\section{Introduction}
\label{sec:Introduction}

The theoretical prediction of gravitational waves (GWs) originates
in 1893 when Heaviside first discussed the possibility of their
existence. In 1916 Einstein predicted their existence
\cite{Einstein:1916cc,Einstein:1918btx} in the context of general
relativity. In the linearized weak-field approximation, he found
that his equations had transverse wave solutions travelling at the
speed of light \cite{Ellis:2016rrr} produced by the time
dependence of the mass quadrupole moment of the source
\cite{LeTiec:2016sgy}. Einstein realized that GW amplitudes would
be small and up until 1957, there was debate about the physical
relevance of their existence \cite{Saulson:2010zz}.\

Nevertheless, the discovery of the binary pulsar system PSR
B1913+16 by Hulse and Taylor \cite{Hulse:1974eb} and subsequent
observations of its energy loss by Taylor and Weisberg
\cite{Taylor:1982zz} demonstrated indirectly the existence of GWs.
This discovery, along with subsequent related analysis
\cite{Press:1972am}, led to the recognition that a possible direct
detection and analysis of GWs could reveal interesting properties of various relativistic systems and could also provide new tests of
general relativity, especially in
the strong-field regime.\\

Recently, Abbot et al
\cite{TheLIGOScientific:2016agk,Abbott:2016blz}, reported the
first direct detection of GWs emitted by
a binary black hole (BBH) system merging to form a single black hole (BH).
Their observation provides a direct window to the properties of
space time in the strong-field limit and is consistent with
predictions of general relativity for the nonlinear dynamics of
highly disturbed BHs. The announced beautiful discovery is the
result of great efforts for a century by several scientists
\cite{Abbott:2016blz} (and references therein). It is a great
investigation, because we have now one more window to the Universe
and one more confirmation of the theory of general relativity.
Such GW observations can be used to test the equivalence principle
\cite{Kahya:2016prx,Wu:2016igi,Liu:2016edq,Yunes:2016jcc}, test
the propagation of GWs
\cite{Yunes:2016jcc,Branchina:2016gad,Schreck:2016qiz,Arzano:2016twc,Bicudo:2016pps,Collett:2016dey,Blas:2016qmn,Garcia-Bellido:2016zmj},
test the validity of general relativity
\cite{Konoplya:2016pmh,Moffat:2016gkd,Vainio:2016qas}, constrain
early cosmological phase transitions
\cite{Dev:2016feu,Jaeckel:2016jlh}, probe the quantum structure of
black holes \cite{Giddings:2016tla}
or the connection between dark matter and primordial black holes \cite{Sasaki:2016jop,Bird:2016dcv,Clesse:2016vqa}. \\

The recent direct discovery of the GWs has been achieved by the
LIGO/Virgo collaboration associating the GW 150914 event
\cite{TheLIGOScientific:2016agk,Abbott:2016blz,Maselli:2016ekw}
to the coalescence of a BBH. This binary detection suggests that
BBH masses and merging rates may be higher than estimated
previously. The rates however, are in agreement with more recent
estimates, obtained with a population synthesis approach,
predicting the early formation of detectable BBH
\cite{Dominik:2012kk,Dominik:2014yma}. Thus, the
stochastic gravitational waves background (SGWB) produced by merging cosmological BBH sources
could be larger than previously assumed \cite{Maselli:2016ekw} (and references therein) and
may be detectable by advanced detectors
\cite{TheLIGOScientific:2016wyq}.\\

A
stochastic background of relic gravitational waves (RGWs) is predicted by inflationary models  \cite{Nakama:2015nea,Nakama:2016enz} and
has been well studied \cite{Starobinsky:1979ty,Abbott:1985cu}. The power spectrum of
relic gravitational wave background reflects the physical
conditions in the early Universe thus providing valuable
information for cosmology \cite{Dufaux:2007pt}. This spectrum
is determined by the early stage of inflation as well as by the expansion properties of the subsequent
epochs, including the current one. The calculation of the spectrum
\cite{Allen:1987bk,Henriques:2003ga} was initially performed for a
currently decelerating universe. However, it is now well known
that the universe expansion is currently accelerating
\cite{Riess:1998cb,Perlmutter:1998np} and since the evolution of
RGWs depends on the expanding background space time,  the spectrum of RGWs should be modified accordingly. This
modification was confirmed and studied in Refs
\cite{Zhang:2005nw,Izquierdo:2004gk} using the well-known
formulation of GWs in an expanding Universe
\cite{Grishchuk:1974ny} and an approximation of the scale
factor $a(\tau)$ in the context of a sequence of successive expansion epochs,
including the current stage of accelerating expansion. It was
found that the current accelerating expansion induces
modifications in both the shape and the amplitude of the RGW
spectrum.

Since existence of RGWs is a key prediction of the
inflationary models, their detection could provide
 evidence that inflation actually took place. Thus,
it is important to accurately calculate the expected detailed form
of the RGW spectrum. Calculations related to RGWs in an
accelerating Universe have been performed \cite{Zhao:2006eb} and a
numerical method has been developed to calculate the power
spectrum of the RGWs. Late evolution of RGWs in coupled dark
energy models has been examined extensively in Ref. \cite{Almazan:2013bea}.\\

Even though the effects of cosmological expansion on GWs have been
investigated mainly in the context of RGWs, these effects are
relevant in all cases when the source is located at cosmologically
large distances from the observer (redshift $z\geq 0.1$). The GW
150914 ($z=0.09$) event is in the limit of such distances and
therefore, the effects of cosmic expansion may be relevant.  Thus
a wide range of studies have investigated the effects of
cosmological expansion of GWs from a variety of viewpoints
including the effects of expansion on the GW group and phase
velocities  \cite{Balek:2007hh,Brillouin}, mathematical aspects
and exact solutions
\cite{Fabris:1998ak,Alekseev:1995uw,Tauber:1985yp,Waylen:1978dx,Tamayo:2015qla},
quantum and thermodynamic properties of GWs
\cite{Arzano:2016twc},\cite{IzquierdoSaez:2005ja}, general
properties \cite{Hartnett:1900zzc,Waylen:1978dx,Jackson:2008yv}
nonlinear effects \cite{Ikeda:2015hqa}, properties of the GW
energy momentum tensor \cite{Su:2012hf}, collision of GWs with
electromagnetic waves
\cite{Alekseev:2015wbn}, evolution of GWs in gravitational plasma \cite{Baptista:1982dc} etc .\\

Even though these studies have properly taken into account the
expansion of the background metric, they have not taken into
account the effects of the gravitational field of  mass
distributions on the evolution of the GWs. Such a gravitational
field combined with the expanding background may induce new
observable effects on the spectrum of propagating gravitational
waves affecting the amplitude and the frequency of such waves (due
to gravitational time delay) \cite{Zeldovich:1992if}.

Assuming spherical symmetry, the background metric around a point
mass embedded in an expanding Friedmann-Lemaitre-Robertson-Walker(FLRW) cosmological background
is well approximated by the McVittie \cite{McVittie:1933zz,9780511535185} spacetime. Such a metric is
further simplified in the Newtonian limit and has been used as the background metric for the investigation of bound system geodesics
in phantom and quintessence cosmologies \cite{Nesseris:2004uj,Antoniou:2016obw,Faraoni:2007es,Nolan:2014maa}.
This metric can also be used as a background for the propagation of GWs in order to investigate the influence of a mass
distribution of a GW propagating in an expanding cosmological background.

In the present analysis we address the following question: 'What
are the weak field effects of a point mass on a multipole spherical wave
component of a GW evolving in an expanding background in the
vicinity of the mass?'. In particular we numerically solve the
dynamical equation for the evolution of GWs in the background of
the Newtonian McVittie metric and identify the effects induced by
the point mass on the amplitude and frequency of the evolving
GW as a function of the parameters determining the mass for fixed
background expansion rate. As a test of our analysis, in the zero
mass limit, our numerical solution reduces to the well known
analytic solution of a GW evolving in an expanding background.

The structure of this paper is the following: In the next section
we review the wave equation and the behavior of the GW in a
homogeneous-isotropic expanding background. We also derive the
gravitational wave equation in a background metric corresponding
to the Newtonian limit of the McVittie metric. In section $III$ we
solve numerically the wave equation in the Newtonian McVittie
background and identify the new features induced in the GW by the
presence of the point mass. Finally in section $IV$ we conclude,
summarize and discuss possible extensions on this analysis.

\section{Gravitational Waves in Expanding Universe in the Presence of Point Mass}
We first briefly review the propagation evolution of a plane GW in the
$\hat{z}$ direction (the direction of the wavevector $\vec{k}$)
with tensor perturbations in the $x-y$ plane. The perturbations to
the metric are described by two functions, $h_+$ and $h_\times$,
assumed small. We use the FRW metric in cartesian coordinates with
the components $g_{00}=1$, zero space-time components $g_{0i}=0$,
 and set $c=1$. The spatial part of the
metric is of the form: \be g_{ij}= -a^2(t)\begin{pmatrix}
  1+h_+ & h_\times & 0 \\
  h_\times & 1-h_+ & 0 \\
  0 & 0 & 1\\
\end{pmatrix}
\label{metric1}\ee

The perturbation tensor $H_{ij}$ is symmetric, divergenceless,
traceless and has the form:  \be H_{ij}= \begin{pmatrix}
  h_+ & h_\times & 0 \\
  h_\times & -h_+ & 0 \\
  0 & 0 & 1\\
\end{pmatrix}
\label{tensor}\ee From the Einstein equations for tensor
perturbations, it is easy to derive a set of equations governing
the evolution of the tensor variables $h_+$ and $h_\times$. We
write the FRW metric in cartesian coordinates and in conformal time $\tau$  (defined by $dt=a d\tau$), in the form \be
ds^2=a^2(\tau)[d\tau^2-(\delta_{ij}+H_{ij})dx^{i}dx^{j}]
\label{metrconf}\ee The dynamical equation determining the
evolution of the GWs is of the form: \be
 \Box H_{ij}= \partial_{\mu}(\sqrt{-g}\partial^{\mu}H_{ij}(\vec{r},\tau)) =0 \label{waveeq}\ee
Since all components of the tensor perturbations evolve in accordance with the same wave equation (\ref{waveeq}) we may set $H_k\equiv H_{ij}$.
Without loss of generality we assume propagation in the $z$
direction and thus we use the ansatz:
 \be  H_{k}(\tau,z)=h_{k}(\tau) e^{\pm ikz} \label{waveans}\ee  Using eq. (\ref{waveans}) in (\ref{waveeq}) we find the dynamical equation for the evolution of gravitational waves in conformal time in an FRW background as
\be h''_{k}+2\frac{a'}{a}h'_{k}+k^2h_{k}=0 \label{waveeq1}\ee
where the prime $'$ denotes the derivative with respect to
conformal time. Notice that  all of the perturbation tensor
components obey the same equation. We introduce a rescaling of
conformal time as $\bar{\tau}=k\tau$ and thus it becomes clear
that \be h_k(\tau)=h(k\tau) \label{hresc} \ee The rescaling
expressed by eq. (\ref{hresc}) can only be made in conformal time
provided that the scale factor is a power law $a(\tau)\sim
\tau^\alpha$. In the radiation dominated epoch we have $\alpha=1$
and during the matter dominated era $\alpha=2$. The wave solution
(\ref{waveans}) can be written in spherical coordinates as: \be
H_k(\tau,\rho,\theta)=h(k\tau)e^{\pm ik\rho\cos\theta}
\label{hsphercoord}\ee The spectrum of the GWs may be obtained as
~\cite{Zhang:2006mja}: \be P(k,\tau)=\frac{4\; l_{Pl}}{\sqrt{\pi}}k
\mid h(k\tau)\mid \label{powersp}\ee
where $l_{Pl}\sim \sqrt{G}$ is the Planck length.
The plane wave of equation (\ref{hsphercoord}) can be expanded in
spherical waves as: \be e^{i k \rho\cos{\theta}}=\sum_{l=0}^{\infty}i^l(2l+1)j_{l}(k\rho)P_l(\cos{\theta})\label{planeexp}\ee where $j_l(x)$ are the
spherical Bessel functions and $P_l(\cos{\theta})$
are Legendre's polynomials. Thus, the partial spherical GW is (at
order $l$) \be H_{ij}(\tau,\rho,\theta) \sim
h(k\tau)j_{l}(k\rho) P_l(\cos{\theta})\label{planeordl}\ee

After rescaling, the dynamical equation (\ref{waveeq1}) is written
as: \be
 h''(k\tau)+2\frac{a'}{a}h'(k\tau)+h(k\tau)=0 \label{rescwaveeq}\ee
where the prime $'$ now denotes differentiation with respect to
the rescaled conformal time $k\tau$.

