Dynamics of dark energy in collapsing halo of dark matter
We investigate the non-linear evolution of spherical density and velocity perturbations of dark matter and dark energy in the expanding Universe. For this we have used the conservation and Einstein equations to describe the evolution of gravitationally coupled inhomogeneities of dark matter, dark en...
Gespeichert in:
| Veröffentlicht in: | Advances in Astronomy and Space Physics |
|---|---|
| Datum: | 2015 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Головна астрономічна обсерваторія НАН України
2015
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/119826 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Dynamics of dark energy in collapsing halo of dark matter / M. Tsizh, B. Novosyadlyj // Advances in Astronomy and Space Physics. — 2015. — Т. 5., вип. 1. — С. 51-56. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-119826 |
|---|---|
| record_format |
dspace |
| spelling |
Tsizh, M. Novosyadlyj, B. 2017-06-09T21:12:09Z 2017-06-09T21:12:09Z 2015 Dynamics of dark energy in collapsing halo of dark matter / M. Tsizh, B. Novosyadlyj // Advances in Astronomy and Space Physics. — 2015. — Т. 5., вип. 1. — С. 51-56. — Бібліогр.: 15 назв. — англ. 2227-1481 DOI: 10.17721/2227-1481.5.51-56 https://nasplib.isofts.kiev.ua/handle/123456789/119826 We investigate the non-linear evolution of spherical density and velocity perturbations of dark matter and dark energy in the expanding Universe. For this we have used the conservation and Einstein equations to describe the evolution of gravitationally coupled inhomogeneities of dark matter, dark energy and radiation from the linear stage in the early Universe to the non-linear stage at the current epoch. A simple method of numerical integration of the system of non-linear differential equations for evolution of the central part of halo is proposed. The results are presented for the halo of cluster (k = 2 Mpc⁻¹ ) and supercluster scales (k = 0.2 Mpc⁻¹ ) and show that a quintessential scalar field dark energy with a low value of effective speed of sound cٍ < 0.1 can have a notable impact on the formation of large-scale structures in the expanding Universe. This work was supported by the projects of Ministry of Education and Science of Ukraine (state registration numbers 0115U003279 and 0113U003059). en Головна астрономічна обсерваторія НАН України Advances in Astronomy and Space Physics Dynamics of dark energy in collapsing halo of dark matter Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Dynamics of dark energy in collapsing halo of dark matter |
| spellingShingle |
Dynamics of dark energy in collapsing halo of dark matter Tsizh, M. Novosyadlyj, B. |
| title_short |
Dynamics of dark energy in collapsing halo of dark matter |
| title_full |
Dynamics of dark energy in collapsing halo of dark matter |
| title_fullStr |
Dynamics of dark energy in collapsing halo of dark matter |
| title_full_unstemmed |
Dynamics of dark energy in collapsing halo of dark matter |
| title_sort |
dynamics of dark energy in collapsing halo of dark matter |
| author |
Tsizh, M. Novosyadlyj, B. |
| author_facet |
Tsizh, M. Novosyadlyj, B. |
| publishDate |
2015 |
| language |
English |
| container_title |
Advances in Astronomy and Space Physics |
| publisher |
Головна астрономічна обсерваторія НАН України |
| format |
Article |
| description |
We investigate the non-linear evolution of spherical density and velocity perturbations of dark matter and dark energy in the expanding Universe. For this we have used the conservation and Einstein equations to describe the evolution of gravitationally coupled inhomogeneities of dark matter, dark energy and radiation from the linear stage in the early Universe to the non-linear stage at the current epoch. A simple method of numerical integration of the system of non-linear differential equations for evolution of the central part of halo is proposed. The results are presented for the halo of cluster (k = 2 Mpc⁻¹
) and supercluster scales (k = 0.2 Mpc⁻¹ ) and show that a quintessential scalar field dark energy with a low value of effective speed of sound cٍ < 0.1 can have a notable impact on the formation of large-scale structures in the expanding Universe.
