From Global to Local Dynamics:
Effects of the Expansion on Astrophysical Structures
A. Balaguera-Antoĺınez
Max Planck Institut für Extraterrestrische Physik, Garching, Gliessenbachstrasse 1
D-85748, Garching, Germany
E-mail: a-balagu@uniandes.edu.co
M. Nowakowski
Departamento de F́ısica, Universidad de los Andes,A.A. 4976, Bogotá, D.C.,
Colombia.
E-mail: mnowakos@uniandes.edu.co
Abstract. We explore the effects of background cosmology on large scale structures
with non-spherical symmetry by using the concept of quasi-equilibrium which allows
certain internal properties (e.g. angular velocity) of the bodies to change with time.
In accordance with the discovery of the accelerated phase of the universe we model
the cosmological background by two representative models: the ΛCDM Model and the
Chaplygin Gas Model. We compare the effects of the two models on various properties
of large astrophysical objects. Different equations of state are also invoked in the
investigation.
PACS numbers: 95.30.Sf, 98.62.Dm, 98.80.Jk, 98.52.Eh, 98.56.Ew
http://arxiv.org/abs/0704.1871v2
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 2
1. Introduction
The interest in the impact of the cosmological expansion on bound astrophysical system
can be traced back to the paper by Einstein and Straus [1]. Over the subsequent years
different approaches have been used to gain some insight into the interplay between
cosmology and astrophysical structures [2, 3, 4, 5, 6]. With the advent of the discovery of
the accelerated Universe [7], it seems timely to pick up the topic once again and contrast
the effects of different models which can explain the current accelerated phase. Indeed,
in [8] such an examination was performed for phantom [9] and quintessence cosmologies
[10]. In the present article we will use the same approach as [2] and [8] and apply
it to models with a positive cosmological constant and Chaplygin gas models. These
two models will serve us as representatives for cosmological models which can explain
the current stage of acceleration. In addition we will also invoke a general model with
variable equation of state of the form ωx = px/ρx (DECDM). The approach mentioned
above consists in generalizing the Newtonian limit by allowing terms which are specific
to the background cosmology. This is not limited to the inclusion of the cosmological
constant. Notice that the effects of the background cosmology on bound systems are now
in general time dependent. This is, however not in contradiction with the fact that we
rightly consider most of the astrophysical structures to be in gravitational equilibrium.
The time dependent background cosmology will not tear the system apart, unless its
density is diluted, but change only, over cosmological time, the internal properties of
the system (like inner velocities or angular velocity etc). As a mathematical tool we will
make use of the virial theorem ([11] for different methods) by allowing certain internal
properties to be time dependent. The virial theorem can be easily derived even in the
presence of the cosmological background as the latter enters only the Newtonian limit
(i.e. the Poisson equation for the gravitational potential which includes now a time
dependent part).
As shown in [12] Dark Energy can affect certain astrophysical static properties
which have to do with either the virialization of the system [13] or the motion of
test bodies [2]. For instance, the cosmological constant sets itself scales of distance
(rΛ = Λ
−1/2) ‡, time and mass, which are of the same order of magnitude as the radius
of the observed universe, the age of the universe and the total mass of the universe (the
so called coincidence problem). At the first glance it looks hopeless to expect any effects
from Λ at astrophysical scales. This is so indeed as long as no other scale enters the
theory. If the latter appears in the theory a combination of the large cosmological and
the small internal scale can yield values of astrophysical relevance (e.g. (rsrΛ)
1/3 where
rs is the Schwarzschild radius) [14]. Furthermore, general conditions for detecting the
effects of variations of ’constants’ represented by scalar fields (just like the cosmological
constant and quintessence models) at scales ranging from terrestial scales have been well
established in [15, 16, 17]. Another enhancement mechanism of background cosmology
emerges when we are dealing with non-spherically symmetric objects. In such case
‡ We work in the so-called geometrized units, i.e. GN = c = 1
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 3
the ratio of two length scales to some positive power (bigger than one) goes often
hand in hand with Λ. For this reason, in examining time dependent effects, we will
mostly employ ellipsoidal configurations. For general consideration of equilibrium in
the spherical case see [18] and [19], the quasi spherical case has been discussed in [20].
In this paper, we will entirely concentrate on effects of Dark Energy followed over
cosmological times, past and future. We do not expect these effects to be large, however,
we think that it is of some importance to probe into these matter, especially in view of
the accelerated universe. In [21] we restricted ourselves to effects over small time scales.
In this sense these two articles complement each other.
2. Local dynamics with background cosmology
In standard cosmology, the universe is described as an ideal fluid which implies the
properties of homogeneity and isotropy. Such properties are mathematically represented
in Friedman-Robertson-Walker line element, which is written for a flat universe as ds2 =
−dt2 + R2(t) (dr2 + r2dΩ2). Einstein field equations and the FRW line element yields
the acceleration equation and the well known Friedman equation, written respectively
Ṙ(t)
= H(t)2 =
πρb(t),
R̈(t)
π [ρb(t) + 3pb(t)] , (1)
In conventional cosmology, the total energy density ρb is a contribution of a radiation,
cold dark matter and dark energy. By neglecting interaction among these components,
each one evolves through the conservation equation ρ̇ = −3H(ρ+ p), so that radiation
scales as ρrad = ρrad(R = 1)R
−4, cold dark matter scales as ρcdm = ρcdm(R = 1)R
while the Dark Energy component associated with the cosmological constant is ρvac =
constant. The major contribution in connection to the cosmological constant is
approximately 70% of total energy density [7].
2.1. Dark energy
As pointed before, almost the 70% of the content of energy density in the unverse is
ruled by a dark energy component, and astronomical observations imply that at the
present time this dark energy component might be represented by the cosmological
constant. That is, at the present time, the equation of state that displays the best
fit with astronomical observation is simply pvac = −ρvac, i.e, ω = −1. Nevertheless,
several models has been proposed in order to reproduce the present value of dark
energy but maintaining a time dependent energy density. One generalizes the equation
of state for the dark energy by px(R) = ωx(R)ρx(R). Dark energy then scales as
ρx(R) = ρx(R = 1)a
−f(R), where the function f(R) is defined as
f(R) ≡ 3
′) + 1
dR′, (2)
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 4
The model is constrained by requiring that at the present time the dark energy density
reproduces the vacuum energy density associated with the cosmological constant, that is,
ρx(R = 1) = ρvac. As pointed before, the case ωx = −1 corresponds to the cosmological
constant ρx = ρvac = Λ/8π, while for 0 > ωx > −1 one stands in the Dark Energy
realms. Models with ωx < −1 lead to future singularities in the so called Phantom regime
[9]. The description of a dynamical dark energy is compatible with the description of
inflationary models.
