Can quintessence and phantom cause the late time acceleration of the Universe?

0.14 MB·English

by  Alvina Burgazli

Can quintessence and phantom cause
the late time acceleration of the Universe?
Alvina Burgazlia, Maxim Eingornb,c, Alexander Zhukc
3
1 aDepartment of Theoretical Physics, Odessa National University,
0 Street Dvoryanskaya 2, Odessa 65082, Ukraine
2 bDepartment of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University,
n
Shevchenko av. 1, Odessa 65044, Ukraine
a cAstronomical Observatory, Odessa National University,
J
Street Dvoryanskaya 2, Odessa 65082, Ukraine
3

]
O
C
Abstract
.
h
In this Letter, we consider the Universe at the late stage of its evolution and
p
- deepinside the cell ofuniformity. At these scales,the Universe is filled with in-
o
homogeneously distributed discrete structures (galaxies, groups and clusters of
r
t galaxies). Supposing that the Universe contains also the cosmological constant
s
a and quintessence/phantom fluids with the constant equation of state parame-
[ ter ω, we investigate scalar perturbations of the FRW metrics due to inhomo-

geneities. Our analysis shows that, to be compatible with the theory of scalar
1
v perturbations, the quintessence/phantom, first, should be clustered and, sec-
8 ond, should have the equation of state parameter ω = 1/3. Therefore, this
−
1 quintessenceneitheracceleratesnordeceleratestheUniverse. Wealsoobtainthe
4
equationforthenonrelativisticgravitationalpotentialcreatedbyasystemofin-
0
homogeneities. Duetothequintessence,thephysicallyreasonablesolutionstake
.
1 place for flat, open and closed Universes. Quintessence is concentrated around
0
the inhomogeneities and results in screening of the gravitationalpotential.
3
1 Keywords: quintessence, phantom, cosmologicalconstant, late time
:
v acceleration, inhomogeneous Universe, scalar perturbations
i
X
r
a 1. Introduction
The accelerated expansion of the Universe at late stages of its evolution,
found little more than a decade ago [1, 2], is one of the most intriguing puzzles
of modern physics and cosmology. Recognition of this fact was the awarding of
the Nobel Prize in 2011 to Saul Perlmutter, Adam Riess and Brian Schmidt.
After their discovery, there were numerous attempts to explain the nature of
Email addresses: [email protected] (AlvinaBurgazli),[email protected]
(MaximEingorn),[email protected] (AlexanderZhuk)
Preprint submitted toElsevier December 11, 2013
such acceleration. Unfortunately, there is no satisfactory explanation so far
(see, e.g., the state of art in [3]). According to the recent observations [4, 5],
the ΛCDM model is the preferable one. Here, the accelerated expansion is
due to the cosmological constant. However, there is a number of problems
associated with the cosmological constant. Maybe, one of the main of them
consists in the adjustment mechanism which could compensate originally huge
vacuum energy down to the cosmologically acceptable value and to solve the
coincidence problem of close magnitudes of the non-compensated remnants of
vacuum energy and the energy density of the Universe at the present time [6].
On the other hand, it is well known that perfect fluids with the equation of
state parameters ω < 1/3 can also cause the accelerated expansion of the
−
Universe. Such fluids are called quintessence [7, 8, 9] and phantom [10, 11] for
1<ω <0 and ω < 1, respectively. Therefore, they can be an alternative to
− −
the cosmological constant. It is of interest to investigate the viability of these
models.
InourLetter,weconsiderthe compatibilityofthese modelswithscalarper-
turbationsofthe Friedmann-Robertson-Walker(FRW) metrics atlatestagesof
theUniverseevolution. We exploreasimplifiedmodelwithaconstantequation
of state parameter ω = const. We demonstrate that, first, these fluids must be
clustered (i.e. inhomogeneous) and, second, ω = 1/3 is the only parameter
−
which is compatible with the theory of scalar perturbations. Therefore, this
quintessence neither accelerates nor decelerates the Universe. We also obtain
formulas for the nonrelativistic gravitational potential created by a system of
inhomogeneities (galaxies, groups and clusters of galaxies). We show that due
to the quintessence, the physically reasonable expressions take place for flat,
open and closed Universes. If quintessence is absent, the hyperbolic space is
preferred [12]. Hence, quintessence can play an important role.
