Evolution of density and velocity profiles of matter in large voids
We analyse the evolution of cosmological perturbations which leads to the formation of large voids in the distribution of galaxies. We assume that perturbations are spherical and all components of the Universe - radiation, matter and dark energy - are continuous media with ideal fluid energy-momentu...
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| Цитувати: | Evolution of density and velocity profiles of matter in large voids / M. Tsizh, B. Novosyadlyj // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 28-33. — Бібліогр.: 8 назв. — англ. |
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Tsizh, M. Novosyadlyj, B. 2017-06-10T13:14:24Z 2017-06-10T13:14:24Z 2016 Evolution of density and velocity profiles of matter in large voids / M. Tsizh, B. Novosyadlyj // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 28-33. — Бібліогр.: 8 назв. — англ. 2227-1481 DOI:10.17721/2227-1481.6.28-33 https://nasplib.isofts.kiev.ua/handle/123456789/119947 We analyse the evolution of cosmological perturbations which leads to the formation of large voids in the distribution of galaxies. We assume that perturbations are spherical and all components of the Universe - radiation, matter and dark energy - are continuous media with ideal fluid energy-momentum tensors, which interact only gravitationally. Equations of the evolution of perturbations in the comoving to cosmological background reference frame for every component are obtained from equations of conservation and Einstein's ones and are integrated by modified Euler method. Initial conditions are set at the early stage of evolution in the radiation-dominated epoch, when the scale of perturbation is mush larger than the particle horizon. Results show how the profiles of density and velocity of matter in spherical voids with different overdensity shells are formed en Головна астрономічна обсерваторія НАН України Advances in Astronomy and Space Physics Evolution of density and velocity profiles of matter in large voids Article published earlier |
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Evolution of density and velocity profiles of matter in large voids |
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Evolution of density and velocity profiles of matter in large voids Tsizh, M. Novosyadlyj, B. |
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Evolution of density and velocity profiles of matter in large voids |
| title_full |
Evolution of density and velocity profiles of matter in large voids |
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Evolution of density and velocity profiles of matter in large voids |
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Evolution of density and velocity profiles of matter in large voids |
| title_sort |
evolution of density and velocity profiles of matter in large voids |
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Tsizh, M. Novosyadlyj, B. |
| author_facet |
Tsizh, M. Novosyadlyj, B. |
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2016 |
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English |
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Advances in Astronomy and Space Physics |
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Головна астрономічна обсерваторія НАН України |
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Article |
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We analyse the evolution of cosmological perturbations which leads to the formation of large voids in the distribution of galaxies. We assume that perturbations are spherical and all components of the Universe - radiation, matter and dark energy - are continuous media with ideal fluid energy-momentum tensors, which interact only gravitationally. Equations of the evolution of perturbations in the comoving to cosmological background reference
frame for every component are obtained from equations of conservation and Einstein's ones and are integrated by modified Euler method. Initial conditions are set at the early stage of evolution in the radiation-dominated epoch, when the scale of perturbation is mush larger than the particle horizon. Results show how the profiles of density and velocity of matter in spherical voids with different overdensity shells are formed
|
| issn |
2227-1481 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119947 |
| citation_txt |
Evolution of density and velocity profiles of matter in large voids / M. Tsizh, B. Novosyadlyj // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 28-33. — Бібліогр.: 8 назв. — англ. |
| work_keys_str_mv |
AT tsizhm evolutionofdensityandvelocityprofilesofmatterinlargevoids AT novosyadlyjb evolutionofdensityandvelocityprofilesofmatterinlargevoids |
| first_indexed |
2025-11-26T08:17:46Z |
| last_indexed |
2025-11-26T08:17:46Z |
| _version_ |
1850618237774987264 |
| fulltext |
Evolution of density and velo
ity pro�les of matter
in large voids
M.Tsizh
∗
, B.Novosyadlyj
Advan
es in Astronomy and Spa
e Physi
s, 6, 28-33 (2016) doi: 10.17721/2227-1481.6.28-33
© M.Tsizh, B.Novosyadlyj, 2016
Ivan Franko National University of Lviv, Kyryla i Methodia str., 8, Lviv, 79005, Ukraine
We analyse the evolution of
osmologi
al perturbations whi
h leads to the formation of large voids in the dis-
tribution of galaxies. We assume that perturbations are spheri
al and all
omponents of the Universe � radiation,
matter and dark energy � are
ontinuous media with ideal �uid energy-momentum tensors, whi
h intera
t only
gravitationally. Equations of the evolution of perturbations in the
omoving to
osmologi
al ba
kground referen
e
frame for every
omponent are obtained from equations of
onservation and Einstein's ones and are integrated by
modi�ed Euler method. Initial
onditions are set at the early stage of evolution in the radiation-dominated epo
h,
when the s
ale of perturbation is mush larger than the parti
le horizon. Results show how the pro�les of density
and velo
ity of matter in spheri
al voids with di�erent overdensity shells are formed.