In the DECDM model, the Friedmann equation and the acceleration equation are
written as
Ṙ(t)
= H20Ωvach1(R)
R̈(t)
= −H20Ωvach2(R) = −
πρvach2(R).(3)
The functions h1,2(R) are given as
h1(R) ≡
R−3 +R−f(R), (4)
h2(R) ≡
R−3 +R−f(z) (1 + 3ωx(R))
2.2. Chaplygin Gas
The main stages the universe has passed through in the standard cosmology scenario can
be reproduced by the introduction of the equation of state of an exotic ideal relativistic
gas written as
pch = κ1ρch −
, (5)
The relevance of this equation of state lies in three different facts: a) it violates the
strong energy condition, which is necessary to obtain an accelerated phase at the present
time b) it generates a well definite speed of sound, which is relevant for the process of
structure formation and c) it unifies the early radiation or dark-matter behavior with
the late dark energy dominance. The so-called pure Chaplygin gas is obtained with
κ1 = 0 and γ = 1.
The time evolution of the Chaplygin Gas-energy density is given from the integration
of mass-energy conservation equation as
ρch(R) =
A+BR−nβ
, (6)
with n = 3(1 + κ1), β = γ + 1 and where A and B are integration constants given as
B = ρch(R = 1)
β − A, A =
0 , (7)
with ρch(R = 1) = ρcrit as the Chaplygin Gas energy density at the present time (only
for a flat universe). In order to write the integration constants as we have done, we
have re-parametrized the Chaplygin Gas equation of state with κ2 = αnH
0 such that a
finite age and a present accelerated phase for the Chaplygin gas universe is reached for
. (8)
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 5
This bound can be derived after integration of the Friedmann equation
H(R)2 =
A+BR−nβ
. (9)
in order to determine the age of the universe. The age of the universe is given in terms
of hypergeometric function as
× 2F1
2nβ + n+ 4
with q ≡ 3ω + 7. For the pure Chaplygin gas one then obtains a finite age and the
observed acceleration at the present time only if the parameter α is such that α3 < 10
which in turn implies a bound on the Chaplygin parameter κ2 < ρ
crit. This warranties
that the equation of state is ωch(today) > −1.
The solution given in Eq.(6) allows us to describe the behavior of Chaplygin gas at
different ages. For early times the κ1-term dominates and Chaplygin Gas behaves as
ρch = B
1/βR−n. This scales as radiation for n = 4 (κ1 = 1/3). Hence we could
in principle relate the integration constant B with cosmological parameters of the
concordance model as B = ρ
rad(R = 1). For κ1 = 0, the model displays a matter
dominated epoch at early times with ρch ≈ B1/βR−3. In this case one can write the
integration constant B = ρ
mat(R = 1) and also the constant A as A = ρ
crit(1 − Ω
mat).
Finally for large times, Chaplygin gas acquires a vacuum-like behavior with
ρch(t → ∞) = A1/β =
H20 . (11)
This would lead to an effective vacuum energy density ρeffvac which can be associated to
the current vacuum energy density ρvac via
ρeffvac = ρvac
With ρeffvac > ρvac, the effects of this effective vacuum energy density may be relevant in
the Newtonian limit when we explore the equilibrium conditions of large scale structures.
Let us concentrate on the case κ1 = 0, which does not reproduce an early radiation
dominated era but still unifies an early dark matter dominance and a vacuum dominated
era at large scale factors. The acceleration equation for this model reads as
R̈(t)
= − 4
πρch(R)ηch(R) (13)
= − 4
A +BR(t)−3β
1− 3κ2
A +BR(t)−3β
where ηch ≡ 1 + 3ωch and
ωch = ωch(R) ≡
= −κ2ρch(R)−β (14)
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 6
1 1.5 2 2.5 3 3.5 4
Scale Factor
β = 2 , α = 0.001
2 , 0.0006
2 , 0.0001
1 1.5 2 2.5 3 3.5 4 4.5 5
Scale factor
−1e−14
−5e−15
5e−15
1e−14
1.5e−14
2e−14
Figure 1. Left: Equation of state for Chaplygin gas ωch = pch/ρch for different
Chaplygin parameters. Right: Acceleration equation as a function of the (future)
scale factor for different Chaplygin parameters.
is the time dependent equation of state. In Fig. 1 we show the behavior of the equation
of state ωch and the acceleration equation for different values of Chaplygin parameters.
2.3. Newtonian description
The effects of the background can be explored through the Newtonian limit of field
equations [22], from which one can derive a modified Poisson’s equation [2]
∇2Φ = 4πδρ− 3
, (15)
where δρ is the overdensity that gives rise to the “gravitational contribution” of the
potential. The total energy density within the clustered configuration ρ is a contribution
of the background ρb and the collapsed fraction δρ. The background energy density
depends on the model we are interested in:
ρb = ρ(R)− δρ =
ρcdm(R) DECDM
ρch(R) Chaplygin
Note that the last expression implies that in the DECDM model one has assumed that
only the cold dark matter component is what collapses and forms vitalized structures,
while in Chaplygin gas the perturbation is done on the total energy density. The solution
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 7
for the potential can be simplified as
Φ(r, R) = Φgrav(r)−
r2, Φgrav(r) = −
δρ(r′)
|r− r′|
d3r′, (17)
where one has neglected the terms associated to boundary conditions (the potential
cannot be zero at infinity) [22]. The last term of Eq.(15) reduces to ρvac for ωx = −1 and
negligible contribution from the cold dark matter component, that is, for low red-shifts.
This limit represents the Newton-Hooke space time, which aside from the gravitational
interaction has an external force of the Hooke form ∼ Λr2.
Gravitational equilibrium is represented through Euler equation ρu̇ = −∇p − ρ∇Φ,
where ρ is given by Eq.(16). The next generation of equilibrium equations comes from
taking moments on Euler equation and derive the second order virial equation [12, 22, 23]
d2Iik
= 2Kik +Wgravik + 3Πik +
Iik, (18)
where Wgravik is the gravitational potential energy tensor, whose trace corresponds to
the gravitational potential energy, Kik is the kinetic energy tensor, Iik is the moment
of inertia tensor and Πik is the dispersion tensor. The trace of these quantities lead to
the well known forms for gravitational potential energy:
Wgrav = −
ρri∂iΦgravd
ρΦgravd
3r, (19)
together with I ≡
2d3r and Π ≡
3r. As usual, the set of equilibrium equations
is closed with the equation for mass conservation, energy conservation and an equation
of state p = p(δρ, s).
