Difference between revisions of "New problems"

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(Problem 17)
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=== Problem 17 ===
 
=== Problem 17 ===
 
<p style= "color: #999;font-size: 11px">problem id: SSC_13</p>
 
<p style= "color: #999;font-size: 11px">problem id: SSC_13</p>
'''(after Tiberiu Harko, arXiv:1310.7167)'''
 
 
Calculate the deceleration parameter for flat Universe filled with the scalar field in form of quintessence.
 
Calculate the deceleration parameter for flat Universe filled with the scalar field in form of quintessence.
 
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     <p style="text-align: left;">There are several ways to obtain the result:
 
     <p style="text-align: left;">There are several ways to obtain the result:
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'''1)''' \[q=-\frac{\ddot a }{aH^2};\quad \frac{\ddot a}{a}=-\frac16(\rho+3p);\quad H^2=\frac13\rho;\]
 
'''1)''' \[q=-\frac{\ddot a }{aH^2};\quad \frac{\ddot a}{a}=-\frac16(\rho+3p);\quad H^2=\frac13\rho;\]
 
\[q=\frac{\dot\varphi^2-V}{\frac{\dot\varphi^2}2+V};\]
 
\[q=\frac{\dot\varphi^2-V}{\frac{\dot\varphi^2}2+V};\]
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<div id="gmudedm_1"></div>
 
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=== Problem 1 ===
 
=== Problem 1 ===
 
<p style= "color: #999;font-size: 11px">problem id: gmudedm_1</p>
 
<p style= "color: #999;font-size: 11px">problem id: gmudedm_1</p>

Revision as of 10:38, 20 December 2013


NEW Problems in Dark Energy Category

Single Scalar Cosmology

The discovery of the Higgs particle has confirmed that scalar fields play a fundamental role in subatomic physics. Therefore they must also have been present in the early Universe and played a part in its development. About scalar fields on present cosmological scales nothing is known, but in view of the observational evidence for accelerated expansion it is quite well possible that they take part in shaping our Universe now and in the future. In this section we consider the evolution of a flat, isotropic and homogeneous Universe in the presence of a single cosmic scalar field. Neglecting ordinary matter and radiation, the evolution of such a Universe is described by two degrees of freedom, the homogeneous scalar field $\varphi(t)$ and the scale factor of the Universe $a(t)$. The relevant evolution equations are the Friedmann and Klein-Gordon equations, reading (in the units in which $c = \hbar = 8 \pi G = 1$) \[ \frac{1}{2}\, \dot{\varphi}^2 + V = 3 H^2, \quad \ddot{\varphi} + 3 H \dot{\varphi} + V' = 0, \] where $V[\varphi]$ is the potential of the scalar fields, and $H = \dot{a}/a$ is the Hubble parameter. Furthermore, an overdot denotes a derivative w.r.t.\ time, whilst a prime denotes a derivative w.r.t.\ the scalar field $\varphi$.


Problem 1

problem id: SSC_0

Show that the Hubble parameter cannot increase with time in the single scalar cosmology.


Problem 2

problem id: SSC_1

Obtain first-order differential equation for the Hubble parameter $H$ as function of $\varphi$ and find its stationary points.


Problem 3

problem id: SSC_2

Consider eternally oscillating scalar field of the form $\varphi(t) = \varphi_0 \cos \omega t$ and analyze stationary points in such a model.


Problem 4

problem id: SSC_3

Obtain explicit solution for the Hubble parameter in the model considered in the previous problem.


Problem 5

problem id: SSC_4

Obtain explicit time dependence for the scale factor in the model of problem #SSC_2.


Problem 6

problem id: SSC_5

Reconstruct the scalar field potential $V(\varphi)$ needed to generate the model of problem #SSC_2.


Problem 7

problem id: SSC_6_00

Describe possible final states for the Universe governed by a single scalar field at large times.


Problem 8

problem id: SSC_6_0

Formulate conditions for existence of end points of evolution in terms of the potential $V(\varphi)$.


Problem 9

problem id: SSC_6_1

Consider a single scalar cosmology described by the quadratic potential \[ V = v_0 + \frac{m^2}{2}\, \varphi^2. \] Describe all possible stationary points and final states of the Universe in this model.


Problem 10

problem id: SSC_7

Obtain actual solutions for the model of previous problem using the power series expansion \[ H[\varphi] = h_0 + h_1 \varphi + h_2 \varphi^2 + h_3 \varphi^3 + ... \] Consider the cases of $v_0 > 0$ and $v_0 < 0$.


