# A modified dyon solution in a non-Abelian gauge model

A.Yu. Loginov<EMAIL_ADDRESS>Laboratory of Applied Mathematics and
Theoretical Physics, Tomsk State University of Control Systems and
Radioelectronics, 634050 Tomsk, Russia

###### Abstract

A modified $SU(2)$ Georgi-Glashow model is considered here, and it is shown
that besides the well-known Julia-Zee dyon solution, the model can also have a
modified dyon solution. The properties of the modified dyon solution are
studied using analytical and numerical methods. A comparative analysis of the
modified dyon and the Julia-Zee dyon shows that their properties are
significantly different. In particular, except for the BPS case, the energy
and electric charge of the modified dyons exceed considerably those of the
Julia-Zee dyons. At the same time, the energy and electric charge of the
modified dyon are bounded for all admissible parameter values, whereas those
of the Julia-Zee dyon can be arbitrarily large in the BPS case.

###### keywords:

magnetic monopole , dyon , electric charge , scalar field

##  1 Introduction

Among the various soliton solutions of field theory, there are those that have
an electric charge. These soliton solutions can exist in both
$(2+1)$-dimensional [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and
$(3+1)$-dimensional [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]
gauge models. To ensure the finiteness of the energy of two-dimensional
electrically charged solitons, $(2+1)$-dimensional gauge models must include
the Chern-Simons term [27, 28, 29], whereas the Maxwell term may be missing.
Due to the Chern-Simons term, the gauge field in these models is topologically
massive, and therefore the electric field of the solitons is short-range. In
contrast, the electric field of three-dimensional solitons is long-range,
since there is no Chern-Simons term in $(3+1)$ dimensions and the gauge field
models include only the Maxwell term.

The electrically charged solitons can be divided into two classes: topological
solitons [13, 14, 15, 16, 17] and nontopological solitons [18, 19, 20, 21, 22,
23, 24, 25, 26]. The properties of the topological and nontopological solitons
are considerably different. Because of topological triviality, the existence
of nontopological solitons is entirely due to dynamical factors. In
particular, a conserved Noether charge must exist in the corresponding field
model, and the self-interaction potential of the scalar fields has to be of a
special form [30, 31]. For a fixed value of the Noether charge, the field
configuration of a nontopological soliton is a stationary point of the energy
functional [32, 33, 34]. As a consequence, the energy and the Noether charge
of the nontopological soliton are related by a differential relation, which
determines a number of the soliton’s properties.

In contrast, topological solitons have topologically nontrivial field
configurations, which prevents them from transitioning into lower energy
states [35]. Hence, the presence of a potential term and Noether charge is not
a necessary condition for the existence of three-dimensional topological
solitons, as can be seen in the example of the BPS monopole [14]. The best
known example of an electrically charged three-dimensional topological soliton
is the dyon solution [13] in the $SU(2)$ Georgi-Glashow model [36]. Similar to
nontopological solitons, a dyon solution is a stationary point (minimum) of
the energy functional for a fixed value of the Noether (electric) charge. It
follows that the the energy and the electric charge of the dyon are also
related by a differential relation [37], as is the case for the nontopological
solitons. As a result, except for the BPS case, the energy and electric charge
of the dyon cannot be arbitrarily large [16].

In [38, 39], a soliton system consisting of vortex and Q-ball components
interacting via an electric field is described. Being a source of an electric
field, a dyon can also interact with another charged field, as a result of
which a new soliton system can arise. In this paper, we consider a soliton
system (modified dyon) arising from the interaction of the Julia-Zee dyon [13]
and an isospinor self-interacting scalar field. The modified dyon consists of
a core (deformed Julia-Zee dyon) surrounded by a cloud of a charged scalar
field, and its properties differ significantly from those of the Julia-Zee
dyon.

Throughout this paper, we use the natural units $c=1$, $\hbar=1$.

##  2 Lagrangian and field equations of the model

We consider a modified Georgi-Glashow model with a non-Abelian gauge group
$SU(2)$. The model contains two types of self-interacting scalar fields, one
of which is transformed according to the adjoint representation of the gauge
group $SU(2)$, and the other according to the fundamental representation. The
Lagrangian of the model takes the form

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\,\mu\nu}+\frac{1}{2}\left(D_{\mu}\phi^{a}\right)\left(D^{\mu}\phi^{a}\right)$
$\displaystyle+\left(D_{\mu}\Phi\right)^{{\dagger}}\left(D^{\mu}\Phi\right)-V\left(\phi^{a}\phi^{a}\right)-U\left(\Phi^{{\dagger}}\Phi\right),$
(1)

where

$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}-g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c}$
(2)

is the non-Abelian field strength,

$D_{\mu}\phi^{a}=\partial_{\mu}\phi^{a}-g\epsilon^{abc}A_{\mu}^{b}\phi^{c}$
(3)

is the covariant derivative of the isovector scalar field $\boldsymbol{\phi}$,
and

$D_{\mu}\Phi=\partial_{\mu}\Phi+igA_{\mu}^{a}\frac{\tau_{a}}{2}\Phi$ (4)

is the covariant derivative of the isospinor scalar field $\Phi$. The self-
interaction potentials of the isovector and isospinor scalar fields are

$V\left(\phi^{a}\phi^{a}\right)=\frac{\lambda}{4}\left(\phi^{a}\phi^{a}-v^{2}\right)^{2}$
(5)

and

$U\left(\Phi^{{\dagger}}\Phi\right)=M^{2}\Phi^{{\dagger}}\Phi-\frac{G}{2}\left(\Phi^{{\dagger}}\Phi\right)^{2}+\frac{H}{3}\left(\Phi^{{\dagger}}\Phi\right)^{3},$
(6)

respectively, where $M$ is a mass parameter, and $\lambda$, $G$, and $H$ are
coupling constants. Note that the six-order potential in Eq. (6) is of the
form widely used in models having Q-ball solutions.

Eq. (5) tells us that the potential $V$ reaches a zero minimum value at non-
zero $\left|\boldsymbol{\phi}\right|=v$. At the same time, we assume that the
potential $U$ has a global minimum at $\Phi=0$. For this to hold, the
parameters of the potential $U$ in Eq. (6) must satisfy the inequality
$3G^{2}<16HM^{2}$. The nonzero vacuum value of the isovector scalar field
$\boldsymbol{\phi}$ leads to a spontaneous violation of the gauge group
$SU(2)$ to an Abelian (electromagnetic) subgroup $U(1)$. As a result of the
Higgs mechanism, the spectrum of small fluctuations of fields relative to the
vacuum contains one neutral massless vector field, two charged massive vector
fields with mass $m_{V}=gv$, one neutral massive Higgs field with mass
$m_{H}=\sqrt{2\lambda}v$, and two charged massive scalar fields with mass
$m_{\Phi}=M$.

Using standard methods of field theory, we find the field equations of model:
(1)

$\displaystyle D_{\mu}F^{a\,\mu\nu}=J^{a\,\nu},$ (7) $\displaystyle
D_{\mu}D^{\mu}\phi^{a}=-\lambda\left(\phi^{b}\phi^{b}-v^{2}\right)\phi^{a},$
(8) $\displaystyle
D_{\mu}D^{\mu}\Phi=-\left[M^{2}-G\left(\Phi^{{\dagger}}\Phi\right)+H\left(\Phi^{{\dagger}}\Phi\right)^{2}\right]\Phi,$
(9)

where the covariantly conserved current

$\displaystyle J^{a\,\nu}$
$\displaystyle=g\epsilon_{abc}\phi_{b}D^{\nu}\phi_{c}+i2^{-1}g\left[\Phi^{{\dagger}}\tau^{a}D^{\nu}\Phi\right.$
$\displaystyle\bigl{.}-\left(D^{\nu}\Phi\right)^{{\dagger}}\tau^{a}\Phi\bigr{]}.$
(10)

Later on, we shall also need the expression for the energy-momentum tensor of
the model,

$\displaystyle T_{\mu\nu}$
$\displaystyle=-F_{\mu\rho}^{a}F_{\nu}^{a\,\rho}+\left(D_{\mu}\phi^{a}\right)\left(D_{\nu}\phi^{a}\right)$
$\displaystyle+D_{\mu}\Phi^{{\dagger}}D_{\nu}\Phi+D_{\nu}\Phi^{{\dagger}}D_{\mu}\Phi$
$\displaystyle+\eta_{\mu\nu}\left[\frac{1}{4}F_{\rho\tau}^{a}F^{a\,\rho\tau}-\frac{1}{2}\left(D_{\rho}\phi^{a}\right)\left(D^{\rho}\phi^{a}\right)\right.$
$\displaystyle\biggl{.}-D_{\mu}\Phi^{{\dagger}}D^{\mu}\Phi+V\left(\phi^{a\ast}\phi^{a}\right)+U\left(\Phi^{{\dagger}}\Phi\right)\biggr{]},$
(11)

where the metric tensor $\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$.