The tensor virial equation can be thought as a differential equation for the moment
of inertia, which in turns is converted to a differential equation for the parameters
determining the geometrical properties of the configuration. In order to explore the
contribution from the background, let us write the virial equation (18) as
d2Iik
R2H2(R)
2Tik +Wgravik +
Iik −
. (20)
3. Dynamical equilibrium
If we assume equilibrium via Ï ≈ 0, we obtain the virial theorem
|Wgrav| = 2K +
I, K = T + 3
Π. (21)
Strictly speaking, a formal equilibrium configuration is never reached since the energy-
like terms in (18) are time dependent. Nevertheless, we can use equation (21) by
assuming that as long as R evolves in time, the configuration evolves through successive
states of equilibrium.
The expressions derived in the last section, specially (15) and (21) can be used for
testing dark energy models on configurations in equilibrium. In this section, we will
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 8
concentrate on possible effects of the two cosmological models described above on
cosmological structures with non-spherical symmetry, such as low density galaxies and
galactic clusters.
For ellipsoidal homogeneous configurations, the gravitational potential tensor and the
moment of inertia tensor are written as
Wgravik = −
π2ρ(ρ− ρb)a1a2a3a2iAiδik = −2πδρAiIik,
Iik =
πρa1a2a3a
i δik, (22)
where the quantities Ai are functions of the eccentricities of each case: for an oblate
configuration we have a1 = a2 > a3, e
oblate = 1 − q23, while for prolate a1 = a2 < a3,
e2prolate = 1− q−23 , with qi ≡ ai/a1. The functions Ai are given as [24]:
A1 = A2 =
1− e2
arcsin e
1− e2
Oblate
1− e2
1− e2
1 + e
Prolate
1− e2
1− e2
arcsin e
Oblate
1− e2
1 + e
Prolate
In the limit e → 0 one has Ai → 2/3 for the prolate and oblate configurations. Using
Eqs.(22), the potential energies are then written for the models of interest as
W ik = Wgravik +
Iik (25)
π2ρ2a1a2a3a
iAiδik −
πηxρxIik + 2πρcdm
π2ρ2a1a2a3a
iAiδik + 2πρch
for DECDM and Chaplygin Gas respectuvely, with ηx = 1+3ωx and ρx = ρvac(1+z)
f(z).
Note that the last term on the DECDM model vanishes in two different situations:
the first is in the case of spherical symmetry, and when we neglect the contribution
from the cold dark matter component with respect to the proper density of the system
and the dark energy contribution. This approximation is only valid for very low red-
shifts. At high red shifts we approach a matter dominated universe and hence the term
proportional to ρcdm is relevant.
3.1. Oblate systems
When considering oblate configurations (a1 = a2 > a3), we assume that the kinetic
energy is due to constant rotation along the minor axis. We then may write the kinetic
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 9
energy tensor as
Tik =
Ω2rotIik − ΩrotiIkjΩrotj
, (26)
In order to determine the angular velocity, one may consider a non-zero isotropic
dispersion tensor Πik = δikΠ. We then have
Ω2rot =
|Wgravxx | − |Wgravzz |
1− Izz
, (27)
which follows from eliminating the trace of the dispersion tensor from the tensor virial
equations. Using Eq.(25), we can explicitly write for the angular velocity
Ω2rot
A1 −A3q23
× (28)
1− q23
2πδρ(A1 −A3q23)
H20Ωvach2(R) DECDM
πρch(R)ηch(R) Chaplygin
In the Newton-Hooke space time with ωx = −1 (h2 → −1) this reduces to
Ω2rot
≈ A1 − A3q23 −
(1− q23). (29)
which in turn reproduces the Maclaurin formula for ρvac = 0.
3.2. Prolate systems
For prolate configurations (a1 = a2 < a3), we solve for the velocity dispersion of main
components rather than for an angular velocity (rotation of prolate configurations,
although rare, has been observed and some properties of this situation has been explored
in [21]). Hence we use the virial theorem (21) with a the kinetic energy given by
K = 1
ρ〈v2〉 d3r = 2
πρa21a3〈v2〉. (30)
In analogy with Eq.(28), the virial theorem then implies for the velocity dispersion
2A1 + q
2 + q23
2πδρ(2A1 + q
H20Ωvach2(R)
πρch(R)ηch(R)
for the DECDM and Chaplygin Gas model respectively. As in (28), one obtains a scale
factor (or red-shift) dependence of the velocity dispersion through the contribution of
the cosmological background. In the Newton-Hooke space-time one reduces to
a21(2A1 + q
2 + q23
2A1 + q
, (32)
which in turn reduces for spherical symmetry to the known expression [12, 25]
. (33)
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures10
0.6 1.6 2.6
Scale factor
0.985
0.995
0.6 1.6 2.6
Scale factor
0.9985
0.999
0.9995
e = 0.5
 δρ = 100 ρvac
 δρ = 1000 ρvac
Figure 2. Equation (34) for ∆v, for different densities and eccentricities in a ΛCDM
model.
The structure of the resulting expressions for the angular velocity and the velocity
dispersion can be written in a similar way. Let ∆ measures the ratio of each velocities
with respect to the velocities when there is not background contribution. We then may
write
∆Ω,v ≡
Ω2rot
Ω2rot(ρb = 0)
〈v2〉(ρb = 0)
= 1− GΩ,v
, (34)
where the geometrical factor G is written for each case as
1− q23
A1 − q23A3
Oblate, rotational velocity
2 + q23
2A1 + q
Prolate, velocity dispersion
Returning to the oblate case, in a generalized Newton-Hooke space-time (i.e, with
−1 < ω < 0 ), one observes that the Dark Energy component decreases the angular
velocity (or the velocity dispersion in the prolate case) in order for the system to
maintain equilibrium (since 1 + 3ωx < 0). At the present time (R = 1, z = 0) we
get ∆Ω ∼ 1− 0.8g(e)(ρvac/δρ), where
g(e) ≡ 4
(1− e2)1/2(3− 2e2) arcsin e− 3e(1− e2)
. (36)
An extreme situation is reached in the limit ∆Ω → 0, which implies Ω2 → 0. For e ∼ 0.8
this would require ζ ≡ 2ρvac/δρ ∼ 0.5, which is a very diluted configuration. For higher
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures11
0.6 1.6 2.6 3.6
Scale factor
e = 0.5 
0.6 1.6 2.6 3.6
Scale factor
0.995
0.997
0.999
1.001
1.003
1.005
1.007
1.009
δρ = 100ρcrit
β = 2
α = 0.005
δρ = 1000ρcrit
β = 2
α = 0.005
Figure 3. Equation (34) for ∆v, for different densities and eccentricities in a Chaplygin
Gas model for typical values of Chaplygin parameters.
