Dynamical Forms of Dark Energy

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The Quintessence

The cosmological constant represents nothing but the simplest realization of the dark energy - the hypothetical substance introduced to explain the accelerated expansion of the Universe. There is a dynamical alternative to the cosmological constant - the scalar fields, formed in the post-inflation epoch. The most popular version is the scalar field $\varphi$ evolving in a properly designed potential $V(\varphi)$. Numerous models of such type differ by choice of the scalar field Lagrangian. The simplest model is the so-called quintessence. In antique and medieval philosophy this term (literally "the fifth essence", after the earth, water, air and fire) meant the concentrated extract, the creative force, penetrating all the material world. We shall understand the quintessence as the scalar field in a potential, minimally coupled to gravity, i.e. feeling only the influence of space-time curvature. Besides that we restrict ourselves to the canonic form of the kinetic energy. The action for fields of such type takes the form \[S=\int d^4x \sqrt{-g}\; L=\int d^4x \sqrt{-g}\left[\frac12g^{\mu\nu}\frac{\partial\varphi}{\partial x^\mu} \frac{\partial\varphi}{\partial x^\nu}-V(\varphi)\right].\] The equations of motion for the scalar field are obtained as usual, by variation of the action with respect to the field (see Chapter "Inflation").


Problem 1

Obtain the Friedman equations for the case of flat Universe filled with quintessence.


Problem 2

Obtain the general solution of the Friedman equations for the Universe filled with free scalar field, $V(\varphi)=0$.


Problem 3

Show that in the case of Universe filled with non-relativistic matter and quintessence the following relation holds: \[\dot H=-4\pi G(\rho_m+\dot\varphi^2).\]


Problem 4

Show that in the case of Universe filled with non-relativistic matter and quintessence the Friedman equations \[H^2=\frac{8\pi G}{3}\left[\rho_m+\frac12\dot\varphi^2+V(\varphi)\right],\] \[\dot H =-4\pi G(\rho_m+\dot\varphi^2)\] can be transformed to the form \[\frac{8\pi G}{3H_0^2}\left(\frac{d\varphi}{dx}\right)^2=\frac{2}{3H_0^2x}\frac{d\ln H}{dx}-\frac{\Omega_{m0}x}{H^2};\] \[\frac{8\pi G}{3H_0^2}V(x)=\frac{H^2}{H_0^2}-\frac{x}{6H_0^2}\frac{d H^2}{dx}-\frac12\Omega_{m0}x^3;\] \[x\equiv1+z.\]


Problem 5

Show that the conservation equation for quintessence can be obtained from the Klein-Gordon equation \[\ddot\varphi+3H\dot\varphi+\frac{dV}{d\varphi}=0.\]


Problem 6

Find the explicit form of Lagrangian describing the dynamics of the Universe filled with the scalar field in potential $V(\varphi)$. Use it to obtain the equations of motion for the scale factor and the scalar field.


Problem 7

In the flat Universe filled with scalar field $\varphi$ obtain the isolated equation for $\varphi$ only. (See S.Downes, B.Dutta, K.Sinha, arXiv:1203.6892)


Problem 8

What is the reason for the requirement that the scalar field's evolution in the quintessence model is slow enough?


Problem 9

Find the potential and kinetic energies for quintessence with the given state parameter $w$.


Problem 10

Find the dependence of state equation parameter $w$ for scalar field on the quantity \[x=\frac{\dot\varphi^2}{2V(\varphi)}\] and determine the ranges of $x$ corresponding to inflation in the slow-roll regime, matter-dominated epoch and the rigid state equation ($p\sim\rho$) limit correspondingly.


Problem 11

Show that if kinetic energy $K=\dot\varphi^2/2$ of a scalar field is initially much greater than its potential energy $V(\varphi)$, then it will decrease as $a^{-6}$.


Problem 12

Show that the energy density of a scalar field $\varphi$ behaves as $\rho_\varphi\propto a^{-n}$, $0\le n\le6$.


