and consequently the density of states is a sum of delta functions,

$\nu(\omega)=\pi\nu_{0}\sum_{k=1}^{\infty}P_{k}\left[\delta(\omega-
E_{k})+\delta(\omega+E_{k})\right],$ (148)

where $E_{k}$ are solutions of $\cos\left[\varphi(E_{k})\right]=0$, or,
equivalently, of $\varphi(E_{k})=\pi(k-1/2)$. Imaginary part of $\nu(\omega)$
comes from poles of $\tan\varphi(\omega)$ at $\omega=\pm E_{k}-i0^{+}$. It is
straightforward to show using Eq. (142) that $A(\omega)$ vanishes as
$1/\omega^{2}$ and $B(\omega)\to 1$ as $\omega\to\infty$. At $\Omega=0$ and
large $\omega$, Eq. (146) takes the form $\varphi^{\prime}=2\omega/\pi$.
Therefore, $\varphi(\omega)\approx\omega^{2}/\pi\equiv\varphi_{0}(\omega)$
which implies the leading large $k$ asymptotic behavior (82) of $E_{k}$.

Let us analyze Eq. (146) at large $\omega$ and small but finite $\Omega$.
Corrections to the right hand side due to finite $\Omega$ are suppressed by a
factor of $1/\omega^{2}$. In the zeroth order in $\Omega$,
$\varphi^{\prime}\sim\omega$ and $\varphi^{\prime\prime}\sim 1$. Therefore,
the term containing $\varphi^{\prime\prime}$ is negligible and we have

$\varphi^{\prime}-\frac{\Omega}{2}{\varphi^{\prime}}^{2}\tan\varphi=\frac{2\omega}{\pi}.$
(149)

The $\tan\varphi$ term is important near $\omega=E_{k}$ where $\tan\varphi$
diverges. Near these points Eq. (149) becomes

$y^{\prime}+\frac{\Omega}{2}\frac{\phantom{a}{y^{\prime}}^{2}}{y}=\frac{2E_{k}}{\pi},$
(150)

where $y=\varphi(\omega)-\varphi_{0}(E_{k})=\varphi(\omega)-E_{k}^{2}/\pi$.
Solving for $y^{\prime}$, we find

$y^{\prime}=-\frac{1}{\Omega}\left(y+\sqrt{y^{2}+by}\right),\quad
b=\frac{8\Omega E_{k}}{\pi}.$ (151)

The plus sign is dictated by the requirement that for $\Omega\to 0$ we recover
the zeroth order equation $y^{\prime}\approx 2E_{k}/\pi$. In zeroth order in
$\Omega$, $y=2\omega E_{k}/\pi$ is real and the density of states (147) is
zero except at $\omega=E_{k}$ In the next order in $\Omega$, $y$ and therefore
$\varphi(\omega)$ acquire an imaginary part proportional to $\Omega$. We see
this from Eq. (151) – the square root is imaginary and of the order $\Omega$
for $-b<y<0$. Upon integration over $\omega$, it gives rise to an imaginary
part of $y$ of the order $\Omega$. The density of states (147) is therefore
nonzero in the interval $(\omega_{1},\omega_{2})$ of $\omega$ for which $y$
falls in between $-b$ and 0. The length of this interval is the bandwidth we
are after.

Equation (151) integrates by variable separation method to

$\displaystyle\frac{b}{u(y)}-\frac{1}{2}\ln
u^{2}(y)=\frac{2\omega}{\Omega}+\mbox{const},$ (152) $\displaystyle
u(y)=b+2y+\sqrt{y^{2}+by}.$ (153)

To determine $\omega_{1}$ and $\omega_{2}$, we set $y=-b$ and $y=0$,
respectively, in the left hand side of Eq. (152). We find
$\omega_{2}-\omega_{1}=\Omega$. Thus, the discrete level $E_{k}$ splits into a
narrow band of width $\Omega$ similar to how atomic energy levels split into
bands when atoms form a lattice and atomic orbitals hybridize.