eccentricities (say e ∼ 0.97) one would need a system with ζ ∼ 0.2 which is still a
rather low value. In the ΛCDM model Ω = 0 is reached at a red-shift given by [26]
zc = [(2Ωvac(ζg(e)− 2))/(ζg(e)Ωcdm)]1/3−1. For realistic examples as δρ ∼ 200ρcdm the
value zc could be reached for e very close to 1. That is, the value zc requires extreme
flat objects.
In figures 2 and 3 we show the behavior of the ratio ∆v for different values of eccentricities
and proper densities in the DECDM case with ωx = −1 and in the Chaplygin Gas
model, respectively. The larger effects appears for low densities and large eccentricities,
as expected. However, the fractional change is very low. For δρ = 102ρvac, the larger
effect occurs for scale factors ≥ 2.6 with a change of 1%.
In Fig. 4 we show the behavior of ∆Ω − 1 as a function of the red-shift for different
densities and e = 0.9, 0.95 for three different values of the equation of state for Dark
Energy. It is clear that the effects associated with a cosmological constant are stronger
than the ones associated with other Dark Energy models, but in general those effects
are small for realistic values of ζ as used in the figure. We would have to measure the
angular velocity at different red-shifts very exactly to see an effect over a range of z.
However, the difference between the models is more significant.
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures12
0 0.1 0.2 0.3 0.4
redshift z
−0.002
−0.001
0.001
0.002
ω x = −1/3
ω x = −2/3
ω x = −1
0 0.2 0.4
redshift z
−0.25
−0.15
−0.05
e = 0.9
δρ = 200 ρvac
e = 0.95
δρ = 20 ρvac
Figure 4. Equation (34) for ∆Ω − 1 in terms of the red-shift (up to a ∼ 0.7), for
different densities and eccentricities in a DECDM model for typical values of equation
of state. [26]
4. Dynamical evolution
In this section we want to explore the virial equation with the cosmological background
without insisting on the equilibrium condition (dynamical equilibrium). Even though
we will not make extensive use of these new equations, we want to demonstrate that the
equilibrium condition is not a necessary ingredient while studying the effects of different
background cosmologies. The results are differential equation for the geometrical
parameters of a given configuration. Our starting point is equation (20). In the first
approach, we will concentrate on pressurless systems with homogeneous density. The
problem is the reduced to determine a1 = a1(R) and a3 = a3(R). As pointed out before,
Eq.(20) represents a set of two non-linear coupled second order equations for i = x, z
d2Iii
R2H2(R)
2Tii +Wgravii +
Iii −
. (37)
where we have used Eq.(25). The derivatives of the moment of inertia tensor are given
as follows
πρa21a3ai
= 2a−1i
Iii, (38)
d2Iii
πρa21a3ai
 = 2a−1i
 Iii,
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures13
These expressions enclose our approximation: the volume of the system is constant even
if the semi-axes may change, so that one can write the moment of inertia tensor as
Iik =
Ma2i δik, (39)
where the mass M is
M = Mb + δM =
πa21a3ρb(R) + δM, δM =
δρ d3r. (40)
In order to deal with the kinetic energy tensor, we follow [27] and transform from
proper coordinates to fixed co-moving coordinates via ri = Aiαxα, where Aiα is such
that Aiα = aiδiα and hence the ellipsoid is characterized by the constraint δαβxαx
β = 1.
We have
Tik =
xαxβAd
3x, (41)
where A = det(Aαβ) = a1a2a3. Using the definition of the moment of inertia tensor one
Iik = ρAiαAkβA
xαxβd
3x , (42)
and hence
Tik =
A−1iα A
Iik. (43)
We then may write
Tii =
Iii = a−2i R2
Iii ×
H20Ωvach1(R)
πρch(R)
for the DECDM and Chaplygin Gas respectively. Equation (37) together with (38) are
the differential equation to be solved. The functions Ai depend on the eccentricity of
the system and hence they are also function of the semiaxis Ai(z) = Ai(a1(R), a3(R)).
The differential equation can then be cast into the following form
R2H2(R)
2Tii +Wii
. (45)
5. Conclusion
In this article we have examined the effects of the expansion of the Universe on certain
quasi-static properties of large astrophysical bodies. We have done this by invoking a
dynamical equilibrium i.e. the time dependent response of the astrophysical objects to
the time dependent cosmological background is taken into account by allowing some
internal properties to become epoch dependent. In spite of the fact that the expansion
of the Universe is accelerated the effects are rather small, at the most few per cents over
a large cosmological time stretch in the case of the Chaplygin gas model. However, a
qualitative investigation, done in the present paper, seems mandatory to establish the
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures14
size of the effects. Different equations of state can be clearly distinguished theoretically,
however, in practice this would require a good knowledge of the angular velocities at
different redshifts. In spite of the smallness of the effects, the results clearly demonstrate
that background cosmology affects, in principle, local properties of astrophysical bodies
and furthermore, the details of these effects are model-dependent. In this paper, we
focused on the the dynamical equilibrium. One can enlarge the concept of the influence
of background cosmology by dropping the equilibrium condition and go over to a fully
dynamical scenario. A first step in this direction was done on section 4. Supplementing
these equation with Euler equation for the angular velocity, one has a full set of
differential equations determining the time evolution of the parameters during (and
after) virialization. We will come back to this point in future publications.
