Main results

The m odel with decaying dark energy belongs to the class involv­

ing interaction in the dark sector. For the late-tim e approximation

o f the m odel (a 2 / t 2 ), the deviation from the standard scale law of

the energy density of dark matter is noticeable. However, the pro­

duction o f dark matter is no longer an effective process. Note that this modification for the early Universe is independent on time.

From the statistical analysis, we get for a 2 ^{/ t} 2 m odel the decay of particles of dark matter instead of the creation of one.

The analysis indicates also that the present value of dark en­

ergy is not sensitive to the value o f a and E 0 parameters.

This m odel can solve the cosm ological constant problem, be­

cause it involves the mechanism of jum ping from the initial value of dark energy E 0 ⁼ 10 12Q to the present value of the cosmological constant.

The characteristic feature of the m odel are oscillations of the density of dark energy occuring for 0 < a < 0.4.

The radioactive-like decaying m odel of dark energy for the late-tim e Universe (t = 2T 0 ) has three different forms of decay of dark energy: radioactive, damping oscillating, and power-law.

In the beginning, the radioactive type of decay dominates up to

2.2 x 10 4 ^T 0 . After the radioactive type of decay, damping oscillating

type of decay appears, which is later superseded by a power-law

type of decay (1 / t 2 ).

Chapter 4

Diffusion dark matter-dark energy interaction model

4.1 Relativistic diffusion interacting of dark matter with dark energy

This section is based on JCAP1607 (2016) no. 07, 0 2 4 1471 and Phys.Rev. D94 (2016) no. 4, 0435211481.

We consider a particular m odel of energy-m om entum exchange between dark matter and dark energy, where baryonic matter is preserved. In this approach, it is assumed that the total number o f particles is conserved and the relativistic version o f the energy- m omentum tensor:

= (

p

+ - g^p. (4.1)

In this model, the energy-m om entum tensor consists of two parts:

T ßV = Tde + Tmv, (4.2)

where T^V is the energy-m om entum tensor for dark energy and Tmv is the energy-m om entum tensor for matter.

We assume the conservation of the total energy m omentum in the following form:

- V^TdV = V ^ = 3

k

2 J V, (4.3) where

k

2 is the diffusion constant and J V is the current which de­

scribes a flow of particles.

This m odel provides that the dark matter is transferred by a dif­

fusion mechanism in an environment corresponding to the perfect fluid, while predicting a unique diffusion which is relativistically in­

variant and preserves the mass m o f a particle [49].

The Friedmann equation is given here as:

3 H 2 = Pb + Pdm + Pde, (4.4) where pb is the density of baryonic matter, pdm is the density of dark matter, pde is the density o f dark energy, and pm = pb + pdm. The densities pm and pde are given by:

Pm = Pb,oa -3 ^{+ pdm,oa} -3 + Y(t - to)a - 3 ^{, (4.5)} Pde = Pde(0) - Y J a -3 dt, (4.6) where

y

is a positive m odel parameter.

If we choose t Q as zero, then we get a modified scale law for the energy density of dark matter:

Pdm = Pdm,o a -3 + Yta -3 . (4.7) The current J ß is conserved [5 0 , 51, 52]. In result, we get:

V J = 0. (4.8)

For the FRW metric from the above equation, we obtain:

JQ = y /3 k 2 «- 3 . (4.9)

From Eq. (4.3), we get the following conservation equations:

pm = —3Hpm + Ya 3 ^{, (4.10)}

Pde = -Y a - 3 , (4.11)

where we assume that the equation of state for dark energy is pde = - p de and for matter is pm = 0. Here, ' = d .

This m odel of diffusion interaction in the dark sector is free from

the difficulties affiicting Alho et al.'s models with diffusion [53]. It

involves no non-physical trajectories crossing the boundary set

pm = °.

4.2 Diffusive DM-DE interaction:

coincidence problem

This section is based on Phys.Rev. D94 (2016) no. 4, 0435211481.