Problem 13

Show that dark energy density with the state equation $p=w(a)\rho(a)$ can be presented as a function of scale factor in the form \[\rho=\rho_0 a^{-3[1+\bar w(a)]},\] where $\bar w(a)$ is the parameter $w$ averaged in the logarithmic scale $$ \bar w(a) \equiv \frac{\int w(a)d\ln a}{\int {d\ln a} }. $$


Problem 14

Consider the case of Universe filled with non-relativistic matter and quintessence with the state equation $p=w\rho$ and show that the first Friedman equation can be presented in the form \[H^2(z)=H_0^2\left[\Omega_{m0}(1+z)^3+(1-\Omega_{m0})e^{3\int_0^z\frac{dz'}{1+z'}(1+w(z'))}\right].\]


Problem 15

Show that for the model of the Universe considered in the previous problem the state equation parameter $w(z)$ can be presented in the form \[w(z)=\frac{\frac23(1+z)\frac{d\ln H}{dz}-1}{1-\frac{H_0^2}{H^2}\Omega_{m0}(1+z)^3}.\]


Problem 16

Show that the result of the previous problem can be presented in the form \[w(z)=-1+(1+z)\frac{2/3E(z)E'(z)-\Omega_{m0}(1+z)^2}{E^2(z)-\Omega_{m0}(1+z)^3},\quad E(z)\equiv\frac{H(z)}{H_0}.\]


Problem 17

Show that decreasing of the scalar field's energy density with increasing of the scale factor slows down as the scalar field's potential energy $V(\varphi)$ starts to dominate over the kinetic energy density $\dot\varphi^2/2$.


Problem 18

Express the time derivative $\dot\varphi$ through the quintessence' density $\rho_\varphi$ and the state equation parameter $w_\varphi$.


Problem 19

Estimate the magnitude of the scalar field variation $\Delta\varphi$ during time $\Delta t$.


Problem 20

Show that in the radiation-dominated or matter-dominated epoch the variation of the scalar field is small, and the measure of its smallness is given by the relative density of the scalar field.


Problem 21

Show that in the quintessence $(w>-1)$ dominated Universe the condition $\dot{H}<0$ always holds.


Problem 22

Consider simple bouncing solution of Friedman equations that avoid singularity. This solution requires positive spatial curvature $k=+1$, negative cosmological constant $\Lambda<0$ and a "matter" source with equation of state $p=w\rho$ with $w$ in the range \[-1<w<-\frac13.\] In the special case $w=-2/3$ Friedman equations describe a constrained harmonic oscillator (a simple harmonic Universe). Find the corresponding solutions.
(Inspired by P.Graham et al. arXiv:1109.0282)


Problem 23

Derive the equation for the simple harmonic Universe (see previous problem), using the results of problem #DE04.


Problem 24

Barotropic liquid is a substance for which pressure is a single--valued function of density. Is quintessence generally barotropic?


Problem 25

Show that a scalar field oscillating near the minimum of potential is not a barotropic substance.


Problem 26

For a scalar field $\varphi$ with state equation $p=w\rho$ and relative energy density $\Omega_\varphi$ calculate the derivative \[w'=\frac{dw}{d\ln a}.\]


Problem 27

Calculate the sound speed in the quintessence field $\varphi(t)$ with potential $V(\varphi)$.


Problem 28

Find the dependence of quintessence energy density on redshift for the state equation $p_{DE}=w(z)\rho_{DE}$.


Problem 29

The equation of state $p=w(a)\rho$ for quintessence is often parameterized as $w(a)=w_0 + w_1(1-a)$. Show that in this parametrization energy density and pressure of the scalar field take the form: $$ \rho(a) \propto a^{-3[1+w_{\it eff}(a)]},\quad p(a) \propto (1+w_{\it eff}(a))\rho(a), $$ where $$ w_{\it eff}(a)=(w_0+w_1)+(1-a)w_1/\ln a. $$


Problem 30

Find the dependence of Hubble parameter on redshift in a flat Universe filled with non-relativistic matter with current relative density $\Omega_{m0}$ and dark energy with the state equation $p_{DE}=w(z)\rho_{DE}$.


Problem 31

Show that in a flat Universe filled with non--relativistic matter and arbitrary component with the state equation $p=w(z)\rho$ the first Friedman equation can be presented in the form: \[w(z)=-1+\frac13\frac{d\ln(\delta H^2/H_0^2)}{d\ln(1+z)},\] where \[\delta H^2 = H^2 - \frac{8\pi G}{3}\rho_m\] describes the contribution into the Universe's expansion rate of all components other than matter.


