# The interface of gravity and dark energy

Kristen Lackeos  Max-Planck-Institut für Radioastronomie (MPIfR), Auf dem
Hügel 69, 53121, Bonn, Germany
NASA Postdoctoral Program Fellow, NASA Marshall Space Flight Center,
Huntsville, AL 35812
Richard Lieu  Department of Physics and Astronomy, University of Alabama,
Huntsville, AL 35899
<EMAIL_ADDRESS>
###### Abstract

At sufficiently large radii dark energy modifies the behavior of (a) bound
orbits around a galaxy and (b) virialized gas in a cluster of galaxies. Dark
energy also provides a natural cutoff to a cluster’s dark matter halo. In (a)
there exists a maximum circular orbit beyond which periodic motion is no
longer possible, and orbital evolution near critical binding is analytically
calculable using an adiabatic invariant integral. The finding implicates the
study of wide galaxy pairs. In (b), dark energy necessitates the use of a
generalized Virial Theorem to describe gas at the outskirts of a cluster. When
coupled to the baryonic escape condition, aided by dark energy, the results is
a radius beyond which the continued establishment of a hydrostatic halo of
thermalized baryons is untenable. This leads to a theoretically motivated
virial radius. We use this theory to probe the structure of a cluster’s
baryonic halo and apply it to X-ray and weak-lensing data collected on cluster
Abell 1835. We find that gas in its outskirts deviates significantly from
hydrostatic equilibrium beginning at $\sim 1.3\ {\rm Mpc}$, the ‘inner’ virial
radius. We also define a model dependent dark matter halo cutoff radius to
A1835. The dark matter cutoff gives an upper limit to the cluster’s total mass
of $\sim 7\times 10^{15}M_{\odot}$. Moreover, it is possible to derive an
‘outer’ hydrostatic equilibrium cutoff radius given a dark matter cutoff
radius. A region of cluster gas transport and turbulence occurs between the
inner and outer cutoff radii.

##  1 Introduction

The large scale structures we observe today are believed to have originated
from seed density enhancements of the early universe that underwent linear
gravitational growth until they ‘condensed’ out of the Hubble flow and grew at
an accelerated non-linear pace during the era of matter domination. As dark
energy started to dominate, the region where a structure ‘merges’ with (or
‘interfaces’) the cosmic substratum becomes one of particular interest and
importance, for at least two reasons.

The first is the need to better understand the large scale dynamics of
structures, in particular their mutual ‘action at a distance’ – at what point
does gravity finish and the repulsive force of dark energy take over? If the
universe has non-vanishing $\Lambda$, there will be a clear ‘zone of
influence’ of a structure, akin to the Debye shielding effect of
electrostatics. More precisely because gravity and dark energy act in
opposition a ‘standoff’ radius exists at which the two forces cancel each
other, _i.e._ despite the long range effect of gravity (when it is the only
force) a boundary must not only be in place but it must also be very sharp and
well-defined. In view of this, the evolution of galaxy pairs, specifically the
so-called ‘wide pairs’ at low redshifts (see _e.g._ Table 5 of Nordgren et al
(1998), see also Chengalur et al (1993)), is especially affected by the cutoff
and could even be used as a test bed of $\Lambda$, see Benisty et al (2023).

The second reason, which is what mooted this paper, is about the largest bound
structures, viz. clusters and groups of galaxies; in particular the
distribution of baryons within them. Although the total mass of one such
system is dominated by the relatively cold and collisionless dark matter, the
baryons are hot and virialized (or partially virialized) interacting
particles. The shape of the density and temperature profile at the outskirts
of a cluster or group is a topic of much current debate and investigation (see
Gouin et al (2023) and references therein). Such studies are concerned with
issues like the ‘gas fraction’ (defined as the mass ratio $f_{\rm gas}$ of
baryon to dark matter), as observations seem to indicate that this fraction is
beneath the cosmic mean value in the interior of clusters, yet increasing
monotonically towards the outskirts, pointing therefore to the phenomenon of
baryonic escape, into regions outside the ‘grip’ of the gravitational field of
the structure, regions in which a hydrostatic halo of thermalized baryons
cannot be established.

The structure of the paper is as follows. We begin in Section 2 by examining
the evolution of bound orbits under the influence of the Hubble expansion in
the dark energy dominated era. We consider three critical orbit scenarios for
a test particle orbiting around a cluster or galaxy where the following
condition holds: (a) the total energy per unit mass of a test particle is
equivalent to the minimum effective potential and (b) $\lesssim$ the effective
potential. Scenarios corresponding to circular and marginally bound orbits,
respectively. Scenario (c) is the condition for the largest possible quasi-
circular orbit defined for an unstable saddle point in the effective
potential. We compare turnaround radii estimates from the literature, which
can be defined for any epoch of the Universe, to the critical radius, $R_{\rm
saddle}$, defined under scenario (c). In Section 3 we expand on these three
critical orbit scenarios by exploring changes to the total energy per unit
mass and effective potential as the Universe evolves. Bound orbits in general
are explored through the use of an adiabatic invariant, treating the dark
energy component of the effective potential as a perturbation. From this we
derive the perihelion and aphelion drift rates. The virial theorem is enlisted
in Section 4, to apply the findings of the previous two sections to baryons at
the outskirts of galaxy clusters, and an application is made to cluster Abell
1835. Lastly, we conclude with a discussion and motivation to search for
widely separated galaxy pairs and circumgalactic emission as a new means to
experimentally test the dark energy paradigm.

##  2 The effect of Hubble expansion on the evolution of bound orbits

###  2.1 Birkhoff Theorem

We begin with a heuristic treatment of the motion of a small test particle of
unit mass within some physical distance $R$ from the center of a large
spherically symmetric mass concentration of mass excess $M$ in an otherwise
homogeneous universe. Since the environment experienced by the test mass is
isotropic, Birkhoff Theorem can be applied, so that if one is in the weak
gravity (large $R$) limit one may write down the energy equation with $J$
being the angular momentum as

$\dot{R}^{2}+\frac{J^{2}}{R^{2}}=\frac{2GM}{R}+\frac{8\pi
G}{3}\bar{\rho}R^{2}+{\rm constant};$ (1)

_i.e._ only the mass ‘underneath’ the particle is relevant and the constant
vanishes in the case of a flat universe (the value of this constant is
unimportant to our work here). Now (1) assumes that test particle is located
at a sufficiently large $R$ for the central clump $M$ to be regarded as a
point mass. If the density at the position of the particle is above the mean
density of the clump at that point, however, (1) will have to be modified to
take account of the internal mass profile of the clump.

Turning to the mean background density, if the universe comprises non-
relativistic matter of $\rho_{m}\sim 1/R^{3}$ and dark energy
$\rho_{\Lambda}=$ constant, one may write
$\bar{\rho}=\bar{\rho}_{m}+\rho_{\Lambda}$ and differentiate (1) w.r.t. the
time $t$ whilst holding the angular momentum $J$ fixed by virtue of space
isotropy, to obtain

$\ddot{R}=-\frac{GM}{R^{2}}-\frac{4\pi G}{3}\bar{\rho}_{m}R+\frac{8\pi
G}{3}\rho_{\Lambda}R+\frac{J^{2}}{R^{3}}.$ (2)

The relative sign difference between the matter and dark energy terms leads to
two opposing forces acting upon the test particle, one attractive and the
other repulsive. At the critical radius where they balance, there is no radial
acceleration of the particle, and bound orbits cannot exist beyond this
radius.

