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\DOI{10.5802/crphys.241}
\datereceived{2024-10-04}
\daterevised{2025-01-15}
\dateaccepted{2025-02-11}
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%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{Simulating gravitational problems with condensed matter analog models: a special issue in memory of RenaudParentani (1962--2020)}
%\def\cdr@specialtitle@french{Simuler des probl\`emes gravitationnels avec des mod\`eles analogues en mati\`ere condens\'ee : un num\'ero sp\'ecial en m\'emoire de Renaud Parentani (1962--2020)}
%\makeatother

\CDRsetmeta{articletype}{research-article}

\title{Looking for {traces} of Hawking radiation in correlation
functions of BEC acoustic black holes}

\alttitle{\`A la recherche des traces de la radiation de Hawking dans
les fonctions de corr\'elation des trous noirs acoustiques}

\shortrunauthors

\author{\firstname{Paul Raymond} \lastname{Anderson}\CDRorcid{0000-0002-9882-001X}}
\address{Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, USA}
\email[P. R. Anderson]{anderson@wfu.edu}

\author{\firstname{Roberto} \lastname{Balbinot}\CDRorcid{0000-0003-2741-699X}}
\address{Dipartimento di Fisica dell'Universit\`a di Bologna and INFN sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy}
\email[R. Balbinot]{Roberto.Balbinot@bo.infn.it}

\author{\firstname{Richard Allen} \lastname{Dudley}\CDRorcid{0000-0003-1771-5072}}
\address{Department of Physics and Optical Sciences, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, USA}
\email[R. A. Dudley]{Richard.A.Dudley@charlotte.edu}

\author{\firstname{Alessandro}\nobreakauthor\lastname{Fabbri}\CDRorcid{0000-0003-2340-9338}\IsCorresp}
\address{Departamento de F\'isica Te\'orica and IFIC, Universidad de Valencia-CSIC, C. Dr. Moliner 50, 46100 Burjassot, Spain}
\email[A. Fabbri]{afabbri@ific.uv.es}

\begin{abstract}
Renaud Parentani was one of the leading figures in Quantum Field Theory
in curved spacetime, in particular concerning its applications to
Hawking-like radiation in analogue models.
In this paper dedicated to
him, we discuss the characteristic features appearing in the correlation
functions in an acoustic black hole formed by a Bose--Einstein
condensate, considered as signature of the presence of Hawking
radiation in this system.
\end{abstract}

\begin{altabstract}
{Renaud Parentani \'etait l'un des principaux leaders de la Th\'eorie
des Champs en espace-temps courbes, en particulier sur les applications
\`a la radiation de type Hawking dans les mod\`eles analogues. Dans ce
papier qui lui est d\' edi\'e nous discutons les \'el\'ements
caract\'eristiques qui apparaissent dans les fonctions de
corr\'elations dans un trou noir acoustique form\'e par un condensat de
Bose--Einstein, consid\'er\'es comme la signature de la pr\'esence de la
radiation de Hawking dans ce syst\`eme}.
\end{altabstract}

\keywords{\kwd{Analog Hawking radiation}\kwd{Bose--Einstein 
condensate}\kwd{Quantum Field theory in curved space}}

\altkeywords{\kwd{Radiation de Hawking analogue}\kwd{Condensat de Bose--Einstein}\kwd{Th\'eorie des champs en espace-temps courbes}}

\thanks{National Science Foundation under grant No.
PHY-2309186, Spanish Grants PID2020-116567GB-C21, 
PID2023-149560NB-C21 funded by MCIN/AEI/10.13039/501100011033,
Severo Ochoa Excellence Grant CEX2023-001292-S}

\maketitle

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\newcommand{\la}{\langle}
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\def\rme{\mathrm{e}}
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\section{Introduction}
\renewcommand{\theequation}{\thesection.\arabic{equation}}

In 1981 Unruh~\cite{unruh81} suggested that Hawking's black hole
(hereafter BH) radiation  can have an analogue in a fluid whose flow
undergoes a transition from a subsonic regime to a supersonic one. The
locus where this happens is the so called ``sonic horizon'', since
sound waves are trapped inside the supersonic region and cannot
propagate upstream; they are trapped by the flow and dragged
downstream. Unruh showed that in this situation one should expect an
emission in the subsonic region of thermal phonons at a temperature
proportional to the surface gravity of the sonic horizon exactly as
predicted by Hawking for gravitational BHs~\cite{hawking74, hawking75}.

In the following years many systems were proposed to experimentally
detect this analogue Hawking radiation~\cite{Barcelo:2005fc}. The most
promising appeared to be the ones constructed by Bose--Einstein
condensates (BECs), since in this case one can arrange the experimental
setup so that the associated Hawking temperature is expected to be just
one order of magnitude smaller than the BEC background temperature
(100~nK). Nevertheless even this difference has so far prevented any
direct detection of these thermal phonons.