Problem 32

Express the time derivative of a scalar field through its derivative with respect to redshift $d\varphi/dz.$


Problem 33

Show that the particle horizon does not exist for the case of quintessence because the corresponding integral diverges (see Chapter 2(3)).


Problem 34

Show that in a Universe filled with quintessence the number of observed galaxies decreases with time.


Problem 35

Let $t$ be some time in the distant past $t\ll t_0$. Show that in a Universe dominated by a substance with state parameter $w>-1$ the current cosmic horizon (see Chapter 3) is \[R_h(t_0)\approx\frac32(1+\langle w\rangle)t_0,\] where $\langle w\rangle$ is the time-averaged value of $w$ from $t$ to the present time \[\langle w\rangle\equiv\frac{1}{t_0}\int\limits_t^{t_0} w(t)dt.\]


Problem 36

From WMAP$^*$ observations we infer that the age of the Universe is $t_0\approx13.7\cdot10^9$ years and cosmic horizon equals to $R_h(t_0)=H_0^{-1}\approx13.5\cdot10^9$ light-years. Show that these data imply existence of some substance with equation of state $w<-1/3$, - "dark energy".
$^*$ Wilkinson Microwave Anisotropy Probe is a spacecraft which measures differences in the temperature of the Big Bang's remnant radiant heat - the cosmic microwave background radiation - across the full sky.


Problem 37

The age of the Universe today depends upon the equation-of-state of the dark energy. Show that the more negative parameter $w$ is, the older Universe is today.


Problem 38

Consider a Universe filled with dark energy with state equation depending on the Hubble parameter and its derivatives, \[p=w\rho+g(H,\dot H, \ddot H,\ldots,;t).\] What equation does the Hubble parameter satisfy in this case?


Problem 39

Show that taking function $g$ (see the previous problem) in the form \[g(H,\dot H, \ddot H)=-\frac{2}{\kappa^2}\left(\ddot H + \dot H + \omega_0^2 H + \frac32(1+w)H^2-H_0\right),\ \kappa^2=8\pi G\] leads to the equation for the Hubble parameter identical to the one for the harmonic oscillator, and find its solution.


Problem 40

Find the time dependence of the Hubble parameter in the case of function $g$ (see problem #DE64) in the form \[g(H;t)= -\frac{2\dot f(t)}{\kappa^2f(t)}H,\ \kappa^2=8\pi G\] where $f(t)=-\ln(H_1+H_0\sin\omega_0t)$, $H_1>H_0$ is arbitrary function of time.


Problem 41

Show that in an open Universe the scalar field potential $V[\varphi(\eta)]$ depends monotonically on the conformal time $\eta$.


Problem 42

Reconstruct the dependence of the scalar field potential $V(a)$ on the scale factor basing on given dependencies for the field's energy density $\rho_\varphi(a)$ and state equation parameter $w(a)$.


Problem 43

Find the quintessence potential providing the power law growth of the scale factor $a\propto t^p$, where the accelerated expansion requires $p>1$.


Problem 44

Let $a(t)$, $\rho(t)$, $p(t)$ be solutions of Friedman equations. Show that for the case $k=0$ the function $\psi_n\equiv a^n$ is the solution of "Schr\"odinger equation" $\ddot\psi_n=U_n\psi_n$ with potential [see A.V.Yurov, arXiv:0905.1393] \[U_n=n^2\rho-\frac{3n}{2}(\rho+p).\]


Problem 45

Consider flat FLRW Universe filled with a scalar field $\varphi$. Show that in the case when $\varphi=\varphi(t)$, the Einstein equations with the cosmological term are reduced to the "Schrödinger equation" \[\ddot\psi=3(V+\Lambda)\psi\] with $\psi=a^3$. Derive the equation for $\varphi(t)$ (see A.V.Yurov, arXiv:0305019).