|
| issn |
2227-1481 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119826 |
| citation_txt |
Dynamics of dark energy in collapsing halo of dark matter / M. Tsizh, B. Novosyadlyj // Advances in Astronomy and Space Physics. — 2015. — Т. 5., вип. 1. — С. 51-56. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT tsizhm dynamicsofdarkenergyincollapsinghaloofdarkmatter AT novosyadlyjb dynamicsofdarkenergyincollapsinghaloofdarkmatter |
| first_indexed |
2025-11-24T23:41:18Z |
| last_indexed |
2025-11-24T23:41:18Z |
| _version_ |
1850498116046815232 |
| fulltext |
Dynamics of dark energy in collapsing halo of dark matter
M.Tsizh∗, B.Novosyadlyj
Advances in Astronomy and Space Physics, 5, 51-56 (2015)
© M.Tsizh, B.Novosyadlyj, 2015
Ivan Franko National University of Lviv, Kyryla and Methodia str., 8, Lviv, 79005, Ukraine
We investigate the non-linear evolution of spherical density and velocity perturbations of dark matter and dark
energy in the expanding Universe. For this we have used the conservation and Einstein equations to describe the
evolution of gravitationally coupled inhomogeneities of dark matter, dark energy and radiation from the linear stage
in the early Universe to the non-linear stage at the current epoch. A simple method of numerical integration of
the system of non-linear di�erential equations for evolution of the central part of halo is proposed. The results
are presented for the halo of cluster (k = 2Mpc−1) and supercluster scales (k = 0.2Mpc−1) and show that a
quintessential scalar �eld dark energy with a low value of e�ective speed of sound cs < 0.1 can have a notable
impact on the formation of large-scale structures in the expanding Universe.
Key words: dynamical dark energy, large scale structure of the Universe
introduction
Dark energy is the mysterious dark component
responsible for accelerated expansion of the Uni-
verse. It became an object of numerous studies in
the past two decades. There are many possible ex-
planations [1, 9] of the nature of dark energy. One
of the most promising among them is scalar �eld
dark energy, which can be modelled as a perfect
�uid. In fact, with the knowledge of just a few
parameters (the density ρDE = ρcrΩDE, the equa-
tion of state (EoS) parameter wDE = ρ/p and ef-
fective speed of sound c2s = δp/δρ), this model per-
fectly corresponds with the latest cosmological ob-
servational data [10, 13], while keeping a large num-
ber of possible parameter variations. Therefore, it
would be interesting to study the behaviour of per-
turbations of such dark energy on non-cosmological
scales. This may yield more constrains on parame-
ters of the model.
It was already shown [2, 15] that non-linear per-
turbations of scalar �eld dark energy can in principle
in�uence structure formation even on galaxy scales.
Finally, in the past decade there was a number of
papers (see [3] for review), which documented the
studies of dark energy accretion in compact objects
(black holes mainly) using the hydrodynamical ap-
proach. In our works [8, 14] we have shown that such
stationary accretion does not change the gravitating
mass of the central object, and in principle, it can
notably a�ect the dynamics of bodies in its vicin-
ity, which would be possible to extract from current
observations.
Here we analyse the dynamics of dark energy in
the collapsing halo of dark matter. In the next sec-
tion we present the system of equations, which we use
to solve the problem of evolution of spherical scalar
perturbations in a 3-component expanding Universe
from an early linear stage to a highly non-linear one,
when a dark matter halo forms. In the last section we
present numerical solutions and discuss their main
features.