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http://arxiv.org/abs/astro-ph/0112320
http://arxiv.org/abs/astro-ph/0409481
http://arxiv.org/abs/hep-th/0505215
http://arxiv.org/abs/astro-ph/0507322
http://arxiv.org/abs/astro-ph/0507299
http://arxiv.org/abs/astro-ph/0410680
http://arxiv.org/abs/astro-ph/9901388
http://arxiv.org/abs/astro-ph/0603624
	Introduction
	Local dynamics with background cosmology
	Dark energy
	Chaplygin Gas
	Newtonian description
	Dynamical equilibrium
	Oblate systems
	Prolate systems
	Dynamical evolution
	Conclusion

  We explore the effects of background cosmology on large scale structures with
non-spherical symmetry by using the concept of quasi-equilibrium which allows
certain internal properties (e.g. angular velocity) of the bodies to change
with time. In accordance with the discovery of the accelerated phase of the
universe we model the cosmological background by two representative models: the
$\Lambda$CDM Model and the Chaplygin Gas Model. We compare the effects of the
two models on various properties of large astrophysical objects. Different
equations of state are also invoked in the investigation.

Introduction
The interest in the impact of the cosmological expansion on bound astrophysical system
can be traced back to the paper by Einstein and Straus [1]. Over the subsequent years
different approaches have been used to gain some insight into the interplay between
cosmology and astrophysical structures [2, 3, 4, 5, 6]. With the advent of the discovery of
the accelerated Universe [7], it seems timely to pick up the topic once again and contrast
the effects of different models which can explain the current accelerated phase. Indeed,
in [8] such an examination was performed for phantom [9] and quintessence cosmologies
[10]. In the present article we will use the same approach as [2] and [8] and apply
it to models with a positive cosmological constant and Chaplygin gas models. These
two models will serve us as representatives for cosmological models which can explain
the current stage of acceleration. In addition we will also invoke a general model with
variable equation of state of the form ωx = px/ρx (DECDM). The approach mentioned
above consists in generalizing the Newtonian limit by allowing terms which are specific
to the background cosmology. This is not limited to the inclusion of the cosmological
constant. Notice that the effects of the background cosmology on bound systems are now
in general time dependent. This is, however not in contradiction with the fact that we
rightly consider most of the astrophysical structures to be in gravitational equilibrium.
The time dependent background cosmology will not tear the system apart, unless its
density is diluted, but change only, over cosmological time, the internal properties of
the system (like inner velocities or angular velocity etc). As a mathematical tool we will
make use of the virial theorem ([11] for different methods) by allowing certain internal
properties to be time dependent. The virial theorem can be easily derived even in the
presence of the cosmological background as the latter enters only the Newtonian limit
(i.e. the Poisson equation for the gravitational potential which includes now a time
dependent part).
As shown in [12] Dark Energy can affect certain astrophysical static properties
which have to do with either the virialization of the system [13] or the motion of
test bodies [2]. For instance, the cosmological constant sets itself scales of distance
(rΛ = Λ
−1/2) ‡, time and mass, which are of the same order of magnitude as the radius
of the observed universe, the age of the universe and the total mass of the universe (the
so called coincidence problem). At the first glance it looks hopeless to expect any effects
from Λ at astrophysical scales. This is so indeed as long as no other scale enters the
theory. If the latter appears in the theory a combination of the large cosmological and
the small internal scale can yield values of astrophysical relevance (e.g. (rsrΛ)
1/3 where
rs is the Schwarzschild radius) [14]. Furthermore, general conditions for detecting the
effects of variations of ’constants’ represented by scalar fields (just like the cosmological
constant and quintessence models) at scales ranging from terrestial scales have been well
established in [15, 16, 17]. Another enhancement mechanism of background cosmology
emerges when we are dealing with non-spherically symmetric objects. In such case
‡ We work in the so-called geometrized units, i.e. GN = c = 1
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 3
the ratio of two length scales to some positive power (bigger than one) goes often
hand in hand with Λ. For this reason, in examining time dependent effects, we will
mostly employ ellipsoidal configurations. For general consideration of equilibrium in
the spherical case see [18] and [19], the quasi spherical case has been discussed in [20].
In this paper, we will entirely concentrate on effects of Dark Energy followed over
cosmological times, past and future. We do not expect these effects to be large, however,
we think that it is of some importance to probe into these matter, especially in view of
the accelerated universe. In [21] we restricted ourselves to effects over small time scales.
In this sense these two articles complement each other.
2. Local dynamics with background cosmology
In standard cosmology, the universe is described as an ideal fluid which implies the
properties of homogeneity and isotropy. Such properties are mathematically represented
in Friedman-Robertson-Walker line element, which is written for a flat universe as ds2 =
−dt2 + R2(t) (dr2 + r2dΩ2). Einstein field equations and the FRW line element yields
the acceleration equation and the well known Friedman equation, written respectively
Ṙ(t)
= H(t)2 =
πρb(t),
R̈(t)
π [ρb(t) + 3pb(t)] , (1)
In conventional cosmology, the total energy density ρb is a contribution of a radiation,
cold dark matter and dark energy. By neglecting interaction among these components,
each one evolves through the conservation equation ρ̇ = −3H(ρ+ p), so that radiation
scales as ρrad = ρrad(R = 1)R
−4, cold dark matter scales as ρcdm = ρcdm(R = 1)R
while the Dark Energy component associated with the cosmological constant is ρvac =
constant. The major contribution in connection to the cosmological constant is
approximately 70% of total energy density [7].
2.1. Dark energy
As pointed before, almost the 70% of the content of energy density in the unverse is
ruled by a dark energy component, and astronomical observations imply that at the
present time this dark energy component might be represented by the cosmological
constant. That is, at the present time, the equation of state that displays the best
fit with astronomical observation is simply pvac = −ρvac, i.e, ω = −1. Nevertheless,
several models has been proposed in order to reproduce the present value of dark
energy but maintaining a time dependent energy density. One generalizes the equation
of state for the dark energy by px(R) = ωx(R)ρx(R). Dark energy then scales as
ρx(R) = ρx(R = 1)a
−f(R), where the function f(R) is defined as
f(R) ≡ 3
′) + 1
dR′, (2)
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 4
The model is constrained by requiring that at the present time the dark energy density
reproduces the vacuum energy density associated with the cosmological constant, that is,
ρx(R = 1) = ρvac. As pointed before, the case ωx = −1 corresponds to the cosmological
constant ρx = ρvac = Λ/8π, while for 0 > ωx > −1 one stands in the Dark Energy
realms. Models with ωx < −1 lead to future singularities in the so called Phantom regime
[9]. The description of a dynamical dark energy is compatible with the description of
inflationary models.