To make contact with the mechanism underlying the apparent phenomenon of
accelerated expansion, we must observe the effects of dark energy in our local
universe. With a general relativistic treatment, we will see below that
assuming a flat universe and using the Newtonian approximation are sufficient
for our purposes. As first demonstrated by McCrea & Milne (1934), ‘the
relativistic and Newtonian theories are indistinguishable in their predictions
of local phenomena’. They are the first to derive (2) using Newtonian physics
only111Specifically, their equation of motion gives the path of a particle in
a homogeneous universe $\rho_{m}=\rho_{m}(t)$, where its motion satisfies the
hydrodynamical equation of continuity. and point out a repulsive force due to
a cosmological constant, proportional to physical coordinate $R$. We extend
this understanding to observable phenomena at the outskirts of large scale
structures.

It is also possible to rewrite (2) as

$\ddot{R}=-\frac{GM}{R^{2}}+\frac{J^{2}}{R^{3}}+H_{0}^{2}[\Omega_{\Lambda}-\frac{1}{2}\Omega_{m}(1+z)^{3}]R,$
(3)

where $H_{0}$ is the present Hubble constant and $\Omega_{m}\approx 0.3$ and
$\Omega_{\Lambda}\approx 0.7$ are the normalized densities. Note that a proper
derivation by Nesseris & Perivolaropoulos (2004), who used the Friedmann-
Robertson-Walker metric as starting point, also yielded (3).

Other metrics describing the geometry of a ‘massive structure in an expanding
universe’ have been shown to produce (3) in the Newtonian limit. Baker (2001)
demonstrates this using a metric originally derived by McVittie (1933). To the
same end, Nandra et al (2012) (NLH1) derive a metric using the tetrad-based
approach in general relativity, providing an exposition of the full general
relativistic equations of motion, for all three curvature cases. In this way
NLH1 point out errors in McVittie’s open and closed universe solutions. And
for a flat universe, they derive a simple coordinate transformation showing
the metric they derive describes the same spacetime as McVittie’s and
reproduce (3), on the further assumption of low velocities.

###  2.2 Bound orbits

During earlier times $z\sim 0.67$ when the quantity inside the square
parantheses of (3) is negative, there was another attractive force in
additional to that of the central mass $M$, due to the cosmic substratum being
matter dominated. A bound orbit remains bound in those times. In the recent
dark energy era, however, the same quantity increases monotonically until it
saturates to become a positive constant. In this era, marginally bound orbits
can become free.

One can gain a considerable amount of further insight on how orbits evolve by
integrating (3) over a period in which the last term has not changed too much
(such an interval ranges from many orbits to a segment of an orbit, depending
on the size of the orbit and $M$) to obtain the energy equation

$\frac{1}{2}\dot{R}^{2}+V=E,$ (4)

where $E=E(t)$ is the total energy per unit mass of the satellite (test)
particle and

$V(R,t)=\frac{J^{2}}{2R^{2}}-\frac{GM}{R}-\frac{1}{2}H_{0}^{2}[\Omega_{\Lambda}-\frac{1}{2}\Omega_{m}(1+z)^{3}]R^{2}\\\
=\frac{J^{2}}{2R^{2}}-\frac{GM}{R}-\gamma R^{2}$ (5)

is the effective potential. The conservation of the ‘total energy’ $E$ is of
course only approximate. Defining the symbols

$q=\frac{\Omega_{m}(1+z)^{3}}{2}-\Omega_{\Lambda},~{}{\rm
and}~{}\gamma=-\frac{1}{2}qH_{0}^{2},$ (6)

with the former being the usual deceleration parameter and the latter a
positive quantity during dark energy domination. Note that $E$ changes at the
rate

$\dot{E}=-\dot{\gamma}R^{2},$ (7)

and this relation is exact.

The behavior of $V(R,t)$ is plotted in Figure 1 for cluster and galaxy-sized
central bodies. Several features are immediately apparent. First, in the
matter era all orbits are bound because the universe was collapsing upon
itself. Second, in the dark energy era the standard criterion of $E<0$ for
bound orbits is no longer sufficient, as there is a maximum for $V$ at large
$R$ which may be negative, and which in general is determined by the
quantities $J$, $M$, $z$, and the two normalized densities $\Omega_{\Lambda}$
and $\Omega_{m}$. The radius $R$ of extremum $V$ is a solution of the quartic

$GMR-J^{2}-2\gamma R^{4}=0.$ (8)

For small angular momentum $J$ and large central mass excess $M$, or more
precisely

$J^{2}\ll(GM)^{4/3}\gamma^{-1/3},$ (9)

the minimum and maximum of $V$ occur at approximately

$R_{\rm min}\approx\frac{J^{2}}{GM}+\frac{2\gamma J^{8}}{G^{5}M^{5}};~{}R_{\rm
max}\approx\left(\frac{GM}{2\gamma}\right)^{1/3},$ (10)

with

$V_{\rm min}=-\frac{G^{2}M^{2}}{2J^{2}}-\frac{\gamma J^{2}}{G^{2}M^{2}},\\\
V_{\rm max}=-3\left(\frac{\gamma G^{2}M^{2}}{4}\right)^{1/3}.$ (11)

Chernin (2001) derives an $R_{\rm max}=(3M/(8\pi\rho_{\Lambda}))^{1/3}$, for a
de Sitter universe, by equating a repulsive force, due to the total effective
vacuum mass $M_{\Lambda}$ in a sphere of radius $R$, where
$M_{\Lambda}\propto\rho_{\Lambda}R^{3}$, to the attractive force of a central
mass $M$. Chernin (2001) estimates in this way the radius for the Local Group
where these two opposing forces equal. In the literature this radius defines
the ‘zero gravity surface’ of a large scale structure. Analytical and
numerical solutions for orbits in a de Sitter universe are presented by
Emelyanov & Kovalyov (2013) and Chernin (2015), respectively. We shall see
below that the saddle point in the effective potential defines a cutoff for
orbital boundedness in a $\Lambda$-dominated universe, which theoretically
occurs sooner than $R_{\rm max}$, at $R_{\rm max}/2^{2/3}$.

Figure 1: Effective potential as given by (5) plotted against the physical
radius $R$ for a test mass orbiting a 10${}^{15}~{}M_{\odot}$ cluster (top)
and a 10${}^{12}~{}M_{\odot}$ galaxy (bottom). In each case, the epochs at
which the potential is evaluated are $z=2$ (matter dominated) and $z=0$
(transition from matter to vacuum domination). It can be seen that the
potential exhibits at maximum as $z\to 0$. The angular momentum for each graph
is set by the tangential physical velocity at $R=R_{\rm max}$, which is 350 km
s-1 (top) and 35 km s-1 (bottom).

We may now discuss critical orbits. If (a) the energy $E=V_{\rm min}$, the
orbit will be circular222Strictly circular orbits are forbidden for effective
potential (5), so in the following ‘circular’ implies ‘quasi-circular’., with
speed and period given respectively by

$v_{\perp}^{2}=\frac{GM}{R_{\rm min}}-\frac{2\gamma J^{8}}{G^{4}M^{4}R_{\rm
min}}\approx\frac{GM}{R_{\rm min}};~{}P=\frac{2\pi R_{\rm min}}{v_{\perp}}.$
(12)

On the other hand, if (b) $E=V_{\rm max}$, the orbital period of this most
elongated orbit (_i.e._ one with the aphelion at $R_{\rm max}$) will be

$P=\frac{2\pi}{3\sqrt{6\gamma}}=\frac{2\pi}{3\sqrt{-3q}H_{0}}$ (13)

and is always comparable to the Hubble time.