##  3 Some properties of the modified dyon solution

From Eqs. (5) and (11), it follows that to possess finite energy, a field
configuration of model (1) must satisfy the asymptotic condition
$\underset{r\rightarrow\infty}{\lim}\left|\boldsymbol{\phi}\right|=v$. From a
topological point of view, this condition is equivalent to the existence of a
mapping of the infinitely distant space sphere $S^{2}_{\infty}$ to the vacuum
sphere $S_{\text{vac}}^{2}\\!:\left|\boldsymbol{\phi}\right|=v$. Since the
second homotopy group $\pi_{2}\left(S^{2}\right)=\mathbb{Z}$, the mapping
$S^{2}_{\infty}\rightarrow S^{2}_{\text{vac}}$ is topologically nontrivial and
splits into different topological classes, which are characterised by the
integer winding number $n$.

In the absence of the isospinor scalar field $\Phi$, model (1) turns into the
$SU(2)$ Georgi–Glashow model. It is well known that in the topological sector
with the winding number $n=1$, the $SU(2)$ Georgi–Glashow model has two
topological soliton solutions: the ’t Hooft–Polyakov monopole [40, 41] and the
Julia–Zee dyon [13]. Both the ’t Hooft–Polyakov monopole and the Julia–Zee
dyon have the same magnetic charge $Q_{M}$, but the dyon also possesses an
electrical charge. The electric charge of the dyon can lie in the range from
zero to some maximum value, which tends to infinity in the BPS case.

The inclusion of the self-interacting isospinor scalar field $\Phi$ modifies
the Georgi–Glashow model leading to model (1). Our aim is to ascertain the
possibility of the existence of a dyon solution in model (1) and to
investigate the properties of this solution. To find a dyon solution, we use
the following ansatz:

$\displaystyle A^{a0}$ $\displaystyle=n^{a}vj\left(r\right),$ (12a)
$\displaystyle A^{ai}$
$\displaystyle=\epsilon^{aim}n^{m}\frac{1-u\left(r\right)}{gr},$ (12b)
$\displaystyle\phi^{a}$ $\displaystyle=n^{a}vh\left(r\right),$ (12c)
$\displaystyle\Phi$
$\displaystyle=vf\left(r\right)\left(in^{a}\tau_{a}\right)\Phi_{0},$ (12d)

where the unit vector $n^{a}=x^{a}/r$, and $\Phi_{0}$ is a constant isospinor
normalized to unity: $\Phi_{0}^{\dagger}\Phi_{0}=1$. The ansatz (12)
corresponds to the most general spherically symmetric (modulo a global gauge
transformation) field configuration that is odd with respect to the inversion
$\mathbf{x}\rightarrow-\mathbf{x}$. Note that Eqs. (12a) – (12c) describe the
field configuration of the Julia–Zee dyon, while Eqs. (12b) and (12d) coincide
in form with the ansatz for the sphaleron solution [42] in the Standard Model.

We now introduce the dimensionless radial variable $\rho=m_{V}r$, where
$m_{V}=gv$ is the mass of the electrically charged gauge bosons. Substituting
ansatz (12) into field equations (7)–(9), we obtain a system of nonlinear
differential equations for the ansatz functions:

$\displaystyle
j^{\prime\prime}\left(\rho\right)+\frac{2}{\rho}j^{\prime}\left(\rho\right)-\left(\frac{1}{2}f\left(\rho\right)^{2}+\frac{2}{\rho^{2}}u(\rho)^{2}\right)j(\rho)=0,$
(13) $\displaystyle
u^{\prime\prime}\left(\rho\right)-\frac{u(\rho)\left(u(\rho)^{2}-1\right)}{\rho^{2}}-\bigl{(}2^{-1}f\left(\rho\right)^{2}\bigr{.}$
$\displaystyle\bigl{.}+h(\rho)^{2}-j(\rho)^{2}\bigr{)}u(\rho)-2^{-1}f\left(\rho\right)^{2}=0,$
(14) $\displaystyle
h^{\prime\prime}\left(\rho\right)+\frac{2}{\rho}h^{\prime}\left(\rho\right)-\frac{2}{\rho^{2}}u(\rho)^{2}h(\rho)$
$\displaystyle+\kappa\left(1-h(\rho)^{2}\right)h(\rho)=0,$ (15) $\displaystyle
f^{\prime\prime}\left(\rho\right)+\frac{2}{\rho}f^{\prime}\left(\rho\right)-\frac{\left(1+u(\rho)\right)^{2}}{2\rho^{2}}f\left(\rho\right)$
(16)
$\displaystyle-\\!\left[\left(\tilde{M}^{2}\\!-\\!\tilde{G}f(\rho)^{2}\\!+\\!\tilde{H}f(\rho)^{4}\right)\\!-\\!2^{-2}j(\rho)^{2}\right]f\left(\rho\right)\\!=\\!0,$

where the dimensionless parameters are $\kappa=\lambda/g^{2}$,
$\tilde{M}=M/m_{V}$, $\tilde{G}=G/g^{2}$, and $\tilde{H}=Hv^{2}/g^{2}$. We
next obtain the expression for the energy $E=\int T_{00}d^{3}x$ of a field
configuration in terms of the ansatz functions:

$\displaystyle
E=m_{M}\int\limits_{0}^{\infty}\left[\frac{u^{\prime}(\rho)^{2}}{\rho^{2}}+\frac{1}{2}j^{\prime}(\rho)^{2}+\frac{1}{2}h^{\prime}(\rho)^{2}+f^{\prime}(\rho)^{2}\right.$
$\displaystyle\left.+\frac{\left(1-u(\rho)^{2}\right)^{2}}{2\rho^{4}}+\frac{\left(j(\rho)^{2}+h(\rho)^{2}\right)}{\rho^{2}}u(\rho)^{2}\right.$
$\displaystyle\left.+\frac{\left(1+u(\rho)\right)^{2}}{2\rho^{2}}f(\rho)^{2}+\frac{1}{4}j(\rho)^{2}f(\rho)^{2}+\frac{\kappa}{4}\left(1-h(\rho)^{2}\right)^{2}\right.$
$\displaystyle\biggl{.}+\tilde{M}^{2}f(\rho)^{2}-\frac{\tilde{G}}{2}f(\rho)^{4}+\frac{\tilde{H}}{3}f(\rho)^{6}\biggr{]}\rho^{2}d\rho,$
(17)

where $m_{M}=4\pi vg^{-1}$ is the mass of the BPS monopole [14].

The dyon solution must be regular at the origin and also must have finite
energy. These two conditions together with Eqs. (12) and (17) lead us to the
boundary conditions for the ansatz functions:

$\displaystyle
j(0)=0,\qquad\underset{\rho\rightarrow\infty}{\lim}j(\rho)=j_{\infty},$ (18a)
$\displaystyle u(0)=1,\qquad\underset{\rho\rightarrow\infty}{\lim}u(\rho)=0,$
(18b) $\displaystyle
h(0)=0,\qquad\underset{\rho\rightarrow\infty}{\lim}h(\rho)=1,$ (18c)
$\displaystyle f(0)=0,\qquad\underset{\rho\rightarrow\infty}{\lim}f(\rho)=0,$
(18d)

where $j_{\infty}$ is a finite value.

