text:
\section{Introduction}

Interaction of charged fields with the electromagnetic background of a charged black hole is an active area of research in the contexts of black hole physics \cite{Bekenstein:1973,HodPiran,Hod:2012-2013,Konoplya:2002wt,Li:2012rx,Konoplya:2002ky,Konoplya:2007zx,Li:2013jna,Konoplya:2013rxa,Uchikata:2011zz,Konoplya:2008au,Kokkotas:2010zd} and gauge/gravity duality description of superconductors with high critical temperature \cite{Horowitz:2010nh,Horowitz:2010gk,Gubser:2008px,Son:2013xra,Yao:2013sha,Konoplya:2009hv}. An essential difference of propagation of \emph{charged} fields in the vicinity of charged black holes is that the electromagnetic energy of the black hole can be extracted and carried out by a wave with increased amplitude \cite{Bekenstein:1973}. This effect is called superradiance and is mainly known for rotating black holes \cite{Starobinsky}, for which it allows a wave to carry out part of the black hole's momentum. Frequently, the stability of a black hole or a field in its background cannot be tested analytically, so that a full analysis of quasinormal spectrum of a black hole is necessary (see \cite{Konoplya:2011qq,Ishibashi:2011ws} for recent reviews of black hole stability).  When the field has a nonzero mass, the corresponding bound states (with perfectly reflecting boundary conditions) may be unstable \cite{Herdeiro:2013pia,Degollado:2013bha}, but no real instability occurs upon imposing the correct, \emph{quasinormal} boundary conditions \cite{Konoplya:2013rxa}. Quasinormal modes are proper modes characterizing evolution of perturbation at later times and requiring purely ingoing wave at the event horizon and purely outgoing wave at infinity (or de Sitter horizon).

First study of evolution of perturbation of a charged scalar field in the background of a charged asymptotically flat black hole was performed by S. Hod \cite{HodPiran}, who found asymptotic tails at late times. Further study of the earlier, \emph{quasinormal}, stage of the charged fields perturbation for Reissner-Nordstr\"om \cite{Konoplya:2002wt},  Reissner-Nordstr\"om -anti-de Sitter \cite{Konoplya:2002ky} and Kerr-Newman(-de Sitter) space-times \cite{Konoplya:2007zx,Konoplya:2013rxa} showed that:
\begin{itemize}
  \item Massless conformally coupled charged scalar field and charged Dirac field in the background of Kerr-Newman and Kerr-Newman-de Sitter show no instability in their quasinormal spectrum \cite{Konoplya:2007zx}.
  \item A minimally coupled massive charged scalar field must be stable in the background of an asymptotically flat charged black hole, such as Reissner-Nordstr\"om or Kerr-Newman \cite{Konoplya:2013rxa}.
\end{itemize}

The Reissner-Nordstr\"om-de Sitter is stable against gravitational and electromagnetic perturbations in 4-dimensional space-time, showing the instability only in higher dimensions \cite{Konoplya:2013sba,Konoplya:2014sna}. However, a new interesting instability has been found recently for a minimally coupled charged scalar field in the Reissner-Nordstr\"om-de Sitter background by Z. Zhu and collaborators \cite{Zhu:2014sya}. This instability has a number of peculiar properties:
\begin{enumerate}
  \item It takes place only for the most symmetric $s$-mode of the perturbation, while higher multipoles are stable.
  \item It occurs for small values of $e Q$ coupling, but surprisingly, large values of $e Q$ stabilize the system.
  \item It develops slowly, so that considerable growth is achieved only at times comparable with cosmological scales.
  \item Even relatively small mass of the field acts as a stabilizing factor.
\end{enumerate}

In the present paper we make further investigation of properties of the above instability. First, we confirm the presence of the instability not only by reproducing the time-domain profiles of \cite{Zhu:2014sya}, but also detecting the growing modes in the frequency domain with the help of Frobenius expansion. Next, we analyze the parametric region of instability. Finally, we show that all the growing modes have nonzero real part of $\omega$, which satisfy the condition of superradiance. We prove analytically that the condition of superradiance is necessary but not sufficient for the instability of the charged scalar field in the background of the Riessner-Nordstr\"om-de Sitter space-time.

The paper is organized as follows. Sec. II introduces the basic formula on the background space-time and the equation of motion for a massive charged scalar field. In Sec. III we analyze the connection between superradiance and instability. Sec. IV is devoted to numerical methods used in the paper, while Sec. V is a summary of obtained numerical results.

\section{The basic equations}

The Reissner-Nordstr\"om-de Sitter black hole background can be described by the metric
\begin{eqnarray}
ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta d\phi^2)\;,
\end{eqnarray}
where
\begin{eqnarray}
f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\Lambda r^2}{3}\,.
\end{eqnarray}
and $M$ and $Q$ are the black hole mass and charge, respectively, $\Lambda$ is the cosmological constant.

We shall designate the Cauchy, event, and cosmological horizons as $r_-$, $r_+$, and $r_c$ respectively ($r_- < r_+ < r_c$).
Then, the metric function $f(r)$ can be written as,
\begin{eqnarray}
f(r)=\frac{\Lambda}{3r^2}(r-r_+)(r-r_-)(r_c-r)(r-r_o)\;.
\end{eqnarray}

A charged, massive scalar field $\psi$ in curved space-time obeys the Klein-Gordon equation
\begin{eqnarray}
[(\nabla^\nu-ieA^\nu)(\nabla_\nu-ieA_\nu)-\mu^2]\psi=0\;,
\end{eqnarray}
where $e$ and $\mu$ are, respectively, the charge and mass of
the field and $A_\mu=-\delta^0_\mu Q/r$ is the electromagnetic 4-potential of the black hole.

After the standard separation of angular variables with the help of spherical harmonics and introduction a new wave function $\Psi$, the above equation of motion can be reduced to the following form:
\begin{eqnarray}
-\frac{\partial^2\Psi}{\partial t^2}+\frac{\partial^2\Psi}{\partial r_*^2}-2i\Phi\frac{\partial\Psi}{\partial t}+(\Phi^2-V)\Psi=0\;,\label{radial_eq}
\end{eqnarray}
where $dr_*=\frac{dr}{f(r)}$ is the tortoise coordinate, and
\begin{eqnarray}
\Phi(r)=\frac{eQ}{r},\quad V(r)=f(r)\bigg(\mu^2+\frac{l(l+1)}{r^2}+\frac{f'(r)}{r}\bigg)\;.\label{V}
\end{eqnarray}

\section{Superradiance and instability}

In order to analyze superradiance around the Reissner-Nordstr\"om-de Sitter black hole, we consider the scattering problem for a charged scalar field.
Owing to the $\partial_t$ Killing vector for stationary space-times, the following ansatz $\Psi(r,t)=\exp[-\imo\omega t]\Psi(r)$ transforms (\ref{radial_eq}) to the Sch\"ordinger-like form
\begin{eqnarray}
{\frac{d^2\Psi}{dr_*^2}}+(\omega-\Phi(r))^2\Psi-V(r)\Psi=0\;.\label{radial}
\end{eqnarray}
Due to the $\re{\omega}\rightarrow-\re{\omega}$, $\Phi(r)\rightarrow-\Phi(r)$ symmetry of (\ref{radial}), consideration of the $eQ>0$ case only is sufficient.

We shall be using the standard scattering boundary conditions, implying that the wave coming from the cosmological horizon will be partially reflected from the potential barrier and come back to the cosmological horizon, while at the event horizon purely in-coming wave is always required:
\begin{eqnarray}
\Psi\sim\left\{
  \begin{array}{ll}
    Be^{-i(\omega-\Phi(r_+))r_*}, & r\rightarrow r_+, \\
    e^{-i(\omega-\Phi(r_c))r_*}+A e^{+i(\omega-\Phi(r_c))r_*}, & r\rightarrow r_c.
  \end{array}
\right.
\end{eqnarray}
Since the Wronskian of the complex conjugated solutions is constant, one can derive the following relation
\begin{eqnarray}
|A|^2=1-{\frac{\omega-\Phi(r_+)}{\omega-\Phi(r_c)}}|B|^2\;.
\end{eqnarray}
Amplification of the incident wave, that is, a \emph{superradiance}, occurs when
\begin{eqnarray}
\Phi(r_c)=\frac{eQ}{r_c}<\omega<\frac{eQ}{r_+}=\Phi(r_+).\label{regime}
\end{eqnarray}
When the cosmological constant $\Lambda$ approaches zero, the above superradiant condition Eq.~(\ref{regime}) is reduced to the one for asymptotically flat space-times \cite{Bekenstein:1973,Hod:2012-2013} ($\omega$ is the real oscillation frequency).