Third, in the dark energy era there can be a scenario (c) under which the only
bound orbit is circular. This occurs when the inequality (9) is violated, and
is instead replaced by the equation

$J^{2}=\frac{3}{8}\left(\frac{G^{4}M^{4}}{\gamma}\right)^{1/3},$ (14)

an equation that represents also the condition for the largest possible quasi-
circular orbit, which is imminently unstable.333Relations similar to (14) and
(15) are derived by Balaguera-Antolénez et al (2006), where they do the
calculation for a de Sitter universe instead. This occurs at the radius where
$V$ reaches the saddle point $V^{\prime}=V^{\prime\prime}=0$. In fact, such a
critical radius is

$R_{\rm
saddle}=\frac{4J^{2}}{3GM}=\frac{3GM}{4v_{\perp}^{2}}=\frac{1}{2}\left(\frac{GM}{\gamma}\right)^{1/3}.$
(15)

Thus this ‘special’ orbit is unlike the usual circular orbits that satisfy the
balance of centrifugal force and gravity, viz. (12a), and the difference by
the factor of $3/4$ is due to the role played by the repulsive force of the
cosmic substratum. Its period

$P=\pi\sqrt{\frac{2}{3\gamma}}=\frac{2\pi}{\sqrt{-3q}H_{0}}$ (16)

is comparable to the Hubble time as well.

Relations (14) and (15) are derived by Nandra et al (2012) (NLH2), where they
describe $R_{\rm saddle}$ as the radius of the largest stable circular orbit.
Rather, this outermost ‘nearly’ circular orbit is imminently unstable, as
explained in Section 3.1. NLH2 also derive (10); however, our respective
interpretations of $R_{\rm max}$ differ when it comes to the radial extent of
hydrostatic equilibrium (H.E.), as shown in Section 4.

The instability of the outermost circular orbit provides us a testable
hypothesis of a $\Lambda$-dominated universe. Dark energy disengages wide-pair
galaxy systems, so one should find a statistical minority of interacting wide-
pairs with projected separations $\sim$1 Mpc. The observed number of
interacting wide-pairs should be no more than the statistical number
calculated, assuming wide-pairs are coincident because the two incidentally
passed within the other’s ‘radius of attraction’, _i.e._ by chance coincidence
and not by tidal interaction. The interaction of galaxies in a pair may be
determined from line-of-sight peculiar velocities, which is an observable in
low redshift wide-pairs.

To gain a numerical feeling of the various quantities, one may calculate
representative values of the critical radius, or radius of the ‘outermost
bound orbit’. We do it for galaxy and cluster sized central mass excess. In
the former, one finds

$R=0.7\left(\frac{M}{10^{12}M_{\odot}}\right)^{1/3}\left(-\frac{q_{0}}{0.55}\right)^{-1/3}\left(\frac{h}{0.7}\right)^{-2/3}~{}{\rm
Mpc},$ (17)

and

$v_{\perp}=70~{}\left(\frac{M}{10^{12}M_{\odot}}\right)^{1/3}\left(-\frac{q_{0}}{0.55}\right)^{1/6}\left(\frac{h}{0.7}\right)^{1/3}~{}{\rm
km}~{}{\rm s}^{-1};$ (18)

while in the latter case one substitutes $M=10^{15}M_{\odot}$ to the above
formulae to obtain $R\approx$ 7 Mpc and $v_{\perp}\approx$ 670 km s-1. NLH2
give comparable estimates for $R_{\rm saddle}$, for a galaxy and cluster.

A satellite orbiting around a cluster or galaxy cannot under any circumstance
be gravitationally bound to either of them if the perihelion exceeds the value
given by (17) for the appropriate $M$. For clusters the issue is academic in
practice, because the timescale of a full orbital cycle exceeds the Hubble
time for orbits of size at or near the scale in question (_i.e._ over such
long periods (4) no longer follows from (3). Instead, the relevance of dark
energy to clusters is its effect on diffuse virialized gas – a topic to be
discussed below.

There is another way to estimate the finite extent of large scale
gravitationally bound structures, even for a matter-dominated universe.
Solving the equation of motion for the velocity of a test mass moving radially
outward from the central mass clump gives the ‘surface of zero velocity’ or
‘turnaround radius’ to a clump. Turnaround may be defined for any epoch and
depends on the initial energy $E(T)$ of the test mass, where $T$ is any
particular time. Using the age of the Universe $T\approx 1/H_{0}$ allows one
to estimate an ‘interface’ radius between the gravitational influence of the
clump and the Hubble flow expansion at the present epoch. This is defined as a
definite boundary to the structure today. (Lynden-Bell (1981); Sandage (1986))

Extending this ‘zero velocity’ method to a dark energy universe, Karachentsev
et al (2007) estimate the boundary radius and total mass of nearby groups and
clusters of galaxies, using observed peculiar velocities of galaxies at the
outskirts and Hubble flow data. With observational data on galaxies around the
Local Group (LG), a boundary444Karachentsev et al (2009) show in Figure 1 of
their paper radial velocities of galaxies in the LG and in the neighboring
outskirts as a function distance with respect to the LG centroid. There
appears to be a boundary to the LG at $\lesssim$ 1 Mpc. to the LG is estimated
to be $\lesssim$ 1 Mpc. Using this estimated radius, Karachentsev et al (2009)
calculate the total mass of the LG to be $\sim 2\times 10^{12}M_{\odot}$. As
expected, this boundary is numerically close to that predicted by (15) above;
their estimated mass for the LG gives $R_{\rm saddle}\approx 0.9$ Mpc. In
another recent study, Nasonova et al (2011) used peculiar velocity data of
more than 1000 galaxies at the Fornax-Eridanus complex outskirts (with radii
$r\lesssim$ 30 Mpc) to estimate the boundary radius and mass of the Fornax-
Eridanus complex, obtaining $\lesssim$ 5 Mpc and ([1.30$-$3.93]$\times
10^{14}M_{\odot}$) respectively. Again using (15), their mean estimated mass
for the Fornax-Eridanus complex gives $R_{\rm saddle}\approx 4.5$ Mpc.

##  3 The evolution of bound orbits

The purpose of this work is to examine how bound orbits evolve in a dark
energy universe (which necessarily means these orbit live in the present epoch
$H=H_{0}$). Therefore we will ignore such other dark energy induced phenomena
as orbital precession (Kerr et al (2003)) which, though equally interesting,
are not directly relevant.

We begin with the critical orbits, viz. the scenarios of (a) to (c) of the
last section.

###  3.1 Critical orbits

In (a), if an orbit is initially circular, _i.e._ $R=R_{\rm min}$, by (7) and
(5) $\dot{E}$ will to lowest order be constant and equal to $\partial
V/\partial t|_{R}=-\dot{\gamma}R_{\rm min}^{2}$ as the satellite moves. Thus,
after one orbital period of $P=2\pi J^{3}/(GM)^{2}$ the position of
$\dot{R}=0$ will be at the same radius as before, except that by then $V$ will
no longer be minimized there because by (10) $R_{\rm min}$ displaced by the
amount $4\pi\dot{\gamma}J^{11}/(GM)^{7}$, _i.e._ the orbit has become slightly
elliptical, with eccentricity $e=4\pi J^{9}\dot{\gamma}/(GM)^{6}$ and hence an
average aphelion advancement rate of $\Delta R_{\rm
aph}/P=2\dot{\gamma}J^{8}/(GM)^{5}$, the latter being of order
$H_{0}^{3}R_{\rm min}^{3}/v_{\perp}^{2}$ which is very small (even for $R_{\rm
min}=1$ Mpc and $v_{\perp}=300$ km s-1 the rate is only $\approx 10$ km s-1).

