# Stochastic motion in an expanding noncommutative fluid

M. A. Anacleto<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  C. H. G. Bessa<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900
Campina Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal
da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil  E. J.
B. Ferreira<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  Departamento de Física, Universidade Federal da Paraíba,
Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil  E. Passos
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Campina
Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil

###### Abstract

A model for an expanding noncommutative acoustic fluid analogous to a
Friedmann-Robertson-Walker geometry is derived. For this purpose, a
noncommutative Abelian Higgs model is considered in a (3+1)-dimensional
spacetime. In this scenario, we analyze the motion of test particles in this
fluid. The study considers a scalar test particle coupled to a quantized
fluctuating massless scalar field. For all cases studied, we find corrections
due to the noncommutativity in the mean squared velocity of the particles. The
nonzero velocity dispersion for particles that are free to move on geodesics
disagrees with the null result found previously in the literature for
expanding commutative fluid.

##  I Introduction

Classical and quantum fluids have been considered in the literature as
theoretical or experimental models where it is possible to mimic some effects
present in theories of classical and quantum gravity, as well as, in quantum
field theory in curved spacetime — see Refs. unruh81 ; Visser:1997ux ;
Barcelo2011 ; faccio2013 ; kms10 ; fs09 ; wf20 ; doc12 ; bcl04 ; jwvg07 ;
pfl10 ; blv11 ; blv03 ; ff03 ; Cha:2016esj ; bffr08 ; foglizzo2012 ; tprw19 ;
tprw20 ; pcrw18 ; pw20 ; ekjsc18 ; lp19 ; sll19 ; Vieira:2014rva . Some of
these effects are related to Hawking radiation, particle production by
cosmological expansion, superradiance, and so on. Moreover, experimental
studies using classical and quantum fluids as table-top experiments have been
done in recent years with remarkable advances. As examples, we mention the
possible observation of classical superradiance Torres2017  and Hawking
radiation steinhauer2016 ; wtpul11 ; ngks19 . Another variety of analog model
scenarios with remarkable experimental advances are also present in fiber-
optics and slow light models lp00 ; l02 ; us03 ; pkrhkl08 ; su05 ; drbsl19 .
In addition, based on the Abelian Higgs model, relativistic acoustic metrics
Bilic:1999sq ; Ge:2010wx ; Visser:2010xv  were found in Anacleto2012  for a
noncommutative background, in Anacleto:2010cr  for a Lorentz-violating
background and in Anacleto:2013esa  with terms of high derivatives in the
bosonic sector. Hence, studies related to Hawking radiation, entropy and
superradiance were performed in Anacleto:2011tr ; Rinaldi:2011nb ;
Rinaldi:2011aa ; Giovanazzi:2011az ; Zhang:2011zzh ; Anacleto:2012ba ;
Zhao:2012zz ; Anacleto:2012du ; Anacleto:2014apa ; Anacleto:2015awa ;
Anacleto:2016qll ; Anacleto:2018acl ; Anacleto:2019rfn .

A distinct effect also mentionable is the stochastic motion of a particle
under quantum fluctuations. This is a non-trivial quantum effects and it is
shown that this motion can be induced by the presence of reflective plates
gs99 ; jr92 ; YuFord2004 ; YuChen2004 ; Seriu2009 ; Seriu2008 ; hwl08 ; whl12
; Lorenci2019 ; clrr19 ; pf11 ; lrs16 ; clrrs18 , non-trivial topology cy04 ;
zy05 ; br20 , or even by a time-dependent expanding universe Bessa2009 ;
Bessa2017 ; mm19  without boundaries. In particular, in Ref. Bessa2009 ,
different types of classical particles111One example of the particles
considered are point electric charges that could be under the influence of a
classical external and non-fluctuating force, $f_{ext}$. They were named bound
particles. Another example are particles that are free to follow their own
geodesics with $f_{ext}=0$, they were named free particles. coupled to a
fluctuating quantum electromagnetic field were considered in a spatially flat
Friedmann-Robertson-Walker (FRW) universe. In this scenario, it was shown that
the bound electric particle can undergo stochastic (Brownian) motion with a
non-zero mean squared velocity (velocity dispersion) while the free electric
particle had a null velocity dispersion. In a different scenario, see Ref.
Bessa2017 , it was proposed the possibility to observe this effect in an
analog cosmological model by the use of a Bose-Einstein condensate (BEC) that
simulates a spatially flat FRW geometry. To establish this analogy, a
linearized perturbation in the field, which is present in the equation of
motion of the BEC, was done and it was found that these perturbations describe
the same equation of motion for a massless scalar field in curved spacetime
222Recently, a model using BEC, which reproduces some behavior of an expanding
universe was performed experimentally in Ref. ekjsc18 .. The main results
calculated in that paper was that the velocity dispersion for a bound scalar
particle (atoms that constitute the fluid) was non-null and for a free scalar
particle a null dispersion was found, exactly to what was calculated in Ref.
Bessa2009  for the charged particle.

Another type of system considered in recent years is a noncommutative fluid
brito2016 ; Barosi:2008gx ; Balachandran:2007ua ; Chu:2000ww ;
Alexander:2001dr ; Jackiw:2001dj ; Das:2016hmc ; Mitra:2018ezo ; Das:2018puw .
In this case, the fluid description is valid only in energy scales much
smaller than the Planck scale. In general the analogy is made with a
Schwarzschild or Kerr-like geometry, and the corrections to the Hawking
temperature are in many cases derived Anacleto2012 . One of the purposes of
this paper, Sec. II, is to show that with the same noncommutative Lagrangian
considered in Ref. Anacleto2012 , it is possible to acquire an analogous to a
FRW geometry by the use of a linearized perturbation in the field. Thus, a
natural question to ask is whether it is possible to observe the influence of
the noncommutativity in the velocity dispersion found in Ref. Bessa2017 . For
this purpose, both the free and bound particles described above will be
considered and no boundaries will be taken into account in the current study.
Corrections in the velocity dispersion due to the noncommutative parameter
($\theta$) will be found for both particles. The most important result is that
the velocity dispersion, for the free particles, will be different from zero
in this noncommutative fluid. This reveals that free particles can possess
stochastic motion induced by quantum fluctuations and that the
noncommutativity of space plays a fundamental role in the model.

The rest of the paper is outlined as follows: in Sec. III, we calculate the
formal expressions of the velocity dispersions in terms of integrals that
depend on the scale factor of the analogous FRW geometry for both free and
bound particles. The scale factor that describes the way the fluid expands is
introduced in Sec. IV and, as proposed by many authors jwvg07 ; pfl10 ;
Bessa2017 , we consider an asymptotic expanded scale factor. Thus, the
velocity dispersions for the free and bound particles are finally expressed.
The possible effects of metric fluctuations on the motion of the particles
will be briefly discussed in Sec. V. In Sec. VI the main results are
summarized and some interpretations are mentioned. In this paper we use units
where $\hbar=c=1.$

##  II Acoustic metric for a noncommutative geometry

To obtain an analog Friedman-Robertson-Walker (FRW) geometry from a
noncommutative fluid, we start with the Lagrangian for a noncommutative
Abelian Higgs model in flat spacetime modified in the scalar and gauge sector
Anacleto2012 ; Ghosh:2004wi ; sw99

$\displaystyle{\mathcal{\hat{L}}=-\dfrac{1}{4}\hat{F}_{\mu\nu}\ast\hat{F}^{\mu\nu}+(D_{\mu}\hat{\phi})^{\dagger}\ast
D^{\mu}\hat{\phi}+m^{2}{\hat{\phi}}^{\dagger}\ast\hat{\phi}-b\hat{\phi}^{\dagger}\ast\hat{\phi}\ast\hat{\phi}^{\dagger}\ast\hat{\phi}},$
(1)

where the Moyal product was used. In what follows, we apply a Seiberg-Witten
map sw99

$\displaystyle{{\hat{A}}_{\mu}=A_{\mu}+\theta^{\nu\rho}A_{\rho}(\partial_{\nu}A_{\mu}-\frac{1}{2}\partial_{\mu}A_{\nu})},$
$\displaystyle{{\hat{F}}_{\mu\nu}=F_{\mu\nu}+\theta^{\rho\alpha}(F_{\mu\rho}F_{\nu\alpha}+A_{\rho}+\partial_{\alpha}F_{\mu\nu})},$
(2)
$\displaystyle{{\hat{\phi}}=\phi-\frac{1}{2}\theta^{\rho\alpha}A_{\rho}\partial_{\alpha}\phi},$

where, only the lowest order terms in $\theta^{\mu\nu}$ are taken into
account. From Eqs. (1) and (II) we see that the noncommutativity is coupled
via the electromagnetic field. So we obtain the following Lagrangian