The Letter is structured as follows. In Sec. 2, we consider scalar pertur-
bations in the Friedmann Universe filled with the cosmological constant, pres-
sureless dustlike matter (baryon and dark matter) and quintessence/phantom
fluids. Here, we get the equation for the nonrelativistic gravitational poten-
tial. In Sec. 3, we find solutions of this equation for an arbitrary system of
inhomogeneities for flat, open and closed Universes. These solutions have the
Newtonian limit in the vicinity of inhomogeneities and are finite at any point
outside inhomogeneities. The main results are summarized in concluding Sec.
4.
2. Scalar perturbations of FRW Universe
Homogeneous background.
To start with, we consider a homogeneous isotropic Universe described by
the FRW metrics
ds2 =a2 dη2 γ dxαdxβ =a2 dη2 dχ2 Σ2(χ)dΩ2 , (1)
− αβ − − 2
(cid:0) (cid:1) (cid:0) (cid:1)
2
where
sinχ, χ [0,π] for =+1
∈ K
Σ(χ)= χ, χ [0,+ ) for =0 (2)
 ∈ ∞ K
sinhχ, χ [0,+ ) for = 1

∈ ∞ K −
and = 1,0,+1 for open, flat and closed Universes, respectively. As matter
K − 
sources, we consider the cosmological constant Λ, pressureless dustlike matter
(inaccordancewiththe currentobservations[4,5],weassumethatdarkmatter
(DM)givesthemaincontributiontothismatter)andanadditionalperfectfluid
with the equation of state p = ωε where 1 < ω < 0 for the quintessence and
−
ω < 1 for the phantom fluid. In the present Letter, ω = const. The overline
−
denotes homogeneous quintessence/phantom fluids. It can be easily seen from
the conservation equation that in the case of the homogeneous perfect fluid
a3(1+ω)
ε=ε 0 , (3)
0a3(1+ω)
where a is the scale factor at the present time and ε is the current value of
0 0
the energy density ε.
BecauseweconsiderthelatestagesoftheUniverseevolution,weneglectthe
contribution of radiation. Therefore, the Friedmann equations read
3 2+
0
H K =κT +Λ+κε (4)
a2 0
(cid:0) (cid:1)
and
2 ′+ 2+
H H K =Λ κωε, (5)
a2 −
where a′/a (da/dη)/a and κ 8πG /c4 (c is the speed of light and
N
H ≡ ≡ ≡
ik
G istheNewton’sgravitationalconstant). Here,T istheenergy-momentum
N
tensor of the average pressureless dustlike matter. For such matter, the energy
density T0 = ρc2/a3 is the only nonzero component. ρ = const is the average
0
rest mass density [12]. It is worth noting that in the case = 0 the comoving
K
coordinates xα may have a dimension of length, but then the conformal factor
a isdimensionless,andvice versa. However,in the cases = 1the dimension
K ±
of a is fixed. Here, a has a dimension of length and xα are dimensionless. For
consistency, we shall follow this definition for =0 as well. For such choice of
K
the dimension of a, the rest mass density has a dimension of mass.
It is well known that quintessence can provide the accelerated expansion of
theUniverseiftheequationofstateparametersatisfiestheconditionω < 1/3.
−
We can easily see it if we rewrite equations (4) and (5) as follows:
a¨ κρc4 Λc2 κc2ε a3(1+ω)
= + 0 0 (1+3ω), (6)
a − 6a3 3 − 6 a3(1+ω)
where overdots denote the differentiation with respect to synchronous time t
connected with conformal time η: cdt=adη.
Scalar perturbations.