equations for evolution
of perturbations
We suppose that the Universe is spatially �at and
�lled by matter (m), dark energy (DE) and radia-
tion (r), the metric of the background space-time is
that of Friedmann-Robertson-Walker (FRW). Each
component is described in the perfect �uid approx-
imation by energy density ε, pressure p and four-
velocity ui. The equation of state for each compo-
nent can be presented as p = wε, with wm = 0 for
matter, wr = 1/3 for radiation and wDE < −1/3 for
dark energy. We assume that dark energy is a scalar
�eld with wDE = const. The goal of the paper is
to analyse the evolution of a spherical halo from the
linear stage in the early radiation-dominated epoch,
through the turnaround point to the highly non-
linear stage, infall of matter before virialization, at
dark energy-dominated epoch. The local spherical
perturbation distorts the FRW metric, so that it be-
comes
ds2 = eν(t,r)dt2 − a2(t)eµ(t,r)×
×
[
dr2 + r2
(
dθ2 + sin2 θdϕ2
)]
, (1)
where the metric functions ν(t, r) and µ(t, r) van-
ish into the cosmological background. At the lin-
∗zzviri@gmail.com
51
Advances in Astronomy and Space Physics M.Tsizh, B.Novosyadlyj
ear stage, when ν(t, r) ∼ µ(t, r) � 1 the metric
(1) becomes the metric of conformal-Newtonian (lon-
gitudinal) gauge [4] in spherical coordinates, since
eν(t,r) ≈ 1 + ν, eµ(t,r) ≈ 1 + µ. We rewrite the
energy-momentum tensor of dark energy, T k
i (DE) =
(εDE+pDE)ui (DE)u
k
DE−δki pDE, in terms of proper 3-
velocity of �uid vDE (measured in local frame), which
only has a radial component. The relation between
components of 4-velocity uiDE and 3-velocity vDE of
�uid are as follows
ui (DE) =
{
eν/2√
1− v2DE
,− avDEe
µ/2√
1− v2DE
, 0, 0
}
,
ui
DE =
{
e−ν/2√
1− v2DE
,
a−1vDEe
−µ/2√
1− v2DE
, 0, 0
}
,
where vDE is in the units of speed of light. The non-
zero components of energy-momentum tensor are
T 0
0 =
εDE + v2DEpDE
1− v2DE
,
T 1
0 =
εDE + pDE
1− v2DE
a−1vDEe
(ν−µ)/2,
T 1
1 = −
v2DEεDE + pDE
1− v2DE
, T 2
2 = T 3
3 = −pDE.
We decompose both density and pressure into back-
ground averaged and perturbed parts as εDE =
ε̄DE(1 + δDE), pDE = ε̄DE
[
wDE + c2sδDE − 3ȧ(1 +
wDE)(c
2
s − wDE)
∫
e(µ−ν)/2vDEdr
]
, where the den-
sity perturbation of each component is de�ned as
δ(t, r) ≡
[
ε(t, r) − ε̄(t)
]
/ε̄(t). We study the model
of dark energy, for which both wDE and c2s are con-
stant. The integral term in the pressure decompo-
sition comes from a non-adiabatic part of pressure
perturbation of scalar �eld dark energy (details can
be found in papers [5, 6, 11]). The presence of this
term makes our equation of state non-barotropic, as
we work in a frame di�erent from that of proper dark
energy and hence, the relation pDE = wDEρDE holds
only for averaged parts of this component. Taking
into account this decomposition, and keeping terms
with v0DE, v
1
DE and v2DE only, we obtain the follow-
ing energy-momentum tensor components of dark
energy:
T 0
0 = ε̄DE(1+δDE)+ε̄DE(1+wDE+(1+c2s)δDE)v
2
DE,
T 1
0 = ε̄DE
[
1 + wDE + (1 + c2s)δDE − 3ȧ(1 + wDE)×
×(c2s − wDE)
∫
e(µ−ν)/2vDEdr
]
a−1vDEe
(ν−µ)/2,
T 0
1 = −ε̄DE
[
1+wDE+(1+c2s)δDE−3ȧ(1+wDE)×
×(c2s − wDE)
∫
e(µ−ν)/2vDEdr
]
avDEe
(µ−ν)/2,
T 1
1 = −ε̄DE
[
wDE + c2sδDE − 3ȧ(1 + wDE)×
×(c2s − wDE)
∫
e(µ−ν)/2vDEdr
]
−
− ε̄DE
[
1 + wDE + (1 + c2s)δDE
]
v2DE,
T 2
2 = T 3
3 = −ε̄DE
[
wDE + c2sδDE − 3ȧ(1 + wDE)×
×(c2s − wDE)
∫
e(µ−ν)/2vDEdr
]
. (2)
Hereafter, one can get the corresponding equations
and expressions for dark matter and radiation com-
ponents from equations and expressions for dark en-
ergy just by putting wDE = c2s = 0 for matter and
wDE = c2s = 1/3 for radiation.
