius of configurations
obtained using the approximation, tend to be smaller than those obtained numerically. If
we concentrate on the baryon numbers greater than 105 so as to ensure our approximation,
B R( 2
) W MADM (
) Smin
100 8.3063 3.1286 1.1023 0.9999
500 18.6031 3.1386 1.1160 0.9997
1× 103 26.313 3.1397 1.1195 0.9996
1× 104 83.206 3.1396 1.1254 0.9987
1× 105 262.94 3.1357 1.1266 0.9960
1× 106 829.60 3.1230 1.1251 0.9872
1× 107 2604.2 3.0825 1.1186 0.9595
1× 108 8032.8 2.9512 1.0972 0.8713
1× 109 22899 2.4837 1.0272 0.5772
2× 109 29121 2.1092 0.9818 0.3645
2.8× 109 29098 1.6623 0.9505 0.1380
2.83× 109 28514 1.6066 0.9495 0.1119
2.839× 109 28024 1.5671 0.94922 0.09373
2.8397× 109 27869.3 1.5556 0.94924 0.08845
2.83975× 109 27869.8 1.5524 0.94925 0.08699
2.839752× 109 27822 1.5521 0.94925 0.08687
Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon
number at fixed α = 1× 10−6
B R( 2
) W MADM (
) Smin
100 5.7924 3.0428 1.0692 0.9999
1× 103 18.5788 3.1305 1.1047 0.9995
1× 104 58.8202 3.1380 1.1201 0.9983
1× 105 185.8420 3.1332 1.1246 0.9944
1× 106 585.7950 3.1153 1.1233 0.9820
1× 107 1833.0500 3.0578 1.1143 0.9428
1× 108 5587.3600 2.8688 1.0840 0.8172
9× 108 14147.1782 2.1859 0.9900 0.4065
9.764724× 108 14472.3851 2.1276 0.9837 0.3746
1× 109 14560.5000 2.1092 0.9818 0.3646
1.4× 109 14549.0000 1.6623 0.9505 0.1381
1.41963× 109 13994.0523 1.5644 0.9492 0.0926
1.419635134× 109 13993.2000 1.5643 0.9492 0.0925
Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon
number at fixed α = 1× 10−6
that the width is much smaller than the radius, is valid, then at worst we find a discrepancy
in the ADM mass of 11% and in the radius of 7%.
In general then, the data seems to confirm the reliability of the ramp-profile approxi-
mation. In fact the approach will be even more reliable at the extremely high values of the
baryon number that we are interested in. This is because the radius of solutions is of orders
of magnitudes greater than the width in such a regime, consistent with the approximations
we have made.
Moreover, whilst searching for minima of the energy does not allow us to probe both
branches of solutions, it does allow us to locate the value of αcrit. We again obtain the
approximate trend αcrit ∝ B−
2 , for large B.
Now in order to say anything about the possibility of baryon stars in the Skyrme model
we need to be able to verify that the decrease in the ADM mass per baryon we observed
at low and moderate baryon numbers, extends to baryon numbers of order 1058 for α =
7.3× 10−40.
Table. 5 summarizes our solutions in such a regime. Firstly we consider constructing
a single layer self-gravitating Skyrmion with these values. We do indeed see that the con-
figuration is bound. This is verified by checking that the ADM mass is lower (even at this
significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon
stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.
The Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as
a tight lattice over this. However a hollow shell is clearly not a realistic construction for a
neutron star. This fact manifests itself in the extremely high radius of the configuration.
Transferring to standard units, the single layer B = 1058 gravitating Skyrmion has a radius
of 2.42× 1010km !
To address this issue, we can use a large number of layered Skyrmions as discussed
earlier. This has several benefits. Firstly, as we are distributing the baryon number through
a larger volume, then at a given baryon density the necessary radius can decrease. Similar
to what we see in the double layer results. On top of this, we expect the radius to decrease
further due to extra gravitational compression, as the outer layers of the Skyrmions feel
the attraction of inner layers. Finally, the many layer approach is also a more realistic
construction of a solid baryon star.
The results for using more and more layers in the construction (for fixed B and α), are
also presented in Fig. 5. We note that not only does the radius decrease significantly, but
the added gravitational binding further improves the energies of the configurations, reflected
in the low ADM masses obtained. There appears to be a critical number of layers that can
be used before there ceases to be any solutions and although the value of Smin is close to
zero at this point, the star still has not collapsed to form a black hole. Finally, we note that
the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This
is comparable to a real neutron star, with a typical radius of 10km.
We reemphasise here that our approach to embedding shells of baryons is quite crude.
For few shells and large baryon number, we might reasonably believe that baryon number
does not chance significantly from one shell to the next. However, when we embed many
shells we should really consider that the baryon number of the inner most shells would likely
be significantly less than the that of the outer shells. Nevertheless, our naive embedding has
produced some interesting properties. In a future work we hope to improve our multi-layer
construction to obtain a more realistic description of a baryon star.