For large $\rho$, Eqs. (13) – (16) are linearized, which makes it possible to
study the asymptotics of the dyon solution. As a result, we conclude that in
Eq. (18a), the limiting value $j_{\infty}$ must satisfy the condition

$\left|j_{\infty}\right|<\min\bigl{(}2\tilde{M},1\bigr{)}.$ (19)

Otherwise, the asymptotics for large $\rho$ of at least one of the ansatz
functions $u(\rho)$ and $f(\rho)$ will be oscillatory, resulting in an
infinite energy of the field configuration.

Eq. (12c) tells us that the Higgs field $\boldsymbol{\phi}$ is invariant under
gauge transformations of the $U(1)$ subgroup corresponding to rotations around
the unit vector $n^{a}=x^{a}/r$ in the isospace. As a result, the dyon
solution possesses a long-range gauge (electromagnetic) field with the field
strength tensor $F_{\mu\nu}=v^{-1}\phi^{a}F_{\mu\nu}^{a}$. The intensities of
the electric and magnetic fields of the dyon solution are expressed in terms
of the ansatz functions as

$E_{i}=v^{-1}F_{0i}^{a}\phi^{a}=-gv^{2}h\left(\rho\right)j^{\prime}\left(\rho\right)n_{i}$
(20)

and

$\displaystyle B_{i}$
$\displaystyle=-\left(2v\right)^{-1}\epsilon_{ijk}F_{jk}^{a}\phi^{a}$
$\displaystyle=-gv^{2}\frac{h\left(\rho\right)}{\rho^{2}}\left(1-u\left(\rho\right)^{2}\right)n_{i},$
(21)

respectively. Using Eqs. (20) and (20), we obtain the expressions for the
electric and magnetic charges of the dyon solution:

$\displaystyle
Q_{E}=\oint_{S_{\infty}^{2}}d^{2}S_{i}E_{i}=-\frac{4\pi}{g}\underset{\rho\rightarrow\infty}{\lim}\rho^{2}j^{\prime}\left(\rho\right)$
(22)

and

$Q_{M}=\oint_{S_{\infty}^{2}}d^{2}S_{i}B_{i}=-\frac{4\pi}{g},$ (23)

where the negative magnetic charge for the field configuration with the
positive winding number $n=1$ is due to the choice of the sign before the
gauge coupling constant $g$ in Eqs.(2) and (3) which is similar to that used
in [35, 43]. From Eq. (22), we then obtain the large $\rho$ asymptotics of the
ansatz function $j(\rho)$ in terms of the electric charge of the dyon

$j\left(\rho\right)\sim j_{\infty}+\frac{g}{4\pi}\frac{Q_{E}}{\rho}.$ (24)

We can write the dyon’s energy (17) as the sum of five terms

$E=E^{\left(E\right)}+E^{\left(T\right)}+E^{\left(B\right)}+E^{\left(G\right)}+E^{\left(P\right)},$
(25)

where

$E^{\left(E\right)}=m_{M}\int\limits_{0}^{\infty}\left[\frac{1}{2}j^{\prime}\left(\rho\right)^{2}+\frac{j\left(\rho\right)^{2}u\left(\rho\right)^{2}}{\rho^{2}}\right]\rho^{2}d\rho$
(26)

is the energy of the electric field,

$E^{\left(T\right)}=m_{M}\int\limits_{0}^{\infty}\left[\frac{1}{4}j\left(\rho\right)^{2}f\left(\rho\right)^{2}\right]\rho^{2}d\rho$
(27)

is the kinetic energy of the isospinor scalar field $\Phi$,

$E^{\left(B\right)}=m_{M}\int\limits_{0}^{\infty}\left[\frac{u^{\prime}\left(\rho\right)^{2}}{\rho^{2}}+\frac{\bigl{(}1-u\left(\rho\right)^{2}\bigr{)}^{2}}{2\rho^{4}}\right]\rho^{2}d\rho$
(28)

is the energy of the magnetic field,

$\displaystyle
E^{\left(G\right)}=m_{M}\int\limits_{0}^{\infty}\left[\frac{1}{2}h^{\prime}\left(\rho\right)^{2}+f\left(\rho\right)^{\prime
2}\right.$
$\displaystyle\biggl{.}+\frac{u\left(\rho\right)^{2}h\left(\rho\right)^{2}}{\rho^{2}}+\frac{\left(1+u\left(\rho\right)\right)^{2}}{2\rho^{2}}f\left(\rho\right)^{2}\biggr{]}\rho^{2}d\rho$
(29)

is the gradient part of the energy, and

$\displaystyle
E^{\left(P\right)}=m_{M}\int\limits_{0}^{\infty}\biggl{[}\frac{\kappa}{4}\bigl{(}1-h\left(\rho\right)^{2}\bigr{)}^{2}+\tilde{M}^{2}f\left(\rho\right)^{2}\biggr{.}$
$\displaystyle\biggl{.}-\frac{\tilde{G}}{2}f\left(\rho\right)^{4}+\frac{\tilde{H}}{3}f\left(\rho\right)^{6}\biggr{]}\rho^{2}d\rho$
(30)

is the potential part of the energy. Using Gauss’s law (13), we then can
express the sum $E^{\left(E\right)}+E^{\left(T\right)}$ in terms of the
electric charge of the dyon

$E^{\left(E\right)}+E^{\left(T\right)}=-2^{-1}vj_{\infty}Q_{E}.$ (31)

Similar to the energy in Eq. (25), the Lagrangian $L=\int\mathcal{L}d^{3}x$
can also be written as a linear combination of Eqs. (26) – (30)

$L=E^{\left(E\right)}+E^{\left(T\right)}-E^{\left(B\right)}-E^{\left(G\right)}-E^{\left(P\right)}.$
(32)

Since the Lagrangian density (1) does not depend on time for field
configurations (12), any solution of system (13) – (16) is a stationary point
of the Lagrangian (32). Let $j(\rho)$, $u(\rho)$, $h(\rho)$, and $f(\rho)$ be
a solution of system (13) – (16) satisfying the boundary conditions in Eq.
(18). After the scale transformation $\rho\rightarrow\varkappa\rho$ of the
argument of the solution, the Lagrangian $L$ becomes a function of the scale
parameter $\varkappa$. From the above, it follows that the function
$L(\varkappa)$ has a stationary point at $\varkappa=1$, and hence its
derivative vanishes at this point:
$\left.dL/d\varkappa\right|_{\varkappa=1}=0$.

We can easily find the transformation laws of the constituent parts of the
dyon’s energy under the rescaling $\rho\rightarrow\varkappa\rho$:
$E^{\left(E\right)}\rightarrow\varkappa^{-1}E^{\left(E\right)}$,
$E^{\left(T\right)}\rightarrow\varkappa^{-3}E^{\left(T\right)}$,
$E^{\left(B\right)}\rightarrow\varkappa E^{\left(B\right)}$,
$E^{\left(G\right)}\rightarrow\varkappa^{-1}E^{\left(G\right)}$, and
$E^{\left(P\right)}\rightarrow\varkappa^{-3}E^{\left(P\right)}$. Using Eq.
(32) and these transformation laws, we obtain the virial relation for the dyon
solution

$E^{\left(E\right)}+E^{\left(B\right)}+3E^{\left(T\right)}-E^{\left(G\right)}-3E^{\left(P\right)}=0.$
(33)

Eqs. (25), (31), and (32) tell us that the Lagrangian, the energy, and the
electric charge of the dyon are connected by the relation

$-L=E+vj_{\infty}Q_{E}.$ (34)

We know that the dyon solution is an unconditional extremum (stationary point)
of the Lagrangian $L$. Then, Eq. (34) tells us that the dyon solution is also
a conditional extremum of the energy functional $E$ at a fixed value of the
electric charge $Q_{E}$, with $vj_{\infty}$ being the Lagrange multiplier. We
define the Noether charge of the dyon as $Q_{N}=g^{-1}Q_{E}$. Varying Eq. (34)
and equating the variation to zero, we obtain the important differential
relation

$dE/dQ_{N}=g\,dE/dQ_{E}=\Omega,$ (35)

where

$\Omega=-gvj_{\infty}=-m_{V}j_{\infty}.$ (36)

Eqs. (35) and (36) tell us that the derivative of the dyon’s energy with
respect to its Noether (electric) charge is proportional to the limiting value
of the ansatz function $j(\rho)$ at infinity. Note that a differential
relation similar to Eq. (35) is a characteristic property of nontopological
solitons and, in particular, of Q-balls [30].

