# 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 1

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

### Problem 1

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 1

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 1

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 1

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 1

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 1

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

### Problem 1

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

### Problem 1

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 1

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 1

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

### Problem 1

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 1

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 1

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 1

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 1

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 1

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

### Problem 1

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

### Problem 1

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 1

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

### Problem 1

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 1

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

### Problem 1

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

### Problem 1

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

### Problem 1

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 1

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

### Problem 1

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

### Problem 1

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 1

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 1

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 1

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

### Problem 1

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

### Problem 1

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

### Problem 1

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 1

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 1

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 1

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 1

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 1

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 1

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

### Problem 1

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 1

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 1

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 1

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 1

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 1

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

### Problem 1

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 1

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 1

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 1

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

### Problem 1

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 1

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

### Problem 1

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

### Problem 1

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

### Problem 1

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 1

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

### Problem 1

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

### Problem 1

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 1

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 1

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

### Problem 1

Find the energy density and pressure of the phantom field.

### Problem 1

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

### Problem 1

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

### Problem 1

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

### Problem 1

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 1

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 1

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

### Problem 1

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

### Problem 1

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 $$\label{disintegration} -\frac{4\pi}{3}(\rho+3p)R^3\simeq M.$$

### Problem 1

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

### Problem 1

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

### Problem 1

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

### Problem 1

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 1

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 $w=-3/2$.