D o c t o r a l T h e s is
Cosmological models with running cosmological
constant
Author:
Aleksander Stachowski Supervisor:
Prof. dr hab. Marek Szydłowski
A thesis subm itted in fu lfillm e n t o f the requirements fo r the degree o f D octor o f Philosophy
in the
Faculty of Physics, Astronomy and Applied Computer Science
October 2018
W ydział Fizyki, Astronomii i Informatyki Stosowanej Uniwersytet Jagielloński
Oświadczenie
Ja niżej podpisany Aleksander Stachowski (nr indeksu: 1016516) doktorant W ydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiellońskiego oświadczam, że przedłożona przeze mnie rozprawa doktorska pt. „Cosmological models with running cosmological constant" je st oryginalna i przedstawia wyniki badań wykonanych przeze mnie osobiście, pod kierunkiem prof. dr. hab.
Marka Szydłowskiego. Pracę napisałem samodzielnie.
Oświadczam, że moja rozprawa doktorska została opracowana zgodnie z Ustawą o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z później
szymi zmianami).
Jestem świadom, że niezgodność niniejszego oświadczenia z prawdą ujawniona w dowolnym czasie, niezależnie od skutków prawnych wynikających z ww. ustawy, może spowodować unie
ważnienie stopnia nabytego na podstawie tej rozprawy.
Kraków, dnia
Abstract
The thesis undertakes an attem pt to solve the problems of cos
m ological constant as w ell as o f coincidence in the models in which dark energy is described by a running cosm ological con
stant. Three reasons to possibly underlie the constant's change
ability are considered: dark energy's decay, diffusion between dark energy and dark matter, and modified gravity. It aims also to pro
vide a parametric form of density of dark energy for the models that involve a running cosm ological constant, which would de
scribe inftation in the early Universe.
The principal method used in my investigations was the dy
namical analysis. The cosm ological equations were accordingly recast as a dynamical system, which enabled me to draw up the relevant phase portrait, much useful in considering the possible evolutionary paths o f the Universe.
The cosm ological models were estimated by taking into ac
count a number o f astronomical data, such as observations of type la supernovae, cosmic microwave background, baryon acoustic oscillations, measurements of the Hubble function for galaxies, and the Alcock-Paczynski test.
The results of my investigations have been published in eleven papers.
Keywords : cosmology, dark energy, dark matter
Streszczenie
Rozprawa podejmuje próbę rozwiązania problemu stałej kos
mologicznej i problemu koincydencji w modelach kosmologicz
nych, gdzie ciemna energia je st opisywana zmienną stałą kosmo
logiczną. Są rozważane trzy możliwe przyczyny zmienności stałej kosmologicznej: rozpadająca się ciemna energia, dyfuzja pomię
dzy ciemną materią a ciemną energią oraz zmodyfikowana gra
witacja. Celem jest także wprowadzenie parametryzacji gęstości ciemnej energii w podejściu ze zmienną stałą kosmologiczną, która opisuje inftację we wczesnym Wszechświecie.
Najważniejszą metodą użytą w tych badaniach była analiza dy
namiczna. Równania kosmologiczne były przepisane do postaci układu dynamicznego, który umożliwiał narysowanie odpow ied
niego portretu fazowego przydatnego przy badaniu możliwych ścieżek ewolucji Wszechświata.
Modele kosmologiczne były estymowane z uwzględnieniem obserwacji astronomicznych takich jak: obserwacje supernowych typu la, mikrofalowego promieniowania tła, barionowych oscylacji akustycznych, pomiarów wartości funkcji Hubble'a dla galaktyk i testu Alcocka-Paczyńskiego.
Wyniki badań zostały zamieszczone w jedenastu opublikowa
nych pracach.
Słowa kluczowe : kosmologia, ciemna energia, ciemna materia
This dissertation has been based on the scientific results previ
ously reported in the following articles:
• Marek Szydłowski, Aleksander Stachowski, Cosmology with decaying cosmological constant - exact solutions and model testing, JCAP 1510 (2015) no. 10, 066,
doi:10.1088/1475-7516/2015/10/0 66.
• Marek Szydłowski, Aleksander Stachowski, Cosmological models with running cosmological term and decaying dark m at
ter, Phys. Dark Univ. 15 (2017) 96-104, doi:10.1016/j.dark.2017.01.002.
• Aleksander Stachowski, Marek Szydłowski, Krzysztof Urba
nowski, Cosmological implications o f the transition from the false vacuum to the true vacuum state, Eur. Phys. J. C77 (2017) no. 6 , 357,
doi:10.1140/epjc/s10052-017-4 934-2.