To �nd the evolution of density and velocity per-
turbations we use two conservation equations, which
have general covariant form
T k
i;k =
1√
−g
∂(
√
−gT k
i )
∂xk
− 1
2
∂gkk
∂xi
gkkT k
k = 0
(for i = 0 it is continuity equation, for i = 1 it is mo-
tion equation). Substituting our energy-momentum
tensor (2) one obtains:
δ̇DE
[
1 + (1 + c2s)δDEv
2
]
+ 3
ȧ
a
δDE(c
2
s − wDE)+
+
[
1 + wDE + (1 + c2s)δDE
] [3µ̇
2
+ (1− 3wDE+
+ 2µ̇)v2DE + 2v̇DEvDE
]
+ a−1e(ν−µ)/2×
×
[
(1 + c2s)δ
′
DEvDE +
{
1 + wDE + (1 + c2s)δDE
}
×
×
(
vDE(ν
′ + µ′ +
2
r
) + v′DE
)]
+ 3ȧ(1 + wDE)×
× (c2s − wDE)
[{
3ȧ
a
+
3µ̇
2
+ a−1e(ν−µ)/2 ×
×
(
vDE(ν
′ + µ′ +
2
r
) + v′
)}
×
×
∫
e(µ−ν)/2vDEdr − v2DE
]
= 0, (3)
52
Advances in Astronomy and Space Physics M.Tsizh, B.Novosyadlyj
v̇DE + vDE
(
ȧ
a
(1− 3c2s) + 2µ̇
)
+
δ′DEe
(ν−µ)/2
a(1 + wDE)
×
×
(
c2s +
1 + c2s
1 + wDE
v2DE
)
+
e(ν−µ)/2
a
(
1 + δDE
1 + c2s
1 + wDE
)
×
×
[
ν′
2
+ (ν′ + µ′)v2DE + 2vDE
(
v′DE +
vDE
r
)]
+
1 + c2s
1 + wDE
×
×
[
δ̇DEv + v̇DEδ + 2µ̇vDEδDE + (1− 3wDE)
ȧ
a
vDEδDE
]
−
− 3
ȧ
a
(c2s − wDE)
[(
v̇DE + (1− 3wDE)
ȧ
a
vDE + 2µ̇vDE+
+
ν′
2a
e(ν−µ)/2
)∫
e(ν−µ)/2vDEdr+
+vDE
∫
e(ν−µ)/2
(
v̇DE +
ä
ȧ
vDE +
µ̇− ν̇
2
vDE
)
dr
]
= 0.
(4)
The continuity equation for background density,
which we also use, is ˙̄εDE + 3 ȧ
a(1 + wDE)ε̄DE = 0.
To �nd the metric functions we exploit the Ein-
stein equations
Ri
j −
1
2
δijR = κ
(
T i
j (DE) + T i
j (m) + T i
j (r)
)
.