Let us now understand how the condition of superradiance is related to instability, that is, to the possible presence of growing quasinormal modes in the black hole's spectrum. For this, let us consider the quasinormal (and not scattering) boundary conditions and imply that frequencies, generally, are complex:

\begin{eqnarray}\label{QNBC}
\Psi\sim\left\{
  \begin{array}{rcl}
    e^{-i(\omega-\Phi(r_+))r_*}, \quad & r\rightarrow r_+, &\quad (r_* \rightarrow -\infty), \\
    e^{+i(\omega-\Phi(r_c))r_*}, \quad & r\rightarrow r_c, &\quad (r_* \rightarrow +\infty).
  \end{array}
\right.
\end{eqnarray}

We shall prove that the real part of $\omega$ satisfying (\ref{regime}) is the necessary, but not sufficient, condition for the instability.

First, let us multiply (\ref{radial}) by the complex conjugated $\Psi^\star$ and integrate the first term by parts,
\begin{eqnarray}
\Psi^\star(r_\star)\Psi^\prime(r_\star)\Biggr|_{-\infty}^\infty + \intop_{-\infty}^\infty (\omega-\Phi)^2|\Psi(r_\star)|^2dr_\star \\\nonumber=\intop_{-\infty}^\infty\left(V|\Psi(r_\star)|^2 + |\Psi^\prime(r_\star)|^2\right)dr_\star.
\end{eqnarray}
The righthand side is real and positive since the effective potential $V$ is positive. Because $\Phi(r)$ is a monotonically decreasing function, taking imaginary part of both sides, we find that if either $\re{\omega}\geq\Phi(r_+)\geq\Phi(r)$ or $\re{\omega}\leq\Phi(r_c)\leq\Phi(r)$ is satisfied, then we have $\im{\omega}<0$. This proves that the instability can take place only if
\begin{eqnarray}\label{superradiant}
\Phi(r_c)=\frac{eQ}{r_c}<\re{\omega}<\frac{eQ}{r_+}=\Phi(r_+).
\end{eqnarray}

\begin{figure}
\resizebox{\linewidth}{!}{\includegraphics*{fig5.eps}}
\caption{Real and imaginary part of the signal for $eQ=0.4$, $r_-=0.07r_+$, $r_+=0.186r_c$ ($Q=0.5M$, $\Lambda M^2=0.023$).
The dominant frequency is $\omega r_+ = 0.089160+0.000754\imo$.}\label{fig5}
\end{figure}

Indeed, in the time-domain profiles (fig. \ref{fig5}) we observe that the real part for all the unstable modes we found satisfy (\ref{superradiant}). For $eQ$ larger than the threshold of instability the real part of the dominant mode also satisfies (\ref{superradiant}), being stable. This is qualitatively different from the higher-dimensional asymptotically anti-de Sitter black holes, for which the necessary condition is also the sufficient one \cite{Kodama:2009rq,Wang:2014eha}.

\section{Time-domain and frequency-domain analysis}

Here in order to integrate the wave equation (\ref{radial_eq}) and analyze the spectrum of the perturbation, we shall use, first of all, the time-domain integration, which includes contribution from all modes.
The discretization scheme which we shall use was proposed in \cite{Abdalla:2010}.
Defining $\Psi(r_*,t)=\Psi(j\Delta r_*, i\Delta t)=\Psi_{j,i}$, $V(r(r_*))=V(j\Delta r_*)=V_j$ and $\Phi(r(r_*))=\Phi (j\Delta r_*)=\Phi_j$, we can write down (\ref{radial_eq}) as
\begin{eqnarray}
&&-\frac{(\Psi_{j,i+1}-2\Psi_{j,i}+\Psi_{j,i-1})}{\Delta
t^2}-2i\Phi_j\frac{(\Psi_{j,i+1}-\Psi_{j,i-1})}{2\Delta t} \nonumber \\ \nonumber
&&+\frac{(\Psi_{j+1,i}-2\Psi_{j,i}+\Psi_{j-1,i})}{\Delta
r_*^2}-V_j\Psi_{j,i}={\cal O}(\Delta t,\Delta
r_*).
\end{eqnarray}
The initial Gaussian wave-package has the form
$\Psi(r_*,t=0)=\exp\bigg[-{(r_*-a)^2/2b^2}\bigg]$, $\Psi(r_*,t<0)=0$.
Then, the evolution of $\Psi$ can be described by the following expression
\begin{eqnarray}
\Psi_{j,i+1}=-{\frac{(1-i\Phi_j\Delta t)\Psi_{j,i-1}}{1+i\Phi_j\Delta t}}+{\frac{\Delta t^2}{\Delta r_*^2}}
{\frac{\Psi_{j+1,i}+\Psi_{j-1,i}}{1+i\Phi_j\Delta t}} \nonumber\\\nonumber
+\bigg(2-2{\frac{\Delta t^2}{\Delta r_*^2}}-\Delta t^2V_j\bigg){\frac{\Psi_{j,i}}{1+i\Phi_j\Delta t}}\;.
\end{eqnarray}
Following \cite{Abdalla:2010}, we choose the parameters $a=0$ and $b=\sqrt{10}$ in the Gaussian wave package and use $${\frac{\Delta t}{\Delta r_*}}=\frac{1}{2}<1,$$ making sure that $\Delta t$ is small enough to achieve the required precision of the profile.

Real part of $\omega$ cannot be extracted from a time-domain profile obtained with the help of the above integration for $|\Psi(r=const,t)|$ because charged field perturbations are complex. In \cite{Zhu:2014sya} it was erroneously reported that the growing modes have zero real part and, therefore, cannot be superradiant.
It turns out, that while absolute value of $\Psi$ does not oscillate at the late times, its phase continues changing, implying nonzero real part of $\omega$. In order to calculate $\re{\omega}$ we used the Prony method of fitting the time-domain profile data by superposition of damping exponents \cite{Berti:2007dg}
\begin{equation}\label{damping-exponents}
\Psi(r,t)\simeq\sum_{i=1}^pC_ie^{-\imo\omega_i (t-t_0)}.
\end{equation}
We consider a late time period, which starts at $t_0$ and ends at $t=N\Delta t+t_0$, where $N$ is an integer and $N\geq2p-1$. Then the formula (\ref{damping-exponents}) is valid for each value from the profile data:
\begin{equation}
x_n\equiv\Psi(r,n\Delta t+t_0)=\sum_{j=1}^pC_je^{-\imo\omega_j n\Delta t}=\sum_{j=1}^pC_jz_j^n.
\end{equation}
The Prony method allows us to find $z_i$ in terms of the known $x_n$ and, since $\Delta t$ is also known, to calculate the quasinormal frequencies $\omega_i$.

In the frequency domain we shall use the well-known method of Frobenius expansion \cite{Leaver:1985ax}. We express $\Psi$ in (\ref{radial}) as a product of the series, convergent in $r_+\leq r<r_c$, and a singular factor, which is fixed by the quasinormal boundary conditions (\ref{QNBC}) in two regular singular points, $r_+$ and $r_c$. This choice implies that the series are convergent in $r=r_c$ if and only if the corresponding solution to (\ref{radial}) satisfies the quasinormal boundary conditions. One can also find the recurrence relation for the series coefficients, which in our case has 7 terms. Using the Gaussian elimination we reduce the order of the recurrence relation. Finally, from the condition for the series convergence at $r=r_c$ we find an equation with the infinite continued fraction, which can be solved numerically with respect to the quasinormal frequency $\omega$. For more details we address the reader to our earlier paper \cite{Konoplya:2007zx} where we considered the conformally coupled charged scalar field in the same background.