• Marek Szydłowski, Aleksander Stachowski, Krzysztof Urba
nowski, Quantum mechanical look at the radioactive-like decay o f metastable dark energy, Eur. Phys. J. C77 (2017) no. 12, 902, doi:10.1140/epjc/s10052-017-5471-8.
• Zbigniew Haba, Aleksander Stachowski, Marek Szydłowski, Dynamics o f the difusive DM-DE interaction - Dynamical system approach, JCAP 1607 (2016) no. 07, 024,
doi:10.1088/1475-7516/2016/07/024.
• Marek Szydłowski, Aleksander Stachowski, Does the difusion dark matter-dark energy interaction model solve cosmological puzzles?, Phys. Rev. D94 (2016) no. 4, 043521,
doi:10.1103/PhysRevD.94.043521.
• Aleksander Stachowski, Marek Szydłowski, Dynamical system approach to running A cosmological models, Eur. Phys. J. C76 (2016) no. 11 , 606,
doi:10.1140/epjc/s10052-016-4439-4.
• Aleksander Stachowski, Marek Szydłowski, Andrzej Borowiec, Starobinsky cosmological model in Palatini form alism , Eur.
Phys. J. C77 (2017) no. 6 , 406,
doi:10.1140/epjc/s10052-017-4981-8.
• Marek Szydłowski, Aleksander Stachowski, Andrzej Borowiec, Emergence o f running dark energy from polynomial f(R) theory in Palatini form alism , Eur. Phys. J. C77 (2017) no. 9, 603,
doi:10.1140/epjc/s10052-017-5181-2.
• Marek Szydłowski, Aleksander Stachowski, Simple cosmologi
cal model with inûation and late times acceleration, Eur. Phys. J.
C78 (2018) no. 3, 249,
doi:10.1140/epjc/s10052-018-5722-3.
• Marek Szydłowski, Aleksander Stachowski, Polynomial f (R) Palatini cosmology: Dynamical system approach, Phys. Rev. D97 (2018) 103524,
doi:10.1103/PhysRevD.97.103524.
Acknowledgements
I am most grateful to my Supervisor, Prof. Marek Szydłowski, for his help and support during preparation of this thesis as w ell as
careful supervision over the past years.
I would like to thank Adam Krawiec, Marek Krośniak, and Aleksander Kurek for remarks, comments and stimulating
discussion.
Contents
1 Introduction 3
2 Statistical analysis of cosmological models 7 3 Cosmology with decay of metastable dark energy 11
3.1 Decay of metastable dark energy from quantum vac
uum ... 11
3.2 Late-time approximation of decaying metastable dark energy ... 13
3.3 Model te s tin g ... 16
3.4 Modified scaling law of matter d e n s ity ... 17
3.5 Cosmological implications of transition from false to true vacuum state ... 18
3.6 Radioactive-like decay of metastable dark energy . . 20
3.7 Main results ... 21
4 Diffusion dark matter-dark energy interaction model 23 4.1 Relativistic diffusion interacting o f dark matter with dark energy ... 23
4.2 Diffusive DM-DE interaction: coincidence problem . . 25
4.3 Diffusive DM-DE interaction: non-relativistic case and statistical analysis ... 27
5 Dynamical system approach to running A cosmological models 29 5.1 A (H )C D M c o s m o lo g ie s ... 29
5.2 A(R)CDM c o s m o lo g ie s ... 31
5.3 Non-canonical scalar field c o s m o lo g y ... 32
5.4 Cosmology with emergent A (a ) relation ... 33
6 Starobinsky cosmological model in Palatini formalism 37 6.1 Palatini formalism in Jordan frame ... 37
6.2 Palatini formalism in Einstein frame ... 39
6.3 Starobinsky cosm ological m odel in Palatini form al ism: dynamical system approach ... 41
6.4 Extended Starobinsky cosm ological m odel in Palatini
formalism ... 43
6.5 Inftation in Starobinsky cosm ological m odel in Pala
tini fo rm a lis m ... 44 6.6 Einstein frame vs Jordan frame ... 45
7 Conclusions 47
Bibliography 51
Appendix 59
A.1 Cosmology with decaying cosm ological constant -
exact solutions and m odel testing ... 61 A.2 Cosmological models with running cosm ologicalterm
and decaying dark matter ... 85 A.3 Cosmological implications of the transition from the
false vacuum to the true vacuum s t a t e ... 95 A.4 Quantum mechanical look at the radioactive-like de
cay of metastable dark e n e rg y ...109 A.5 Dynamics of the diffusive DM-DE interaction - Dy
namical system a p p ro a c h ...123 A .6 Does the diffusion dark matter-dark energy interac
tion m odel solve cosm ological p u z z le s ? ...147 A.7 Dynamical system approach to running A cosm olog
ical models ... 159 A .8 Starobinsky cosm ological m odel in Palatini formalism 181 A.9 Emergence of running dark energy from polynomial
f (R) theory in Palatini formalism ...199 A.10 Simple cosm ological m odel with inftation and late
times acceleration ... 209 A.11 Polynomial f (R) Palatini cosmology: Dynamical sys
tem a p p ro a ch ...219
Chapter 1
Introduction
The main aim of the thesis is to address the following question: can the cosmological constant problem and the coincidence prob
lem be solved? In the investigations, it is assumed that the cos
m ological constant evolves in time.