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# Investigating Dark Energy with Black Hole Binaries

Laura Mersini-Houghton
Adam Kelleher [UNCCH][DAMTP]  Department of Physics and Astronomy, The
University of North Carolina at Chapel Hill, Phillips Hall, CB # 3255, Chapel
Hill, NC 27599-3255, USA  DAMTP, Center for Mathematical Sciences, University
of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

###### Abstract

The accelerated expansion of the universe is ascribed to the existence of dark
energy. Black holes accretion of dark energy induces a mass change
proportional to the energy density and pressure of the background dark energy
fluid. The time scale during which the mass of black holes changes
considerably is too long relative to the age of the universe, thus beyond
detection possibilities. We propose to take advantage of the modified black
hole masses for exploring the equation of state $w[z]$ of dark energy, by
investigating the evolution of supermassive black hole binaries on a dark
energy background. Deriving the signatures of dark energy accretion on the
evolution of binaries, we find that dark energy imprints on the emitted
gravitational radiation and on the changes in the orbital radius of the binary
can be within detection limits for certain supermassive black hole binaries.
In this talk I describe how binaries can provide a useful tool in obtaining
complementary information on the nature of dark energy, based on the work done
with A.Kelleher.

##  1 INTRODUCTION

Dark energy drives the universe into an accelerated expansion. A decade after
its discovery, the nature and origin of dark energy remain as elusive as ever.
The detection of time variations in the equation of state $w[z]$ of dark
energy, given by the ratio $w[z]=p/\rho$ of its pressure $p$ to the energy
density $\rho$, is notoriously hard to constrain. A precise measurement of
$w[z]$ will guide our theoretical exploration and shed light into the type of
dark energy, thereby helping us understand our present universe and its future
evolution. For these reasons, a precision measurement of $w[z]$ is of the
highest priority in cosmology. Our current bounds are derived by a combination
of data from cosmic microwave background radiation (CMBR), Large Scale
Structure (LSS) surveys, baryon acoustic oscillation (BAO) and Supernova $1A$
(Sn1A) data. Current data analysis constrains the dark energy equation of
state to $-0.14<1+w<0.12$ [2]. While this represents progress, we would like
to know if $1+w$ is positive, negative or $0$, and whether or not w changes
with time. Each case corresponds to quite different types of this mysterious
energy and, it leads to dramatically different predictions for the future
evolution of the universe.

Dark energy evolves slowly with time for the cases when $1+w$ is positive or
negative. However, since the bounds on its equation of state are close to a
pure cosmological constant, $w=-1$, then unlike matter dark energy does not
cluster. For these reasons a universe filled with dark energy can be
reasonably assumed to be a background of a perfect cosmic fluid. T.Jacobson
and other authors [14, 6] took dark energy to be a cosmic ’fluid’ and showed
that the mass of a black hole changes due to accretion of this background
’fluid’. The mass of the black hole increases or decreases, depending on the
sign of $1+w$. This effect would be hard to observe in a single black hole, as
the time scale for this phenomena is quite slow relative to the age of the
universe. For example the evolution time scale $\tau$ for a solar mass black
hole is about $10^{32}yrs$. We proposed in [1] to use the evolution of
supermassive black hole binaries instead of single black hole accretion to
probe dark energy. The reason for our proposed method relies on the fact that,
instead of measurements of the evolution time scale of a single black hole,
the ’footprints’ of $w[z]$ for supermassive black hole binaries can be
observed and tracked down through the modifications introduced by dark energy
accretion in the the orbital separation of the binary and its emitted
gravitational waves. For certain binaries dark energy modifications can be
detectable by gravitational waves or X-ray and radio measurements. The
information obtained by these modifications should help increase our bounds on
$w[z]$. The use of binaries for probing dark energy is a different approach
from our current methods of large scale experiments, since the observation
utilizes localized systems, thereby avoiding noise inherited by propagation of
signals through the vast structures of the universe.

##  2 Dark Energy Accretion by Black Holes

###  2.1 Evolution of a Single Black Hole on the Dark Energy Background:
Review

Let us assume that dark energy corresponds to a perfect fluid with an energy
density $\rho$, pressure $p$ and equation of state $w[z]$ as a function of the
redshift $z$, related by $p=w[z]\rho$. The case of a pure vacuum energy would
have $\rho=-p$. As long as $w[z]$ is close to -1 and varies slowly with time,
the metric solution from Einstein’s equations is taken to be approximately
that of a De Sitter geometry, i.e. for $w=-1$.

