Cosmological Evolution in R+R2 modified gravity тема диссертации и автореферата по ВАК РФ 01.04.02, кандидат наук Раджниш Сингх

• Раджниш Сингх
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Раджниш Сингх

Contents

Declaration of Authorship i

Acknowledgements ii

Abstract iv

1 Introduction

1.1 Modified Gravity Theories

1.1.1 Einstein-Cartan Theory

1.1.2 Gauss-Bonnet Gravity

1.1.3 Brans-Dicke Theory

1.2 f (R)-gravity

1.2.1 Metric f (R) gravity

1.2.2 Palatini f (R) gravity

1.2.3 Metric-affine f (R) gravity

1.2.4 Starobinsky Inflation

4

1.2.5 The (R - ^) model

2 Evolution in R + R2 gravity

2.1 The Modified Einstein's Field Equations

2.2 Solution from the beginning to gt

2.2.1 Solution at inflationary epoch

2.2.2 Numerical solutions at post-inflationary epoch

2.2.3 Asymptotic behavior of the solution at t » 1 and w = 1/3

2.2.4 Asymptotic solution at t » 1, gt < 1, and w =

2.2.5 Energy influx to cosmological plasma from the scalaron decay

2.2.6 Features of the distinctive cosmological evolution at t < 1/g

2.3 Solution at gt &

3 Dark Matter

3.1 Galaxy Rotation Curves

3.2 Possible Candidates

3.3 LSP as dark matter in R + R2 gravity

3.3.1 R2 -gravity versus General Relativity

3.3.2 LSP density for the scalaron decay into scalars

3.3.3 Decay into fermions or conformal scalars

3.4 Anomalous decay into gauge bosons

4 Dark matter in R + R2 cosmology with conformal anomaly

4.1 X-particle production through the scalaron decay

4.2 Production of X-particles in thermal plasma

4.3 Possible Observations

5 Conclusion

List of Figures

1.1 The Einstein Frame potential V (f) [44]

2.1 Evolution of h(t) at the inflationary stage with the initial values of dimensionless curvature r = 300 (left) and 600 (right). Initially h is taken to be zero, hin = 0, but it quickly reaches the value given by Eq. (2.2.2), h(0) = \J—r0/12. The numbers of e-foldings, according to Eq. (2.2.1), are respectively 75 and

2.2 Evolution of the dimensionless energy density of matter during inflation for w = 0 (solid line) and w = 1/3 (dotted line). Left panel: initially yn = 0 and right panel: yin = 0.1. The initial fast rise of p from zero in the left panel during short time is generated by the S[r]-term (2.1.18) taken as S[r] = (r')2/288n. The results are not sensitive to the form S[r] because at inflation y(t) quickly vanishes anyhow

2.3 Evolution of the dimensionless curvature scalar for rn = —300 (solid) and rn = —600 (dotted). Left panel: shows evolution during inflation and right panel: shows evolution after the end of inflation when curvature scalar starts to oscillate

2.4 Evolution of the curvature scalar tr(t) in post-infationary epoch. Left panel (w=1/3): initially rin = —300 (solid), rn = —600 (dotted). There is absolutely no difference between the curves. Right panel (rin = —300): w = 1/3 (solid) and w = 0 (dotted). The difference is minuscule. The source term (2.1.18) here is taken as S[r] = (r')2/1152n.

The results are not sensitive to its form

2.5 Evolution of the Hubble parameter, ht, in post-inflationary epoch for w = 1/3 (solid

line) and w = 0 (dotted)

2.6 Evolution of the energy density of matter yt at small t (left) and at large t (right). Parameter w = 1/3 (solid) and w = 0 (dotted)

2.7 Left panel: comparison of numerical solution for ht (solid) with analytic estimate (2.2.20) (dotted). Right panel: the same for numerically calculated rt with analytic result (2.2.21). The difference between the solid and dotted curves is not observable

2.8 Comparison of the numerical solution for the dimensionless energy density 120nt y(t) (solid) for w = 1/3 with the asymptotic expression (2.2.29) (dotted) for moderately large t (left panel) and very large t (right panel). The agreement is very good

2.9 Numerical solution of Eq. (2.2.7) for 120nty(t) in different time intervals with w =

1/3

2.10 Comparison of the solutions of the differential Eq. (2.2.32) for w = 0: integral solution (2.2.33) (solid) and asymptotic solution (2.2.34) (dotted). The dimensionless energy density 72 nt y(t) is presented for moderately large t (left panel) and very large t (right panel). The agreement is very good

3.1 Sketch of the observed HI rotation curve of the dwarf galaxy M33(red solid line) and

of that expected from the stellar distribution(blue dashed line) [69]

4.1 Log of ratio of the calculated number density of X-particles to the equilibrium number density (4.2.5) calculated in the limit nx ^ nei?;

left panel: y;n = 0.1 and right panel: y;n =

4.2 Log of the ratio of the calculated number density of X-particles to the equilibrium number density (4.2.5) calculated in the limit nx ^ nei?;

left panel: yin = 20 and right panel: yin =

4.3 Evolution of u(y) for y = 100 (left) and y = 50 (right)

4.4 Evolution of u(y) for y = 30 (left) and y = 20 (right)

4.5 Log of the ratio of the energy density of X-particles (4.2.15) to the observed energy density of dark matter as a function of y = Mr /Mx

Введение диссертации (часть автореферата) на тему «Cosmological Evolution in R+R2 modified gravity»

Introduction

In this chapter, the reason why alternatives to General Relativity are needed is briefly explored and then we lightly scratch the surface of some of the modified gravity theories present out there. For a comprehensive review on almost all kind of modified gravity theories readers can refer to [1, 2, 3, 4]. Later, we discuss f(R) gravity in detail, which is the main framework for this thesis.