Key words:
osmology: dark energy, large-s
ale stru
ture of Universe
introdu
tion
Large voids in the spatial distribution of galax-
ies are elements of the large s
ale stru
ture, study
of whi
h
an give important information about the
hidden
omponents of the Universe � dark matter
and dark energy. Usually it is assumed, that dark
energy is unperturbed in the voids or, at least, im-
pa
t of its density perturbations on the pe
uliar mo-
tion and spatial distribution of galaxies is negligibly
small. In this paper we investigate the evolution of
perturbations of density and velo
ity of matter to-
gether with dark energy ones from the early stage,
when the s
ale of initial perturbation is mu
h larger
than parti
le horizon, up to
urrent epo
h. We anal-
yse the in�uen
e of dynami
al dark energy on su
h
evolution and its dependen
e on initial
onditions.
We point attention to the evolution of density and
velo
ity pro�les of matter during void formation. For
this we have developed the program for integrating
the system of equation, obtained for des
ription of
evolution of spheri
al perturbation in 3-
omponent
medium � radiation, matter and dark energy, �
from the equations of relativisti
hydrodynami
s and
gravitation [3℄. The
omponent �matter�
onsists of
dark matter (25% of total density) and typi
al bary-
oni
matter (5%) the dynami
s of whi
h is well de-
s
ribed by the dust-like medium approa
h at large
s
ales.
model of spheri
al void
and initial
onditions
We assume that voids in spatial distribution of
galaxies are formed as the result of the evolution of
osmologi
al density perturbations with a negative
initial amplitude. It is believed that su
h pertur-
bations are the result of quantum �u
tuations of
spa
e-time metri
in the in�ationary epo
h. They
are randomly distributed in amplitude with normal
distribution and are symmetri
al by sign of density
perturbation from the average in di�erent regions
of spa
e. We
onsider only s
alar mode of pertur-
bations, in whi
h perturbations of density δN (t, r)
and velo
ity vN (t, r) in every
omponent N are
or-
related be
ause of survival of the growing solution
only at the stage when the s
ale of perturbation
was larger than parti
le horizon. Positive perturba-
tions lead to the formation of galaxies and galaxy
lusters and negative ones � to the formation of
voids. Formation of stru
tures with positive per-
turbation is well des
ribed by Press-S
he
hter for-
malism, theory of Gaussian peaks and halo theory
of stru
ture formation and their modern modi�
a-
tions based on the numeri
al N-body simulations.