$\displaystyle\mathcal{\hat{L}}=$ $\displaystyle-$
$\displaystyle\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}\left(1+\dfrac{1}{2}\theta^{\alpha\beta}F_{\alpha\beta}\right)+\left(1-\dfrac{1}{4}\theta^{\alpha\beta}F_{\alpha\beta}\right)(|D_{\mu}\phi|^{2}+m^{2}|\phi|^{2}-b|\phi|^{4})$
(3) $\displaystyle+$
$\displaystyle\dfrac{1}{2}\theta^{\alpha\beta}F_{\alpha\mu}\left[(D_{\beta}\phi)^{\dagger}D^{\mu}\phi+(D^{\mu}\phi)^{\dagger}D_{\beta}\phi\right],$

where the operator $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Here, $F_{\mu\nu}$
is the Maxwell tensor, $A_{\mu}$ is the 4-potential, and $e$, $b$ are coupling
constants. The term $\theta^{\alpha\beta}$ is the real constant noncommutative
parameter with dimensions of length squared represented by an anti-symmetric
$D$-dimensional square matrix Szabo:2001kg ; Nicolini:2008aj . The field
$\phi$ can be decomposed by
$\phi=\sqrt{\rho(\bar{x},\bar{t})}e^{iS(\bar{x},\bar{t})}$, where
$\rho(\bar{x},\bar{t})$ is the fluid density and $S(\bar{x},\bar{t})$ is a
phase. We consider that the noncommutative effect is absent in the time
coordinate. Following Ref. Anacleto2012  let us apply the perturbations in
$\rho=\rho_{0}+\rho_{1}$, $S=S_{0}+S_{1}$ and $\phi=\phi_{0}+\phi_{1}$ in Eq.
(3), where $\rho_{1}\ll\rho_{0},S_{1}\ll S_{0}$ and consequently
$\phi_{1}\ll\phi_{0}$ . When we compute the equations of motion with these
perturbations, it can be viewed as a Klein-Gordon equation in curved spacetime
and the following noncommutative relativistic metric can be given by Eq. (31)
of Ref. Anacleto2012  :

$\displaystyle
d{\bar{s}}^{2}=\dfrac{b\rho_{0}}{2c_{s}\sqrt{f}}\left[-\mathcal{F}(v)d{t^{\prime}}^{2}+\Lambda\left(\dfrac{v^{i}v^{j}\Gamma+\Sigma^{ij}}{\Lambda\mathcal{F}(v)}+\delta^{ij}\right)d{\bar{x}}^{i}d{\bar{x}}^{j}\right],$
(4)

with the following terms

$\displaystyle dt^{\prime}$ $\displaystyle=$ $\displaystyle
d\bar{t}+\dfrac{\vec{\xi}(v)\cdot d\vec{\bar{x}}}{2\mathcal{F}(v)},$
$\displaystyle f$ $\displaystyle=$
$\displaystyle[(1-2\vec{\theta}\cdot\vec{B})(1+c_{s}^{2})-(1+4\vec{\theta}\cdot\vec{B})v^{2}]-3(\vec{\theta}\times\vec{E})\cdot\vec{v}+2(\vec{B}\cdot\vec{v})(\vec{\theta}\cdot\vec{v}),$
$\displaystyle\mathcal{F}(v)$ $\displaystyle=$
$\displaystyle(1-3\vec{\theta}\cdot\vec{B})c_{s}^{2}-(1+3\vec{\theta}\cdot\vec{B})v^{2}-(\vec{\theta}\times\vec{E})\cdot\vec{v}+2(\vec{\theta}\cdot\vec{v})(\vec{B}\cdot\vec{v}),$
$\displaystyle\Lambda(v)$ $\displaystyle=$
$\displaystyle(1+\vec{\theta}\cdot\vec{B})(1+c_{s}^{2}-v^{2})-(\vec{\theta}\times\vec{E})\cdot\vec{v},$
(5) $\displaystyle\vec{\xi}(v)$ $\displaystyle=$
$\displaystyle[2(1+2\vec{\theta}\cdot\vec{B})-(\vec{\theta}\times\vec{E})\cdot\vec{v}]\vec{v}+(1+c_{s}^{2})(\vec{\theta}\times\vec{E})-(\vec{B}\cdot\vec{v})\vec{\theta}-(\vec{\theta}\cdot\vec{v})\vec{B},$
$\displaystyle\Gamma(v)$ $\displaystyle=$ $\displaystyle
1+4\vec{\theta}\cdot\vec{B}+(1+2\vec{\theta}\cdot\vec{B})c_{s}^{2}-(1+4\vec{\theta}\cdot\vec{B})v^{2}-2(\vec{\theta}\times\vec{E})\cdot\vec{v}+2(\vec{\theta}\cdot\vec{v})(\vec{B}\cdot\vec{v}),$
$\displaystyle\Sigma^{ij}(v)$ $\displaystyle=$
$\displaystyle[(1+c_{s}^{2})(\vec{\theta}\times\vec{E})^{i}-(\vec{B}\cdot\vec{v})\theta^{i}-(\vec{\theta}\cdot\vec{v})B^{i}]v^{j},$

where $c_{s}^{2}=b\rho_{0}/2\mathcal{W}_{0}^{2}$ is the local sound velocity
in the fluid and $\vec{v}=\vec{v}_{0}/\mathcal{W}_{0}$ is the velocity flux,
with $\mathcal{W}_{0}=-\dot{S}+eA_{t}$ and $\vec{v}_{0}=\nabla S_{0}+e\vec{A}$
(the local velocity field) and $b$ is a coupling constant with $\vec{E}$ and
$\vec{B}$ being the electric and magnetic fields, respectively.

To simplify our expressions, let us choose a null electric field
($\vec{E}=0$). Now the noncommutativity is coupled only to the magnetic field,
and considering the constant coupling $e=0$ with a phase $S_{0}$ being time-
dependent only, $S_{0}=S_{0}(t)$, the velocity flux ($\vec{v}$) is now equal
to zero. Thus, considering that $c_{s}^{2}\ll 1$, we obtain the non-
relativistic acoustic metric

$d\overline{s}^{2}=\dfrac{b\rho_{0}}{2}\left[-\dfrac{(1-3\vec{\theta}\cdot\vec{B})}{(1-2\vec{\theta}\cdot\vec{B})^{\frac{1}{2}}}{{c_{s}^{2}}}d\overline{t}^{2}+\dfrac{(1+\vec{\theta}\cdot\vec{B})}{(1-2\vec{\theta}\cdot\vec{B})^{\frac{1}{2}}}c_{s}^{-1}\delta^{ij}d\overline{x}^{i}d\overline{x}^{j}\right].$
(6)

By defining the following change in the spatial coordinates
$d\overline{x}^{i}=\left(\frac{H}{W}\right)dx^{i}$, we obtain

$d\overline{s}^{2}=\dfrac{b\rho_{0}}{2}H^{2}\left[{{-c_{s}^{2}}}d\overline{t}^{2}+c_{s}^{-1}\delta^{ij}dx^{i}dx^{j}\right],$
(7)

where $H$ and $W$ were defined by

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\dfrac{(1-3\vec{\theta}\cdot\vec{B})}{(1-2\vec{\theta}\cdot\vec{B})^{\frac{1}{2}}},$
$\displaystyle W^{2}$ $\displaystyle=$
$\displaystyle\dfrac{(1+\vec{\theta}\cdot\vec{B})}{(1-2\vec{\theta}\cdot\vec{B})^{\frac{1}{2}}}.$
(8)

The upper bar in the previous expressions has been added to distinguish the
variables in the mathematical manipulations. Note that, in this model, the
magnetic field ${\vec{B}}$ plays the role to turn on and off the
noncommutativity. If the noncommutativity is off or null, the terms $H$ and
$W$ are equal to the unity. In fact the term $\theta$ must be very small which
in turn makes $H\approx W\approx 1$.