In the paper [48], we recast cosm ological equations of the diffu­

sion cosm ological m odel as a dynamical system. By inserting Eqs (4.5) and (4.6) into the Friedman equation (4.4), we get:

3H2 = pb,oa 3 + pdm,oa 3 + y (t - t°)a 3 + pde(0) - y J a -'dt. (4.12)

Now let x = Qm, y = Qde, 6 = H t^ and ' = dint is a differen­

tiation with respect to the reparametrized time lna(t). Equations (4.10), (4.11) and (4.12) can be rewritten as the dynamical system in variables x, y and z with respect to time ln a(t). Thus we get the following dynamical system:

x' = x ( —3 + 6 + 3x), (4.13)

y' = x (—6 + 3y), (4.14)

3

6' = 6 (-6 + - x). (4.15)

From Eq. (4.12), we have that 3H2 + 3H2 = 1. In result, we get that x + y = L Accordingly, dynamical system (4.13)-(4.15) is reduced to a two-dimension dynamical system.

In order to analyse this system in the infinity, we use the rewrit­

ten forms of Eqs (4.13) and (4.15) in variables

x 6

X = — = ^ = , A = - = ^ = . (4.16) V x 2 + 62 V x 2 + 62

Ultimately, we get the following dynamical system:

X ' = X - A 2 ^ 3X - A ^ + ( 1 - X 2)(3X + A - 3 -1 - X 2 - A 2) , (4.17) A ' = A (1 - A 2) ( 3 X - A ^ - X 2(3X + A - 3 -1 - X 2 - A 2) ,

(4.18)

where ' = V 1 - X 2 - A 2dlndt(t). The critical points of the system

(4.17) and (4.18) are collected in Table I in Phys.Rev. D94 (2016) no. 4,

043521 [48].

We considered also the case when the equations of state for baryonic and dark matter and dark energy are in a generalized form:

Pde = wpde, (4.19)

Pdm = wpdm, (4.20)

Pb = 0, (4.21)

where w and w are constant coefficients for dark energy and matter respectively.

Now the continuity equations are:

pdm = - 3(1 + w )H Pdm + Ya 3, (4.22) Pde = -3 (1 + w)Hpde - Ya 3, (4.23)

pb = - 3Hpb. (4.24)

From the above equations and Eq. (4.10), we get the following dynamical system in the analogous way like (4.13)-(4.15):

dx r z "

= 3x (1 + w)(x - 1) + (1 + w)y + Z , (4.25)

d ln a L 3J

,dy = 3y[(1 + w)(y - 1) + (1 + w)x] - xz, (4.26) d ln a

dz 3

—— = z 3w - z + - [( 1 + w)x + (1 + w)y] . (4.27)

d ln a 2

As x + y = 1, the above system is reduced to a two-dimensional one. The critical points of this m odel are collected in Table II in Phys.Rev. D94 (2016) no. 4, 043521 [48]. The critical point {x o = - 31W +_W W ) ■ zo = 1 + 3w} represents a scaling solution pdm = pde, thus providing a mechanism to solve the coincidence problem.

We considered the special case of Eqs (4.25) and (4.27) when dark matter is relativistic (w = 1/3) and w = -1 . Then they sim plify to the following form:

x' = x ( - 4 + z + 4x), (4.28)

z' = z(1 - z + 2x). (4.29)

For the purpose of examining Eqs (4.28) and (4.29) in the infinity, we choose variables X = Vx2+52 , A = Vx2+52 . Thus we get: 3

X ' = X - A 2 ^ V i - X 2 - A 2 + 3X - A ^ +

(1 - X 2)(3X + A - 4V i - X 2 - A 2) , (4.30)

A ' = A ( i A 2) ^ V i X 2 A 2 + 3X A ^

-X 2(3-X + A - 4 V i - -X 2 - A 2) , (4.31)

where ' = V i - X 2 - A 2 JT.

The critical points of system (4.30)-(4.31) are collected in Ta­

ble III in Phys.Rev. D94 (2016) no. 4, 043521 [48].

4.3 Diffusive DM-DE interaction:

non-relativistic case and statistical analysis

This section is based on JCAP1607 (2016) no. 07 024 [471.

In the paper [47], we examine two cases of the diffusion interaction in the dark sector: relativistic and non-relativistic. The relativistic case was considered in the previous sections. The other one uses the non-relativistic limit of the above energy-m om entum tensor:

pdm = r 00 = Vg(2n)-3 y dpp0H = g- 2Zm + Vg(2n)- ^ dp 0 ^ H

= Zma-3 + a-2pnr, (4.32)

where 2

pnr = Vg(2n)- 3 / dp Ha42 ^ (4.33) where H is the concentration o f mass, p is the momentum and m is the mass of the particle of dark matter. The constant Z is given

Z = 3K2 = g / (2 p 3 H ^

where

k

2 is the diffusion constant.