Although the evolution of voids in the distribu-
tion of galaxies is mu
h simpler than evolution of
galaxy
lusters, sin
e it is des
ribed by the quasi-
linear theory, there is no
omplete theory of voids
formation. Here we analyse the development of
negative
osmologi
al density perturbations, whi
h
form the voids. The mathemati
al base of their de-
s
ription is the system of 7 di�erential equations
in partial derivatives for 7 unknown fun
tions of 2
independent variables δDE(a, r), δM (a, r), δR(a, r),
vDE(a, r), vM (a, r), vR(a, r), ν(a, r), whi
h were ob-
tained in [3℄ (equations (17)-(22)). Here Ω-s denote
the mean densities of the
omponents in the unit of
the
riti
al one at the
urrent epo
h, w ≡ pDE/ρDE
∗
tsizh�astro.franko.lviv.ua
28
Advan
es in Astronomy and Spa
e Physi
s M. Tsizh, B. Novosyadlyj
is the equation of state parameter of dark energy,
cs is the e�e
tive speed of sound of dark energy
in its proper frame, H(a) ≡ d ln a/dt is the Hub-
ble parameter, whi
h de�nes the rate of the expan-
sion of the Universe and is known fun
tion of time
for given
osmology and the model of dark energy
(
H(a) = H0
√
ΩRa−4 +ΩMa−3 +ΩDEa−3(1+w)
)
and H0 is its today value (Hubble
onstant). The
independent variables are s
ale fa
tor a and radial
omoving
oordinate r, whi
h de�ne the interval in
Friedman-Robertson-Walker 4-spa
e:
ds2 = eν(t,r)dt2 − a2(t)e−ν(t,r)[dr2 + r2(dθ2 +sin2 θdϕ2)].
It is assumed that geometry of 3-spa
e of the
Universe (unperturbed
osmologi
al ba
kground) is
Eu
lidean. The metri
fun
tion ν(t, r) at the late
stages, when the s
ale of perturbation is mu
h
smaller than the parti
le horizon, is the doubled
gravitational potential in the Newtonian approxima-
tion of Eq. (17) in the paper [3℄. The density and
3-velo
ity perturbations δN and vN are de�ned in
oordinates, whi
h are
omoving to the unperturbed
osmologi
al ba
kground (see paragraph 2.2 in [3℄).
Thus, the velo
ity perturbation
oin
ide with de�ni-
tion of pe
uliar velo
ity of galaxies (see, e. g., [4℄).
To solve the system of equations (17)�(22)
from [3℄ the initial
onditions must be set. Let
us relate the initial amplitude of given perturba-
tion with mean-square one given by power spe
trum
of
osmologi
al perturbations. For this we de�ne
the initial
onditions in the early Universe, when
ρR ≫ ρM ≫ ρDE, and physi
al size of the pertur-
bation aλ ≫ ct. In that time the perturbations are
linear (δ, v, ν ≪ 1), so without loss of generality
the solution
an be presented in the form of sepa-
rated variables:
ν(a, r) = ν̃(a)f(r), δN (a, r) = δ̃N (a)f(r),
vN (a, r) = ṽN (a)f ′(r),
where f(0) = 1 and f ′(r) ∝ r near the
entre
r = 0. Ordinary di�erential equations for ampli-
tudes ν̃(a), δ̃N (a), ṽN (a) are obtained from general
system of equations (17)�(22) from [3℄ by their ex-
pansion in Taylor series near the
entre. The analyt-
i
al solutions of equations for the amplitudes for the
radiation-dominated epo
h (matter and dark energy
an be treated as test
omponents) in the �superhori-
zon� asymptoti
give the simple relation for them:
δ̃initR =
4
3
δ̃initM =
4
3(1 + w)
δ̃initDE = −ν̃ init = C,
ṽinitR = ṽinitM = ṽinitDE =
C
4a
init
H(a
init
)
, (1)
where C is an integration
onstant, whi
h is de-
�ned by initial
onditions. We set the value of C
in the units of mean-square amplitude of perturba-
tions, whi
h is implied from modern observations.
The Plan
k + HST + WiggleZ + SNLS3 data (see
[6℄ and referen
es therein) tell that amplitude As and
spe
tral index ns of power spe
trum of initial per-
turbations of
urvature PR(k) = As(k/0.05)
ns−1
are
the following [6℄: As = 2.224 · 10−9, ns = 0.963.