Assuming a power law for the background scale factor as
$a(\tau)\sim \tau^\alpha$, the solution of the wave equation
(\ref{rescwaveeq}) can be written in terms of incoming and outgoing waves (Hankel functions) as \be
h(k\tau)=\frac{1}{a(\tau)}(\tilde{A}_{k}\sqrt{k\tau}H^{(1)}_{\alpha-1/2}(k\tau)+\tilde{B}_{k}\sqrt{k\tau}H^{(2)}_{\alpha-1/2}(k\tau))\label{wavesol1}\ee
where $H^{(1)}$, $H^{(2)}$ are the Hankel functions and
$\tilde{A}_{k}$, $\tilde{B}_{k}$ are arbitrary constants which may
depend on $k$ and are determined by the initial conditions. This
solution may also be written in terms of standing waves as \be
h(k\tau)=(k\tau)^{\frac{1}{2}-\alpha}(A_{k}J_{\alpha-1/2}(k\tau)+B_{k}Y_{\alpha-1/2}(k\tau))\label{wavesol2}\ee
where $J$,$Y$ are the Bessel functions of first and second kind
respectively and $A_k=\tilde{A}_k k^\alpha$, $B_k=\tilde{B}_k
k^\alpha$. Thus, for a power law scale factor the spherical GW is
\begin{eqnarray}
&&H_k(\tau,\rho,\theta)=(k\tau)^{\frac{1}{2}-\alpha}(A_{k}J_{\alpha-1/2}(k\tau)+B_{k}Y_{\alpha-1/2}(k\tau))\nonumber\\&&j_{l}(k\rho)P_{l}(cos\theta)\label{wave}\end{eqnarray}

As a warm up exercise before the introduction of a point mass in
the metric, we now rederive the solution (\ref{wavesol2}),
(\ref{wave}) starting from the FRW metric in spherical coordinates
\begin{equation}\label{frwsphercoords}
ds^2=a(\tau)^2\{d\tau^2-[d\rho^2+\rho^2(d\theta^2+\sin^2{\theta}d\phi^2)]\}
\end{equation}

Because of the azimuthal symmetry, the solution doesn't depend on
the variable $\phi$ we will seek solutions of the Eq.
(\ref{waveeq}) of the form
\begin{equation}\label{waveansspher} H_k(\tau,\rho,\theta,\phi)=
f(\tau)R(\rho)P_{l}(cos\theta) \end{equation} From the Eqs.
(\ref{waveeq}),(\ref{waveansspher}), after separation of variables
we find
\begin{equation}\label{rdiffeqsep}
\frac{1}{R}[\frac{d^2 R}{d \rho^2}+\frac{2}{\rho}\frac{d R}{d
\rho}-\frac{l(l+1)}{\rho^2}]=-k^2
\end{equation}
and
\begin{equation}\label{fdiffeqsep}
\frac{1}{f}[\frac{d^2 f}{d \tau^2}+\frac{2a'}{a}\frac{d f}{d
\tau}]=-k^2
\end{equation}
where $k^2$ is arbitrary constant. As expected eq.
(\ref{fdiffeqsep}) is identical to eq. (\ref{waveeq1}) while eq.
({\ref{rdiffeqsep}) is the spherical Bessel equation  with
acceptable solution
\begin{equation}\label{reqjl}
R_{l}(\rho)=A_{l}j_{l}(k\rho)\end{equation}  We thus reobtain
the general solution in spherical coordinates (\ref{wave}).

We can now generalize the above analysis to investigate the
behavior of GWs when they interact with a point mass $M$. In the
presence of a point mass and cosmological expansion, the
appropriate background metric is the McVittie metric. In the
Newtonian limit, using comoving coordinates the McVittie metric is
\cite{Antoniou:2016obw,Nesseris:2004wj}:
\begin{equation}\label{nmcvitttiemetric} ds^2=(1-\frac{R_s}{\rho
a(t)})dt^2-a(t)^2[d\rho^2+\rho^2(d\theta^2+\sin^2{\theta}d\phi^2)]
\end{equation}
where  $R_s=2GM$. The angular variable $\theta$ separates and thus
we use the perturbation ansatz
\begin{equation}\label{hanzmass} H_k(t,\rho,\theta,\phi)=
Q(t,\rho)P_{l}(cos\theta)
\end{equation}

\begin{figure*}[ht] \centering
\begin{center}
$\begin{array}{@{\hspace{-0.10in}}c@{\hspace{0.0in}}c}
\multicolumn{1}{l}{\mbox{}} & \multicolumn{1}{l}{\mbox{}} \\
[-0.2in] \epsfxsize=3.4in \epsffile{fig0a.eps} & \epsfxsize=3.4in
\epsffile{fig0b.eps} \\
\end{array}$
\end{center}
\vspace{0.0cm} \caption{\small a: A superposition of the analytic
solution with the numerical simulation initial condition taken at
$\tau=1$. The spherical wave with $l=6$ and scale factor
$a(\tau)=\tau$ is shown.   b: The numerically evolved solution (dashed red line) for
the gravitational wave $Q(\tau,\rho)$ at $\tau=16.5$ with $R_s=0$, is in
excellent agreement with the corresponding analytic evolved
solution (blue line). This is a test of the quality of the numerical
solution.} \label{fig1}
\end{figure*}

Using the background metric (\ref{nmcvitttiemetric}) and the
ansatz (\ref{hanzmass}) in the gravitational wave equation
(\ref{waveeq}) we obtain the dynamical equation for $Q(t,\rho)$ as
\begin{eqnarray}\label{qpde}
&&\frac{\partial^2 Q}{\partial \rho^2}+\frac{2}{\rho}(1-\frac{3 R_s}{4a\rho})(1-\frac{R_s}{a\rho})^{-1}\frac{\partial Q}{\partial \rho}-\frac{l(l+1) Q}{\rho^2}=\nonumber\\
&&\frac{a^2}{1-\frac{R_s}{a\rho}}[\frac{\partial^2 Q}{\partial
t^2}+\frac{3\dot{a}}{a}(1-\frac{7
R_s}{6a\rho})(1-\frac{R_s}{a\rho})^{-1}\frac{\partial Q}{\partial
t}] \end{eqnarray}
 Assuming that
\begin{equation}\label{smallrs}
\frac{R_s}{a\rho}\ll 1
\end{equation}
and keeping terms $R_s/a\rho$ only in first order we can write eq.
(\ref{qpde}) in conformal time as
\begin{eqnarray}\label{qpdeconf}
&&(1+\frac{R_s}{a\rho})\frac{\partial^2 Q}{\partial
\tau^2}+\frac{2a'}{a}(1+\frac{3R_s}{4a\rho})\frac{\partial
Q}{\partial \tau}\nonumber\\
&&=\frac{\partial^2 Q}{\partial
\rho^2}+\frac{2}{\rho}(1+\frac{R_s}{4a\rho})\frac{\partial
Q}{\partial \rho}-\frac{l(l+1) Q}{\rho^2}
\end{eqnarray}

Eq. (\ref{qpdeconf}) is not separable and it is not tractable
analytically in a simple manner. As expected, in the limit of zero
mass it separates and reduces to eqs. (\ref{rdiffeqsep}) and
(\ref{fdiffeqsep}).

In the next section we integrate eq. (\ref{qpdeconf}) numerically
and investigate the dependence of the solution on the values of
the parameter $R_s$. It will be seen that as the wave approaches
the point mass it experiences two types of distortion
\begin{itemize}
\item gravitational time delay and increase of its period in conformal cosmological time. \item
Its amplitude increases in comparison to the amplitude it would
have in the absence of the point mass. \end{itemize}

According to general relativity the expected period of the wave at
a comoving distance $\rho$ from the point mass, as measured by an
observer at infinity, is
 \be
T=\frac{T_0}{\sqrt{1-\frac{R_s}{a\rho}}}\label{periodatrho}\ee
where $T_{0}$ is the corresponding period at infinity (or in the
absence of the mass). For small mass or large distance from the
source the increase of the period is \be \frac{\Delta
T}{T_0}=\frac{1}{2a\rho}R_s\label{periodincrease}\ee where $\Delta
T=T-T_0$ is the difference of the wave periods with and without
the presence of the mass. The validity of eq.
(\ref{periodincrease}) for the GW in the vicinity of a point mass
will be demonstrated numerically in the next section.

\section{Numerical Analysis}
\label{sec:Section 3}

In order to keep the analogy with the massless case $R_s=0$ we
rescale the dynamical equation (\ref{qpdeconf}) to dimensionless
form using the wavenumber $k$ defining $k\rho=\bar{\rho}$,
$k\tau=\bar{\tau}$. In this case we have an additional physical
dimensionless parameter: \be \bar{R_s}=k R_s \label{rsdim}\ee In
the numerical analysis that follows we only use dimensionless
quantities even though we will omit the bar in what follows.

\begin{figure*}[ht]
\centering
\begin{center}
$\begin{array}{@{\hspace{-0.10in}}c@{\hspace{0.0in}}c}
\multicolumn{1}{l}{\mbox{}} & \multicolumn{1}{l}{\mbox{}} \\
[-0.2in] \epsfxsize=3.4in \epsffile{fig1a.eps} & \epsfxsize=3.4in
\epsffile{fig1b.eps} \\
\epsfxsize=3.4in \epsffile{fig1c.eps} & \epsfxsize=3.4in
\epsffile{fig1d.eps}
\end{array}$
\end{center}
\vspace{0.0cm} \caption{\small The evolution of the profile of a
partial spherical gravitational wave with $l=6$, $R_s=5$ (red
dashed line) in comparison with the corresponding free solution
($R_s=0$, blue continuous line).
 The free wave reaches its maximum first (Fig. 1b) while the wave in the presence of the point mass shows a
  delay in reaching its maximum (Fig. 1c) due to gravitational redshift. The wave in the presence of the mass
  has a higher amplitude (compare 1b with 1c) in the vicinity of the mass and the phase difference increases
  with time in the vicinity of the mass (Fig. 1d).}
\label{fig2}
\end{figure*}

We solve numerically eq. (\ref{qpdeconf}) with initial conditions
corresponding to a standing gravitational wave evolving in a homogeneous
FRW spacetime ($R_s=0$) using eq. (\ref{wavesol2}) with $B_k=0$
starting the evolution at $\tau=1$. This is equivalent to assuming
that the point mass appears at $\tau=1$. For definiteness we set
$\alpha=1$ or $a(\tau)\sim \tau$ corresponding to an expanding
background in the radiation era. The boundary conditions are
imposed for $Q$ and for its first derivative at large $\rho$ where
the effects of the point mass are negligible and also correspond
to a standing GW evolving in a homogeneous FRW spacetime ($R_s=0$) using
eq. (\ref{wavesol2}) with $B_k=0$.

