# Cosmic voids are emptier in the presence of symmetron’s domain walls

Bahar Nosrati<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, G.C., Evin, Tehran 19839, Iran  Nima Khosravi
<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti University, G.C.,
Evin, Tehran 19839, Iran

###### Abstract

The symmetron field has an environment (density) dependent behavior which is a
common feature of the models with the screening mechanism and results in a
rich phenomenology. This model can produce domain walls between regions with
different densities. We consider this aspect and study the physics of domain
walls in between (underdensity) voids and (overdensity) halo structures. The
(spherical) domain walls exert a repulsive force on a test mass outside of the
wall while a test mass inside of the wall sees no force. This makes the
structures outside the voids go further to a larger radius. Effectively, this
means the voids are becoming larger in this scenario in comparison to the
standard model of cosmology. Interestingly, this makes voids emptier which may
shed light on Peebles’ void phenomenon.

##  I Introduction

The field of cosmology is in one of its interesting milestones. The standard
model of cosmology, $\Lambda$CDM, is very successful in describing many of the
different observations including the cosmic microwave background, the large
structure formations, etc. But meanwhile, some of the new observation hint at
anomalies and tensions in $\Lambda$CDM. These anomalies can be categorized
roughly to i) discrepancies between the observations of the early universe and
the late one e.g. the Hubble tension Riess:2019cxk ; Bernal:2016gxb ;
Riess:2018byc , $\sigma_{8}$ tension Heymans:2020gsg  and CMB lensing
Planck:2018vyg , ii) some anomalies are spatial anomalies e.g. the CMB dipole
modulations Eriksen:2003db ; Hansen:2008ym  and quadrupole-octopole alignment
Copi:2006tu ; Copi:2005ff ; Copi:2010na ; Copi:2013jna ; deOliveira-
Costa:2003utu ; Schwarz:2004gk  and iii) the third category is in the physics
of (nonlinear) cosmological structures e.g. the core-cusp problem
deBlok:2009sp ; Flores:1994gz  and the void phenomenon Peebles:2001nv . We do
not claim these different tensions are not related and there are proposals in
the literature to study some of them altogether Banihashemi:2018has . On the
other hand, there are some theoretical issues in $\Lambda$CDM: the
cosmological constant problem and the physics behind both dark matter and dark
energy. The current situation convinced the community to explore different
paths both in the observations and theoretical models to make our
understanding deeper.

Among all of the observational probes, the cosmological voids are the less
studied ones but they have become more interesting recently Pisani:2019cvo .
The physics (and evolution) of voids is correlated to the overdensity
structures, both are given by one model e.g. $\Lambda$CDM model. Void physics
is, however, linear, while (overdensed) structures are nonlinear. This fact
makes voids an interesting cosmological probe in addition to the large scale
(overdense) structures and CMB. Though there are few tensions concerning voids
when we assume $\Lambda$CDM. The void phenomenon mentioned by Peebles
Peebles:2001nv  says that the observed voids are emptier than the prediction
of $\Lambda$CDM. This fact can be interpreted as less structure in the voids
and may hint at an environment dependent structure formation physics
Peebles:2001nv . There is another issue concerning voids: Tavasoli reported
that the structures in voids are more concentrated in the center of them which
is not consistent with $\Lambda$CDM’s prediction Tavasoli:2021reo . Is there
any explanation for these observations?

In the theory of $\Lambda$CDM, the physics of dark energy was/is always a
matter of question. For many different reasons, the modified gravity models
Joyce:2014kja ; Amendola:2016saw ; Clifton:2011jh  are suggested instead of
just having a constant, the cosmological constant. In this work, for our
purposes, we focus on the symmetron model Hinterbichler:2011ca . This model
suggests that the gravitational force behaves differently according to the
density. The change in its behavior is realized in a way very similar to the
phase transition. When a system experiences phase transition, it has undergone
symmetry breaking and in a model such as symmetron model $\rm Z_{2}$ symmetry,
a discrete symmetry, breaks, and as a consequence domain walls (DWs) are
created. This is also a natural prediction in the GLTofDE model
Banihashemi:2018has . One should be careful that these DWs are produced in
very small energy scales at late times which can satisfy the observational
constrains111Note that the DWs which are produced by the electroweak phase
transition at very high energy scales cannot satisfy the observational tests.
It is one of the main reasons for cosmological inflation Mukhanov:2005sc ..
The symmetron DWs should be placed in the region between high and low
densities. Consequently, we expect to have them at the border between the
cosmological overdensities (nonlinear structures) and underdensities (voids).

In this paper, we study the effects of such DWs and their effects on the
(size) distribution of cosmic voids. In the next section, we briefly review
the symmetron mechanism and the production of DWs in it. Then we study the
behavior of a spherical DW and especially consider the movement of a test mass
in the presence of a wall. We close our paper with conclusions and future
perspectives.