Sin
e for perturbations with ak−1 ≫ ct the power
spe
trum perturbations of
urvature PR ≡< ν · ν >
is
onstant in time in the matter- and radiation-
dominated epo
hs, in the range of s
ales 0.01 ≤
k ≤ 0.1 the initial amplitude whi
h
orrespondent
to mean-square one is: σk ≡
√
As ≈ 4.7 ·10−5
. Here-
after we put in our
omputations C = −1·10−4 ≈ 2σ
at a
init
= 10−6
.
numeri
al integration
For numeri
al integration of the system of equa-
tions (17)�(22) from [3℄ with initial
onditions (1)
we have
reated a
omputer
ode npdes.f, whi
h
implements the modi�ed Euler method taking into
a
ount the derivatives from the forth
oming step
and improving the results by iterations. This s
heme
of integration is the most resistant to the numeri
al
spurious os
illations, is the fast and pre
ise enough.
For example, the Hamming method of predi
tion and
orre
tion of 4-order of pre
ision with 5 iterations at
ea
h step need 3 times more pro
essor time for the
same pre
ision of �nal result. The step of integra-
tion was posed as variable: da = a/Na, where num-
ber Na was pi
ked up so that the numeri
al pre
ision
of the result of integration at a = 1 was not worse
than 0.1%. In all
al
ulations presented here we took
Na = 3 · 106.
The numeri
al derivatives with respe
t to r in
the grid with
onstant step dr = RM/NR, where
RM is radius of spatial region of integration, were
evaluated with help of 3-rd order polynomial by
method of Savitzky-Golay
onvolution [5℄: y′i =
[3(yi+1−yi−1)/4−(yi+2−yi−2)/12]/dr. The method
was tested by
omparing the derivatives of analyti
al
fun
tions of the initial pro�les of density and velo
ity
perturbations. The value of step dr was estimated so
that the di�eren
e between numeri
al and analyti
al
derivatives do not ex
eed ∼ 10−5
of their values.
To take into a
ount the Silk damping e�e
t for
radiation we have added the terms δRkD/H/a2 and
vRkD/H/a2 into equations of evolution of δR and
vR, respe
tively, where the s
ale of damping kD was
omputed by formula (10) from [1℄.
If the values of e�e
tive speed of sound in dark
energy are cs > 0.01c, then the spurious os
illations
with growing amplitude appear in this
omponent.
Their
ause
onsist in no perfe
t s
heme of integra-
tion by time, the numeri
derivatives on spatial
oor-
dinates and a
umulation of numeri
al errors. To re-
move them we used the Savitzky-Golay
onvolution
29
Advan
es in Astronomy and Spa
e Physi
s M. Tsizh, B. Novosyadlyj
�lter [5℄ with parameters nl = 12, nR = 12, m = 6,
by whi
h the spa
e-dependen
es of derivatives δ̇DE
and v̇DE were smoothing at ea
h step of integration
by a. Su
h smoothing pra
ti
ally does not in�uen
e
on the �nal result of integration, whi
h is
on�rmed
by
omparison of the results with smoothing and
without it for
ase of the dark energy model with
cs = 0, for whi
h spurious os
illations do not ap-
pear. The maximum di�eren
e is less than 4% for
density perturbation and 1% for velo
ity perturba-
tion of dark energy in the region of maximum ampli-
tude of velo
ity perturbation.
The input parameters of the program are: the
Hubble
onstant H0, the density parameters of all
omponents ΩR, ΩDE, ΩM = 1−ΩDE−ΩR, the equa-
tion of state parameter of dark energy w, the speed
of sound of dark energy cs, the initial amplitude of
perturbation C, the parameters of pro�le f(r) of ini-
tial perturbation, the parameter of step Na in a, the
size of integration region RM , and number of steps
of the spatial grid NR.