In (b), a marginally bound orbit with $E\lesssim V$ has, by (13), a period
$P\approx 1/H_{0}$, hence it is reasonable to assume that the satellite spends
most of its time at or near the aphelion, _i.e._ $R\approx R_{\rm aph}$ is a
constant, in which case $\dot{E}\approx-\dot{\gamma}R_{\rm aph}^{2}$, where
$R_{\rm aph}\lesssim R_{\rm max}$ with the latter defined by (10). This means,
in turn, that $|\dot{E}|<\dot{\gamma}R_{\rm
max}^{2}=2^{-2/3}\dot{\gamma}(GM/\gamma)^{2/3}=-\dot{V}_{\rm max}$ where the
last step is taken with the aid of (11). Thus it is evident that $E$ will
eventually reach the level $V_{\rm max}$ from below. At this point $E$ and
$V_{\rm max}$ will go down at the same rate, since $R_{\rm aph}=R_{\rm max}$,
while $R_{\rm max}$ continues to decrease by (10) and the fact that
$\dot{\gamma}$ is positive. As time elapses further the satellite will find
itself on the other side of the potential barrier, and its orbit has become
unbound.

In (c), the only bound orbit possible is the largest circular orbit with
$R=R_{\rm saddle}$. From here onwards, $E$ and $V(R_{\rm saddle})$ go down
with time in tandem while $R_{\rm saddle}$ also decreases, _i.e._ $R$ will
soon become $R>R_{\rm saddle}$ and the satellite has escaped.

The following is a discussion on the largest bound initially circular orbits
around galaxies. This topic was first explored by NLH2, for orbits about
clusters. Here we extend their discussion, for theoretical completeness only.
The time scales involved are much longer than the age of the universe and
involve unobservable future evolution. In Appendix I, we use our methodology
to provide rough upper estimates of the largest initially circular orbits to
remain bound indefinitely We do so for a test mass set in motion at various
redshifts. This is only an estimate, as the rate of decrease of $V_{\rm max}$
is greater than the orbital energy, so an orbit becomes unbound with less
angular momentum than is required if the orbit were to remain exactly circular
as the universe evolved. Namely, the test mass becomes effectively free at the
time satisfying the condition $|E(t)|\leq|V_{\rm max}(t)|$, see Figure 6. For
initial radii equal to critical this condition is met prior to the
deceleration parameter saturating to $-0.7$. And for initial radii less than
critical this condition is never satisfied, and the orbit remains bound
indefinitely.

We find the largest initially circular orbits, that are bound indefinitely,
occur for an angular momentum less than the maximum at the saddle point
angular momentum (14) (the maximum occurs for $q_{\rm min}=-0.7$). That is,
the maximum angular momentum allowed for boundedness is slightly less than
that given by (14) and is found by numerically evolving the orbit, giving the
exact critical radius for any starting time. NLH2 find this critical radius
for a test mass orbiting a cluster, beginning at the present epoch. In
Appendix I we estimate, and calculate numerically, critical radii for test
mass motion about a galaxy initiated at various epochs.

###  3.2 Bound orbits in general, adiabatic invariant

For an arbitrary bound orbit of $V_{\rm min}\ll E\ll V_{max}$, or more
explicitly from (11),

$(G^{2}M^{2}\gamma)^{1/3}\ll|E|\ll\left(\frac{GM}{J}\right)^{2},$ (19)

quantifying its evolution is a more arduous task because both the time
dependence of $\dot{\gamma}$ and $R$ have to be taken into account (in
particular, the change of $R$ with time as the satellite moves along such an
‘intermediate’ orbit cannot be ignored) when integrating (7) to produce
$E(t)$.

Nevertheless, since the period of this class of orbits is $\ll 1/H_{0}$ it is
possible to take advantage of the adiabatic invariant

$I=\int_{R_{\rm per}}^{R_{\rm aph}}\sqrt{E-V(R)}~{}dR,$

which may be evaluated by treating the $\gamma R^{2}$ term of the effective
potential of (5) as a perturbation (Landau & Lifshitz (1976)), viz.

$I_{R}=\int_{R_{\rm per}}^{R_{\rm
aph}}dR~{}\Bigg{[}\Big{(}{E+\frac{GM}{R}-\frac{J^{2}}{2R^{2}}}\Big{)}^{1/2}\\\
+\frac{\gamma
R^{2}}{2}\Big{(}E+\frac{GM}{R}-\frac{J^{2}}{2R^{2}}\Big{)}^{-1/2}\Bigg{]}.$
(20)

The first term of the integrand leads to the standard result for the radial
action variable $p_{R}$ of the Kepler problem, _i.e._

$\int_{R_{\rm per}}^{R_{\rm
aph}}dR~{}\left({E+\frac{GM}{R}-\frac{J^{2}}{2R^{2}}}\right)^{1/2}\\\
=\pi\left(\frac{GM}{2\sqrt{-E}}-\frac{J}{\sqrt{2}}\right).$ (21)

The second term may be handled by the substitution

$R=\frac{J^{2}}{GM}\frac{1}{1+e\cos\phi}$

which converts the integral to one of the form $\int d\phi/(1-e\cos\phi)^{4}$,
resulting altogether in

$I_{R}=\frac{\pi J}{\sqrt{2}}\left(\frac{1}{\sqrt{1-e^{2}}}-1\right)\\\
+\frac{\pi\gamma J^{7}}{2\sqrt{2}G^{4}M^{4}}(2+3e^{2})(1-e^{2})^{-7/2},$ (22)

where

$e=\left[1+2E\left(\frac{J}{GM}\right)^{2}\right]^{1/2}$ (23)

is the lowest order contribution to the orbit eccentricity.

One can rewrite (22) in terms of yet another variable
$y=1/\sqrt{1-e^{2}}=GM/(J\sqrt{-2E})$, so that the statement of $I_{R}$ as an
adiabatic invariant may be expressed as

$\frac{\pi\gamma J^{7}\dot{y}}{2\sqrt{2}G^{4}M^{4}}(35y^{6}-15y^{4})+\frac{\pi
J}{\sqrt{2}}\dot{y}=-\frac{\pi\dot{\gamma}J^{7}}{2\sqrt{2}G^{4}M^{4}}(5y^{7}-3y^{5}).$

Under the criterion (19) for intermediate orbits $y\gg 1$, so that only the
3rd term on the left side and the 1st term on the right of the equation need
to be kept, leading to the solution

$E=-\frac{15^{1/3}}{2}(GM)^{2/3}(\gamma+\gamma_{0})^{1/3},$ (24)

where $\gamma_{0}\gg\gamma$ is a constant. This results in the rate of change
of energy

$\dot{E}=-\frac{5}{8}\left(\frac{GM}{E}\right)^{2}\dot{\gamma},$ (25)

which expectedly puts

$\left(\frac{J^{2}}{2GM}\right)^{2}\dot{\gamma}=\dot{\gamma}R_{\rm
per}^{2}\ll|\dot{E}|<\dot{\gamma}R_{\rm
aph}^{2}=\left(\frac{GM}{E}\right)^{2}\dot{\gamma},$ (26)

where in arriving at (26) use was made of (19).

It is now possible to calculate the speed and direction at which the aphelion
and perihelion drift. Consider first the aphelion. After one orbital period
$P=\pi GM/\sqrt{-2E^{3}}$, the effective potential changes by $\Delta
V=-\dot{\gamma}R_{\rm aph}^{2}P$ while $E$ changes by $\Delta E=\dot{E}P$. The
displacement in the aphelion therefore given by the relation
$V^{\prime}(R_{\rm aph})\Delta R_{\rm aph}=\Delta E-\Delta V$. Using (5), (6),
(25) and (26) one then obtains the aphelion drift rate, to lowest order,

$\frac{\Delta R_{\rm aph}}{P}=\frac{3G^{3}M^{3}}{8E^{4}}\dot{\gamma},$ (27)

which indicates that the aphelion drifts outwards with the rate of order the
product of the Hubble velocity at the distance $R_{\rm aph}$ (viz.
$H_{0}R_{\rm aph}$) and the square of the ratio of this velocity to that of
the orbit. For $R_{\rm aph}\approx 1$ Mpc and orbital velocity $\approx$ 300
km s-1, the rate is then of order 10$h^{3}$ km s-1. A similar calculation for
the perihelion leads to the rate

$\frac{\Delta R_{\rm per}}{P}=\frac{5J^{4}}{32GME^{2}}\dot{\gamma},$ (28)

which is negligible compared with the aphelion by virtue of (19).