\section{Numerical results}

\begin{figure}
\resizebox{\linewidth}{!}{\includegraphics*{fig1.eps}}
\caption{Time-domain profile for $eQ=0.001$, $\mu=0$, $r_-=0$, $r_+=0.2r_c$ ($Q \rightarrow 0$, $\Lambda M^2\approx0.023$).}\label{fig1}
\end{figure}

The question which could be addressed first is for which values of $e Q$ and $\Lambda$ the instability takes places. We conclude that the onset of instability occurs for \emph{arbitrarily small values of} $e Q$ and $\Lambda$. An illustration of this can be seen on fig.~\ref{fig1}, where the instability with very small growth rate $e Q=0.001$ is shown. For smaller $e Q$ the unstable mode has even smaller growth rate, which vanishes when one goes to the limit $e Q \rightarrow 0$. For the exact value $e Q =0$, there is no static mode and the normal fall-off takes place instead. For neutral fields the quasinormal modes of asymptotically de Sitter black holes approach quasinormal modes of asymptotically flat black holes in the limit $\Lambda \rightarrow 0$ \cite{Konoplya:2004uk}. As we can see here, for a charged scalar field this is not true anymore, because a new unstable mode appear for however small $\Lambda$.

\begin{table}
\caption{Crossing the threshold of instability for $r_-=0.07r_+$, $r_+=0.186r_c$ ($Q\approx0.5M$, $\Lambda M^2\approx0.023$).}\label{tabl1}
\begin{tabular}{|c|c|c|}
\hline
$eQ$&$\omega r_+$&$eQr_+/r_c$\\
\hline
$0.40$&$0.08916+0.00075\imo$&$0.07440$\\
$0.41$&$0.09149+0.00067\imo$&$0.07626$\\
$0.42$&$0.09383+0.00058\imo$&$0.07812$\\
$0.43$&$0.09617+0.00048\imo$&$0.07998$\\
$0.44$&$0.09850+0.00037\imo$&$0.08184$\\
$0.45$&$0.10083+0.00023\imo$&$0.08370$\\
$0.46$&$0.10317+0.00011\imo$&$0.08556$\\
$0.47$&$0.10551-0.00002\imo$&$0.08742$\\
$0.48$&$0.10784-0.00017\imo$&$0.08928$\\
$0.49$&$0.11017-0.00033\imo$&$0.09114$\\
\hline
\end{tabular}
\end{table}

\begin{figure}
\resizebox{\linewidth}{!}{\includegraphics*{fig2.eps}}
\caption{Time-domain profiles for $eQ=0.43$ (red, growing) $eQ=0.44$ (green) $eQ=0.45$ (blue, decaying), $\mu=0$, $r_-=0$, $r_+=0.2r_c$
($Q \rightarrow 0$, $\Lambda M^2\approx0.023$).}\label{fig2}
\end{figure}

\begin{figure}
\resizebox{\linewidth}{!}{\includegraphics*{fig3.eps}}
\caption{Time-domain profiles for $eQ=0.51$ (red, growing) $eQ=0.52$ (green) $eQ=0.53$ (blue, decaying), $\mu=0$, $r_-=0.837r_+$, $r_+=0.106r_c$ ($Q\approx M$, $\Lambda M^2\approx0.023$).}\label{fig3}
\end{figure}

On figs.~\ref{fig2}~and~\ref{fig3} one can see the time-domain profiles for various values of $e Q$ near the onset of instability: for
almost neutral black hole $Q \rightarrow 0$, but nonzero $e Q$ (fig.~\ref{fig2}) and for highly charged black hole $Q \approx M$ (fig.~\ref{fig3}). Comparison of these two plots shows that the black hole charge $Q$ only softly increases the threshold value of $e Q$ for the onset of instability.
On the contrary, larger cosmological constant $\Lambda$ significantly diminishes the region of instability as can be seen from fig.~\ref{instreg}.

\begin{table}
\caption{Dominant modes for $r_-\approx0.0717r_+$, $r_+\approx0.0108r_c$ ($Q\approx0.50M$, $\Lambda M^2\approx0.0001$).}\label{tabl2}
\begin{tabular}{|c|c|c|c|}
\hline
$eQ$&$\omega r_+$ (time domain)&$\omega r_+$ (frequency domain)&$eQr_+/r_c$\\
\hline
$0.10$&$0.001099\!\!+\!0.000044\imo$&$0.00109917\!\!+\!0.000044298\imo$&$0.00108$\\
$0.50$&$0.00638+0.00104\imo$&$0.00637557+0.00102996\imo$&$0.00541$\\
$0.75$&$0.01080+0.00049\imo$&$0.0107901+0.000490746\imo$&$0.00813$\\
$0.80$&$0.01150+0.00023\imo$&$0.0115066+0.000224642\imo$&$0.00867$\\
$0.85$&$0.01215-0.00002\imo$&$0.0121627-0.000025554\imo$&$0.00921$\\
$0.90$&$0.01277-0.00024\imo$&$0.0127799-0.000248131\imo$&$0.00975$\\
\hline
\end{tabular}
\end{table}

From Table I and II we can see that all the growing modes satisfy the superradiance condition (\ref{regime}), which is necessary but not sufficient condition for the instability. That is why, some decayed modes in Table I and II in the stable sector also satisfy the superradiance condition (\ref{regime}). The fact that all growing modes satisfy (\ref{regime}) is a strong indication that the found instability is of superradiant nature.
From Table II we can also see that the time-domain and frequency-domain calculations are in excellent concordance. Note, that in \cite{Konoplya:2007zx} no instability for the massless charged scalar field in the Kerr-Newman black hole was observed, because the scalar field was taken in its conformal form (despite eq.~(4) in \cite{Konoplya:2007zx} is written for the minimally coupled field).

\begin{figure}
\resizebox{\linewidth}{!}{\includegraphics*{fig4.eps}}
\caption{Time-domain profiles for charged $eQ=0.4$ massless (black) and massive: $\mu r_+=0.01$ (blue), $\mu r_+=0.02$ (green), and $\mu r_+=0.03$ (red, stable) scalar field in Reissner-Nordstr\"om-de Sitter background: $r_-=0.07r_+$, $r_+=0.186r_c$ ($Q=0.5M$, $\Lambda M^2=0.023$).}\label{fig4}
\end{figure}

The smaller $\Lambda$, the smaller is the instability growth rate, so that in the real world the found instability should have so small growth rate
that it should practically look like a kind of scalar ``hair''. The exponential growth could be significant only at cosmological times. Moreover, even a very small but nonzero mass of the field $\mu$ would stabilize the scalar field, though for large $\Lambda$ one needs much larger $\mu$ for stabilization. In addition, the instability requires $e Q \gg \mu M$ which means that the electromagnetic repulsion must much be larger than the gravitational attraction of a charged particle to the black hole. Thus, the instability is far beyond the accretion limit for a charged black hole (see fig.~\ref{fig4}). Fig.~\ref{instreg} shows that for small $\Lambda$, the critical value of $e Q$ at which the stabilization occurs quickly diminishes. This certainly does not contradict to the fact that the instability takes place for however small $\Lambda$.

\begin{figure}
\resizebox{\linewidth}{!}{\includegraphics*{instreg.eps}}
\caption{The instability region for a weakly charged Reissner-Nordstr\"om-de Sitter black hole ($Q \rightarrow 0$, $eQ\neq0$).}\label{instreg}
\end{figure}

\section{Conclusions}

The main result of this work is \emph{observation of a superradiant nature of the recently found instability} of a charged scalar field in the background of the 4-dimensional Reissner-Nordstr\"om-de Sitter black hole. It is evident that the distribution of the extracted from the black hole electromagnetic energy between the event horizon and cosmological horizon leads to such a bizarre instability. A further interesting step in this direction could be a nonlinear analysis of collapse of a charged field in the de Sitter space.

The instability takes place for the spherically symmetric perturbation $\ell =0$, while the superradiant condition is hold also for higher $\ell$. Thus, once the instability has superradiant origin, it would be natural to suspect that there are growing modes for some range of parameters also at higher multipoles. We tested a few first $\ell$ at moderately large values of $e Q$ and found no unstable modes, though one should not exclude this possibility \emph{ad hoc}. Apparently, higher multipole moment $\ell$ increases the height of the potential barrier and makes the field more stable.