The cosm ological constant problem is a consequence of inter
preting dark energy as a vacuum energy. The presently observed value of the constant is by 120 orders of magnitude smaller than the value resulting from quantum physics [ 1 ].
The coincidence problem [2] consists in finding an explanation why the cosm ological constant has the same order o f magnitude as the energy density of matter today.
Another goal o f the dissertation is to provide a description of inftation in the early Universe involving an assumption of the running cosmological constant.
For the purpose of proving the main thesis of my dissertation that the running cosmological constant is actually able to ex
plain the cosmological problems, three hypotheses concerning the cosm ological constant's changebility are put forward:
• it is a consequence of the decay of metastable dark energy,
• it is a result of diffusion interaction of dark energy with dark mat
ter,
• it is an intrinsic attribute of dark energy in the Starobinsky m odel using the Palatini formalism.
We neglected the inftuence o f radiation on the evolution of the Universe. The baryonic matter is treated as dust (the equation of state is pm = 0 , where pm is the pressure of matter) in our pa
pers. Throughout the thesis, dark matter is generally treated as
dust too. It was assumed that dark energy has the equation of
state pde = - p de, where pde is the pressure and pde is the energy
density. In most cases, we do not take into account the curvature
effect in the cosm ological equations. In the thesis, the following
convention is used: c = 8nG = 1 .
One of the models investigated is one with a decaying m eta
stable dark energy. The vacuum energy decay was considered in the following papers: [3, 4, 5, 6 , 7, 8 ]. It is assumed that the process of dark energy decay is a transition the false vacuum state to the true vacuum. The Fock-Krylov theory of quantum unstable states is applied here [5]. Next, the Breit-Wigner energy distribution func
tion is used for the m odel of the quantum unstable systems [9]. In this context, we examined the radioactive-like m odel of the decay of the false vacuum. The late-tim e approximation of this m odel are also considered (A(t) = Abare + ^2). It assumes an interaction in the dark sector (i.e. dark matter and dark energy). The models are analysed also by the statistical analysis methods.
Another m odel investigated in the thesis is one with a diffusion between dark matter and dark energy. The natural result o f this interaction is a modification o f the standard scale law o f the dark matter energy density. This m odel is also statistically analysed us
ing astronomical data.
The third one is the Starobinsky cosm ological m odel in the Palatini formalism, which we examined in both the Jordan and Ein
stein frames, looking for differences between them by the dynam ical methods. In the case of the Einstein frame, the m odel belongs to the class involving an interaction between dark matter and dark energy. The special point o f these investigations is to search for singularities within the models, while the models' fitting is done through statistical analysis.
The important method o f investigation of the evolution o f the Universe consists in making conclusions on the basis o f phase por
traits. Accordingly, we recast cosm ological equations into the form of the dynamical system, which allows for drawing the phase por
trait. The analysis of trajectories that represent the particular paths of the evolution of the Universe as w ell as critical points gives us the most interesting scenarios o f the evolution o f the Universe.
Such a method can be helpful in solving of the problems of the cosm ological constant and of coincidence.
The methods of dynamical analysis are used in most of my pa
pers, while my main article on dynamical systems in cosm ology is Eur.Phys.J. C76 (2016) no. 11, 606 [10], dealing with the dynamics of cosm ological models in the different parametrization of the run
ning cosm ological constant. In this paper, five classes o f models are investigated:
• A(H)CDM, where H is the Hubble constant,
• A(a), where a is the scale factor,
• A(R)CDM, where R is the Ricci scalar,
• A(0) with diffusion, where 0 is a scalar field,
• A (X ), where X = 2gaßV aVß0 is a kinetic part of density o f the scalar field.
The structure of the thesis is as follows. The statistical analysis used in my papers is described in Chap. 2. In Chap. 3, models with decaying dark energy and my papers pertaining to this subject are contemplated. In Chap. 4 , the m odel with diffusion in the dark sec
tor and my papers about this m odel are considered. In Chap. 5,
my paper Eur.Phys.J. C76 (2016) no. 11, 606 [10] is discussed. The
Starobinsky cosm ological m odel in the Palatini formalism and my
papers on this m odel are discussed in Chap. 6 . The conclusions of
the thesis are provided in Chap. 7.
