Extended Starobinsky cosm ological m odel in Palatini

6 Starobinsky cosmological model in Palatini formalism 37

6.4 Extended Starobinsky cosm ological m odel in Palatini

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

H2 , (6.47)

—5 = 9£H0 4, (6.48)

-

= 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(

+ 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

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 in the first phase and

_ 1 - 16 y A - —1 - 32y A (6C3)

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

16 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 .

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 approach to the running A are:

• In A (H ) cosmoLogy, the ALcaniz and Lima's soLution represents the scaLing type pA(a) ~ pm(a) [55].

• Trajectories within the phase space for which pA(a) ~ pm(a) rep

resent scaLing soLutions, which couLd soLve the cosmic coinci

dence probLem.

• The non-covariant A(a) parametrization can be obtained from the covariant action for the scaLar fieLd as an emergent parame- trization.

• We have found a strong evidence for tuning A term in A(a) cos

moLogy: n A> 0 < 3.19 x 10 - 7 .

The concLusions reLated to the Starobinsky cosmological model in the Palatini formalism are:

• In the Einstein frame, there occurs interaction between dark matter and dark energy, as contrary to the Jordan frame.

• In the Jordan frame, new types o f singuLarities appear, such as a sewn freeze singuLarity for the positive vaLue of

and a sewn typicaL sudden singuLarity for the negative vaLue o f

.

• In the Jordan frame, the phase portrait is topoLogicaLLy equiva

Lent to the phase portrait of the ACDM modeL for the positive

parameter.

• In the Einstein frame, in the case when matter is negLigibLe as compared to dark energy, inflation sets in when modeL param

eter y is cLose to zero (y ~ 1.16 x 10 - 69 s 2 ).

• In the Einstein frame, for the positive vaLue o f

, there is a gen

eraLized sudden singuLarity instead of the Big Bang.

• The phase portraits in the Einstein frame and the Jordan frame are not equivaLent, which Leads to the Lack of physicaL equiva

Lence of the modeL considered within these frames.

• The extension of Starobinsky modeL f (R) = R + yR 2 + £R 3 in the Jordan frame can generate an additionaL sewn freeze singuLar

ity and a typicaL sudden singuLarity instead o f the Big Bang.

One interesting phenomenon to appear often in the modeLs considered in the thesis, is the interaction between dark energy and dark matter, which can be treated as an energy transfer in the dark sector. Its obvious effect is a modification o f the standard scaL

ing Law of the energy density of dark matter (pdm(i) = pdm, 0 a(£)-3).

From observations, in the modeL of 1 / t 2 type of decay of dark en

ergy, we have a transfer energy from the dark matter to dark en

ergy sector. Such a mechanism is at work in the Starobinsky modeL in the PaLatini formaLism within the Einstein frame, the diffusion dark matter-dark energy interaction modeL as weLL as in the de

caying dark energy modeL.

A pLausibLe soLution o f the probLem of the cosmoLogicaL con

stant is the modeL with decaying metastabLe dark energy, which offers the mechanism decreasing the vaLue of dark energy in the earLy Universe (pde = 10 120) to its present vaLue. In this case, the transition consists in an onset o f osciLLation behaviour of the den

sity of dark energy.

The phenomenon o f inflation in the evoLution o f the Universe is given by the Starobinsky modeL in the PaLatini formaLism within the Einstein frame, which determines inflation when matter is negLigi

bLe as compared to the density of dark energy. Its characteristic feature is the creation o f matter throughout process o f inflation.

The statisticaL anaLysis indicates that none of the three hypothe

ses put forward in the Introduction cannot be rejected. Accord

ingLy, whiLe abiding by the vaLidity o f the main thesis of this dis

sertation (see Introduction), we are not in a position to teLL deci

siveLy which of the mechanisms considered here actuaLLy under

Lies changeabiLity of the cosmoLogicaL constant.

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Appendix

The Appendix contains aLL the eLeven journaL articLes that have con

tributed to this dissertation in the respective formats of thejournaLs

in which they originaLLy appeared.

Cosmology with decaying cosmological constant — exact solutions and model testing

M arek Szydłowski 0’6 and Aleksander Stachowskia aAstronomical Observatory, Jagiellonian University.