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Obviously,theinhomogeneitiesintheUniverseresultinscalarperturbations
of the metrics (1). In the conformal Newtonian gauge, such perturbed metrics
is [13, 14]
ds2 a2 (1+2Φ)dη2 (1 2Ψ)γ dxαdxβ , (7)
αβ
≈ − −
where scalar perturbation(cid:2)s Φ,Ψ 1. Following the standa(cid:3)rd argumentation,
≪
we can put Φ = Ψ. We consider the Universe at the late stage of its evolution
when the peculiar velocities of inhomogeneities (both for dustlike matter and
quintessence/phantom fluids) are much less than the speed of light:
dxα dxα1 vα
=a 1. (8)
dη dt c ≡ c ≪
We should stress that smallness of the nonrelativistic gravitational potential Φ
and peculiar velocities vα are two independent conditions (e.g., for very light
relativistic masses the gravitational potential can still remain small). Under
these conditions, the gravitational potential Φ satisfies the following system of
equations (see [12] for details):
1 1
∆Φ 3 (Φ′+ Φ)+3 Φ= κa2δT0+ κa2δε, (9)
− H H K 2 0 2
∂
′
(Φ + Φ)=0, (10)
∂xβ H
1
Φ′′+3 Φ′+(2 ′+ 2)Φ Φ= κa2δp, (11)
H H H −K 2
where the Laplace operator
1 ∂ ∂
= √γγαβ (12)
△ √γ∂xα ∂xβ
(cid:18) (cid:19)
and γ is the determinant of γ . Following the reasoning of [12], we took into
αβ
account that peculiar velocities of inhomogeneities are nonrelativistic and the
contribution of δT0 is negligible compared to that of δT0 both for dustlike
β 0
matter and quintessence/phantom. In other words, account of δT0 is beyond
β
the accuracy of the model. This approach is completely consistent with [15]
where it is shown that the nonrelativistic gravitational potential is defined by
thepositionsoftheinhomogeneitiesbutnotbytheirvelocities(seeEq. (106.11)
in this book). In the case of an arbitrary number of dimensions, a similar
result was obtained in [16]. On the other hand, the motion of nonrelativistic
inhomogeneities is defined by the gravitational potential (see, e.g., [17]). The
perturbedDMremainsnonrelativistic(pressureless)thatresultsinthecondition
δTα = 0. For the quintessence/phantom we have δTα = δpδα. In Eq. (9),
β β − β
δε is afluctuationof the energydensity forquintessence/phantom,while δT0 is
0
related to the fluctuation of the energy density of dustlike matter and has the
form [12]:
δρc2 3ρc2Φ
δT0 = + , (13)
0 a3 a3
4
where δρ is the difference between real and average rest mass densities:
δρ=ρ ρ. (14)
−
From Eq. (10) we get
ϕ(r)
Φ(η,r)= , (15)
c2a(η)
where ϕ(r) is a function of all spatial coordinates and we have introduced c2
in the denominator for convenience. Below, we shall see that ϕ(r) 1/r in
∼
the vicinity of aninhomogeneity, andthe nonrelativistic gravitationalpotential
Φ(η,r) 1/(ar)=1/R,whereR=aristhephysicaldistance. Hence,Φhasthe
∼
correct Newtonian limit near the inhomogeneities. Substituting the expression
(15) into Eqs. (9) and (11), we get the following system of equations:
1 1 1
(∆ϕ+3 ϕ)= κc2δT0+ κc2δε, (16)
a3 K 2 0 2
1 1
( ′ 2 )ϕ= κc2δp. (17)
a3 H −H −K 2
From the Friedmann equations (4) and (5) we obtain
1 ′ 2 = 1 κT0 κ(1+ω)ε . (18)
a3 H −H −K 2a − 0−
(cid:0) (cid:1) (cid:16) (cid:17)
Then, Eq. (17) reads
ρc2 ε a4+3ω
κ κ(1+ω) 0 0 ϕ=κc2ωδε. (19)
− a4 − a a4+3ω
(cid:18) 0 (cid:19)
It should be noted that we consider quintessence/phantom fluids without ther-
malcouplingtoanyothertypeofmatter. Itmeans,inparticular,thatevolution
of its homogeneous backgroundas well as scalar perturbations occurs adiabati-
cally or, in other words, without change of entropy. Therefore, we preserve the
same linear equation of state δp=ωδε with the same constantparameter ω for
the scalar perturbations δp and δε of pressure and energy density respectively,
as for their background values p and ε.