The
omputer
ode npdes.f has been tested
by
omparison of the results of the integration by
ode with 1) known analyti
al solutions for density
and velo
ity perturbations in
onformal-Newtonian
frame for radiation- and matter-dominated Uni-
verses [8℄, 2) results of integration of linear perturba-
tion by CAMB
ode
1
[2℄, and 3) results of integration
by dedmhalo.f
ode [3℄, developed on the basis of
dverk.f
2
for perturbation in the
entral region of the
spheri
al perturbation. In all
ases deviations did
not ex
eed a few tenths of a per
ent, whi
h means,
that pre
ision of the integration is better then 1%,
and hen
e is high enough for our studies.
formation of voids
in the
osmologi
al models
with dark energy
From our previous studies and studies of other au-
thors we know that the values of density parameter
and the equation of state parameter of dark energy
are well
onstrained by
urrent observational data,
while the value of e�e
tive speed of sound of dark en-
ergy is not
onstrained (see, e. g., [6℄ and referen
es
therein). That is why in this work we analyse the
formation of voids in the
osmologi
al models with
dark energy with ΩDE = 0.7, w = −0.9 and di�erent
values of cs ∈ [0, 1]. Other
osmologi
al parameters
in
omputations are �xed too: ΩR = 4.17 · 10−5
,
ΩM = 0.3− ΩR, H0 = 70 km/s·Mp
.
In this work we study the formation of the spher-
i
al voids with initial pro�le f(r) = (1 − αr2)e−βr2
,
where α gives the size of the void rv = 1/
√
α and
β de�nes the initial amplitude of shell overdensity
around the void: δe = −αβ−1Ce−1−β/α
. For
om-
parison of the results of this paper with the results
of a
ompanying one [7℄, let us set α = (k/π)2 and
β = 3α/4. This is a proto-void, whi
h is surrounded
with overdensity shell with δe ≈ δ(r = 0)/8. For
omparison we will also analyse the evolution of the
void with shells with smaller amplitudes of overden-
sity in 2 and 4 times.
In Figure 1 we show the formation of the spheri-
al void with rv = 31.4Mp
(k = 0.1Mp
−1
) in the
matter and dark energy with c2s = 0: δM,DE(ai, r)
and vM,DE(ai, r) for ai = a
init
, . . . , a30 = 1. Thi
k
solid lines denote the initial pro�les of density and
velo
ity perturbations of both
omponents, dashed
lines denote the �nal ones. The �gure on the right de-
pi
ts the evolution of absolute values of amplitudes of
perturbations in the
entral point of spheri
al void.
Velo
ity perturbation (
entral panel) are given for
the �rst maximum (thi
k lines) and �rst minimum
(thin lines). Dotted line denotes the radiation
om-
ponent. One
an see, that in this dark energy model
the perturbations of matter and dark energy grow
monotoni
ally after entering the horizon: the thi
k
solid lines are internal, the thi
k dashed lines are ex-
ternal. We also note, that the amplitude of the den-
sity perturbation of dark energy is approximately 40
times smaller than the matter one. The values of
velo
ity perturbations of matter and dark energy in
this model of dark energy are the same throughout
the evolution of the void. They in
rease monotoni-
ally from a
init
to a ≈ 0.56. It is easy to see that the
latter value
orresponds to the moment of
hange
from de
elerated expansion of the Universe to the
a
elerated one. The evolution of the absolute val-
ues of density and velo
ity perturbations of matter
and dark energy in the overdensity shell is similar to
those in the
entre.
Similar results of modelling of the void formation
in the matter and dark energy with c2s = 0.1 are
shown in Figure 2. �The pi
ture� of the evolution
of the matter density and velo
ity perturbations has
not
hanged, while for dark energy it has
hanged
drasti
ally. The �nal pro�les of dark energy pertur-
bations are lying on the zero line now. The right
�gure explains su
h behaviour of dark energy during
the void formation: the velo
ity perturbation after
the entering into horizon de
rease qui
kly, and den-
sity perturbation slightly
hanges during all stages
and in the
urrent epo
h does not di�er pra
ti
ally
from the ba
kground value: δDE(1, 0) ≈ −2 · 10−5
.
The matter density perturbation in the
entral part
of this void at the
urrent epo
h is δM (1, 0) ≈ −0.7.
We see also that the evolution of the absolute values
of density and velo
ity perturbations of dark energy
in the overdensity shell slightly di�er from the evo-
lution of ones in the
entre of the void.