##  II Symmetron Model

As it was discussed in the previous section one of the models which can
produce DWs due to symmetry breaking is the symmetron model. We will introduce
some of its parameters in this section. Symmetron action was presented by
Hinterbichler:2011ca  as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\sqrt{-g}\bigg{[}\frac{M^{2}_{Pl}}{2}\,R-\frac{1}{2}\,g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi-V(\phi)\bigg{]}$
(1) $\displaystyle+$ $\displaystyle\int
d^{4}x\sqrt{-\tilde{g}}\mathcal{L}_{m}(\psi,\tilde{g}_{\mu\nu})\,,$

where $\tilde{g}_{\mu\nu}\equiv A^{2}(\phi)\,g_{\mu\nu}$. The conformal
coupling factor is

$\displaystyle A^{2}(\phi)$ $\displaystyle=$ $\displaystyle
1+\frac{\phi^{2}}{2\,M^{2}}\,,$ (2)

and the potential has the following form

$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\mu^{2}\,\phi^{2}+\frac{1}{4}\lambda\,\phi^{4}\,.$
(3)

We can introduced the effective potential as Hinterbichler:2011ca

$\displaystyle
V_{eff}(\phi)=\frac{1}{2}\,(\frac{\rho}{M^{2}}-\mu^{2})\,\phi^{2}+\frac{1}{4}\,\lambda\,\phi^{4}\,.$
(4)

Vacuum expectation value before the symmetry breaking would be $\phi_{0}=0$
and after the symmetry breaking would be:

$\displaystyle\phi_{0}$ $\displaystyle=$
$\displaystyle\pm\frac{\mu}{\sqrt{\lambda}}\,.$ (5)

Also Llinares:2014zxa  has introduced another form for the effective
potential;

$\displaystyle V_{eff}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{4\lambda_{0}^{2}}\,\Big{(}\frac{\rho}{\rho_{SSB}}-1\Big{)}\,\phi^{2}+\frac{1}{4}\lambda\,\phi^{4}\,.$
(6)

where $\lambda_{0}=\frac{1}{\sqrt{2}\,\mu}$ is the Compton wavelength, and
$\rho_{SSB}=\mu^{2}\,M^{2}$ is the symmetry breaking density. We assume the
field becomes tachyonic around the symmetry breaking redshift, $z_{SSB}$ i.e.
$\mu^{2}\,M^{2}\sim\alpha\,M_{pl}^{2}\,H_{SSB}^{2}$ where $H_{SSB}$ is the
Hubble parameter at $z_{SSB}$ , $M_{pl}$ is the Planck mass and $\alpha$ is a
free parameter. This assumption seems natural since it means the symmetry
breaking (roughly) depends only on the local (environmental) temporal and
spatial parameters. We will see that different values of the free parameter
$\alpha$ plays an important role in the results by giving larger values for
the DW’s surface tension (which plays a key role in the effect of DW on a test
particle).

So far we have presented the important parameters of the symmetron model.
Since the symmetron model is a screening mechanism, there are strong
constraints on its effect by local tests of gravity. Here are some constraints
set by Hinterbichler:2010es

$\displaystyle M$ $\displaystyle\lesssim$ $\displaystyle 10^{-3}M_{pl}\,,$ (7)
$\displaystyle\beta=\frac{\phi_{0}M_{pl}}{M^{2}}$ $\displaystyle\approx$
$\displaystyle 1\,,$ (8) $\displaystyle\lambda_{0}$ $\displaystyle\lesssim$
$\displaystyle 10^{-3}\,c\,\alpha^{-1}\,H_{0}^{-1}\,(1+z_{SSB})^{-3/2}\,,$ (9)

where $\beta$ is a dimensionless coefficient. Note that this is only valid in
a matter dominated universe which is the regime of our interests. From
relations (5), (8), and our assumption ($\mu^{2}\,M^{2}\sim
M_{pl}^{2}\,H_{SSB}^{2}$), one is lead to

$\displaystyle\phi_{0}$ $\displaystyle=$ $\displaystyle
6\,\beta\,\lambda_{0}^{2}\,M_{pl}\,\alpha^{2}H_{0}^{2}\,\Omega_{0}\,(1+z_{SSB})^{3}.$
(10)

where $\Omega_{0}$ is the background density parameter. It is evident from (6)
that the symmetron’s potential undergoes a symmetry breaking in under-dense
regions, $\rho<\rho_{SSB}$. Breaking of symmetron’s discrete symmetry will
result in the existence of DWs. In Llinares:2014zxa  it is suggested that the
surface energy density of a DW is the difference between the energy density of
a universe with a DW and without one,

$\displaystyle\sigma=\frac{2\,\phi_{0}^{2}}{3\,c\,\hbar\,\lambda_{0}}\,\sqrt{1-\frac{(1+z)^{3}}{(1+z_{SSB})^{3}}}\,\bigg{[}1+\frac{(1+z)^{3}}{2(1+z_{SSB})^{3}}\bigg{]}.$
(11)

Figure 1: In this figure, we illustrate our idea with a cartoon. The symmetron
field changes its behavior with respect to the environment density. In high
densities (i.e. structures which are shown in a yellow-ish stripe),
$\rho_{SSB}<\rho$, the effective potential (6) has a minimum at $\phi_{0}=0$
as it is shown at the lower-right plot. But in low densities (i.e. voids which
are dark in the simulation), $\rho_{SSB}>\rho$, the potential experiences a
symmetry breaking resulting in equilibrium at $\phi_{0}\neq 0$ (shown at the
upper-right plot). These different values of the field cause a DW between the
structures and voids (shown as the white circle). It is obvious that in the
real world we cannot assume a spherical DW but in this work we keep this
approximation Springel:2005nw .