The system of differential equations (13) – (16) together with boundary
conditions (18) is a boundary value problem on the semi-infinite interval
$\rho\in\left[0,\infty\right)$. This problem admits the trivial solution
$f(\rho)=0$; in this case the solution to the remaining Eqs. (13) – (15) of
the system coincides with the Julia-Zee dyon solution [13] of the $SU(2)$
Georgi-Glashow model [36]. The ansatz function $f(\rho)$ enters Eqs. (13) –
(15) only via the quadratic combination $f(\rho)^{2}$. We can therefore
suppose that if $f(\rho)$ is small enough, its backreaction on Eqs. (13) –
(15) is all the smaller, and the solution to the boundary value problem will
also exist for non-zero $f(\rho)$.

A qualitative analysis of the boundary value problem reveals that the
amplitude of $f(\rho)$ is roughly determined by the position $f_{\min}$ of the
non-zero minimum of the potential
$U\left(f\right)=\tilde{M}^{2}f^{2}-2^{-1}\tilde{G}f^{4}+3^{-1}\tilde{H}f^{6}$.
In addition, $U(f_{\min})$ must be sufficiently close to zero, otherwise
$f(\rho)$ increases indefinitely as $\rho\rightarrow\infty$. These two
conditions allow us to obtain approximate estimates for the parameters
$\tilde{G}$ and $\tilde{H}$: $\tilde{G}\lessapprox
4\tilde{M}^{2}f_{\min}^{-2}$ and $\tilde{H}\lessapprox
3\tilde{M}^{2}f_{\min}^{-4}$. In addition, the smallness condition for
$f(\rho)$ can be written as $f_{\min}^{2}\ll 1$, since for the Julia-Zee dyon,
the maximum values of the ansatz functions $u(\rho)$ and $h(\rho)$ are equal
to unity.

##  4 Numerical results

In this section, we present numerical results concerning the modified dyon
solution of model (1). To obtain them, we use the numerical methods provided
in the Maple package [44]. To check the correctness of the numerical results,
we use Eqs. (31), (33), and (35). It follows from Eqs. (13) – (16) and (18)
that the dimensionless ansatz functions $u(\rho)$, $j(\rho)$, $h(\rho)$, and
$f(\rho)$ depend on the five dimensionless parameters: $\kappa$, $\tilde{M}$,
$\tilde{G}$, $\tilde{H}$, and $\tilde{\Omega}=-j_{\infty}$. Furthermore, from
Eqs. (17) and (22) it follows that the dimensionless combinations
$\tilde{Q}_{E}=gQ_{E}$ (rescaled electric charge) and
$\tilde{E}=g^{2}m_{V}^{-1}E=gv^{-1}E$ (rescaled energy) also depend only on
these five parameters. The numerical results presented here correspond to the
parameters $\tilde{M}=0.3$, $\tilde{G}=12.5$, and $\tilde{H}=326$. These
parameters correspond to self-interaction potential (6) having an almost zero
positive minimum at sufficiently small $f\approx 0.17$. In accordance with the
above, this property of the potential (6) makes possible the existence of dyon
solutions in model (1). Note that for $v=240\,\text{GeV}$ and $g=1/3$, which
are typical Standard Model values corresponding to $m_{V}=80\,\text{GeV}$, we
obtain the following values of the dimensional parameters: $M=24\,\text{GeV}$,
$G=1.39$, and $H=6.3\times 10^{-4}\,\text{GeV}^{-2}$.

Figure 1:  Dependence of the rescaled energy $\tilde{E}=g^{2}m_{V}^{-1}E$ on
the parameter $\tilde{\Omega}=m_{V}^{-1}\Omega=-j_{\infty}$ for different
values of the parameter $\kappa=g^{-2}\lambda$. The solid lines correspond to
the modified dyons, and the dashed lines correspond to the Julia-Zee dyons.

Figure 1 presents the dependence of the energy $\tilde{E}$ for the modified
dyon solution of model (1) on the parameter $\tilde{\Omega}$ for different
values of the parameter $\kappa=g^{-2}\lambda$. In addition, Fig. 1 shows
similar dependencies for the Julia-Zee dyon. Firstly, we see that the modified
dyon solution exists only in a narrow range
$\bigl{[}\tilde{\Omega}_{\min},2\tilde{M}\bigr{]}$ of values of the parameter
$\tilde{\Omega}$. The right boundary of the interval
$\bigl{[}\tilde{\Omega}_{\min},2\tilde{M}\bigr{]}$ is in agreement with Eq.
(19), while the left boundary decreases monotonically with an increase in
$\kappa$ and tends to a finite limit as $\kappa\rightarrow\infty$. The width
of this interval is quite small and is only $0.02-0.025$, whereas the Julia-
Zee dyon exists in a much wider wide interval $\left[0,1\right]$ of
$|\tilde{\Omega}|$.

It follows from Fig. 1 that the $\tilde{E}(\tilde{\Omega})$ curves for the
modified and Julia-Zee dyons behave completely differently. Indeed, we see
that for the modified dyon, the $\tilde{E}(\tilde{\Omega})$ curves consist of
two (lower and upper) branches that join at
$\tilde{\Omega}=\tilde{\Omega}_{\min}$. For a given $\kappa$, the energy of
the modified dyon reaches a minimum (maximum) value on the lower (upper)
branch at the maximum value of $\tilde{\Omega}=2\tilde{M}$. Such a behaviour
of the $\tilde{E}(\tilde{\Omega})$ curves is similar to their behaviour for an
electrically charged Q-ball and completely different from that for the Julia-
Zee dyon. Note that in Fig. 1, the $\tilde{E}(\tilde{\Omega})$ curves
corresponding to the Julia-Zee dyon are practically indistinguishable from
straight lines. This is because in Fig. 1, the width of the interval of
$\tilde{\Omega}$ is much less than unity.

Figure 2:  Dependence of the rescaled electric charge $\tilde{Q}_{E}=gQ_{E}$
on the parameter $\tilde{\Omega}=m_{V}^{-1}\Omega=-j_{\infty}$ for different
values of the parameter $\kappa=g^{-2}\lambda$. The solid lines correspond to
the modified dyons, and the dashed lines correspond to the Julia-Zee dyons.

Figure 2 presents the dependence of the electric charge $\tilde{Q}_{E}$ of the
modified dyon solution on $\tilde{\Omega}$ for different values of $\kappa$.
We see that the behavior of the $\tilde{Q}_{E}(\tilde{\Omega})$ curves is
similar to that of the $\tilde{E}(\tilde{\Omega})$ curves for both the
modified and Julia-Zee dyon solutions. The difference is that for the modified
dyon solution, the $\tilde{E}(\tilde{\Omega})$ curves intersect each other,
whereas the $\tilde{Q}_{E}(\tilde{\Omega})$ curves do not. As a result, for
the modified dyon solution, the minimum and maximum energies are reached at
$\tilde{\Omega}=2\tilde{M}$, and both of them increase with an increase in
$\kappa$. But the minimum (maximum) electric charge of the modified dyon
solution decreases (increases) with an increase in $\kappa$.

Figure 3:  The $\tilde{E}(\tilde{Q}_{E})$ dependences for different $\kappa$.
The solid lines correspond to the modified dyons, and the dotted lines
correspond to the Julia-Zee dyons. The black segments on the dotted curves
correspond to the ranges of the Julia-Zee dyon solutions presented in Figs. 1
and 2.