The Einstein's theory of gravitation [5, 6] had been really successful after its importance was realized in 1920 [7]. It provided a new dimension to physics and especially to cosmology. It described the basic properties of the Universe with very good agreement with the observations. The theory of General Relativity (GR) is based on the Einstein-Hilbert action given as:

here R is the curvature scalar, mpi = 1.22 • 1019 GeV is the Planck mass, which is related to the gravitational coupling constant as mp = G—1 and g is the determinant of the metric tensor g^v. However even after the massive success of GR, some astonishing observations in the late 1990s made it quite clear that some alternative to GR is needed to explain the new observations. So, a natural choice was higher order extension of GR and modified gravity theories.

Заключение диссертации по теме «Теоретическая физика», Раджниш Сингх

Conclusion

In chapter 2 it was found that the cosmological evolution in R2-gravity is drastically different from that of GR based cosmology. The history of cosmological evolution in R2-modified gravity can be separated into four distinct epoch. At first, there was the exponential expansion, the curvature scalar R(t) was very large but as the universe expanded it slowly started to decrease. At this stage the universe was void and dark with slowly decreasing curvature scalar R(t). To ensure sufficiently long inflation, it is necessary that the initial value of R should be quite large, R > 300m2 , such that the number of e-folding exceeded 70.

The next epoch began when R(t) dropped down to zero and started to oscillate as R ~ m cos(mt)/t. The oscillations of R led to particle production and this moment can be considered as Big Bang. The universe expansion at this stage is described by very simple, but unusual law with the Hubble parameter periodically reaching (almost) zero, H = (2/3t)[1 + sin(mt)] (2.2.20). Such a regime was realised asymptotically for large time, mt ^ 1, but Gt . 1.

Later, when time becomes so large that Gt exceeds unity, the oscillations of all relevant quantities exponentially damps down and the particle production by curvature switches off, becoming negligible. This marks the next epoch which is the transition period from scalaron domination to the relativistic matter domination, and it takes place when Gt becomes larger than unity by the logarithmic factor, ln(mpi /m). Finally, after scalaron had decayed completely we arrive to the standard cosmology which is governed by General Relativity. A comparison of theoretical prediction with CMB fluctuation data gives the value of m, later indicated as mr — 3 1013GeV. The unusual cosmological evolution during early period, t < 1/G, lead to modification of the cosmological baryogenesis scenarios, which can affect the probability of the formaltion of primordial black holes, and the frozen number density of dark matter particles.

Usually the cosmological abundance, nx, is inversely proportional to the Planck mass squared. However, for the regime Gt . 1, nx depends on the mass of scalar mR and is independent of mvi. This fact opened the possibility for LSPs with mass larger than TeVs to form the cosmological dark matter.

This possibility was explored in chapter 3. First it was discovered that due to the continuous and slow production of matter by scalaron field, R(t), the temperature

of the ordinary matter in the universe drops much slower than in the usual FLRW cosmology. This led to a new modified canonical relation between temperature of matter and cosmological time which is T4t = Cr2 where as it is usually T21 = Cgr. Here Cgr is a universal constant which is proportional to Planck mass, whereas Cr2 is a constant which depends entirely on the model.

Next to study the number density of LSP, we used the Zeldovich equation and studied two cases i.e. scalaron decays into a pair of scalars or into a pair of fermions. If scalaron decays into a pair of scalars, then LSP can play the role of dark matter particle if it is considerably heavier than the scalaron. For the next case, when scalaron decays into fermions or conformally coupled scalars the mass of LSP may be at the level of 103 TeV, which can open the possibility for their direct detection in experiments. However, such energies are still beyond the reach of present accelerators. The search for such dark matter particles in low background experiments looks presently more feasible. If they are discovered, it would be an interesting confirmation of R2 inflationary model.

However, the analysis presented in Chapter 3 was done without the consideration of conformal anomaly. This restiction was lifted in Chapter 4, and the model was studied with strength of conformal anomaly. In the presence of conformal anomaly, the thermalization of the cosmological plasma is caused by the creation of gauge bosons and the reactions between them led to creation of all other particles.

Again two processes were considered for the production of X-particle, first being the direct decay of scalaron into a pair XX and secondly, by thermal production of X-particles in the plasma. To restrict the density of X-particles produced by the direct decay the observed value Mx should be below 107 GeV but it leads to very strong thermal production of these particles. This inconsistency was resolved by considering that the X particles are Majorana fermions, therefore the production of X particles by direct decay of scalaron is suppressed which allowed us to have larger a value for Mx and the thermal production of X particle is favourable. It opens the possibility for X-particles to make proper amount of dark matter, if their mass is about 5 ■ 1012 GeV.

So a supersymmetric type of dark matter particles seems to be possible only if their mass is quite high i.e. from 106 up to 5 x 1012 GeV, or even higher than the scalaron mass, mR = 3 x 1013 GeV. There is no possibility in near future to discover these particles, but they may be observable through cosmic rays from their annihilation in high density clumps of dark matter, or from annihilation in their gravita-tionally bound two-body states, or through the products of their decays, since they naturally should be unstable.

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