A particularly interesting case is that of a Schwarzschild black hole in the
background of the dark energy cosmic fluid. This case was studied in [14, 3,
6, 7] and the fluid accretion flowing through the black hole horizon was
solved analytically. The perfect fluid energy-momentum tensor with equation of
state $p=w\rho$ is assumed to be

$T^{\mu\nu}=\rho(1+w)u^{\mu}u^{\nu}+w\rho g^{\mu\nu}$ (1)

where $g^{\mu\nu}$ is the inverse of the Schwarzschild metric,
$g_{\mu\nu}=diag(-(1-2M/r),(1-2M/r)^{-1},r^{2},r^{2}sin^{2}\theta)$, and for
the fluid 4-velocity, $u^{\mu}u_{\mu}=-1$. Integrating the energy-momentum
conservation equation and the projection of the fluid 4-momentum into it leads
to the expression

$u\left(\frac{M}{r}\right)^{2}exp\left[\int_{\rho_{\infty}}^{\rho}\frac{d\rho^{\prime}}{\rho^{\prime}+p(\rho^{\prime})}\right]=-A$
(2)

and

$\displaystyle\left(\rho+p\right)\\!\left(1-\frac{2M}{r}+u^{2}\right)^{1/2}\\!\\!$
(3) $\displaystyle\times
exp\left[-\int_{\rho_{\infty}}^{\rho}\frac{d\rho^{\prime}}{\rho^{\prime}+p(\rho^{\prime})}\right]=C$

Following Babichev et al [7], these expressions are manipulated into one for
$r^{2}T_{0}^{r}$. Then, integrating the conservation law over the volume
within the event horizon yields

$\dot{M}=-4\pi r^{2}T_{0}^{r}=4\pi
AM^{2}\left[\rho_{\infty}+p(\rho_{\infty})\right]$ (4)

This expression can be integrated to give M as a function of time, neglecting
the cosmological time evolution of $\rho_{\infty}$:

$M(t)=\frac{M(0)}{1-\frac{t}{\tau}}.$ (5)

The timescale for the accretion of dark energy Eq. 5 is given by the parameter
$\tau$, and is $\tau=\frac{1}{(4\pi AM(0)[\rho_{\infty}+p(\rho_{\infty})])}$.
For a black hole of a mass $m=am_{s}$ which is $a$ times larger than a solar
mass $m_{s}$, the evolution timescale is roughly $\tau=10^{32}/a$ years. This
is usually much longer than the age of the universe, thus beyond observational
feasibility. But the detection possibilities of the dark energy accretion
improve dramatically for the case of black hole binaries. The reason is that
the evolution of the black hole binaries in the background of dark energy is
different from that of a single hole. Therefore, as described below, detection
of the modifications to the orbital radius and the emission of gravitational
radiation from these binaries in the background of the dark energy fluid is
within reach, [1] .

###  2.2 Evolution of Black Hole Binaries in the Background of Dark Energy

The Hulse-Taylor effect predicts a decrease of the orbital radius of the
binary due the energy lost by the emission of gravitational radiation. This
prediction has been succesfully tested. The evolution of black hole binaries
in the background of dark energy is modified relative to the Hulse-Taylor
effect, due to the accretion of dark energy by the stars in the binary. Since
the change in the mass of the stars, Eq.5, has a direct dependence on the
parameters of the dark energy being accreted, specifically on $w[z]$, then
this information is carried out on the amplitude and the power of
gravitational radiation produced [1] by the binary. The dark energy accretion
also imprints modifications in the orbital separation and merging time.

The modifications induced from the background dark energy fluid onto the
evolution of the binary, namely on the frequency $\omega$ of gravitational
waves emitted by the binary and, on the orbital separation $R$, can be derived
as follows: the change of the gravitational energy of the binary is equal to
the power lost due to gravitational radiation[5]:

$\frac{d}{dt}\left(m_{1}+m_{2}-\frac{1}{2}\frac{m_{1}m_{2}}{R}\right)=P_{GW}$
(6)

where

$P_{GW}=\frac{-32G^{4}}{5c^{5}}\left[\frac{m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})}{R^{5}}\right]$
(7)

This is a temporal equation. Power losses via emission of gravitational
radiation drives the binary’s configuration to a new gravitational equlibrium
separation. As a result the orbital radius $R$ decreases and eventually the
stars inspiral and merge. For binaries immersed in the dark energy fluid, it
should be noticed that the point of gravitationally stable configurations of
the binary at each moment is now driven by two effects: the usual loss of
energy via gravitational waves emission; and, the stars changing mass (leading
to a change of the gravitational energy of the binary) due to dark energy
accretion. The masses of the two black holes in the binary, $m_{1}$ and
$m_{2}$, are increasing or decreasing with time, Eq.5, depending on $(1+w)$
being positive or negative. Therefore the two terms that induce changes in the
orbital radius and period, namely: modifications due to dark energy accretion,
and modifications due to energy losses from gravitational radiation compete
with each other and determine the evolution of the orbit.