Taking into account the expression (15), we get that in the right hand side
of Eq. (13) the second term is proportional to 1/a4 and should be dropped
because we consider the nonrelativistic matter. This is the accuracy of our
approach, i.e. for the terms of the form of 1/an, we drop ones with n 4
≥
and leave terms with n < 4. Obviously, 4+3ω < 4 for ω < 0. Hence, we can
drawtheimportantconclusionregardingthepurelyhomogeneousnon-clustered
quintessence/phantom fluids with δp,δε = 0. For these fluids, we arrive at a
contradictionbecause in Eq. (19) the right hand side is equal to zero while the
left hand side is nonzero. Therefore, such fluids are forbidden. Quintessence
and phantom should be capable of clustering. In the papers [9, 18], it was also
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pointedoutthatthequintessencehastobeinhomogeneous. Forinhomogeneous
quintessence/phantom we get from Eq. (19) that
1+ω a3+3ω
δε= ε 0 ϕ. (20)
− c2ω 0a4+3ω
It is worth noting that in our model neither the nonrelativistic gravitational
potential Φ 1/a nor the quintessence/phantom density contrast δε/ε 1/a
∼ ∼
diverge with time (with the scale factor a) in spite of the negative sign of the
ratio δp/δε which is often treated as the speed of sound squared. Substituting
(20) into (16), we obtain within our accuracy
1 1 δρc2 1 1+ω a3+3ω
(∆ϕ+3 ϕ)= κc2 κc2 ε 0 ϕ
a3 K 2 a3 − 2 c2ω 0a4+3ω
1 1+ω a1+3ω
∆ϕ+3 ϕ= κc4δρ κε a2 0 ϕ. (21)
⇒ K 2 − 2ω 0 0a1+3ω
Inthisequation,alltermsexceptthelastonedonotdependontime. Therefore,
ω = 1/3 is the only possibility to avoid this problem. Hence, we arrive at the
−
following important conclusion. At the late stage of the Universe evolution,
quintessence/phantom fluids are compatible with the scalar perturbations only
if, first, they are inhomogeneous, and, second, they have the equation of state
parameter ω = 1/3. Such fluid is quintessence which cannot provide the late
−
time acceleration of the Universe (i.e. this quintessence neither accelerates nor
decelerates the Universe (see Eq. (6))).
For ω = 1/3, the equation for the gravitational potential and the fluctua-
−
tion of the energy density of the quintessence read, respectively:
8πG
∆ϕ+ 3 Nε a2 ϕ=4πG (ρ ρ) (22)
K− c4 0 0 N −
(cid:18) (cid:19)
and
2ε a2
δε= 0 0ϕ. (23)
c2a3
Inthenextsection,weshallinvestigateEq. (22)dependingonthecurvature
parameter . Weshallshowthatreasonableexpressionsoftheconformalgravi-
K
tationalpotentialϕexistforanysignof . Thistakesplaceduetothepresence
K
of the quintessence with ω = 1/3. If quintessence is absent, the hyperbolic
−
model = 1 is preferred [12]. Therefore, the positive role of quintessence is
K −
that its presence gives a possibility to consider models for any .
K
3. Gravitational potentials
It is convenient to rewrite Eq. (22) as follows:
∆φ λ2φ=4πG ρ, (24)
N
−
6
where the truncated gravitationalpotential is
4πG ρ
N
φ=ϕ , λ=0, (25)
− λ2 6
and
8πG
λ2 Nε a2 3 . (26)
≡ c4 0 0− K
On scales smaller than the cell of uniformity size (which is of the order of 150
Mpc)andonlatestagesofevolution,theUniverseisfilledwithinhomogeneously
distributed discrete structures (galaxies, groups and clusters of galaxies) with
dark matter concentratedaroundthese structures. Then, the rest mass density
ρ reads [12]
1
ρ= m δ(r r ), (27)
i i
√γ −
i
X
where m is the mass of i th inhomogeneity. Therefore, Eq. (24) satisfies the
i
−
very important principle of superposition. It is sufficient to solve this equation
for one gravitating mass m and obtain its gravitational potential φ . The
i i
gravitational potential for all system of inhomogeneities is equal to a sum of
potentials φ . It is worth recalling that the operator ∆ is defined by Eq. (12).
i
As boundary conditions, we demand that, first, the gravitational potential of
a gravitating mass should have the Newtonian limit near this inhomogeneity
φ 1/rand,second,thispotentialshouldconvergeatanypointoftheUniverse
i
∼
(except the gravitating mass position).
It seems reasonable to assume also that the total gravitational potential
averagedover the whole Universe is equal to zero (see, e.g., [12]):
4πG ρ 1
N
ϕ=φ+ =0, φ= φ dV , (28)
λ2 V i
i ZV
X
where V is the volume of the Universe. This demand results in another physi-
cally reasonable condition: δε=0 (see Eq. (23)).
