Diffusive DM-DE interaction: non-relativistic case and

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 SR3 into the f (R) formula in the Jordan frame. In this case, some additional singularities appear in the model. For example, in the case when

y

parameter is positive and S parameter is negative, an additional sewn freeze singularity and a typical sudden singularity appear during the evolution of the Universe.

6.5 Inflation in Starobinsky cosmological model in Palatini formalism

This section is based on Eur.Phys.J. C78 (2018) no. 3, 2 4 9 1631.

The main aim o f the paper [63] is the analysis of inflation in the Starobinsky cosm ological m odel in Palatini formalism within the Einstein frame. We found that inflation appears when matter is negligible with comparison to the = U. The evolution o f the Universe during inflation in this m odel consists of four phases:

• In the first phase, matter is negligible and the density of matter grows due to the interaction between matter and the dark en­

ergy. In the inflation process, the production of matter disturbs inflation beginning from the point when matter can no longer be neglected. In consequence, in the first phase inflation be­

comes unstable and the second phase sets in.

• During the second phase, the effects of matter are not negligi­

ble and the density of matter grows further.

• In the third phase, the density of matter decreases but is still

not negligible. During the second and third phases the process

of inflationary behaviour of the Universe is terminated.

• In the fourth phase, the effects of matter become negligible and so inflation reappears. During that phase, the Universe follows the AO D M model.

In the first and fourth phase p $ has constant values:

_ 1 - 16ya + —1 - 32ya (6 p$ = --- 77 --- (6.52)

16

y

in the first phase and

_ 1 - 16 y A - —1 - 32y A (6C3)

p$ = --- 77 --- (6.53)

16

y

in the last phase.

If w e assume that N e (50, 60) [73], then y parameter belongs to the interval (1.16 x 10-69 , 1.67 x 10-69 ).

6.6 Einstein frame vs Jordan frame

This section is based on Phys.Rev. D97 (2018) 1035241641.

In the paper [64], w e consider differences in the Einstein and Jor­

dan frames as applied to the Starobinsky cosmological model in Palatini formalism, finding that the topological structures of the phase space depend on the choice of the frame.

In the case of the Einstein frame, H and R were chosen as vari­

ables of the dynamical system. Eqs (6.24) and (6.26) can be then rewritten as a dynamical system:

H(t) = --- 1 . _ 6 ( 1 + 2 y R (£))2

( 6 A - 6 H(t ) 2(1 + 2 YR(t ) ) 2 + R(i ) ( - 1 + 24yA + y (1 + 24yA)Ê(î))) , (6.54)

R = 3 H (t ) ( 1 + 2 YiR(t)) (1 - YR(t))

^4A + R(f) ^ - 1 + 16yA + 16Y2AR(f)jj , (6.55) where ' = 4 .

at

In the Jordan frame case, Eqs (6.10) and (6.11) can be rewritten

as

H « = - 6 6 (2A + H (t )2 ) + R M + 18(1|+ 8;^)<A H/.f}

6 v - 1 - 12;A + ;R(t)

- 18(1+ 8; a ) h (t)2 (656) 1 + 2 ;R(t) J , ' RR(t) = -3H(t)(R(t) - 4A), (6.57) where ' = J . dt

In this paper, we consider the behaviour of the trajectories in the phase portrait for both the frames when ; parameter has a positive value and find that the different types of singularities appear in the models. In the Jordan frame, the sewn freeze singularity appears beyond the Big Bang, while in the Einstein frame, the freeze sin­

gularity is substituted by the generalized sudden singularity and there is a bounce instead of the Big Bang. In consequence, the models in both the frames are not qualitatively equivalent to the A CDM model. The main result of this paper consists in showing that the models in the Jordan and Einstein frame are not equiva­

lent due to the different types o f singularities included in them.

Chapter 7

Conclusions

The main aim of the thesis is to point out that the running cos­

moLogicaL constant can soLve two major probLems of present cos­

moLogy. We consider many types of cosmoLogicaL modeLs, such as modeLs with f ( R) gravity, diffusion in the dark sector, decaying dark energy, and with the different parametrization of the density o f dark energy, in an attem pt to address the cosmoLogicaL constant probLem and the coincidence probLem.

The concLusions reLated to decay of metastable dark energy are:

• From statisticaL anaLysis, we get that the present vaLue of den­

sity dark energy is independent of the modeL parameters a and E 0 .

• This modeL provides the mechanism of jum ping from the ini­

tiaL vaLue of dark energy E 0 = 10120 to the present vaLue of the cosmoLogicaL constant.

• The osciLLation of dark energy density occurs for 0 < a < 0.4.

• In the radioactive-Like decay modeL o f dark energy, for the Late­

tim e Universe (t = 2T0), there are three different forms of de­

caying: radioactive, damping osciLLating, and power-Law type.

• In the radioactive-Like decay modeL o f dark energy, this type of decay dominates to 2.2 x 10 4 T0.

• In the radioactive-Like decay modeL of dark energy, after the radioactive type of decay the damping osciLLating type sets in, which Later is repLaced by the power-Law decay ( 1 / i 2).

• In the modeL o f 1 ^{/ i} 2 type o f decay of dark energy, there is in­

teraction in the dark sector, which modifies the standard scaLe Law of dark matter: pdm = Pdm, 0 a (i)-3+A(t).

• For the earLy Universe, the A(t) function can be regarded as a

constant.

• In the modeL of 1 / i 2 type o f decay of dark energy, statisticaL anaLysis favours the negative vaLue of a 2 parameter. In resuLt, we get the decay of particLes of dark matter.

The concLusions reLated to the diffusion dark matter-dark en­

ergy interaction model are:

• In this modeL, the standard scaLe Law of energy density of dark matter is modified to pdm = p dm,0 a (i)- 3 + Yia(t)- 3 .

• This modeL is free from the difficuLties present in ALho et aL.'s modeLs with diffusion [53], since there are not any non-physicaL trajectories crossing the boundary set pm = 0 .

• This modeL invoLves a mechanism soLving the coincidence probLem.

The concLusions of this thesis reLated to the dynamical system

The concLusions of this thesis reLated to the dynamical system

W dokumencie Cosmological models with running cosmological constant (Stron 39-0)