NShell R(
) W MADM/(6π
2B) Smin
1× 102 8.3236× 1027 3.1416 1.1285 1.0000
1× 103 2.6321× 1027 3.1416 1.1285 1.0000
1× 104 8.3236× 1026 3.1416 1.1285 1.0000
1× 105 2.6321× 1026 3.1416 1.1285 1.0000
1× 106 8.3236× 1025 3.1416 1.1285 1.0000
1× 107 2.6321× 1025 3.1416 1.1285 1.0000
1× 108 8.3236× 1024 3.1416 1.1285 1.0000
1× 109 2.6321× 1024 3.1415 1.1285 1.0000
1× 1010 8.3234× 1023 3.1415 1.1285 0.9999
1× 1011 2.6319× 1023 3.1412 1.1285 0.9997
1× 1012 8.3216× 1022 3.1402 1.1283 0.9991
1× 1013 2.6301× 1022 3.1373 1.1278 0.9971
1× 1014 8.3034× 1021 3.1280 1.1263 0.9907
1× 1015 2.6118× 1021 3.0986 1.1213 0.9705
1× 1016 8.1147× 1020 3.0037 1.1057 0.9063
1× 1017 2.4001× 1020 2.6810 1.0552 0.6977
5× 1017 8.2066× 1019 1.7888 0.9552 0.2036
5.3× 1017 7.4172× 1019 1.6227 0.9491 0.1247
5.33× 1017 7.1871× 1019 1.5625 0.94866 0.0971
5.3306× 1017 7.1597× 1019 1.5549 0.94868 0.0936
5.33065× 1017 7.1525× 1019 1.5528 0.948694 0.0927
5.330657× 1017 7.1506× 1019 1.5523 0.948692 0.0924
Table 5: Table of properties of the step ansatz configurations for varying the number of embedded
shells at fixed B = 1058 and α = 7.3× 10−40.
7 Conclusions
Previous work on the Einstein-Skyrme model highlighted a considerable problem with using
the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were
simply not energetically favourable states. We have shown that this is simply a consequence
of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map
ansatz, we have generated energetically favourable configurations of multibaryons.
We also observe the interesting property that near the critical coupling, the Skyrmions
can decrease in radius as we add more baryons. This hints towards the similar behaviour
exhibited by real neutron stars.
Although the rational map ansatz does not have an exact radial symmetry, at large scale
it does. The anisotropy only appears at the nucleon scale.
Finally, since we started with the motivation of studying baryon stars within the Skyrme
model, it is interesting to compare the features of our configurations with those of neutron
stars. For realistic values, B = 1058 and α = 7.3 × 10−40 we find a minimal energy single
layer configuration with radius=2.42 × 1010km. This is clearly too large for a neutron star
(which is of order 10km. in radius). This is to be expected however due to the shell model
we have taken. Firstly, as we are distributing the baryons over the surface area rather
than throughout the volume of the star we naturally must require a much larger star for
a given baryon number. This effect is two-fold in that if we were distributing the baryons
throughout the volume, outer layers would feel the attraction of inner layers and enhanced
radial compression would occur. The loss of such an effect is pronounced when we are
considering realistically small values of the coupling.
It seems therefore that the way to construct baryon stars in the Skyrme model is to
consider embedding shells of baryons within shells. This gives rise to more appropriate
specifications for the star and is also more realistic. We do indeed observe such improvements
for a many layered configuration. In fact the radius of B = 1058 gravitating Skyrmion (at
realistic α), can be decreased in this manner to approximately 20.91km.
We note however that this approach to shell embedding has only be done naively thus far.
We have only considered the case where the baryon number is equal for each shell. We really
should allow the baryon number(and hence the rational map quantities) to vary over the
shells. One approach towards this would be to assume that the baryon density is a constant
over the shells. An even better approach would be to allow this to be a smoothly varying
function that must be determined by minimising the energy. This will give a more realistic
description of baryon stars within the Einstein-Skyrme model, as traditional descriptions
of neutron stars also involve many strata, of differing neutron density. We are currently
investigating such configurations.
8 Acknowledgements
GIP is supported by a PPARC studentship.
References
[1] P. Bizon & T. Chmaj “Gravitating Skyrmions” Phys. Lett. B 297 (1992), 55-62
[2] T. H. R. Skyrme, “A Non-Linear Field Theory” Proc. Roy. Soc. A 260 (1961), 127-138
[3] T. H. R. Skyrme, “A Unified Theory of Mesons and Baryons” Nucl. Phys.31 (1962)
[4] E. Witten “Global Aspects of Current Algebra” Nucl. Phys. B223 (1983), 422-433
[5] N. K. Glendenning, T. Kodama & F. R. Klinkhamer “Skyrme Topological Soliton
Coupled to Gravity” Phys. Rev. D 38 Number 10 (1988), 3226-3230
[6] M. S. Volkov & D. V. Gal’tsov “Gravitating Non-Abelian solitons and Black Holes with
Yang-Mills Fields” Physics Reports, 319, Numbers 1-2, 1-83 (1999)
[7] H. Luckock & I. Moss “Black Holes HAve Skyrmion Hair” Phys. Lett B176 (1986),341-
[8] S. Droz, M. Heusler & N. Straumann “New Black Hole Solutions with Hair” Phys.
Lett. B268 (1991), 371-376
[9] R.A. Battye, P.M. Sutcliffe “ MULTI - SOLITON DYNAMICS IN THE SKYRME
MODEL.” Phys.Lett. B391 (1997), 150-156
[10] C. Houghton, N. Manton & P. Sutcliffe “Rational MAps, Monopoles and Skyrmions”
Nucl. Phys. B510 (1998), 507-537
[11] R. A. Battye & P. M. Sutcliffe “Skyrmions, Fullerenes and Rational Maps” Rev. Math.