In our model, we take the sound velocity to be time dependent,
$c_{s}=c_{s}(t)$, and it can be represented in terms of the parameter
$\chi(t)=\left[\frac{c_{s}(t)}{c_{0}}\right]^{2}$, which is interpreted as a
scale factor of the expanding fluid. Note that, one can admit, without loss of
generality, that for a given initial time $t_{0}$, $\chi(t_{0})=1$. The line
element given by Eq. (7) becomes

$\displaystyle
d\overline{s}^{2}=\dfrac{b\rho_{0}}{2}H^{2}\left[-\chi^{\frac{1}{2}}(t){{c_{0}^{2}}}d\overline{t}^{2}+\chi^{-\frac{1}{2}}(t)c_{0}^{-1}\delta^{ij}dx^{i}dx^{j}\right],$
(9)

that defining $\Omega_{0}^{2}=\frac{b\rho_{0}}{2c_{0}}$ and
$ds^{2}=\Omega_{0}^{-2}d\overline{s}^{2}$, by performing a change in the time
coordinate, such that $dt=\chi^{\frac{1}{4}}(t)d\overline{t}$, we have

$\displaystyle
ds^{2}=H^{2}\left[-c_{0}^{2}dt^{2}+a^{2}(t)\delta^{ij}dx^{i}dx^{j}\right],$
(10)

where

$\displaystyle a^{2}(t)=\chi^{-\frac{1}{2}}(t)=\dfrac{c_{0}}{c_{s}(t)}.$ (11)

Apart from the constant $H$, these definitions gives an effective metric that
mimics a FRW geometry for a certain coordinate time $t$. Since
$a(\eta)=c_{0}/c_{s}(t)$, it is important to note that $c_{s}(t=t_{f})<c_{0}$,
where $t_{f}$ is the final time when the expansion ends, is a necessary
condition to the expansion occur, that is, for $a(t)$ assuming increasing
values. In order to get a conformal effective metric, one defines the
following transformation in the time coordinate $dt=ad\eta$, where $\eta$ is
the conformal time and the effective conformal metric is

$\displaystyle
ds^{2}=H^{2}a^{2}(\eta)\left[-c_{0}^{2}d\eta^{2}+\delta^{ij}dx^{i}dx^{j}\right],$
(12)

with $a^{2}(\eta)H^{2}$ being the conformal factor. When $H$ is equal to the
unity, we recover the usual forms of Eqs. (10) and (12). This form of the
metric is conformal to Minkowski spacetime and it will be useful when we
evaluate the two point function for a scalar field in the next section.

##  III Random motion of particles in an analog FRW noncommutative geometry

The motion of a point scalar particle with mass $m$ in a conformal curved
space-time is represented by the following equation

$\displaystyle{f}^{\mu}=m\dfrac{Du^{\mu}}{d\tau}=qg^{\mu\nu}\nabla_{\nu}\phi,$
(13)

where $f^{\mu}$ is the 4-force, $u^{\mu}$ is the 4-velocity of the scalar
particles, $q$ is the charge of the scalar particle interacting with a
massless scalar field and the operator $D/d\tau$ is the covariant derivative.
This equation is valid for a commutative space. However, in a noncommutative
space it must be modified.

For the case studied here, we consider the motion of free and bound scalar
test particles described before. They are the constituents of a noncommutative
fluid that expands according to the metric given by Eq. (12). Thus, these
point-like massive particles interact with a massless scalar field of the
Abelian-Higgs model performed by the Lagrangian (3). The acoustic
perturbations of this system (i.e., phonons) are described by this field, and
the equation of motion of a single particle with mass $m$ in such conformal
curved space-time (12) is given by

$\displaystyle
f^{\mu}=q\left[\left(1-\dfrac{1}{4}\theta^{\alpha\beta}F_{\alpha\beta}\right)g^{\mu\nu}+\Theta^{\mu\nu}\right]\nabla_{\nu}\phi,$
(14)

where the Lagrangian given by Eq. (3) was used. This is the equation of motion
to the scalar particle in the noncommutative space with the metric
$g^{\mu\nu}$ given by Eq. (10) or (12) and

$\displaystyle\Theta^{\mu\nu}=\theta^{\alpha\mu}F_{\alpha}^{\phantom{a}\nu}.$
(15)

Note that when $\theta=0$ we recover Eq. (13).

Adopting a particular $i$-direction and considering only non-commutative
effects on spatial part, i.e., $\theta^{0j}=\theta^{i0}=0$, Eq. (14) becomes

$f^{i}=q\left[\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)g^{ip}+\Theta^{ip}\right]\nabla_{p}\phi,$
(16)

where $i,p=x,y,z,$

$\Theta^{ip}=\theta^{ji}F_{j}^{\ p},$ (17)

and

$\displaystyle\theta^{ij}=\epsilon^{ijk}\theta^{k},$ $\displaystyle
F^{ij}=\epsilon^{ijl}B^{l},$ (18)

where a sum is adopted in repeated indices. Thus, if $B^{i}=0$ or $\theta=0$,
Eq. (13) is recovered.

Following the same procedure of Refs. Bessa2017  and Bessa2009 , with the
effective metric given by Eq. (10), we obtain

$\displaystyle\dfrac{1}{m}f^{i}=\frac{Du^{\mu}}{dt}=\dfrac{du^{i}}{dt}+2\dfrac{\dot{a}}{a}u^{i},$
(19)

where $\dot{a}=da/dt$ and we have assumed non-relativistic motion for the
particles, which implies that the time coordinate $t$ is their proper time
$\tau$.

In the sequence, we will consider that the force $f^{i}$ can be split into two
parts, the first one is originated by a non-fluctuating classical external
force ($f^{i}_{ext}$) and the second one is a fluctuating force ($f^{i}_{q}$)
associated with the quantized scalar field. Thus, we have

$\dfrac{1}{m}\left(f^{i}_{ext}+f^{i}_{q}\right)=\dfrac{du^{i}}{dt}+2\dfrac{\dot{a}}{a}u^{i}.$
(20)

In the equation above we can study two distinct situations. In the first case,
we consider free particles which are characterized by a null external force
($f^{i}_{ext}=0$). Thus, they can move freely following their geodesic in the
expanding background. In the second case, they are named ‘bound particles’.
Now, they are influenced by an external force given by
$f^{i}_{ext}=2m\dfrac{\dot{a}}{a}u^{i}$. This force cancels out locally the
effects of the expansion. Both cases will be study in the next two sections.
We will find that the non-commutativity can give relevant contribution to the
stochastic motion of the particles in this expanding background.

###  III.1 Free particle

In this section we consider that no external, classical force, is acting on
the particle (i.e., $f^{i}_{ext}=0$). We implement this into Eq. (20) to
obtain the following equation333In what follows, the sub-indexes $q$ from
$f_{q}$ will be omitted.

$\displaystyle\dfrac{1}{m}f^{i}=\dfrac{1}{a^{2}}\dfrac{d}{dt}(a^{2}u^{i}),$
(21)

that by integrating once and admitting that the particle is initially at rest
($u^{i}(t_{0})=0$), we have

$\displaystyle u^{i}(t_{f},r)$ $\displaystyle=$
$\displaystyle\dfrac{1}{ma^{2}(t_{f})}\int^{t_{f}}_{t_{0}}a^{2}(t)f^{i}(t,r)dt.$
(22)

From the above results, the correlation function for the velocity of the
particle is

$\displaystyle\langle(\Delta
u^{i})^{2}\rangle=\dfrac{1}{m^{2}a^{4}(t_{f})}\iint
dt_{1}dt_{2}a^{2}(t_{1})a^{2}(t_{2})\langle
f^{i}(t_{1},r_{1})f^{i}(t_{2},r_{2})\rangle_{{}_{FRW}}.$ (23)

Now we follow the standard procedure YuFord2004  which consists in assuming
that the field $\phi$ can be decomposed into a classical and a quantum part,
i.e., $\phi=\phi_{c}+\phi_{q}$ and the conditions
$\langle\phi(t_{1},r_{1})\rangle=0$,
$\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle\neq 0$ are satisfied. In
this way, one can use the fact that $f^{i}$ is related to a scalar field by
Eq. (16). Consequently, the equation above becomes