In 2008 it was shown that,  since the Hawking effect is a genuine
process of entangled pair creation  in which for each thermal phonon in
the subsonic region there is a corresponding negative energy partner
inside the horizon,  a characteristic {correlation band} in the in-out 
density--density correlation function should appear~\cite{paper1,
cfrbf}.  This is the smoking gun of the Hawking effect. This band was
observed in a series of experiments performed by Steinhauer and his
group~\cite{jeff2016, jeff2019, jeff2021}.  This represents the best
evidence to date for the presence of Hawking radiation in sonic BHs.

Renaud Parentani suggested that besides this main {band}, two other
minor {bands} should appear in the density--density correlation function
because of backscattering effects on the modes~\cite{mp}. While there
is as yet no experimental evidence for these bands, numerical
calculations have confirmed their presence~\cite{paper2, rpc,
paper2011}.  

In this paper, dedicated to Renaud Parentani for his invaluable
contributions to this field, we shall investigate, using the framework
of Quantum Field Theory in curved space (see for example~\cite{bd, fu,
pt}), the significative features of the density--density correlation
function connected to Hawking radiation for two flow profiles having
the same asymptotic sound speed limits and horizon surface gravity,
outlining similarities and differences.


\section{The setting}

In a BEC the phase fluctuation $\hat\theta$ on top of the condensate
in the hydrodynamical approximation obeys an
equation which is formally identical to a wave equation for a massless
scalar field propagating in a fictitious curved
space-time\footnote{More details of the review in this section can be
obtained in~\cite{Barcelo:2005fc}.}   described by the line element
\def\theequation{\thesection.\arabic{equation}}
\setcounter{equation}{0}
{\begin{equation} \label{dueuno} 
\mathrm{d} s^2=\frac{n}{mc}[-c^2\mathrm{d} T^2+(\mathrm{d} \vec x - \vec{v} \mathrm{d} T)(\mathrm{d} \vec{x} - \vec{v} \mathrm{d} T)], 
\end{equation}}\unskip
where $n$ is the condensate density, $c$ the local speed of sound,
$\vec{v}$ the velocity field and $m$ the mass of a single atom. The
wave equation reads
{\begin{equation}\label{duedue}
\Box \hat\theta = 0,
\end{equation}}\unskip
where $\Box\equiv \nabla_\mu\nabla^\mu$ is the covariant d'Alembertian
calculated with the metric (\ref{dueuno}).


This system can be treated by using the methods of Quantum Field Theory
in curved space-time. This has been done in a paper written in
collaboration with  Parentani~\cite{paper2013}. Here we just outline
the main points.

We shall consider for simplicity a stationary unidimensional flow
directed along the $\hat x$ axis with a constant velocity $\vec v$, the
density  $n$  is also constant. By an appropriate rescaling of the
phase operator  $\hat{\theta}=\sqrt{{mc}/{n\hbar L_\perp^2 }}
{\hat{\theta}^{(2)}}$, where $L_\perp$ is the size of the transverse
direction with $L_\perp\ll {\hbar}/{mc}$ so that excitations with
transverse momenta are frozen, Equation~(\ref{duedue}) can be rewritten as
{\begin{equation} \label{duetre}
\left(\Box^{(2)} - V \right)\hat \theta^{(2)}=0,
\end{equation}}\unskip
where $V$ is given by
{\begin{equation}
V= -\frac{1}{2} \left(1-\frac{v^2}{c^2}\right) \frac{\mathrm{d}^2c}{\mathrm{d} x^2} +\left(\frac{1}{4c}-\frac{5v^2}{4c^3} \right)\left(\frac{\mathrm{d}c}{\mathrm{d}x}\right)^2    \label{duequattro}
\end{equation}}\unskip
and $\Box^{(2)}$ is the two dimensional (2D) d'Alembertian associated with the 2D section of the line element (\ref{dueuno}), namely
{\begin{equation} \label{duecinque}
\mathrm{d}s^2=\frac{n}{m}\left[-\frac{c(x)^2-v^2}{c(x)}\mathrm{d}t^2+\frac{c(x)}{c(x)^2-v^2}\mathrm{d}x^2\right] .
\end{equation}}\unskip
Here we have introduced a ``Schwarzschild'' time $t$ such that
{\begin{equation}\label{duesei}
    t=T-\int^x \mathrm{d} y \frac{v}{c(y)^2-v^2} .
    \end{equation}}\unskip
 By considering the coordinate $x^*$, given by
 {\begin{equation} \label{dueotto}
 x^*=\int^x \frac{c(y)\mathrm{d} y}{c(y)^2-v^2},
 \end{equation}}\unskip
 we can rewrite (\ref{duetre}) in the form
 {\begin{equation} \label{eqep}
\left(\frac{\partial^2}{\partial t^2}+\frac{\partial}{\partial x^{*2}}+V_{\rm eff}\right)\theta^{(2)}=0,
\end{equation}}\unskip
with effective potential
{\begin{equation} \label{ep}
V_{\rm eff}=\frac{c^2-v^2}{c}V. 
\end{equation}}\unskip
  