In the DECDM model, the Friedmann equation and the acceleration equation are
written as
Ṙ(t)
= H20Ωvach1(R)
R̈(t)
= −H20Ωvach2(R) = −
πρvach2(R).(3)
The functions h1,2(R) are given as
h1(R) ≡
R−3 +R−f(R), (4)
h2(R) ≡
R−3 +R−f(z) (1 + 3ωx(R))
2.2. Chaplygin Gas
The main stages the universe has passed through in the standard cosmology scenario can
be reproduced by the introduction of the equation of state of an exotic ideal relativistic
gas written as
pch = κ1ρch −
, (5)
The relevance of this equation of state lies in three different facts: a) it violates the
strong energy condition, which is necessary to obtain an accelerated phase at the present
time b) it generates a well definite speed of sound, which is relevant for the process of
structure formation and c) it unifies the early radiation or dark-matter behavior with
the late dark energy dominance. The so-called pure Chaplygin gas is obtained with
κ1 = 0 and γ = 1.
The time evolution of the Chaplygin Gas-energy density is given from the integration
of mass-energy conservation equation as
ρch(R) =
A+BR−nβ
, (6)
with n = 3(1 + κ1), β = γ + 1 and where A and B are integration constants given as
B = ρch(R = 1)
β − A, A =
0 , (7)
with ρch(R = 1) = ρcrit as the Chaplygin Gas energy density at the present time (only
for a flat universe). In order to write the integration constants as we have done, we
have re-parametrized the Chaplygin Gas equation of state with κ2 = αnH
0 such that a
finite age and a present accelerated phase for the Chaplygin gas universe is reached for
. (8)
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 5
This bound can be derived after integration of the Friedmann equation
H(R)2 =
A+BR−nβ
. (9)
in order to determine the age of the universe. The age of the universe is given in terms
of hypergeometric function as
× 2F1
2nβ + n+ 4
with q ≡ 3ω + 7. For the pure Chaplygin gas one then obtains a finite age and the
observed acceleration at the present time only if the parameter α is such that α3 < 10
which in turn implies a bound on the Chaplygin parameter κ2 < ρ
crit. This warranties
that the equation of state is ωch(today) > −1.
The solution given in Eq.(6) allows us to describe the behavior of Chaplygin gas at
different ages. For early times the κ1-term dominates and Chaplygin Gas behaves as
ρch = B
1/βR−n. This scales as radiation for n = 4 (κ1 = 1/3). Hence we could
in principle relate the integration constant B with cosmological parameters of the
concordance model as B = ρ
rad(R = 1). For κ1 = 0, the model displays a matter
dominated epoch at early times with ρch ≈ B1/βR−3. In this case one can write the
integration constant B = ρ
mat(R = 1) and also the constant A as A = ρ
crit(1 − Ω
mat).
Finally for large times, Chaplygin gas acquires a vacuum-like behavior with
ρch(t → ∞) = A1/β =
H20 . (11)
This would lead to an effective vacuum energy density ρeffvac which can be associated to
the current vacuum energy density ρvac via
ρeffvac = ρvac
With ρeffvac > ρvac, the effects of this effective vacuum energy density may be relevant in
the Newtonian limit when we explore the equilibrium conditions of large scale structures.
Let us concentrate on the case κ1 = 0, which does not reproduce an early radiation
dominated era but still unifies an early dark matter dominance and a vacuum dominated
era at large scale factors. The acceleration equation for this model reads as
R̈(t)
= − 4
πρch(R)ηch(R) (13)
= − 4
A +BR(t)−3β
1− 3κ2
A +BR(t)−3β
where ηch ≡ 1 + 3ωch and
ωch = ωch(R) ≡
= −κ2ρch(R)−β (14)
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 6
1 1.5 2 2.5 3 3.5 4
Scale Factor
β = 2 , α = 0.001
2 , 0.0006
2 , 0.0001
1 1.5 2 2.5 3 3.5 4 4.5 5
Scale factor
−1e−14
−5e−15
5e−15
1e−14
1.5e−14
2e−14
Figure 1. Left: Equation of state for Chaplygin gas ωch = pch/ρch for different
Chaplygin parameters. Right: Acceleration equation as a function of the (future)
scale factor for different Chaplygin parameters.
is the time dependent equation of state. In Fig. 1 we show the behavior of the equation
of state ωch and the acceleration equation for different values of Chaplygin parameters.
2.3. Newtonian description
The effects of the background can be explored through the Newtonian limit of field
equations [22], from which one can derive a modified Poisson’s equation [2]
∇2Φ = 4πδρ− 3
, (15)
where δρ is the overdensity that gives rise to the “gravitational contribution” of the
potential. The total energy density within the clustered configuration ρ is a contribution
of the background ρb and the collapsed fraction δρ. The background energy density
depends on the model we are interested in:
ρb = ρ(R)− δρ =
ρcdm(R) DECDM
ρch(R) Chaplygin
Note that the last expression implies that in the DECDM model one has assumed that
only the cold dark matter component is what collapses and forms vitalized structures,
while in Chaplygin gas the perturbation is done on the total energy density. The solution
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 7
for the potential can be simplified as
Φ(r, R) = Φgrav(r)−
r2, Φgrav(r) = −
δρ(r′)
|r− r′|
d3r′, (17)
where one has neglected the terms associated to boundary conditions (the potential
cannot be zero at infinity) [22]. The last term of Eq.(15) reduces to ρvac for ωx = −1 and
negligible contribution from the cold dark matter component, that is, for low red-shifts.
This limit represents the Newton-Hooke space time, which aside from the gravitational
interaction has an external force of the Hooke form ∼ Λr2.
Gravitational equilibrium is represented through Euler equation ρu̇ = −∇p − ρ∇Φ,
where ρ is given by Eq.(16). The next generation of equilibrium equations comes from
taking moments on Euler equation and derive the second order virial equation [12, 22, 23]
d2Iik
= 2Kik +Wgravik + 3Πik +
Iik, (18)
where Wgravik is the gravitational potential energy tensor, whose trace corresponds to
the gravitational potential energy, Kik is the kinetic energy tensor, Iik is the moment
of inertia tensor and Πik is the dispersion tensor. The trace of these quantities lead to
the well known forms for gravitational potential energy:
Wgrav = −
ρri∂iΦgravd
ρΦgravd
3r, (19)
together with I ≡
2d3r and Π ≡
3r. As usual, the set of equilibrium equations
is closed with the equation for mass conservation, energy conservation and an equation
of state p = p(δρ, s).