In this case, the conservation equation for dark matter is:

/Pdm + 5Hpdm = 3ZK2a _ 3 + 2ZmHa _ 3. (4.35) Let

Pdm Pde (2Zm)a _ 3 Ya _3

x = 3H ■ y = 3H ■ u = --- S = w— pdm Hpdm (4.36) and

t

= ln a is a reparametized time. Then we get the following dynamical system:

x' = x ^ - 5 + S + u - 2— j , (4.37)

y' = - x(S + u) - 2y-H , (4.38) H 2

u' = u(2 - S - u), (4.39)

S' = S ^ 2 - S - u - ■ (4.40)

where ' = JT and J2 = - 1 x(5 - u).

Since —dm + —de = 1 we have x + y = 1 . In effect, the above dynamical system reduces to a three-dimensional dynamical sys­

tem. The system (4.37)-(4.40) has the invariant submanifold { J2 = 0} determined by the equations x = 0 or u = 5. Its other subman­

ifold is S = 0. For this invariant submanifold, the system reduces to

x' = x(u + 5(x - 1) - xu), (4.41)

u' = u (2 - u). (4.42)

In the statistical analysis of the m odel parameters of relativis­

tic and non-relativistic case, we used the following astronomical

observations: supernovae of type Ia (SNIa, Union 2.1 dataset [13]),

BAO data [14, 15, 16], measurements of the Hubble parameter H (z)

of galaxies [17, 18, 19], the Alcock-Paczynski test [2 0 , 21] (data from

[22, 23, 24, 25, 26, 27, 28, 29, 3 0 ]) and measurements of CMB and

lensing [31]. The value of the best fit and errors are given in Table 3

and 4 in JCAP1607 (2016) no. 07, 024 [47]. The BIC criterion gives

a strong evidence in favour of the A CDM m odel in comparison to

these models. However, this is not sufficient for rejecting of the

diffusion models.

Chapter 5

Dynamical system approach to running A cosmological

models

This chapter is based on Eur.Phys.J. C76 (2016) no. 11, 6 0 6 1101.

In the paper [10], we investigate cosm ological models in which the cosm ological constant term is a tim e-dependent function, exam­

ining the following parametrization of cosm ological parameter A:

A (H ), A(a) as w ell as three covariant ones: A(R), A(0) - cosm olo­

gies with diffusion, and A (X ), where X = 2gaßV aVß0 is the kinetic part of density of the scalar field. We also considered an emer­

gent relation A(a) obtained from the behaviour of trajectories in a neighbourhood of an invariant submanifold. In the thesis, we limit to A (H ), A(R), A (X ), and A(a).

5.1 A(H )C D M cosmologies

We take the parametrization A (H ) in the form of the Taylor series:

œ 1 dn œ

A (H ) = £ n dHHn A (H )I°H n = £ anHn (S.1>

n=1 n=1

Here, a reflection sym metry H ^ - H is additionally assumed.

Only terms of type H 2n in the above expansion series have this symmetry, thus [54]:

A (H ) = Abare + a2 H 2 + a4H4 + • • • . (5.2)

The cosm ological equations with the parametrization A (H ) are:

H = - H 2 - - pm + i A (H ), (5.3)

6 3

pm = - 3Hpm - A' (H) ^ - H 2 - g pm +

Let x = H 2, y = pm and

t

= ln a is a new parametrization of time.

Then system (5.3)-(5.4) gives the following dynamical system:

dx i i 2

x = —— = 2 - x - - y + - ( A + a 2 x + a 4 x +---) , (5.5)

d ln a L 6 3

y' = d = -3 y - ö (a2 + 2a4x + ■ d ln a 3 ■ ■)

x - x - — y + - (A + a 2 ^{x + a} 4 ^{+ ■ ■ ■ ) . (5.6)} 6 3

As 3H2 = pm + A (H ), we get an additional equation:

y — 3x = —(A + a 2 x + a 4 x + ■ ■ ■ ). (5.7) This equation lets us reduce the system (5.5)-(5.6) to one dim en­

sion.