In summary, because both the perihelion and aphelion drift outwards with the
latter at a much higher rate than the former, the orbit becomes more
elliptical, _i.e._ intermediate orbits also evolve in the direction of
becoming unbound.

##  4 Virial Theorem in a dark energy universe, an application to cluster
Abell 1835

Apart from orbits, a gravitationally bound system also can also manifest
itself as a spatially confined region of dark matter and thermalized baryons.
Notable examples are clusters and groups of galaxies that harbor a 106-8 K gas
accounting for a good fraction of the baryonic mass of such structures. The
bulk of the X-ray and Sunyaev-Zeldovich measurements to date (Vikhlinin et al
(2006), Afshordi et al (2007), Arnaud et al (2007), Sun et al (2009), and
Ettori et al (2009)), with the notable exception of Landry et al (2013) and
Simionescu et al (2011), reveal that the mass ratio of baryons-to-total matter
falls below the value predicted by the standard cosmological model, with a
cluster’s dark matter more centrally condensed than the baryonic matter, as
the latter has a higher temperature. Specifically, for a small radius $R$ the
enclosed baryonic to dark matter mass ratio rises with $R$, but fails to reach
the cosmic value of $f_{\rm gas}\approx 1/7$ even at the limiting radius of
X-ray sensitivity, which for a rich cluster is of order the so-called ‘virial
radius’ of $R_{\rm virial}=2$ Mpc.

###  4.1 Clusters bounded by purely gravitational effect

We begin by addressing the question of what binds the baryons. From numerical
hydrodynamic simulations there emerged a widely used spherically symmetric
‘universal’ profile for the dark matter density distribution of clusters and
groups, _i.e._ the matter that shapes the gravitational potential of these
systems. It is known as NFW profile, see Navarro et al (1995), (1996), (1997),
and also the earlier work of Dubinski & Carlberg (1991), and has the form

$\rho_{m}(R)=\frac{\rho_{s}R_{s}^{3}}{R(R+R_{s})^{2}}$ (29)

where $R$ is a physical radius, $R_{s}$ is a constant scale radius, and
$\rho_{s}$ is the central density parameter. But since at the outer radii
$R\gg R_{s}$ the density scales as $\rho_{m}(R)\sim 1/R^{3}$, and there is no
further change of functional form with increasing distance, _i.e._
hierarchical structure formation codes do not seem to reveal the surface
radius of a clump, the total integrated mass is divergent unless an upper
limit (or cutoff) radius $R$ is ‘manually’ assigned via the ‘tapered NFW
models’ (Springel & White (1998), Lokas & Mamon (2001)).

In the following we investigate the ‘lowest order’ effect of dark energy on
baryonic gas at the cluster outskirts. We develop a theory that produces a
cutoff radius to hydrostatic equilibrium $R_{\rm H.E.}=R_{\rm virial}$ and a
separate cutoff to the dark matter enveloping the baryonic matter of a
cluster. The latter cutoff radius is used to estimate the total mass of a
cluster. We apply our theory to A1835, and with observations of the cluster we
estimate its ‘true’ virial radius and the extent of its dark matter halo and
total mass.

###  4.2 The outskirts of clusters in a dark energy universe

For a cluster in Minkowski spacetime, up to a certain scale height, the
‘atmosphere’ of baryons in its potential well could be approximated as an
ideal gas in hydrostatic equilibrium with temperature profile $T(R)$ such that
the mean kinetic energy $K$ equals half the mean potential energy $V$, or

$3kT=\frac{GM_{c}m}{R},$ (30)

where $m=\mu m_{p}\approx 0.6m_{p}$, and $M_{c}=M_{c}(R)$ is the mass
‘underneath’ a thin shell of radii $R$ and $R+dR$ . This equation of state
ignores the external pressure. When we derive the Virial Theorem for
structures in a dark energy universe below, the contribution to the mean
kinetic energy from the external pressure is included.

With a constant amount of underlying mass $M_{c}$, the mean kinetic energy per
unit mass of the thermal gas is $\bar{K}=3k_{B}T/(2m)=GM_{c}/(2R)$ for
$R>R_{\rm virial}$, while at such radii the extra kinetic energy per unit
mass, above the mean, required to enable an ‘average’ particle to escape to
infinity, the place where $V=0$, is $K=GM_{c}/R$. Thus a particle with kinetic
energy satisfying the relation $K\geq 2\bar{K}$ can escape the gravitational
pull of the cluster. To calculate the extent of the hydrostatic halo of
baryons in the outskirts of a cluster, we must model a realistic scenario: we
take account of the cosmology dependent background density, and we include
mass and pressure profiles. The addition of these profiles allow us to include
the effects of both the dark matter and gas at the outskirts of the cluster.

The escape condition is different in a $\Lambda$-dominated universe. A
particle with kinetic energy in excess of its potential energy $V(R)$, in
absolute value, minus $|V_{\rm max}|$ will escape: the condition to escape the
cluster is $K\geq|V|-|V_{\rm max}|$. More generally, the criterion for escape
is $K\geq\eta\bar{K}$, where $\eta$ becomes scale dependent:

$\eta(R)=\frac{|V|-|V_{\rm max}|}{\bar{K}}.$ (31)

From (5) we see that the potential per unit mass is now $V=V_{1}+V_{2}$ (the
angular momentum term in (5) plays no role for virialized systems). The dark
energy contribution to $V$ is

$V_{2}=-\frac{1}{2}|q|H_{0}^{2}R^{2}.$ (32)

Owing to the presence of the $V_{2}$ term in the effective potential and to
the external pressure from the outermost shell of baryons (immediately outside
the hydrostatic halo) the Virial Theorem is different from its usual version
for gas trapped in a pure gravitational field. Instead it is generalized to
become

$2\bar{K}=\frac{3P(R)}{\rho_{b}(R)}-\bar{V}_{1}+2\bar{V}_{2},$ (33)

which is the same as the equation of hydrostatic equilibrium in a dark energy
universe. See Appendix II for derivation of (33) from H.E.. The kinetic energy
per unit mass is $K$, where $\bar{X}$ denotes the average of $X$ over a
timescale $\ll 1/H_{0}$. We assume $q=-|q_{0}|$ to find the largest $R$
satisfying (33) at the present epoch.

We estimate the external pressure term in (33) with a temperature profile,
where $P(R)/\rho_{b}(R)=k_{B}T(R)/(\mu m_{p})$. We use temperature profile
$T(R)$ from Burns et al (2010), with average temperature and fitting
parameters from Zhang et al (2007) and Ichikawa et al (2013) respectively, for
cluster A1835

$k_{B}T(R)=k_{B}T_{\rm avg}\times A\Big{[}1+B(R/R_{\rm 200})\Big{]}^{\beta}\
{\rm keV}.$ (34)

Where $k_{B}T_{\rm avg}\sim 7.67\pm 0.21$ keV $A=1.74\pm 0.03$, $B=0.64\pm
0.10$, $\beta=-3.2\pm 0.4$ and $R_{200}\approx 2.2$ Mpc (Ichikawa et al
(2013)).