The tensor virial equation can be thought as a differential equation for the moment
of inertia, which in turns is converted to a differential equation for the parameters
determining the geometrical properties of the configuration. In order to explore the
contribution from the background, let us write the virial equation (18) as
d2Iik
R2H2(R)
2Tik +Wgravik +
Iik −
. (20)
3. Dynamical equilibrium
If we assume equilibrium via Ï ≈ 0, we obtain the virial theorem
|Wgrav| = 2K +
I, K = T + 3
Π. (21)
Strictly speaking, a formal equilibrium configuration is never reached since the energy-
like terms in (18) are time dependent. Nevertheless, we can use equation (21) by
assuming that as long as R evolves in time, the configuration evolves through successive
states of equilibrium.
The expressions derived in the last section, specially (15) and (21) can be used for
testing dark energy models on configurations in equilibrium. In this section, we will
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 8
concentrate on possible effects of the two cosmological models described above on
cosmological structures with non-spherical symmetry, such as low density galaxies and
galactic clusters.
For ellipsoidal homogeneous configurations, the gravitational potential tensor and the
moment of inertia tensor are written as
Wgravik = −
π2ρ(ρ− ρb)a1a2a3a2iAiδik = −2πδρAiIik,
Iik =
πρa1a2a3a
i δik, (22)
where the quantities Ai are functions of the eccentricities of each case: for an oblate
configuration we have a1 = a2 > a3, e
oblate = 1 − q23, while for prolate a1 = a2 < a3,
e2prolate = 1− q−23 , with qi ≡ ai/a1. The functions Ai are given as [24]:
A1 = A2 =
1− e2
arcsin e
1− e2
Oblate
1− e2
1− e2
1 + e
Prolate
1− e2
1− e2
arcsin e
Oblate
1− e2
1 + e
Prolate
In the limit e → 0 one has Ai → 2/3 for the prolate and oblate configurations. Using
Eqs.(22), the potential energies are then written for the models of interest as
W ik = Wgravik +
Iik (25)
π2ρ2a1a2a3a
iAiδik −
πηxρxIik + 2πρcdm
π2ρ2a1a2a3a
iAiδik + 2πρch
for DECDM and Chaplygin Gas respectuvely, with ηx = 1+3ωx and ρx = ρvac(1+z)
f(z).
Note that the last term on the DECDM model vanishes in two different situations:
the first is in the case of spherical symmetry, and when we neglect the contribution
from the cold dark matter component with respect to the proper density of the system
and the dark energy contribution. This approximation is only valid for very low red-
shifts. At high red shifts we approach a matter dominated universe and hence the term
proportional to ρcdm is relevant.
3.1. Oblate systems
When considering oblate configurations (a1 = a2 > a3), we assume that the kinetic
energy is due to constant rotation along the minor axis. We then may write the kinetic
From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 9
energy tensor as
Tik =
Ω2rotIik − ΩrotiIkjΩrotj
, (26)
In order to determine the angular velocity, one may consider a non-zero isotropic
dispersion tensor Πik = δikΠ. We then have
Ω2rot =
|Wgravxx | − |Wgravzz |
1− Izz
, (27)
which follows from eliminating the trace of the dispersion tensor from the tensor virial
equations. Using Eq.(25), we can explicitly write for the angular velocity
Ω2rot
A1 −A3q23
× (28)
1− q23
2πδρ(A1 −A3q23)
H20Ωvach2(R) DECDM
πρch(R)ηch(R) Chaplygin
In the Newton-Hooke space time with ωx = −1 (h2 → −1) this reduces to
Ω2rot
≈ A1 − A3q23 −
(1− q23). (29)
which in turn reproduces the Maclaurin formula for ρvac = 0.
3.2. Prolate systems
For prolate configurations (a1 = a2 < a3), we solve for the velocity dispersion of main
components rather than for an angular velocity (rotation of prolate configurations,
although rare, has been observed and some properties of this situation has been explored
in [21]). Hence we use the virial theorem (21) with a the kinetic energy given by
K = 1
ρ〈v2〉 d3r = 2
πρa21a3〈v2〉. (30)
In analogy with Eq.(28), the virial theorem then implies for the velocity dispersion
2A1 + q
2 + q23
2πδρ(2A1 + q
H20Ωvach2(R)
πρch(R)ηch(R)
for the DECDM and Chaplygin Gas model respectively. As in (28), one obtains a scale
factor (or red-shift) dependence of the velocity dispersion through the contribution of
the cosmological background. In the Newton-Hooke space-time one reduces to
a21(2A1 + q
2 + q23
2A1 + q
, (32)
which in turn reduces for spherical symmetry to the known expression [12, 25]
. (33)
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures10
0.6 1.6 2.6
Scale factor
0.985
0.995
0.6 1.6 2.6
Scale factor
0.9985
0.999
0.9995
e = 0.5
 δρ = 100 ρvac
 δρ = 1000 ρvac
Figure 2. Equation (34) for ∆v, for different densities and eccentricities in a ΛCDM
model.
The structure of the resulting expressions for the angular velocity and the velocity
dispersion can be written in a similar way. Let ∆ measures the ratio of each velocities
with respect to the velocities when there is not background contribution. We then may
write
∆Ω,v ≡
Ω2rot
Ω2rot(ρb = 0)
〈v2〉(ρb = 0)
= 1− GΩ,v
, (34)
where the geometrical factor G is written for each case as
1− q23
A1 − q23A3
Oblate, rotational velocity
2 + q23
2A1 + q
Prolate, velocity dispersion
Returning to the oblate case, in a generalized Newton-Hooke space-time (i.e, with
−1 < ω < 0 ), one observes that the Dark Energy component decreases the angular
velocity (or the velocity dispersion in the prolate case) in order for the system to
maintain equilibrium (since 1 + 3ωx < 0). At the present time (R = 1, z = 0) we
get ∆Ω ∼ 1− 0.8g(e)(ρvac/δρ), where
g(e) ≡ 4
(1− e2)1/2(3− 2e2) arcsin e− 3e(1− e2)
. (36)
An extreme situation is reached in the limit ∆Ω → 0, which implies Ω2 → 0. For e ∼ 0.8
this would require ζ ≡ 2ρvac/δρ ∼ 0.5, which is a very diluted configuration. For higher
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures11
0.6 1.6 2.6 3.6
Scale factor
e = 0.5 
0.6 1.6 2.6 3.6
Scale factor
0.995
0.997
0.999
1.001
1.003
1.005
1.007
1.009
δρ = 100ρcrit
β = 2
α = 0.005
δρ = 1000ρcrit
β = 2
α = 0.005
Figure 3. Equation (34) for ∆v, for different densities and eccentricities in a Chaplygin
Gas model for typical values of Chaplygin parameters.
eccentricities (say e ∼ 0.97) one would need a system with ζ ∼ 0.2 which is still a
rather low value. In the ΛCDM model Ω = 0 is reached at a red-shift given by [26]
zc = [(2Ωvac(ζg(e)− 2))/(ζg(e)Ωcdm)]1/3−1. For realistic examples as δρ ∼ 200ρcdm the
value zc could be reached for e very close to 1. That is, the value zc requires extreme
flat objects.