After cutting the second term out o f the series (5.2) we get the following equations:

d r = x(a2 - 3 ) + A, dx (5.8) dT

y = (3 - a2)x - A. (5.9)

The system has the critical point:

xo = — ^ - , y = 0. (5.10) 3 - a2

Now we introduce a new variable x ^ X = x - x0, obtaining

dX = (a2 - 3)X. (5.11)

dT

The above equation has an exact solution in the form:

X = X oeT (a2-3) = X 0a-3+“ 2, (5.12) which can be interpreted as the Alcaniz-Lima solution [55]:

x =

h

2 = ^ a-3+“ 2 + ^ , (5.13)

3 3

where pm;0 = g - ^ pm,°. This constitutes the scaling solution pA(a) ~ pm(a), which provides a way to solve the coincidence problem.

5.2 A(R)CDM cosmologies

We investigate the parametrization of A(R) in the form pA = - f R = 3a(H + 2H2 + ) [56], where k = - 1 , 0, +1, getting the following cosm ological equations:

H = - H 2 - “ (pm + PA), (5.14) 6

p = - 3Hpm (5.15)

with the first integral o f the form

H 2 = 1 ( - — pm,°a-3 + f°a 2^ , (5.16)

3 \ a2 2 - a J

where f° is an integration constant.

We can rewrite the above equations as a dynamical system in the variables a, x = a

a = x, (5.17)

x = a-2+

2 - a

+ ^ a - ^ ( QA, 0 ^{- Qm,} 0 2----^ a“ 3. (5.18) We can analyse the system (5.17)-(5.18) in the infinity, using vari­

ables A = t , X = x . Then we get the following dynamical system:

A = - X A , (5.19)

X = A - Qm,0~--- 1 L ’ 2 - a

+ ( ^ ^ r ) (Qa>° - Q m ,o ^ a ) A 3-2 ] - X 2. (5.20)

We can use also the Poincaré sphere to investigate critical

points at the infinity. If we take B = ^ 1+t~2+x2, Y = ^ 1+t;2 2, then

we obtain a dynamical system in the following form:

B ' = Y B 2(1 - B 2)

- B Y [ - —m,o— ^ ( 1 - B 2 - Y 2)3/2 L 2 - a

+ ( ^ -

t

) ( —Ao - —m'°2 - a ) B _1+2/a

x (1 - B 2 - Y 2)2 _ 1/a] , (5.21) Y ' = [ - —m , ^ - ^ ( 1 - B 2 - Y 2)3/2

L 2 - a

+ f ^ ) ( —A.o- —m,o 2 - a ) B _1+2/a

(1 - B 2 - Y 2)2_ 1/a (1 - Y 2) - Y 2B 3, (5.22) where ' = B 2 J . dt

5.3 Non-canonical scalar field cosmology

The dark energy can be also parameterized by a non-canonical scalar field 0 [57]. In the canonical scalar field approach, the pres­

sure p^ is given by the form ulap^ = ^ - V (0), w here' = J and V (0) is the potential o f the scalar field. In the non-canonical scalar field, the pressure is described by the formula p^ = - V (0), where a is a parameter. Note that when a is equal to 1, then the pres­

sure o f the non-canonical scalar field corresponds to the canonical case.

The cosm ological equations for this m odel are the Friedmann equation:

3h2 = Pm + (2a - 1) ^ 02“ ^ + V(0) - C2 , (5.23)

where k = -1 , 0, +1 and the Klein-Gordon equation:

0 + J H 1 + ( / ' (0> V I 1 = 0 . (5.24) 2a - 1 \a (2 a - 1 )/ \ 0 2)

The above equations can be rewritten as a dynamical system. We

choose a and x = a as variables, obtaining from Eqs (5.23) and

a' = xa 2 , (5.25)

/ pm ,0 a + i —3— A 3 ^{n r \}

x' = - — --- a i- 2“ +— a 3 ^{, (5.26)}

6 3 3 ’

where ' = a 2 J . dt

For the purpose o f analysing critical points in the infinity, we choose the coordinates: A = 1 a , X = - and B = a, Y = a - 1 - .