If we construct the equation G1
1−G2
2 = κ
(
T 1
1 − T 2
2
)
then it becomes apparent that at the linear stages
µ = −ν. At the non-linear stage, the right-hand side
of this equation equals zero at the centre of perturba-
tion, which again gives a reduction to only one poten-
tial. In this paper we analyse the dynamics of dark
matter and dark energy in the central part of spher-
ical overdensity only, therefore we accept µ = −ν
approximation, which gives us the possibility to use
only one (00) Einstein equation for determination of
one metric function ν(t, r):
− 3
ȧ
a
ν̇ +
3ν̇2
4
+ 3
ȧ2
a2
(1− eν) +
1
a2
[
ν′′ +
2
r
ν′ +
ν′2
4
]
=
= 3
ȧ2
a2
eν
Ωra
−1δr +ΩDEa
−3wDEδDE +Ωmδm
Ωm +Ωra−1 +ΩDEa−3wDE
. (5)
Here we used the notations:
ΩN ≡ ε̄0N/
(
ε̄0m + ε̄0DE + ε̄0r
)
,
where N denotes the type of �uid and �0� marks the
value at current epoch. In the �at three component
universe Ωm+Ωr+ΩDE = 1. Eqs. (3)�(5) have non-
relativistic limit (Appendix), which in case of dark
matter (c2s = wDE = 0) coincide with well known
classical hydrodynamic equations and Poisson equa-
tion accordingly.
We are interested in the cluster and supercluster
scales of perturbations, for which ν � 1, vm � 1
in the bulk of object, while δm � 1 in their halos.
So, the equations can be essentially reduced by ne-
glecting the terms like, ν2, νvm, νvDE and higher
order terms. The terms with v∇v must be kept,
since they are important during the highly non-linear
stage. The �nal reduced form of these equations,
which we use in the code, can be found in our up-
coming paper [11].
Therefore, we have seven 1st-order partial di�er-
ential equations for seven unknown functions
δm(t, r), vm(t, r), δDE(t, r), vDE(t, r),
δr(t, r), vr(t, r), ν(t, r), (6)
which can be solved numerically for given initial con-
ditions.
At the early epoch the amplitudes of cosmologi-
cal perturbations of space-time metric, densities and
velocities are low and the Eqs. (3)�(5) can be lin-
earized for all components. Moreover, we can present
each function of (t, r) as a product of its amplitude,
which depends on t only, and some function of radial
coordinate r, which describes the initial pro�le of
spherical perturbation, which can be expanded into
series of some orthogonal functions, e. g. spherical
ones in our case. In particular, we can present the
perturbations of the metric, density and velocity of
N-component as follows
ν(t, r) = ν̃(t)
sin kr
kr
, δN (t, r) = δ̃N (t)
sin kr
kr
,
vN (t, r) = ṽN (t)
(
sin kr
kr
)′
= ṽN (t)k
(
cos kr
kr
− sin kr
k2r2
)
.
In the analysis of the evolution of the central part of
a spherical halo we can decompose r-function in the
Taylor series and only keep the leading terms:
fk(r) ≈ 1, f ′
k(r) ≈ −1
3
k2r,∫
f ′(r)dr ≈ 1, f ′′
k (r) +
2fk(r)
r
≈ −k2,
where fk(r) = sin kr/kr. It gives the possibility to
reduce the system of seven partial di�erential equa-
tions for unknown functions (6) to the system of
seven ordinary di�erential equations for their am-
plitudes
δ̃m(t), ṽm(t), δ̃DE(t), ṽDE(t),
δ̃r(t), ṽr(t), ν̃(t),
which is presented in [11].
We set the adiabatic initial conditions in the fol-
lowing way. To �nd the relations between amplitudes
at some ainit � 1 when the scales of gravitation-
ally bound systems were essentially larger than the
horizon scale, ainitk
−1 � t we used the lineariza-
tion of system (3)�(5), which has an exact analyt-
ical solution for a one-component Universe. Evi-
dently for cluster and supercluster scales the epoch
53
Advances in Astronomy and Space Physics M.Tsizh, B.Novosyadlyj
was the early radiation-dominated one when ε̄r �
ε̄m � ε̄DE, thus, the matter and dark energy can
be treated as test components. The amplitude of
the metric function ν̃ is de�ned by density pertur-
bations of the relativistic component. The non-
singular solution of the corresponding equations has
asymptotic values at ainitν̃
init = −C, δ̃initr = C,
ṽinitr = C/[4ainitH(ainit)], where C is some constant
(see for details [7]). The solutions of the equation
for matter and dark energy as test components give
the asymptotic values for superhorizon perturbations
at ainit: δ̃initm = 3C/4, ṽinitm = C/ [4ainitH(ainit)],
δ̃initDE = 3(1+w)C/4, ṽinitDE = C/[4ainitH(ainit)]. These
relations contain only a single constant C, the value
of which speci�es the initial amplitudes of perturba-
tions in all components. Below we put C = 2.6 ·10−5
for cluster scales k = 2Mpc−1 and C = 6.5 · 10−5
for supercluster scales k = 0.2Mpc−1 at time when
a = 10−10.