1
http://
amb.info
2
http://www.
s.toronto.edu/NA/dverk.f.gz
30
Advan
es in Astronomy and Spa
e Physi
s M. Tsizh, B. Novosyadlyj
The perturbation of dark energy with larger val-
ues of e�e
tive speed of sound after entering the par-
ti
le horizon is smoothed out even faster. Therefore,
the ratio of densities of dark energy and matter in
the
entre of the void is
ρDE(1, 0)
ρM (1, 0)
=
1 + δDE(1, 0)
1 + δM (1, 0)
ΩDE
ΩM
,
and in the
ase of evolution with
onsidered initial
ondition this ratio is 3 time larger than on
osmo-
logi
al ba
kground. This points to the importan
e
of studying of the voids for establishing the nature
of dark energy.
Study of the evolution of spatial pro�les of matter
density and velo
ity perturbations is important for
understanding of the formation of voids. They
an
be obtained by normalization of every
urve in the
left
olumns in Fig. 1�2 by its amplitude. The re-
sult is given in Figure 3 for initial pro�les with three
di�erent values of parameter β.
They show, that in our model the perturbation
with initial density pro�le δ
init
(r) = −1 · 10−4[1 −
(r/rv)
2]e−βr2
with rv = 31.4Mp
and β = 3r−2
v /4
(left
olumn) leads to the formation of void with ra-
dius in
omoving
oordinates ≈ 38Mp
with ampli-
tude of density perturbation in the
entre δM (1, 0) ≈
−0.68 and the overdensity shell around it (δM > 0)
with thi
kness ≈ 36Mp
and amplitude of density
perturbation δe ≈ 0.33. In the
ase of β = 3r−2
v /2
(
entral
olumn) the radius of the
entral void is
≈ 31Mp
, the amplitude of density perturbation in
the
entre is δM (1, 0) ≈ −0.69, the shell of over-
density has thi
kness ≈ 30Mp
and the amplitude
of density perturbation δe ≈ 0.16. In the
ase of
β = 3r−2
v (right
olumn) the void has the following
parameters: the radius ≈ 25Mp
, the amplitude of
density perturbation in the
entre δM (1, 0) ≈ −0.73,
the maximum of overdensity in the shell δe ≈ 0.09
is at distan
e ≈ 31Mp
from the
entre, external
bound of shell, where sign of perturbation
hanges
from �+� to �−� is absent. Important is the de-
penden
e of pro�les of pe
uliar velo
ity of matter
in the void and around it on the model parameters
and initial perturbation. From Figs. 1�3 one
an see,
that the �rst positive peak of pe
uliar velo
ity (from
the
entre) is at the edge of the void and the se
-
ond negative (velo
ity towards the
entre) is at the
edge of the overdensity shell. The values of velo
i-
ties in the
ase of β = 3r−2
v /4 are vm−v(30Mp
) ≈
200 km/s, vm−e(65Mp
) ≈ −70 km/s, in the
ase
of β = 3r−2
v /2: vm−v(25Mp
) ≈ 170 km/s,
vm−e(60Mp
) ≈ −66 km/s, and in the
ase of β =
3r−2
v : vm−v(20Mp
) ≈ 150 km/s and the se
ond neg-
ative peak is absent.
Note, that �nal value of the amplitude of the per-
turbation in the shell is the greater, the greater is its
initial value (smaller value of β), for the same value
of initial amplitude in the
entre. One
an see also,
that overdensity shell appears in the pro
ess of evo-
lution of void even if its amplitude was very small
in the initial pro�le (�gure on the right), or absen
e
at all (α = 0, Gaussian initial pro�le). The evo-
lution of matter density and velo
ity pro�les points
that for interpretation of the observational data on
the distribution of void galaxies in the phase spa
e
the non-linear theory should be used (see also table
in [7℄).