Problem 46

Consider FLRW space-time filled with non-interacting matter and dark energy components. Assume the following forms for the equation of state parameters of matter and dark energy \[w_m=\frac{1}{3(x^\alpha+1)},\quad w_{DE}=\frac{\bar{w}x^\alpha}{x^\alpha+1},\] where $x=a/a_*$ with $a_*$ being some reference value of $a$, $\alpha$ is some positive constant and $\bar{w}$ is a negative constant. Analyze the dynamics of the Universe in this model. (see S.Kumar,L.Xu, arXiv:1207.5582)

Tracker Fields

A special type of scalar fields - the so-called tracker fields - was discovered at the end of the nineties. The term reflects the fact that a wide range of initial values for the fields of such type rapidly converges to the common evolutionary track. The initial values of energy density for such fields may vary by many orders of magnitude without considerable effect on the long-time asymptote. The peculiar property of tracker solutions is the fact that the state equation parameter for such a field is determined by the dominant component of the cosmological background.
It should be stressed that, unlike the standard attractor, the tracker solution is not a fixed point (in the sense of a solution corresponding to the fixed point in a system of autonomous differential equations ): the ratio of the scalar field energy density to that of background component (matter or radiation) continuously changes as the quantity $\varphi$ descends along the potential. It is well desirable feature because we want the energy density $\varphi$ to exceed ultimately the background density and to transfer the Universe into the observed phase of the accelerated expansion.
Below we consider a number of concrete realizations of the tracker fields.


Problem 47

Show that initial value of the tracker field should obey the condition $\varphi_0=M_{Pl}$.


Problem 48

Show that densities of kinetic and potential energy of the scalar field $\varphi$ in the potential of the form \[V(\varphi)=M^4\exp(-\alpha\varphi M),\quad M\equiv\frac{M_{PL}^2}{16\pi}\] are proportional to the density of the concomitant component (matter or radiation) and therefore it realizes the tracker solution.


Problem 49

Consider a scalar field potential \[V(\varphi)=\frac A n\varphi^{-n},\] where $A$ is a dimensional parameter and $n>2$. Show that the solution $\varphi(t)\propto t^{2/(n+2)}$ is a tracker field under condition $a(t)\propto t^m$, $m=1/2$ or $2/3$ (either radiation or non-relativistic matter dominates).


Problem 50

Show that the scalar field energy density corresponding to the tracker solution in the potential \[V(\varphi)=\frac A n\varphi^{-n}\] (see the previous problem #DE73) decreases slower than the energy density of radiation or non-relativistic matter.


Problem 51

Find the equation of state parameter $w_\varphi\equiv p_\varphi/\rho_\varphi$ for the scalar field of problem #DE73.


Problem 52

Use explicit form of the tracker field in the potential of problem #DE73 to verify the value of $w_\varphi$ obtained in the previous problem.

The K-essence

Let us introduce the quantity $$X\equiv \frac{1}{2}{{g}^{\mu \nu }}\frac{\partial \varphi }{\partial {{x}^{\mu }}}\frac{\partial \varphi }{\partial {{x}^{\nu }}}$$ and consider action for the scalar field in the form $$ S=\int{{{d}^{4}}x\sqrt{-g}}\; L\left( \varphi ,X \right), $$ where Lagrangian $L$ is generally speaking an arbitrary function of variables $\varphi$ and $X.$ The dark energy model realized due to modification of the kinetic term with the scalar field, is called the $k$-essence. The traditional action for the scalar field corresponds to $$ L\left( \varphi ,X \right)=X-V(\varphi ). $$ In the problems proposed below we restrict ourselves to the subset of Lagrangians of the form $$ L\left( \varphi ,X \right)=K(X)-V(\varphi ), $$ where $K(X)$ is a positively defined function of kinetic energy $X$. In order to describe a homogeneous Universe we should choose $$ X=\frac{1}{2}{\dot{\varphi}^{2}}. $$


Problem 53

Find the density and pressure of the $k$-essence.


Problem 54

Construct the equation of state for the $k$-essence.


Problem 55

Find the sound speed in the $k$-essence.


Problem 56

The sound speed $c_s$ in any medium must satisfy two fundamental requirements: first, the sound waves must be stable and second, its velocity value should be low enough to preserve the causality condition. Therefore \[0\le c_s^2\le1.\] Reformulate the latter condition in terms of scale factor dependence for the equation of state parameter $w(a)$ for the case of the $k$-essence.


Problem 57

Find the state equation for the simplified $k$-essence model with Lagrangian $L=F(X)$ (the so-called pure kinetic $k$-essence).


Problem 58

Find the equation of motion for the scalar field in the pure kinetic $k$-essence.


Problem 59

Show that the scalar field equation of motion for the pure kinetic $k$-essence model gives the tracker solution.