Figure 3 shows the dependences of the energy $\tilde{E}$ on the electric
charge $\tilde{Q}_{E}$ for both the modified and Julia-Zee dyons. Note that in
accordance with [16], the energy $\tilde{E}$ and the electric charge
$\tilde{Q}_{E}$ of the Julia-Zee dyon can take arbitrarily large values only
when $\kappa=0$ (the BPS case), whereas there are maximum possible values for
both $\tilde{E}$ and $\tilde{Q}_{E}$ when $\kappa>0$. We see that the
$\tilde{E}(\tilde{Q}_{E})$ curves are substantially different for the modified
and Julia-Zee dyons. Firstly, except for the case $\kappa=0$, the energies and
electric charges of the modified dyon solutions exceed the maximum possible
values of the corresponding Julia-Zee dyon solutions. Secondly, for the
modified dyon solutions, the curves $\tilde{E}(\tilde{Q}_{E})$ are visually
indistinguishable from straight lines, which is not the case for the Julia-Zee
dyon solutions. This is because in Figs. 1 and 2, the interval of variation of
$\tilde{\Omega}$ is much less than $\tilde{\Omega}$ itself:
$\Delta\tilde{\Omega}/\tilde{\Omega}\sim 1/30$, and the above property of the
solid curves in Fig. 3 follows from Eq. (35) rewritten as

$d\tilde{E}/d\tilde{Q}_{E}=\tilde{\Omega}.$ (37)

Thirdly, unlike the Julia-Zee dyon, the energy and electric charge of the
modified dyon cannot be arbitrarily large even when the parameter $\kappa=0$.

Figure 3 allows us to address the subject of stability of the modified dyon
solution. Consider the family of the half-lines $a+b\tilde{Q}_{E}$ coming from
the points of the dotted curve that correspond to a given value of $\kappa$.
It is understood that the parameter $a$ depends on the origin point of the
half-line. Then, it follows from Fig. 3 that for all the points $p$ of the
dotted curves, the half-lines $a(p)+\tilde{Q}_{E}$ pass over the corresponding
solid lines. Hence, the modified dyon is stable to the transition into the
Julia-Zee dyon accompanied by emission of the charged vector bosons of mass
$m_{V}=gv$. In contrast, for all the points $p$ of the dotted curves
corresponding to $\kappa\geq 0.1$, the half-lines
$a(p)+2\tilde{M}\tilde{Q}_{E}$ pass under the corresponding solid lines, where
the appearance of the factor $2$ is due to the fact that the electric charges
of the components of the scalar isodublet $\Phi$ are $\pm g/2$. The same is
true for points $p$ of the dotted curve corresponding to $\kappa=0$ provided
that $\tilde{Q}_{E}\left(p\right)\lesssim 25$. It follows that the modified
dyon is unstable to the transition into the Julia-Zee dyon accompanied by
emission of the charged scalar bosons of mass $M$.

Figure 4:  Ansatz functions of the dyon solutions corresponding to the
parameters $\tilde{\Omega}=-j_{\infty}=0.58$ and $\kappa=g^{-2}\lambda=2$. The
solid lines correspond to the modified dyon solution, and the dashed lines
correspond to the Julia-Zee dyon solution.

The two possibilities described above are idealised. In reality, the
transition of the modified dyon into the Julia-Zee dyon is accompanied by the
emission of both vector and scalar bosons. Calculations show that the modified
dyon is stable to the transition into the Julia-Zee dyon if the fraction of
electric charge carried by the charged vector bosons exceeds $0.33$ for
$\kappa=0$, and $0.24$ for $\kappa=100$.

In Fig. 4, we can see the ansatz functions corresponding to the two types of
dyon solutions. The most significant differences are observed for $j(\rho)$.
According to Eq. (24), the large $\rho$ asymptotics of this ansatz function is
determined by the electric charge of the dyon. From Fig. 4, it follows that
for a given $\tilde{\Omega}$ and $\kappa$, the electric charge of the modified
dyon is substantially larger than that of the Julia-Zee dyon.

The ansatz function $u(\rho)$ is also different for the two dyon solutions,
although not as much as $j(\rho)$. Nevertheless, there is an important
difference between the two dyon solutions. From the subplot in Fig. 4, it
follows that for the modified dyon, $u(\rho)$ crosses zero from above at
finite $\rho$, reaches a minimum, and then tends to zero from below. At the
same time, for the Julia-Zee dyon, $u(\rho)$ does not cross zero at finite
$\rho$, but tends monotonically to zero from above. In contrast to $j(\rho)$
and $u(\rho)$, the ansatz function $h(\rho)$ is practically the same for the
two dyon solutions.

From Fig. 4, we see that the maximum of the ansatz function $f(\rho)$ is
outside of the core of the Julia-Zee dyon. Hence, the modified dyon consists
of a core (deformed Julia-Zee dyon) surrounded by a spherical cloud of the
charged scalar condensate. Note that in contrast to the electrically charged
Q-ball, the density of the scalar condensate vanishes at the center of the
modified dyon, which is caused by the boundary condition (18d).

##  5 Conclusion

We have shown that a modified dyon solution exists in a non-Abelian gauge
model, which is essentially a modified Georgi-Glashow model. The modification
consists in the inclusion of a self-interacting scalar isospinor field in the
Lagrangian. The sixth-order self-interaction potential of the scalar field is
of the Q-ball type. This modification of the Georgi-Glashow model leads to the
existence of modified dyon solutions in addition to the usual Julia-Zee dyon
solutions. The modified dyons exist in a much narrower parametric domain
compared to the Julia-Zee dyons, and their properties differ significantly.

Firstly, except for the case $\kappa=0$, the energy and electric charge of the
modified dyons significantly exceed those of the Julia-Zee dyons. Secondly,
there is a non-zero minimum possible value of the electric charge for the
modified dyon, whereas this is equal to zero for the Julia-Zee dyon. And
thirdly, both the minimum and maximum value of the energy (electric charge) of
the modified dyon are achieved at the maximum value of the parameter $\Omega$,
whereas the energy (electric charge) of the Julia-Zee dyon increases
monotonically with $\Omega$. As for the solutions themselves, the greatest
differences are observed for the ansatz function $j(\rho)$, whose derivative
determines the electric field strength of the dyon.

Finally, we note that various cosmological models tell us that the evolution
of the early Universe proceeded through a sequence of phase transitions. Both
electrically charged magnetic monopoles (dyons) and clouds of a charged scalar
condensate can arise as a result of these phase transitions. The interaction
of a long-range electric field of the dyon with the charged scalar condensate
can lead to the formation of modified dyons similar to those considered in
this paper.

## Acknowledgements

This work was supported by the Russian Science Foundation, grant No
23-11-00002.

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# Vector dark energy and high-$z$ massive clusters

Edoardo Carlesi,1 Alexander Knebe,1 Gustavo Yepes,1 Stefan Gottlöber,3 Jose
Beltrán Jiménez,4,5 Antonio L. Maroto,2

1Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049,
Cantoblanco, Madrid, Spain
2Departamento de Física Teórica, Universidad Complutense de Madrid, 28040,
Madrid, Spain
3Leibniz Institut für Astrophysik, An der Sternwarte 16, 14482, Potsdam,
Germany
4Institute de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, 24 quai E. Ansermet, 1211 Genève, Switzerland
5Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway
E-mail<EMAIL_ADDRESS>
(Accepted XXXX . Received XXXX; in original form XXXX)

###### Abstract

The detection of extremely massive clusters at $z>1$ such as SPT-CL
J0546-5345, SPT-CL J2106-5844, and XMMU J2235.3-2557 has been considered by
some authors as a challenge to the standard $\Lambda$CDM cosmology. In fact,
assuming Gaussian initial conditions, the theoretical expectation of detecting
such objects is as low as $\leq 1\%$. In this Letter we discuss the
probability of the existence of such objects in the light of the Vector Dark
Energy (VDE) paradigm, showing by means of a series of $N$-body simulations
that chances of detection are substantially enhanced in this non-standard
framework.

###### keywords:

methods:$N$-body simulations – galaxies: haloes – cosmology: theory – dark
matter

††pagerange: Vector dark energy and high-$z$ massive
clusters–References††pubyear: 2011

##  1 Introduction

Present day cosmology is still failing to explain satisfactorily the nature of
dark energy, which is supposed to dominate the energetic content of the
universe today and to be responsible for the current accelerated expansion. In
the standard $\Lambda$CDM model, this cosmic acceleration is generated by the
presence of a cosmological constant. However, the required value for that
constant turns out to be tiny when compared to the natural scale of gravity,
namely the Planck scale. Thus, the gravitational interaction would hence be
described by two dimensional constants differing by many orders of magnitude,
and this poses a problem of naturalness. This is the so-called “cosmological
constant problem” and it motivated to consider alternative explanations for
the current acceleration of the universe by either modifying the gravitational
interaction at large distances or introducing a new dynamical field.