Observationally, we expect to have these DWs at the boundary of cosmological
voids and (overdensity) structures as shown in figure 1. Since in voids, we
can have the negative value for the first term in (6) while it is positive for
the structures. This means the field does (not) see the symmetry breaking in
voids (structures). Consequently, the region between voids and structures has
potential to host a DW, To study the effects of such DWs on the void
environment, we study the geometrical properties of DWs in the next section.

##  III Spherical domain wall’s metric

For our purposes, we focus on the spherical DWs, they are able to give us a
clue about what we should expect in observation. In the first step, we find
the metric for a moving spherical DW and then we switch to the static ones.

###  III.0.1 Moving Spherical Domain Wall

In order to find the metric of spherical DW, we have considered the general
form of spherically symmetric metric

$\displaystyle ds^{2}|_{+}$ $\displaystyle=$ $\displaystyle
e^{A(r,t)}\,dt^{2}-e^{B(r,t)}\,dr^{2}-r^{2}\,d\Omega^{2}\,,$ $\displaystyle
ds^{2}|_{-}$ $\displaystyle=$ $\displaystyle
e^{D(r,t)}\,dt^{2}-e^{C(r,t)}\,dr^{2}-r^{2}\,d\Omega^{2}\,,$ (12)

where the signs $+$ and $-$ refer to the outside and inside of the DW.

As a starting point for solving Einstein’s equations, we have assumed the
following constraints: First, we are in the thin-wall limit and the wall is
moving, which means; it can either collapse or expand. Moreover, the wall is
assumed to be in a vacuum, and the outside metric is asymptotically flat at
infinite distances. Solutions to the Einstein equations yield the metric

$\displaystyle ds^{2}|_{+}$ $\displaystyle=$ $\displaystyle
e^{A(t)}\,\big{(}1+\frac{F}{r}\big{)}\,dt^{2}-(1+\frac{F}{r})^{-1}\,dr^{2}-r^{2}\,d\Omega^{2}\,,$
$\displaystyle ds^{2}|_{-}$ $\displaystyle=$ $\displaystyle
e^{D(t)}\,\big{(}1+\frac{H}{r}\big{)}\,dt^{2}-(1+\frac{H}{r})^{-1}\,dr^{2}-r^{2}\,d\Omega^{2}\,.$

Here $F$ and, $H$ are the constant of integration. Since we should prevent the
inside metric from becoming singular, $H$ should be zero. Consequently; the
metric will have the following form:

$\displaystyle ds^{2}|_{+}$ $\displaystyle=$ $\displaystyle
e^{A(t)}\,\big{(}1+\frac{F}{r}\big{)}\,dt^{2}-(1+\frac{F}{r})^{-1}\,dr^{2}-r^{2}\,d\Omega^{2}\,,$
$\displaystyle ds^{2}|_{-}$ $\displaystyle=$ $\displaystyle
e^{D(t)}\,dt^{2}-dr^{2}-r^{2}\,d\Omega^{2}\,.$ (13)

To find $F$, $A(t)$, and $D(t)$, one should satisfy the two Israel junction
conditions. The first junction condition implies that the intrinsic three-
metric, $h_{ab}$, for the hypersurface (in this case the DW) be the same both
inside and outside of the hypersurface ($h_{ab}|_{+}=h_{ab}|_{-}$). The second
condition that was defined by Israel Israel:1966rt  is the equality of the
following equations

$\displaystyle\gamma_{ab}$ $\displaystyle=$ $\displaystyle
K_{ab}|_{+}-K_{ab}|_{-}\,,$ (14) $\displaystyle\gamma_{ab}$ $\displaystyle=$
$\displaystyle 8\pi\,G\,\big{(}S_{ab}-\frac{1}{2}h_{ab}\,S_{c}^{c}\big{)}\,,$
(15)

where $K_{ab}$ is the extrinsic curvature and, $S_{ab}$, the stress-energy
tensor, is

$\displaystyle S^{ab}$ $\displaystyle=$ $\displaystyle\sigma
u^{a}u^{b}-\tau(h^{ab}+u^{a}u^{b})\,.$ (16)

$\sigma$ and, $\tau$ are respectively the surface energy density and tension
of the wall measured by any observer whose world line lies on the hypersurface
(DW) and sees no energy flux in her local frame, and $u^{a}$ is the four-
velocity of that observer. For the choice of equation (16), equation (15)
becomes

$\displaystyle\gamma_{ab}$ $\displaystyle=$
$\displaystyle-4\,\pi\,G\big{[}\sigma\,h_{ab}+2\,(\sigma-\tau)\,u_{a}\,u_{b}\big{]}\,.$
(17)

For DWs, $\sigma=\tau$, implying that

$\displaystyle\gamma_{ab}$ $\displaystyle=$
$\displaystyle-4\,\pi\,G\,\sigma\,h_{ab}\,.$ (18)

As a result, for a DW to satisfy the second junction condition, we must
equalize (14) and (18).

To determine the intrinsic metric of the DW and its extrinsic curvature, we
should calculate the unit spacelike normal for both metrics.