Chapter 2
Statistical analysis of cosmological models
The cosm ological models considered in the thesis are analysed by the statistical methods in order to find the best fit for the values o f parameters and their errors. In the statistical analysis, I used my own CosmoDarkBox script for estimating m odel parameters.
In order to find errors o f the best fit, this code uses the Monte-Carlo methods — the Metropolis-Hastings algorithm [11, 12].
For the purpose of statistical analysis in my papers, I used the following astronomical data:
• supernovae o f type Ia (SNIa; Union 2.1 1 dataset [13]),
• Baryon Acoustic Oscillations (BAO) data from:
- Sloan Digital Sky Survey Release 7 (SDSS DR7 )2 dataset at z = 0.275 [14],
- 6 dF Galaxy Redshift Survey 3 measurements at redshift z = 0.1 [15],
- W iggleZ 4 measurements at redshift z = 0.44,0.60,0.73 [16],
• measurements of the Hubble parameter H(z) of galaxies [17, 18, 19],
• the Alcock-Paczynski test [2 0 , 21] (AP; data from [22, 23, 24, 25, 26, 27, 28, 29, 3 0 ])
• measurements of Cosmic Microwave Background (CMB) and lensing by Planck satellite 5 [31] and low i polarization from WMAP.
The overall likelihood function is expressed by the following formula:
Ltot = ^SNIaLBAoLAP^H (z)LCMB+lensing, (2.1)
1 h ttp ://s u p e rn o v a .lb l.g o v /u n io n / 2 h ttp s://classic.sd ss.o rg /d r7/
3 h ttp ://w w w .6d fg s.n e t
4 h ttp ://w ig g le z .s w in .e d u .a u /s ite /
5 h ttp s ://w w w .e s a.in t/O u r_ A ctiv ities /S p a ce _ S cien c e/P la n c k
where the likelihood functions LsNia, Lbao, Lap, L n{z), LoMB+iensing are for SNIa, BAO, AP, measurements of H (z) and OMB respectively, which are defined in the following way:
LsNia = exp - 2 [A - B 2/C + log(C/(2n))] , (2.2) where A = (pobs- p th)C-1 (pobs- p th), B = C-1(pobs- p th), C = Tr C-1 and C is a covariance matrix for SNIa, pobs is the observed distance modulus and pth is the theoretical distance modulus,
L = exp 1 ( dobs r's(zd ) \ C-1 ( dobs r's(zd ) \ (2 0) Lbao = exp 2 ( d - D V (Z ^j C ( d -
dV (Ż)) J ' (23) where r s(zd) is the sound horizon in the drag epoch [32, 33], 1 /d obs is the observed value of the acoustic-scale distance ratio,
( \ 1/3
D V = ((1 + z2)DA(z)H(zy) , where D A is the angular diameter distance,
r r 1 A ( H (zi )obs - H(zi )th \ 2] (21)
L «(z) = exp - I , (2 )
i=1 V * 7
where a is the measurement error,
r r 1 ^ ( A P (z,)»bs - A P (z*)thV ] (25)
L
a p= exp - a ) , (2 5)
i=1 _
where A P (z)th = HZz) fZ idZ) and AP(zj)obs are observational data and
LcMB+lensing = exp - ^ ( x th - x obs)C-1(xth - x obs) , (2 . 6) where C is the covariance matrix with the errors, x is a vector of the acoustic scale lA, the shift parameter R and Qb0, where lA = r-(z^)c f Z , R ^ ^ m . 0 Ho2 f cz* H(Zö, and Qb0 = 3H2, where z* is the redshift in the recombination epoch [32], r s is the sound hori
zon, pb 0 is the present value of the energy density of baryonic mat
ter, H0 is the present value o f the Hubble function, and Qm0 = PmO, where pm 0 is the present value o f the energy density o f matter. 3H0
In my paper Phys.DarkUniv. 15(2017)96-104, the likelihood func
tion for OMB is given by
r I" 1 ^ (
d$ «
w- D i : i ( / y y 2] (22)
L
omb= exp - 2 0 ---j , (27)
where DTT(i) is the value of the temperature power spectrum of CMB and i is a multipole. The temperature power spectrum is de
termined for i in the interval of (30,2508).
In my analysis o f cosm ological models, I used the Akaike infor
mation criterion (AIC) and the Bayesian information criterion (BIC) [34, 35]:
AIC = — 2 ln L + 2d, (2.8)
BIC = — 2 ln L + d ln n, (2.9)
where L is the value of the likelihood function in the best fit, d is the
number of m odel parameters, and n is the number of data points
involved in the estimation.