The BEC flows along the $\hat x$ direction from right to left (i.e.\ 
$\vec v = -v_0\hat x$, with $v_0>0$). By varying $c(x)$ one can
engineer the flow so that it is subsonic for $x>0$ ($R$ region) and
supersonic for $x<0$ ($L$ region). $x=0$ is the sonic horizon. $c(x)$
can be further assumed to approach constant values asymptotically;
namely $\lim_{x\to +\infty} c(x)=c_R>v_0$ and $\lim_{x\to -\infty}
c(x)=c_L<v_0$. Note that the effective potential $V_{\rm eff}$
(\ref{ep}) vanishes asymptotically and on the horizon. The Penrose
diagram of the spacetime is depicted in Figure~\ref{figuno}.

\begin{figure}[t!]
\includegraphics{fig01}
\caption{\label{figuno}Penrose diagram of the spacetime of Equation~(\ref{duecinque}).
$H^+$ is the future horizon, $H^-$ the past one, $I^\pm_{R,L}$ are null
infinities. $R$ is the region outside the horizon, $L$ is the one
inside.}
\end{figure}

For further use we introduce the retarded ($u$) and advanced ($v$) null
Eddington--Finkelstein like coordinates as
{\begin{eqnarray} 
u &=& t-x^* \label{duesettea}, \Seqnsplit
v &=& t+x^*, \label{duesetteb} 
\end{eqnarray}}\unskip
and the Kruskal one
{\begin{equation}
 U = \pm \frac{1}{\kappa} \mathrm{e}^{-\kappa u}, \label{duenovea} 
 \end{equation}}\unskip
 where $\kappa$ is the surface gravity of the horizon
{\begin{equation} 
\kappa=   \left. \frac{\mathrm{d}c}{\mathrm{d}x}\right|_{x=0} \label{duedieci} 
 \end{equation}}\unskip
and in Equation (\ref{duenovea}) the plus sign is for the $L$ region
and the minus sign is for the $R$ region.


 The quantum state of our field $\hat\theta^{(2)}$, as is well known,
can be approximated at late times after the formation of the BH by the
Unruh state $|U\mathop{\rangle}$~\cite{unruh76}. This corresponds to an
expansion of the quantum operator as
 {\begin{equation} \label{dueundici}
\hat{\theta}^{(2)}=\int_0^\infty \mathrm{d} \omega_K \left[\hat{a}_K(\omega_K) u_H^{K}+
\mathrm{h.c.}
\right]+\int_0^\infty \mathrm{d} \omega \left[\hat{a}_I(\omega) u_I^{R}+
\mathrm{h.c.}
\right],
\end{equation}}\unskip
 where the form of the mode $u_I^R$ on $I_R^-$ is
  {\begin{equation} \label{duedodici}
 u_I^{R}(\omega, x)=\frac{\mathrm{e}^{-\mathrm{i}\omega v}}{\sqrt{4\rmpi \omega}},
 \end{equation}}\unskip
 while that for the mode $u_H^K$ on $H^-$ is
  {\begin{equation} \label{duetredici}
 u_H^K(\omega_K, x)=\frac{\mathrm{e}^{-\mathrm{i}\omega_K U}}{\sqrt{4\rmpi \omega_K}}.
 \end{equation}}\unskip
 The Unruh state $|U\mathop{\rangle}$ is defined as
 {\begin{equation} 
 \hat{a}_K(\omega_K)|U\mathop{\rangle} = 0 = \hat{a}_I(w) |U\mathop{\rangle} \label{duequattordici}
 \end{equation}}\unskip
for every $\omega, \omega_K$. In Figure~\ref{figdue} we illustrate
these modes on the Cauchy surface $H^-\cup I_R^-$.

\begin{figure}[t!]
\includegraphics{fig02}
\caption{\label{figdue}Modes for the Unruh state.}
\end{figure}

 
 Note that while $u_I^R$ is a positive frequency mode with respect to
Schwarzschild time, $u_H^K$ is positive frequency with respect to
Kruskal time. The Unruh state describes a situation in which one has no
incoming radiation on $H^- \cup I_R^-$, while at $I_R^+$ one has a
thermal flux at the Hawking temperature
${T_H={\hbar\kappa}/{2\rmpi k_{\mathrm{B}}}}$, where $k_{\mathrm{B}}$ is
Boltzmann's constant.