$\displaystyle n_{\mu}|_{+}$ $\displaystyle=$
$\displaystyle\Big{(}-e^{\frac{A(t)}{2}}\dot{R},\,\big{(}1+\frac{F}{r}\big{)}^{-1}\sqrt{1+\frac{F}{r}+\dot{R}^{2}},\,0,\,0\Big{)}\,,$
$\displaystyle n_{\mu}|_{-}$ $\displaystyle=$
$\displaystyle\Big{(}-e^{\frac{D(t)}{2}}\,\dot{R},\sqrt{1+\dot{R}^{2}},\,0,\,0\Big{)}\,,$
(19)

where $R$ is the radius of the DW. Be aware that dot denotes a derivative with
respect to proper time and prime denotes a derivative with respect to $r$.

Now we can calculate the intrinsic metric of the DW using the outside and
inside metric

$\displaystyle h_{ab}|_{+}$ $\displaystyle=$ $\displaystyle
diag\Big{(}e^{A(t)}\,\big{(}1+\frac{F}{R}-\dot{R}^{2}\big{)},\,-R^{2},\,-R^{2}\sin^{2}{\theta}\Big{)}\,,$
$\displaystyle h_{ab}|_{-}$ $\displaystyle=$ $\displaystyle
diag\Big{(}e^{D(t)}(1-\dot{R}^{2}),\,-R^{2},\,-R^{2}\sin^{2}{\theta}\Big{)}\,.$
(20)

The first junction condition indicates that $h_{ab}|_{+}=h_{ab}|_{-}$,
therefore this will lead us to

$\displaystyle\frac{e^{D(t)}}{e^{A(t)}}$ $\displaystyle=$
$\displaystyle\frac{1+\frac{F}{R}-\dot{R}^{2}}{1+\dot{R}^{2}}\,.$ (21)

Here we have found a relation between the metrics coefficients.

In order to satisfy the second junction condition, we need to find the
extrinsic curvature for both metrics from
$K_{ab}=h_{a}^{c}\bigtriangledown_{c}n_{b}$. The extrinsic curvature for the
outside metric has the following form

$\displaystyle K_{tt}|_{+}$ $\displaystyle=$
$\displaystyle-e^{\frac{-A(t)}{2}}\,\ddot{R}-\frac{F}{2R^{2}}\,e^{A(t)}\sqrt{1+\frac{F}{R}+\dot{R}^{2}}\,,$
$\displaystyle K_{\theta\theta}|_{+}$ $\displaystyle=$ $\displaystyle
R\,\sqrt{1+\frac{F}{R}+\dot{R}^{2}}\,,$ (22) $\displaystyle K_{\phi\phi}|_{+}$
$\displaystyle=$ $\displaystyle
R\,\sin^{2}{\theta}\,\sqrt{1+\frac{F}{R}+\dot{R}^{2}}\,,$

and for the inside metric is

$\displaystyle K_{tt}|_{-}$ $\displaystyle=$
$\displaystyle-e^{\frac{D(t)}{2}}\,\ddot{R}\,,$ $\displaystyle
K_{\theta\theta}|_{-}$ $\displaystyle=$ $\displaystyle
R\,\sqrt{1+\dot{R}^{2}}\,,$ (23) $\displaystyle K_{\phi\phi}|_{-}$
$\displaystyle=$ $\displaystyle R\,\sin^{2}{\theta}\,\sqrt{1+\dot{R}^{2}}\,.$

Hence to compare the extrinsic curvature with stress-energy tensor relation
one finds $\gamma_{ab}$ from (14),

$\displaystyle\gamma_{tt}$ $\displaystyle=$
$\displaystyle-e^{\frac{-A(t)}{2}}\,\ddot{R}-\frac{F}{R^{2}}\,e^{A(t)}\sqrt{1+\frac{F}{R}+\dot{R}^{2}}+e^{\frac{D(t)}{2}}\ddot{R}\,,$
$\displaystyle\gamma_{\theta\theta}$ $\displaystyle=$ $\displaystyle
R\,\Big{(}\sqrt{1+\frac{F}{R}+\dot{R}^{2}}-\sqrt{1+\dot{R}^{2}}\,\Big{)}\,,$
(24) $\displaystyle\gamma_{\phi\phi}$ $\displaystyle=$ $\displaystyle
R\,\sin^{2}\theta\,\Big{(}\sqrt{1+\frac{F}{R}+\dot{R}^{2}}-\sqrt{1+\dot{R}^{2}}\,\Big{)}\,.$

$\gamma_{ab}$ is also given by Eq. (18)

$\displaystyle\gamma_{tt}$ $\displaystyle=$
$\displaystyle-4\,\pi\,G\,\sigma\,e^{D(t)}(1-\dot{R}^{2})\,,$
$\displaystyle\gamma_{\theta\theta}$ $\displaystyle=$ $\displaystyle
4\,\pi\,G\,\sigma\,R^{2}\,,$ (25) $\displaystyle\gamma_{\phi\phi}$
$\displaystyle=$ $\displaystyle 4\,\pi\,G\,\sigma\,R^{2}\,\sin^{2}\theta\,.$

From equalizing $\gamma_{tt}$ ((III.0.1) and (III.0.1)) we will obtain

$\displaystyle-e^{\frac{-A(t)}{2}}\,\ddot{R}$ $\displaystyle-$
$\displaystyle\frac{F}{R^{2}}\,e^{A(t)}\,\sqrt{1+\frac{F}{R}+\dot{R}^{2}}+e^{\frac{D(t)}{2}}\,\ddot{R}$
(26) $\displaystyle=$
$\displaystyle-4\,\pi\,G\,\sigma\,e^{D(t)}\,(1-\dot{R}^{2})\,,$

and from $\gamma_{\theta\theta}$ (or $\gamma_{\phi\phi}$) we will have

$\displaystyle
F=R\,\big{(}4\pi\,G\,\sigma\,R+\sqrt{1+\dot{R}^{2}}\,\big{)}^{2}-(1+\dot{R}^{2})\,R\,,$
(27)

which they have satisfied the second junction condition.