Problem 11

problem id: SSC_8

Estimate main contribution to total expansion factor of the Universe.


Problem 12

problem id: SSC_9_0

Explain difference between end points and turning points of the scalar field evolution.


Problem 13

problem id:

Show that the exponentially decaying scalar field \[ \varphi(t) = \varphi_0 e^{-\omega t} \] can give rise to unstable end points of the evolution.


Problem 14

problem id: SSC_10

Analyze all possible final states in the model of previous problem.


Problem 15

problem id: SSC_11

Express initial energy density of the model of problem #SSC_9 in terms of the $e$-folding number $N$.


Problem 16

problem id: SSC_12

Estimate mass of the particles corresponding to the exponential scalar field considered in problem #SSC_9.


Problem 17

problem id: SSC_13

Calculate the deceleration parameter for flat Universe filled with the scalar field in form of quintessence.



Generalized models of unification of dark matter and dark energy

(see N. Caplar, H. Stefancic, Generalized models of unification of dark matter and dark energy (arXiv: 1208.0449))

Problem 1

problem id: gmudedm_1

The equation of state of a barotropic cosmic fluid can in general be written as an implicitly defined relation between the fluid pressure $p$ and its energy density $\rho$, \[F(\rho,p)=0.\] Find the speed of sound in such fluid.


Problem 2

problem id: gmudedm_2

For the barotropic fluid with a constant speed of sound $c_s^2=const$ find evolution of the parameter of EOS, density and pressure with the redshift.



Hybrid Expansion Law

In problems #SSC_18 - #SSC_19_0 we follow the paper of Ozgur Akarsu, Suresh Kumar, R. Myrzakulov, M. Sami, and Lixin Xu4, Cosmology with hybrid expansion law: scalar field reconstruction of cosmic history and observational constraints (arXiv:1307.4911) to study expansion history of Universe, using the hybrid expansion law---a product of power-law and exponential type of functions \[a(t)=a_0\left(\frac{t}{t_0}\right)^\alpha\exp\left[\beta\left(\frac{t}{t_0}-1\right)\right],\] where $\alpha$ and $\beta$ are non-negative constants. Further $a_0$ and $t_0$ respectively denote the scale factor and age of the Universe today.

Problem 1

problem id: SSC_18

Calculate Hubble parameter, deceleration parameter and jerk parameter for hybrid expansion law.


Problem 2

problem id: SSC_18_2

For hybrid expansion law find $a, H, q$ and $j$ in the cases of very early Universe $(t\to0)$ and for the late times $(t\to\infty)$.


Problem 3

problem id: SSC_18_3

In general relativity, one can always introduce an effective source that gives rise to a given expansion law. Using the ansatz of hybrid expansion law obtain the EoS parameter of the effective fluid.


Problem 4

problem id: SSC_19

We can always construct a scalar field Lagrangian which can mimic a given cosmic history. Consequently, we can consider the quintessence realization of the hybrid expansion law. Find time dependence for the the quintessence field $\varphi(t)$ and potential $V(t)$, realizing the hybrid expansion law. Obtain the dependence $V(\varphi)$.


Problem 5

problem id: SSC_19_1

Quintessence paradigm relies on the potential energy of scalar fields to drive the late time acceleration of the Universe. On the other hand, it is also possible to relate the late time acceleration of the Universe with the kinetic term of the scalar field by relaxing its canonical kinetic term. In particular this idea can be realized with the help of so-called tachyon fields, for which \[\rho=\frac{V(\varphi)}{\sqrt{1-\dot\varphi^2}},\quad p=-V(\varphi)\sqrt{1-\dot\varphi^2}.\] Find time dependence of the tachyon field $\varphi(t)$ and potential $V(t)$, realizing the hybrid expansion law. Construct the potential $V(\varphi)$.


Problem 6

problem id: SSC_19_2

Calculate Hubble parameter and deceleration parameter for the case of phantom field in which the energy density and pressure are respectively given by \[\rho =-\frac{1}{2}\dot{\varphi}^2+V(\varphi),\quad p =-\frac{1}{2}\dot{\varphi}^2-V(\varphi).\]


Problem 7

problem id: SSC_19_0

Solve the problem #SSC_19 for the case of phantom field.



Tutti Frutti

Problem 1

problem id: TF_1

Construct planck units in a space of arbitrary dimension.


Problem 2

problem id: TF_2

Show that for power law $a(t)\propto t^n$ expansion slow roll inflation occurs when $n\gg1$.


Problem 3

problem id: TF_3

Find the general condition to have accelerated expansion in terms of the energy densities of the darks components and their EoS parameters