It is convenient for the calculations of the 2-point function to
express the modes $u_H^K$ in terms of modes (Boulware modes~\cite{bo})
that on $H^-$ behave as
 {\begin{eqnarray}
  u_H^{R} &=& \frac{\mathrm{e}^{-\mathrm{i}\omega u}}{\sqrt{4\rmpi \omega}},\quad x>0,
\label{duequindicia} \Seqnsplit
  u_H^{L} &=& \frac{\mathrm{e}^{\mathrm{i}\omega u}}{\sqrt{4\rmpi \omega}},\quad x<0 \label{duequindicib} . 
  \end{eqnarray}}\unskip
Note the $+$ sign in front of the exponential in Equation (\ref{duequindicib}). $u_H^L$ has negative frequency (while having positive norm), it is associated with the negative (Killing) energy partners.
  We have
  {\begin{equation} \label{duesedici}
 u_H^{K}(\omega_K,x)=\int_0^\infty \mathrm{d} \omega \left[\alpha_{\omega_K \omega}^R u_H^{R}+\beta^{R}_{\omega_K \omega} u_H^{R*}\right]+ R \leftrightarrow L .
 \end{equation}}\unskip
The Bogoliubov coefficients are given in
Ref.~\cite{paper2013}\footnote{In Ref.~\cite{paper2013} there is a
misprint in Equations~(4.14b)--(4.14e). $\omega_K$ should be replaced
by ${\omega_K}/{\kappa}$.} and are summarised in
Appendix~\ref{appendixA}. Because of the presence of the effective
potential $V_{\rm eff}$ in (\ref{ep}),  the incoming modes will be
modified from their asymptotic forms in Equations~(\ref{duedodici}),
(\ref{duequindicia}), (\ref{duequindicib}) due to backscattering
effects. In Ref.~\cite{paper1}, $V$ was neglected and the modes
maintained their expressions (\ref{duedodici}), (\ref{duequindicia}),
(\ref{duequindicib}) throughout the space-time. In
Figures~\ref{figtre}--\ref{figcinque} we schematically describe the
backscattering of each mode.

 
 \begin{figure}[t!]
\includegraphics{fig03}
\caption{\label{figtre}Backscattering of the $u_I^R$ modes.}
\end{figure}

 \begin{figure}[t!]
\includegraphics{fig04}
\caption{\label{figquattro}Backscattering of the $u_H^R$ modes.}
\end{figure}

 \begin{figure}[t!]
\includegraphics{fig05}
\caption{\label{figcinque}Backscattering of the $u_H^L$ modes.}
\end{figure}

In any case, since $V_{\rm eff}$ vanishes asymptotically at future null
infinity each incoming mode will be a linear combination of
$\mathrm{e}^{-\mathrm{i}\omega v}$ and $\mathrm{e}^{\pm \mathrm{i}\omega u}$.

We will consider two profiles of the sound velocity. The first one has
been proposed in a numerical simulation fully based on the Bogoliubov
theory of a BEC~\cite{cfrbf}  which confirmed the presence of a peak in
the in-out correlation function as predicted by~\cite{paper1} using
only QFT in curved space methods. It is
{\begin{eqnarray}  
c(x)&=&\sqrt{c_L^2 +
\frac{1}{2}(c_R^2-c_L^2)\left[1+\frac{2}{\rmpi}\tan^{-1}
\left(\frac{x+b}{\sigma_v}\right)\right]} , \label{duediciassette}\Seqnsplit
      b &=& \sigma_v \tan \left[\frac{\rmpi}{c_R^2-c_L^2} \left( v_0^2 - 
      \frac{1}{2} (c_R^2+c_L^2)\right) \right], \label{duediciotto} 
\end{eqnarray}}\unskip
where $\sigma_v$ is an arbitrary positive constant. The horizon is at $x=0$. The surface gravity for this profile is
 {\begin{equation} 
 \kappa = \left. \frac{\mathrm{d} c}{\mathrm{d}x} \right \vert_{x=0} = \frac{1}{2 \rmpi v_0 \sigma_v}(c_R^2-c_L^2) \sin^2 \left[\frac{\rmpi(c_R^2-v_0^2)}{c_R^2-c_L^2}\right] \label{kappa-2013-profile}.  
 \end{equation}}\unskip
We call this the ``original'' profile. In this case the equation for
the modes Equation (\ref{duetre}) has to be solved numerically. The
explicit construction of the modes has been given in
Ref.~\cite{paper2013}, written in collaboration with Renaud Parentani,
where all the details can be found. The second profile, which we call
``analytical'', is
{\begin{equation}
c(x) =  \frac{c_R}{\sqrt{1 + \left(\frac{c_R^2}{v_0^2} - 1 \right) 
\exp \left[-\frac{2 \frac{c_R^2}{v_0^2} k x}{\left(\frac{c_R^2}{v_0^2}
- 1\right)}\right]}} \theta(x) +
 \frac{c_L}{\sqrt{1 - \left(1 -\frac{c_L^2}{v_0^2} \right) 
 \exp \left[-\frac{2 k \frac{c_L^2}{v_0^2} x}{\left(\frac{c_L^2}{v_0^2}-1\right)}\right]}}\theta(-x), \label{duediciannove}
\end{equation}}\unskip
where $x=0$ is the horizon and $k$ is a positive constant of dimension
$L^{-1}$ and the corresponding surface gravity is
{\begin{equation}
\kappa=  \left. \frac{\mathrm{d}c}{\mathrm{d}x} \right|_{x=0}=k \, v_0. \label{kappa-analytical-profile} 
\end{equation}}\unskip
This profile has been introduced in Ref.~\cite{fba2016}. The advantage
of this profile is that the modes can be computed analytically in terms
of hypergeometric functions.  