Without loss of generality, this expression can be algebraically simplified by
taking the two stars to be of equal mass, $m_{1}=m_{2}=m$ with $m_{0}=m(0)$.
Replacing the constant mass of a black hole with the new expression, the mass
rate of change of the black holes induced by the dark energy accretion from
Eq.5, leads to a differential equation for the evolution of the binary R(t).

$\displaystyle{}R^{3}\frac{dR}{dt}=-\frac{64}{5}\frac{2m_{0}^{3}}{\left[1-2Ltm_{0}\right]^{3}}$
(8)
$\displaystyle{}-\\!\\!\left[\frac{-4LR^{4}m_{0}}{\left[1-2Ltm_{0}\right]}+8LR^{6}\right]$

where the parameter $L$ denotes $L=\frac{c^{3}}{2G^{2}m_{0}\tau}$. This
parameter contains all the modifications induced by the dark energy background
and the new modification terms due to dark energy to the orbital radius can be
tracked down from all the terms in Eqs.9 that contain $L$. It should be noted
that $L\simeq(1+w)$ is positive for quintessence type fluids; $L$ is negative
for phantom type fluids ($1+w<0$); and, it becomes identically zero for
$w=-1$. In the latter case, all the modifications due to dark energy on the
orbital radius vanish. The sign of $L\neq 0$ determines the behaviour of the
orbit, i.e. whether it grows or decreases with time for the cases when the
’$L-terms$’ dominate over the conventional gravitational waves term.

An approximate solution to Eq.2.2 for the case when dark energy changes
adiabatically is [1]

$\displaystyle R(t,w)=\\!R_{0}[1+16Lm_{0}\left(\frac{G^{2}}{c^{3}}\right)t$
(9)
$\displaystyle-32LR_{0}\left(\frac{G}{c}\right)t-\frac{64}{5}\left(4\frac{G^{3}}{c^{5}}\right)\left[\frac{8tm_{0}^{3}}{R_{0}^{4}}\right]]^{1/4},$

The fourth term in the expression Eq.9 corresponds to the conventional Hulse-
Taylor term. As a consistency check, the Hulse-Taylor equation is recovered in
the limit when our universe approaches a DeSitter geometry $w\rightarrow-1$,
(i.e. $L\rightarrow 0$). The Hulse-Taylor term describes the changes in the
orbital radius that result for the energy losses of a binary from the emitted
gravitational radiation. The terms proportional to $L$ are the new
modification terms to the evolution of black hole binaries. They account for
the effects of dark energy accretion by the system. It should be noticed that
one of the dark energy modification terms is of opposite sign to the Hulse-
Taylor term; and, the type of dark energy with $(1+w)$ positive or negative
leads to different types of evolution for the binary. An analysis of the
solution for $R[z]$ shows that the dark energy terms can dominate the
evolution of the orbital radius, Eq.9, for certain cases of supermassive black
hole binaries with large separation, quantified below. The different time
evolution of the two terms, dark energy and the conventional Hulse-Taylor one,
in Eq.9 on $R$ and $m$, especially the linear dependence of $\dot{R}$ on the
equation of state of dark energy $1+w$, allow us to discriminate among the
modifications to the orbit induced by dark energy and for probing the dark
energy equation of state $w[z]$ by observing the rate at which the orbit
change $\dot{R}$.

In order to quantify the analysis of the above expression Eq.9 and discuss the
interplay between the dark energy and Hulse-Taylor types of modifications in
the orbital radius we can parameterize the binary as follows: let the initial
mass of the star, (before modifications due to accretion), be $m_{0}=am_{s}$
where $m_{s}$ is a solar mass and $a$ a parameter; and the orbital radius be
$\beta$ times larger than the Schwarzchild radius of each star
$R_{0}=2m_{0}\beta\frac{G}{c^{2}}$. Then, the ratio of the two correction
terms to the binary’s orbit $R$ in Eq.9, the Hulse-Taylor correction due to
the emission of gravitational waves (GW), and corrections due to dark energy
(DE) accretion (terms containing $L$), is

$\frac{GWcorrection}{DEcorrection}=\frac{10^{45}}{(2\beta)^{5}(1+w)a^{2}}\geq
1$ (10)

We can use Eq.10 to quantify the classes of binaries for which the dark energy
correction terms dominate over the Hulse-Taylor term. Clearly for large enough
separations ($\beta$) or masses of the black holes ($a\simeq 10^{18}$) the
corrections due to dark energy can dominate over gravitational radiation.