Phys. 14 (2002), 29-86
[12] N. S. Manton & B. M. A. G. Piette “Understanding Skyrmions using Rational
Maps” hep-th/0008110 Understanding Skyrmions Using Rational Maps: Proceedings
of the European Congress of Mathematics, Barcelona 2000, eds. C.Casacuberta et al.,
Progress in Mathematics, Birkhauser, Basel Vol. 201 (2001) 469-479
http://arxiv.org/abs/hep-th/0008110
[13] V. B. Kopeliovich “The Bubbles of Matter from MultiSkyrmions” JETP Lett. 73
(2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671
[14] V. B. Kopeliovich “MultiSkyrmions and Baryonic Bags” J.Phys. G28 (2002), 103-120
	Introduction
	The Einstein-Skyrme Model
	Gravitating Hedgehog Skyrmions
	The Rational Map Ansatz
	Hollow Skyrmion Shells
	The Ramp-profile Approximation
	Conclusions
	Acknowledgements

  We investigate the large baryon number sector of the Einstein-Skyrme model as
a possible model for baryon stars. Gravitating hedgehog skyrmions have been
investigated previously and the existence of stable solitonic stars excluded
due to energy considerations. However, in this paper we demonstrate that by
generating gravitating skyrmions using rational maps, we can achieve
multi-baryon bound states whilst recovering spherical symmetry in the limit
where B becomes large.

Introduction
The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in
a series of papers as a non-linear field theory of pions [2], [3]. Skyrme’s initial idea was to
think of baryons (in particular the nucleons) as secondary structures arising from a more
fundamental mesonic fluid. The key property of the model was that the baryons arose as
solitons in a topological manner and thus possessed a conserved topological charge identified
with the baryon number.
The lowest energy stable solutions of the model are termed Skyrmions and can be thought
of as baryonic solitons. The Skyrme model has been very successful in modelling the struc-
tures of various nuclei and has been shown by Witten et al. [4] to possess the general
features of a low energy effective field theory for QCD.
Some studies of the Skyrme model coupled to gravity have previously been undertaken
[1], [5], [6], mainly with the motivation of a comparison of its features with those of other non-
linear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory,
in which gravitationally bound configurations of non-abelian gauge fields are produced.
Other reasons for studying the Einstein-Skyrme model are cosmological and astrophysi-
cal ones. Various authors have studied black hole formation in the model, with the conclu-
sion that the so-called no-hair conjecture may not hold [7], [8].
The purpose of this paper is to study large baryon number Skyrmions or configurations
of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable
solitonic stars could exist within the model and to compare their properties to those of
neutron stars.
Preliminary studies of Skyrmion stars have predicted instability to single particle decay
[1]. However this was done using the hedgehog ansatz for baryon number larger than 1
which is known to lead to unstable solutions even for the usual Skyrme model. Since then,
it has been shown that the Skyrme model has stable shell-like solutions[9] which can be well
approximated by the so called rational map ansatz [10].
In this paper we use the rational map ansatz and its extension to multiple shells to
construct configurations in the Gravitating Skyrme model that have a very large number
of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound
even for very large baryon numbers.
To construct configurations that have a baryon number comparable to that of neutron
star, we have to introduce a further approximation, which we call the ramp ansatz. We show
that this anstaz introduces further errors of only a few percent and we use it to compute
very large Skyrmion configurations.
The paper is organised as follow: first we outline the Einstein-Skyrme model and discuss
the main features of the results on static gravitating SU(2) hedgehogs obtained by Bizon
and Chmaj [1]. We then use the rational map ansatz to construct shell like gravitating
multibaryon configurations and show that for a fixed value of the coupling constant, the
configurations exist only when the baryon number is below a certain critical value. Finally
we introduce a ramp profile approximation to construct solutions with extremely high baryon
numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.
2 The Einstein-Skyrme Model
The action for gravitating Skyrmions is formed from the standard Skyrme action for the
matter field and the Einstein-Hilbert action for the gravitational field.
LSk −
x. (1)
Here LSk is the Lagrangian density for the Skyrme model defined on the manifold M :
LSk =
Tr(∇µU∇µU−1) +
Tr[(∇µU)U−1, (∇νU)U−1]2, (2)
where U belongs to SU(2). As we eventually wish to study baryon stars, we take a spheri-
cally symmetric metric, such as associated with the line element
= −A2(r)
1− 2m(r)
1− 2m(r)
+ sin
), (3)
where A(r) and m(r) are two profile functions that must be determined by solving the
Einstein equations for the model. Our choice of ansatz is motivated by the fact that although
in some cases we will be studying non-spherical Skyrmion configurations, the regime we are
primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to
admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational
interaction is small compared to the Skyrme interaction and thus the use of a spherical
metric even with non-spherical configurations, is not a great problem.
From (3), it can be shown that the Ricci scalar is
−A′′r2 − 2A′r + 2A′′rm+ A′m+ 3A′rm′ + Arm′′ + 2Am′
which, after integrating various terms by parts and noting that asymptotic flatness requires
both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational
part of the action to
Sgr =
−m′(r)
. (5)
For what follows, it will be convenient to scale to dimensionless variables by defining x =
eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the
model, α = πF 2πG. We note that taking Fπ = 186Mev and G = 6.72 × 10−45Mev−2, then
the physical value of the coupling is α = 7.3× 10−40.
As the Skyrme field is an SU(2) valued scalar field, at any given time one can think of it
as a map from R3 to the SU(2) manifold. Finite energy considerations impose that the field
at spatial infinity should map to the same point on SU(2), say the identity. Thus, one can
simply think of the Skyrme field as a map between three-spheres. All such maps fall into
disjoint homotopy classes characterised by their winding number. This winding number is
a conserved topological charge because no continuous deformation of the field and thus no
time evolution, can allow transitions between homotopy classes. It is this topological charge
that is interpreted as the baryon number.