We are curious to obtain these equations for a static wall as well. To solve
the first equation (26) for a static wall, one should put all $\dot{R}$ to
zero, but in this equation, the $\ddot{R}$ will remain which tells us that
static DWs are unstable. If we convert (27) to SI units, this part
$1+(\frac{\dot{R}}{c})^{2}$ will tell us that the speed of the DW does not
play a key role in this equation. We will see in the next section that
$1+(\frac{\dot{R}}{c})^{2}$ can be used as a correction amplitude for a static
wall solution.

Though we have mentioned that static walls are unstable, for sake of
simplicity we are going to ignore the tiny effects of DW’s movement and find
the metric of static DW.

###  III.0.2 Static Spherical Domain Wall

For a static wall, we can put $\dot{R}=0$ and re-write the above equations.
You can find detailed calculations in the Appendix. So if the metric of the
wall is

$\displaystyle ds^{2}|_{+}$ $\displaystyle=$ $\displaystyle
e^{A}\,\big{(}1+\frac{F}{r}\big{)}\,dt^{2}-(1+\frac{F}{r})^{-1}\,dr^{2}-r^{2}\,d\Omega^{2}\,,$
$\displaystyle ds^{2}|_{-}$ $\displaystyle=$ $\displaystyle
e^{D}\,dt^{2}-dr^{2}-r^{2}\,d\Omega^{2}\,.$ (28)

Since wall is static, we have $A(t)=A$ and $D(t)=D$. We will get the following
relation between the parameters due to the junction conditions

$\displaystyle\frac{e^{D}}{e^{A}}$ $\displaystyle=$ $\displaystyle
1+\frac{F}{R}\,,$ (29) $\displaystyle F$ $\displaystyle=$ $\displaystyle
R\,\big{(}4\pi\,G\,\sigma\,R+1\,\big{)}^{2}-R\,.$ (30)

As it was explained for the moving DW, these equations explain the relation
between the metric’s coefficients and give us $F$. It is clear that equation
(27) will be reduced to (30) when $\dot{R}$ is zero or even when the wall is
moving slowly since the sentence $1+(\frac{\dot{R}}{c})^{2}$ was presumed to
be a correction amplitude. Therefore we conclude that it is possible to
neglect the movement of the wall. Here we have not written the counterpart of
equation (26) for static DW, since it cannot be trusted due to having
$\ddot{R}$. We will explain in the next section that ignoring the instability
of static walls will not cause us any problems.

##  IV Geodesic of a test particle near a static domain wall

Now we want to find the effects of the static wall on a test mass. Since
solving geodesic equations for (III.0.2) is difficult we will use a shortcut
and solve $u^{\mu}\,u_{\mu}=+1$ using the Killing vectors

$\displaystyle K_{\mu}$ $\displaystyle=$
$\displaystyle\Big{(}e^{A}\,(1+\frac{F}{r}),\,0,\,0,\,0\Big{)}\,,$
$\displaystyle R_{\mu}$ $\displaystyle=$
$\displaystyle\Big{(}0,\,0,\,0,-r^{2}\,\sin^{2}\theta\Big{)}\,.$ (31)

Hence by solving the four-velocity equation,

$\displaystyle e^{A}$
$\displaystyle\,(1+\frac{F}{r})\,(\frac{dt}{d\tau})^{2}-(1+\frac{F}{r})^{-1}\,(\frac{dr}{d\tau})^{2}$
(32) $\displaystyle-$ $\displaystyle
r^{2}\,(\frac{d\theta}{d\tau})^{2}-r^{2}\,\sin^{2}\theta\,(\frac{d\phi}{d\tau})^{2}=+1\,,$

neglecting the test particle’s movement in $(\theta,\phi)$ plane due to
symmetries, and using the following conserved quantities,

$\displaystyle E$ $\displaystyle=$ $\displaystyle
K_{\mu}\,\frac{dx^{\mu}}{d\lambda}=e^{A}\,(1+\frac{F}{r})\,\frac{dt}{d\tau}\,,$
$\displaystyle L$ $\displaystyle=$ $\displaystyle-
R_{\mu}\,\frac{dx^{\mu}}{d\lambda}=r^{2}\,\sin^{2}\theta\,\frac{d\phi}{d\tau}\,,$
(33)

we will have the radial equation as

$\displaystyle\frac{d^{2}r}{d\tau^{2}}$ $\displaystyle=$
$\displaystyle\frac{F}{2r^{2}}\,.$ (34)

Since from (30) we can see that $F$ is positive, it is without question that a
test particle near a spherical DW will experience repulsion. From equations
(30) and (34) we understand that the radial movement of the test particle is
only influenced by the surface energy density and radius of the DW.