The Bessel function boundary condition (\ref{wavesol2})-(\ref{wave}) we
have used for the wave equation at large distance from the source, describes a standing
GW which however can be expressed as a superposition of two propagating modes (Hankel functions). The asymptotic behaviour of Hankel functions, which is proportional to $e^{i\tau}$, corresponds to a propagating GW, while the asymptotic behavior of Bessel functions is proportional to $\cos(\tau)$ and corresponds to a standing GW.

We stress that since we have made the Newtonian approximation our results are reliable in
regions where the weak field condition (\ref{smallrs}) is
satisfied.

We thus construct numerically the solution $
Q(\rho,\tau,R_s,l,\alpha)$ and compare with the corresponding
analytical solution  $ Q(\rho,\tau,R_s=0,l,\alpha)$. We have
tested our numerical evolution by verifying that the numerical
solution for $R_s=0$ agrees with the corresponding analytical
solution at a level better that $1\%$ (Fig. \ref{fig1}).

Following the above comments about the boundary condition we fix
$Q$ (and its derivative) at the boundary $\rho_{end}>500$ using eq. (\ref{wave}) with $\alpha=1$, $A_k=1$ and $B_k=0$ as \be
Q(\tau,\rho_{bound})=\sqrt{\frac{2}{\pi}}\frac{\sin(\tau)}{\tau}j_{l}(\rho_{bound})\label{boundary}\ee
Similarly, the initial conditions set at $\tau_i=1$ are: \be
Q(\tau_i,\rho)= \sqrt{\frac{2}{\pi}}\frac{\sin(\tau_i)}{\tau_i}j_{l}(\rho)\label{initial1}\ee
and \be \frac{\partial Q(\tau_i,\rho)}{\partial
\tau}=\frac{\partial}{\partial \tau}[\sqrt{\frac{2}{\pi}}\frac{\sin(\tau)}{\tau}j_{l}(\rho)]|_{\tau=\tau_i}\label{initial2}\ee
In addition to the test of the validity of numerical solution
presented in Fig. \ref{fig1} we have performed other tests
including the verification of the independence of the numerical
solution from the location of the boundary for $\rho_{bound}>200$.

We have solved the partial differential equation (\ref{qpdeconf})
for various values of $l$ with results that are qualitatively
similar. For definiteness we present in Fig. \ref{fig2} the
solution corresponding to $l=6$ for $R_s=5$ superposed with the
corresponding solution for $R_s=0$ in order to identify the new
features introduced in the evolution of the GW by the presence of
the point mass.

There are three main features to observe in Fig.
\ref{fig2}. First, the waves are practically identical far away
from the point mass as expected. Second, there is a time delay for
the wave in the presence and in the vicinity of the point mass
(Fig. 2b upper right). Third, the amplitude of the wave in the
presence and in the vicinity of the mass increases (compare Fig.
2b (upper right) with Fig. 2c (lower left)).

The main effect of the expansion is to reduce the amplitude of the
gravitational wave by a factor proportional to the scale factor in the absence of the
mass. This is shown in Fig. \ref{fig3} which shows that the amplitude multiplied by the
scale factor remains constant in the absence of the mass (blue oscillating line has
constant amplitude) for the particular time dependence of the scale factor considered
($a(\tau)\sim \tau$). In the presence of the mass however, the decrease of the amplitude due to
the expansion is less efficient (red line) and the product of the amplitude times the
scale factor increases slowly with time.

The gravitational wave time evolution shown in Fig. \ref{fig3} corresponds to $\rho=7.9$ (closest
maximum amplitude to the mass for $l=6$) for $R_s=5$ (red dashed
line) and is superposed with the corresponding evolution for $R_s=0$
(blue continuous line). This plot demonstrates the relative
(linear) increase of the  amplitude with time, as well as the
increased period of the wave in the presence of the mass. It also
demonstrates (as discussed above) the well known fact that the wave amplitude in the
absence of the mass ($R_s=0$) is inversely proportional to the
scale factor (the blue wave has a constant amplitude).

\begin{figure}[!t]
\centering
\vspace{0cm}\rotatebox{0}{\vspace{0cm}\hspace{0cm}\resizebox{0.49\textwidth}{!}{\includegraphics{fig2.eps}}}
\caption{ The time evolution of the first spatial maximum (at
$\rho=7.9$) of the  partial spherical wave with $l=6$, $R_s=5$
(red line)  in comparison with the corresponding free solution
($R_s=0$, blue continuous line) at the same spatial point. The
free wave reaches its maximum first (Fig. 2b) while the wave in
the presence of the point mass shows a delay in reaching its
maximum (Fig. 2c) due to gravitational redshift. The wave in the
presence of the mass has an amplitude that increases with time as
indicated with the dashed red line that in tangent to the
gravitational wave maxima. As expected, the product $a(\tau) Q(\tau)$ is constant for the free wave in an expanding background. } \label{fig3}
\end{figure}

The effects of the gravitational time delay on the evolution of
the wave may also be demonstrated by plotting the power spectrum
obtained by a Fourier series expansion of the evolving in
conformal time numerical solution at $\rho=7.9$ in harmonic waves.

The finite time interval power spectrum may be defined through the expansion
\be
Q(\tau,\rho)=\frac{a_0}{2}+\sum \limits_{i=1}^n(a_n
\cos(n\tau)+b_n\sin(n\tau))\label{coef}\ee
as
\be P(n)\equiv \log\sqrt{a_n^2+b_n^2}\label{spectrum}\ee

We used a time interval of approximately  two
complete oscillations which corresponds to a time interval
$\tau \in [1,20]$ ($\tau_i=1, \tau_{max}=20$ as shown in Fig. \ref{fig4}).

As shown in Fig. \ref{fig4} the presence of the mass (red continuous line)
leads to an increase of the amplitude of low harmonics and
decrease of the amplitude of higher harmonics which is consistent
with the effects of gravitational time delay.  The exact
form of the spectrum clearly depends on the time interval
considered, however the qualitative feature of higher amplitudes
for lower frequencies persists for all time intervals. This
feature is more prominent for lower values of $\rho$.

\begin{figure}[!b]
\centering
\vspace{0cm}\rotatebox{0}{\vspace{0cm}\hspace{0cm}\resizebox{0.49\textwidth}{!}{\includegraphics{fig3.eps}}}
\caption{The time power spectra of the gravitational wave in the
presence (red line) and in the absence (blue line) of the mass. Notice that
lower frequencies have a higher amplitude for the wave in the
presence of the mass as expected due to the gravitational time
delay.} \label{fig4}
\end{figure}

In accordance with eq. (\ref{periodincrease}) the increase of the
period of the wave at a given distance from the mass is
proportional to the mass in the weak field approximation. This is
consistent with our numerical solution as shown in Fig. \ref{fig5} where we
show the relative increase of the period of the wave $\Delta
T/T_0$ at given distances $\rho$
from the mass ($\rho=7.9$ and $\rho=15.89$) for various values the parameter $R_s$ (points in
plot). In order to evaluate the relative change of the period $\Delta
T/T_0$ we use the time evolution of the wave perturbation as shown in Fig. \ref{fig3} to obtain the period of the wave in the
presence of the mass and the corresponding period in the absence of the mass. Superposed in Fig. \ref{fig5}
is the best fit straight line in each case. As
is theoretically expected there is a linear relationship in
accordance with eq. (\ref{periodincrease}). The correlation
coefficients of the points with the corresponding best fit straight line are
equal to $0.99$ indicating an excellent quality of fit.

The theoretically predicted slope is $\frac{1}{2a\rho}$ where the scale factor can
be taken as approximately constant and equal to its average value
during the wave period used to evaluate $\Delta
T/T_0$.

\begin{figure}[!t]
\centering
\vspace{0cm}\rotatebox{0}{\vspace{0cm}\hspace{0cm}\resizebox{0.49\textwidth}{!}{\includegraphics{fig5.eps}}}
\caption{The relative difference of the wave periods $\Delta
T/T_0$, where $T$ is the period in the presence of mass and $T_0$
is the period in the absence of mass, as a function of $R_s$. It
is clear that as the value of the variable $\rho$ increases, the
statistical slope of the curve decreases. This is an anticipated
result due to theoretical slope of the curve $\frac{1}{2a\rho}$.}
\label{fig5}
\end{figure}

In order to estimate the theoretical value of the scale factor, we
calculate the mean value $\bar{a}(\tau)$, in the time interval
$\tau_{1}-\tau_{2}$ of a single period, through the formula: \be
\bar{a}(\tau)=\frac{1}{\tau_{2}-\tau_{1}}\int_{\tau_1}^{\tau_2}a(\tau)d\tau=\frac{\tau_2+\tau_1}{2}\label{bara}\ee
 The observed
deviations by about $20\%$ between theoretically expected slope
and numerically obtained can be attributed to the approximations
we have made which include, the weak field assumption  ($R_s \ll
a\rho$ while in the cases considered $\frac{R_s}{a\rho} \leq
0.1$), the assumed constant scale factor for
the evaluation of the slope etc. As shown in Figs 2 and 3 the amplitude of the wave also increases
as the point mass is approached. A quantitative estimate of this
effect is shown in Fig. \ref{fig6} where we show the ratio of the
amplitudes of the waves $A/A_0$ in the presence of a mass ($A$)
and in the absence of the mass ($A_0$) for various values of the
parameter $R_s$, when $\rho=7.9$ and $\rho=15.89$. The best fit
straight line is also superposed on the points showing that a
linear relationship between $A/A_0$ and $R_s$ is a good
approximation.
\begin{figure}[!b]
\centering
\vspace{0cm}\rotatebox{0}{\vspace{0cm}\hspace{0cm}\resizebox{0.49\textwidth}{!}{\includegraphics{fig6.eps}}}
\caption{The ratio of the amplitudes of the waves $A/A_0$ in the
presence of a mass ($A$) and in the absence of the mass ($A_0$) as a function of
the parameter $R_s$, when $\rho=7.9$ and $\rho=15.89$.}
\label{fig6}
\end{figure}
The amplitude increases up to $10\%$ when $\rho=7.9$ and $R_s=10$,
while for $\rho=15.89$ and the same value of $R_s$, the increase is about $5\%$. Thus the
amplitude increase appears to vary inversely proportional with
$\rho$ which is consistent with the fact that the GW gains energy
as it enters regions of space with higher curvature.

\section{Conclusion}
\label{sec:Section 4}

The effects of a point mass on a GW evolving in an expanding
universe are determined by the mass $M$ and the physical distance
$a \rho$ of the wave from the mass through the expression
$\frac{R_s}{a\rho}\equiv\frac{2GM}{a\rho}$. In the context of a
perturbative weak field analysis we have demonstrated that a point
mass tends to increase the amplitude and the period of the GW
linearly with respect to $\frac{R_s}{a\rho}$. This result is
consistent with expectations based on gravitational time delay and
energy considerations.

Even though our numerical results were presented for the special
case of a radiation dominated cosmological background ($a(\tau)\sim \tau$) and a
specific multipole component of the wave ($l=6$) we have checked that their qualitative features persist for
all multipole components and cosmological backgrounds provided that
the weak field condition (\ref{smallrs}) is respected. Thus, even though we have considered specific spherical waves in this analysis, we anticipate that our
results can also describe a plane wave when expressed as a superposition of spherical
waves.

The time slicing we considered corresponds to the coordinate
time of the particular metric we used. This coordinate time is particularly interesting
and generic as it corresponds to the proper time of a static observer located far away
from the point mass or in the absence of the point mass. This is the standard cosmic
observer whose observations are consistent with the cosmological principle. Clearly a
different choice of time slicing would correspond to a different observer and would lead to a different metric and thus different
results.