Phantom Energy

The full amount of available cosmological observational data shows that the state equation parameter $w$ for dark energy lies in a narrow range near the value $w=-1$. In the previous subsections we considered the region $-1\le w\le-1/3$. The lower bound $w=-1$ of the interval corresponds to the cosmological constant, and all the remainder can be covered by the scalar fields with canonic Lagrangians. Recall that the upper bound $w=-1/3$ appears due to the necessity to provide the observed accelerated expansion of Universe. What other values of parameter $w$ can be used? The question is very hard to answer for the energy component we know so little about. General Relativity restricts possible values of the energy - momentum tensor by the so-called "energy conditions" (see Chapter 2). One of the simplest among them is the so-called Null Dominant Energy Condition (NDEC) $\rho+p\ge0$. The physical motivation of the latter is to avoid the vacuum instability. Applied to the dynamics of Universe, the NDEC requires that density of any allowed energy component cannot grow with the expansion of the Universe. The cosmological constant with $\dot\rho_\Lambda=0$, $\rho_\Lambda=const$ represents the limiting case. Because of our ignorance concerning the nature of dark energy it is reasonable to question whether this mysterious substance can differ from the already known "good" sources of energy and if it can violate the NDEC. Taking into account that dark energy must have positive density (it is necessary to make the Universe flat) and negative pressure (to provide the accelerated expansion of Universe), the violation of the NDEC must lead to $w<-1$. Such substance is called the phantom energy. The phantom field $\varphi$ minimally coupled to gravity has the following action: \[S=\int d^4x \sqrt{-g}L=-\int d^4x \sqrt{-g}\left[\frac12g^{\mu\nu}\frac{\partial\varphi}{\partial x_\mu} \frac{\partial\varphi}{\partial x_\nu}+V(\varphi)\right],\] which differs from the canonic action for the scalar field only by the sign of the kinetic term.


Problem 60

Show that the action of a scalar field minimally coupled to gravitation \[S=\int d^4x\sqrt{-g}\left[\frac12(\nabla\varphi)^2-V(\varphi)\right]\] leads, under the condition $\dot\varphi^2/2<V(\varphi)$, to $w_\varphi<-1$, i.e. the field is phantom.


Problem 61

Obtain the equation of motion for the phantom scalar field described by the action of the previous problem.


Problem 62

Find the energy density and pressure of the phantom field.


Problem 63

Show that the phantom energy density grows with time. Find the dependence $\rho(a)$ for $w=-4/3$.


Problem 64

Show that the phantom scalar field violates all four energety conditions.


Problem 65

Show that in the phantom scalar field $(w<-1)$ dominated Universe the condition $\dot{H}>0$ always holds.


Problem 66

As we have seen in Chapter 3, the Friedman equations, describing spatially flat Universe, possess the duality, which connects the expanding and contracting Universe by appropriate transformation of the state equation. Consider the Universe where the weak energetic condition $\rho\ge0,\ \rho+p\ge0$ holds and show that the ideal liquid associated with the dual Universe is a phantom liquid or the cosmological constant.


Problem 67

Show that the Friedman equations for the Universe filled with dark energy in the form of cosmological constant and a substance with the state equation $p=w\rho$ can be presented in the form of nonlinear oscillator (see M.Dabrowski arXiv:0307128 ) \[\ddot X-\frac{D^2}{3}\Lambda X+D(D-1)kX^{1-2/D}=0\] where \[X=a^{D(w)},\quad D(w)=\frac32(1+w).\]


Problem 68

Show that the Universe dual to the one filled with a free scalar field, is described by the state equation $p=-3\rho$.


Problem 69

Show that in the phantom component of dark energy the sound speed exceeds the light speed.


Problem 70

Construct the phantom energy model with negative kinetic term in the potential satisfying the slow-roll conditions \[\frac 1 V \frac{dV}{d\varphi}\ll1\] and \[\frac 1 V \frac{d^2V}{d\varphi^2}\ll1.\]

Disintegration of Bound Structures

Historically the first criterion for decay of gravitationally bound systems due to the phantom dark energy was proposed by Caldwell, Kamionkowski and Weinberg (CKW) (see arXiv:astro-ph/0302506v1). The authors argue that a satellite orbiting around a heavy attracting body becomes unbound when total repulsive action of the dark energy inside the orbit exceeds the attraction of the gravity center. Potential energy of gravitational attraction is determined by the mass $M$ of the attracting center, while the analogous quantity for repulsive potential equals to $\rho+3p$ integrated over the volume inside the orbit. It results in the following rough estimate for the disintegration condition \begin{equation}\label{disintegration} -\frac{4\pi}{3}(\rho+3p)R^3\simeq M. \end{equation}


Problem 71

Show that for $w\ge-1$ a system gravitationally bound at some moment of time (Milky Way for example) remains bound forever.