Indeed, one of the main challenges of observational cosmology is exactly to
devise new tests which could help discriminating between the constant or
dynamic nature of dark energy. In this regard, several authors have recently
pointed out that the observation of extremely massive clusters at high
redshift, such as SPT-CL J2106-5844 (Foley et al. (2011), $z\simeq 1.18$,
$M_{200}=(1.27\pm 0.21)\times 10^{15}M_{\odot}$), SPT-CL J0546-5346 (Brodwin
et al. (2010), $z\simeq 1.07$, $M_{200}=(7.95\pm 0.92)\times
10^{14}M_{\odot}$), and XMMU J2235.3-2557 (Jee et al. (2009), $z\simeq 1.4$,
$M_{200}=(7.3\pm 1.3)\times 10^{14}M_{\odot}$) may represent a major
shortcoming of the $\Lambda$CDM paradigm, where the presence of such objects
should be in principle strongly disfavoured (see, for example, Baldi &
Pettorino, 2011; Mortonson et al., 2011).

While, on the one hand, this tension could be solved keeping the standard
scenario and relaxing the assumption of Gaussianity in the initial conditions
(as proposed in Hoyle et al. (2011) and Enqvist et al. (2011)), it could be as
well possible to use this observations as a constraint for different
cosmological models. In this work we look at the VDE model, where the role of
the dark energy is played by a cosmic vector field (Beltrán Jiménez & Maroto,
2008). By means of a series of $N$-body simulations, we study the large scale
clustering properties of this cosmology, computing the cumulative halo mass
functions at different redshifts and comparing them to the predictions of the
standard model. In this way, we are able to show that the VDE cosmology does
indeed predict a higher abundance of massive haloes at all redshifts, thus
enhancing the probability of observing such objects with respect to
$\Lambda$CDM.

##  2 Vector Dark Energy

The action of the vector dark energy model (see Beltrán Jiménez & Maroto
(2008)) can be written as:

$\displaystyle S=\int d^{4}x\sqrt{-g}\left[-\frac{R}{16\pi
G}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right.$
$\displaystyle\left.-\frac{1}{2}\left(\nabla_{\mu}A^{\mu}\right)^{2}+R_{\mu\nu}A^{\mu}A^{\nu}\right].$
(1)

where $R_{\mu\nu}$ is the Ricci tensor, $R=g^{\mu\nu}R_{\mu\nu}$ the scalar
curvature and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. This
action can be interpreted as the Maxwell term for a vector field supplemented
with a gauge-fixing term and an effective mass provided by the Ricci tensor.
It is interesting to note that the vector sector has no free parameters nor
potential terms, being $G$ the only dimensional constant of the theory.

For a homogeneous and isotropic universe described by the flat Friedmann-
Lemaître-Robertson-Walker metric:

$ds^{2}=dt^{2}-a(t)^{2}d\vec{x}^{2}$ (2)

we have $A_{\mu}=(A_{0}(t),0,0,0)$ so that the corresponding equations read:

$\ddot{A}_{0}+3H\dot{A}_{0}-3\left[2H^{2}+\dot{H}\right]A_{0}=0$ (3)
$H^{2}=\frac{8\pi G}{3}\left[\rho_{R}+\rho_{M}+\rho_{A}\right]$ (4)

with $H=\dot{a}/a$ the Hubble parameter and

$\rho_{A}=\frac{3}{2}H^{2}A_{0}^{2}+3HA_{0}\dot{A}_{0}-\frac{1}{2}\dot{A}_{0}^{2}$
(5)

the energy density associated to the vector field, while $\rho_{M}$ and
$\rho_{R}$ are the matter and radiation densities, respectively. During the
radiation and matter eras in which the dark energy contribution was
negligible, we can solve Eq. (3) with $H=p/t$, where $p=1/2$ for radiation and
$p=2/3$ for matter eras respectively, that is equivalent to assume that
$a\propto t^{p}$. In that case, the general solution is:

$A_{0}(t)=A_{0}^{+}t^{\alpha_{+}}+A_{0}^{-}t^{\alpha_{-}},$ (6)

with $A_{0}^{\pm}$ constants of integration and $\alpha_{\pm}=-(1\pm 1)/4$ in
the radiation era, and $\alpha_{\pm}=(-3\pm\sqrt{33})/6$ in the matter era.
After dark energy starts dominating, the equation of state abruptly falls
towards $w_{DE}\rightarrow-\infty$ as the Universe approaches $t_{end}$, and
the equation of state can cross the so-called phantom divide line (Nesseris &
Perivolaropoulos (2007)), so that we can have $w_{DE}(z=0)<-1$.

Using the growing mode solution in Eq. (6) we can obtain the evolution for the
energy density as:

$\rho_{A}=\rho_{A0}(1+z)^{\kappa},$ (7)

with $\kappa=4$ in the radiation era and $\kappa=(9-\sqrt{33})/2\simeq-1.63$
in the matter era. Thus, the energy density of the vector field scales like
radiation at early times so that the ratio $\rho_{A}/\rho_{R}$ is constant
during such a period. Moreover, the value of the vector field $A_{0}$ during
that era is also constant hence making the cosmological evolution insensitive
to the time at which we impose the initial conditions (as long as they are set
well inside the radiation dominated epoch). Also, the required constant values
of such quantities in order to fit observations are $\rho_{A}/\rho_{R}|_{\rm
early}\simeq 10^{-6}$ and $A_{0}^{\rm early}\simeq 10^{-4}M_{p}$ which can
arise naturally during the early universe, for instance, as quantum
fluctuations. Furthermore, they do not need the introduction of any unnatural
scale, thus, alleviating the naturalness or coincidence problem. On the other
hand, when the Universe enters the era of matter domination, $\rho_{A}$ starts
growing relative to $\rho_{M}$ eventually overcoming it at some point so that
the dark energy vector field becomes the dominant component.

Figure 1: Equation of state of the vector dark energy model for the best fit
to SNIa data, shown in the range of our simulations.

Once the present value of the Hubble parameter $H_{0}$ and the constant
$A_{0}^{early}$ during radiation (which fixes the total amount of matter
$\Omega_{M}$) are specified, the model is completely determined. In other
words, this model contains the same number of parameters as $\Lambda$CDM, i.e.
the minimum number of parameters of any cosmological model with dark energy.
Notice however, that in the VDE model the present value of the equation of
state parameter $w_{0}=-3.53$ is radically different from that of a
cosmological constant (cf. Fig. 1, where the redshift evolution of $\omega(z)$
is shown our the range of our simulations). Despite this fact, VDE is able to
simultaneously fit supernovae and CMB data with comparable goodness to
$\Lambda$CDM (Beltrán Jiménez & Maroto, 2008; Beltrán Jiménez et al., 2009).

##  3 The Data

###  3.1 Simulations

We wanted to estimate the probability of finding massive clusters at $z>1$ in
the VDE scenario compared to the $\Lambda$CDM one by means of CDM only
$N$-body simulations. For this purpose, we chose to use a suitably modified
version of the publicly available GADGET-2 tree-PM code (Springel, 2005),
which had to take into account the different expansion history that
characterizes the two cosmologies. In Table 1 we show the cosmological
parameters used in the different simulations. For the VDE model, we have used
the value of $\Omega_{M}$ provided by the best fit to SNIa data; the remaining
cosmological parameters have been obtained by a fit to the WMAP7 CMB data of
the model. $w_{0}$ denotes the present value of the equation of state
parameter of dark energy. For $\Lambda$CDM we used the Multidark Simulation
(Prada et al., 2011) cosmological parameters with a WMAP7 $\sigma_{8}$
normalization (Larson et al., 2011). In addition, we also simulated a
$\Lambda$CDM-vde model, which implements the VDE values for $\Omega_{M}$ and
$\sigma_{8}$ in an otherwise standard $\Lambda$CDM picture. Although this
model is certainly ruled out by current cosmological constraints, it provides
nonetheless an interesting case study that allows us to disentangle the
effects of these two cosmological parameters on structure formation in the VDE
model.
We chose to run a total of eight $512^{3}$ particles simulations summarized in
Table 2 and explained below:

  * •

a VDE (and a $\Lambda$CDM started with the same seed for the phases of the
initial conditions) simulation in a 500 $h^{-1}\,{\rm Mpc}$ box,

  * •

a second VDE (and again corresponding $\Lambda$CDM) simulation in a 1
$h^{-1}$Gpc box,

  * •

two more VDE simulations with a different random seed, one in a 500
$h^{-1}\,{\rm Mpc}$ and one in a 1 $h^{-1}$Gpc box, as a check for the
influence of cosmic variance,

  * •

two $\Lambda$CDM-vde simulations in a 500 $h^{-1}\,{\rm Mpc}$ and a 1000
$h^{-1}\,{\rm Mpc}$ box.