\section{Correlation functions}
  
  As mentioned in the Introduction, the only experimental support for
the existence of Hawking-like radiation in an acoustic BH formed by a
BEC comes from the observation of a correlation band appearing in the
in-out (one point inside the horizon and the other outside) equal time
density-density correlation function, in  agreement with the
theoretical prediction~\cite{paper1}.

  
 Defining the operator $\hat n$ as the quantum density fluctuation on
top of the condensate, the density--density correlation function is
\def\theequation{\thesection.\arabic{equation}}
\setcounter{equation}{0}
 {\begin{equation}
G_2\left(T, x;T^\prime, x^\prime\right)=
\mathop{\langle} U|\hat{n}(T, x)\hat{n}(T^\prime, x^\prime)|U \mathop{\rangle}. \label{treuno}
\end{equation}}\unskip
 In the hydrodynamical approximation we have~\cite{Barcelo:2005fc}  
 {\begin{equation}
\hat{n}= \frac{\hbar n}{mc^2}\left[v_0\partial_x\hat{\theta}-\partial_T  \hat{\theta}\right] . \label{eqtredue}
\end{equation}}\unskip
One finds that
{\begin{equation}
G_2\left(T, x;T, x^\prime \right) =
\frac{\hbar n}{2 m L_\perp^2 c^{2}(x)c^{2}(x^\prime)} D 
\sqrt{c(x)c(x')}\mathop{\langle} U| 
\{\hat{\theta}^{(2)}(t, x), \hat{\theta}^{(2)}(t^\prime, x^\prime)\}|U\mathop{\rangle}, \label{tretre}
\end{equation}}\unskip
where
{\begin{equation}
D \equiv \partial_T\partial_{T^\prime}-v_0\partial_x\partial_{T^\prime}-v_0\partial_T\partial_{x^\prime}+v_0^2\partial_x\partial_{x^\prime}
 . \label{trequattro}
 \end{equation}}\unskip
 Using Equations~(\ref{duedodici}) and (\ref{duesedici}) and integrating over $\omega_K$  (see~\cite{paper2013}) the two-point function entering (\ref{tretre}) can be written as~\cite{paper1}
 {\begin{equation} 
 \mathop{\langle} U| \left\{ \hat{\theta}^{(2)}(t, x), \hat{\theta}^{(2)}(t^\prime, x^\prime) \right\} |U\mathop{\rangle} = I + J,  \label{trecinque}
 \end{equation}}\unskip
 where
{\begin{eqnarray}
 I &=& \int_0^\infty \mathrm{d}  \omega \frac{1}{\sinh\left(\frac{\rmpi \omega}{\kappa}\right)} \left\{ u^{L}_H(\omega,t,x) \, u^{R}_H(\omega,t',x') +
   u^{L *}_H(\omega,t,x) \, u^{R *}_H(\omega,t',x') \right. \nonumber \\
   & &  \left. +\, u^{R}_H(\omega,t,x) \, u^{L}_H(\omega,t',x') +
   u^{R *}_H(\omega,t,x) \, u^{L *}_H(\omega,t',x') \right. \nonumber \\
   & &  \left.  +\, \cosh \left(\frac{\rmpi \omega}{\kappa}\right) \left[ u^{L}_H(\omega,t,x) \, u^{L *}_H(\omega,t',x')
   + u^{L *}_H(\omega,t,x) \, u^{L}_H(\omega,t',x') \right. \right. \nonumber \\
  & &  \left. \left. +\, u^{R}_H(\omega,t,x) \, u^{R *}_H(\omega,t',x') + u^{R *}_H(\omega,t,x) \, u^{R }_H(\omega,t',x') \right]
   \right\} , \label{tresei} \Seqnsplit
   J &=& \int_0^\infty \mathrm{d}  \omega \, \left[ u^{R}_I(\omega,t,x) \, u^{{R} \, *}_I(\omega,t',x')
   +  u^{{R} \, *}_I(\omega,t,x) \, u^{R}_I(\omega,t',x') \right]  \label{tresette}
 \end{eqnarray}}\unskip
 and the relation between Schwarzschild time $t$ and $T$ is given by  Equation (\ref{duesei}).
 
\begin{figure}[t!]
\includegraphics{fig06}
\caption{\label{figsei}Plot for the profiles of Equations~(\ref{duediciassette})
(blue curve), (\ref{duediciannove}) (orange curve) arranging them to
have the same surface gravity and asymptotic behaviour.}
\end{figure}

\begin{figure}[t!]
\includegraphics{fig07}
\caption{\label{figsette}Plot of the effective potential (\ref{ep}), (\ref{duequattro})
for the profiles of Equations~(\ref{duediciassette}) (blue curve),
(\ref{duediciannove}) (orange curve) with the same asymptotic behaviour
and surface gravity.}
\end{figure}

\begin{figure}[t!]
\includegraphics{fig08}
\caption{\label{figotto}Density--density correlation function for
the two sound speed profiles. Left: Original profile
(\ref{duediciassette}). Right:  Result using the analytical profile
(\ref{duediciannove}). Each figure is oriented so that the bottom left
quadrant where both points are in the interior and the upper right
quadrant where both points are in the exterior of the analog black
hole.  {The green-dashed lines highlight the main band (i) by showing
the locations of two parts of its negative correlation peak.  The
black-dashed and brown-dashed lines do the same for the secondary
positive band (ii) and the secondary negative band (iii) respectively. 
Regions shaded in gray are outside of the ranges of the
plots.}}
\end{figure}   

\begin{figure}[t!]
\includegraphics{fig09}
\caption{\label{Figi}Correlations between $u,u'$ modes on both sides of the
horizon.}
\end{figure}