$\displaystyle\langle(\Delta u^{i})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}H^{-4}}{m^{2}a^{4}(t_{f})}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)^{2}\iint
dt_{1}dt_{2}\partial_{i_{1}}\partial_{i_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
(24) $\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{-2}\Theta^{is}}{m^{2}a^{4}(t_{f})}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
dt_{1}dt_{2}a^{2}(t_{2})\partial_{i_{1}}\partial_{s_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{-2}\Theta^{ip}}{m^{2}a^{4}(t_{f})}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
dt_{1}dt_{2}a^{2}(t_{1})\partial_{p_{1}}\partial_{i_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}\Theta^{ip}\Theta^{is}}{m^{2}a^{4}(t_{f})}\iint
dt_{1}dt_{2}a^{2}(t_{1})a^{2}(t_{2})\partial_{p_{1}}\partial_{s_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}},$

where we have assumed that $H$ is a constant as well as $\vec{\theta}$ and
$\vec{B}$, and the subscript $FRW$ in
$\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{FRW}$ indicates that
the vacuum expectation value in the FRW-geometry was taken. In this case,
there are no distinction between these parameters at different times, e.g.,
$H_{1}=H_{2}$, ${\theta}_{1}={\theta}_{2}$, and ${B}_{1}={B}_{2}$. Note that
to obtain the proper velocity from Eq. (24) we use the relation between the
coordinate $x^{i}$ and the proper distance $l^{i}$, in which
$l^{i}=a(t_{f})x^{i}$, where $a(t_{f})$ is the scale factor in certain final
time. Thus, the proper velocity $v^{i}$ is related to $u^{i}$ by
$u^{i}a(t_{f})=v^{i}$. In this way we obtain

$\langle(\Delta u^{i})^{2}\rangle=\frac{1}{a^{2}(t_{f})H^{2}}\langle(\Delta
v^{i})^{2}\rangle,$ (25)

and the proper velocity dispersion of the particle is

$\displaystyle\langle(\Delta v^{i})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}H^{-2}}{m^{2}a^{2}(t_{f})}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)^{2}\iint
dt_{1}dt_{2}\partial_{i_{1}}\partial_{i_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
(26) $\displaystyle+$
$\displaystyle\dfrac{q^{2}\Theta^{is}}{m^{2}a^{2}(t_{f})}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
dt_{1}dt_{2}a^{2}(t_{2})\partial_{i_{1}}\partial_{s_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}\Theta^{ip}}{m^{2}a^{2}(t_{f})}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
dt_{1}dt_{2}a^{2}(t_{1})\partial_{p_{1}}\partial_{i_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{2}\Theta^{ip}\Theta^{is}}{m^{2}a^{2}(t_{f})}\iint
dt_{1}dt_{2}a^{2}(t_{1})a^{2}(t_{2})\partial_{p_{1}}\partial_{s_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}.$

Using the relation present in Ref. Birrell1984 , we obtain

$\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{FRW}}=H^{-2}a^{-1}(\eta_{1})a^{-1}(\eta_{2})\langle\phi_{1}(\eta_{1},r_{1})\phi_{2}(\eta_{2},r_{2})\rangle_{{}_{M}},$
(27)

which relates the two-point (Hadamard) function of a massless scalar field in
the conformal FRW spacetime to the two-point function in Minkowski spacetime.

Substituting (27) into (26) in terms of the conformal time
($dt=a(\eta)d\eta$), we obtain

$\displaystyle\langle(\Delta v^{i})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}H^{-4}}{m^{2}a_{f}^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)^{2}\iint
d\eta_{1}d\eta_{2}\partial_{i_{1}}\partial_{i_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
(28) $\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{-2}\Theta^{is}}{m^{2}a_{f}^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
d\eta_{1}d\eta_{2}a^{2}(\eta_{2})\partial_{i_{1}}\partial_{s_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{-2}\Theta^{ip}}{m^{2}a_{f}^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
d\eta_{1}d\eta_{2}a^{2}(\eta_{1})\partial_{p_{1}}\partial_{i_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}\Theta^{ip}\Theta^{is}}{m^{2}a_{f}^{2}}\iint
d\eta_{1}d\eta_{2}a^{2}(\eta_{1})a^{2}(\eta_{2})\partial_{p_{1}}\partial_{s_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}},$

where $a(t_{f})\equiv a_{f}$ is the scale factor in a final time when the
expansion ends and the subscript $M$ indicates that the vacuum state is the
Minkowski vacuum state. However, the equivalent Hadamard function for a
massless scalar field is given by

$\displaystyle\langle\phi_{1}(\eta_{1},r_{1})\phi_{2}(\eta_{2},r_{2})\rangle_{M}=\dfrac{1}{4\pi^{2}}\left[\dfrac{1}{-c_{0}^{2}(\eta_{1}-\eta_{2})^{2}+r^{2}}\right].$
(29)

Here $c_{0}$ is the speed of sound when the expansion starts and because no
boundary is present, the spatial separation $r$ is given by

$\displaystyle r^{2}$ $\displaystyle=$ $\displaystyle\Delta x^{2}+\Delta
y^{2}+\Delta z^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}.$
(30)

In Sec. IV we will evaluate the velocity dispersion given by Eq. (28) for a
noncommutative fluid that expands asymptotically.

###  III.2 Bound particles

In this section let us admit that the particle is subject to a classical
external non-fluctuating force of the type

$f^{i}_{ext}=2m\dfrac{\dot{a}}{a}u^{i}.$ (31)

By substituting this into Eq. (20) and taking an integral, we obtain

$u^{i}(t_{f},r)=\dfrac{1}{m}\int f^{i}(t,r)dt,$ (32)

where we considered a null velocity for the initial time $t_{0}$. Thus, the
velocity dispersion of the bound particles is

$\displaystyle\langle(\Delta u^{i})^{2}\rangle=\dfrac{1}{m^{2}}\iint
dt_{1}dt_{2}\langle f^{i}(t_{1},r_{1})f^{i}(t_{2},r_{2})\rangle_{{}_{FRW}}.$
(33)

Now using Eq. (16) and the metric (10), we obtain

$\displaystyle\langle(\Delta u^{i})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}H^{-4}}{m^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)^{2}\iint
dt_{1}dt_{2}a^{-2}(t_{1})a^{-2}(t_{2})\partial_{i_{1}}\partial_{i_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
(34) $\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{-2}\Theta^{is}}{m^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
dt_{1}dt_{2}a^{-2}(t_{1})\partial_{i_{1}}\partial_{s_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}H^{-2}\Theta^{ip}}{m^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
dt_{1}dt_{2}a^{-2}(t_{2})\partial_{p_{1}}\partial_{i_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}$
$\displaystyle+$ $\displaystyle\dfrac{q^{2}\Theta^{ip}\Theta^{is}}{m^{2}}\iint
dt_{1}dt_{2}\partial_{p_{1}}\partial_{s_{2}}{{\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle_{{}_{FRW}}}}.$

Following the same steps of the previous section we apply a conformal
transformation in time and using Eqs. (25) and (27) we get

$\displaystyle\langle(\Delta v^{i})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}a^{2}_{f}H^{-4}}{m^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)^{2}\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})a^{-2}(\eta_{2})\partial_{i_{1}}\partial_{i_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
(35) $\displaystyle+$
$\displaystyle\dfrac{q^{2}a^{2}_{f}H^{-2}\Theta^{is}}{m^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})\partial_{i_{1}}\partial_{s_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}a^{2}_{f}H^{-2}\Theta^{ip}}{m^{2}}\left(1-\dfrac{1}{2}\vec{\theta}\cdot\vec{B}\right)\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{2})\partial_{p_{1}}\partial_{i_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}a^{2}_{f}\Theta^{ip}\Theta^{is}}{m^{2}}\iint
d\eta_{1}d\eta_{2}\partial_{p_{1}}\partial_{s_{2}}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}.$

In the next section, we will evaluate the velocity dispersion for free and
bound particles for a fluid that expands asymptotically. However, it is worth
noting that, when we quantize the scalar field the fluctuations related to
this field could imply fluctuations in the effective metric (10).These metric
fluctuations are viewed in general as linearized perturbations
($\gamma_{\mu\nu}$) upon the effective metric, see for instance Refs. f95 ;
fs97 . So in this context, Eq. (10) should read,

$ds^{2}=H^{2}(g_{\mu\nu}dx^{\mu}dx^{\nu}+\gamma_{\mu\nu}dx^{\mu}dx^{\nu})$
(36)

where $g_{\mu\nu}=diag[-c_{0},a^{2},a^{2},a^{2}],x^{\mu}\in\\{t,x,y,z\\}$ and
the entries of the matrix $\gamma_{\mu\nu}$ are $0<|\gamma_{\mu\nu}|\ll 1$. A
complete treatment to our case is not an easy task once it involves the
integration of Eq. (20). In section V, we will give a simple example where the
metric fluctuations are considered in the context of the analog models
program, in which these fluctuations can induce sound cone fluctuations.We
will see that such fluctuations should contribute as a minor or null deviation
in the random motion of the particles.