Before showing our results we must arise the issue of the instability of
static DWs. In Llinares:2014zxa , it has been stated that when spherical DWs
are pinned to matter, they become stable. Therefore by trusting their results
we will examine the effect of a static spherical DW on a test particle.

To find the effect of symmetron model DWs, we should change equation (34) so
that it will be connected to cosmological parameters. First, One should change
proper time $\tau$ to cosmological time via $d\tau^{2}=g_{00}\,dt^{2}$, and
$t$ should be converted into redshift. Therefore, equation (34) becomes

$\frac{d^{2}r}{dz^{2}}=\frac{F}{2r^{2}}\,\big{(}1+\frac{F}{r}\big{)}\,\frac{1}{H_{0}^{2}\,\Omega_{0}\,(1+z)^{3}}\,.$
(35)

for a matter dominant universe. We have solved equation (35) for a test
particle near a spherical DW, with the following parameters

$\displaystyle H_{0}$ $\displaystyle\approx$ $\displaystyle
10^{-10}\,\frac{1}{yr}\,,$ $\displaystyle M_{pl}$ $\displaystyle\approx$
$\displaystyle 10^{-8}\,kg\,,$ $\displaystyle G$ $\displaystyle\approx$
$\displaystyle 10^{-64}\,\frac{Mpc^{3}}{kg\,yr^{2}}\,,$ $\displaystyle\hbar$
$\displaystyle\approx$ $\displaystyle 10^{-72}\,\frac{Mpc^{2}\,kg}{yr}\,,$
$\displaystyle\Omega_{0}$ $\displaystyle=$ $\displaystyle 1\,,$ $\displaystyle
z_{0}$ $\displaystyle=$ $\displaystyle 0\,.$

In figure 2, we have shown the changes in void’s radius as a function of
redshift for a typical void with an initial radius $10Mpc$ and free parameter
$\alpha=30$. As we expected for earlier $z_{SSB}$ we see a larger effect. It
is worth mentioning that the effect of DW can make a typical void with radius
$10Mpc$ to be larger by $\sim 10\%$ depending on the symmetry breaking
redshifts. In figure 3, we have the same plot but for different initial sizes
for voids while $z_{SSB}=10$ and free parameter $\alpha=30$. It is obvious
from this plot that for smaller voids, the effect is larger. This can be
confirmed, mathematically, by rewriting equation (35) as

$\displaystyle\frac{d^{2}x}{dz^{2}}$ $\displaystyle\propto$
$\displaystyle\frac{1}{R^{2}\,x^{2}}\,$ (36)

where we have defined $x\equiv r/R$ and omit non-relevant terms. This means
for larger $R$’s we have less changes in the $x$ which is the normalized
displacement.

Figure 2: For a typical initial size of a void, $10Mpc$, we have plotted the
change in its radius as a function of redshift (where we fixed $\alpha=30$).
As we could expect, earlier symmetry breaking causes more effect i.e. larger
changes in the void size. Figure 3: For a fixed $z_{SSB}=10$ and $\alpha=30$
we have plotted the change in the void’s radius as a function of redshift for
the different initial sizes of voids. Interestingly, we see that the relative
change is larger for the smaller voids. Figure 4: In this plot we have shown
the affects of our free parameter (i.e. $\alpha$ which is introduced below
relation (6) as $\mu^{2}\,M^{2}\sim\alpha\,M_{pl}^{2}\,H_{SSB}^{2}$) on the
final results for a typical case of $R=10Mpc$ and $z_{SSB}=10$. The case
$\alpha=1$ is the case which is considered in Llinares:2014zxa  which has a
very (unobservable) tiny effect. But it is interesting that for one order of
magnitude larger values of $\alpha$ we can get around $10\%$ increasing in the
radius of voids.

In figure 4 we have shown the different values of the free parameter and it is
evident that the slight change in $\alpha$ (one order of magnitude) can result
in a larger voids (about $10\%$).

The interpretation of these results can be interesting. The voids are
underdense regions which are surrounded by the (overdense) structures. It
means when (symmetron’s) DW pushes the structures away from the center of the
voids, then the voids seem larger. This can be a smoking gun for the symmetron
mechanism. More interestingly, this fact can be a solution for the void
phenomenon. Since the DW’s effect is a late time effect (via dark energy) then
their effects on the nonlinear halo structures can be very small. This means
while the voids are getting larger, the (number) distribution of the halos
does not change. Consequently, we have bigger voids with the same number of
halos in them i.e. the voids seem emptier in this model than in the standard
model of cosmology. Note that our proposal is opposed to usual (cosmological)
proposals for the void phenomenon. Usually, the proposals look for less number
of halo structures inside the voids but we claim that voids are larger.

##  V Concluding remarks and future perspectives

We studied the cosmological effects of DWs which are produced by the symmetron
mechanisms. This is important as a test for these kinds of models beyond the
standard probes e.g. the late time acceleration. The existence of these DWs
can be consistent with the observational constraints on the parameters of the
symmetron model. These DWs are placed in between the voids and structures and
can affect their relative positions. We showed that there is an additional
repulsive force on the matter (particles) outside the DW while the metric
inside the DW is ineffective. This makes voids look larger than their
predicted size in $\Lambda$CDM. In other words, the voids are emptier in
comparison to the standard ($\Lambda$CDM) voids. It is a very interesting
result since it can be a solution for the famous void phenomenon
Peebles:2001nv . Usually, to solve this phenomenon, models try to lessen the
population of the structures inside the voids Peebles:2001nv . But our
proposal opens another path to look at this problem: larger voids make less
dense voids. This way of looking at the void phenomenon has another very
specific prediction which makes it distinguishable from the other scenarios.
Since in our proposal there is no modification for the structure formation
inside the voids then we expect the same distribution of the structures inside
the voids as the standard scenario. However, since the voids are getting
bigger these structures should seem to appear more at the center of the voids.
This fact is in the direction of recent observations Tavasoli:2021reo .