If one neglects the effective potential in Equation (\ref{eqep}), the
modes maintain the form given by Equations~(\ref{duedodici}),
(\ref{duequindicia}), (\ref{duequindicib}) throughout the entire
space-time. In this case one can obtain an analytical expression for
$G_2(T,x;T',x')$ which, taking the point $x$ in the asymptotic $R$
region where $c(x)\sim c_R$ and the point $x'$ in the asymptotic $L$
region where $c(x)\sim c_L$, can be written as
{\begin{eqnarray} 
G_2(T,x;T,x') &=&  \frac{\hbar n}{2 m L_\perp^2 c_R^{3/2}c_L^{3/2}} 
\Bigg\{  -\frac{1}{(c_R-v_0)(v_0-c_L)}
\frac{\kappa^2}{\cosh^2 \frac{\kappa}{2}(u-u')}\nonumber \\
  &&+\, \frac{1}{(c_R+v_0)(c_L+v_0)}\frac{1}{(v-v')^2}\Bigg\}  . \label{treotto} 
\end{eqnarray}}\unskip
We see that this function has a negative minimum peaked along $u=u'$
which corresponds (in the geometrical optics approximation) to the
trajectory of the Hawking quanta ($u=\mathrm{const}$) and its partner
($u'=\mathrm{const}$).  Beside this no other structure is present. This feature
is the one observed by Steinhauer's group~\cite{jeff2019, jeff2021}. We
see that the no-backscattering asymptotic correlation function
(\ref{treotto}) has the same form for all profiles having the same
$c_R,c_L,v_0$ and surface gravity $\kappa$; in particular the height,
width, and location of the minima are identical. We shall impose this
to be the case for our two profiles introduced in the previous section.
In Figure~\ref{figsei} we have plotted the two profiles  for
$v_0={3}/{4}, c_L={1}/{2}, c_R=1, \sigma_v=8$ and numerically
matched the two surface gravities.   Even with all these parameters
matched, there are noticeable differences in the comparison of the
sound profiles. This leads to differences in the scattering of the
modes as can be seen in Figure~\ref{figsette}, where the effective
potential for the two profiles is plotted.  The extrema of the
potential appear to be higher and narrower for the original profile. 
All this has a significant signature in the correlation \mbox{functions} as we
shall see. In Figure~\ref{figotto} we have represented the correlation
function Equation~(\ref{tretre}) at equal time $T=T'$ for the original
and the analytical profile respectively. This function {is symmetric
under the exchange $x\leftrightarrow x'$} and diverges  when the points
come together and thus the region $x=x'$ is cutoff for this reason. {In
each figure one can clearly see the large negative correlation band,
(i), when one point is in the interior and the other point is in the
exterior region. This is the one predicted by the no-backscattering
asymptotic expression (\ref{treotto})  and corresponds to the
correlation between the modes depicted in Figure~\ref{Figi}. One can
also see two much smaller bands predicted by R. Parentani: a positive
one, labeled (ii), when one point is inside the horizon and the other
outside (correlations between the modes represented in
Figure~\ref{Figii}) and a negative one when both points are inside the
horizon, labeled (iii) (correlation corresponding to the modes in
Figure~\ref{Figiii}).} These two secondary correlation bands exist
because of the backscattering of the modes.\looseness=-1

 \begin{figure}[t!]
{\vspace*{-2pt}}
\includegraphics{fig10}
{\vspace*{-2pt}}
\caption{\label{Figii}Correlation between $u,v'$ modes  modes across the
horizon.}
\end{figure}

\begin{figure}[t!]
\includegraphics{fig11}
{\vspace*{-2pt}}
\caption{\label{Figiii}Correlation between $u,v'$ modes  modes inside the
horizon.}
\end{figure}

\begin{figure}[t!]
{\vspace*{-2pt}}
\includegraphics{fig12}
{\vspace*{-2pt}}
\caption{\label{fignove}The density--density correlation
function at $x^\prime=-250$ when one point is in the interior and one
point in in the exterior of the analog black hole. Blue dot: Original
profile (\ref{duediciassette}). Orange diamond:  Analytic profile
(\ref{duediciannove}).}
{\vspace*{-2pt}}
\end{figure}