3 Gravitating Hedgehog Skyrmions
Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties
of static spherically symmetric gravitating SU(2) skyrmions. Taking the Hedgehog Ansatz
for the Skyrme field
U = exp(i−→σ .r̂F (r)) (6)
subject to the boundary conditions
F (r = 0) = Bπ (7)
F (r = ∞) = 0 (8)
where B is the Baryon number associated with the Skyrmion configuration, they derived
the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the
model admit two branches of global solitonic solutions at each given baryon number, which
annihilate at a critical value of the coupling parameter. Above αcrit no further solutions
were found. In particular the value of the critical coupling decreased quite considerably
with increasing baryon number as αcrit ≈ 0.040378/B2 . It appears that the existence of a
critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact
the metric factor S(x) = (1 − 2µ(x)
) is non-zero at αcrit; there simply ceases to be any
stationary points of the action above the critical coupling.
The major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any
given value of α, MADM (B = N) > NMADM (B = 1). This is actually the case for the pure
Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the
lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1
are known not to be spherically symmetric[11] but are stable i.e. E(B = N) < N ∗E(B = 1).
It was actually shown by Houghton et al [10], [12] that the multi-baryon solutions of
the pure Skyrme model can be well approximated by the so called rational maps ansatz
which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz
separates its radial and angular dependence through a profile function and a rational map
respectively.
In the following sections we will generalise the construction of Houghton et al to approx-
imate the solution of the Einstein-Skyrme model.
4 The Rational Map Ansatz
The rational map ansatz introduced by Houghton et al.[10] works by decomposing the
field into angular and radial parts. Using the polar coordinates in R3 and defining the
stereographic coordinates z = tan(θ/2) expiφ the ansatz reads [10]
U = exp (i~σ · n̂RF (r, t)) (9)
where
n̂R =
1 + |R|2
2ℜ(R), 2ℑ(R), 1− |R|2
is a unit vector where R is a rational function of z.
It can be shown that the baryon number for Skyrmions constructed in this way, is equal
to the degree of the rational map providing we take the boundary conditions
F (r = 0) = π
F (r = ∞) = 0. (11)
Substituting the ansatz (9) into the action for the model and scaling to dimensionless
variables as earlier, we obtain the following reduced Hamiltonian
16πFπ
S(x)F (x)
+ Bsin2F (x)(1 + S(x)F (x)′2)
Isin4F (x)
where
S(x) = 1− 2µ(x)
From which one obtains the following field equations
S(x)x
F (x)
+ B sin2 F (x) + S(x)BF (x)′2 sin2 F (x) + I sin
4 F (x)
F (x)
S(x)V (x)
sin 2F (x)
B + S(x)BF (x)′2 + I sin
2 F (x)
− αS(x)F (x)
′3V (x)2
− S(x)′F (x)′V (x)− S(x)F (x)′V (x)′
= αA(x)F (x)
x+ 2B sin
2 F (x)
where, for convenience, we have defined V (x) as
V (x) = x
+ 2B sin2 F (x). (17)
B is the baryon number and
I = 1
1 + |z|2
1 + |R|2
2idzdz
(1 + |z|2)2
Its value depends on the chosen rational map R. To compute low energy configurations for
a given baryon charge B one must find the rational map R or degree B that minimize I.
This has been done in [10] and [11] for several values of B. Moreover when b is large, one
can use the approximation[11] I ≈ 1.28B2 . The value of I so obtained is then used as a
parameter and one can solve equations (14) - (16) for the radial profiles F (x), A(x) and
µ(x).
We should point out here that for the pure Skyrme model the rational map ansatz
produce very good approximation to the multi skyrmion solutions [10]: the energies are only
3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by
very little. All the solutions computed by Battye and Sutcliffe[11], when B is not too small,
have somehow the shape of a hollow shell. The baryon density is very small everywhere
outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.
5 Hollow Skyrmion Shells
Using the rational map ansatz, we will now solves the field equations (14) - (16) to compute
some low action configurations. These solutions will correspond, initially, to a hollow shell
of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure
Skyrme model. In the following sections we will show how our ansatz can be generalised to
allow for more realistic configuration made out of embedded shells.
The first thing to note about our solutions is that we again obtain two branches of
solutions at each baryon number (Fig. 1). Obtaining this same qualitative behaviour is not
surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual
B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered
for the rational map generated configurations. Namely, we observe that it decreases as
approximately 0.040378/B
2 (Fig. 2). In particular this means that for a given value of the
coupling, the rational map generated skyrmions can possess a much higher topological charge
than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively
if Bhedgehog is the maximum baryon number for which hedgehog solutions can be found at
a given value of the coupling, then the highest baryon number rational map solution found
at the same value of α will be approximately B4hedgehog . Again we observe that the metric
function S(x) is non-zero at the critical coupling for all the solutions we have found and as
such a horizon has not formed.
In Table 1 we present the radius, ADM mass per baryon and minimum value of the
metric function, S(x), for configurations up to the maximum baryon number allowed at
α = 1× 10−6. These values were obtained by direct numerical solution of equations (14) -
(16), where we have used the boundary data as specified in (11).
We didn’t didn’t use the physical value of α (7.3×10−40) because for this value, the ratio
between the width of the shell and its radius is so small when we reach the maximum value
of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10−6
is small enough to allow for a shell with a large baryon number to exist but large enough
to make it possible to compute these solution nears the critical value of B for a single shell
configuration.