##  IV Asymptotic expansion for a noncommutative fluid

In this section we investigate the behavior of the particle under the
scenarios discussed in the previous section. We consider that the
noncommutative fluid is taking an asymptotic expansion according to the
following scale factor:

$a^{2}(\eta)=a_{0}^{2}+a_{1}^{2}\tanh\left(\dfrac{\eta}{\eta_{0}}\right),$
(37)

which describes an asymptotically flat spacetime in extreme regions, where the
constant $a_{0}$ produces a vertical displacement of the point on the
$a^{2}(\eta)$ axis, and $a_{1}$ modifies the spacing between asymptotic limits
(maximum and minimum points of the $a^{2}(\eta)$). The parameter $\eta_{0}$
modifies the smoothness of the transition between the asymptotic regions but
without changing the spacing between the maximum and minimum points.

Based on the asymptotic behavior of the scale factor we can write

$\displaystyle a_{0}^{2}=\dfrac{a_{f}^{2}+a_{i}^{2}}{2},$ (38)

and

$\displaystyle a_{1}^{2}=\dfrac{a_{f}^{2}-a_{i}^{2}}{2},$ (39)

where $a_{i}$ and $a_{f}$ are the scale factor at the beginning and end of the
expansion. Note that, for $\chi(t=t_{0})=1$ in Eq. (11), we obtain
$a_{i}=a(\eta=\eta_{i})=1$. Next, we will use this scale factor to evaluate
the dispersion velocity for the free and bound particles, respectively.

###  IV.1 Free particle in an expanding noncommutative fluid

According to our choices in Eqs. (II), (17), and (18), the magnetic field is
responsible to turn on the noncommutativity. To simplify our expressions, let
us consider that it is turn on in just one direction. Let us choose, for
instance, $\vec{B}=B^{z}\hat{k}$. So, the dispersion in the $z$-direction is
now parallel to the field vector. Thus, when the field is off, all terms
proportional to $\theta$ disappear in Eq. (28) and only the first integral in
the right hand side remains. This recover the result found in Ref. Bessa2017 .
However, as the scale factor does not appeas in the integrand, this integral
gives infinite contribution and must be renormalized. This procedure consists
in subtracting the Minkowski contribution from the Minkowski two-point
function. Thus, the first integral gives null contribution.

Now when the field $\vec{B}$ is on, the terms proportional to $\theta$ become
important and we have to consider all terms with at least one $a^{2}(\eta)$ in
the integrand of (28) to obtain

$\displaystyle\langle(\Delta v^{z})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}}{m^{2}a_{f}^{2}}\left(1+\dfrac{3}{2}(\vec{\theta}\cdot\vec{B})\right)\iint
d\eta_{1}d\eta_{2}a^{2}(\eta_{2})\left[\Theta^{zx}\partial_{z_{1}}\partial_{x_{2}}+\Theta^{zy}\partial_{z_{1}}\partial_{y_{2}}\right.$
(40) $\displaystyle+$
$\displaystyle\left.\Theta^{zz}\partial_{z_{1}}\partial_{z_{2}}\right]{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}+\dfrac{q^{2}}{m^{2}a_{f}^{2}}\left(1+\dfrac{3}{2}(\vec{\theta}\cdot\vec{B})\right)\iint
d\eta_{1}d\eta_{2}a^{2}(\eta_{1})\left[\Theta^{zx}\partial_{x_{1}}\partial_{z_{2}}\right.$
$\displaystyle+$
$\displaystyle\left.\Theta^{zy}\partial_{y_{1}}\partial_{z_{2}}+\Theta^{zz}\partial_{z_{1}}\partial_{z_{2}}\right]{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}$
$\displaystyle+$ $\displaystyle\dfrac{q^{2}}{m^{2}a_{f}^{2}}\iint
d\eta_{1}d\eta_{2}a^{2}(\eta_{1})a^{2}(\eta_{2})\\{\Theta^{zx}[\Theta^{zx}\partial_{x_{1}}\partial_{x_{2}}+\Theta^{zy}\partial_{x_{1}}\partial_{y_{2}}+\Theta^{zz}\partial_{x_{1}}\partial_{z_{2}}]$
$\displaystyle+$
$\displaystyle\Theta^{zy}[\Theta^{zx}\partial_{y_{1}}\partial_{x_{2}}+\Theta^{zy}\partial_{y_{1}}\partial_{y_{2}}+\Theta^{zz}\partial_{y_{1}}\partial_{z_{2}}]+\Theta^{zz}[\Theta^{zx}\partial_{z_{1}}\partial_{x_{2}}+\Theta^{zy}\partial_{z_{1}}\partial_{y_{2}}$
$\displaystyle+$
$\displaystyle\Theta^{zz}\partial_{z_{1}}\partial_{z_{2}}]\\}{{\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{{}_{M}}}}.$

Because the integrals above has at least one $a^{2}(\eta)$ factor in their
integrands, they give finite contributions. We also use the fact that $\theta$
is small and the following Taylor expansion was made:

$H^{-2}\left(1-\frac{1}{2}{\vec{\theta}\cdot\vec{B}}\right)\approx
1+\frac{3}{2}\left(\vec{\theta}\cdot\vec{B}\right)+\frac{9}{2}\left(\vec{\theta}\cdot\vec{B}\right)^{2}+\cdots,$
(41)

where $H$ is given by Eq. (II) and terms up to second order were considered.
Note that the derivatives above obey the relation,

$\partial_{k_{1}}\partial_{\ell_{2}}\langle\phi_{1}\phi_{2}\rangle_{{}_{M}}=\left\\{\begin{array}[]{cc}\textrm{If}\
k=\ell,&\dfrac{1}{2\pi^{2}}[f_{2}(\eta,r)+4\Delta k^{2}f_{3}(\eta,r)]\\\
\textrm{If}\ k\neq\ell,&\dfrac{2\Delta k\Delta\ell
f_{3}(\eta,r)}{\pi^{2}}\end{array}\right.,$ (42)

where $\Delta k=k_{1}-k_{2}$, $\Delta\ell=\ell_{1}-\ell_{2}$, with
$k,\ell=x,y,z$, and

$f_{n}(\eta,r)=\dfrac{1}{[c_{0}^{2}(\eta_{1}-\eta_{2})^{2}-r^{2}]^{n}},$ (43)

with $n=2,3$ and $r$ defined by Eq. (30).

Although the number of terms is significantly large, from the generalization
given by Eq. (42), we note that some terms will be null in the coincidence
limit ($r\rightarrow 0$). Thus, applying this limit we have

$\displaystyle\langle(\Delta v^{z})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}\Theta^{zz}}{\pi^{2}m^{2}a_{f}^{2}c_{0}^{4}}\left(1+\dfrac{3}{2}(\vec{\theta}\cdot\vec{B})\right)\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\dfrac{1}{(\eta_{1}-\eta_{2})^{4}}$
(44) $\displaystyle+$
$\displaystyle\dfrac{q^{2}}{2\pi^{2}m^{2}a_{f}^{2}c_{0}^{4}}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\left[\dfrac{\Theta^{zx}\Theta^{zx}}{(\eta_{1}-\eta_{2})^{4}}\right.$
$\displaystyle+$
$\displaystyle\left.\dfrac{\Theta^{zy}\Theta^{zy}}{(\eta_{1}-\eta_{2})^{4}}+\dfrac{\Theta^{zz}\Theta^{zz}}{(\eta_{1}-\eta_{2})^{4}}\right].$

Using the scale factor given by Eq. (37) into Eq. (44) we first integrates by
parts and then uses the residue theorem boas2006 , to obtain the velocity
dispersion of the particles in the $z$-direction

$\displaystyle\langle(\Delta v^{z})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}a_{1}^{2}}{3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}(\theta^{x}B^{x}+\theta^{y}B^{y})\left(1+\dfrac{3}{2}(\vec{\theta}\cdot\vec{B})\right)$
(45) $\displaystyle\times$
$\displaystyle\left\\{7\zeta(3)+Re\left[\frac{1}{2}\Psi\left(2,\dfrac{\pi+2wi}{2\pi}\right)\right]\right\\}$
$\displaystyle+$
$\displaystyle\dfrac{2q^{2}a_{1}^{4}}{\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}\zeta(3)\left[(\theta^{x}B^{z})^{2}+(\theta^{y}B^{z})^{2}+(\theta^{x}B^{x}+\theta^{y}B^{y})^{2}\right],$

where we have used the definition $w=\eta_{f}/\eta_{0}$, which represents a
relation between the final conformal time $\eta_{f}$ and the constant
parameter $\eta_{0}$. Here, $\zeta(x)$ is the zeta function and $\Psi(n,x)$ is
the nth-polygamma function.