From the results shown in Fig. \ref{fig5} and Fig. \ref{fig6}, we conclude that $T=T_0(1+\mu R_s)$ and $A=A_0(1+\nu R_s)$ where $\mu$ and $\nu$ are the slopes of the curves which are approximately equal. Thus we have demonstrated that the energy density of GWs which is proportional to $\omega^2 A^2$ has a weak dependence on $R_s$ in the context of our weak field approximation as long as the slopes $\mu$ and $\nu$ are approximately equal.

Our result has interesting implications for the calculation of the
RGW spectrum which currently assumes
\cite{Koh:2009cy,Corda:2009bx,Grishchuk:2007uz,Zhang:2008zx,Gogoberidze:2007an,Buonanno:1996xc}
a smooth homogeneous cosmological background and ignores the
presence of mass concentrations which as shown in the present
analysis would tend to modify both the magnitude and the shape of
this spectrum. A proper stochastic analysis including the effects
of mass concentrations on the relic GW spectrum is therefore an
interesting extension of the present work.

A distortion of the RGW spectrum is expected due to the presence
of point masses on various scales due to the increase of each mode amplitude and
decrease of each mode frequency. The effect will be stronger in regions of higher
mass concentrations. On scales larger than the galactic scales the role of the point
mass could be played by a galaxy while on scales of the solar system the role of the point mass could be played by a planet.
In the solar system the effect is expected to be rather weak and beyond the sensitivity of current experiments.

An additional interesting extension could be the drop of the weak
field approximation and the use of the full McVittie metric
\cite{McVittie:1933zz} for the study of GW evolution in an
expanding background and in the vicinity of a black hole allowing
for strong gravitational field.

Even though our numerical analysis has been well tested and
provides detailed quantitative information on the GW evolution
in the presence of expansion and a point mass, an analytical
perturbative solution describing this evolution would provide
further physical insight and appears to be a tractable useful
extension of the present work.

\textbf{Numerical Analysis Files}: The mathematica files used for the production of the figures, as well as the figures may be downloaded from \href{https://drive.google.com/open?id=0B7rg6X3QljQXck5YSmQ5Rl9HeUU}{here} or upon request from the authors.

\section*{Acknowledgements}
D. Papadopoulos would like to thank the Department of Physics of
the University of Ioannina for hospitality during the period when
part of this work was in progress. We also thank K. Kleidis for useful comments.

\raggedleft


\section{Introduction}
\label{sec:intro}

The development of technologies to trap and cool neutral atoms has led to
the emergence of cold atom physics as a new subfield of atomic physics.
The atoms can be cooled to temperatures that are orders of magnitude smaller
than the tiny differences between the hyperfine energy levels of the atoms.
Many of the most important loss processes for ultracold atoms involve deeply inelastic reactions,
which produce atoms with kinetic energies much larger than those of the cold trapped atoms.
One such process is three-body recombination,
in which a collision of three low-energy atoms results in the binding of two of the atoms
into a diatomic molecule.
If the binding energy of the molecule is large compared to the energy scale of the cold atoms,
the momenta of the molecule and the recoiling atom
are often large enough for them to escape from the trapping potential for the atoms.

Ultracold atoms can be described by a local nonrelativistic effective field theory
for which the coupling constant is  the scattering length \cite{Braaten:2004rn}.
Local effective field theories can be applied most rigorously to a system in which there is an energy gap
separating the low-energy particles described explicitly by the effective field theory
from the higher-momentum particles \cite{Georgi:1994qn,Kaplan:1995uv,Manohar:1996cq,Shankar:1996vk,Burgess:2007pt}.
In a system consisting of low-energy atoms,
conservation of energy ensures that a high-momentum atom
can only appear through a virtual state that, by the time-energy uncertainty principle, has a short lifetime.
During that short lifetime, the high-momentum atom can propagate only over a short distance.
Its effects on low-energy atoms are therefore restricted to short distances.
These effects can be reproduced by local Hermitian operators in the
Hamiltonian for the effective field theory.

Local effective field theories have also been applied to systems
with deeply inelastic reactions that produce particles with momenta too large to be
described accurately within the effective field theory.
For example,  a deeply inelastic three-body recombination
event produces a molecule and an atom whose momenta
may be outside the domain of validity of the effective theory.
The large energy release from the  inelastic reaction comes from the  conversion of rest energy
into kinetic energy.
The standard argument for the locality of the effective field theory does not apply.
The particles with large momenta produced by the inelastic reaction can propagate over arbitrarily
long distances, so their effects on low-energy particles are not
obviously restricted to short distances.
Nevertheless, there are general arguments based on the uncertainty principle
that indicate that their effects can be taken into account systematically through
local anti-Hermitian operators  in the effective  Hamiltonian \cite{BHL:1607}.
The effective Hamiltonian can be expressed as  $H_{\rm eff} = H -iK$,
where $H$ and $K$ are Hermitian.

The dynamics of a multi-atom system with deeply inelastic reactions
is conveniently described by a density matrix.
A system that starts as a pure quantum state with $n$ low-energy atoms evolves
into a mixed quantum state
that is a superposition of a state with $n$ low-energy atoms and states with fewer low-energy atoms,
as the inelastic reactions shift probability from the low-energy atoms into
the high-momentum atoms. An effective density matrix $\rho$
can be defined by tracing the density matrix over states containing high-momentum atoms.
Naively we might expect  the time evolution equation for $\rho$ to be
  \begin{eqnarray}
 i\hbar \frac{d\ }{dt}\rho &\overset{?}{=} &  H_\mathrm{eff} \rho- \rho H_\mathrm{eff}
=  \left[H, \rho\right] - i \left\{K,\rho\right\}.
\label{eq:drhodt-naive}
\end{eqnarray}
As we will demonstrate in this paper, the correct evolution equation
for the effective density matrix is the {\it Lindblad equation}
\cite{Lindblad:1975ef,Gorini:1975nb}, which has an additional term.
The Lindblad equation arises in the quantum information theory
of  {\it open quantum systems}.
An  open quantum system
consists of all the degrees of freedom
of both the subsystem of interest and the environment.
Under special circumstances, the density matrix for the subsystem evolves
in time according to the Lindblad equation.
In the Lindblad equation for the density matrix of an effective field theory
obtained by integrating out deeply  inelastic reactions,
the additional Lindblad term is local, and it
can be deduced from the local anti-Hermitian terms
in the effective Hamiltonian.

An open quantum system in which the subsystem of interest is a  field theory
is called an {\it open effective field theory} \cite{Grozdanov:2013dba,Burgess:2014eoa}.
Grozdanov and Polonyi have proposed that an open effective field theory for the hydrodynamic modes
of a quantum field theory can be used as a framework for deriving
dissipative hydrodynamics \cite{Grozdanov:2013dba}.
Burgess, Holman, Tasinato, and Williams have applied open effective field theory
to the super-Hubble modes  of primordial quantum fluctuations
in the early universe \cite{Burgess:2014eoa,Burgess:2015ajz}.
In the stochastic inflation framework, the master equation is  the Lindblad equation.
Since the density matrix for an effective field theory in which deeply inelastic reaction products
have been integrated out satisfies the Lindblad equation,
this system can also be regarded as an open effective field theory \cite{BHL:1607}.
In this case, the environment consists of the high-momentum particles
produced by the deeply inelastic reactions.

The paper is organized as follows.
In Section~\ref{sec:DensityMatrix}, we summarize the basic properties of the density matrix
and we introduce the Lindblad equation.
In Section~\ref{sec:Effective Theory}, we explain how integrating out deeply inelastic reaction
products results in local operators in the effective Lagrangian density.
We also explain why the effective density matrix obtained by tracing over states that include
deeply inelastic reaction products must satisfy the Lindblad equation.
In Section~\ref{sec:AtomLoss}, we apply the Lindblad equation to the mean
number of low-energy atoms.
We derive the universal relation for the two-atom inelastic loss rate for fermions with two spin states
and  the universal relation for the three-atom inelastic loss rate for identical bosons.
Our results  are summarized briefly in Section~\ref{sec:Summary}.
In an Appendix, we demonstrate how the Lindblad equation for the density matrix emerges from
the diagrammatic analysis of a simple field theory model with a deeply
inelastic two-atom scattering reaction.

\section{Density Matrix}
\label{sec:DensityMatrix}

The density matrix can be used to describe  pure quantum states, mixed quantum states,
and statistical ensembles of quantum states.
The dynamics of a multi-atom system with atom losses
must be described by the density matrix in order to be able to track
the losses in the different $n$-atom sectors.
To lay the foundation for the rest of the paper,
we summarize the key properties of the density matrix
and we introduce the Lindblad equation.

\subsection{General Properties}

A pure state in a quantum system
can be represented by a vector $|\psi\rangle$ in a complex vector space.
The time evolution of the quantum state is
described by the Schr\"odinger equation:
$i \hbar (d/dt)|\psi\rangle =  H |\psi\rangle$,
where the Hamiltonian $H$ is a Hermitian operator.
The time evolution preserves the norm $\langle \psi | \psi \rangle$ of a state.

An arbitrary statistical ensemble of quantum states
can be represented by a density matrix $\rho$.
The density matrix has the following basic properties:
\begin{itemize}
\item
It is Hermitian:  $\rho^\dagger = \rho$.
\item
It is positive: $\langle \psi | \rho | \psi \rangle \ge 0$ for  all nonzero states $|\psi\rangle$.
\item
It can be normalized to have unit trace: ${\rm Tr}(\rho) = 1$.
\end{itemize}
The density matrix can also describe a pure quantum state,
in which case $\rho^2 = \rho$.

The time evolution of the density matrix is described by
the von Neumann equation:
\begin{equation}
i \hbar \frac{d\ }{dt} \rho =  [ H, \rho],
\label{evol:Schrodinger}
\end{equation}
which follows from the Schr\"odinger equation for $|\psi\rangle$.
This evolution equation has the following properties:
\begin{itemize}
\item
It is linear in $\rho$.
\item
It preserves the trace of $\rho$.
It therefore respects the normalization condition ${\rm Tr}(\rho) = 1$.
\item
It is {\it Markovian}:    the future is determined by the present only.
There is no additional dependence on the past history.
\end{itemize}
The system average of an operator $O$
can be expressed as the trace  of its product with the density matrix:
\begin{equation}
\langle O \rangle = {\rm Tr}(\rho O ).
\label{d<O>}
\end{equation}
If the operator has no explicit time dependence,
the time evolution of the system average is determined by
the evolution equation of $\rho$ in Eq.~\eqref{evol:Schrodinger}.

\subsection{Lindblad equation}
\label{sec:Lindblad}

In the field of  quantum information theory, the full system
is often separated into a {\it subsystem} of interest and its {\it environment}  \cite{Preskill}.
Of particular interest is the decoherence of the subsystem due to the effects of the environment.
The basis for the quantum states of the full system can be chosen to be
direct products of  basis states for the subsystem and those for the environment.
A  density matrix $\rho$ for the subsystem can be obtained
from the density matrix $\rho_\mathrm{full}$ for the full system
by the partial trace over the states of the environment:
    \begin{equation}
    \label{eq:rho_eff-qi}
        \rho = \mathrm{Tr}_{\rm environment} \left(\rho_\mathrm{full} \right).
    \end{equation}
The density matrix for the full system evolves according to the
von Neumann equation in Eq.~\eqref{evol:Schrodinger}.
Given the initial density matrix $ \rho(t=0)$ for the subsystem
and a specified initial state of the environment, the von Neumann equation
can in principle be solved to determine $ \rho(t)$ at future times.
It is possible in principle to construct a self-contained differential equation
for $ \rho(t)$, but its evolution is non-Markovian  \cite{Preskill}.
The time derivative of $\rho(t)$  is determined
by the present density matrix and by its history from time 0 to $t$.
The previous history is needed to take into account the flow of information
between the subsystem and the environment.