The full set of simulations will be presented and analyzed in more detail in
an upcoming companion paper; in this Letter, instead, we chose to focus on
some of them only in order to gather information on large scale structures and
massive cluster at high redshift, respectively, in the two cosmologies. To
this extent, the use of the same initial seed for generating the initial
conditions in the coupled $\Lambda$CDM vs. VDE simulations allows us to
directly compare the structures identified by the halo finder.

Table 1: Cosmological parameters for $\Lambda$CDM $\Lambda$CDM-vde and VDE.  Model | $\Omega_{m}$ | $\Omega_{de}$ | $w_{0}$ | $\sigma_{8}$ | h
---|---|---|---|---|---
$\Lambda$CDM | 0.27 | 0.73 | -1 | 0.8 | 0.7
$\Lambda$CDM-vde | 0.388 | 0.612 | -1 | 0.83 | 0.7
VDE | 0.388 | 0.612 | -3.53 | 0.83 | 0.62
Table 2: $N$-body settings used for the GADGET-2 simulations, the two 500$h^{-1}\,{\rm Mpc}$ and the two 1$h^{-1}$Gpc have the same initial random seed and starting redshift $z_{\rm start}=60$ in order to allow for a direct comparison of the halo properties. The number of particles in each was fixed at $512^{3}$.The box size $B$ is given in $h^{-1}\,{\rm Mpc}$ and the particle mass in $h^{-1}{\rm{M_{\odot}}}$. Simulation | $B$ | $m_{p}$
---|---|---
$2\times$VDE-0.5 | 500 | $1.00\times 10^{11}$
$2\times$VDE-1 | 1000 | $8.02\times 10^{11}$
$\Lambda$CDM-0.5 | 500 | $6.95\times 10^{10}$
$\Lambda$CDM-1 | 1000 | $5.55\times 10^{11}$
$\Lambda$CDM-0.5vde | 500 | $1.00\times 10^{11}$
$\Lambda$CDM-1vde | 1000 | $8.02\times 10^{11}$

As a final remark, we underline here that the choice of the boxes was made in
order to allow the study of clustering on larger scales, without particular
emphasis on low mass objects, e.g. dark matter haloes with
$M<10^{14}$$h^{-1}{\rm{M_{\odot}}}$. This means that even though our halo
finder has been able to identify objects down to $\sim
10^{12}{{h^{-1}{\rm{M_{\odot}}}}}$ in the 500$h^{-1}\,{\rm Mpc}$ box and $\sim
10^{13}$ $h^{-1}{\rm{M_{\odot}}}$ in the 1$h^{-1}$Gpc one (which corresponds
to a lower limit of 20 particles, see below), we are not comparing the mass
spectrum at this far end. Therefore, since we are only interested in studying
the behaviour of the mass function of these models at the very high mass end,
in the following sections we will mostly refer to the $\Lambda$CDM-1,
$\Lambda$CDM-vde and VDE-1 simulations, where we have a larger statistics for
the supercluster scales.

###  3.2 Halo Finding

In order to identify halos in our simulation we have run the MPI+OpenMP hybrid
halo finder AHF described in detail in Knollmann & Knebe (2009). AHF is an
improvement of the MHF halo finder (Gill et al., 2004), which locates local
overdensities in an adaptively smoothed density field as prospective halo
centres. The local potential minima are computed for each of these density
peaks and the gravitationally bound particles are determined. Only peaks with
at least 20 bound particles are considered as haloes and retained for further
analysis, even though here we focus on the most massive objects only.

The mass of each halo is then computed via the equation
$M(r)=\Delta\rho_{c}4\pi r^{3}/3$ where we applied $\Delta=200$ as the
overdensity threshold. Using this relation, particular care has to be taken
when considering the definition of the critical density $\rho_{c}=3H^{2}/8\pi
G$ because it involves the Hubble parameter, that differs substantially at all
redshifts in the two models. This means that, identifying the halo masses, we
have to take into account the fact that the value of $\rho_{c}$ changes from
$\Lambda$CDM to VDE. This has been incorporated into and taken care of in the
latest version of AHF where $H_{VDE}(z)$ is being read in from a precomputed
table.

We would like to mention that we checked that the objects obtained by this
(virial) definition are in fact in equilibrium. To this extent we studied the
ratio between two times kinetic over potential energy $2T/|U|$ confirming that
at each redshift under investigation here this relation is equally well
fulfilled for the $\Lambda$CDM and – more importantly – the VDE simulations
(not presented here though). We therefore conclude that our adopted method to
define halo mass in the VDE model leads to unbiased results and yields objects
in equilibrium – as is the case for the $\Lambda$CDM haloes.

##  4 The Results

Figure 2: Mass functions (and their ratios) as computed for the VDE-1,
$\Lambda$CDM-1 and $\Lambda$CDM-1vde simulations at $z=1.4,1.2,1.1,$ and $0$.
These redshifts have been chosen in order to overlap with the aforementioned
observed massive clusters.

###  4.1 Mass Function

With the halo catalogues at our disposal, we computed the cumulative mass
functions $n(>M)$ at various redshifts. We show in Fig. 2 the results for the
1$h^{-1}$Gpc simulations at redshifts $z=1.4$, $z=1.2$, $z=1.1$ and $z=0$.
This plot is accompanied by Table 3 where we list the masses of the most
massive haloes found in each model and the redshifts under consideration.

We notice that the mass function for objecst with
$M>10^{14}{{h^{-1}{\rm{M_{\odot}}}}}$ is several times larger in VDE than in
$\Lambda$CDM at all redshifts, i.e. the number of high mass haloes in this
non-standard cosmological model is significantly increased. In particular, at
this mass scale the VDE mass function is about three times larger at the
relevant redshifts $z=1.4,1.2,$ and $1.1$ – and even larger at today’s time.

In order to verify that this feature of the VDE model is not a simple
reflection of cosmic variance (which should affect in particular the high mass
end, where the statistics is small) we compared the results presented in Fig.
2 to the mass functions of the set of two additional simulations started from
a different random seed for the initial conditions confirming aforementioned
results.

An interesting remark we would like to add here, is that the physical mass
(obtained dividing by the corresponding $h$ values the values quoted in
$h^{-1}{\rm{M_{\odot}}}$units) of the largest haloes in the VDE-1 simulation
at $z=1.4$, $z=1.2$ and $z=1.1$ are perfectly compatible with the ones of the
above clusters referred to in the Introduction, whereas the corresponding
$\Lambda$CDM candidates are outside the $2\sigma$ compatibility level. And
again, similar massive clusters have also been found in the duplicate VDE-1
simulation with a different initial seed.