Chapter 3
Cosmology with decay of metastable dark energy
3.1 Decay of metastable dark energy from quantum vacuum
This section is based on Eur.Phys.J. C77 (2017) no. 6, 3 5 71361 and Eur.Phys.J. C77 (2017) no. 12, 9 0 2 1371.
The quantum unstable systems are characterized by their survival probability (decay law). The survival probability P (t) of a state |0) of vacuum equals P (t) = |A(t)|2, where A(t) is the probability am pli
tude (A(t) = (0|0(t))) and |0(t)) is the solution of the Schrödinger equation:
d
%h~dt ^ (t)) = H ^ ( t) ) (3 1 ) where H is the Hamiltonian. The am plitude A(t) can be expressed as the following Fourier integral:
A(t) = A (t — * , ) = / w(E) e— h E (t — t0) dE, (3.2)
“ E m in
where w(E) > 0 (see: [5, 6 , 7]).
From the Schrödinger equation (3.1), we can obtain that:
d
%h~dt (^ (t)) = (^ |H|^ (t))- (3.3) This relation gives us the am plitude A(t) as a solution of the fo l
lowing equation:
dA(t)
ih d ^ tZ = h(t) A(t), (3.4) where
h(t) = ^ (3.5)
v ' A(t) A(t) dt
and h(t) is the effective Hamiltonian. In result, we get:
h(t) = E*(t) - 2 r * ( i) , (3.6) where E^(t) = ft [h(t)], /^ (t) = - 2 S [h(t)] are the instantaneous energy (mass) E^(t) and the instantaneous decay rate /^ (t) [38, 39, 4 0 1. We interpret the expression (t) = - 2 S [h(t)] as the decay rate, because it satisfies the definition of the decay rate used in
quantum theory: == - .
From the form of the effective Hamiltonian h(t), we get the fo l
lowing solutions of Eq. (3.4):
A(t) = e- i t h(t) = e- i t (EV(t) - 2r ^(t)), (3.7) where h(t) is the average effective Hamiltonian h(t) for the time interval [0,t]: h(t) == \ /0 h(x) dx (averages E^(t), /^ (t) are defined analogously).
We assume that w(E) is given in the Breit-Wigner (BW) form:
w(E) = ^bw (E) d=f N 0 (E - Emin) (E_Eor °+(r°)2, where N is a nor
malization constant and 0 (E ) = 1 for E > 0 and 0 (E ) = 0 for E < 0. E0 is the energy of the system, r o is a decay rate, while E min is the minimal energy of the system. Inserting uBW(E ) into formula (3.2), we get:
A(t) = - to) = 2n e- * Eot Iß ( r0<<t- t0) ) ■ (3.8) where
I
- 2 + T e- in T dn. (3.9)
■ß n2 + 4
Here t = r °(t_to) and ß = E°_ppmin > 0. It is assumed that t0 = 0.
Using A(t), as given by Eqs (3.8), and the effective Hamiltonian (3.5), we find the Breit-Wigner m odel as:
J (— )
h(() = Eo + r o ■ (3.10)
1ß(— ) where
/ œ - X T e- iX T dx. (3.11)
■ß X + 4
In result, the instantaneous energy E^(t) has the following form:
J ( rot )
E*(t) = ft [h(t)] = Eo + r o ft . (3.12)
.
ß(r r )
_The simplest way to extend the classical m odel of the decay:
Pde(t) = Pde(to) X exp [ - T (t - t o)] = pde(t - t o) (3.13)
is to replace the classical decay rate r by the decay rate r ^ (t)/h
appearing in quantum theoretical considerations. In consequence, we get:
pde(t) = - j r 4>(t) Pde(t) (3.14) instead o f the classical fundam ental equation of the radioactive decay theory
Ultimately, the formula for the decay is:
Pde(t) = Pde(to) X exp - j /^ (t)
r i f ' ]
= pde(t0) X exp - — r ^(x) dx . (3.15)
L to
This relation, superseding the classic decay formula, contains quantum corrections resulting from the use of the quantum theory decay rate. Using (3.8), we can rewrite the relation (3.15) as:
Pde(t) = £ 2 Pde(to) Iß ( F 0 itl- t0) ) . (3.16) The m odel can be expressed in a more general form of the en
ergy density:
pde(t) = pde(t) - pba^ (3.17) where pbare = const is the minimal value o f the dark energy density.
When t ^ <x), the density pde(t) tends to pbare.
3.2 Late-time approximation of decaying metastable dark energy
This section is based on JCAP1510 (2015) no. 10, 0 6 6 1411 and Phys.Dark Univ. 15 (2017) 96-1041421.