There are situations in which the time evolution of the subsystem
can be described approximately by a self-contained Markovian differential equation.
The time during which the subsystem is observed
must be much larger than the time scale for correlations between
the subsystem and the environment.
We must also restrict our consideration to the low-frequency behavior of the subsystem,
which can be accomplished by smoothing out over times larger than the correlation time scale.
The density matrix $\rho$ for the subsystem necessarily satisfies
the three basic properties listed before Eq.~\eqref{evol:Schrodinger}:
it is Hermitian, it is positive, and it can be normalized.
It would be desirable for its time evolution to also be in accord with the three
basic properties listed after Eq.~\eqref{evol:Schrodinger}:
it should be linear in $\rho$, it should preserve the trace of $\rho$, and it should be Markovian.
In 1976, Lindblad and, independently, Gorini, Kossakowski, and Sudarshan
showed that these conditions together with one additional technical condition require the
time evolution equation to have the form \cite{Lindblad:1975ef,Gorini:1975nb}

\begin{equation}
i \hbar \frac{d\ }{dt} \rho =
 [ H, \rho] - \frac{i}{2} \sum_n
\left( L_n^\dagger L_n \rho + \rho L_n^\dagger L_n
-2 L_n \rho L_n^\dagger \right),
\label{evol:Lindblad}
\end{equation}
where  $H$ is a Hermitian operator
and the $L_n$'s are an additional set of operators called {\it Lindblad operators}.
The evolution equation in Eq.~\eqref{evol:Lindblad} is called the {\it Lindblad equation}.
The linear operator acting on $\rho$ defined by the right side of
Eq.~\eqref{evol:Lindblad} is called the {\it Lindbladian}.
Thus the Lindbladian is the generator of the time evolution of the density matrix.

The additional technical condition required to derive the Lindblad equation is that
the linear operator that determines the time evolution of $\rho$
should be completely positive  \cite{Lindblad:1975ef,Gorini:1975nb}.
For any complex vector space, this linear operator
has a natural extension to an operator acting on the tensor product
of the quantum-state spaces for the subsystem and the other complex vector space.
Complete positivity requires the extension of the operator to be positive
in the larger space.
An accessible derivation of the Lindblad equation can be found in lecture notes
on quantum information and computation by John Preskill \cite{Preskill}.

The Lindblad equation in Eq.~\eqref{evol:Lindblad} can be expressed in the form
\begin{equation}
i \hbar \frac{d\ }{dt} \rho =
 [ H, \rho] - i \left\{K,\rho\right\}
 + i \sum_n L_n \rho L_n^\dagger ,
\label{evol:Lindblad-2}
\end{equation}
where the Hermitian operator $K$ is
\begin{equation}
K =\frac12\sum_n L_n^\dagger L_n.
\label{eq:K-Ln}
\end{equation}
Comparison with the naive evolution equation in Eq.~\eqref{eq:drhodt-naive}
reveals that $H - i K$ can be interpreted as a non-Hermitian effective Hamiltonian
for the subsystem. The additional {\it Lindblad term} in Eq.~\eqref{evol:Lindblad-2}
is necessary to preserve the trace of $\rho$.

\section{Effective Theory from Deeply Inelastic Reactions}
\label{sec:Effective Theory}

In this section, we review how effective  field theories for cold atoms
are constructed, with particular focus on the treatment of deeply inelastic
reactions. We first discuss the construction of the effective
Lagrangian, and how integrating deeply inelastic reaction products out
of a theory results in new local operators in the effective Lagrangian
density. We then discuss how such a formalism
can be used to study multi-particle
states by means of an effective density matrix obtained by tracing over
states that include deeply inelastic reaction products.

\subsection{Locality}
\label{sec:Locality}

\begin{figure}[t]
\centerline{ \includegraphics*[width=10cm,clip=true]{fig_atoms1.pdf} }
\caption{
The amplitude for the elastic scattering process $\psi \psi \to \psi \psi$
has contributions from intermediate states $\phi \phi$, where $\phi$ is a
different hyperfine state. If  $\phi$ is a higher hyperfine state,
the virtual atoms can propagate only over short distances,
so their effects can be mimicked by local $2 \to 2$ operators.
If  $\phi$ is a lower hyperfine state,
the virtual atoms can propagate over long distances,
but the imaginary part of the amplitude is still local
and  can again be mimicked by local $2 \to 2$ operators.
}
\label{fig:2to2-elastic}
\end{figure}

An effective field theory is obtained by removing (``integrating out'')
states from a field theory. The simplest applications involve removing
particles with much higher energies.
We consider atoms with mass $M$ in a specific hyperfine state labelled $\psi$
and with kinetic energies much smaller than the splitting $\Delta$ between hyperfine states.
As illustrated in Fig.~\ref{fig:2to2-elastic},
there is a contribution to the elastic scattering amplitude from scattering
into a virtual pair of particles in a higher hyperfine state labelled $\phi$,
which then rescatter into atoms in the original hyperfine state.
The virtual atoms are off their energy shells by large amounts of order $\Delta$.
By the time-energy uncertainty principle, the virtual states have short lifetimes of order $\hbar/\Delta$,
during which the $\phi$ atoms can propagate only over short distances
much smaller than the wavelengths of the $\psi$ atoms.
The effects of the higher hyperfine states can be modeled in the effective field theory
by a contact interaction of the form $\psi^\dagger \psi^\dagger \psi \psi$,
where $\psi$ is the quantum field that annihilates an atom in the specific hyperfine state $\psi$.
(We use the same symbol for the label of the hyperfine state and for its quantum field.)
The $2 \to 2$ contact interaction is the leading term in a series of
local operators in the effective Lagrangian that mimic the  effects of the higher hyperfine states
to arbitrary precision. This series is obtained at tree-level by
Taylor expanding the scattering amplitude in the momentum transfer
and then writing down terms in the effective Lagrangian density that reproduce this expansion:
\begin{equation}
  \delta\mathcal{L} =
  g\, \psi^\dagger \psi^\dagger \psi \psi
  + h \,\nabla(\psi^\dagger\psi)
  \cdot \nabla(\psi^\dagger\psi) + \ldots,
  \label{eq:deltaL}
\end{equation}
where  the coefficients $g$, $h$, \ldots\ are real valued.
The individual operators are renormalized
when loop corrections are included, but the effective theory
is still capable of reproducing the original theory to arbitrary
precision provided operators with a sufficient number of gradients are
retained. Operators with $n$ gradients correct the theory at
order $(q/P)^n$, where $q$~is the momentum transfer and
$P = (M \Delta)^{1/2}$ is the momentum scale associated with the
energy splitting $\Delta$ between the hyperfine states.
We have integrated the higher hyperfine states out of the theory by replacing them
by the interactions terms in Eq.~\eqref{eq:deltaL}.

\begin{figure}[t]
\centerline{ \includegraphics*[width=15cm,clip=true]{fig_locatoms.pdf} }
\caption{
The optical theorem relates the imaginary part of the
    amplitude for $\psi \psi \to \phi \phi \to \psi \psi$,
    where $\phi$ is a lower hyperfine state,
to the rate for the deeply inelastic scattering process $\psi \psi \to \phi \phi$.
}
\label{fig:2to2-inelastic}
\end{figure}

A less obvious opportunity to remove states from the theory arises when atoms in the hyperfine state $\psi$
scatter inelastically into atoms in a  lower hyperfine state $\phi$.
Such a scattering process is an example of a \emph{deeply inelastic reaction},
in which the difference in the rest energies of the initial-state and final-state atoms is
converted into large kinetic energies of the final-state atoms.
The rate for the deeply inelastic scattering process $\psi \psi \to \phi \phi$ is
related by the optical theorem to the imaginary part of the
amplitude for $\psi \psi \to \phi \phi \to \psi \psi$ (see
Fig.~\ref{fig:2to2-inelastic}). That amplitude is analytic in the momentum transfer $\bm{q}$,
because the nearest non-analyticity in the amplitude is at
the threshold energy for the $\phi \phi$~state,
which is lower by an amount of order $ \Delta$. As a result, we can Taylor expand the amplitude in
powers of~$\bm{q}/(M \Delta)^{1/2}$ and reproduce this expansion by terms in the
effective Lagrangian as in Eq.~\eqref{eq:deltaL}.
In this case, the coefficients $g$, $h$, \ldots\ are complex valued.
We are particularly interested in their imaginary parts,
which come from the deeply inelastic scattering reaction.
These terms mimic the effects of the lower hyperfine states in the effective theory.

It may seem nonintuitive
that the effects of inelastic scattering to on-shell particles
can be mimicked by local operators, because once an atom in the lower hyperfine state $\phi$
is created, it can propagate over long distances. In fact, as far as low-energy atoms
in the hyperfine state $\psi$ are concerned, the inelastic scattering  process is quite
local. This is because the reaction region in which the low-energy  atoms disappear can be reconstructed
by tracking the inelastic scattering  products back to their origin. The inelastic reaction products
have high momenta of order~$(m \Delta)^{1/2}$ and therefore
short wavelengths of order~$\hbar/(m \Delta)^{1/2}$,  so they
can locate the decay with a resolution of order~$\hbar/(m \Delta)^{1/2}$.
Therefore the reaction is localized over a
region of size~$\hbar/(m \Delta)^{1/2}$, which is very small compared to the
wavelengths of the incoming low-energy atoms.

\subsection{Effective density matrix for decaying atoms}

Some aspects of the effective field theory obtained by integrating out deeply inelastic reactions
are most easily understood by considering a deeply inelastic 1-body reaction,
namely the decay of an atom.
The atom could be a metastable excited state, such as the $2 ^3S_1$ state of $^4$He,
which was one of the earliest atoms for which Bose-Einstein condensation was achieved
\cite{Aspect:2001,Cohen-Tannoudji:2001}. The radiative decay of the atom into its ground state
is a deeply inelastic reaction. For simplicity, we
assume the ground-state atoms interact weakly with the metastable atoms
and that, once they are produced, the ground-state atoms quickly escape from  the system.
The Hamiltonian $H - iK$ for the effective theory is the sum of the Hermitian
Hamiltonian for the metastable atoms
and an anti-Hermitian piece $-iK$.
The Hermitian operator $K$ can be expressed as $\frac12 \Gamma\, N$,
where $\Gamma$ is the decay rate of the metastable atom
and $N$ is the number operator that counts the metastable atoms:
\begin{equation}
  N = \int d^3\bm{r} \, \psi^\dagger \psi .
  \label{eq:Nmuhat}
\end{equation}

The quantum mechanics of such a theory is unconventional.
The effective Hamiltonian $H - iK$ commutes with the number operator $N$,
so the time evolution generated by $H - iK$ does \emph{not}
change the number of atoms in a state.
Instead the effects of atom decay
must be taken into account by transferring
the probability carried by the $n-$atom component
to states containing fewer atoms. An $n-$atom state
evolves into a superposition of states with $n$ and fewer atoms.
The norm of a state containing $n$~atoms
decreases to zero exponentially with the decay rate~$n \Gamma$.
This is the correct result:  if the probability for one atom to
still be an atom after
time~$t$ is $\exp(-\Gamma t)$, the probability for $n$~atoms to still
be $n$~atoms is $\exp(-n\Gamma t)$.