Flat space: =0.
K
In the case ε >0 λ2 = 8πGNε a2 >0, the solutionof (24) for a separate
0 → c4 0 0
mass m satisfying the mentioned above boundary conditions reads
i
G m
N i
φ = exp( λr), λ>0, 0<r <+ . (29)
i
− r − ∞
It can be easily seen that this truncated potential has the Newtonian limit
for r 0. This expression shows that quintessence results in the screening
→
of the Newtonian potential. A similar effect for the Coulomb potential takes
place in plasma. In our case, the screening originates due to specific nature
of quintessence. It is worth mentioning that the exponential screening of the
gravitationalpotentialwasintroduced”byhand”inanumberofmodelstosolve
7
thefamousSeeligerparadox(see,e.g.,thereview[19]). Inourmodel,weresolve
this paradox in a natural way due to the presence of quintessence.
For a many-particle system, the total gravitationalpotential takes the form
m 4πG ρ
i N
ϕ= G exp( λr r )+ . (30)
− N r r − | − i| λ2
i
i | − |
X
Substituting(30)into(23),wegetforthefluctuationofthequintessenceenergy
density the following expression:
2ε a2 m c4ρ
δε= 0 0 G i exp( λr r )+ . (31)
c2a3 − N i |r−ri| − | − i| 2ε0a20!
X
Therefore, we arrive at a physically reasonable conclusion that these fluctua-
tionsareconcentratedaroundthe matter/darkmatterinhomogeneitiesandthe
corresponding profile is given by Eq. (31).
The averaged value of the i-th component of the truncated potential over
some finite volume V is
0
4π r0 exp( λr)
φ = G m − r2dr (32)
i V − N i r
0 Z0 (cid:20) (cid:21)
4πG m exp( λr ) 1 1
N i 0
= − r + + .
− V − λ 0 λ λ2
0 (cid:20) (cid:18) (cid:19) (cid:21)
Then, letting the volume go to infinity (r + V + ) and taking all
0 0
→ ∞⇒ → ∞
gravitating masses, we obtain
4π
φ= G ρ , (33)
− N λ2
where ρ = lim m /V . Therefore, the averaged gravitational potential
i 0
V0→+∞ i
(28) is equal to zeroP: ϕ=0. Consequently, δε=0.
The case ε < 0 λ2 µ2 < 0 is not of interest. Here, we get the
0
⇒ ≡ −
expression φ = (G m /r)cos(µr) which does not have clear physical sense.
i N i
−
Additionally, this expression does not allow the procedure of averaging.
Spherical space: =+1.
K
Let us consider, first, the case λ2 = 8πGNε a2 3 µ2 < 0. This case is
c4 0 0− ≡ −
of interest because it allows us to perform the transition to small values of the
energydensityofquintessence: ε 0. Here,thesolutionof(24)foraseparate
0
→
mass m is
i
sin (π χ) µ2+1
−
φ = G m , 0<χ π. (34)
i − N isinhπ µ2+p1 sinχi ≤
(cid:16) p (cid:17)
8
For µ2+1 = 2,3,... (we would remind that µ2 = 0), this formula is finite
6 6
at any point χ (0,π] and has the Newtonian limit for χ 0. In the case of
p ∈ →
absence of quintessence ε = 0 µ2+1 = 2, this expression is divergent at
0
→
χ = π. We demonstrated this fact in our paper [12]. Therefore, quintessence
p
gives a possibility to avoid this problem for the models with = +1. It can
K
be easily verified that for the total system of gravitating masses, the averaged
value of the total truncated potential has the form of (33) that results in ϕ =
0 δε=0.
⇒
In the caseλ2 >0,the formulascanbe easily foundfrom(34) withthe help
of analytical continuationµ iµ. In other words,it is sufficient in Eq. (34) to
→
replace µ2 by λ2. The obtained expression is finite for all χ (0,π] and the
− ∈
averagedgravitationalpotential is equal to zero: ϕ=0 δε=0.
⇒
Hyperbolic space: = 1.