The dynamical system for variables A and X is:

A' = - X A 2 , (5.27)

X ' = A 4 ^ - p m0 - A 20-1^ +

a

^ A - X ^ , (5.28)

where ' = A J . We can then obtain the following dynamical system based on variables B and Y:

B = B Y [b + f pY 3 ^{+ B i - a Y} 20 ^- 1 - A B ^ l , (5.29)

[ \6 3 3

^{) \}

’

ÿ = Y 2 ^{f p Y} 3 ⁺ B i - a Y ^ A B 3 ^{^ , (5.30)}

\6 3 3 7 ’

where ' = B 2 ^{Y J .} _dt

5.4 Cosmology with emergent A (a) relation

We consider cosm ology with a scalar field which is non-minimally coupled to gravity. In this case, the cosm ological equations are:

0 + 3 H 0 + £R0 + V'(0) = 0, (5.31) where ' = , 0 is a scalar field, V (0) is a potential of the scalar field and

3H 2 = pm + 2 0 2 + 3£H 2 0 2 + 6£H00 + V (0) (5.32) and

pm = -3Hpm. (5.33)

We introduce the following variables [58]:

x = _ t , , = = -0=. (5.34)

>/6H > /3 ^

(5.24):

In result, we get:

( H \ 2

( — J = + Qm = (1 - 6£)x2 + y2 + 6£(x + z)2 + Qm,°a 3. (5.35) Our aim is to generalize the AODM m odel by including a contri­

bution beyond Abare in the above equation. In our further analysis we w ill call it ‘the em ergent A term'. Thus,

Q

a

,emergent = (1 - 6£)x2 + y2 + 6£(x + z)2. (5.36) The dynamical system which describes the evolution in phase space has the form:

dx dx 1 2 ^ . H lr. __

dcima) = ^ = - 3x - 12£z+ 2 Ay - ( x + 6£z)

h

■ <5.37>

dy dy ^ H ,r ops

d(ina) = dT = - 2 Axy - ■ <5.38>

d7ddz^ = £ = x, (5.39)

d(ln a) dT

= dT = - A2(r(-V) - 1)x, (5.40) d(ln a) dT

where r = v:V 8V r1 , A = - - 6 W and

h

1 r 1/ \ 1 _3"

H = H 2 - 2 (p^ + p^) - 2pm,0a

= 6^z2(1 -1 6£) - 1 I-12£(1 - 6£)z2 - 3£Ay2z

3 3 3 "

+3(1 - 6£)x2 + 3£(x + z)2 + 2 - 3y2 . (5.41) For the sake of illustrating the emergent A(a) relation, we con­

sider two cosmologies for which we derive A = A(a): V = const or A = 0, if £ = 0 (minimal coupling), and V = const, if £ = 1 (conformal coupling). For the above cases, the system (5.37)-(5.40) reduces to

dx dx H

“TT

ï

r = y = - 3 x - ^772, (5.42)

d(ln a) dT H 2

dy _ dy _ H ,c /i

“TTj 7 = = - y Tr2 , (5.43)

d(ln a) dT H 2

dz dz

= TT = x, (5.44)

d(ln a) dT

where

H 3 2 3 3 2 ,r

H2 = - 2 ^{•’ - 2 +} 2 ^{y2 (545)} and

dx H . ...

— = - 3x - 2z — — (x + z), (5.46)

dT H 2

dy H / c / m

dT = - y H , <Ł47>

= x, (5.48)

dT where

H = - + z)2 - - + - y2. ^ For the minimal coupling case(£ = 0, V = const), the dynamical system (5.42)-(5.44) is expressed by:

dx dx

-jTi— t = ~T = - 3x, (5.50) d(ln a) dT

= d y = 0 , (5.51)

d(ln a) dT d z d z

TTn— V = T = x (5.52)

d(ln a) dT with the condition

0 = x2 - y2 + 1. (5.53)

The solution of the above system is x = C1a_3, y = const and z = - 3 C1a _3 + C2.

Accordingly, — A,emergent for this case is:

—A,emergent = —A,emergent,0a + — A,0. (5.54) For the conform al coupling case, the system (5.42)-(5.44) is:

— = - 3x - 2z, dx (5.55)

dT

^ = 0 ^ y = const, (5.56)

dT y

^ = x (5.57)

dT

with the condition

0 = (x + z) 2 — 3y 2 + 3. (5.58) The solution of the above dynamical system is x = - 2 C 1 a- 2 - C^a 1 , y = const and z = C 1 ^a 2 ^{+ C^a} 1 ^.

In consequence, n A,emergent is:

QA,emergent = QA,0 + ^A,emergent,0a . (5.59)

The m odel with £ = 1 / 6 (conformal coupling) and V = const

involves the early constant ratio dark energy Qde = const during

the radiation epoch. In this case, we can use the fractional early

dark energy parameter Q = 1 - 1 ^ , where Qtot is the sum of the

densities of both matter and dark energy [59, 6 0 ]. For the fractional

early dark energy parameter, there is a strong observational upper

limit (Q < 0.0036) [31]. Accordingly, we obtain the following limit on

the running A parameter in the present epoch: Qem, 0 < 3.19 x 10- 7 .