In figures 2 and 3 we show the behavior of the ratio ∆v for different values of eccentricities
and proper densities in the DECDM case with ωx = −1 and in the Chaplygin Gas
model, respectively. The larger effects appears for low densities and large eccentricities,
as expected. However, the fractional change is very low. For δρ = 102ρvac, the larger
effect occurs for scale factors ≥ 2.6 with a change of 1%.
In Fig. 4 we show the behavior of ∆Ω − 1 as a function of the red-shift for different
densities and e = 0.9, 0.95 for three different values of the equation of state for Dark
Energy. It is clear that the effects associated with a cosmological constant are stronger
than the ones associated with other Dark Energy models, but in general those effects
are small for realistic values of ζ as used in the figure. We would have to measure the
angular velocity at different red-shifts very exactly to see an effect over a range of z.
However, the difference between the models is more significant.
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures12
0 0.1 0.2 0.3 0.4
redshift z
−0.002
−0.001
0.001
0.002
ω x = −1/3
ω x = −2/3
ω x = −1
0 0.2 0.4
redshift z
−0.25
−0.15
−0.05
e = 0.9
δρ = 200 ρvac
e = 0.95
δρ = 20 ρvac
Figure 4. Equation (34) for ∆Ω − 1 in terms of the red-shift (up to a ∼ 0.7), for
different densities and eccentricities in a DECDM model for typical values of equation
of state. [26]
4. Dynamical evolution
In this section we want to explore the virial equation with the cosmological background
without insisting on the equilibrium condition (dynamical equilibrium). Even though
we will not make extensive use of these new equations, we want to demonstrate that the
equilibrium condition is not a necessary ingredient while studying the effects of different
background cosmologies. The results are differential equation for the geometrical
parameters of a given configuration. Our starting point is equation (20). In the first
approach, we will concentrate on pressurless systems with homogeneous density. The
problem is the reduced to determine a1 = a1(R) and a3 = a3(R). As pointed out before,
Eq.(20) represents a set of two non-linear coupled second order equations for i = x, z
d2Iii
R2H2(R)
2Tii +Wgravii +
Iii −
. (37)
where we have used Eq.(25). The derivatives of the moment of inertia tensor are given
as follows
πρa21a3ai
= 2a−1i
Iii, (38)
d2Iii
πρa21a3ai
 = 2a−1i
 Iii,
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures13
These expressions enclose our approximation: the volume of the system is constant even
if the semi-axes may change, so that one can write the moment of inertia tensor as
Iik =
Ma2i δik, (39)
where the mass M is
M = Mb + δM =
πa21a3ρb(R) + δM, δM =
δρ d3r. (40)
In order to deal with the kinetic energy tensor, we follow [27] and transform from
proper coordinates to fixed co-moving coordinates via ri = Aiαxα, where Aiα is such
that Aiα = aiδiα and hence the ellipsoid is characterized by the constraint δαβxαx
β = 1.
We have
Tik =
xαxβAd
3x, (41)
where A = det(Aαβ) = a1a2a3. Using the definition of the moment of inertia tensor one
Iik = ρAiαAkβA
xαxβd
3x , (42)
and hence
Tik =
A−1iα A
Iik. (43)
We then may write
Tii =
Iii = a−2i R2
Iii ×
H20Ωvach1(R)
πρch(R)
for the DECDM and Chaplygin Gas respectively. Equation (37) together with (38) are
the differential equation to be solved. The functions Ai depend on the eccentricity of
the system and hence they are also function of the semiaxis Ai(z) = Ai(a1(R), a3(R)).
The differential equation can then be cast into the following form
R2H2(R)
2Tii +Wii
. (45)
5. Conclusion
In this article we have examined the effects of the expansion of the Universe on certain
quasi-static properties of large astrophysical bodies. We have done this by invoking a
dynamical equilibrium i.e. the time dependent response of the astrophysical objects to
the time dependent cosmological background is taken into account by allowing some
internal properties to become epoch dependent. In spite of the fact that the expansion
of the Universe is accelerated the effects are rather small, at the most few per cents over
a large cosmological time stretch in the case of the Chaplygin gas model. However, a
qualitative investigation, done in the present paper, seems mandatory to establish the
From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures14
size of the effects. Different equations of state can be clearly distinguished theoretically,
however, in practice this would require a good knowledge of the angular velocities at
different redshifts. In spite of the smallness of the effects, the results clearly demonstrate
that background cosmology affects, in principle, local properties of astrophysical bodies
and furthermore, the details of these effects are model-dependent. In this paper, we
focused on the the dynamical equilibrium. One can enlarge the concept of the influence
of background cosmology by dropping the equilibrium condition and go over to a fully
dynamical scenario. A first step in this direction was done on section 4. Supplementing
these equation with Euler equation for the angular velocity, one has a full set of
differential equations determining the time evolution of the parameters during (and
after) virialization. We will come back to this point in future publications.
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http://arxiv.org/abs/astro-ph/0112320
http://arxiv.org/abs/astro-ph/0409481
http://arxiv.org/abs/hep-th/0505215
http://arxiv.org/abs/astro-ph/0507322
http://arxiv.org/abs/astro-ph/0507299
http://arxiv.org/abs/astro-ph/0410680
http://arxiv.org/abs/astro-ph/9901388
http://arxiv.org/abs/astro-ph/0603624
	Introduction
	Local dynamics with background cosmology
	Dark energy
	Chaplygin Gas
	Newtonian description
	Dynamical equilibrium
	Oblate systems
	Prolate systems
	Dynamical evolution
	Conclusion