The total mass of the cluster inside radius $R$, (38), is given by the full
NFW profile, where we use parameters from the weak-lensing analysis of cluster
A1835 (Okabe et al (2010)), taken in their current form from Ichikawa et al
(2013). ‘Virial mass’ $M_{\rm\Delta}$ and halo concentration
$c_{\Delta}=R_{\Delta}/R_{s}$ are

$M_{\rm\Delta}=1.37^{+0.37}_{-0.29}\times 10^{15}M_{\odot}h^{-1}$ (35)
$c_{\rm\Delta}=3.35^{+0.99}_{-0.79},$ (36)

with Hubble parameter $h=0.7$. Up to $R_{\rm\Delta}$ we define the total
cluster mass as follows, using (29).555One may consult Okabe et al (2010)’s
equations (11), (13), and (14) to see how we defined the central density
parameter $\rho_{s}$ and scale radius $R_{s}$ in (38), using only
$M_{\rm\Delta}=M_{\rm virial}$, $c_{\Delta}=c_{\rm virial}$,
$\Delta=\Delta_{\rm virial}=112$, and $z=0.253$.

$M(R)=4\pi\int^{R}_{0}\frac{\rho_{s}R_{s}^{3}}{r(R_{s}+r)^{2}}r^{2}dr.$ (37)

For a cluster at low redshift, using the NFW density profile (29), we define
the mass profile by integrating (37). This becomes

$M(R)=4\pi\rho_{s}R_{s}^{3}\Bigg{[}{\rm
Log}\Bigg{(}1+\frac{R}{R_{s}}\Bigg{)}-\frac{R}{R+R_{s}}\Bigg{]}.$ (38)

For $R=R_{\rm\Delta}$ in (37) $M(R)$ equals $M_{\rm\Delta}$, of (35), and the
potential reduces to

$V_{1}(R)=-G\frac{M_{\rm\Delta}}{R}.$ (39)

We use (39) to represent the inner region of the cluster. Otherwise we have
the general NFW potential,

$V_{1}(R)=-4\pi G\rho_{s}R_{s}^{3}\frac{{\rm Log}(1+\frac{R}{R_{s}})}{R}.$
(40)

The former potential satisfies Laplace’s equation, and the latter (40)
satisfies Poisson’s equation, and the two, (39) and (40), will be used to
define the criterion for escape,(31).

We use Sigmoid function

$f=\frac{1}{{\rm exp}[\big{(}\frac{R}{{\rm
Mpc}}\big{)}-\big{(}\frac{R_{\rm\Delta}}{{\rm Mpc}}\big{)}]+1}$ (41)

to define $\eta(R)$, and require two conditions to be satisfied: $\eta(R)$
approaches 2 as $R\rightarrow 0$ and is continuous at $R=R_{\rm\Delta}=2.8$
Mpc.

$\eta(R)=\eta(R\leq R_{\rm\Delta})\times f+\eta(R\geq
R_{\rm\Delta})\times(1-f),$ (42)

with

$\eta(R\leq R_{\rm\Delta})=\eta(R,M_{\rm\Delta})$ (43) $\eta(R\geq
R_{\rm\Delta})=\eta(R,M_{R\geq R_{\rm\Delta}}).$ (44)

We have from (31) and (33) that

$\eta(R,M)=2\frac{|V|-|V_{\rm max}|}{-V_{1}+2V_{2}+3k_{b}T/(\mu m_{p})}.$ (45)

Next we estimate the extent of the dark matter profile, which we assume cuts
off at $R_{\rm max}$. The maximum of the cluster potential $V_{\rm max}$
occurs where the force of dark energy balances the gravitational attraction of
the cluster’s mass. We use (40) and (32) to define the effective potential,
$V_{\rm eff}(R)=V_{1}(R)+V_{2}(R)$ and use (38) for the total mass. We find
the maximum of the effective potential $V_{\rm max}$ is located at $R_{\rm
max}\approx$ 17 Mpc. This gives an upper limit to the cluster’s mass of
$M_{c}(R_{\rm max})\lesssim 7\times 10^{15}M_{\odot}$. Using $R_{\rm max}$,
and $M_{c}(R_{\rm max})$, we note $V_{\rm max}$ is constant in (45).

The condition $\eta(R_{\rm H.E.})=1$ equates the kinetic energy of an average
baryon, $K$, to the mean kinetic energy of a baryon, $\bar{K}$, given by the
modified Virial Theorem (33). So baryons with kinetic energy equal to the mean
at $R_{\rm H.E.}$ are not virialized and escape the cluster, hence $R_{\rm
H.E.}=R_{\rm virial}$. Using the escape condition, $K\geq\eta\bar{K}$, we find
$\eta=1$ at $R=R_{\rm H.E.}=1.3$ Mpc. See Figure 2. Since baryons lack
sufficient kinetic energy to remain virialized beyond $R_{\rm H.E.}$, they
escape the gravitational well (with $K\geq|V|-|V_{\rm max}|$).

Figure 2: A plot of Equation (42), $\eta(R)$. The function approaches 1.9 as
$R\rightarrow 0$, but is unphysical at zero, because $\eta$ is infinite at
zero exactly. It is the contribution of $\eta(R\geq R_{\rm virial})$ that
makes (42) unphysical at small r, as it diverges at zero; however, $\eta(R\leq
R_{\rm virial})$ (43) converges to 2 as $R\rightarrow 0$ as expected.

We see in Figure (2) there is a complete cutoff at $R_{\rm H.E.}=1.3$ Mpc.
Ichikawa et al (2013), Kawaharada et al (2010), and Bautz et al (2009) lend
observational support to our theory, as each propose deviations from H.E.
beginning from radii 1.4 Mpc, 1.7 Mpc, 1.3 Mpc, their respective values for
$R_{500}$. In Figure (10) of Ichikawa et al (2013) they compare hydrostatic
and weak lensing masses, one sees that a deviation from H.E. appears to begin
as early as 1 Mpc even.

So with a cutoff to hydrostatic equilibrium at $R_{\rm H.E.}=1.3$ Mpc, we find
the hydrostatic halo of baryons cuts off many times sooner than the dark
matter profile at 24 Mpc.666NLH2 suggest the condition of H.E. in a dark
energy persists out to a radius where the total pressure due to gas and dark
matter equals zero, which they find occurs at $R_{\rm max}$. However, this may
not be physically well-motivated at $R_{\rm max}$, because H.E. is
inconsistent beyond $R_{\rm H.E.}$.

To check the results of Figure 2 we evaluate the energy contribution of the
external pressure term in (33) in a different way, this time using the
pressure and electron number density profiles of (Nagai et al (2007)) and
(Vikhlinin et al (2006)) respectively, $P(R)/(2\mu m_{p}n_{e}(R))$. We assume
$n_{b}=2n_{e}$ for a gas of thermalized baryons, with equal numbers of
oppositely charged particles. We take from Ade et al (2013) and Bonamente et
al (2012) best fitting X-ray pressure and number density profile parameters
obtained from their respective observations of A1835. We reevaluate equations
(38)-(44) with parameters from Bonamente et al (2012) (Tables 3 and 4): the
overdensity, total mass and halo concentration are $\Delta=100$,
$M_{\rm\Delta}\approx 8.34\times 10^{14}M_{\odot}$ (giving $c_{\rm\Delta}=5.8$
and $R_{\rm\Delta}=2.2$ Mpc). We alter (45) to the following, substituting
$P(R)/\rho_{b}(R)$ in place of $k_{B}T(R)/(\mu m_{p})$ now.