K −
Here,the mostinterestingcasecorrespondstoλ2 = 8πGNε a2+3>0. This
c4 0 0
choiceof signgivesa possibility to performthe transitionto smallvalues of the
energy density of quintessence: ε 0. Then, the desired solution of Eq. (24)
0
→
for a mass m is
i
G m
φ = N i exp χ λ2+1 , 0<χ<+ . (35)
i
−sinhχ − ∞
(cid:16) p (cid:17)
If quintessence is absent (ε = 0), then we reproduce the formula obtained in
0
[12]. Ontheotherhand,theexpression(35)showsthatforε >0 λ2+1>4,
0
→
quintessenceenhancesthescreeningofthegravitatingmass. Foramany-particle
system, the total gravitational potential takes the form
exp( l √λ2+1) 4πG ρ
i N
ϕ=−GN mi −sinhl + λ2 , (36)
i
i
X
where l denotes the geodesic distance between the i-th mass m and the point
i i
of observation. Similarly, using Eq. (34), we can write the expression for the
total potential in the case of the spherical space.
Taking into account that the averaged total truncated potential has again
the form (33), the procedure of averaging leads to the physically reasonable
result: ϕ=0 δε=0.
⇒
Concerning the case λ2 < 0, the truncated gravitational potential is finite
in the limit χ + . However,the procedure ofaveragingdoes not existhere.
→ ∞
Therefore, this case is not of interest for us.
To conclude this section, we discuss briefly the case λ2 =0. For =0, 1,
K −
theprincipleofsuperpositionisabsentnow. Tomakethegravitationalpotential
finiteatanypointincludingthespatialinfinity,weneedtocutoffitsmoothlyat
somedistancesfromeachgravitatingmass. If =0,thenquintessenceisabsent
K
and this case was described in detail in [12]. It was shown that the averaged
gravitational potential is not equal to zero. This is a disadvantage of such
9
models. In the case = +1, the principle of superposition can be introduced
K
due to the finiteness of the total volume of the Universe. Here, the comoving
averaged rest mass density can be split as follows: ρ = m /(2π2) ρ .
i i ≡ i i
Then, Eq. (22) can be solved separately for each combination (m ,ρ ). As a
i i
P P
result, the gravitationalpotential of the i-th mass is
G m cosχ χ
N i
ϕ = G m 1 , 0<χ π. (37)
i N i
2π − sinχ − π ≤
(cid:16) (cid:17)
This potential is convergent at any point χ = 0, including χ = π. It is not
6
difficulttoseethatϕ =0. Therefore,thetotalaveragedgravitationalpotential
i
is also equal to zero: ϕ= ϕ =0 δε=0.
i i ⇒
P
4. Conclusion
In our Letter, we have investigated the role of quintessence and phantom
field for the Universe at late stages of its evolution. It is well known that these
fieldscanbeanalternativetothecosmologicalconstantexplainingthelatetime
acceleration of the Universe. It happens if their parameter of the equation of
state ω < 1/3. To check the compatibility of these fluids with observations,
−
we consider our Universe at scales much less than the cell of homogeneity size
which is approximately 150 Mpc. At such distances, our Universe is highly
inhomogeneous and the averagedFriedmann approach does not work here. We
need to take into account the inhomogeneities in the form of galaxies, groups
and clusters of galaxies. All of them perturb the FRW metrics.
To clarify the role of quintessence and phantom, we endowed the Universe
with these fluids with a constant equation of state parameter ω. We have
shown that quintessence and phantom are compatible with the theory of scalar
perturbationsiftheysatisfytwoconditions. First,thesefluidsmustbeclustered
(i.e. inhomogeneous). Second, the parameter of the equation of state ω should
be 1/3. Therefore, this quintessence neither accelerates nor decelerates the
−
Universe.
Then,wehaveobtainedtheequationforthenonrelativisticgravitationalpo-
tential. We haveshownthatduetoquintessencethe physicallyreasonablesolu-
tionstakeplaceforflat,openandclosedUniverses. Thepresenceofquintessence
helps to resolve the Seeliger paradox [19] for any sign of the spatial curvature
parameter . If quintessence is absent, the hyperbolic space is preferred [12].
K
Hence, quintessence can play an important role. Quintessence is concentrated
around the inhomogeneities and results in screening of the gravitationalpoten-
tial.
Acknowledgements
Thisworkwassupportedinpartbythe”Cosmomicrophysics-2”programme
of the Physics and Astronomy Division of the National Academy of Sciences of
Ukraine.
10