 0  0.005  0.01  0.015  0.02  0.025  0.03  0.035
Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the
rational map ansatz.
The major difference between these configuration and the solutions of Bizon and Chmadj
is that the rational map ansatz configurations become more bound when the baryon number
increases This suggests the possibility that giant gravitating Skyrmions can be bound and
consequently, that the Skyrme model can be used to study baryon stars.
Another interesting feature of the data is the observed change in the radius of the solu-
tions with increasing baryon number. We note that the radius grows as approximately B
However there are two main deviations from this. Firstly, the constant of proportionality
relating the radius to the square root of the baryon number decreases slightly but persis-
tently as we increase the baryon number, indicating the gravitational interaction becoming
more important as the number of baryons increases.
As we approach the maximum baryon charge that can exist at α = 1 × 10−6, we also
notice that the radius of the skyrmion actually decreases as we add more baryons. This
shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.
This is a tantalising property when one considers that generally a neutron star’s radius must
decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
 0.035
 0.04
 0.045
 0.05
 2  4  6  8  10  12  14  16  18  20
Figure 2: Plot of the decrease in αcrit with increasing baryon number, for configurations generated
with the rational map ansatz. +: αcr for the minimum value of I; curve: αcr = 0.0404B
−1/2.
to support the star.
To motivate the further approximation that we will introduce in the next section, we
now look at the profiles of the configuration that we have computed. First of all, we observe
that the profile function F (x) stays approximately at its boundary value, π, for a finite
radial distance before decreasing monotonically over some small region and finally attaining
its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x)
and the metric field A(x) (see Fig. 3). Furthermore, as we increase the baryon number
the structure becomes more pronounced, with the distance before the fields change (shell
radius) increasing significantly, whilst the distance over which the fields change (shell width)
settles to a constant size. We conclude that at large baryon numbers, those configurations
correspond to hollow shells where the baryons are distributed on a tight lattice over the
shell. As such the, structures are nearly spherical, validating our choice of radial metric.
Such structures immediately pose an interesting question. Can the gravitating Skyrmions
exist as shells with more than one layer? To investigate this we note that it is possible to
B R( 2
) MADM (
2etopconv
) Smin
1 0.8763 1.2315 1.0000
4 1.7728 1.1365 1.0000
8 2.5065 1.1180 1.0000
100 8.6829 1.0845 0.9999
500 19.3994 1.0827 0.9998
1× 103 27.4314 1.0825 0.9997
1× 104 86.7192 1.0821 0.9989
1× 105 274.0397 1.0814 0.9963
1× 106 864.6968 1.0792 0.9883
1× 107 2715.0729 1.0722 0.9628
1× 108 8377.4601 1.0500 0.88192
1× 109 23585.5315 0.9743 0.6107
1.5× 109 26860.2040 0.9463 0.5020
1.8× 109 27470.2449 0.9302 0.4256
1.81× 109 27456.5804 0.9296 0.4225
1.85× 109 27357.9201 0.9274 0.4090
1.9× 109 27078.6014 0.9246 0.3886
1.95× 109 26126.5508 0.9217 0.3517
1.951× 109 26050.7695 0.9217 0.3495
1.952× 109 25937.4210 0.9216 0.3463
Table 1: Properties of the one shell low energy configuration for α = 1× 10−6
 1210  1215  1220  1225  1230  1235
 1210  1215  1220  1225  1230  1235
 0.99
 0.995
 1.005
 1.01
 1210  1215  1220  1225  1230  1235
Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 106 and
α = 1× 10−6.
modify the boundary condition (11) to read
F (r = 0) = Nπ (19)
F (r = ∞) = 0 (20)
whilst still ensuring that the Skyrme field is well defined at the origin. This idea was first
used in[12] to construct two shell configurations for the pure Skyrme model.
The baryon charge is now N times the degree of the rational map. Fig. 4 shows the
structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion
now exists as a N-layered structure. This is exhibited in the form of the profile, mass and
metric functions which interpolate between the boundary values in N distinct steps of equal
 1200  1205  1210  1215  1220  1225  1230
 1200  1205  1210  1215  1220  1225  1230
 0.995
 1.005
 1.01
 1.015
 1.02
 1200  1205  1210  1215  1220  1225  1230
Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations
(F (0) = 2π) when B = 2× 106 and α = 1× 10−6.
size stacked next to each other.
We can therefore think of this as a naive way of constructing a gravitating Skyrmion.
Instead of using the boundary conditions as in (11) and a rational map of degree B we
consider constructing the B-Skyrmion using a rational map of degree B/N (with the asso-
ciated value of I) and the boundary condition (20). This is a crude construction as we are
effectively considering N adjacent shells of baryons, all with the same baryon number. We
might realistically expect that the baryon number per shell and distribution of shells may
vary significantly for the minimum energy configuration. Nevertheless we shall study the
properties of such structures. In fact, in the case where the baryon number is large and the
number of shells is small, we expect this crude construction to be quite valid. That is, we
do not expect the baryon number to change significantly over the few shells at large radius.