Orla 171, 30-244 Krakow, Poland

bMark Kac Complex Systems Research Centre, Jagiellonian University, ul. Lojasiewicza 11, 30-348 Krakow, Poland

E-mail: marek.szydlowski@uj.edu.pl, aleksander.stachowski@uj.edu.pl Received July 9, 2015

Revised September 25, 2015 Accepted October 9, 2015 Published October 30, 2015

Abstract. We study dynamics of A(t) cosmological models which are a natural generalization

of the standard cosmological model (the ACDM model). We consider a class of models: the

ones with a prescribed form of A(t) = Abare + a?. This type of a A(t) parametrization 2

is motivated by different cosmological approaches. We interpret the model with running

Lambda (A(t)) as a special model of an interacting cosmology with the interaction term

-dA (t)/dt in which energy transfer is between dark matter and dark energy sectors. For the

A(t) cosmology with a prescribed form of A(t) we have found the exact solution in the form

of Bessel functions. Our model shows that fractional density of dark energy Qe is constant

and close to zero during the early evolution of the universe.

Abstract. We study dynamics of A(t) cosmological models which are a natural generalization

of the standard cosmological model (the ACDM model). We consider a class of models: the

ones with a prescribed form of A(t) = Abare + a?. This type of a A(t) parametrization 2

is motivated by different cosmological approaches. We interpret the model with running

Lambda (A(t)) as a special model of an interacting cosmology with the interaction term

-dA (t)/dt in which energy transfer is between dark matter and dark energy sectors. For the

A(t) cosmology with a prescribed form of A(t) we have found the exact solution in the form

of Bessel functions. Our model shows that fractional density of dark energy Qe is constant

and close to zero during the early evolution of the universe.

W dokumencie Cosmological models with running cosmological constant (Stron 55-200)

Extended Starobinsky cosm ological m odel in Palatini

6 Starobinsky cosmological model in Palatini formalism 37

6.4 Extended Starobinsky cosm ological m odel in Palatini

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

H2 , (6.47)

—5 = 9£H0 4, (6.48)

-

= 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(

+ 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

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

• 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

in the first phase and

_ 1 - 16 y A - —1 - 32y A (6C3)

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

16

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

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 .

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

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

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.

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

• 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

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

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

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

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

moLogicaL constant can soLve two major probLems of present cos

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

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

• 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

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

The concLusions reLated to the diffusion dark matter-dark en

• Trajectories within the phase space for which pA(a) ~ pm(a) rep

resent scaLing soLutions, which couLd soLve the cosmic coinci

• We have found a strong evidence for tuning A term in A(a) cos

• In the Jordan frame, the phase portrait is topoLogicaLLy equiva

• In the Einstein frame, in the case when matter is negLigibLe as compared to dark energy, inflation sets in when modeL param

, there is a gen

• The phase portraits in the Einstein frame and the Jordan frame are not equivaLent, which Leads to the Lack of physicaL equiva

• The extension of Starobinsky modeL f (R) = R + yR 2 + £R 3 in the Jordan frame can generate an additionaL sewn freeze singuLar

One interesting phenomenon to appear often in the modeLs considered in the thesis, is the interaction between dark energy and dark matter, which can be treated as an energy transfer in the dark sector. Its obvious effect is a modification o f the standard scaL

From observations, in the modeL of 1 / t 2 type of decay of dark en

ergy, we have a transfer energy from the dark matter to dark en

ergy sector. Such a mechanism is at work in the Starobinsky modeL in the PaLatini formaLism within the Einstein frame, the diffusion dark matter-dark energy interaction modeL as weLL as in the de

A pLausibLe soLution o f the probLem of the cosmoLogicaL con

stant is the modeL with decaying metastabLe dark energy, which offers the mechanism decreasing the vaLue of dark energy in the earLy Universe (pde = 10 120) to its present vaLue. In this case, the transition consists in an onset o f osciLLation behaviour of the den

The phenomenon o f inflation in the evoLution o f the Universe is given by the Starobinsky modeL in the PaLatini formaLism within the Einstein frame, which determines inflation when matter is negLigi

The statisticaL anaLysis indicates that none of the three hypothe

ses put forward in the Introduction cannot be rejected. Accord

ingLy, whiLe abiding by the vaLidity o f the main thesis of this dis

sertation (see Introduction), we are not in a position to teLL deci

siveLy which of the mechanisms considered here actuaLLy under

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