To see the differences between the two profiles, we have taken in
Figure~\ref{fignove} a slice at $x'=-250$ of the in-out region
($x'<0,x>0$) of Figure~\ref{figotto}. {In this and in the following
figures the extrema of the correlations bands will appear as peaks}.
The large main peak corresponding to the (i) {correlation band} is
clearly visible but the peak height and location are offset for the two
profiles. The main negative peak for the analytic profile appears
slightly smaller and shifted to the left as compared to that of the
original one. The opposite occurs for the smaller secondary peak
corresponding to the {band} (ii) as seen in Figure~\ref{figdieci}, where
we have magnified the scale to better appreciate this point. More
striking is the relative difference appearing in the negative peak
corresponding to the {band} (iii) in the in-in region, see
Figure~\ref{figundici}. The backscattering also affects the main peak.
In Figure~\ref{figdodici}a comparison is made for the two profiles
with the no-backscattering approximation Equation (\ref{treotto}). The
differences are more significant for the original profile.

\begin{figure}[t!]
{\vspace*{2pt}}
\includegraphics{fig13}
{\vspace*{2pt}}
\caption{\label{figdieci}A comparison of the secondary peak at
$x^\prime=-250$ when one point is in the interior and one point in in
the exterior of the analog black hole.  Blue dot: Original profile
(\ref{duediciassette}). Orange diamond:  Analytic profile
(\ref{duediciannove}).}
\end{figure}

\begin{figure}[t!]
{\vspace*{2pt}}
\includegraphics{fig14}
{\vspace*{2pt}}
\caption{\label{figundici}Density--density correlation function for both
points in the interior at a fixed ${x^\prime =400}$. Blue dot: Original
profile (\ref{duediciassette}). Orange diamond:  Analytic profile
(\ref{duediciannove}).}
{\vspace*{2pt}}
\end{figure}

\begin{figure}[t!]
\includegraphics{fig15}
\caption{\label{figdodici}A comparison of the main peak at
$x^\prime=-250$ when one point is in the interior and one point in in
the exterior of the analog black hole. Left: Blue(Solid): Curve for the
analytic expression~(\ref{treotto}). Blue dot: Numeric data for
original profile (\ref{duediciassette}). Right: Orange (solid): Curve
for the analytic expression~(\ref{treotto}). Orange diamond:  Numeric
result for analytic profile~(\ref{duediciannove}).}
\end{figure}

Finally, it is interesting to compare the ratios of the heights of each
of the two minor peaks to that of the main peak.   The ratio of the
height of the positive minor peak with one point inside and one point
outside the horizon to that of the main peak for the original profile
is $0.0293$.   For the analytical profile it is $0.0302$.  The ratio of
the height of the negative peak when both points are inside the horizon
to the main peak for the original profile is $0.261$.  For the
analytical profile it is $0.191$.  In both cases there is agreement in
the first digit only, so differences in the profiles lead to relatively
significant differences between the two profiles.

\section{Scaling}

There is a scaling related to the surface gravity $\kappa$ that works
for both sound speed profiles used in this paper,
(\ref{duediciassette}) and (\ref{duediciannove}).  It is
\def\theequation{\thesection.\arabic{equation}}
\setcounter{equation}{0}
{\begin{subequations}
\begin{eqnarray} \bar{\w} &=& \frac{\w}{\kappa}, \Seqnsplit
    \tau &=& \kappa t,  \Seqnsplit
     \xi &=& \kappa x . \label{xi-def}
\end{eqnarray} \label{scaled-quantities} 
\end{subequations}}\unskip
It is easy to see that for this scaling both sound speed profiles,
written in terms of $\xi$, are independent of $\kappa$.

In general, for any sound speed profile that, when written in terms of
$\xi$, is independent of $\kappa$ one can
substitute~(\ref{scaled-quantities}) into the Boulware and Kruskal modes of
(\ref{dueundici}) and show that they both scale as $\kappa^{-1/2}$.
Using these results, one can show that the two-point function
(\ref{trecinque}) is independent of $\kappa$. Then one finally has from
(\ref{tretre}) that
{\begin{equation} 
G_2(T,x;T',x') = \kappa^2 \; G_2(\bar{T},\xi;\bar{T}',\xi') \label{scaled-G2}. 
\end{equation}}\unskip


This means that the heights and depths of the correlation peaks are
larger for larger values of $\kappa$.  Since $x = {\xi}/{\kappa}$,
the widths of the correlation peaks in terms of the space coordinate
$x$ are narrower for larger values of $\kappa$. See
Figures~\ref{figtredici}, \ref{figquattordici}.

\begin{figure}[t!]
\includegraphics{fig16}
\caption{\label{figtredici}Scaling present in the
density--density correlation function: Left: Result using the original
profile (\ref{duediciassette}) with $\sigma_v=8$ with axis limits at
$-400\leq x \leq 400 $. Right:  Result using the Original profile
(\ref{duediciassette}) with $\sigma_v=4$    with axis limits at $-200\leq
x \leq 200 $.  {The green-dashed lines highlight the main band (i) by
showing the locations of two parts of its negative correlation peak. 
The black-dashed and brown-dashed lines do the same for the secondary
positive band (ii) and the secondary negative band (iii) respectively. 
Regions shaded in gray are outside of the ranges of the
plots}.}
\end{figure}

\begin{figure}[t!]
\includegraphics{fig17}
\caption{\label{figquattordici}Comparison of the main peak in analytic
and numerical results for  with $\sigma_v=8$ and $\sigma_v=4$ along a
slice at $x=-100$. Blue (solid ): Theoretical curve (\ref{treotto}) for
the original profile (\ref{duediciassette}) with $\sigma_v=8$.  Blue
diamond: Numerical result for the original profile
(\ref{duediciassette}) with $\sigma_v=8$. Red (solid): Theoretical
curve for original profile (\ref{duediciassette}) with $\sigma_v=4$.
Red square: Numerical result for the original profile
(\ref{duediciassette}) with $\sigma_v=4$.}
\end{figure}