As we are dealing with $\vec{B}=B^{z}\hat{k}$ only, the above expression
simplifies to

$\displaystyle\langle(\Delta
v^{z})^{2}\rangle=\dfrac{2q^{2}a_{1}^{4}}{\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}\zeta(3)\left[(\theta^{x}B^{z})^{2}+(\theta^{y}B^{z})^{2}\right],$
(46)

which corresponds to the dispersion parallel to the magnetic field. Note that,
the velocity dispersion in the $z$-direction is constant and different from
zero. The dominant noncommutativity contribution is a second order term and
when $\theta^{i}=0$ or $B^{z}=0$ the dispersion is null which is the same
result found in Ref. Bessa2017  for a commutative expanded fluid.

Now let us apply the same methodology for the perpendicular dispersion. For
this we consider $i=x$ in Eq. (28), to obtain444The case $i=y$ is also
perpendicular to the field and it gives similar results.

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}\Theta^{xx}}{\pi^{2}m^{2}a_{f}^{2}c_{0}^{4}}\left(1+\dfrac{3}{2}(\vec{\theta}\cdot\vec{B})\right)\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\dfrac{1}{(\eta_{1}-\eta_{2})^{4}}$
(47) $\displaystyle+$
$\displaystyle\dfrac{q^{2}}{2\pi^{2}m^{2}a_{f}^{2}c_{0}^{4}}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\left[\dfrac{\Theta^{xx}\Theta^{xx}}{(\eta_{1}-\eta_{2})^{4}}\right.$
$\displaystyle+$
$\displaystyle\left.\dfrac{\Theta^{xy}\Theta^{xy}}{(\eta_{1}-\eta_{2})^{4}}+\dfrac{\Theta^{xz}\Theta^{xz}}{(\eta_{1}-\eta_{2})^{4}}\right].$

Note that we have used the fact that the two first integrals are equal. Since
$\Theta$ is constant, the integrals of Eq. (47) are the same as the ones
obtained in Eq. (45). The difference appears in the $\Theta$ factors, which by
the use of Eq. (18), they can be expressed in terms of $\vec{\theta}$ and
$\vec{B}$. Thus, using the scale factor (37), we get

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}a_{1}^{2}}{3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}(\theta^{y}B^{y}+\theta^{z}B^{z})\left(1+\dfrac{3}{2}(\vec{\theta}\cdot\vec{B})\right)$
(48) $\displaystyle\times$
$\displaystyle\left\\{7\zeta(3)+Re\left[\frac{1}{2}\Psi\left(2,\dfrac{\pi+2wi}{2\pi}\right)\right]\right\\}$
$\displaystyle+$
$\displaystyle\dfrac{2q^{2}a_{1}^{4}}{\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}\zeta(3)\left[(\theta^{y}B^{y}+\theta^{z}B^{z})^{2}+(\theta^{y}B^{x})^{2}+(\theta^{z}B^{x})^{2}\right],$

which is the expression for the proper velocity dispersion in $x$ direction
and $w=\eta_{f}/\eta_{0}$. The fact that the magnetic field
$\vec{B}=B^{z}\hat{k}$ results in

Figure 1: The contribution produced by noncommutativity in the equation (50)
as a function of the $w$. Here, $w=\eta_{f}/\eta_{0}>0$ and
$G(w)\equiv\langle(\Delta
v^{x})^{2}\rangle\times[3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}/2q^{2}a_{1}^{2}(\theta^{z}B^{z})]$.

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}a_{1}^{2}}{3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}(\theta^{z}B^{z})\left(1+\dfrac{3}{2}(\theta^{z}B^{z})\right)$
(49) $\displaystyle\times$
$\displaystyle\left\\{7\zeta(3)+Re\left[\frac{1}{2}\Psi\left(2,\dfrac{\pi+2wi}{2\pi}\right)\right]\right\\}$
$\displaystyle+$
$\displaystyle\dfrac{2q^{2}a_{1}^{4}}{\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}\zeta(3)(\theta^{z}B^{z})^{2},$

or considering up to the first order terms in $\theta$,

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle\approx$
$\displaystyle\dfrac{2q^{2}a_{1}^{2}}{3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}(\theta^{z}B^{z})\left\\{7\zeta(3)+Re\left[\frac{1}{2}\Psi\left(2,\dfrac{\pi+2wi}{2\pi}\right)\right]\right\\}.$
(50)

Note that now the velocity dispersion in the $x$ direction has a time
dependence. This dependence in time did not appear in the $z$-direction. It is
given in terms of the dimensionless parameter $w=\eta_{f}/\eta_{0}$. Fig. 1
shows curves for Eq. (50) as function of the $w$ parameter.

From Fig. 1 we note that for large values of $w$, corresponding to large
values of time $\eta_{f}$, the noncommutativity effect approaches a constant
value. Moreover, the effect has a fast behavior to short values of $w$, that
is, small time separation between $\eta_{f}$ and $\eta_{0}$. However, the most
important result here is to find an asymptotic solution for $\langle(\Delta
v^{x})^{2}\rangle$. Thus for large $w$ we find

$\displaystyle\langle(\Delta
v^{x})^{2}\rangle_{Asymptotic}\approx\dfrac{14q^{2}a_{1}^{2}}{3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}}(\theta^{z}B^{z})\zeta(3).$
(51)

In the Fig. 2 we show the plots of the asymptotic result (51) (dashed line),
together with (50) (solid line).

Figure 2: The contribution produced by noncommutativity in the equation (50)
(represented by $G(w)$) and (51) (represented by $F(w)$) as a function of $w$.
Here, $w=\eta_{f}/\eta_{0}>0$ and $G(w)\equiv\langle(\Delta
v^{x})^{2}\rangle\times[3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}/2q^{2}a_{1}^{2}(\theta^{z}B^{z})]$
and $F(w)\approx\langle(\Delta
v^{x})^{2}\rangle_{Asymptotic}\times[3\pi^{4}m^{2}a_{f}^{2}c_{0}^{4}\eta_{0}^{2}/2q^{2}a_{1}^{2}(\theta^{z}B^{z})]$
.

In summary, Eqs. (45) and (48) are the general expressions for the velocity
dispersion of the massive scalar particles that form the noncommutative fluid
in the $z$ and $x$ directions, respectively. When we assume that the magnetic
field is nonzero only in the $z$-direction we obtain Eqs. (46), (49). So,
these equations show a non-null velocity dispersion. This result disagrees
with the one found in Ref. Bessa2017 , where the stochastic motion of scalar
particles with mass m for a commutative fluid was studied. In this case, the
fluid considered was a BEC and in a similar way to what was done in Sect. II
of the present paper, linear perturbations in the fluid parameters
$(\rho,\phi,S)$ were applied, and a metric similar to the one found in Eq.
(12) was obtained with the same scale factor present in Eq. (37). Thus, due to
the null result found in Ref. Bessa2017 , the non-null result found in the
present paper, suggests a relevant contribution coming from the
noncommutativity in the stochastic motion of the particles.

###  IV.2 Bound particle in an expanding noncommutative fluid

Similarly to the previous case, we will now analyze the velocity dispersion in
two distinct directions. First, let us verify the dispersion in $z$ direction.
From equation (35) we obtain

$\displaystyle\langle(\Delta v^{z})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}a^{2}_{f}}{2\pi^{2}m^{2}}(1+3\vec{\theta}\cdot\vec{B})\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})a^{-2}(\eta_{2})[f_{2}(\eta,z)+4\Delta
z^{2}f_{3}(\eta,z)]$ (52) $\displaystyle+$
$\displaystyle\dfrac{2q^{2}a^{2}_{f}}{\pi^{2}m^{2}}\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})\bigg{\\{}2\Theta^{zx}\Delta x\Delta
zf_{3}(\eta,z)+2\Theta^{zy}\Delta y\Delta zf_{3}(\eta,z)$ $\displaystyle+$
$\displaystyle\dfrac{1}{2}\Theta^{zz}[f_{2}(\eta,z)+4\Delta
z^{2}f_{3}(\eta,z)]\bigg{\\}},$

where we have used the definition of $f_{n}(\eta,z)$ given by Eq. (43), and
the fact that the second and third integrals in the right hand side of Eq.
(35) are equal and a Taylor expansion in the noncommutative parameter
($\theta$), up to first order, was made. Furthermore, in the fourth term in
the right hand side of Eq. (35) the Minkowski vacuum divergent term must be
subtracted during the renormalization procedure.