The stability of higher multipoles is closely related also to the perturbations of charged nonzero spin fields in the Reissner-Nordstr\"om background.   As the first dynamical mode of the charged Dirac field  starts not from $\ell =0$, but effectively from $\ell = 1/2$, while for higher spins the lowest value of $\ell$ is larger, one should not expect any instability of this kind for charged fields with nonzero spin. Indeed, careful search in the frequency domain \cite{Konoplya:2007zx} for a charged Dirac field shows no growing modes in the spectrum. However, an additional check with the help of  the time-domain integration would guarantee us that no unstable mode was lost in the frequency-domain search.

\begin{acknowledgments}
A.~Z. was supported by the Alexander von Humboldt Foundation, Germany and Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior (CAPES), Brazil.
\end{acknowledgments}


text:
\section{Introduction}\label{sec:Introduction}

The $q$--theory description of the quantum vacuum provides
a natural cancellation mechanism for the vacuum energy
density~\cite{KlinkhamerVolovik2008a,KlinkhamerVolovik2008b,KlinkhamerVolovik2008c}.
The basic idea is to consider the macroscopic equations of a  \emph{conserved}
microscopic variable $q$, whose precise nature need not be known.
For a particular realization of $q$, it was found~\cite{KlinkhamerVolovik2008b}
that, if the vacuum energy density has initially a large Planck-scale value,
$\rho_{V}\sim E_\text{Planck}^4$, it relaxes according
to the following power-law modulation:
\begin{subequations}\label{eq:VacuumEnergyDecay}
\begin{eqnarray}
 \rho_{V}(t)\;\Big|^\text{nondissipative}
 \propto
 \frac{\omega^2}{t^2}\;\sin^2 \omega\,  t\,,
\label{eq:VacuumEnergyOscillating-dimensionfull}
\end{eqnarray}
with $\hbar=c=k=1$ in natural units and
a frequency $\omega$ of the order of the reduced Planck-energy scale
$E_\text{Planck} \equiv 1/\sqrt{8\pi G_{N}} \approx
2.44\times 10^{18}\:\text{GeV}$. Quantum
dissipative effects have not been taken into account in the above result.
Indeed, matter field radiation (matter quanta emission)
by the oscillations of the vacuum can be expected to lead to
faster relaxation~\cite{Starobinsky1980,Vilenkin1985},
\begin{eqnarray}
 \rho_{V}(t)\;\Big|^\text{dissipative}
 \propto \Gamma^4  \exp(-\Gamma\, t)\,,
\label{eq:VacuumEnergyQuantum-dimensionful}
\end{eqnarray}
\end{subequations}
with a decay rate $\Gamma\sim \omega \sim E_\text{Planck}$.

In the present article, we consider what happens during the electroweak
crossover~\cite{Csikor-etal1999} of a spatially flat
Friedmann--Robertson--Walker (FRW) universe~\cite{Weinberg2008} at cosmic time
\begin{equation}
t_\text{ew}\sim
E_\text{Planck}/E_\text{ew}^2\,,
\label{eq:t_ew}
\end{equation}
where $E_\text{ew}\sim 10^{3}\:\text{GeV}$ is the effective electroweak
energy scale. In the epoch before the crossover, the vacuum energy
density has already relaxed to zero, according to
\eqref{eq:VacuumEnergyQuantum-dimensionful}. The classical equations of
$q$--theory demonstrate that during the epoch when only ultrarelativistic
matter (``radiation'') is present,
i.e., when the matter equation-of-state (EOS) parameter $w_{M}\equiv
P_{M}/\rho_{M}$ is exactly 1/3,  the vacuum energy density remains
strictly zero. But $w_{M}(t)$ deviates from 1/3 during the electroweak
crossover and the subsequent period when massive particles annihilate.
This implies, as will be shown in the present article, that
the vacuum energy density moves away from zero and acquires,
at $t \sim t_\text{ew}$, a positive value of order
\begin{equation} \label{eq:vac_energy_at_ew}
\rho_{V,0}(t)  \sim \big( w_{M}(t) - 1/3 \big)^2\;H^4(t)\,,
\end{equation}
where the suffix 0 will be explained later and $H(t)$ is the Hubble parameter
of the spatially flat FRW universe considered.

After the electroweak crossover, the value $w_{M}=1/3$ is restored
and, if no other effects are operative,
the vacuum energy density smoothly returns to a zero value.
If, however, quantum relaxation effects are taken into account, the vacuum
energy density does not return to zero, but approaches a constant
value, which is of the order of the vacuum energy density
\eqref{eq:vac_energy_at_ew} at $t\sim t_\text{ew}$.
This remnant vacuum energy density corresponds to the measured value of
the cosmological constant (see, e.g., Refs.~\cite{Weinberg2008,Komatsu2008}
and other references therein):
\begin{eqnarray}
\label{eq:vac_energy_at_present}
\Lambda &\equiv& \lim_{t \to\infty}\,\rho_{V}(t)
\sim  \rho_{V,0}(t_\text{ew})
\sim H^4(t_\text{ew})
          \sim
t_\text{ew}^{-4} \sim  \big( E_\text{ew}^2/E_\text{Planck}\big)^4
\sim \big( 10^{-3}\:\text{eV} \big)^4  \,,
\end{eqnarray}
for the energy scales $E_\text{Planck}$ and $E_\text{ew}$ defined
under \eqref{eq:VacuumEnergyOscillating-dimensionfull} and \eqref{eq:t_ew},
respectively.
The several steps in \eqref{eq:vac_energy_at_present} will be detailed
in the following, with the most important intermediate steps
collected in \eqref{eq:VacuumEnergySmoothEWtime} and \eqref{eq:FinalrhoV}.

The scenario outlined above differs from that of a cosmological phase
transition, for which the vacuum energy density may only decrease (changing
to a negative value if it was originally zero), and resembles the scenario in
which the vacuum energy density is generated by the conformal anomaly. In fact,
it has been suggested in Refs.~\cite{Schutzhold2002,KlinkhamerVolovik2009}
that the conformal anomaly of quantum chromodynamics (QCD)
gives rise to the vacuum energy density
$\rho_{V}(t)\propto |H(t)|\, E_\mathrm{QCD}^3$, where
$E_\mathrm{QCD} \sim 10^2\;\text{MeV}$
is the QCD energy scale (see also Ref.~\cite{Bjorken2004} for related remarks).
The rigorous microscopic derivation of this nonanalytic term has not yet been
given, as it requires the detailed behavior of QCD in the infrared.
For the moment, the main motivation of this particular nonanalytic
term is that it naturally provides the correct order of magnitude for the
present vacuum energy density and appears to give a good description of the
late evolution of the Universe~\cite{Klinkhamer2009}.
We remark also that part of the contribution
of the conformal anomaly to the vacuum energy density
has been estimated~\cite{Thomas-etal2009} as $\rho_{V}(t)\propto H^4(t)$,
which has the same $H$ dependence as \eqref{eq:vac_energy_at_ew}.
But the mechanisms of Ref.~\cite{Thomas-etal2009} and the present article
are different, as will be explained later.

The scenario with the emergence of a
positive vacuum energy density  \eqref{eq:vac_energy_at_present}
triggered by the electroweak crossover confirms the earlier suggestion
by Arkani-Hamed \emph{et al.}~\cite{ArkaniHamed-etal2000}
that electroweak physics is at the origin of a
``triple cosmic coincidence'' for the matter, radiation, and
vacuum energy densities in the present Universe
(see also the general discussion in Ref.~\cite{Chernin2008}).
While the coincidence among the matter and radiation energy densities appears to
be justified by the electroweak scenario~\cite{ArkaniHamed-etal2000},
the coincidence of these two ingredients
with the remnant vacuum energy density (effective cosmological constant) $\Lambda$
requires a particular relation in terms of the electroweak energy scale
$E_\text{ew}$  and the ultraviolet energy scale
$E_\text{Planck}$, namely, $\Lambda  \sim    E_\text{ew}^8/E_\text{Planck}^4$.
In order to explain this particular relation,
the authors of Ref.~\cite{ArkaniHamed-etal2000} suggested a
phenomenological model but had to assume
(page 4436, right column of the cited reference)
that ``an unknown mechanism canceled the vacuum energy density
at the global minimum of the potential.''
In our scenario, this mechanism is natural.

\section{Dynamical equations}
\label{sec:Dynamical_equations}

The present discussion starts from the theory outlined in
Ref.~\cite{KlinkhamerVolovik2008b}. We introduce a special conserved
quantity, the  vacuum ``charge''  $q$, to describe the statics and dynamics
of the quantum vacuum.
An example of this vacuum variable is given by the four-form field
strength~\cite{DuffNieuwenhuizen1980,Aurilia-etal1980,Hawking1984,Duff1989,
DuncanJensen1989,BoussoPolchinski2000,Aurilia-etal2004,Wu2008},
expressed in terms of $q$ as
$F_{\alpha\beta\gamma\delta}(x)$ $=$
$q(x)\,\sqrt{-g(x)}\, \epsilon_{\alpha\beta\gamma\delta}$.
But the dynamic equations for the vacuum variable $q$ and
the metric $g_{\alpha\beta}$
are \emph{universal}, that is, they do not depend on the particular realization of $q$. For
example, in the four-form realization, the generalized Maxwell equation for
the $F$--field is reduced to the following generic equation for the charge $q$:
\begin{equation}
\frac{\partial\epsilon(q)}{\partial q} + R\,\frac{\partial K(q)}{\partial q}=\mu\,,
\label{eq:genMaxwellSolution}
\end{equation}
where $\epsilon(q)$ is the vacuum energy density expressed in terms of $q$
[the possible dependence on other fields is kept implicit],
$R$ the Ricci curvature scalar,
$K(q)$ the gravitational coupling parameter which depends on the vacuum state,
and $\mu$  an integration constant.
The latter quantity $\mu$ plays the role of a Lagrange multiplier
related to the conservation of the charge $q$ and corresponds to the chemical
potential in thermodynamics~\cite{KlinkhamerVolovik2008a,KlinkhamerVolovik2008b}.