Chapter 6

Starobinsky cosmological model in Palatini formalism

6.1 Palatini formalism in Jordan frame

This section is based on Eur.Phys.J. C77 (2017) no. 6, 406 1611, Eur.Phys.J. C77 (2017) no. 9, 603 1621, Eur.PhysJ. C78 (2018) no. 3, 2 4 9 1631, and Phys.Rev. D97(2018) 1035241641.

In this section, we consider the Starobinsky cosm ological m odel ( f (R) = R +

y

R2) in the Palatini formalism. This m odel can be for­

mulated either in the Jordan frame or in the Einstein frame.

First, the m odel w ill be considered in the Jordan frame. Then its action has the following form:

S = Sg + Sm = 1 J V = g f (R)d4x + Sm, ( 6 . 1 ) where R = R^v(r) is the generalized Ricci scalar and R^v(r) is the Ricci tensor o f a torsionless connection r [65, 66 ]. Since we assume that the equation of state for matter is given in the form P = p(p), the action for matter Sm is [67]:

Sm = J - V=gp ^ 1 + J p(p) dp^ d 4 x. ( 6 . 2 ) After varying Eq. (6.1) with respect to the metric and the con­

nection r, we get the equations of motion:

f'( R ) J V - 2 f ( R ) f t - = r *v, < 6 ^■3>

'7 „ ( ^ = ? / '( R ) < r ) = 0 , (6.4)

where V« is the covariant derivative obtained with respect to the

connection r and jL “1 is the energy-m om entum tensor

for which V mTmv = 0 .

The structural equation is obtained from the trace o f Eq. (6.3) as:

/'(R )R - 2 / (R) = T. (6.5) For the Starobinsky model's case, Eq. (6.5) simplifies to

- R = T. (6.6)

As we consider the perfect ftuid, the energy-m om entum tensor is given by:

TV = diag (-p ,p ,p ,p ), (6.7) where p is the pressure of matter. In this case, the equation of state has the form p = wp, where w is a constant, which equals zero for dust, i/3 for radiation and - i for dark energy. The trace o f the energy-m om entum tensor is:

T = ^p i,o (3 W i - i)a (t)-3(1+Wi). (6.8) i

Since V

m

T

mv

= 0, the density of matter p is equal to pm,0a-3(1+w). For the case of dust, we get p = pm,0a-3 and for the case of radiation p — pm,0a . 4

We assume that matter has the form o f dust and dark energy is described by the cosm ological constant A, so the trace of the ten­

sor energy-m om entum T is pm,0a-3 + A. In consequence, Eq. (6.6) gives the relation between the Ricci scalar R and the scale factor a:

R = pm ,0 a-3 + 4A. (6.9)

In the Palatini formalism in the Jordan frame for the FRW metric, we get the Friedmann equation from Eq. (6.3):

H - b2 f o m „-3 + — ^ (K - 3)(K + i) H0 = (b + d)2 P ’ + ’ 2b

o a — 4

+(Hm0a-3 + Hao) + - i,0— + Hk , (6.10) b

where Hk = - —Ofl, -r,o = 3HH-, -m,o = PH|, -a,o = 3Hf} , - y = 3yH2, K = ^ s j ■ b = / ' ( R) = i + 2 - y (- m.ca—3 + W o ) , d = H I = - 2 - y(Hm0a—3 + —A,o)(3 - K ), Ho is the Hubble constant, pr;0 is the present value o f the energy density o f radiation and pm 0 is the present value of the density of matter.

As V

m

T

mv

= 0, we get the following continuity equation:

pm = -3 H p m. (6.11)

When

y

is zero, then the m odel is equivalent to the ACDM model.

6.2 Palatini formalism in Einstein frame

This section is based on Eur.Phys.J. C77 (2017) no. 6, 406 1611, Eur.Phys.J. C77 (2017) no. 9, 603 1621, Eur.Phys.J. C78 (2018) no. 3, 2 4 9 1631, and Phys.Rev. D97(2018) 1035241641.

In the Einstein frame, if f (R) = 0 the action ( 6 ^. 1 ) is equivalent to the Palatini gravitational action [ 68 ^]:

S(guv, ^ , X) = 2 y ' d 4 x V ^ ( f ' (X)(RR - X) + f w ) + Sm(gMv, ^ ).