We investigated the late-tim e approximation o f the m odel with de
caying dark energy as the A(t)CDM model, where A(t) = A bare + , where a is a m odel parameter. We assume that a2 > 0 or a2 < 0.
a can be imaginary. This parametrization of dark energy is a late
tim e approximation of Eq. (3.16). This m odel is an example of the
m odel involving interaction between dark matter and dark energy.
In result, we get the following Friedmann equation:
a2
3H (t)2 = pm(t)+Abare + ^ . (3.18) Due to the assumption that the energy-m om entum tensor for all ftuids satisfies the conservation condition:
T aß = 0, (3.19)
we get the conservation equation:
pm + 3Hpm = - — . dA (3.20) dt
This form of the conservation equation guarantees that the inter
action in the dark sector is actual.
From Eqs (3.18) and (3.20), we obtain that:
HT= - 1 pm. (3.21)
This equation can be rewritten as:
H = 2 (Abare + ^ - 3H (t)2) . (3.22) The above form ula has the following solution:
I / 3^/H
a,
oH
oA 1 2n ,____In - M 2 t
h(t) = + v / ^ --- > _ / , (3.23) 3Hot
i/ 3^/nA,oHo t j
where h = — , H0 is the present value o f the Hubble constant, 0 =
t???, I n is the modified Bessel function o f the first kind, and
’ 0 _____________
n = 2 V i + 9n«2;oTo2Ho2, where f i^ o = ^-2Tr and To is the present age of the Universe.
The formula for the scale factor a can be derived from the Eq. (3.23) and after the calculations we get:
a(t) = C V t ( ^ I n (^3^ n A °H° t j j . (3.24)
Constant C is equal to V T 0 ^ I n ^ 3V ^ ° HoTo^ , as it is
assumed that a(T0) = i. Since a(t) function is monotonic, one can
obtain formula for t(a) function from Eq. (3.24):
2 S _, / y w - A o H ^ i c ï 3 j (325)
(a) 3 * ^ —A Ä n_1 ^ 2 VC) j ■ .
where Sn(x) is a Riccati-Bessel function Sn(x) = Jn+1 (x). J is the Bessel function of the first kind.
As the form ula for H (t) is known, the equation for pm can be derived from Eqs (3.18) and (3.23):
— T 2 Pm(t) = -3Ho2 —
a,
o+ - a J - !o--
( T f 3V Qa,°h°+A \
1 - 2n I ™_1 I 2 t
- - 3HT + V/-A;0---/ _ / . (3.26) 3Hot r / 3v/nA7fl° A
V H 2 ) / _
Since we assume the interaction is between dark matter and dark energy only, the energy density of baryonic matter pb(t) scales as a_3. In result, we get
Pm(t) = Pdm(t) + Pb (t) = Pdm(t) + Pb,0ß (t)_3, (3.27) where pb 0 is the present value the energy density of baryonic mat
ter.
From Eqs (3.24), (3.26) and (3.27), we can obtain a formula for the energy density of dark matter pdm(t):
— T 2 Pdm(t) = -3H 2 — A,o + -
/
i/ 3^Q
a,°h° A \
- W 3Hot + ^ P z - 2- ^v/nA^H° A ^
V V 2 V / _
- Pb,oC_3 3v / - A öHo^ . (3.28)
3.3 Model testing
This section is based on JCAP1510 (2015) no. 10, 0 6 6 1411
.
The paper [41] concerns the cosm ological m odel with the follow -
2
ing parametrization of the dark energy: pde = Abare + ^ . In particu
lar, we investigated the behaviour of the je rk using Sahni et al. [43, 44, 45] Om(z) diagnostic test. We also performed the dynamical and statistical analysis of the model.
From the Eqs (3.23) and (3.24), we find that the je rk function is given by the following equation:
. 1 d3a(t)
j H (t)3a(t) dt3
( I ( 3V qa,0h0 A \
1 - 3JW ¥ Ł \ W Hot3 3Hot + ^ 0 r / r - 2- 3^Q—~oHo \ \ • (3.2Q)
V H 2 ) )
In the present epoch, the je rk function is given by
jo = 1 - H t 0, (3.30)
H0r 0 where T0 is the present age of the Universe.
The evolution of the je rk function is shown in Fig. 5 in JCAP 1510 (2015) no. 10, 066 [41].
The Om(z) diagnostic test measures the deviation from the AODM m odel (Om(z) = Qm for AODM model). The function Om(z) is Om(z) = ^XX--1 , where x = 1 + z. For our model, it has the fo l
lowing form:
Om(t ) =
\ 3H-t + V A’0 !„ ( 3^ U-f
H- 1) )
/ „ ( r ^ ] 2 * ) ) ] " 1 - 1 (3.31) The evolution o f the Om(z) function is shown in Fig. 6 in JCAP 1510 (2015) no. 10, 066 [41].