The metric field $g_{\alpha\beta}$  obeys the generalized Einstein equation
\begin{subequations}\label{eq:genEinsteinEquation-rhoV}
\begin{eqnarray}
2K\,\big( R_{\alpha\beta}-g_{\alpha\beta}\,R/2 \big)&=&
-2\,\big(  \nabla_\alpha\nabla_\beta - g_{\alpha\beta}\, \square\big)\, K(q)
+\rho_{V}(q)\, g_{\alpha\beta} - T_{\alpha\beta} \,,
\label{eq:genEinsteinEquation}\\
\rho_{V}(q) &\equiv&  \epsilon(q)-  \mu\, q \,,
\label{eq:rhoV}
\end{eqnarray}
\end{subequations}
where the metric has signature $(-,+,+,+)$ and $T_{\alpha\beta}$
is the matter energy-momentum tensor with vanishing covariant divergence
$\nabla_\alpha\,T^{\alpha\beta}=0$  from general coordinate invariance.
The particular combination \eqref{eq:rhoV}, and not $\epsilon(q)$,
is seen to determine the cosmological term in \eqref{eq:genEinsteinEquation},
which is perhaps the most important characteristic of our approach.

In what follows, we choose a value $\mu_0$ of the integration
constant $\mu$ in such a way that, in the absence of matter or other types
of perturbations, the solution of the equations corresponds to
the full-equilibrium Minkowski-spacetime vacuum.
The actual value $\mu_0$ and corresponding charge  $q_0$ of
the equilibrium vacuum  are determined by two equations:
\begin{subequations}\label{eq:equil-eqs}
\begin{eqnarray}
\Big[\dd \epsilon(q)/\dd q-  \mu\,    \Big]_{\mu=\mu_0\,,\,q=q_0} &=&0\,,\\[2mm]
\Big[\epsilon(q)    -  \mu\, q\,\Big]_{\mu=\mu_0\,,\,q=q_0} &=&0\,,
\label{eq:equil-eqs-GDcondition}
\end{eqnarray}
\end{subequations}
which follow from \eqref{eq:genMaxwellSolution}
and \eqref{eq:genEinsteinEquation-rhoV},
respectively, for $R_{\alpha\beta}=T_{\alpha\beta}=0$
and spacetime-independent $q_0$.
The equilibrium conditions \eqref{eq:equil-eqs} are supplemented by the
following stability condition:
\begin{equation}
\big(\chi_0\big)^{-1} \equiv
q^2\:\frac{d^2\epsilon(q)}{dq^2}\,\Bigg|_{q=q_0} > 0\,,
\label{eq:chi_0}
\end{equation}
where $\chi$ corresponds to the
vacuum compressibility~\cite{KlinkhamerVolovik2008a}.

The ``cosmological constant problem'' would be completely
solved if we could explain the origin
of this particular value $\mu_0$ for the integration constant $\mu$ appearing
in \eqref{eq:genMaxwellSolution} and \eqref{eq:genEinsteinEquation-rhoV}.
Here, our assumption is that the
Minkowski-spacetime vacuum is a \emph{self-sustained} system,
i.e., an isolated system that can exist without external pressure,
at $P=0$.  In general, the vacuum pressure $P$ and the vacuum
energy density $\epsilon$ are related by the thermodynamic Gibbs--Duhem
equation~\cite{KlinkhamerVolovik2008a}, $P=-\epsilon + \mu\, q$.
The vanishing pressure $P$ allowed for a self-sustained system
(from the assumed absence of external pressure)
then gives the additional condition \eqref{eq:equil-eqs-GDcondition},
which fixes $\mu$ to the value $\mu_0$.
From this viewpoint, cosmology corresponds to
the dynamic process of approach to the equilibrium state with $q=q_0$,
which is natural for any system isolated from the external environment.

Close to equilibrium, at $|q-q_0|\ll |q_0|$,
the dynamics of the system is determined by the coefficients
in the Taylor expansion of $\epsilon(q)$ and $K(q)$ near the equilibrium point $q_0$:
\begin{subequations}\label{eq:expansions}
\begin{eqnarray}
K(q) &=& K(q_0) + K^\prime(q_0)\,(q-q_0)
+\text{O}\big( (q-q_0)^2 \big)\,,
\label{eq:expansions-K}\\[2mm]
\epsilon(q)-\mu_0\, q  &=& \epsilon^{\prime\prime}(q_0)\,(q-q_0)^2/2
+\epsilon^{\prime\prime\prime}(q_0)\,(q-q_0)^3/6
+\text{O}\big((q-q_0)^4 \big)\,.
\label{eq:expansions-epsilon}
\end{eqnarray}
\end{subequations}
All coefficients in these expansions have Planck-scale values, for example,
$K(q_0) = 1/(16\pi G_{N})= (1/2)\,E_\text{Planck}^2$
in terms of Newton's constant $G_{N}$ and the energy scale
$E_\text{Planck}$ defined under \eqref{eq:VacuumEnergyOscillating-dimensionfull}.

We now consider the spatially flat FRW universe~\cite{Weinberg2008}
described by the Hubble expansion parameter $H\equiv (d a/d t)/a$
for scale factor $a(t)$ and
use the dimensionless variables $y\propto (q-q_{0})$ and $h\propto H$,
which have been rescaled with the Planck-scale parameters of the theory.
These two variables $y(\tau)$ and
$h(\tau)$ are governed by the following two coupled ordinary differential
equations (ODEs):
\begin{subequations}\label{eq:Matter excluded1and2}
\begin{eqnarray}

\ddot{y} -\dot{y}\, h +2\, (1+y)\, \dot{h} &=&
-3 \big(1+w_{M}\big)\,\big[\dot{y}\, h
+ (1+y)\, h^2 - r_{V}  \big]\,.
\label{eq:Matter excluded1}
\\[2mm]
\dot{h} +2\, h^2   &=& r_{V}^\prime\,,
\label{eq:Matter excluded2}
\end{eqnarray}
\end{subequations}
with the prime standing for differentiation with respect to $y$ and
the overdot for differentiation with respect to dimensionless
cosmic time $\tau$ (cosmic time $t$ in the corresponding Planckian units).
In the derivation of the above ODEs, the function $K(q)$ has been
assumed~\cite{KlinkhamerVolovik2008b} to be linear in $q$ for simplicity
[in terms of the coefficients of \eqref{eq:expansions-K}, one has
$q_0\,K^\prime(q_0) = K(q_0)$ and $K^{(n)}(q_0) = 0$ for $n \geq 2$].

The dimensionless vacuum energy density $r_{V}$ (vacuum energy
density $\rho_{V}$ in Planckian units) is taken to be given by
\begin{equation}
r_{V}(y)= \frac{1}{2}\,y^2 +\frac{2}{3}\,y^3 +\frac{1}{6}\,y^4\,,
\label{eq:DimensionlessVacEn2}
\end{equation}
which vanishes in the equilibrium state $y=0$, having chosen
$\mu=\mu_0$ in \eqref{eq:genMaxwellSolution}--\eqref{eq:genEinsteinEquation-rhoV}.
Later on, only the quadratic part of $r_{V}(y)$ will be relevant.
Equations \eqref{eq:Matter excluded1and2} and \eqref{eq:DimensionlessVacEn2}
lead to the rapid relaxation
\eqref{eq:VacuumEnergyOscillating-dimensionfull}, if the Universe starts
out with a nonequilibrium value of the charge,
$q_\text{initial}\neq q_0$ or $y_\text{initial} \ne 0$.
These Eqs.~\eqref{eq:Matter excluded1and2} and \eqref{eq:DimensionlessVacEn2}
are, in fact, identical to Eqs.~(5.2) and (5.3) in Ref.~\cite{KlinkhamerVolovik2008b},
to which the reader is referred for all details.