Problem 72

Show that in the phantom energy dominated Universe any gravitationally bound system will dissociate with time.


Problem 73

Show that in a Universe filled with non-relativistic matter a hydrogen atom will remain a bound system forever.


Problem 74

Demonstrate, that any gravitationally bound system with mass $M$ and radius (linear scale) $R$, immersed in the phantom background $\left( {w < - 1} \right)$ will decay in time \[t \simeq P\frac{|1+3w|}{|1+w|}\frac29\sqrt{\frac{3}{2\pi}}\] before Big Rip. Here \[P=2\pi\sqrt{\frac{R^3}{GM}}\] is the period on the circular orbit of radius $R$ around the considered system.


Problem 75

Use the result of the previous problem to determine the time of disintegration for the following systems: galaxy clusters, Milky Way, Solar System, Earth, hydrogen atom. Consider the case $w=-3/2$.


Big Rip, Pseudo Rip, Little Rip

The future finite-time singularity is an essential element of phantom cosmology (see S.Nojiri, S. Odintsov, arXiv:hep-th/0505215). One may classify the future singularities as in the following way (see S.Nojiri, S. Odintsov and S.Tsajikava, arXiv:hep-th/0501025):
1. For $t\to t_s$, $a\to\infty$, $\rho\to\infty$, $|p|\to\infty$ ("Big Rip").
The density of phantom dark energy and scale factor become infinite at some finite time $t_s$.
2. For $t\to t_s$, $a\to a_s$, $\rho\to\rho_s$ or $\rho\to0$, $|p|\to\infty$ ("sudden singularity").
The condition $w<-1$ is necessary for future singularities, but it is not sufficient. If $w$ approaches to $-1$ sufficiently rapidly, then it is possible to have a model in which there are no future singularities. Models without future singularities in which $\rho_{DE}$ increases with time will eventually lead to dissolution of bound systems. This process received the name "Little Rip" (see P.Frampton, K.Ludwick and R.Scherrer, arXiv:1106.4996). In the Big Rip the scale factor and energy density diverge at finite future time. As opposed to Big Rip in the $\Lambda$CDM, there is no such divergence. Little Rip represents an interpolation between these two limit cases.
3. For $t\to t_s$, $a\to a_s\ne0$, $\rho\to\infty$, $|p|\to\infty$.
4. For $t\to t_s$, $a\to a_s\ne0$, $\rho\to\rho_s$ (including $\rho_s=0$), while derivatives of $H$ diverge.
Here $t_s$, $a_s\ne0$ and $\rho_s$ are constants.



Problem 76

For the flat Universe composed of matter $(\Omega_m\simeq0.3)$ and phantom energy $(w=-1.5)$ find the time interval left to the Big Rip.


Immediate consequence of approaching the Big Rip is the dissociation of bound systems due to negative pressure inside them.


Problem 77

Show that all little-rip models can be described by condition $\ddot f>0$ where $f(t)$ is a nonsingular function such that $a(t)=\exp[f(t)]$.


Problem 78

Consider the approach of the following authors (see S. Nojiri, S.D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005); S. Nojiri and S.D. Odintsov, Phys. Rev. D 72, 023003 (2005); H. Stefancic, Phys. Rev. D 71, 084024 (2005)), who expressed the pressure as a function of the density in the form \[p=-\rho-f(\rho).\] Show that condition $f(\rho>0)$ ensures that the density increases with the scale factor.


Problem 79

Find the dependencies $a(\rho)$ and $t(\rho)$ for the case of flat Universe filled by a substance with the following state equation \[p=-\rho-f(\rho).\]


Problem 80

Solve the previous problem in the case of \(f(\rho)A\rho^\alpha,\ \alpha=const.\)


Problem 81

Find the condition for big-rip singularity in the case $p=-\rho-f(\rho).$


Problem 82

Show that taking a power law for $f(\rho)$, namely $f(\rho)=A\rho^\alpha$ a future singularity can be avoided for $\alpha\le1/2$.