We typically want more information about where the probability goes. An $n-$atom state
evolves into a superposition of states with $n$, $n-1$, $n-2$\,\ldots\,atoms.
In the full theory,  the superposition can be described by the density matrix $\rho_\mathrm{full}$.
In the effective theory, we
describe the superposition of states with different atom numbers
using an \emph{effective density matrix} $\rho$, from which
we remove ground-state atoms
by tracing over states that include them:
\begin{equation}
  \label{eq:rho-eff-defn}
  \rho \equiv \mathrm{Tr}_\mathrm{deep} \left(\rho_\mathrm{full} \right).
\end{equation}
The subscript ``deep'' indicates that the partial trace is over states that include
deeply inelastic reaction products.
This effective density matrix, like $\rho_\mathrm{full}$, is
Hermitian, positive, and has unit trace: $\mathrm{Tr}(\rho)=1$. These
properties follow from the definition in Eq.~(\ref{eq:rho-eff-defn}).
Naively we might expect  the time evolution equation for $\rho$ to be
Eq.~\eqref{eq:drhodt-naive}, which reduces to
\begin{equation}
  i\frac{d\ }{dt}\rho =   \left[H, \rho\right]
  - \frac{i}{2} \Gamma \left\{ N, \rho\right\} .
\end{equation}
However this equation does \emph{not} conserve
the total probability~$\mathrm{Tr}(\rho)$.
The correct evolution equation is the Lindblad equation:
\begin{equation}
  i\frac{d}{dt}\rho =
  \left[H, \rho\right]
  - \frac{i}{2} \Gamma \left\{ N, \rho\right\}
  + i \Gamma \int d^3\bm{r}\,\psi(\bm{r})\,\rho\,\psi^\dagger(\bm{r}).
  \label{eq:muon-evol-eq}
\end{equation}
Time evolution preserves the trace of $\rho$, because the trace of
the additional Lindblad term cancels the trace of the anticommutator
term in Eq.~\eqref{eq:muon-evol-eq}.

The role of the Lindblad term is easily understood
if we use the evolution equation
to calculate the rate of change of the
probability $P_n(t)$ for finding $n$~metastable atoms in the system.
This probability can be expressed as the partial trace of $\rho$
over states $ | X_n \rangle$ that contain $n$ atoms:
\begin{equation}
  P_n(t) \equiv \sum_{X_n} \langle X_n | \rho(t) | X_n \rangle.
\end{equation}
The partial trace of the  evolution equation in Eq.~\eqref{eq:muon-evol-eq} gives
\begin{equation}
  \frac{d\ }{dt}  P_n(t)= - n\Gamma P_n(t) + (n+1)\Gamma P_{n+1}(t).
  \label{eq:dPn(t)}
\end{equation}
The partial trace of the commutator term in Eq.~\eqref{eq:muon-evol-eq} is 0,
because the operator $H$ does not change the atom number.
This allows a complete set of $n-$atom states to be inserted between $H$ and $\rho$.
The partial trace of the anticommutator term in Eq.~\eqref{eq:muon-evol-eq}
gives $-n\Gamma P_n$, which is the rate at which
probability leaves the $n-$atom sector because of
the decay of an atom. The partial trace of the Lindblad
term in Eq.~\eqref{eq:muon-evol-eq} gives $+(n+1)\Gamma P_{n+1}$,
which is the rate at which probability enters the $n-$atom sector
from the decay of an atom in the $(n+1)-$atom sector.
This expression can be obtained by inserting complete sets of $(n+1)-$atom states
on the left and right of $\rho$ in the Lindblad term in Eq.~(\ref{eq:muon-evol-eq})
and then rearranging the factors to give
\begin{eqnarray}
  i \Gamma \sum_{X_{n+1}} \sum_{X'_{n+1}}\,
  \langle X'_{n+1}| \rho |X_{n+1} \rangle
  \langle X_{n+1} | N |X'_{n+1} \rangle.
  \label{eq:drho-lindblad}
\end{eqnarray}
The number operator $N$, which is given in Eq.~\eqref{eq:Nmuhat},
was obtained by replacing the sum of $ |X_n \rangle \langle X_n |$
between $ \psi^\dagger(\bm{r})$ and $\psi(\bm{r})$ by the identity operator.
The number operator in Eq.~\eqref{eq:drho-lindblad} can be replaced by its eigenvalue $n+1$.

The Lindblad term in Eq.~(\ref{eq:muon-evol-eq}) is essential to get the correct physical behavior
for the time evolution of the total number of metastable atoms. The expectation value of the atom number is
\begin{equation}
  N(t) \equiv {\rm Tr} \big( N \rho(t)\big) = \sum_n n P_n(t).
  \label{eq:Nmu-t}
\end{equation}
We can use Eq.~\eqref{eq:dPn(t)} to determine the time dependence of $N(t)$:
\begin{equation}
  \frac{d\ }{dt} N(t)= -\Gamma \Big[
    \sum_n n^2 P_n(t) - \sum_n n(n+1) P_{n+1}(t) \Big] .
\end{equation}
After shifting the index of the second term on the right side,
we obtain
\begin{equation}
  \frac{d\ }{dt} N(t)= -\Gamma N .
\end{equation}
This implies that $N(t) = N_0 \exp(-\Gamma t)$, as expected for  a collection of
atoms with decay rate $\Gamma$.

\subsection{Effective density matrix from deeply inelastic reactions}

We now consider a more general effective field theory obtained by integrating out the
reaction products from deeply inelastic reactions.
By the arguments presented in Sec.~\ref{sec:Locality},
the effects of the deeply inelastic reactions
can be reproduced by local anti-Hermitian terms in the effective Hamiltonian.
The effective Hamiltonian can be expressed in the form $H_\mathrm{eff} = H  - i K$,
where $H$ and $K$ are both  Hermitian operators.
The operator  $K$ can be expressed in  the form
\begin{equation}
K = \sum_i \gamma_i \int d^3r\,  \Phi_i^\dagger \Phi_i ,
\label{eq:K-Phi}
\end{equation}
where the local operators $\Phi_i(\bm{r})$  annihilate low-energy atoms
in configurations  corresponding to inelastic reactions.
The real constants $\gamma_i$ can be determined by
matching low-energy observables in the full theory and in the effective theory.

The Hamiltonian $H  - i K$ commutes with the number operator $N$ for low-energy atoms,
so the time evolution generated by $H - iK$ does \emph{not} change the number of low-energy atoms in a state.
Instead the effects of the deeply inelastic reactions
must be taken into account by transferring
the probability carried by the $n-$atom component
to states containing fewer atoms.
The norm of a state containing $n$~atoms
decreases  to zero at a rate that increases with $n$.
An $n-$atom state
evolves into a superposition of states with $n$ and fewer atoms.
We describe this superposition of states using an \emph{effective density matrix} $\rho$
defined by tracing over the deeply inelastic decay products
as in Eq.~\eqref{eq:rho-eff-defn}.   More precisely,
we trace out any state containing an atom with momentum
exceeding the ultraviolet cutoff of the effective field theory.
The effective density matrix, like the underlying density matrix, is
Hermitian, positive, and it has unit trace: $\mathrm{Tr}(\rho)=1$. These
properties follow from the definition in Eq.~(\ref{eq:rho-eff-defn}).

As is familiar in the field of quantum information \cite{Preskill}, the definition
of the effective density matrix $\rho$  as the partial trace in Eq.~(\ref{eq:rho-eff-defn})
implies that $\rho(t)$ satisfies a self-contained time evolution equation
that is completely determined by $\rho$
and by the full density matrix $\rho_\mathrm{full}(0)$ at some initial time.
This evolution equation is however non-Markovian:
the time derivative $(d/dt)\rho(t)$ is determined not only
by $\rho(t)$ but also by $\rho(t')$ at the earlier times $0<t'<t$.
This dependence on the history takes into account the effects
of high-momentum atoms that are produced by inelastic reactions
and subsequently interact with the low-energy atoms.
However the density matrix $\rho(t)$ for the effective theory
does not need to reproduce the full density matrix
on short time scales of order  $\hbar /E_{\rm deep}$.
It is sufficient to reproduce its effects after  time averaging over time intervals
much larger than $\hbar /E_{\rm deep}$.
This time average removes transients associated with high-momentum atoms
that cannot be described accurately within the effective field theory.
It is only after removing these transients that
it is possible to have a density matrix $\rho(t)$ that is Markovian.
Thus the proper definition of the effective density matrix
requires that
the partial trace in Eq.~(\ref{eq:rho-eff-defn})
be supplemented by an appropriate time average
that makes $\rho(t)$ Markovian.

Naively we might expect  the time evolution equation for $\rho$ to be Eq.~\eqref{eq:drhodt-naive},
where $K$ is the Hermitian operator in Eq.~\eqref{eq:K-Phi}.
However, this equation does \emph{not} conserve  the total probability~$\mathrm{Tr}(\rho)$.
The correct evolution equation is the Lindblad equation:
\begin{equation}
i \hbar \frac{d\ }{dt} \rho = \left[H, \rho\right]- i \left\{K,\rho\right\}
+2i \sum_i \gamma_i   \int \!\!d^3r\,
\Phi_i \rho \Phi_i^\dagger.
\label{eq:LindbladPhi}
\end{equation}
As in Eq.~\eqref{eq:muon-evol-eq},
the trace of the additional  Lindblad term cancels the trace of the anticommutator term.
The local operators $\Phi_i(\bm{r})$, which  annihilate
configurations of low-energy atoms that can undergo a deeply inelastic reaction,
are the {\it Lindblad operators}. In quantum information theory,
a Lindblad operator is sometimes called a {\it quantum jump operator}.
In the low-energy effective theory, a Lindblad operator produces a jump to
a state with a different low-energy atom number.
Given the form of~$K$ in Eq.~\eqref{eq:K-Phi}, the Lindblad equation is a necessary
consequence of our physical requirements
on the effective density matrix: $\rho$~is Hermitian, positive,
has unit trace, and it is Markovian.
An explicit diagrammatic illustration of the emergence of the
Lindblad equation from tracing over
states that include deeply inelastic reaction products
is given in Appendix~\ref{sec:Deductive}.

In order to obtain the Lindblad equation in Eq.~\eqref{eq:LindbladPhi},
it is essential that $K$ have the structure shown
in~Eq.~(\ref{eq:K-Phi}). A more generic form for this operator is
\begin{equation}
  K = \int \!\!d^3 r \sum_{ij} c_{ij} \Phi_i^\dagger(\bm{r}) \Phi_j(\bm{r}),
  \label{eq:K-structure-gen}
\end{equation}
where $c_{ij}$ is a positive Hermitian matrix.
It is a Hermitian matrix, because $K$ is Hermitian by definition.
It is guaranteed to be a positive matrix by the optical theorem,
which implies that the $T$ matrix satisfies $-i (T - T^\dagger) = T^\dagger T$.
Matrix elements of $K$ reproduce
the anti-Hermitian parts of scattering amplitudes $\bra{b}T\ket{a}$,
where~$\ket{a}$ and~$\ket{b}$ are states in the effective theory
that are connected by intermediate deeply inelastic  reaction channels.
The Hermitian parts of these amplitudes are reproduced by $H$.
The optical theorem guarantees the positivity of the anti-Hermitian parts.
The double sum in Eq.~\eqref{eq:K-structure-gen} is easily rewritten in the canonical
form in Eq.~\eqref{eq:K-Phi} by expanding $c_{ij}$ in terms of outer products of its eigenvectors.

\section{Atom Loss from Deeply Inelastic Reactions}
\label{sec:AtomLoss}

In this section, we use the Lindblad equation to determine the time evolution
of the mean number of low-energy atoms, which
can be changed by deeply inelastic reactions.
We also derive universal relations for the inelastic two-atom loss rate
for fermions with two spin states
and for the inelastic three-atom loss rate for identical bosons.