\section{Conclusions}

While awaiting the direct detection of the thermal phonons, the
correlations bands and their related peaks in the density--density
correlation function are the major tool to experimentally \mbox{investigate}
the analogue of Hawking radiation in a sonic BH formed by a BEC. Of the
three characteristic bands only one, the main one predicted
in~\cite{paper1}, has been observed so far.  The detection of the other
much smaller two, whose existence was predicted by R. Parentani,
represents the next challenge for the experimentalists in this field.

{In this paper we have investigated the details of the three
characteristic bands for two specific sound velocity profiles.   The
profiles we use have flow speeds that are constant and sound speeds
that vary with position.  To date, the experimental profiles have had
both flow and sound speeds that vary with position.  However, the three
bands are expected to be very robust in the sense that they should
occur for any profiles in which the sound and flow speed profiles are
effectively one dimensional, result in a single sonic horizon, and
approach constant values far from that horizon}. 

{The goal of this work was to show how sensitive the correlations are
to differences in the sound speed profile.  We find that significant
differences occur for each of the characteristic bands even though the
sound speed profiles have the same asymptotic and horizon limits. 
Since the profiles are quite similar, this implies that one would
expect a significant amount of sensitivity to the details of the
experimental profiles as well}. 

{Future improvements in the precision of the experiments, including 
hopefully  the detection of the secondary  bands, will allow for a more
detailed analysis of the experimental results.  This in turn should
allow us to determine the extent to which the gravitational analogy and
the corresponding quantum field theory in curved space calculations can
be trusted to explain the fine structure features of the correlations
in the Hawking radiation for a BEC analogue black hole}.


\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgments}

PRA was supported in part by the National Science Foundation under
grant No. PHY-2309186.  Some of the numerical work was done using the
WFU DEAC Cluster; we thank the WFU Provost's Office and Information
Systems Department for their generous support. AF acknowledges
partial financial support by the Spanish Grants PID2020-116567GB-C21,
PID2023-149560NB-C21 funded by MCIN/AEI/10.13039/501100011033, and by
the Severo Ochoa Excellence Grant CEX2023-001292-S.

\back{}

\appendix{}

\def\appendixlabel{Appendix}

\section{Bogoliubov coefficients relating the Kruskal and Boulware modes}\label{appendixA}
\setcounter{equation}{0}

We report here the Bogoliubov coefficients appaearing in Equation (\ref{duesedici}),
{\begin{eqnarray}
 \label{duesedicibis}
 u_H^{K}(\omega_K,x)&=&\int_0^\infty \mathrm{d} \omega \left[\alpha_{\omega_K \omega}^R u_H^{R}+\beta^{R}_{\omega_K \omega} u_H^{R*}+
 \alpha_{\omega_k \omega}^L u_H^{L}+\beta^{L}_{\omega_k \omega}
u_H^{L*}\right],\Seqnsplit  
      \alpha^R_{\omega_k \omega} &=& \frac{1}{2 \rmpi \kappa} \sqrt{\frac{\w}{\w_K}} \Gamma(-\mathrm{i} \w/\kappa) \; \left(-\mathrm{i} \frac{\w_K}{\kappa} \right)^{\mathrm{i} \w/ \kappa}, \nonumber \\
\beta^R_{\omega_k \omega} &=& \frac{1}{2 \rmpi \kappa} \sqrt{\frac{\w}{\w_K}} \Gamma(\mathrm{i} \w/\kappa) \; \left(-\mathrm{i} \frac{\w_K}{\kappa} \right)^{-\mathrm{i} \w/ \kappa}, \nonumber \\
      \alpha^L_{\omega_k \omega} &=& \frac{1}{2 \rmpi \kappa} \sqrt{\frac{\w}{\w_K}} \Gamma(\mathrm{i} \w/\kappa) \; \left(\mathrm{i} \frac{\w_K}{\kappa} \right)^{-\mathrm{i} \w/ \kappa}, \nonumber \\
\beta^L_{\omega_k \omega} &=& \frac{1}{2 \rmpi \kappa} \sqrt{\frac{\w}{\w_K}} \Gamma(-\mathrm{i} \w/\kappa) \; \left(\mathrm{i} \frac{\w_K}{\kappa} \right)^{\mathrm{i} \w/ \kappa} .   \label{alphas-betas}
\end{eqnarray}}\unskip

\printbibliography
\refinput{crphys20240819-reference.tex}

\end{document}