For the present analysis, it turns out to be useful to define
the following matter EOS parameter:
\begin{equation}
\kappa_{M} \equiv 4 - 3\,(1+w_{M})\,,
\label{eq:EOS}
\end{equation}
where $\kappa_{M}=0$ corresponds to matter with $T_{\alpha}^{\;\;\alpha}=0$,
for example, electromagnetic radiation (photons) or
ultrarelativistic massive particles (e.g., electrons and positrons).
Then, \eqref{eq:Matter excluded1} and \eqref{eq:Matter excluded2}
can be written as
\begin{subequations}\label{eq:Matter excluded1and2new}
\begin{eqnarray}
\ddot{y} +3\, \dot{y}\, h  +2\, (1+y)\, r_{V}^\prime
&=&4\, r_{V} +\kappa_{M}\,\big[\dot{y}\, h
 + (1+y)\, h^2 - r_{V}  \big]\,,
\label{eq:Matter excluded1new}
\\[2mm]
\dot{h} +2\, h^2  - r_{V}^\prime &=& 0\,.
\label{eq:Matter excluded2new}
\end{eqnarray}
\end{subequations}
The crucial observation, now, is that, for $\kappa_{M}(\tau)=0$,
there is a solution of the ODEs \eqref{eq:Matter excluded1new}
and \eqref{eq:Matter excluded2new},
where the vacuum energy density is exactly zero. This solution
corresponds to an FRW universe with ultrarelativistic matter present
but dark energy and cold dark matter (CDM) absent:
\begin{subequations}\label{eq:PureRadiation}
\begin{eqnarray}
y(\tau)&=&0 \,,
\\[1mm]
h(\tau)&=&1/(2\, \tau)  \,,
\end{eqnarray}
\end{subequations}
which, as said, holds for $\kappa_{M}(\tau)=0$.

Next, consider what happens when the model universe described
by \eqref{eq:PureRadiation} enters a phase at $t\sim t_\text{kick}$
for which $\kappa_{M}(t)\neq 0$.
Then, the vacuum variable $y$ becomes nonzero and a nonzero value of
the vacuum energy density emerges continuously.
Specifically, we consider a time $t_\text{kick}\gg t_\text{Planck}$,
so that the corresponding dimensionless time
is large, $\tau_\text{kick}\gg 1$. At large $\tau$, the variable  $y(\tau)$ is always small
and one can make an expansion in terms of powers of $y$.
To first order in $y$ and $h^2$, one obtains the following ODEs
from \eqref{eq:Matter excluded1new} and \eqref{eq:Matter excluded2new}:
\begin{subequations}\label{eq:Linearized1_y1andy2}
\begin{eqnarray}
\ddot y + 3\, h\, \dot{y} +  \omega^2\, y&=& \kappa_{M}\, h^2\,,
\label{eq:Linearized1_y1}
\\[2mm]
\dot{h} +2\, h^2 - y   &=& 0 \,,
\label{eq:Linearized1_y2}
\end{eqnarray}
\end{subequations}
with  an implicit $\tau$ dependence for all three functions $y$, $h$,
and $\kappa_{M}$. Here, $\omega$ is the natural frequency of the
microscopic oscillations~\cite{KlinkhamerVolovik2008b},
which is given by $\omega=\sqrt{2}$  in Planckian units.

\section{Electroweak kick}
\label{sec:Electroweak-kick}

There are different regimes for the behavior of the
vacuum energy density obtained from \eqref{eq:Linearized1_y1andy2},
depending on the sharpness of the profile of the transition,
i.e., the width $\Delta\tau_\kappa$ of the function  $\kappa_{M}(\tau)$.
For the case of a smooth transition (that is, smooth on microscopic time scales,
$\Delta\tau_\kappa \gg 1/\omega \sim 1$), one may neglect the time derivatives
of $y$ in \eqref{eq:Linearized1_y1} to obtain:
\begin{subequations}\label{eq:Linearized2_y1andy2}
\begin{eqnarray}
y&=&\kappa_{M}\, h^2/2\,,
\label{eq:Linearized2_y1}
\\[2mm]
y   &=& \dot{h} +2\, h^2\,,
\label{eq:Linearized2_y2}
\end{eqnarray}
\end{subequations}
where the specific value $\omega^2=2$ has been reinstated in the first equation.
Eliminating $y$ from the above equations gives immediately the following solution
for $h(\tau)$:
\begin{equation}
h(\tau)=
\left[\, 2\int^\tau_0  d\tau'\,\Big(1-\kappa_{M}(\tau')/4 \Big)\right]^{-1}\,,
\label{eq:GeneralSolution}
\end{equation}
which holds for an arbitrary (smooth) function $\kappa_{M}(\tau)$
and has boundary condition $1/h(0)=0$, appropriate for
the standard hot big bang universe. Taking the square of
\eqref{eq:GeneralSolution},
the solution for $y(\tau)$  follows from \eqref{eq:Linearized2_y1}.

Now apply this result to the cosmological epoch of the electroweak
crossover~\cite{Csikor-etal1999}.
During the crossover, the Standard Model particles acquire masses and, as a
result, $w_{M}(t)$ deviates from 1/3. In principle, this deviation may be
enhanced by ``new physics'' at the TeV energy scale, which might be responsible
for the observed cold-dark-matter component of the present Universe by providing
a TeV--scale WIMP (weakly interacting massive particle).
According to the electroweak scenario of Ref.~\cite{ArkaniHamed-etal2000},
this new physics may have many particles ($n=1, \ldots ,N$)
with masses $M_n\sim E_\text{ew} \sim
1\;\text{TeV}$, which are created before and during the electroweak epoch.
Perhaps we will know from future particle-collider experiments
(for example, at the Large Hadron Collider of CERN) whether or not there
exists a TeV-scale WIMP responsible for the observed CDM.

Anyway, massive Standard Model particles
(and possible additional massive particles of new TeV--scale physics)
annihilate during the electroweak-crossover period and, afterwards,
the EOS parameter returns to its standard radiation-dominated value
$w_{M}=1/3$ [or $\kappa_{M}=0$], with the result that
the vacuum energy density is no longer perturbed.
In the epoch after the electroweak period when all perturbations have ceased,
the Hubble parameter \eqref{eq:GeneralSolution} is given by
\begin{subequations}\label{eq:HubbleLateTime}
\begin{eqnarray}
h(\tau)&\approx&\frac{1}{2\,(\tau-\tau_0)}\,,
\\[1mm]
\tau_0&\equiv&\frac{1}{4}\,\int_0^\infty d\tau'~ \kappa_{M}(\tau')\,,
\end{eqnarray}
\end{subequations}
for $\tau \gg \tau_0 \sim \tau_\text{ew}
\sim E_\text{Planck}^2/E_\text{ew}^2 \sim 10^{30}$.

 From \eqref{eq:DimensionlessVacEn2} and \eqref{eq:Linearized2_y1},
the dimensionless and dimensionful vacuum energy densities
during the electroweak crossover  behave as follows:
\begin{subequations}\label{eq:VacuumEnergySmooth}
\begin{eqnarray}
r_{V}(\tau)&=&(1/8)\,\kappa_{M}^2(\tau)\, h^4(\tau)\,,
\label{eq:VacuumEnergySmooth-rV}
\\[1mm]
\rho_{V}(t)&\propto& \kappa_{M}^2(t)\, H^4(t)\,,
\label{eq:VacuumEnergySmooth-rhoV}
\end{eqnarray}
\end{subequations}
where only the quadratic part of \eqref{eq:DimensionlessVacEn2}
has been kept as $|y|\ll 1$ and where the precise numerical constant in
\eqref{eq:VacuumEnergySmooth-rhoV} depends on the microphysics
but can be expected to be of order unity~\cite{KlinkhamerVolovik2008b}.

Even though result \eqref{eq:VacuumEnergySmooth-rhoV} is similar to the
vacuum energy density estimate~\cite{Thomas-etal2009} from the
conformal anomaly,
$\rho_{V}(t)\sim \left<T_{\alpha}^{\;\;\alpha}\right>
\sim H^4(t)$, the mechanism of the emerging vacuum energy density
in \eqref{eq:VacuumEnergySmooth} is different.
The underlying theory~\cite{KlinkhamerVolovik2008b} of
result \eqref{eq:VacuumEnergySmooth} has, in fact, a gravitational
coupling parameter $K$ that depends on the vacuum variable, $K=K(q)$,
with Newton's constant recovered in the  $q=q_0$ equilibrium state,
$G_{N}=1/\big(16\pi\,K(q_{0})\big)$. Precisely this variability
$K(q)$ allows for a time-dependent vacuum energy density,
$\dot{\rho}_{V} \propto \dot{K}\,(\dot{H}+2H^2)$,
provided the expansion differs from that of a
radiation-dominated FRW universe with $H(t)=1/(2\,t)$ and $\dot{H}+2H^2=0$.