on
lusion
The large voids in the spatial distribution of
galaxies are formed from the negative
osmologi
al
density perturbations of matter. The amplitude of
the density perturbation in the
entral part of the
void at the
urrent epo
h is de�ned by the depth of
dip of Gaussian �eld of the initial matter density per-
turbations, the parameters of the
osmologi
al model
and parameters of the initial pro�le. For example, in
the
osmologi
al model with quintessential dark en-
ergy the initial negative density perturbation with
pro�les similar to the Gaussian ones lead to the for-
mation of the voids with the overdensity shells. In
su
h voids with rv ≈ 30 Mp
and δM(1, 0) ≈ −0.7
the maximal values of the pe
uliar velo
ity of galax-
ies are ∼ 150 − 200 km/s (movement from the
en-
tre in the
omoving
oordinates) and are rea
hed
near the boundary. In the shells su
h velo
ity is
dire
ted to the
entre, however its value does not
ex
eed ∼ 70 km/s. With in
reasing the parameter
of initial pro�le β for the same rv the amplitudes of
the density and velo
ity perturbations in the shells
de
rease.
The density and velo
ity perturbations of the
dark energy evolve similarly to the perturbations of
matter at the stage when their s
ales are mu
h larger
than the parti
le horizon. After they enter the par-
ti
le horizon their evolution depends on the value
of the e�e
tive speed of sound cs. If cs = 0, then
similarity is
onserved with the di�eren
e that the
amplitude of density perturbation of dark energy is
smaller in fa
tor 1 + w. At the later epo
h, when
the dark energy density dominates, this di�eren
e
in
reased yet in ≈ 4 − 5 times more. If 0 < cs ≤ 1,
then the amplitude of velo
ity perturbation of dark
energy after entering the horizon de
reases rapidly,
the amplitude of the density perturbation does not
in
rease or even de
reases too. Therefore, in the
voids the density of quintessential dark energy is ap-
proximately the same as in
osmologi
al ba
kground.
The ratio of the densities of dark energy and mat-
ter is in 1/(1 + δM) larger than in the
osmologi
al
ba
kground. The more hollow void is the larger this
ratio is. That is why the large voids are important
elements of large-s
ale stru
ture of the Universe for
testing models of dark energy and gravity modi�
a-
tions.
31
Advan
es in Astronomy and Spa
e Physi
s M. Tsizh, B. Novosyadlyj
Fig. 1: Void formation in dark matter (left
olumn) and dark energy with c2s = 0 (
entral
olumn). On the right �
evolution of absolute values of amplitudes of density (top panel) and velo
ity (bottom panel) perturbations; solid lines
� dark matter, dashed lines - dark energy, point lines � radiation; thi
k solid and dashed lines � for
entral point,
thin ones � for overdensity shell.
referen
es
[1℄ HuW. & Sugiyama N. 1995, ApJ, 444, 489
[2℄ Lewis A., Challinor A. & LasenbyA. 2000, ApJ, 538, 473
[3℄ NovosyadlyjB., TsizhM. & Kulini
hY. 2016, Gen. Relat.
Grav., 48, 30
[4℄ Peebles P. J. E. 1980, `The large s
ale stru
ture of the
Universe', Prin
eton University Press, Prin
eton
[5℄ SavitzkyA. & GolayM. J. E. 1964, Analyti
al Chemistry,
36, 1627
[6℄ SergijenkoO. & Novosyadlyj B. 2015, Phys. Rev. D, 91,
083007
[7℄ TsizhM. & NovosyadlyjB. 2016, Visnyk Kyivskogo Uni-
versytetu. Astronomia, 53, 32
[8℄ Novosyadlyj B. 2007, J. Phys. Studies, 11, 226
32
Advan
es in Astronomy and Spa
e Physi
s M. Tsizh, B. Novosyadlyj
Fig. 2: Void formation in dark matter (left
olumn) and dark energy with c2s = 0.1 (
entral
olumn). On the right �
evolution of absolute values of amplitudes of density (top panel), velo
ity (bottom panel) perturbations in the
entral
point (thi
k lines) and overdensity shell (thin lines).
Fig. 3: Evolution of pro�les of matter density and velo
ity perturbations with initial parameters α = (31.4)−2
Mp
−2
and β = 3α/4 (left), β = 3α/2 (
entral), β = 3α (right). Thi
k solid line is initial pro�le, thi
k dashed one is �nal
pro�le.
33
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