B R( 2
) MADM (
2etopconv
) Smin
4 1.2898 1.6179 1.0000
8 1.7858 1.4072 1.0000
100 6.1754 1.1363 0.9999
1× 103 19.4157 1.0913 0.9996
1× 104 61.3207 1.0833 0.9985
1× 105 193.7006 1.0812 0.9949
1× 106 610.6271 1.0779 0.9835
1× 107 1911.3704 1.0680 0.9475
1× 108 5825.2626 1.0362 0.8325
9.0× 108 13736.9982 0.9302 0.4258
9.7× 108 13263.0853 0.9224 0.3644
9.76× 108 12998.2817 0.9217 0.3480
9.764× 108 12931.5189 0.9216 0.3444
9.7647× 108 12895.6984 0.9216 0.3425
9.76472× 108 12891.4247 0.9216 0.3423
9.764724× 108 12889.8645 0.9216 0.3422
Table 2: Table of properties of double layer solutions obtained numerically at α = 1× 10−6
For the remainder of this section we will restrict ourselves to the case where N = 2.
Table 2 summarises the properties of double layered gravitating skyrmions up to the
maximum baryon charge allowed at α = 1 × 10−6. Briefly, we note the main features.
Firstly, for all baryon numbers, the radius of the double layered solutions is significantly
less than their single layered counterparts. This is not surprising as the baryon charge exists
over a thicker region and so the mean radius can decrease with the baryon density remaining
the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense,
the double layer solutions are energetically favourable when one compares the ADM mass
with the single layer solutions. Finally we note that the maximum baryon number allowed
(at the given coupling) is almost twice as much in the case of the single layer skyrmions.
Of course the results of this section are not really the main regime of interest. We clearly
need to study configurations of extremely high baryon number (of order 1058) relevant for
baryon stars.
We will now discuss this high baryon number regime.
6 The Ramp-profile Approximation
Unfortunately, at very high baryon numbers, eqns. ((14) - (16)) become difficult to handle
numerically. This is largely because the radius of the solutions becomes much larger than
the distance over which the fields change. That is, we need to integrate over a region which
is much less than 10−16radius, and so even double precision data types have insufficient
precision.
Moreover, single shell configurations are not physically relevant and multiple shells will
only yield configuration that looks like a star if the number of layers is very large, typically
well over 1017. With such a large number of layers we won’t be able to solve the equation nu-
merically as we will need at least 10 times as many sampling points for the profile functions.
We must thus resort to another level of approximation: approximate the profile functions
by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14]
except that our ansatz has to be piecewise linear to be able to generate configurations with
a huge number of layers. After defining the ansatz for an arbitrary number of layers, we
will show that for a single layer configuration the ansatz produces configurations that are
in good agreements with the rational map ansatz configuration. Then we will use the new
ansatz to construct configurations that are made out of a very large no of layers.
We have shown, in the previous section, that one can construct shell like structures
with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell
like structures. That is, the fields are constant nearly everywhere except in a small region
corresponding to the shell. In that region, the profile look like linear functions smoothly
linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate
the fields by the ramp-functions
F (x) =
− (x− x0)
, (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (21)
µ(x) =
+ (x− x0)
, (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (22)
A(x) =
(1 + A0)
+ (x− x0)
(1−A0)
, (x0 −NW/2) ≤ x ≤ (x0 +NW/2) (23)
In the above there are four free parameters, namely the central radius x0 of the shell
over which the fields change, the width of the shell W , the mass field at spatial infinity M
and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number
of layers we wish to study and, as such, is treated as an input parameter.
The picture is of a gravitating skyrmion with very high baryon number existing as N
thin layers or shell of small thickness.
The above ansatz, allow us to find an approximation to the integrated energy. To do
this we use the fact that the shell width is much smaller than the radius at large baryon
numbers. In particular to evaluate the action integral we can approximate expressions of
the type
G(x) sinp F (x) for any function G(x) that varies very little over the width of the
shell by
G(x0) sin
p F (x). We then use the fact that
Z x0+NW/2
x0−NW/2
F (x) =
y dy. (24)
This leads to the following expression for the energy:
E = −16πFπ
1 +A0
1 + A0
0 −Mx0 +
1− A0
W 2x0
− MWx0
1 + A0
−W − π
− 3IW
16x20
1 + A0
To find the configurations which minimize this energy we first minimised it with respect to
A0 and M algebraically in order to find an expression for the energy as a function of the
width and radius only. Then we minimised this numerically using Mathematica. We will
now discuss the features of these configurations.
First of all, we must compare the results obtained with the ramp-profile when N = 1
and compare them to the result obtained with the full profile. Tables. 3 and 4 show
the properties of solutions we obtained using the ramp-profile approximation, again at
α = 1 × 10−6. All the general features of the full numerical solutions are reproduced. In
particular, the approximate B
2 scaling and then decrease of the radius, the decreasing ADM
mass and the differences between the double and single layer solutions are all exhibited by
the data obtained using the ramp-profile approximation.