Table 3: The most massive halo found in the three 1$h^{-1}$Gpc simulations (in units of $10^{14}$$h^{-1}{\rm{M_{\odot}}}$) as a function of redshift.  z |  $\Lambda$CDM-1 | VDE-1 |  $\Lambda$CDM-1vde
---|---|---|---
1.4 | 4.16 | 5.63 | 6.47
1.2 | 5.13 | 6.51 | 8.16
1.1 | 6.01 | 7.63 | 10.2
0 | 18.1 | 31.6 | 35.1

Comparing the $\Lambda$CDM-1vde to the VDE-1 simulation at different
redshifts, we note that while the two mass functions are almost
indistinguishable for $M<10^{14}\times{{h^{-1}{\rm{M_{\odot}}}}}$, on the
higher mass end the former even outnumbers the latter by approximately $\sim
3$. In the hierarchical picture of structure formation, we can attribute this
relative difference to dynamical effects caused by the different expansion
histories (based upon different $H(z)$) at later times $z\approx 1$, when the
most massive structures actually start to form. In general, however, the
$\Lambda$CDM-vde analysis shows that the enhancement seen in the VDE mass
function with respect to $\Lambda$CDM is clearly driven by the higher values
of $\Omega_{M}$ and $\sigma_{8}$, a result expected since the abundance of
clusters sensitively depends on the product of these two parameters (cf.
Huterer & White, 2002). On the one hand, this complicates the issue of model
selection, since (although disfavoured by the WMAP7 data) we could invoke a
(slightly) larger $\Omega_{M}$ or a higher $\sigma_{8}$ normalization at $z=0$
for $\Lambda$CDM trying to alleviate the current tension with the high-$z$
massive clusters observations. On the other hand, the distinct expansion
history that characterizes and differentiates between the two $\Lambda$CDM and
VDE models would still leave a clear imprint on structure formation, which
could be detected by, for instance, measuring $\sigma_{8}$’s dependence on
redshift. Such a test would indeed provide invaluable information for the
study of $\Lambda$CDM and for any cosmological model beyond it such as VDE.

###  4.2 Probability

Figure 3: Numerical cumulative number densities of objects with $M>5\times
10^{14}$$h^{-1}{\rm{M_{\odot}}}$ for VDE, $\Lambda$CDM and $\Lambda$CDM-vde.

In order to provide a more quantitative estimate of the the relative
probability of observationally detecting such massive clusters at the
indicated redshifts we used $n(>M,z)$ – the expected cumulative number density
of objects above a threshold mass $M$ as a function of redshift as given by
our simulations – and integrated it over the comoving volume $V_{c}$ of the
survey

$N(>M)=\int_{\Delta z,\Omega_{\rm survey}}n(>M,z)dV_{c}(z)$ (8)

where $\Delta z$ and $\Omega_{\rm survey}$ are the redshift interval and the
fraction of the sky covered by the survey to which we want to compare our
theoretical expectations.

While $n(>M,z)$ can be readily calculated in $\Lambda$CDM cosmologies (e.g.
Press & Schechter, 1974; Sheth & Tormen, 1999; Jenkins et al., 2001; Tinker et
al., 2008), in VDE we have to devise a strategy to compute it based upon our
numerical results only. We chose to adjust the formula of Sheth & Tormen
(1999) as follows:

  * •

we calculated the cumulative number densities in the desired redshift
intervals $\Delta z$ based upon our simulation data,

  * •

we adjusted the parameters of the Sheth-Tormen mass function fitting the
numerical cumulative number densities derived from the VDE-1 and VDE-0.5
simulations,

  * •

we used these best fit estimates to analytically compute $n(>M,z)$ now having
access to masses even outside our numerically limited range to be used with
Eq. (8).

The results of the numerical integration over the comoving volumes (obtained
using the limits quoted in the observational papers by Jee et al. (2009),
Brodwin et al. (2010) and Foley et al. (2011)) are listed in Table 4 for the
VDE, $\Lambda$CDM-vde and $\Lambda$CDM model. We can clearly see that the
chances are substantially larger in the VDE model to find such massive objects
than in $\Lambda$CDM; the numbers for VDE are in fact comparable to our
fiducial $\Lambda$CDM-vde model confirming their relation to the enhanced
values of $\Omega_{M}$ and $\sigma_{8}$. However, note that while the VDE
model is compliant with both SN-Ia and CMB data the $\Lambda$CDM-vde is
obviously ruled out observationally. We complement these results with Fig. 3
where we plot the abundance evolution of clusters with mass $M>5\times
10^{14}$$h^{-1}{\rm{M_{\odot}}}$ computed utilizing above described procedure
again. This plot confirms our previous analysis of the mass functions and
shows that the expectation of massive objects is amplified in VDE by a factor
$\sim 3$ to $\sim 10$ over the considered redshift range, a factor which is
even higher for the discretionary $\Lambda$CDM-vde. We would like to remark
here that while our $\Lambda$CDM estimate for XMMU J2235.3-2557 is in
agreement with the result quoted by Jee et al. (2009) (obtained using the same
approach as here), the calculation done for SPT-CL J2106-5844 leads to an
estimate substantially smaller than the one quoted by Foley et al. (2011),
calculated using a Monte Carlo technique. However, this does not affect our
conclusions, which are based on the comparison of results obtained in a
consistent manner for the two models.

Table 4: Expected number of objects $N(>M)$ in excess of mass $M$ and inside a certain (comoving) volume in the $\Lambda$CDM and VDE for different mass thresholds and survey volumes. Solid angles $\Omega$ are measured in deg2 and masses are measured in $10^{14}$$h^{-1}{\rm{M_{\odot}}}$. $M$ | $\Delta z$ | $\Omega_{\rm survey}$ | $N_{\Lambda\rm CDM}$ | $N_{\rm VDE}$ | $N_{\Lambda\rm-VDE}$
---|---|---|---|---|---
$>10$ | $>1$ | 2500 | 0.007 | 0.02 | 0.04
$>7$ | $>1$ | 2500 | 0.03 | 0.31 | 0.56
$>5$ | $1.38-2.2$ | 11 | 0.005 | 0.06 | 0.07

##  5 Conclusions

The observation of massive clusters at $z>1$ provides an additional, useful
test for the $\Lambda$CDM and other cosmological models beyond the standard
paradigm. In this Letter we have shown that the Vector Dark Energy (VDE)
scenario (Beltrán Jiménez & Maroto, 2008) might account for such observations
better than the $\Lambda$CDM concordance model, since the relative abundance
of extremely massive clusters with $M>5\times 10^{14}$$h^{-1}{\rm{M_{\odot}}}$
is at all redshifts higher in this non-standard cosmology: the expected number
of massive clusters is enhanced in VDE by at least a factor of $\sim 3$ to
find an object such as SPT- CL J2106-5844 at redshift $z\approx 1.2$ (Foley et
al., 2011) and a factor of $\sim 10$ for the other two observed clusters SPT-
CL J0546-5346 (Brodwin et al., 2010) and XMMU J2235.3-2557 (Jee et al., 2009).
Of course, these results might as well simply point in the direction of
modifying the standard paradigm, for example including non-Gaussianities in
the initial conditions or either using a higher $\sigma_{8}$ or $\Omega_{M}$
value for the $\Lambda$CDM as the comparison to the $\Lambda$CDM-vde model
seems to suggest.

Nonetheless, this first results on the large scale clustering in the case of
VDE cosmology point in the right direction, significantly enhancing the
probability of producing extremely massive clusters at high redshift as recent
observations seem to require. For a more elaborate discussion and comparison
of the VDE to the $\Lambda$CDM model (not solely focusing on massive clusters)
we though refer the reader to the companion paper (Carlesi et al., in
preparation).

## Acknowledgements

EC is supported by the MareNostrum project funded by the Spanish Ministerio de
Ciencia e Innovacion (MICINN) under grant no. AYA2009-13875-C03-02 and
MultiDark Consolider project under grant CSD2009-00064. AK acknowledges
support by the MICINN’s Ramon y Cajal programme as well as the grants AYA
2009-13875-C03-02, AYA2009-12792-C03-03, CSD2009-00064, and CAM
S2009/ESP-1496. GY acknowledges support from MICINN’s grants
AYA2009-13875-C03-02 and FPA2009-08958. JBJ is supported by the Ministerio de
Educación under the postdoctoral contract EX2009-0305 and also wishes to
acknowledge support from the Norwegian Research Council under the YGGDRASIL
programme 2009-2010 and the NILS mobility project grant UCM-EEA-ABEL-03-2010.
We also acknowledge support from MICINN (Spain) project numbers FIS
2008-01323, FPA 2008-00592 and CAM/UCM 910309.

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