\subsection{Mean Atom Number}

We consider an open effective field theory with effective Hamiltonian
$H_{\rm eff}=H-iK$,
where $H$ and $K$ are Hermitian.
The Hermitian operator $K$ in the anti-Hermitian part can be expressed
in terms of local Lindblad operators $\Phi_i(\bm{r})$ as in
Eq.~\eqref{eq:K-Phi}.
The time evolution of the density matrix $\rho$ for the effective field theory
is given by the Lindblad equation in Eq.~\eqref{eq:LindbladPhi}.

We reconsider  an atom number operator $N$
that is conserved by the reactions  in the effective field theory
but is violated by deeply inelastic reactions in the full theory.
Since Lindbladian time evolution preserves the trace of $\rho$,
the system average of $N$ can be expressed as
\begin{equation}
\langle N \rangle = {\rm Tr}(\rho N).
\label{N-ave}
\end{equation}
The statement that $N$ is conserved by the reactions in the effective theory
can be expressed as the commutation relation
\begin{equation}
\big[ N,H\big ]=0.
\label{[N,H]}
\end{equation}
The inelastic reaction responsible for the $\Phi_i^\dagger \Phi_i$ term
in $K$ changes the atom number $N$ by some integer $k_i$.
This statement can be expressed as a commutation relation:
\begin{equation}
\big[ N,\Phi_i(\bm{r}) \big]=-k_i \Phi_i(\bm{r}).
\label{[N,Phi]}
\end{equation}

We now consider the time evolution of $\langle N \rangle$.
By taking the time derivative of ${\rm Tr}(\rho N)$,
inserting the time evolution equation in Eq.~\eqref{eq:LindbladPhi},
and rearranging the terms,
we get
\begin{eqnarray}
\frac{d\ }{dt} \langle N \rangle =
- \frac{i}{\hbar} {\rm Tr}(\rho [N,H])
 - \frac{1}{\hbar} \sum_i \gamma_i  \int \!\! d^3r \,
{\rm Tr} \big( \rho (N\Phi_i^\dagger \Phi_i  +\Phi_i^\dagger \Phi_i  N
-2 \Phi_i^\dagger N \Phi_i) \big).
\label{Nevol:Lindblad}
\end{eqnarray}
The first term on the right side vanishes,
because $N$ commutes with $H$.
By using the  commutation relation in Eq.~\eqref{[N,Phi]}
in the second term on the right side, the rate of change of atom number
reduces to
\begin{eqnarray}
\frac{d\ }{dt} \langle N \rangle &=& -\frac{2}{\hbar}
\sum_i k_i \gamma_i \int \!\! d^3r \, \langle \Phi_i^\dagger \Phi_i \rangle.
\label{Nevol:Phi}
\end{eqnarray}
The rate of change in $\langle N \rangle$ has been expressed as
a sum of expectation values of the same terms that appear in
the expression for $K$ in Eq.~\eqref{eq:K-Phi} but multiplied by integers $ k_i$.

\subsection{Three-Body Recombination Rate}
\label{sec:3bodyrecombination}

The atoms that are studied in cold atom experiments
can form deeply bound diatomic molecules whose binding energies are
orders of magnitude larger than the energy scales of the cold trapped atoms.
Three-body recombination, in which three low-energy atoms collide
and two of them bind into a molecule, is
a deeply inelastic reaction if the molecule is deeply bound.
The momenta of the outgoing molecule and the recoiling atom are often
large enough for them to escape from the trapping potential.
In a locally homogeneous gas of atoms in thermal equilibrium,
the rate of decrease in the local number density $n_A$ of low-energy atoms
can be expressed as a rate equation:
\begin{equation}
(d/dt) n_A = - 3 K_3(T) n_A^3,
\label{dn/dt-thermal}
\end{equation}
where $K_3(T)$ is the 3-body recombination event rate constant,
which depends on the temperature $T$.

Bosonic atoms that are all in the same hyperfine state
can be described by an effective field theory with a quantum field $\psi$.
At the  low temperatures in cold atom experiments,
the only relevant interaction between the atoms is
S-wave scattering, whose strength is determined by the scattering length $a$.
The interaction term in the Hamiltonian  is
\begin{equation}
H_{\rm int} = (2 \pi \hbar^2 a/m) \int d^3 r\, \psi^{\dagger 2}\psi^2.
\label{Hint-psi}
\end{equation}
This interaction term can be treated using perturbation theory,
provided the scattering length $a$ is not much larger than the range
of the interactions between the atoms,
which for ultracold atoms is the van der Waals length scale.

We now consider a deeply inelastic three-body recombination reaction
that  produces a diatomic molecule with binding energy much larger than the temperature $T$.
Its effect on low-energy atoms  can be described in the effective field theory
by adding to the effective Hamiltonian the anti-Hermitian term $-i K$,
where the local Hermitian operator $K$ is
\begin{equation}
K = (\hbar K_3/12) \int d^3 r\,(\psi^3)^\dagger  \psi^3,
\label{Kint-psi3}
\end{equation}
and where $K_3$ is a constant that does not depend on the temperature.
The expression for $K$ in Eq.~\eqref{Kint-psi3}
has a single term with Lindblad operator $\Phi_i = \psi^3$
and coefficient $\gamma_i = \hbar K_3/12$.
The expression for the atom loss rate in Eq.~\eqref{Nevol:Phi}
thus has a single term with integer $k_i=3$. In a locally
homogeneous system of atoms, the local version of the loss rate in Eq.~\eqref{Nevol:Phi} is
\begin{eqnarray}
\frac{d\ }{dt} \langle \psi^\dagger  \psi \rangle &=&
- \frac{K_3}{2} \,\langle (\psi^3)^\dagger  \psi^3 \rangle.
\label{Nevol:Phi3}
\end{eqnarray}
The expectation value of $\psi^\dagger  \psi $ is the local number density:
$n_A = \langle \psi^\dagger  \psi \rangle$. In a thermal gas,
the expectation value of $(\psi^3)^\dagger  \psi^3$ is $6 n_A^3$.
Comparing with Eq.~\eqref{dn/dt-thermal},
we see that the constant $K_3$ in Eq.~\eqref{Kint-psi3} is the
$T \to 0$ limit of the 3-body recombination event rate constant $K_3(T)$.
Its temperature dependence is negligible,
because $kT$ is much smaller than the binding energy of the molecule.
In a Bose-Einstein condensate at zero temperature,
the expectation value of $(\psi^3)^\dagger  \psi^3$ can be expressed as
$|\langle  \psi \rangle^3|^2 =  n_A^3$,
where  $\langle  \psi \rangle$ is the mean field.
The atom loss rate is given by Eq.~\eqref{dn/dt-thermal}
except that the prefactor on the right side is smaller than for a thermal gas by
a factor of 1/6.

\subsection{Inelastic Two-Atom Loss Rate}
\label{sec:2atomloss}

In cold atom experiments,
the scattering length $a$ can be controlled experimentally by tuning an external
magnetic field to near a Feshbach resonance.
If $a$ is much larger than the range of the interactions between the atoms,
which for ultracold atoms is the van der Waals length scale,
the interactions must be treated  nonperturbatively.

Fermionic atoms in two hyperfine states 1 and 2 with scattering length $a$
can be described by  an effective field theory
with quantum fields $\psi_1$ and $\psi_2$.
The interaction term in the Hamiltonian can be expressed as
\begin{equation}
 H_{\rm int} = \frac{g_0}{m}\int d^3 r\,
\big(\psi_2 \psi_1\big)^\dagger \big( \psi_2 \psi_1 \big),
\label{Hint-psi0}
\end{equation}
where $g_0$ is the bare coupling constant.
If the ultraviolet cutoff $\Lambda$ is imposed on the
momenta of virtual atoms, the bare coupling constant is
\begin{equation}
g_0 = \frac{4 \pi}{1/a-(2/\pi)\Lambda}.
\label{g0}
\end{equation}

If the pair of atoms in the hyperfine states 1 and 2 has a {\it spin-relaxation} scattering channel
into a pair atoms in lower hyperfine states 3 and 4,
the optical theorem implies
that the scattering length $a$ is complex with a small negative imaginary part.
The energy release from the spin-relaxation reaction
is much larger than the energy scales for ultracold atoms,
so this is a deeply inelastic reaction.
The high-momentum atoms in the hyperfine states 3 and 4 can be integrated out to get
an effective field theory for low-energy  atoms in the hyperfine states 1 and 2 only.
The interaction term in the Hamiltonian is still that in Eq.~\eqref{Hint-psi},
except that now $H_{\rm int}$ has an anti-Hermitian part
because the complex scattering length $a$
makes the bare coupling constant $g_0$ in Eq.~\eqref{g0} complex.

Determining the loss rate of atoms due to the deeply inelastic spin-relaxation
reaction is not trivial, because the large scattering length makes the problem nonperturbative.
However, an exact result for the inelastic two-atom loss rate
in any state has been proposed by Shina Tan \cite{Tan-unpub}.
The rates of decrease in the numbers $N_1$ and $N_2$ of atoms in the two hyperfine states are
given by the universal relation
\begin{equation}
\frac{d\ }{dt} \langle N_1 \rangle= \frac{d\ }{dt} \langle N_2 \rangle =
- \frac{\hbar}{2 \pi m}{\rm Im} \big( 1/a \big)\, C,
\label{uni2loss}
\end{equation}
where $C$ is  a property of the system called the {\it contact}.
The coefficient of $C$, which is proportional to the imaginary part of $1/a$,
is determined by 2-body physics.
The contact was first introduced by Shina Tan in 2005 \cite{Tan:0505,Tan:0508}.
It is the  thermodynamic variable conjugate to $1/a$.
The contact $C$ is an extensive variable:
it is the integral over space of the contact density  ${\cal C}$,
which measures the number of 1-2 pairs per (volume)$^{4/3}$.
(The unusual power of volume arises from an anomalous dimension
associated with a non-Gaussian renormalization group fixed point \cite{Braaten:2010if}.)
Shina Tan derived other  universal relations involving the contact
that hold for any state of the system  \cite{Tan:0505,Tan:0508}.
They include the ``adiabatic relation''
that specifies how the free energy of a system
is affected by a change in the scattering length \cite{Tan:0508}:
\begin{equation}
\frac{d~~~~}{d(1/a)}F =
- \frac{\hbar^2}{4 \pi m} C.
\label{adiabatic}
\end{equation}
Many other universal relations involving the contact were subsequently
derived \cite{Braaten:2010if}.