 From \eqref{eq:VacuumEnergySmooth-rhoV},
the magnitude of the vacuum energy density
at the crossover time \eqref{eq:t_ew} is given by
\begin{equation}
 \rho_\text{V,0}(t_\text{ew})
 \sim  H^4(t_\text{ew})
 \sim t_\text{ew}^{-4}
 \sim E_\text{ew}^8/E_\text{Planck}^4\,,
\label{eq:VacuumEnergySmoothEWtime}
\end{equation}
where $\kappa_{M}(t_\text{ew})$ has been assumed to be of order unity
and where, for later use, a suffix $0$ has been appended to
distinguish the ``classical'' result. This completes the first step
toward establishing a nonzero cosmological constant of the present Universe.
The second step is to make sure that the vacuum energy density generated
at $t\sim t_\text{ew} \sim 10^{-12}\,\text{s}$ is not lost during the
remaining $10^{10}$ years.

\section{Subsequent evolution}
\label{sec:Subsequent_evolution}

The typical value of the vacuum energy
density \eqref{eq:VacuumEnergySmoothEWtime} emerging from
the electroweak crossover is comparable to the presently observed
value~\cite{Weinberg2008,Komatsu2008} of the vacuum energy
density.\footnote{An excellent description of the currently available data is,
in fact, given by the flat--$\Lambda$CDM model
(cf. Refs.~\cite{Weinberg2008,Komatsu2008}),
with an inhomogeneous cold-dark-matter component (EOS parameter $w_\text{CDM}=0$)
and a perfectly homogeneous and time-independent vacuum energy density component
($w_{V}=-1$), which corresponds to Einstein's cosmological
constant $\Lambda$.\label{ftn:LambdaCDM}}
As mentioned in Sec.~\ref{sec:Introduction}, this suggests a possible explanation
of the triple cosmic coincidence according to the electroweak scenario
discussed in Ref.~\cite{ArkaniHamed-etal2000}. But,
for this explanation to work, we need a mechanism to stabilize
the vacuum energy density after the electroweak crossover.

At the moment, we do not have a complete theory which describes the
\emph{irreversible} dynamics of the quantum vacuum. The classical equations
of $q$--theory~\cite{KlinkhamerVolovik2008a} describe only the
\emph{reversible} classical dynamics of the vacuum. One needs to extend
$q$--theory to the quantum domain, in order to incorporate the dissipative
relaxation of the vacuum energy density due to the quantum effect of matter field
radiation (matter quanta emission).

Awaiting the definite theory of the quantum vacuum,
the following model equation can be used for a rough estimate:
\begin{equation}
\dot{\rho}_{V} = - \Gamma(t)\,\big[\rho_{V}(t)- \rho_\text{V,0}(t) \big]\,.
\label{eq:relaxation_equation}
\end{equation}
Here, $\rho_\text{V,0}(t)$  is the ``bare'' vacuum energy density driven
by the kick, which, according to result \eqref{eq:VacuumEnergySmooth-rhoV} of
the classical $q$--theory, is given by
\begin{equation}
 \rho_\text{V,0}(t) \propto   \kappa_{M}^2(t)\, H^4(t)\,,
\label{eq:classical_value}
\end{equation}
and $\Gamma(t)\geq 0$ in \eqref{eq:relaxation_equation} is the rate
at which the ``surplus'' vacuum energy density is dissipated into particles.

Particle production occurs when the background spacetime is changing on a
timescale comparable to the particle Compton time~\cite{BirrellDavies1980},
which implies different particle production rates for different
cosmological epochs. In the epoch before the electroweak crossover,
matter consists of ultrarelativistic particles (radiation)
with EOS parameter $\kappa_{M}=0$ and, thus,
there is no ``external force'' to drive the vacuum energy density.
Rapid oscillations with frequency $\omega \sim E_\text{Planck}$
lead to the decay of the vacuum energy density with the rate
$\Gamma\sim \omega \sim E_\text{Planck}$~\cite{Starobinsky1980,Vilenkin1985}.
As a result, \eqref{eq:relaxation_equation} gives exponential decay
\eqref{eq:VacuumEnergyQuantum-dimensionful} of the
vacuum energy density to a zero value.
The model universe rapidly approaches the stage with pure
radiation, evolving as in \eqref{eq:PureRadiation}.

During the electroweak crossover, the EOS parameter $ \kappa_{M}(t)$
in \eqref{eq:classical_value} deviates from zero, which drives the
vacuum energy density \eqref{eq:relaxation_equation} away from
zero towards a positive value. The change of the vacuum energy density
during the crossover results in the emission of particles.
The radiation rate  $\Gamma(t)$ is concentrated in the crossover period,
because after the crossover the model universe returns to radiation-dominated
expansion without particle production. The decay rate $\Gamma(t)$ is,
therefore, peaked at $t\sim t_\text{ew}$,
\begin{equation}
\Gamma(t)\;\big|_{t \ll t_\text{ew} \;\vee\; t \gg t_\text{ew}}  \ll \Gamma(t_\text{ew}) \sim 1/t_\text{ew}\,,
\label{eq:Gamma-peak-value}
\end{equation}
where the maximal value $1/t_\text{ew}$ will be derived shortly. Note that
the maximal rate $\Gamma(t_\text{ew}) \sim E_\text{ew}^2/(\hbar\,E_\text{Planck})$
goes to infinity for $\hbar\to 0$
and fixed energy $E_\text{ew}^2/E_\text{Planck}$,
so that \eqref{eq:relaxation_equation} reproduces the classical result,
$\rho_{V}(t)\to \rho_\text{V,0}(t)$. In fact, this particular classical
limit corresponds to the hydrodynamic limit in fluid dynamics; cf. the
section on ``second viscosity'' in Ref.~\cite{LandauLifshitz-Fluid-mechanics}.
Further remarks on the heuristics of the vacuum dynamics equation
\eqref{eq:relaxation_equation} will be presented in the
paragraph starting a few lines after \eqref{eq:FinalrhoV}.

The estimate for the maximal value of the decay rate
in \eqref{eq:Gamma-peak-value} can be obtained as follows.
Start from the observation~\cite{ZeldovichStarobinsky1977} that,
for an FRW universe with appropriate boundary conditions~\cite{DobadoMaroto1999},
the number of particles created per unit of time and per unit of volume
is given by $\dot{n} \propto R^2$, where $R$ is the Ricci curvature scalar.
For an FRW universe with pure radiation, the Ricci scalar
$R\propto (\dot{H}+2H^2)$ vanishes and there is no particle production.
As mentioned before,
this is the reason why the radiation rate $\Gamma(t)$ is peaked
in the crossover period.\footnote{Matter radiation must also vanish in a
de-Sitter spacetime, where no relaxation of the vacuum energy density
is expected. For a discussion of the controversies
concerning the stability of de-Sitter spacetime, see, e.g.,
Refs.~\cite{Starobinskii1979,StarobinskyYokoyama1994,GarrigaTanaka2007,
TsamisWoodard2007,Busch2008,Volovik2008}.\label{ftn:deSitter}}
In the period of the electroweak crossover, one
has $R^2( t_\text{ew}) \sim \dot{H}^2( t_\text{ew})  \sim H^4(
t_\text{ew})\sim \rho_{V}( t_\text{ew})$.
Particles created~\cite{BirrellDavies1980} during this period
have a Compton time of order $t_\text{ew}$ and, thus, a
characteristic energy of order $E\propto 1/ t_\text{ew}$.
The only known elementary particles whose energy $E$ can be of order
$1/ t_\text{ew} \sim  E_\text{ew}^2/E_\text{Planck} \sim \text{meV}$
are massless gravitons and massive neutrinos, some of
whose masses~\cite{Amsler-etal2008} may be comparable
with $1/ t_\text{ew}$ (all the other particles of the Standard
Model have larger masses, including the photon which gets an
effective mass in the cosmic plasma).
During the electroweak-crossover period, the radiated energy
per unit of time and per unit of volume
is then $\dot{\rho}_{V}  \propto - E\,\dot{n} \propto -\rho_{V}/
t_\text{ew}$, giving $\Gamma(t_\text{ew}) \sim 1/t_\text{ew}$
for the decay rate entering \eqref{eq:relaxation_equation}
and delivering the announced estimate \eqref{eq:Gamma-peak-value}.