The idea which is explored in this work can be seen as a new way to look for
the effects of modified gravity models in future surveys, especially in
cosmological voids. In addition, late time DWs can have their fingerprints in
gravitational waves if they could evaporate. On the theory side, the
production of DWs in the symmetron model can be studied in more detail
especially by asking about their stabilities (in this paper we assumed the DWs
are stable since they are pinned to matter Llinares:2014zxa ). One other
interesting way to pursue our idea is to check what happens if we do not
assume the voids to be spherically symmetric.

## Acknowledgments

We would like to thank Shant Baghram, Levon Pogosian, and Saeed Tavasoli for
their comments on our manuscript.

## Appendix

To calculate the static spherical DW metric, as it was stated, normal unit
vectors are needed. One finds $n_{a}$ for the outside and inside of the DW as

$\displaystyle n_{\mu}|_{+}$ $\displaystyle=$
$\displaystyle\Big{(}0,\,-\big{(}1+\frac{F}{r}\big{)}^{-\frac{1}{2}},\,0,\,0\Big{)}\,,$
$\displaystyle n_{\mu}|_{-}$ $\displaystyle=$
$\displaystyle\Big{(}0,\,-1,\,0,\,0\Big{)}\,.$ (37)

In addition, intrinsic metric is assumed to have the following form

$\displaystyle h_{ab}|_{+}$ $\displaystyle=$ $\displaystyle
diag\Big{(}e^{A}\,\big{(}1+\frac{F}{R}\big{)},\,-R^{2},\,-R^{2}\,\sin^{2}\theta\Big{)}\,,$
$\displaystyle h_{ab}|_{-}$ $\displaystyle=$ $\displaystyle
diag\Big{(}e^{D},\,-R^{2},\,-R^{2}\,\sin^{2}\theta\Big{)}\,,$ (38)

therefore; the first condition yields (29). Here we have found the ratio of
$\frac{e^{H}}{e^{G}}$. To find $F$ we should obtain the second condition. In
order to do so, we need to calculate (14), and (18). Extrinsic curvature for
the static wall from the outside metric is

$\displaystyle K_{tt}|_{+}$ $\displaystyle=$
$\displaystyle-\frac{F}{2R^{2}}\,e^{A}\,,$ $\displaystyle
K_{\theta\theta}|_{+}$ $\displaystyle=$ $\displaystyle
R\,\sqrt{1+\frac{F}{R}}\,,$ (39) $\displaystyle K_{\phi\phi}|_{+}$
$\displaystyle=$ $\displaystyle R\sqrt{1+\frac{F}{R}}\,\sin^{2}\theta\,,$

and from the inside metric, extrinsic curvature has the following form

$\displaystyle K_{tt}|_{-}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle K_{\theta\theta}|_{-}$ $\displaystyle=$ $\displaystyle R\,,$
(40) $\displaystyle K_{\phi\phi}|_{-}$ $\displaystyle=$ $\displaystyle
R\,\sin^{2}\theta\,.$

We obtain $\gamma_{ab}$ by substituting the above equations in (14)

$\displaystyle\gamma_{tt}$ $\displaystyle=$
$\displaystyle-\frac{F}{2R^{2}}\,e^{A}\,,$
$\displaystyle\gamma_{\theta\theta}$ $\displaystyle=$ $\displaystyle
R\,\big{(}\sqrt{1+\frac{F}{R}}-1\big{)}\,,$ (41)
$\displaystyle\gamma_{\phi\phi}$ $\displaystyle=$ $\displaystyle
R\,\sin^{2}\theta\,\big{(}\sqrt{1+\frac{F}{R}}-1\big{)}\,.$

In addition (18) shows that

$\displaystyle\gamma_{tt}$ $\displaystyle=$
$\displaystyle-4\,\pi\,G\sigma\,e^{D}\,,$ $\displaystyle\gamma_{\theta\theta}$
$\displaystyle=$ $\displaystyle 4\,\pi\,G\,\sigma\,R^{2}\,,$ (42)
$\displaystyle\gamma_{\phi\phi}$ $\displaystyle=$ $\displaystyle
4\,\pi\,G\sigma\,R^{2}\,\sin^{2}\theta\,.$

We can obtaine the constant $F$ by comparing (Appendix) and (Appendix) which
will lead us to (30).