$\eta(R,M)=2\frac{|V|-|V_{\rm max}|}{-V_{1}+2V_{2}+3P/(\rho_{b})}$ (46)

We find $\eta(R)=1$ at $R_{\rm H.E}$ = 2.0 Mpc, see Figure 3. From Figures 2
and 3, we conclude that dark energy affects a well-defined cutoff to H.E.
between 1 and 2 Mpc. Using parameters from Bonamente et al (2012) we find the
maximum of the effective potential $V_{\rm max}$ is located at $R_{\rm
max}\approx$ 17 Mpc. This gives an upper limit to the cluster’s total mass of
$M_{c}(R_{\rm max})\lesssim 2.2\times 10^{15}M_{\odot}$.

Figure 3: A plot of Equation (42), $\eta(R)$. Radius $R_{\rm H.E.}$ moves out
from 1.3 Mpc to 2.0 Mpc when we use $P/\rho_{b}$ instead of $k_{B}T/(\mu
m_{p})$. See (46) in contrast to (45).

The hydrostatic mass of A1835 calculated by Bonamente et al (2012) may be an
underestimate of the total mass at larger radii, see Ichikawa et al (2013)
Figure 10 comparing hydrostatic and weak-lensing masses. Landry et al (2013)
suggest the hydrostatic mass underestimates the true mass also. To account for
this possibility we use equations (38)-(44) with parameters (35) and (36)
(from the weak-lensing analysis of Okabe et al (2010)), as in our first
calculation, but, for our check, continue to use $P(R)/(2\mu m_{p}n_{e}(R))$
in place of $k_{B}T/(\mu m_{p})$. When using the larger lensing mass of Okabe
et al (2010), we find $\eta(R)=1$ at $R_{\rm H.E}$ = 2.1 Mpc, with the same
dark matter cutoff and total cluster mass as in our first calculation of
$R_{\rm H.E.}$.

From Sunyaev-Zel’dovich observations of the Coma cluster by the Planck
satellite (Ade et al (2013)), we note an ‘atmosphere’ of hot gas continues
outside $R_{\rm H.E.}$; the hot gas atmosphere of the Coma cluster is found to
extend out to 4 Mpc, at least. Since a cluster’s dark matter cuts off at
$R_{\rm max}$, we recalculate $\eta(R)$ for A1835. We ignore its inner
structure and represent it as a point mass. This leads to an ‘outer’ virial
radius.

###  4.3 Constraining dark energy by observation of large scale structures

In view of the aforementioned considerations, dark energy may be scrutinized
by observing wide galaxy pairs and the thermal baryon envelope in clusters of
galaxies.

One specific example of the first method is the galaxy pair of the Milky Way
and the Andromeda galaxy. This is a wide pair because the Keplerian orbital
period $\approx 17$ Gyr, which exceeds the age of the Universe. Thus dark
energy plays a crucial role in the evolution of the pair. An upper limit on
$\Omega_{\Lambda}$ based on current mass estimates of both galaxies is
provided by Benisty et al (2023).

On the second method, baryonic gas crossing the cutoff surface defined by
$R_{\rm H.E.}$ may ‘escape’ the cluster to radii greater than $R_{\rm max}$,
by way of a cluster wind for example, but indefinite escape is not a
guarantee. A halo of thermalized baryons may exist farther out too, but inside
$R_{\rm max}$ still, with a terminating surface at an ‘outer’ virial radius
defined analytically next. One way to see this is to recalculate $\eta(R)$
assuming the upper total mass limit of the cluster given by (37) evaluated at
$R_{\rm max}$ (for A1835 $R_{\rm max}\approx 24$ Mpc). In Figure 4 we show
that hydrostatic equilibrium remains an option at larger radii for baryons
transported outward by cluster winds and in general for baryons unsupported by
H.E. at the inner cutoff radii derived above.

Figure 4: A plot of Equation (42), $\eta(R)$. Using the dark matter cutoff of
$R_{\rm max}$ the cluster is modeled as a ‘point mass’, and hydrostatic
equilibrium persists out to $\sim$ 8.4 Mpc, which is comparable to our
analytically derived result (48).

Applying the condition $\eta$ = 1 to A1835, the radius satisfying (31)
($\bar{K}=|V|-|V_{\rm max}|$) and (33) is calculated analytically and compared
with $R_{\rm max}$. The maximum of the effective potential $V_{\rm max}$ is
still given by (11b) for constant $M$. In solving (47) we shall find that
$R_{\rm H.E.}<R_{\rm max}$.

$\frac{GM_{\rm max}}{2R_{\rm H.E.}}+|q|H_{0}^{2}R_{\rm
H.E.}^{2}-\frac{3}{2}|q|^{1/3}(GM_{\rm max}H_{0})^{2/3}=0.$ (47)

By rewriting in terms of the dimensionless variable $x=RH_{0}/c$ (47) becomes

$x^{3}-ux+v=0,$

where $u=3|q|^{-2/3}(GM_{\rm max}H_{0})^{2/3}/(2c^{2})$ and $v=GM_{\rm
max}H_{0}/(2|q|c^{3})$ are also dimensionless. In a cluster environment where
$GM_{\rm max}/R\ll c^{2}$ and $R\ll c/H_{0}$, it can readily be shown that
$u\gg v$. Moreover, in a $\Omega_{\Lambda}=0.7$ and $\Omega_{m}=0.3$ cosmology
where (6a) gives $q=-0.41$ for $z=0.253$, it can further be shown that
$u^{3}\gg v^{2}$. Under these conditions, the $x^{3}$ term of the cubic is of
little importance, _i.e._ $x=v/u$ or

$R_{\rm H.E.}\\\ =8.02\left(\frac{M_{\rm max}}{6.76\times
10^{15}~{}M_{\odot}}\right)^{1/3}\left(\frac{|q|}{0.41}\right)^{-1/3}\left(\frac{h}{0.7}\right)^{-2/3}~{}{\rm
Mpc}.$ (48)

This is to be compared with

$R_{\rm max}=\left(\frac{GM_{\rm max}}{|q|H_{0}^{2}}\right)^{1/3}\\\
=24.06~{}\left(\frac{M_{\rm max}}{6.76\times
10^{15}~{}M_{\odot}}\right)^{1/3}\left(\frac{|q|}{0.41}\right)^{-1/3}\left(\frac{h}{0.7}\right)^{-2/3}~{}{\rm
Mpc}.$ (49)

Thus we see that $R_{\rm H.E.}\approx R_{\rm max}/3$ in general. Using $R_{\rm
max}=17$ Mpc gives an $R_{\rm H.E.}=5.5$ Mpc.

For A1835 the latter is close to the $R_{500}$ radius obtained from Suzaku
X-ray observations of the intracluster medium of cluster A1835 (Ichikawa et al
(2013)) and the circumgalactic medium of the Illustris-TNG simulations (Gouin
et al (2023)). It is now apparent that in a dark energy universe the
atmosphere of thermal gas bounded by the gravitational field of a central mass
excess has a clear boundary.

##  5 Discussion and conclusion

The manner in which dark energy (a) disengages binary systems, and (b) places
a hard upper limit on the extent of any baryonic halo in a cluster, is
demonstrated in some detail. In (a), it is shown that closed orbits can no
longer exist, and all orbits evolve to become less bound. Moreover, the path
of this evolution can be determined by means of an adiabatic invariant in most
cases. The phenomenon provides another way of testing the dark energy
interpretation, via the observation of wide ($\sim$ 1 Mpc) galaxy pairs at low
redshifts, as their line-of-sight velocities should indicate that they are at
best only marginally bound. In (b), we find that the thermal escape velocity
of the ‘last scattering surface’ of the atmosphere of hot baryons in clusters
and groups of galaxies is lowered by dark energy, which causes the potential
function to exhibit a sub-zero maximum. The radius $R_{\rm H.E.}$ that marks
this limit. The results of this paper can be tested by observation of wide
galaxy pairs and circumgalactic emission (see previous section).