2 ( 6 . 12 )

Now we can introduce a new scalar field $ = f'(x ), where x = R.

In this case action (6.12) is given by:

S (g„v, r j „ , $) = 1 / d 4 x V = g ($ R - U ($ )) + , »«. (6.13)

Here, the function U($) is a potential o f the form:

Uf ($) = U ($) = x ($ )$ - f (x($)), (6.14) where $ = and R = x = .

After varying the action (6.13) with respect of the metric guv and the connection r, we get

$ ^ Ruv — 2 g^vR^ + 2 g^vU($ ) — Tuv = 0, (6.15)

V« (V =g$gMv ) = 0 . (6.16)

From Eq. (6.16), we get the connection r for the new metric guv = $guv. A new structural equation can be obtained from the trace of Eq. (6.15):

2U($) - U '($ )$ = T. (6.17) l_et -R^v — Rßv, R — /?^v — $ R and guvR — guvR. Then Eq. (6.15) is:

RMv — 2 guv R = T/uv — 2 ^{guv (6.18)} where U7(0) = U(0 ) / $ 2 and Tmv = $ - 1 TjUv. Because R = x =

then

$rR - ( $ 2 " ( $ ) ) ' = 0. (6.19)

From Eq. (6.18), we get a new structural equation:

$ U '($ ) + T — 0 ^{. (} 6 ^. 2 0 ⁾ In this parametrization, the action (6.13) has the following form:

S (0.V, $) = 2 k / d W = I (R - U ($)) + Sm($-1gMv ,^ ), ( 6 . 21 ) where

9 A

= — — — Sm = (a + p)uMuv + PgMV = $ - 3 T^v , ( 6 . 22 ) V g

and = $ - 1 uM, p = $ - 2 p, p = $ - 2 p, 7)^ = $ -1T^v, T = $ -2T [69, 70].

As we use a new metric , the FRW line elem ent has a new form:

ds 2 = d? — a 2 (t) [dr 2 + r 2 (d0 2 + sin 2 0d02)] , (6.23) where the new cosm ological time d" = $ (t )1 dt and the new scale factor a( 0 ^{) = $ (} 0)1 ^{a (a).}

We assume the barotropic matter (p = wp). Accordingly, the cosm ological equations are:

3H 2 = /9$ + pm, (6.24)

6 à = 2p$ — pm (1 + 3w), (6.25) a

where p$ = 2U ($),pm = p 0 a-3(1+w )$ 1 (3 w-1) and w = pm/p m. In this case, the conservation equation is:

pm + 3Hpm(1 + w) = —p$. (6.26) The Starobinsky m odel ( f (R) = R +

y

R) in cosm ology yields the potential U in the form:

U ($) = ( ^ + 2 ^{A^} -1 ^{- + ^ . (6.27)}

\4 y J $ 2 2 ^{y $ 4y}

From Eq. (6.20), we can obtain the scalar field $(a) as:

$(a) = 1 ^— 8 ^{yA +} 2 ^{Ypm +} 8 ^{yP“ (6.28)} or

$(a) = 1 - 8 yA + (2yp„ + 8 Ypm)$ 2 (a). (6.29)

Ultimately, $ is dependent on pm:

$(a) = 1 + V 1 - 8Y(pm + 4pm)(1 - 8YAI (6.30) 4Y(pm + 4pm )

or ______________________

$ (a) = 1 - \ / 1 - 8

y

(pm + 4pm)(1 - 8

y

A) (6 31) 4Y(/m + 4p^m)

We can obtain the Friedmann equation in form 3H(R)2 from Eqs (6.20) and (6.24), getting:

3H(R)2 = Pm(R) + M + A = R(2 + Y

j

R )2 - 3A. (6.32) 2 2 (1 + 2yR)

6.3 Starobinsky cosmological model in Palatini formalism: dynamical system approach

This section is based on Eur.Phys.J. C77 (2017) no. 6, 4 0 6 1611.

In the paper [61], we consider singularities that can appear in the Starobinsky cosm ological m odel in the Palatini formalism. Inves­

tigating it in the Einstein frame, we found inflation in the m odel when matter is negligible in comparison to = U and the value o f

y

parameter is close to zero. Moreover, when number of e-folds is equal to 60, then the value of

y

parameter is 1.16 x 10 _69 s2.