The behaviour o f th e je rk and Om(z) function provides a test for
the deviation from the AODM model. These tests tools com m only
used to indicate variability of dark energy in time.
After recating the cosm ological equations (3.18) and (3.20) to the form of the dynamical system we have:
X ' = - 3 X + 3X 2 + 2v/3a2Z 3, (3.32)
Y ' = 3 XY, (3.33)
3
Z ' = - v ^ Z 2 + - Z X , (3.34) where
X = 3H5 (3.35)
and the squares of Y and Z are equal to:
Y 2 = Ab ^ ,
z2 = (3.36)
3H2 , 3H2t 2
and ' = dlna. The critical points of the system (3.32)-(3.35) are col
lected in Table 1 in JCAP1510 (2015) no. 10, 066 [41].
In the statistical analysis o f the m odel parameters, we have used the SNIa [13], BAO (SDSS DR7 data) [14], CMB and lensing ob
servations [31], measurements of H(z) [17, 18, 19] and the Alcock- Paczyński test [22, 23, 24, 25, 26, 27, 28, 29, 3 0 ]. The value of the best fit and errors are given in Table 2 and 3 in JCAP 1510 (2015) no. 10, 066 [41]. The analysis shows that the m odel with negative values of the a2 parameter is more favoured than one with positive values.
3.4 Modified scaling law of matter density
This section is based on Phys.Dark Univ. 15 (2017) 96-1041421.
In the paper [42], we consider the cosm ological m odel with the parametrization of the dark energy pde = Abare + ^ . We check how this parametrization modified the scaling law o f the energy den
sity of matter and dark matter. The cosm ological equations (3.18) and (3.20) give us the formula for the energy density of matter (see Eq. (3.26)). We can rewrite Eq. (3.26) as:
pm = pm,oa-3+' (t), (3.37)
where £ = joga / £(t)d log a, where £(t) = t 3 H(2t)pm(t). The evolution of
£(t) function is presented in Fig. 5 in Phys.Dark Univ. 15 (2017) 96
104 [42]. If £(t) is constant, then we get that £(t) is constant too. In
this case, Eq. (3.37) is given by:
Pm = Pm,o a_3+5. (3.38)
When £(t) is a constant, then also:
a = aot 3=7 2 (3.39)
and
Pm = Pm,oa_3+51_2. (3.40)
For the early Universe, £(t) function can be approximated as:
<S(t) = , 9a ---. 9a2 (3.41) W ( V i + 3a2 + 1)2
We can use the same approach in the case of dark matter Pdm rewriting Eq. (3.28) as:
Pdm = Pdm,oa_3+A(t), (3.42)
where A(t) = ^ 1 ^ ) log ^m^0na'0(-_)^b^ob’o. For the early Universe, A(t) = const. In result, Pdm = Pdm,oa_3+A. The evolution of A(t) function is presented in Fig. 6 in Phys.Dark Univ. 15 (2017) 96-104 [42].
The statistical analysis in this paper is based on the astronom
ical observations, such as SNIa [13], BAO [14, 15, 16], observations of the temperature power spectrum of CMB [31], measurements of H(z) [17, 18, 19] and the Alcock-Paczyński test [22, 23, 24, 25, 26, 27, 28, 29, 3 0 ]. The value of the best fit and errors are given in Table 1 in Phys.Dark Univ. 15 (2017) 96-104 [42]. We obtain the decay of particles o f dark matter rather than their creation. The AIC criterion favours this m odel ju st very weakly in comparison to the ACDM model, while the BIC criterion supports positively the ACDM model. However, this is not suffficient for rejecting it.
3.5 Cosmological implications of transition from false to true vacuum state
This section is based on Eur.Phys.J. C77 (2017) no. 6, 3 5 71361.
In the paper [36], we investigate a cosm ological m odel with de
caying metastable dark energy. Here, the m odel of the decaying
metastable dark energy is provided by quantum mechanics. The
parametrization of dark energy is given by Eq. (3.12). Replacing en
ergy by the density o f energy in Eq. (3.12), we obtain:
pde = Abare + E R 1 + “ ^ ( "777)1 , (3.43) L 1 - a VI(t) /J
where E R = E o - Abare and a is a m odel parameter, which belongs to the interval (0, 1). The functions I(t) and T(t) are:
/ ™ T X T e~ "7 dn
■
±=
a an2 + 4 4
= 2 .- » ( - 2 ,. + - E , ( |1 - T )
+ E . ( I - 1 -
t) ) '3 .4 ., and I(t) can be expressed as:
r™ i
I (t)
=T T T e -inT dn
•/-^ n2 + 4
- » ( i + 2 n ( - ' E . ( | 2 - )
+ E ' ( I - 1 - ^ -) ) ) ■ <3-45.
where t = a(f --_Ab)are) Vot and V0 is the volum e o f sphere of radius, which is equal to the Planck length. The function E 1 is the expo
nential integral E 1(z) = f z™ dx.