Problem 83

Solve the previous problem using the condition for absence of future singularities obtained in Problem #RIPS_1.


Problem 84

Formulate the condition for the absence of a finite-time future (Big Rip) singularity in terms of function $\rho(a)$ .


The problems below develop an alternative approach to investigate the singularities in the phantom Universe (see P-H. Chavanis, arXiv:1208.1195)


Problem 85

Consider the polytropic equation of state \[p=\alpha\rho+k\rho^{1+1/n}\equiv-\rho+\rho\left(1+\alpha+k\rho^{1/n}\right)\] under assumption $-1<\alpha\le1$. The case $\alpha=-1$ is treated separately in Problem #RIPS_7_4. The additional assumption $1+\alpha+k\rho^{1/n}\le0$ (and necessary condition $k<0$) guarantees that the density increases with the scale factor. This corresponds to phantom Universe. Find explicit dependence $\rho(a)$ and analyze limits $a\to0$ and $a\to\infty$.


Problem 86

Consider the previous problem with $\alpha=-1$ and $k<0$. This equation of state was introduced by Nojiri and Odintsov (see problem #RIPS_7_3). Chavanis re-derives their results in a more transparent form.


The little-rip dissociates all bound structures, but
the strength of the dark energy is not enough to rip
apart space-time as there is no finite-time singularity
P. Frampton, K. Ludwick1, and R. Scherrer

(see A. Astashenok, S. Nojiri, S. Odintsov, and R. Scherrer, arXiv:1203.1976)


Problem 87

Show that for any bound system the rip always occurs when either $H$ diverges or $\dot H$ diverges (assuming $\dot H>0$ ( expansion of Universe is accelerating)).


Problem 88

Solve the previous problem in terms of function $f(\rho)$.


Problem 89

Perform analysis of possible singularities in terms of characteristics of the scalar field $\varphi$ with the potential $V(\varphi)$.


All the Big Rip, Little Rip and Pseudo-Rip arise from the assumption that the dark energy density $\rho(a)$ is monotonically increasing. Let us investigate what will happen if this assumption is broken and then propose a so-called "Quasi-Rip" scenario, which is driven by a type of quintom dark energy. In this work, we consider an explicit model of Quasi-Rip in details. We show that Quasi-Rip has an unique feature different from Big Rip, Little Rip and Pseudo-Rip. Our universe has a chance to be rebuilt in the ash after the terrible rip. This might be the last hope in the "hopeless" rip.



Problem 90

So-called soft singularities are characterized by a diverging $\ddot a$ whereas both the scale factor $a$ and $\dot a$ are finite. Analyze features of intersections between the soft singularities and geodesics.


Problem 91

(see F.Cannata, A. Kamenshchik, D.Regoli, arXiv:0801.2348) The power law cosmological evolution $a(t)\propto t^\beta$ leads to the Hubble parameter $H(t)\propto 1/t$. Consider a "softer" version of the cosmological evolution given by the law \[H(t)=\frac{S}{t^\alpha},\] where $S$ is a positive constant and $0<\alpha<1$. Analyze the dynamics of such model at $t\to 0$.


Problem 92

Reconstruct the potential of the scalar field model, producing the given cosmological evolution $H(t)$.


Problem 93

Reconstruct the potential of the scalar field model, producing the cosmological evolution \begin{equation}\label{RIPS_68} H(t)=\frac{S}{t^\alpha}, \end{equation} using the technique described in the previous problem.

The Statefinder

In the models including dark energy in different forms it is useful to introduce a pair of cosmological parameters $\{r,s\}$, which is called the statefinder (see V.Sahni, T.Saini, A.Starobinsky, U.Alam astro-ph/0201498): \[r\equiv\frac{\dddot a}{aH^3},\ s\equiv\frac{r-1}{3(q-1/2)}.\] These dimensionless parameters are constructed from the scale factor and its derivatives. Parameter $r$ is the next member in the sequence of the kinematic characteristics describing the Universe's expansion after the Hubble parameter $H$ and the deceleration parameter $q$ (see Chapter "Cosmography"). Parameter $s$ is the combination of $q$ and $r$ chosen in such a way that it is independent of the dark energy density. The values of these parameters can be reconstructed with high precision basing on the available cosmological data. After that the statefinder can be successfully used to identify different dark energy models.