Quantitatively though, there are some differences. The approximation allows a signif-
icant increase in the maximum allowed baryon charge. Also, the radius of configurations
obtained using the approximation, tend to be smaller than those obtained numerically. If
we concentrate on the baryon numbers greater than 105 so as to ensure our approximation,
B R( 2
) W MADM (
) Smin
100 8.3063 3.1286 1.1023 0.9999
500 18.6031 3.1386 1.1160 0.9997
1× 103 26.313 3.1397 1.1195 0.9996
1× 104 83.206 3.1396 1.1254 0.9987
1× 105 262.94 3.1357 1.1266 0.9960
1× 106 829.60 3.1230 1.1251 0.9872
1× 107 2604.2 3.0825 1.1186 0.9595
1× 108 8032.8 2.9512 1.0972 0.8713
1× 109 22899 2.4837 1.0272 0.5772
2× 109 29121 2.1092 0.9818 0.3645
2.8× 109 29098 1.6623 0.9505 0.1380
2.83× 109 28514 1.6066 0.9495 0.1119
2.839× 109 28024 1.5671 0.94922 0.09373
2.8397× 109 27869.3 1.5556 0.94924 0.08845
2.83975× 109 27869.8 1.5524 0.94925 0.08699
2.839752× 109 27822 1.5521 0.94925 0.08687
Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon
number at fixed α = 1× 10−6
B R( 2
) W MADM (
) Smin
100 5.7924 3.0428 1.0692 0.9999
1× 103 18.5788 3.1305 1.1047 0.9995
1× 104 58.8202 3.1380 1.1201 0.9983
1× 105 185.8420 3.1332 1.1246 0.9944
1× 106 585.7950 3.1153 1.1233 0.9820
1× 107 1833.0500 3.0578 1.1143 0.9428
1× 108 5587.3600 2.8688 1.0840 0.8172
9× 108 14147.1782 2.1859 0.9900 0.4065
9.764724× 108 14472.3851 2.1276 0.9837 0.3746
1× 109 14560.5000 2.1092 0.9818 0.3646
1.4× 109 14549.0000 1.6623 0.9505 0.1381
1.41963× 109 13994.0523 1.5644 0.9492 0.0926
1.419635134× 109 13993.2000 1.5643 0.9492 0.0925
Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon
number at fixed α = 1× 10−6
that the width is much smaller than the radius, is valid, then at worst we find a discrepancy
in the ADM mass of 11% and in the radius of 7%.
In general then, the data seems to confirm the reliability of the ramp-profile approxi-
mation. In fact the approach will be even more reliable at the extremely high values of the
baryon number that we are interested in. This is because the radius of solutions is of orders
of magnitudes greater than the width in such a regime, consistent with the approximations
we have made.
Moreover, whilst searching for minima of the energy does not allow us to probe both
branches of solutions, it does allow us to locate the value of αcrit. We again obtain the
approximate trend αcrit ∝ B−
2 , for large B.
Now in order to say anything about the possibility of baryon stars in the Skyrme model
we need to be able to verify that the decrease in the ADM mass per baryon we observed
at low and moderate baryon numbers, extends to baryon numbers of order 1058 for α =
7.3× 10−40.
Table. 5 summarizes our solutions in such a regime. Firstly we consider constructing
a single layer self-gravitating Skyrmion with these values. We do indeed see that the con-
figuration is bound. This is verified by checking that the ADM mass is lower (even at this
significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon
stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.
The Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as
a tight lattice over this. However a hollow shell is clearly not a realistic construction for a
neutron star. This fact manifests itself in the extremely high radius of the configuration.
Transferring to standard units, the single layer B = 1058 gravitating Skyrmion has a radius
of 2.42× 1010km !
To address this issue, we can use a large number of layered Skyrmions as discussed
earlier. This has several benefits. Firstly, as we are distributing the baryon number through
a larger volume, then at a given baryon density the necessary radius can decrease. Similar
to what we see in the double layer results. On top of this, we expect the radius to decrease
further due to extra gravitational compression, as the outer layers of the Skyrmions feel
the attraction of inner layers. Finally, the many layer approach is also a more realistic
construction of a solid baryon star.
The results for using more and more layers in the construction (for fixed B and α), are
also presented in Fig. 5. We note that not only does the radius decrease significantly, but
the added gravitational binding further improves the energies of the configurations, reflected
in the low ADM masses obtained. There appears to be a critical number of layers that can
be used before there ceases to be any solutions and although the value of Smin is close to
zero at this point, the star still has not collapsed to form a black hole. Finally, we note that
the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This
is comparable to a real neutron star, with a typical radius of 10km.
We reemphasise here that our approach to embedding shells of baryons is quite crude.
For few shells and large baryon number, we might reasonably believe that baryon number
does not chance significantly from one shell to the next. However, when we embed many
shells we should really consider that the baryon number of the inner most shells would likely
be significantly less than the that of the outer shells. Nevertheless, our naive embedding has
produced some interesting properties. In a future work we hope to improve our multi-layer
construction to obtain a more realistic description of a baryon star.
NShell R(
) W MADM/(6π
2B) Smin
1× 102 8.3236× 1027 3.1416 1.1285 1.0000
1× 103 2.6321× 1027 3.1416 1.1285 1.0000
1× 104 8.3236× 1026 3.1416 1.1285 1.0000
1× 105 2.6321× 1026 3.1416 1.1285 1.0000
1× 106 8.3236× 1025 3.1416 1.1285 1.0000
1× 107 2.6321× 1025 3.1416 1.1285 1.0000
1× 108 8.3236× 1024 3.1416 1.1285 1.0000
1× 109 2.6321× 1024 3.1415 1.1285 1.0000
1× 1010 8.3234× 1023 3.1415 1.1285 0.9999
1× 1011 2.6319× 1023 3.1412 1.1285 0.9997
1× 1012 8.3216× 1022 3.1402 1.1283 0.9991
1× 1013 2.6301× 1022 3.1373 1.1278 0.9971
1× 1014 8.3034× 1021 3.1280 1.1263 0.9907
1× 1015 2.6118× 1021 3.0986 1.1213 0.9705
1× 1016 8.1147× 1020 3.0037 1.1057 0.9063
1× 1017 2.4001× 1020 2.6810 1.0552 0.6977
5× 1017 8.2066× 1019 1.7888 0.9552 0.2036
5.3× 1017 7.4172× 1019 1.6227 0.9491 0.1247
5.33× 1017 7.1871× 1019 1.5625 0.94866 0.0971
5.3306× 1017 7.1597× 1019 1.5549 0.94868 0.0936
5.33065× 1017 7.1525× 1019 1.5528 0.948694 0.0927
5.330657× 1017 7.1506× 1019 1.5523 0.948692 0.0924
Table 5: Table of properties of the step ansatz configurations for varying the number of embedded
shells at fixed B = 1058 and α = 7.3× 10−40.