## References

  * (1) A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic, Astrophys. J. 876 (2019) no.1, 85 [arXiv:1903.07603 [astro-ph.CO]].
  * (2) J. L. Bernal, L. Verde and A. G. Riess, JCAP 10 (2016), 019 [arXiv:1607.05617 [astro-ph.CO]].
  * (3) A. G. Riess, S. Casertano, W. Yuan, L. Macri, B. Bucciarelli, M. G. Lattanzi, J. W. MacKenty, J. B. Bowers, W. Zheng and A. V. Filippenko, et al. Astrophys. J. 861 (2018) no.2, 126 [arXiv:1804.10655 [astro-ph.CO]].
  * (4) C. Heymans, T. Tröster, M. Asgari, C. Blake, H. Hildebrandt, B. Joachimi, K. Kuijken, C. A. Lin, A. G. Sánchez and J. L. van den Busch, et al. Astron. Astrophys. 646 (2021), A140 [arXiv:2007.15632 [astro-ph.CO]].
  * (5) N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] [arXiv:1807.06209 [astro-ph.CO]].
  * (6) H. K. Eriksen, F. K. Hansen, A. J. Banday, K. M. Gorski and P. B. Lilje, Astrophys. J. 605 (2004), 14-20 [erratum: Astrophys. J. 609 (2004), 1198] [arXiv:astro-ph/0307507 [astro-ph]].
  * (7) F. K. Hansen, A. J. Banday, K. M. Gorski, H. K. Eriksen and P. B. Lilje, Astrophys. J. 704 (2009), 1448-1458 [arXiv:0812.3795 [astro-ph]].
  * (8) C. Copi, D. Huterer, D. Schwarz and G. Starkman, Phys. Rev. D 75 (2007), 023507 [arXiv:astro-ph/0605135 [astro-ph]].
  * (9) C. J. Copi, D. Huterer, D. J. Schwarz and G. D. Starkman, Mon. Not. Roy. Astron. Soc. 367 (2006), 79-102 [arXiv:astro-ph/0508047 [astro-ph]].
  * (10) C. J. Copi, D. Huterer, D. J. Schwarz and G. D. Starkman, Adv. Astron. 2010 (2010), 847541 [arXiv:1004.5602 [astro-ph.CO]].
  * (11) C. J. Copi, D. Huterer, D. J. Schwarz and G. D. Starkman, Mon. Not. Roy. Astron. Soc. 449 (2015) no.4, 3458-3470 [arXiv:1311.4562 [astro-ph.CO]].
  * (12) A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga and A. Hamilton, Phys. Rev. D 69 (2004), 063516 [arXiv:astro-ph/0307282 [astro-ph]].
  * (13) D. J. Schwarz, G. D. Starkman, D. Huterer and C. J. Copi, Phys. Rev. Lett. 93 (2004), 221301 [arXiv:astro-ph/0403353 [astro-ph]].
  * (14) W. J. G. de Blok, Adv. Astron. 2010 (2010), 789293 [arXiv:0910.3538 [astro-ph.CO]].
  * (15) R. A. Flores and J. R. Primack, Astrophys. J. Lett. 427 (1994), L1-4 [arXiv:astro-ph/9402004 [astro-ph]].
  * (16) P. J. E. Peebles, Astrophys. J. 557 (2001), 495-504 [arXiv:astro-ph/0101127 [astro-ph]].
  * (17) A. Banihashemi, N. Khosravi and A. H. Shirazi, Phys. Rev. D 99 (2019) no.8, 083509 [arXiv:1810.11007 [astro-ph.CO]].
  * (18) A. Pisani, E. Massara, D. N. Spergel, D. Alonso, T. Baker, Y. C. Cai, M. Cautun, C. Davies, V. Demchenko and O. Doré, et al. [arXiv:1903.05161 [astro-ph.CO]].
  * (19) S. Tavasoli, Astrophys. J. Lett. 916 (2021) no.2, L24 [arXiv:2109.10369 [astro-ph.CO]].
  * (20) A. Joyce, B. Jain, J. Khoury and M. Trodden, Phys. Rept. 568 (2015), 1-98 [arXiv:1407.0059 [astro-ph.CO]].
  * (21) L. Amendola, S. Appleby, A. Avgoustidis, D. Bacon, T. Baker, M. Baldi, N. Bartolo, A. Blanchard, C. Bonvin and S. Borgani, et al. Living Rev. Rel. 21 (2018) no.1, 2 [arXiv:1606.00180 [astro-ph.CO]].
  * (22) T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513 (2012), 1-189 [arXiv:1106.2476 [astro-ph.CO]].
  * (23) K. Hinterbichler, J. Khoury, A. Levy and A. Matas, Phys. Rev. D 84 (2011), 103521 [arXiv:1107.2112 [astro-ph.CO]].
  * (24) V. Mukhanov, Cambridge University Press, 2005, ISBN 978-0-521-56398-7 doi:10.1017/CBO9780511790553
  * (25) C. Llinares and L. Pogosian, Phys. Rev. D 90 (2014) no.12, 124041 [arXiv:1410.2857 [astro-ph.CO]].
  * (26) K. Hinterbichler and J. Khoury, Phys. Rev. Lett. 104 (2010), 231301 [arXiv:1001.4525 [hep-th]].
  * (27) V. Springel, S. D. M. White, A. Jenkins, C. S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton and J. Helly, et al. Nature 435 (2005), 629-636 [arXiv:astro-ph/0504097 [astro-ph]], https://wwwmpa.mpa-garching.mpg.de/galform/virgo/millennium/.
  * (28) W. Israel, Nuovo Cim. B 44S10 (1966), 1 [erratum: Nuovo Cim. B 48 (1967), 463]