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##  Appendix A Estimating the largest initially circular orbits to remain
bound indefinitely

To estimate the largest circular orbit that remains bound for all time, we
equate the angular momentum of a test particle in an initially circular orbit,
set in motion at any redshift, to the maximum angular momentum found at the
saddle point, _i.e._ at $q_{\rm min}=-0.7$. If the test mass lacks sufficient
energy to escape by the time $q$ saturates to $q_{\rm min}$, the orbit remains
bound indefinitely. The critical radius for a particular initial $q$ is
approximately given by

$GMR_{c}(t)+q(t)H_{0}^{2}R^{4}_{c}(t)=\frac{3}{4}\left(-\frac{G^{4}M^{4}}{4q_{\rm
min}H_{0}^{2}}\right)^{1/3}.$ (A1)

A test mass on an initially circular orbit with initial radius $<R_{c}$ will
remain bound indefinitely. For an initially circular orbit around a
$10^{12}M_{\odot}$ galaxy, starting in the present epoch, $q(t_{0})=-0.55$ and
$H_{0}=70\ \rm km\ s^{-1}Mpc^{-1}$, (39) gives $R_{c}(t_{0})\approx 0.58$ Mpc,
and by evolving the orbit numerically we find $R_{c}(t_{0})\approx 0.57$ Mpc.
Moreover, with (39) we find valid critical radii estimates when initiating the
evolution from any epoch.

###  A.1 Exact critical radii from numerical experiment

We check the validity of our estimated critical radii for a galaxy, by
evolving (6) numerically, as NLH2 have done for a test mass about cluster. We
use the universal expansion factor for a flat, $\Lambda$CDM universe. From
Peebles (1993), we use expansion factor

$a(t)=\Bigg{[}\frac{\Omega_{m}}{1-\Omega_{m}}{\rm
sinh^{2}}\Bigg{(}\frac{3}{2}(1-\Omega_{m})^{1/2}H_{0}t\Bigg{)}\Bigg{]}^{1/3}.$
(A2)

We use equation of motion (3), where the starting time is found for various
$q$’s using

$q(t)=\frac{\Omega_{m}a(t)^{-3}}{2}-\Omega_{\Lambda}.$ (A3)

We take a galaxy sized central mass $M=10^{12}M_{\odot}$,
$\Omega_{\Lambda}=0.7$, $\Omega_{m}=0.3$, and constant (specific) angular
momentum $J=\Big{(}GMR_{i}+q_{i}H_{0}^{2}R_{i}^{4}\Big{)}^{1/2}$, _i.e._ for a
test mass in an initially circular orbit. We start the orbit with an initial
radius near the appropriate critical radius estimated from (39). By numerical
experiment (trial and error) we find the actual $R_{c}=R_{i}$ for which any
initially circular orbit with radius greater than or equal to this will become
free, prior to the saturation of $q$ to $q_{\rm min}$. In Figure (A1) we
compare estimated and numerical critical radii for various initial $q$’s. In
Figures (A2), Figure (A3) and Figure (A4) we illustrate behavior near the
critical initial radius for $q=q_{0}=-0.55$. And in Figure (A5) we compare the
evolution of the orbital energy to the evolving effective potential maximum,
for an initial radius at critical and less than critical.

Figure A1: For various initial $q$’s, corresponding to different starting
times, the largest initially circular orbit to remain bound after $q$
saturates to $q_{\rm min}$ is estimated and calculated numerically. The dashed
curve gives critical radii estimated using (39), and the solid curve is from
numerical evolution.
By increasing $R_{c}$ by one order of magnitude, one obtains an analogous plot
for a cluster sized central mass, $M=10^{15}M_{\odot}$. The same applies to
$r$ in Figures (A1), (A2) and (A3) below. Figure A2: Starting the numerical
evolution at $R_{i}=0.568$ Mpc, for $q_{i}=q_{0}$, the test mass oscillates
about $R=0.642$ Mpc indefinitely. Figure A3: Starting the numerical evolution
at $R_{i}=R_{c}=0.569$ Mpc, for $q_{i}=q_{0}$, the test mass escapes the
potential well. As pointed out by NLH2, its motion, dashed curve, lags behind
the Hubble expansion, solid curve, by a significant margin. Figure A4: Figures
(A1) and (A2) together, comparing test mass evolution for the first 100 Gyr.
Figure A5: Continuing the scenario presented in Figures (A1)-(A3), with the
evolution beginning at $q_{i}=-0.55$, the total energy per unit mass of the
orbit as a function of $q$ is calculated by solving (10) numerically, and is
compared to the exact maximum of the effective potential per unit mass. The
curve with open circles is $V_{\rm max}$, with $R_{i}=R_{c}=0.569$ Mpc, and
the dotted curve is $V_{\rm max}$, with $R_{i}=0.568$ Mpc. The dashed curve is
the energy of the test mass, with $R_{i}=0.569$ Mpc, and the solid curve is
the energy of the test mass, with $R_{i}=0.568$ Mpc. For both starting radii
the test mass is initially bound, having energy ‘below’ the respective $V_{\rm
max}$. When the test mass is set in motion at the critical radius, its energy
goes ‘above’ $V_{\rm max}$, and so is able to escape the potential well,
_i.e._ prior to $q$ saturating to $q_{\rm min}$, though, as shown in Figure
(A2), its subsequent path lags behind the universal expansion. When starting
from smaller radii its energy is below $V_{\rm max}$ indefinitely.

##  Appendix B Generalized Virial Theorem from hydrostatic equilibrium

$\frac{dP}{dR}=-\rho_{b}(R)\frac{GM_{\rm Total}(R)}{R^{2}}$ (B1)

Multiply through by the volume and integrate both sides, where we use
$dM_{b}=4\pi R^{2}\rho_{b}(R)dR$ to affect a change of integration variables
on the RHS.

$\int_{0}^{P(R)}VdP=-\frac{1}{3}\int_{0}^{M_{b}(R)}\frac{GM_{\rm
Total}(R^{\prime})}{R^{\prime}}dM_{b}^{\prime}=\frac{1}{3}\Phi(R)$

Integrate the LHS by parts.

$\left.PV\right|_{0}^{R}-\int_{0}^{V(R)}PdV=\frac{1}{3}\Phi(R)$

$P(R)V(R)-\int_{0}^{M_{b}(R)}\frac{k_{B}T}{\mu
m_{p}}dM^{\prime}_{b}=\frac{1}{3}\Phi(R)$

Substituting the average energy per baryon $\bar{T}=3/2k_{B}T$, we have the
familiar Virial Theorem with an external pressure term:

$3\frac{P(R)}{n_{b}}-2\bar{T}=-\frac{Gm_{b}M_{\rm Total}(R)}{R}.$

Adding the force of dark energy per unit volume $\rho_{b}(R)|q_{0}|H_{0}^{2}R$
to the RHS of (B1) we find the generalized Virial Theorem:

$3\frac{P(R)}{n_{b}}-2\bar{T}=-\frac{Gm_{b}M_{\rm
Total}(R)}{R}+m_{b}|q_{0}|H_{0}^{2}R^{2}.$

Dividing through by the mass of a baryon $m_{b}$ we arrive to (33) of the
text, where the average energy per unit baryon mass is
$\bar{K}=\bar{T}/m_{b}$.

$2\bar{K}=-V_{1}+2V_{2}+3\frac{P(R)}{\rho_{b}}$