results and discussion
In this section we present the results of numer-
ical integrations of the described system of equa-
tions. To achieve this we have designed a Fortran-
95 tool based on open dverk.f package for ODEs,
in which time-dependant functions are replaced by
scale-factor dependant functions, since a(t) is well-
de�ned for given cosmology. The cosmological pa-
rameters are taken as follows Ωr = 4.2 · 10−5, ΩDE =
0.7, Ωm = 1− Ωr − ΩDE, H0 = 70 km/s/Mpc.
The results of numerical integration of the system
of equations for time evolution of amplitudes of dark
energy and dark matter density, velocity perturba-
tions, and potential for di�erent scales and param-
eters of dark energy, are presented in Figs. 1a)�f).
In these �gures, the solid lines correspond to dark
matter perturbations and dashed lines correspond
to dark energy perturbations. The dotted lines at
all panels show the predictions of linear theory. For
such perturbations non-linearity becomes noticeable
already at a ∼ 0.1.
It should be noted that in all cases and for all
sets of parameters analysed here, dark energy is sub-
dominant: the density of its perturbations is several
orders of magnitude lower then that of dark matter.
The reason for this is the large absolute value of pres-
sure which keeps it from collapsing. The high e�ec-
tive speed of sound (including speed of pressure per-
turbation distribution) forces dark energy to oscillate
after the perturbation enters the horizon (Figs. 1a)
and d)). Both velocity and density perturbation of
dark energy are lower throughout the entire course
of evolution, and the di�erence is greater for smaller
scales, as can be seen comparing Fig. 1a), b) and
Fig. 1d), e).
Another interesting observation is how behaviour
of dark energy depends on its parameters. From the
Figs. 1a) and c) it is apparent that the EoS parame-
ter wDE slightly changes the character of evolution of
inhomogeneities, while the initial amplitude of den-
sity perturbation is ∼ (1 +wDE). At the same time,
the e�ective speed of sound cs actually de�nes how
fast the perturbation will grow. Comparing Figs. 1a)
and b), or d), e) and f) we see that the closer c2s
is to zero, the higher amplitude of perturbations is
achieved at the �nal stages of evolution. With lower-
ing of e�ective speed of sound, the pressure gradient
of dark energy lowers the counteraction to gravity.
The oscillations become smaller and even disappear
at very low values of cs, while the perturbations grow
at a higher rate, and the time evolution of the veloc-
ity and density become more and more similar to
that of dark matter. This fact is in good accordance
with the nature of dependence on e�ective speed of
sound in the process of accretion of dark energy on a
compact object, which was studied recently [8]. As
can be seen from Fig. 1f), when cs = 0 the dark en-
ergy and dark matter always have the same velocity,
the gap between their densities is constant, and the
velocity factor in the continuity equation for dark
energy is ∼ (1 + wDE) (3).
These two properties, together with the observa-
tion that dark energy background density is lower
than that of components in the past, imply that
dark energy could signi�cantly in�uence the process
of dark matter halo collapse, but only under the con-
dition of a very low value of e�ective speed of sound.
The current cosmological observational data do not
exclude such models of dark energy ([13]).