7 Conclusions
Previous work on the Einstein-Skyrme model highlighted a considerable problem with using
the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were
simply not energetically favourable states. We have shown that this is simply a consequence
of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map
ansatz, we have generated energetically favourable configurations of multibaryons.
We also observe the interesting property that near the critical coupling, the Skyrmions
can decrease in radius as we add more baryons. This hints towards the similar behaviour
exhibited by real neutron stars.
Although the rational map ansatz does not have an exact radial symmetry, at large scale
it does. The anisotropy only appears at the nucleon scale.
Finally, since we started with the motivation of studying baryon stars within the Skyrme
model, it is interesting to compare the features of our configurations with those of neutron
stars. For realistic values, B = 1058 and α = 7.3 × 10−40 we find a minimal energy single
layer configuration with radius=2.42 × 1010km. This is clearly too large for a neutron star
(which is of order 10km. in radius). This is to be expected however due to the shell model
we have taken. Firstly, as we are distributing the baryons over the surface area rather
than throughout the volume of the star we naturally must require a much larger star for
a given baryon number. This effect is two-fold in that if we were distributing the baryons
throughout the volume, outer layers would feel the attraction of inner layers and enhanced
radial compression would occur. The loss of such an effect is pronounced when we are
considering realistically small values of the coupling.
It seems therefore that the way to construct baryon stars in the Skyrme model is to
consider embedding shells of baryons within shells. This gives rise to more appropriate
specifications for the star and is also more realistic. We do indeed observe such improvements
for a many layered configuration. In fact the radius of B = 1058 gravitating Skyrmion (at
realistic α), can be decreased in this manner to approximately 20.91km.
We note however that this approach to shell embedding has only be done naively thus far.
We have only considered the case where the baryon number is equal for each shell. We really
should allow the baryon number(and hence the rational map quantities) to vary over the
shells. One approach towards this would be to assume that the baryon density is a constant
over the shells. An even better approach would be to allow this to be a smoothly varying
function that must be determined by minimising the energy. This will give a more realistic
description of baryon stars within the Einstein-Skyrme model, as traditional descriptions
of neutron stars also involve many strata, of differing neutron density. We are currently
investigating such configurations.
8 Acknowledgements
GIP is supported by a PPARC studentship.
References
[1] P. Bizon & T. Chmaj “Gravitating Skyrmions” Phys. Lett. B 297 (1992), 55-62
[2] T. H. R. Skyrme, “A Non-Linear Field Theory” Proc. Roy. Soc. A 260 (1961), 127-138
[3] T. H. R. Skyrme, “A Unified Theory of Mesons and Baryons” Nucl. Phys.31 (1962)
[4] E. Witten “Global Aspects of Current Algebra” Nucl. Phys. B223 (1983), 422-433
[5] N. K. Glendenning, T. Kodama & F. R. Klinkhamer “Skyrme Topological Soliton
Coupled to Gravity” Phys. Rev. D 38 Number 10 (1988), 3226-3230
[6] M. S. Volkov & D. V. Gal’tsov “Gravitating Non-Abelian solitons and Black Holes with
Yang-Mills Fields” Physics Reports, 319, Numbers 1-2, 1-83 (1999)
[7] H. Luckock & I. Moss “Black Holes HAve Skyrmion Hair” Phys. Lett B176 (1986),341-
[8] S. Droz, M. Heusler & N. Straumann “New Black Hole Solutions with Hair” Phys.
Lett. B268 (1991), 371-376
[9] R.A. Battye, P.M. Sutcliffe “ MULTI - SOLITON DYNAMICS IN THE SKYRME
MODEL.” Phys.Lett. B391 (1997), 150-156
[10] C. Houghton, N. Manton & P. Sutcliffe “Rational MAps, Monopoles and Skyrmions”
Nucl. Phys. B510 (1998), 507-537
[11] R. A. Battye & P. M. Sutcliffe “Skyrmions, Fullerenes and Rational Maps” Rev. Math.
Phys. 14 (2002), 29-86
[12] N. S. Manton & B. M. A. G. Piette “Understanding Skyrmions using Rational
Maps” hep-th/0008110 Understanding Skyrmions Using Rational Maps: Proceedings
of the European Congress of Mathematics, Barcelona 2000, eds. C.Casacuberta et al.,
Progress in Mathematics, Birkhauser, Basel Vol. 201 (2001) 469-479
http://arxiv.org/abs/hep-th/0008110
[13] V. B. Kopeliovich “The Bubbles of Matter from MultiSkyrmions” JETP Lett. 73
(2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671
[14] V. B. Kopeliovich “MultiSkyrmions and Baryonic Bags” J.Phys. G28 (2002), 103-120
	Introduction
	The Einstein-Skyrme Model
	Gravitating Hedgehog Skyrmions
	The Rational Map Ansatz
	Hollow Skyrmion Shells
	The Ramp-profile Approximation
	Conclusions
	Acknowledgements