Finally, applying the coincidence limit in Eq. (52) we find,

$\displaystyle\langle(\Delta v^{z})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}a^{2}_{f}}{2\pi^{2}m^{2}c_{0}^{4}}(1+3\vec{\theta}\cdot\vec{B})\int_{-\infty}^{+\infty}a^{-2}(\eta_{2})\int_{-\infty}^{+\infty}d\eta_{1}a^{-2}(\eta_{1})\dfrac{1}{(\eta_{1}-\eta_{2})^{4}}$
(53) $\displaystyle+$
$\displaystyle\dfrac{q^{2}a^{2}_{f}\Theta^{zz}}{\pi^{2}m^{2}c_{0}^{4}}\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{+\infty}d\eta_{1}a^{-2}(\eta_{1})\dfrac{1}{(\eta_{1}-\eta_{2})^{4}}.$

Using the scale factor given by Eq. (37) and the residue integration method in
Eq. (53) we obtain

$\displaystyle\langle(\Delta v^{z})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}a^{2}_{f}\sinh^{4}(g)}{\pi^{4}m^{2}\eta_{0}^{2}a_{1}^{4}c_{0}^{4}}\left[\zeta(3)-\dfrac{\pi^{4}}{90}\right](1+3\vec{\theta}\cdot\vec{B})+\dfrac{q^{2}a^{2}_{f}\sinh^{2}(g)}{3\pi^{4}m^{2}\eta_{0}^{2}a_{1}^{2}c_{0}^{4}}(\theta^{x}B^{x}+\theta^{y}B^{y})$
(54) $\displaystyle\times$ $\displaystyle
Re\left[\Psi\left(2,\dfrac{\pi+2gi}{2\pi}\right)-\Psi\left(2,\dfrac{\pi+2(g+w)i}{2\pi}\right)\right],$

where $w=\eta_{f}/\eta_{0}$ and

$\displaystyle
g=\dfrac{1}{2}\ln\left(\dfrac{\alpha^{2}+1}{\alpha^{2}-1}\right)=\dfrac{1}{2}\ln\left(\dfrac{c_{0}}{c_{sf}}\right),$
(55)

with $\alpha^{2}=a_{0}^{2}/a_{1}^{2}>1$. The parameter
$c_{sf}=c_{s}(\eta=\eta_{f})$ represents the final sound velocity in the fluid
at a final time $\eta_{f}$ and $c_{0}$ the initial sound velocity.

The equation (54) is the general expression to the proper velocity dispersion
of the particles in $z$ direction. Applying the same procedure done in the
previous section, let us choose the magnetic field in the $z$ direction, so we
get

$\displaystyle\langle(\Delta
v^{z})^{2}\rangle=\dfrac{2q^{2}a_{f}^{2}\sinh^{4}(g)}{m^{2}\pi^{4}a_{1}^{4}c_{0}^{4}\eta_{0}^{2}}\left[\zeta(3)-\dfrac{\pi^{4}}{90}\right](1+3\theta^{z}B^{z}).$
(56)

Note that, for $\theta^{z}=0$ or $B^{z}=0$ we recover the result found in Ref.
Bessa2017  in the absence of boundaries. Thus, in the present paper, there is
an additional first order contribution due the noncommutativity of space.

Now, for completeness, let us investigate the velocity dispersion in a
perpendicular direction, by making $i=x$ in the equation (35), we obtain

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}a^{2}_{f}}{2\pi^{2}m^{2}}(1+3\vec{\theta}\cdot\vec{B})\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})a^{-2}(\eta_{2})[f_{2}(\eta,x)+4\Delta
x^{2}f_{3}(\eta,x)]$ (57) $\displaystyle+$
$\displaystyle\dfrac{2q^{2}a^{2}_{f}}{\pi^{2}m^{2}}\iint
d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})\bigg{\\{}\dfrac{1}{2}\Theta^{xx}[f_{2}(\eta,x)+4\Delta
x^{2}f_{3}(\eta,x)]$ $\displaystyle+$ $\displaystyle 2\Theta^{xy}\Delta
x\Delta yf_{3}(\eta,x)+2\Theta^{xz}\Delta x\Delta zf_{3}(\eta,x)\bigg{\\}}.$

Since the factors $\Theta^{ij}$ are constants the integrals in the $x$
direction are equal to those obtained to the $z$ direction. Therefore in the
coincidence limit

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}a^{2}_{f}\sinh^{4}(g)}{\pi^{4}m^{2}\eta_{0}^{2}a_{1}^{4}c_{0}^{4}}\left[\zeta(3)-\dfrac{\pi^{4}}{90}\right](1+3\vec{\theta}\cdot\vec{B})+\dfrac{q^{2}a^{2}_{f}\sinh^{2}(g)}{3\pi^{4}m^{2}\eta_{0}^{2}a_{1}^{2}c_{0}^{4}}(\theta^{y}B^{y}+\theta^{z}B^{z})$
(58) $\displaystyle\times$ $\displaystyle
Re\left[\Psi\left(2,\dfrac{\pi+2gi}{2\pi}\right)-\Psi\left(2,\dfrac{\pi+2(g+w)i}{2\pi}\right)\right],$

and once we choose the magnetic field in the $z$ direction

$\displaystyle\langle(\Delta v^{x})^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}a^{2}_{f}\sinh^{4}(g)}{\pi^{4}m^{2}\eta_{0}^{2}a_{1}^{4}c_{0}^{4}}\left[\zeta(3)-\dfrac{\pi^{4}}{90}\right](1+3\theta^{z}B^{z})+\dfrac{q^{2}a^{2}_{f}\sinh^{2}(g)}{3\pi^{4}m^{2}\eta_{0}^{2}a_{1}^{2}c_{0}^{4}}(\theta^{z}B^{z})$
(59) $\displaystyle\times$ $\displaystyle
Re\left[\Psi\left(2,\dfrac{\pi+2gi}{2\pi}\right)-\Psi\left(2,\dfrac{\pi+2(g+w)i}{2\pi}\right)\right].$

In this particular case we have a time dependence given in terms of the
parameter $w$. Now, let us plot the effect of the non-commutativity. For this
purpose, we define a new function $T(w)$

$\displaystyle T(w)$ $\displaystyle=$
$\displaystyle\dfrac{6\sinh^{2}(g)}{a_{1}^{2}}\left[\zeta(3)-\dfrac{\pi^{4}}{90}\right]$
(60) $\displaystyle+$
$\displaystyle\dfrac{1}{3}Re\left[\Psi\left(2,\dfrac{\pi+2gi}{2\pi}\right)-\Psi\left(2,\dfrac{\pi+2(g+w)i}{2\pi}\right)\right],$

this is the term proportional to $\theta^{z}B^{z}$ and it can be written as

$\displaystyle
T=\dfrac{\pi^{4}m^{2}a_{1}^{2}c_{0}^{4}\eta_{0}^{2}}{q^{2}a_{f}^{2}\sinh^{2}(g)(\theta^{z}B^{z})}\left\\{\langle(\Delta
v^{x})^{2}\rangle-\dfrac{2q^{2}a_{f}^{2}\sinh^{4}(g)}{m^{2}\pi^{4}a_{1}^{4}c_{0}^{4}\eta_{0}^{2}}\zeta(3)\right\\}.$
(61)

As we know from (55), the $g$ factor is expressed in terms of the fluid
parameter and similarly we can show that

$\dfrac{\sinh^{2}(g)}{a_{1}^{2}}=\dfrac{1}{2}\left(1-\dfrac{c_{sf}}{c_{0}}\right).$
(62)

Figure 3 shows the plot of Eq. (60) for different values of $c_{sf}$. We
observe that the first order noncommutativity contribution on the velocity
dispersion in the $x$ direction (perpendicular to the magnetic field) is
negative. Note that, $T(w)$ decreases for small values of $w$, but when $w$
assumes large values we note that $T(w)$ tends to a constant. In addition,
when the sound velocity in the fluid $c_{sf}$ assumes large values, the
noncommutative corrections are more negative. It also shows the influence of
noncommutativity on the velocity dispersion of the particles when the sound
velocity in fluid is relatively large, however when the sound velocity in
fluid $c_{sf}$ takes on small values, as can be seen in Fig. 4 in a region of
large $w$ and small $c_{sf}$, the noncomutativity contributions for velocity
dispersion can be positive.