Problem 94

Explain the advantages for the description of the current Universe's dynamics brought by the introduction of the statefinder.


Problem 95

Express the statefinder $\{r,s\}$ in terms of the total density, pressure and their time derivatives for a spatially flat Universe.


Problem 96

Show that for a flat Universe filled with a two-component liquid composed of non--relativistic matter (dark matter + baryons) and dark energy with relative density $\Omega _{DE} = \rho _{DE} /\rho _{cr} $ the statefinder takes the form $$ r = 1 + {\frac92}\Omega _{DE} w(1 + w) - {\frac32}\Omega _{DE} {\frac{\dot w}{H}}; $$ $$ s = 1 + w - {\frac13}{\frac{\dot w}{wH}};\quad w \equiv {\frac{p_{DE} } {\rho _{DE} }}. $$


Problem 97

Express the statefinder in terms of Hubble parameter $H(z)$ and its derivatives.


Problem 98

Find the statefinders
a) for dark energy in the form of cosmological constant;
b) for the case of time--independent state equation parameter $w$;
c) for dark energy in the form of quintessence.


Problem 99

Express the photometric distance $d_L(z)$ through the current values of parameters $q$ and $s$.


Crossing the Phantom Divide

In the quintessence model of dark energy $-1<w<-1/3$. In the phantom model with negative kinetic energy $w<-1$. Recent cosmological data seem to indicate that there occurred the crossing of the phantom divide line in the near past. This means that equation of state parameter $w_{DE}$ crosses the phantom divide line $w_{DE}=-1$. This crossing to the phantom region is possible neither for an ordinary minimally coupled scalar field nor for a phantom field. There are at least three ways to solve this problem. If dark energy behaves as quintessence at early stage, and evolves as phantom at the later stage, a natural suggestion would be to consider a 2-field model (quintom model): a quintessence and a phantom. The next possibility, discussed in the next Chapter, is to consider an interacting model, in which dark energy interacts with dark matter. Yet another possibility would be that General Relativity fails at cosmological scales. In this case quintessence or phantom energy can cross the phantom divide line in a modified gravity theory. We investigate this approach in Chapter 12.



Problem 100

Show that at the point of transition between the quintessence and the phantom phases $\dot H$ vanishes.


Problem 101

Show that the sound speed of a single perfect barotropic fluid is diverges when $w$ crosses the phantom divide line.


Problem 102

Find a dynamical law for the equation of state parameter $w=p/\rho$ in the barotropic cosmic fluid (see N.Caplar, H.Stefancic, arXiv:1208.0449).


Problem 103

Using the results of previous problem, find the functions $w(z)$, $\rho(z)$ and $p(z)$ for the simplest possibility $c_S=const$.


Problem 104

Realize the procedure described in the problem #DE117_11 for the case of a minimally coupled scalar field $\varphi$ with potential $V(\varphi)$ in a spatially flat Universe.


Problem 105

Consider the case of Universe filled with non-relativistic matter and quintessence and show that the condition to cross the phantom divide line $w=-1$ is equivalent to sign change in the following expression \[\frac{dH^2(z)}{dz}-3\Omega_{m0}H_0^2(1+z)^2.\]


Problem 106

Consider a model with the scale factor of the form \[a=a_c\left(\frac{t}{t-t_s}\right)^n,\] where $a_c$ is a constant, $n>0$, $t_s$ is the time of a Big Rip singularity. Show that on the interval $0<t<t_s$ there is crossing of the phantom divide line $w=-1$.


Problem 107

Show, that for the model considered in the previous problem the parameter $H(t)$ and density $\rho(t)$ achieve their minimal values at the phantom divide point. (see K.Bamba, S.Capozziello, S.Nojiri, S.Odintsov, arXiv:1205.3421)


Problem 108

Find condition of intersection with the line $w=-1$ for the quintom Lagrangian \[L=\frac12g^{\mu\nu}\left(\frac{\partial\varphi}{\partial x^\mu}\frac{\partial\varphi}{\partial x^\nu} - \frac{\partial\psi}{\partial x^\mu}\frac{\partial\psi}{\partial x^\nu} \right)-W(\varphi,\psi).\]