We investigate also singularities in the Jordan frame, introduc­

ing the classification of singularities in FRW cosm ology and reduc­

ing dynamics to the dynamical system of the Newtonian type. This classification is given in terms of the geom etry of a potential V(a) if this potential has a pole.

In the standard cosmology, the potential V(a) is expressed by the following equation:

a2 = - 2V (a), (6.33)

where V(a) = - . In consequence, we obtain that:

a = - ^ . (6.34)

da

This leads to the following dynamical system:

• dV (a)

x = --- -— . (6.36)

da

In our model, Eq. (6.10) can be rewritten analogically as a dy­

namical system (6.35)-(6.36):

a' = x, (6.37)

x' = - ^ , (6.38)

da

where v = - a2 ( q 7 Q2h(K-32bK+1) + Qch + Qfc) and ' = - i = H-b d t is a new parametrization of time.

We treat the above dynamical system as a sewn dynamical sys­

tem [71, 72]. Accordingly, we consider two cases. The first one is for a < asing and the second one is for a > asing, where asing is the value of the scale factor in the singularity.

For a < asing, the dynamical system (6.37)-(6.38) can be rewrit­

ten as:

a' = x, (6.39)

x' = - ^V l<à ) , (6.40) da

where V = V (-n (a - asing) + 1) and n(a) denotes the Heaviside function.

For a > asing, we get:

a' = x, (6.41)

x = - ^ V ^ , (6.42)

da where V 2 = Vn(a - asing).

In the Starobinsky cosm ological m odel in Palatini formalism in the Jordan frame, we found two new types of singularities of a fi­

nite scale factor. The first type is the sewn freeze singularity, for which the Hubble function H , pressure p and energy density p are divergent. It appears when

y

parameter has a positive value. The second type is the sewn typical singularity, for which the Hubble function and energy density p are finite and H and pressure p are divergent. It appears when

y

parameter has a negative value. At the sewn singularity which is of a finite scale factor type, the singu­

larity in the past meets the singularity in the future. In the Jordan frame, the phase portrait is topologically equivalent to the phase portrait of the AODM m odel for the positive

y

parameter.

a = x, (6.35)

In order to estimate this m odel through statistical analysis, we used 580 supernovae o f type Ia [13], BAO [14, 15, 16], measure­

ments of H(z) for galaxies [17, 18, 19], Alcock-Paczyński test [22, 23, 24, 25, 26, 27, 28, 29, 3 0 ], measurements of CMB and lensing by Planck, and low i by WMAP [31] finding that the best fit value o f —Y = 3

y

—2 is 9.70

x

i0 —11. The BIC criterion gives a strong evi­

dence in favour o f the ACDM m odel in comparison to this model.

However, we are not able to reject it.

6.4 Extended Starobinsky cosmological model in Palatini formalism

This section is based on Eur.Phys.J. C77 (2017) no. 9, 6 0 3 1621.

In the paper [62], we consider the FRW cosm ological m odel for / (R) = R + yR 2 + £R 3 gravity within the Jordan and Einstein frame in the Palatini formalism. We investigate singularities in this m odel and demonstrate how the Starobinsky m odel is modified by adding a new term in / (R) formula.

By adding of ^R 3 in / (R) expression in the Jordan frame case, the Friedmann formula (6.10) is modified as follows:

H 2 ^{_ b} 2 f —R IV, ,n ,n , H = p f f X [26 p (—R - 4—,ot )

+ 2 — 5 —r ( —r - 3—tot)) + —tot + —k , (6.43)

where

—tot = —m,0a 3 + — A,o, (6.44) b = / '(R) = i + —r [ 2 —y + 3 — 5 —r] , (6.45) d = H t = 6 6w r [- r ( — - —5—R) - 4 - a-o] ■

—Y = 3

y

H2 , (6.47)

—5 = 9£H0 4, (6.48)

-

r

= J * • œ 49)

In the case of the Einstein frame when we insert SR3 in f (R) formula, the potential function (6.27) is substituted by:

U (R) = R 2(

y

+ 2SR) 2. (6.50) ( 1 + 2yR + 3SR2)

In consequence, the Friedmann equation is modified to the form:

3H2 = pm (R) + + A = --- RR(2 + yR)---2 - 3A. (6.51) 2 2 ( 1 + 2yR + 3SR2)

A major qualitative change in the m odel occurs after inserting

A major qualitative change in the m odel occurs after inserting

W dokumencie Cosmological models with running cosmological constant (Stron 33-0)