As this m odel involves interactions between dark matter and dark energy, we have the following cosm ological equations:
3H = 3 ^ = ptot = pb + pdm + pde, (3.46)
pb = - 3H p b, (3.47)
pdm = - 3 H pdm + Q (3.48) and
pde = - Q, (3.49)
where pb is the density of baryonic matter and Q = - is the
interaction between dark matter and dark energy, which actually
consists energy transfer. If Q > 0, then energy ftows from dark
energy to dark matter, while if Q < 0, then energy ftows from dark
matter to dark energy.
In this model, there occurs an intermediate phase of oscillations of the dark energy density between the phases o f constant dark energy. We found also a mechanism to causejum ping of the value of energy density of dark energy from the initial value of E0 (E0 = 10120) to present value of the cosm ological constant.
The oscillations appear when 0 < a < 0.4. Their number, pe
riod, and amplitude, as w ell as the duration o f this intermediate phase, decrease when a parameter grows. For a > 0.4, the oscil
lations disappear altogether.
In the statistical analysis, we use the astronomical observations such as the supernovae of type Ia (SNIa) [13], BAO [14, 15, 16], mea
surements of H (z ) for galaxies [17, 18, 19], the Alcock-Paczyński test [22, 23, 24, 25, 26, 27, 28, 29, 3 0 ] and the measurements OMB [31]. The analysis showed us that independently of the values of the parameters a and E0, we obtain the present value of the en
ergy density of the dark energy. The value of the best fit and errors are given in Table 1 in Eur.Phys.J. C77(2017) no. 6, 357 [36].
3.6 Radioactive-like decay of metastable dark energy
This section is based on Eur.Phys.J. C77 (2017) no. 12, 9021371.
In the paper [37], we consider the m odel with the radioactive-like decay of metastable dark energy. The cosm ological equations are:
3H2 = pm + Pde, (3.50)
pm = -3 H p m - pde, (3.51) where the density of dark energy pde is parametrized as follows:
pde(t) = pbare + e Iß ^ , (3.52) where Iß(
t) is defined as
r ^ 1
Iß (
t) = 2 , 1 e %r>T dn, (3.53) J-ß n + 4
where
t= r t . The parameter e = e(ß) = pdj1°((~pj2are measures the
deviation from the AODM m odel (Iß(0) = N = n + 2 arctan(2ß) and
ß > 0), ß is equal to > 0, while the parameters E° and r 0
correspond to the energy of the system in the unstable state and its decay rate at the exponential (or canonical) regime of the decay process.
For t > t L = - f - |+ t [46], the approximation of (3.52) is given in the following form:
pde(t) ~ pbare+
( 2 r0. 4ne- 1 » * sin (ß ^ 1) 1 \ , x e 4n 2 e- r * + ---- ( 1 --- 0) Vr f ; 2 . (3.54)
\ (1 + ß 2) f ‘ (( 4 + ß'2) f t) V
For the late time, Eq. (3.54) can be approximated as:
pde(t) ~ Pbare + 2 . (3.55)
( ( J + ß 2) f ) 2 t 2
If we use formula (3.54), the Friedmann equation (3.50) is:
3H = ptot = pB + pDM + pbare + prad.dec + pdam.osc + ppow.law, (3.56) where prad.dec = 4n 2 ee- ^ is the radioactive-like decay dark energy, Pdam.osc = 4 ne^ 1+ t) is the damping oscillating dark energy
(4 + ß ) h *
and pDowlaw = —---\ r , 2 is the power-law dark energy. The ra
p .--- ((4+- 1 ) rhot)
dioactive type of decay dominates up to 2.2 x 10 4 T 0 .
We performed also statistical analysis using the following as
tronom ical observations: supernovae o f type Ia (SNIa, Union 2.1 dataset [13]), BAO data (Sloan Digital Sky Survey Release 7 (SDSS DR7)) dataset at z = 0.275 [14], 6 dF Galaxy Redshift Survey mea
surements at redshift z = 0.1 [15], W iggleZ measurements at red
shift z = 0.44,0.60,0.73 [16]), measurements of the Hubble param
eter 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 1 in Eur.Phys.J. C77(2017) no. 12, 902 [37].
We found that the decay half-life tim e T 1 / 2 of dark energy is 8503 Gyr « 616 x T 0 and the radioactive type o f decay is the most effective mechanism of decaying metastable dark energy.
3.7 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
k2 J V, (4.3) where
k2 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
yis 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(3X + 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
k2 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 =
h2 = ^ 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
ïr = 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.