Observing this effect via gravitational waves experiments, we need both
parameters of the binary to be such that they favor the dark energy
corrections over the Hulse-Taylor corrections to $R$, while at the same time
being within observable ranges of frequency windows. The frequency and
amplitude of gravitational radiation from these systems are given by

$f=\frac{10^{5}}{(2\beta)^{3/2}a}$ (11)

and

$h=\frac{1}{r}\frac{2}{\beta}a10^{3},$ (12)

where r is the distance of the binary from the observer. Eq.12 constrains the
second parameter $\beta$ not to be too large. For example LIGO is designed to
detect radiation around 150 Hz optimally, and with an amplitude greater than
$h=10^{-23}Hz^{1/2}$.[8] LISA can detect much lower frequency radiation, down
to $\sim 10^{-5}Hz$.[9]. Since for both GW experiments, the binary radius
$\beta$ can not be too large, then our requirement of Eq.10 for using binaries
to probe dark energy via their modifications on the orbital radius $R$, can be
fulfilled by considering supermassive black holes, $a\gg 1$, (see [1] for
specific examples).

###  2.3 Observation Candidates?

There have now been observations of black hole binary systems, and we can
assess the effects of dark energy accretion on these systems. One example is
galaxy 0402+379, observed in 2007 with VLBA. It has parameters $a=210^{8}$,
$2\beta=10^{6}$, and $r=10^{26}m$. This case, while not observable through
gravitational radiation, is one that is sensitive to the sign of $1+w$, and
it’s merging time is either greatly accelerated (1,000 years instead of 60,000
years) by the effect of dark energy accretion when $1+w>0$, or it is driven
apart when $1+w<0$.

Another example is Radio Galaxy OJ287, observed in 2008 with VLBA [10]. This
system is more complicated, since one of the black holes is more massive than
the other. The parameters for this system are $R_{0}\simeq 10^{20}m$, or
$2\beta\simeq 10^{7}$ and $r\simeq 3.5Mly\simeq 10^{22}m$. While it is
straightforward to treat this system with the above equations for $m_{1}\neq
m_{2}$, the point is well illustrated by a simpler system of comparable
parameters. We will use $a\simeq 10^{9}$. This gives $f\simeq 10^{-9}Hz$ and
$h\simeq 10^{-}20$. Again, this system falls outside of observable ranges for
gravitational radiation, but the merging time is shortened by three orders of
magnitude when $1+w>0$, and the black holes are pulled apart for $1+w<0$.

##  3 Conclusions

It is remarkable that localized systems like black hole binaries can be used
to provide information about dark energy. The method described here provides a
complementary way of probing the dark energy equation of state by using
supermassive black hole binaries. Its strength lies on the fact that it avoids
the noise inherited by the signal propagating through the vastness of
structure in the universe and it takes advantage of existing experiments,
initially designed to investigate gravitational waves or structure.
Supermassive black hole binaries could soon help to shed light on the nature
of dark energy.

By observing gravitational radiation from black hole binaries, we might
distinguish the waveform from a system accreting dark energy from one that
does not. Observing changes in the orbital radius over a fraction of the
binary’s period with X-ray and Radio measurement is entirely possible with our
current experiments and provides a wealth of information on $w[z]$ through the
dark energy correction terms in Eq.8,9. This method should helps us pinpoint
at least whether dark energy is a quintessence or a phantom type, or simply a
cosmological constant. Observing how the waveform differs from the
cosmological constant case gives further information about the sign of $1+w$
[11]. If $1+w>0$, the masses of the black holes will increase, they will
spiral in faster, and this will result also in a faster increase in frequency
of their gravitational radiation. If $1+w<0$ [12], the masses decrease and the
system can be pulled apart by the effects of dark energy accretion.

It is possible we are close to collecting evidence from existing observed
supermassive black hole binaries that the phantom type $(1+w)<0$ which rips
the binary apart may be already disfavored. One such binary of supermassive
black holes was recently observed [15]. An interesting question is whether
this method can be used to test theories of modified gravity and to
discriminate those from the dark energy models. The evolution of binaries on
the background of modified gravity is currently under investigation.

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