The universal relation  for the inelastic two-atom loss rate in Eq.~(\ref{uni2loss})
can be derived by expressing the Hermitian operator $K$
in the effective Hamiltonian $H-iK$ as
\begin{equation}
K = \frac{\hbar^2}{4\pi m}{\rm Im} \big( 1/a \big)
\int\! d^3r\, \Phi^\dagger \Phi,
\label{Kint-psi}
\end{equation}
where the local Lindblad operator is
\begin{equation}
\Phi = g_0 \psi_2 \psi_1.
\label{Phi-psi2}
\end{equation}
The operator product $ \psi_2 \psi_1$ is singular:
its matrix elements diverge linearly with the cutoff $\Lambda$.
The multiplication by $g_0$,
which according to Eq.~\eqref{g0} decreases as $1/\Lambda$ for large $\Lambda$,
makes $\Phi$ a finite operator
whose matrix elements have well-behaved limits as $\Lambda \to \infty$.
In a spin-relaxation scattering reaction, the decreases in the atom numbers
$N_1$ and $N_2$ are both $k_i=1$.  According to Eq.~\eqref{Nevol:Phi},
the rates of decrease in their system averages  are therefore
\begin{eqnarray}
\frac{d\ }{dt} \langle N_1 \rangle= \frac{d\ }{dt} \langle N_2 \rangle =
- \frac{\hbar}{2 \pi m}{\rm Im} \big( 1/a \big)
\int\! d^3r\, \left\langle \Phi^\dagger \Phi \right\rangle.
\label{N12evol:psi}
\end{eqnarray}
The universal loss rate in Eq.~(\ref{uni2loss}) then follows from the identification
of the contact density as  \cite{Braaten:2008uh}
\begin{eqnarray}
{\cal C} =
\left\langle\Phi^\dagger \Phi \right\rangle.
\label{contactdensity}
\end{eqnarray}
This identification can be verified by using the adiabatic relation in Eq.~\eqref{adiabatic}.
The only dependence of the Hamiltonian on the scattering length $a$
is in the  interaction term in the Hamiltonian density ${\cal H}_{\rm int}$ in Eq.~\eqref{Hint-psi}.
If the tiny imaginary part of the scattering length is neglected,
the derivative of the Hermitian part $H$ of the effective Hamiltonian with respect to $1/a$ is
\begin{equation}
\frac{d~~~~}{d(1/a)}H = - \frac{\hbar^2}{4 \pi m} \,
\int\! d^3r\, \Phi^\dagger \Phi.
\label{dHint/da}
\end{equation}
By the Feynman-Hellman theorem, the system average of the left side of this equation
is the left side of the adiabatic relation in Eq.~\eqref{adiabatic}.
With the identification of the contact density in Eq.~\eqref{contactdensity},
the system average of the right side of Eq.~\eqref{dHint/da}
is the right side of the adiabatic relation in Eq.~\eqref{adiabatic}.
This completes the derivation of the inelastic two-atom loss rate  in Eq.~(\ref{uni2loss})
and the adiabatic relation in Eq.~\eqref{adiabatic}.

In Ref.~\cite{Braaten:2013eya},
the authors presented an incorrect derivation of the universal
relation for the inelastic two-atom loss rate
that suggests that there are additional terms on the right side of Eq.~\eqref{uni2loss}.
They assumed  the density matrix $\rho$ satisfies the naive time evolution equation in
Eq.~\eqref{eq:drhodt-naive}.  This equation implies that the number of atoms is conserved,
but that the probability ${\rm Tr}(\rho)$ decreases with time.
In Ref.~\cite{Braaten:2013eya}, the authors assumed that
the mean atom number $\langle N \rangle$
was given by ${\rm Tr}(\rho N)/{\rm Tr}(\rho)$.
The resulting expression for the time derivative of $\langle N \rangle$
can be expressed as a double integral over space of an expression
involving the product of the number density operator and the Hamiltonian density operator.
Upon applying the operator product expansion to the product of these operators,
they obtained an expansion  for $(d/dt)\langle N \rangle$
in terms of increasingly higher-dimension operators
with the leading term given by Eq.~\eqref{uni2loss}.
The derivation of the universal relation  using the
Lindblad equation makes it clear that there are no additional terms
beyond the contact term in Eq.~(\ref{uni2loss}).

\subsection{Inelastic Three-Atom Loss Rate}
\label{sec:3atomloss}

Bosonic atoms that are all in the same hyperfine state
can be described by a nonrelativistic quantum field theory with a quantum field $\psi$ \cite{Braaten:2004rn}.
If the scattering length $a$ is much larger than the range of interactions of the atoms,
two-atom interactions and three-atom interactions must both be  treated nonperturbatively.
The resulting quantum field theory is characterized by a discrete scaling symmetry
in which lengths and time are multiplied by integer multiples of $\lambda_0$ and $\lambda_0^2$,
respectively, where  $\lambda_0= e^{\pi/s_0}$ and $s_0\approx1.00624$ is a universal number.
Three-atom interactions are determined by a parameter $\Lambda_*$ with dimensions of momentum,
upon which physical observables can only depend log-periodically.
If $a$ is positive, two-atom interactions produce a weakly bound diatomic molecule  called the {\it shallow dimer}.
Regardless of the sign of $a$, three-atom interactions produce a sequence of
weakly bound triatomic molecules called {\it Efimov trimers} \cite{Braaten:2004rn}.
In Ref.~\cite{Braaten:2011sz}, several universal relations were derived that hold for any system consisting
of identical bosons.
The universal relations relate various properties of the system to two extensive thermodynamic variables:
the 2-body contact $C_2$, which is the analog of the contact for  fermions with two spin states,
and the 3-body contact $C_3$.

The atoms that are studied in cold atom experiments
form deeply bound diatomic molecules ({\it deep dimers}) whose binding energies are
orders of magnitude larger than the energy scales of the cold trapped atoms.
The deep dimers provide various pathways for deeply inelastic reactions
that result in the loss of three atoms.  They include (a)
three-body recombination, in which three low-energy atoms collide
and two of them bind into a deep dimer,
(b) dimer relaxation, in which a shallow dimer and an atom scatter into a deep dimer and an atom,
and (c) Efimov trimer decay into a deep dimer and an atom.
In Ref.~\cite{Smith:2013eoa}, a universal relation for the three-atom inelastic loss rate
was presented but not derived.
In this section, we use the Lindblad equation to sketch the derivation of this relation.

To derive the universal relation for the three-atom inelastic loss rate,
we follow the formalism laid out in Ref.~\cite{Braaten:2011sz}.
In the absence of deep dimers, the interaction term in the Hamiltonian can be written as
\begin{equation}
H_{\rm int} = \frac{g_2}{4m} \, \int\! d^3r\, (\psi^2)^\dagger\psi^2
+\frac{g_3}{36m} \, \int\! d^3r\, (\psi^3)^\dagger\psi^3.
\label{Hint-psi-3b}
\end{equation}
Since this effective field theory has ultraviolet divergences, it
must be regularized.  The bare coupling constants $g_2$ and $g_3$ depend on
an ultraviolet momentum cutoff $\Lambda$:
\begin{subequations}
\begin{eqnarray}
g_2(\Lambda) &=& \frac{8 \pi}{1/a-(2/\pi)\Lambda}\,,
\label{eq:g2}
\\
g_3(\Lambda) &=& h_0 \,\frac{9g_2^2}{\Lambda^2}\,
\frac{\sin(s_0 \ln(\Lambda/\Lambda_*)-\arctan(1/s_0))}
{\sin(s_0 \ln(\Lambda/\Lambda_*)+\arctan(1/s_0))} ,
\label{eq:g3}
\end{eqnarray}
\end{subequations}
where $a$ is the two-body scattering length of the bosons
and $\Lambda_*$ is a three-body parameter
introduced in Ref.~\cite{Bedaque:1998kg}.
The three-body parameter $\Lambda_*$ can be determined
from any three-body datum, such as the binding energy
of an Efimov trimer or the atom-dimer scattering length.
(See Ref.~\cite{Braaten:2004rn} for explicit relations).
The numerical prefactor $h_0$ in Eq.~\eqref{eq:g3} depends on the ultraviolet
cutoff prescription and has the value $h_0\approx 0.879$
if $\Lambda$ is a sharp momentum cutoff \cite{Braaten:2011sz}.

Three-atom losses from deeply inelastic reactions
involving deep dimers can be taken into account by adding an imaginary part to the
coupling constant $g_3$ in the Hamiltonian in Eq.~\eqref{Hint-psi-3b}.
The resulting anti-Hermitian part of the Hamiltonian is $-i K$, where $K$ is
\begin{equation}
K = -\frac{{\rm Im}(g_3)}{36m}
\int\! d^3r\, (\psi^{3})^\dagger\psi^3.
\label{Hint-imag-3b}
\end{equation}
Physical observables can be expressed particularly conveniently in terms of
$\Lambda_*$ and an additional real three-body
loss parameter $\eta_*$ \cite{Braaten:2003yc,Braaten:2004rn}
defined by analytically continuing the three-body coupling constant $g_3$ in Eq.~\eqref{eq:g3}
to complex values using the substitution
\begin{equation}
\ln\Lambda_* \to \ln\Lambda_* +i\eta_*/s_0.
\end{equation}

Inserting $k_i=3$ into the expression in Eq.~\eqref{Nevol:Phi} for the three-atom
inelastic loss rate,
we obtain the universal relation
\begin{equation}
\frac{d\ }{dt} \langle N \rangle=
- \frac{6\hbar}{m s_0}\sinh(2\eta_*) C_3,
\label{uni3loss}
\end{equation}
where $C_3$ is the three-body contact,
which can be expressed as
\begin{equation}
C_3 = f(\Lambda)  \int d^3r\, \langle (\psi^{3})^\dagger\psi^3 \rangle.
\label{eq:3bcontact}
\end{equation}
The prefactor $f(\Lambda)$ depends on the ultraviolet cutoff \cite{Braaten:2011sz}.
Since its precise form is not very insightful, it will not be given here.
The three-body contact can also be defined
in terms of the expectation value of
the logarithmic derivative with respect to $\Lambda_*$ of the Hermitian part of the
effective Hamiltonian in Eq.~\eqref{Hint-psi-3b} at fixed $a$~\cite{Braaten:2011sz}:
\begin{equation}
\Lambda_*\frac{\partial ~~}{\partial\Lambda_*}\langle H \rangle\bigg|_a =
-\frac{2\hbar^2}{m}C_3.
\end{equation}
The universal relation in Eq.~\eqref{uni3loss},
with the factor  $\sinh(2\eta_*)$ approximated by $2\eta_*$,
was previously given in Ref.~\cite{Smith:2013eoa}.
It was also given previously  in
Ref.~\cite{Werner:2012aa} for the special case of an Efimov trimer state.

\section{Summary}
\label{sec:Summary}

In this paper, we have shown that integrating out the high-momentum atoms produced by
deeply inelastic reactions produces an open effective field theory
in which the time evolution of the  effective density matrix is given by the Lindblad equation.
The effective density matrix can be obtained from the density matrix of the full theory
by taking a partial trace over states that include high-momentum atoms
and then carrying out an appropriate time average to eliminate high-frequency transients.
The Lindblad operators are local, and they can be deduced from the anti-Hermitian terms
in the effective Hamiltonian.
The Lindblad terms in the evolution equation are essential to get the
correct evolution equation for the mean atom number.
We used the Lindblad equation to present the first correct  derivation of the
universal relation for the two-atom inelastic loss rate for fermionic atoms
with two spin states that  interact through a large scattering length.
We also used the Lindblad equation to present the first derivation of the
universal relation for the three-atom inelastic loss rate for identical bosonic atoms
with  a large scattering length.

The Lindblad equation has many other applications to atom loss processes
in cold atom physics.  An obvious extension is to heteronuclear
systems, for which there are two types of Efimov trimers.
Due to the smaller discrete scaling factor associated with one type of Efimov trimer,
atom loss processes in such systems have  recently been of
great theoretical and experimental interest
\cite{Helfrich:2010yr,Tung:2014,Pires:2014,Petrov:2015}.
The Lindblad equation could be applied to fermionic atoms with two spin states
on the upper branch of the two-atom spectrum at scattering lengths for which
three-body recombination into the shallow dimer is a
deeply inelastic reaction \cite{Shenoy:2011}.
It could also be applied to losses of dipolar atoms from three-body
recombination  into deep dimers \cite{Ticknor:2010,Wang:2011a,Wang:2011b}.

\begin{acknowledgments}
E.B.\ and H.-W.H.\ acknowledge useful discussions with Shina Tan.
The research of E.B.\ was supported in part by the Department of Energy
under grant DE-FG02-05ER15715, by the
National Science Foundation under grant PHY-1310862,
and by the Simons Foundation.
The research of H.-W.H.\ was supported in part
by the BMBF under contract 05P15RDFN1,
and by the Deutsche Forschungsgemeinschaft (SFB 1245).
The research of G.P.L.\ was supported in part by the National Science Foundation.
\end{acknowledgments}