It should also be noted that predictions of den-
sity and velocity of the linear (doted lines) and non-
linear theories at a smaller scale become distinguish-
able approximately at the same time (scale factor)
as for a larger scale. However, for predictions of po-
tential, both linear and non-linear theories diverge
only at late times for a smaller scale. The reason
for this is, di�erent initial perturbations were taken
for these scales, in order to have approximately the
same density perturbation at a = 1. For the same
initial amplitudes the perturbations of smaller scales
would collapse �rst.
acknowledgement
This work was supported by the projects of Min-
istry of Education and Science of Ukraine (state reg-
istration numbers 0115U003279 and 0113U003059).
references
[1] Amendola L., Appleby S., BaconD. et al. 2013, Living
Rev. Relativity, 16, 6
[2] ArbeyA., Lesgourgues J. & Salati P. 2001, Phys. Rev. D,
64, 123528
[3] Babichev E.O., Dokuchaev V. I. & Eroshenko Yu.N.
2013, Physics Uspekhi, 56, 1155
[4] Bardeen J.M. 1980, Phys. Rev. D, 22, 1882
54
Advances in Astronomy and Space Physics M.Tsizh, B.Novosyadlyj
[5] HuW. 1998, ApJ, 506, 485
[6] HuW. 2004, [arXiv:astro-ph/0402060].
[7] Novosyadlyj B. 2007, J. Phys. Stud., 11, 226
[8] Novosyadlyj B., KulinichYu. & TsizhM. 2014, Phys.
Rev. D, 90, 063004
[9] Novosyadlyj B., PelykhV., ShtanovYu. & ZhukA., 2013,
`Dark Energy: Observational Evidence and Theoretical
Models', Akademperiodyka, Kyiv
[10] Novosyadlyj B., SergijenkoO., DurrerR. & PelykhV.
2014, JCAP, 05, 030
[11] Novosyadlyj B., TsizhM. & KulinichYu. Submited to
General Relativity and Gravitation
[12] Peebles P. J. E. 1980, `The large-scale structure of the
Universe', Princeton University Press, Princeton
[13] SergijenkoO. & Novosyadlyj B. 2015, Phys. Rev. D, 91,
083007
[14] TsizhM. & Novosyadlyj B. 2014, in `WDS'14 Proceedings
of Contributed Papers, Physics', 21
[15] WetterichC. 2001, Phys. Lett. B., 522, 5
appendix.
newtonian approximation
of equations for evolution
of dark energy perturbations
Conservation equation T k
0;k transforms to classic
continuity equation for dark energy:
δ̇DE + 3
ȧ
a
(c2s − wDE)δDE + a−1
{
(1 + c2s)δ
′
DEv+
+
(
1 + w + (1 + c2s)δ
)(
v′ +
2v
r
)}
= 0.
Combination of conservation equations T k
1;k0 and T k
0;k
transform to classic Euler equation for dark energy:
v̇ +
v
a
(
v′ +
2v
r
)
+
+ v
ȧ
a
(1− 3c2s)(1 + wDE) + (1 + c2s)(1− 3w)δDE
1 + wDE + (1 + c2s)δDE
+
+
ν ′
2a
+
c2sδ
′
DE
a(1 + wDE + (1 + c2s)δDE)
= 0.
If we consider the last two equations for dark matter
(c2s = wDE = 0), we will see that they coincide with
well-known hydrodynamic equation of collapse [12]
(equation 9.17)
The non-relativistic approximation of Einstein
equation (5) gives the Poisson equation for metric
function ν in the coordinates of FRW frame
∆ν = 8πGa2ρ0cr
[
Ωma−3δm+
+Ωde(1 + 3c2s)a
−3(1+wde)δde + 2Ωra
−4δr
]
.
The Newtonian gravitational potential Φ = ν/2.
55
Advances in Astronomy and Space Physics M.Tsizh, B.Novosyadlyj
Fig. 1: Evolution of the amplitudes of density perturbations of matter δ̃m and dark energy δ̃DE (top panels), velocity
of matter Vm and dark energy VDE in the units of Hubble one (middle panels) and gravitational potential ν̃ (bottom
panels) at the central part of spherical halo of cluster scale which is collapsing now. In the top and middle panels solid
lines corresponds to matter, dashed lines to dark energy and dotted lines at all panels show the prediction of linear
theory.
56
|