Figure 3: The contribution of the noncommutativity in equation (60). The three
lines represents the curves for distinct values that the final velocity
$c_{sf}$ take in relation to the initial fluid velocity $c_{0}$. Figure 4: The
contribution of the noncommutativity in equation (60) for small values of
$c_{sf}$.

As it was noted in Ref. Bessa2017  for commutative spaces with bounded
particles the velocity dispersion of the particles is constant and is the same
in all directions, that is, in the absence of boundary it is isotropic.
However, as shown in Eqs. (56) and (59), for noncommutative spaces we obtain
different velocity dispersion for different directions and the isotropy, at
least in first order, is broken. It is also important to note that the
negative contributions that appears in Figs. 3 and 4 are up to the first order
$\theta$-corrections. The dominant contribution is really the zeroth order
term present in Eq. (59) which is positive.

##  V Metric Fluctuations

In this section, we wish to discuss briefly the validity of the methodology
adopted in previous sections where the effects of the metric fluctuations were
not considered. We will see that these effects can be, for the cases treated
in this paper, neglected. So, let us remember that at the end of Sect. III, we
have said that these fluctuations are in general viewed as linearized
perturbations upon a metric. In some analog models for quantum gravity
effects, the metric fluctuations could induce sound cone fluctuations. In
general, a diagonal metric is taken into account, and white or colored noise
is implemented in the equation of motion of the particles kms10 . Thus, we
could write Eq. (10) under the perspective of Eq. (36) in the following
diagonal form,

$ds^{2}=H^{2}(1+\gamma)\left[-c_{0}^{2}dt^{2}+a^{2}(dx^{2}+dy^{2}+dz^{2})\right],$
(63)

where, in general, $\gamma=\gamma(t,r)$ and $\gamma\ll 1$. In what follows,
let us incorporate these fluctuations on the geodesic equation. Thus, the
right hand side (RHS) of Eq. (20) becomes

$\frac{1}{m}f^{i}\cong\frac{du^{i}}{dt}+2\frac{\dot{a}}{a}u^{i}+\dot{\gamma}u^{i}+(\partial_{j}\gamma)u^{i}u^{j},$
(64)

where we have considered up to the first order terms in $\gamma$.

Since, from Eq. (11), $a\geq 1$, and in many analog models kms10 ; akms12 ;
bds12 ; dms13 ; adsbm14 , the tensor $\gamma_{\mu\nu}$ has only the
$\gamma_{00}$ component, which is regarded as white noise with spatial
dependence ($\gamma=\gamma(r)$) only. As we are working in a non-relativistic
limit, $u^{i}\ll 1$, we can see that, from Eq. (64), the dominant contribution
comes from the two first terms. Then, the RHS of Eq. (20) describes the motion
of the particles properly.

Now we wish to see the consistency in considering the fluctuations only in the
scalar field $\phi$ and neglecting the metric fluctuations, as was done in
Sect. III. In general, these fluctuations has the following features:
$\langle\gamma(t,r)\rangle=0$ and
$\langle\gamma(t_{1},r_{1})\gamma(t_{2},r_{2})\rangle\neq 0$, which is similar
to the fluctuations in $\phi$.

The $3$-force acting on the particles is given by Eq. (16). Note that the
metric tensor appears only in the two first terms in the RHS of this equation.
So, instead of taking into account metric (10), let us consider the linearized
metric (63). Thus, the $3$-force, up to the first order in $\gamma$ is,

$f^{i}\cong
qH^{-2}a^{-2}\left(1-\frac{1}{2}\vec{\theta}.\vec{B}-\gamma+\frac{\gamma}{2}\vec{\theta}.\vec{B}\right)\nabla_{i}\phi+\Theta^{ij}\nabla_{j}\phi.$
(65)

Note that the 4th term inside parentheses can be neglected since it is
proportional to $\theta\gamma$ with both very small.

In Sect. IV, to see the consistency with Ref. Bessa2017 , a null magnetic
field was considered. So to make the calculations simpler, let us consider
again that the magnetic field is off ($\vec{B}=0$). Consequently $H=1$ and Eq.
(65) reads,

$f^{i}=qa^{-2}(1-\gamma)\nabla_{i}\phi.$ (66)

The two-point function which must appear in the integrand when the velocity
dispersion is being evaluated is now,

$\displaystyle\langle
f^{i}(t_{1},r_{1})f^{i}(t_{2},r_{2})\rangle\propto\partial_{i_{1}}\partial_{j_{2}}\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle-\partial_{i_{1}}\partial_{j_{2}}\left[\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle\right]\langle\gamma(t_{1},r_{1})\gamma(t_{2},r_{2})\rangle,$
(67)

where we used the fact that
$\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\gamma(t_{1},r_{1})\gamma(t_{2},r_{2})\rangle\propto\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle\langle\gamma(t_{1},r_{1})\gamma(t_{2},r_{2})\rangle$
with $\langle\gamma\phi\rangle=0$. Note that since the second term in the RHS
of Eq. (67) is a product of fluctuations it is much smaller than the first
one.

In what follows in the rest of Sect. IV, when $\vec{B}\neq 0$, it is also
possible to neglect this contribution since, in our model, $\vec{B}$ and
$\vec{\theta}$ are non-fluctuating constants. So the two-point functions that
appear in the integrands of Sect. IV is always dominant against the quadratic
term of Eq. (67) that would appear in these integrals if the metric
fluctuations were considered.

##  VI Conclusions

In this paper, we studied the stochastic motion of a classical scalar particle
coupled to a quantized massless scalar field in an expanding noncommutative
background. We have shown that this expansion is analogous to a Friedmann-
Robertson-Walker (FRW) geometry. To perform this analogy, we considered a
decomposition and a linear expansion of the scalar field that is a solution
consistent with a Lagrangian describing the noncommutative Abelian Higgs model
in a flat spacetime. In order to simplify our expressions, the model admitted
that the electric field present in Lagrangian (3) and the flux velocity were
null, and that the noncommutativity was activated when a magnetic field is
turned on in a given direction. Thus, the study for the stochastic motion of a
scalar particle was implemented for this expanding noncommutative background.
It was found noncommutative correction for the free and bound particles.
Whereas the former were defined as particles that follow geodesics, and the
latter as particles that were under the influence of a classical external
force that cancels locally the effects of the expansion.

In a recent paper Bessa2017  a commutative Bose-Einstein condensate (BEC) was
taken into account to study the same type of motion described above. In this
situation, it was found a non-null velocity dispersion associated with the
bound particles, meaning that these particles undergo stochastic motion due to
quantum fluctuations. In the present paper, when bound particles were
considered, we have found the same result present in Ref. Bessa2017  added by
a factor proportional to the noncommutative parameter $\theta$. In this case,
the noncommutativity contributes as a first-order correction. Such correction
could be negative for short times and positive in the long time regime.
However, the dominant contribution is the zeroth term in $\theta$, which is
positive.

In the same context, for a commutative fluid, when no boundary is present, it
was found in Ref. Bessa2017  that the velocity dispersion associated with the
free particles was zero. This means that free particles has no stochastic
motion due to the quantized field in the expanding commutative background.
However, in the present paper, when the noncommutativity of space is taken
into account, a nonzero velocity dispersion was found for the free particles.
For the dispersion perpendicular to the magnetic field, it was found a term
proportional to $\theta$ and for the dispersion parallel to the magnetic field
a term proportional to $\theta^{2}$ was found. This result could be
interpreted as a direct consequence of the noncommutativity of the space.

The stochastic motion of the particles shown in this paper is a subtle, non-
trivial quantum effect and, although the mechanism presented here to observe
this manifestation is limited, the main interest of our paper is theoretical
once our results show a relevant contribution coming from the noncommutativity
of space.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant nos.
165/2018 and 015/2019), for partial financial support. MAA, FAB and EP
acknowledge support from CNPq (Grant nos. 306962/2018-7 and 433980/2018-4,
312104/2018-9, 304852/2017-1). The authors would like to thank J. P. Spinelly
and F. G. Costa for helpful comments.

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