Now, the solution of \eqref{eq:relaxation_equation} is given by
\begin{equation}
\rho_{V}(t)=  \int_0^{t} dt' \;\Gamma(t')\, \rho_\text{V,0}(t')\;
                   \exp\left[ -\int_{t'}^{t} dt''\;\Gamma(t'')\right]\,,
\label{eq:general-solution-rhoV}
\end{equation}
for boundary condition $\rho_{V}(0)=0$,
which is reasonable for times $t$ well after the Planck era.
Since $\Gamma(t)$ is concentrated in the crossover period and has
peak value \eqref{eq:Gamma-peak-value}, the solution
\eqref{eq:general-solution-rhoV} gives
$\lim_{t \rightarrow \infty}\,\rho_{V}(t)\sim \rho_\text{V,0}(t_\text{ew})$.
For very late times, $t\gg t_\text{ew}$,
one thus obtains that the vacuum energy density approaches
the following positive and time-independent value:
\begin{eqnarray}
  \rho_{V}(t)\;\big|_{t \gg  t_\text{ew}}
  &\sim& \rho_{V}(t_\text{ew})
  \sim \rho_\text{V,0}(t_\text{ew})
\sim E_\text{ew}^8/E_\text{Planck}^4\,,
\label{eq:FinalrhoV}
\end{eqnarray}
where  \eqref{eq:VacuumEnergySmoothEWtime} has been  used in the last step.
The final result \eqref{eq:FinalrhoV} is comparable to the measured value
of the cosmological constant, as shown in \eqref{eq:vac_energy_at_present}.

The heuristics of the obtained nonzero remnant vacuum energy density
is as follows.
The quantity $\Gamma(t)$ in \eqref{eq:relaxation_equation} can be
interpreted as the inverse of the instantaneous response time $\theta(t)$
of the vacuum energy density $\rho_{V}(t)$ to an ``external perturbation.''
Here, the external perturbation \eqref {eq:classical_value} comes from the
``kick'' in $\kappa_{M}(t)$, which is assumed to happen at $t\sim t_\text{ew}$
and to have a full width at half maximum $\Delta t_\kappa\sim t_\text{ew}$.
Moreover,  $\Gamma(t)\equiv 1/\theta(t)$ is taken to have a width
$\Delta t_\Gamma$, which is comparable to or larger than the duration of the kick,
$\Delta t_\Gamma \gtrsim \Delta t_\kappa$.
\mbox{\emph{A priori},} there are then two possibilities.
First, the typical response time $\theta$ is short ($\theta \ll \Delta t_\kappa$),
which implies that the vacuum energy density $\rho_{V}(t)$ can follow
the kick in $\kappa_{M}(t)$ and that
$\rho_{V}(t)$ can recover a near-zero value,
as $\kappa_{M}(t)$ drops to zero for $t\gg t_\text{ew}$.
Second, the typical response time $\theta$ is relatively long
($\theta\gtrsim \Delta t_\kappa$), which implies that the vacuum energy density
$\rho_{V}(t)$ cannot keep up with $\kappa_{M}(t)$, as the latter
drops to zero, and that a nonzero asymptotic value of $\rho_{V}$
remains. According to \eqref{eq:Gamma-peak-value}, this second type
of behavior occurs for the case considered,
with $\theta \sim \Delta t_\kappa \sim t_\text{ew}$,
and a nonvanishing asymptotic value of $\rho_{V}(t)$
follows from the general solution \eqref{eq:general-solution-rhoV}.
In short, the nonzero remnant vacuum energy density
\eqref{eq:FinalrhoV} is a \emph{time-lag effect}, because the response
(relaxation) time of the vacuum energy density is of the same order of
magnitude as the duration of the kick.\footnote{In principle,
the same time-lag (freezing) mechanism may work for the scenario of
Ref.~\cite{Thomas-etal2009}, where a vacuum energy density $\rho_{V}\propto
H^4(t)$ emerges due to the conformal anomaly. During the electroweak crossover,
the number of massless fields contributing to the anomaly changes, which
results in a kick of the vacuum energy density.  In turn, this gives rise to
matter radiation, which leads to the stabilization of the
vacuum energy density at a value of the order of \eqref{eq:FinalrhoV}.}

After the electroweak crossover, further perturbations
of the vacuum energy density occur during the QCD confinement transition at
a typical temperature $T\sim E_\text{QCD} \sim 10^2\;\text{MeV}$
and the epoch following the moment of radiation-matter equality,
when the radiation-dominated effective EOS parameter $w_{M}=1/3$
changes to the matter-dominated parameter $w_{M}=0$.
[The moment of radiation-matter energy density equality happens to be close
to the epoch of recombination with $T \sim T_\text{rec} \sim 10^{-1}\;\text{eV}$
and this energy scale will be used for definiteness.] The first-mentioned
perturbation of the vacuum energy density by the QCD confinement transition
(see, e.g., Fig.~19.3 in Ref.~~\cite{Amsler-etal2008} for the change in the
number of relativistic degrees of freedom)  can be expected to give a
change of the order of $H^4(t_\text{QCD})\sim (E_\text{QCD}^2/E_\text{Planck})^4$,
which is negligible compared to the present value of $\Lambda$
according to \eqref{eq:vac_energy_at_present}.
The second perturbation of the vacuum energy density acts during the whole
matter-dominated era. However, the resulting change of the vacuum energy density
can be expected not to exceed a value of order
$H^4(t_\text{rec})\sim (T_\text{rec}^2/E_\text{Planck})^4$, which is, again, many
orders of magnitude smaller than the present value of $\Lambda$ and can
be neglected.

Turning the argument of the preceding paragraph around, it would seem that
the suggested electroweak explanation \eqref{eq:vac_energy_at_present}
of the present value of $\Lambda$ would rule out
(leave no room for) similar crossover effects at much higher temperature
$T_\star \gg E_\text{ew} \sim $ \text{TeV},
the expected remnant vacuum energy density $H^4(t_\star)$
being much larger than $H^4(t_\text{ew})$.
This conclusion, if correct, may be consistent with the
picture~\cite{ArkaniHamed-etal2000} of having only two fundamental energy scales,
$E_\text{ew}$ and $E_\text{Planck}$, without unification of the
Standard Model gauge group at an intermediate grand-unification
energy scale~\cite{KlinkhamerVolovik2005,Kawamura2009}.

\section{Discussion}\label{sec:Discussion}

The $q$--theory approach~\cite{KlinkhamerVolovik2008a}
to gravitational effects of the quantum vacuum suggests at least two types
of behavior for the evolution of the vacuum energy density,
each based on solutions of the $q$--theory dynamical equations
and their modifications due to dissipative effects from matter radiation.
For the first type of
solution~\cite{KlinkhamerVolovik2008b,KlinkhamerVolovik2008c},
the model universe is vacuum dominated with,
according to  \eqref{eq:VacuumEnergyOscillating-dimensionfull},
the vacuum energy density $\rho_{V}(t)$ relaxing as $1/t^2$
from its  natural Planck-scale value at early times when the system is
far from equilibrium to a naturally small value at late times when the system is
close to equilibrium. [Quantum effects (e.g., the emission of matter quanta
caused by the rapid oscillations of the vacuum state)
make the relaxation even faster,
as shown by \eqref{eq:VacuumEnergyQuantum-dimensionful}.]
This essentially solves the main cosmological problem
(but with the \emph{caveat} mentioned in Sec.~\ref{sec:Dynamical_equations}):
the present vacuum energy density is small compared to Planck-scale values
simply because the age of our Universe happens to be large
compared to Planck-scale values.
However, it leaves the following question: why does not the vacuum energy density
relax completely to zero as $t\rightarrow \infty$?

In order to answer this last question,
we presented a second type of solution in which the vacuum energy
density has already relaxed to  zero after the initial disturbance
in the very early universe and a nonzero value reemerges only  after a ``kick''
generated by nonrelativistic matter during the epoch of the electroweak crossover.
(These nonrelativistic particles consist of Standard Model particles and possibly
thermal relics from new physics at the $\text{TeV}$ scale, as
discussed in Sec.~\ref{sec:Electroweak-kick}.) In the process, a
nonoscillating vacuum energy density is generated, which starts to decay
after the kick. Such a behavior  emerges during the electroweak period, because
in this epoch the matter EOS parameter $w_{M}(t)$ deviates from the
radiative value $w_{M}=1/3$. Quantum effects now lead to a
stabilization of the vacuum energy density at the level indicated
by \eqref{eq:FinalrhoV}, which reproduces the expression suggested
previously by Arkani-Hamed \emph{et al.}~\cite{ArkaniHamed-etal2000}.

It was assumed in the reasoning leading up
to \eqref{eq:FinalrhoV} that there was no real phase
transition at cosmic time $t_\text{ew}$. Instead, there was taken to be a
crossover at a temperature $T_\text{ew} = \text{O}(10^2\;\text{GeV})$,
which does not give a change of order $T_\text{ew}^4$
in the vacuum energy density as a genuine phase transition would do.
The absence of a real electroweak phase transition is by now well
established~\cite{Csikor-etal1999}, at least, in the framework of
the Standard Model of elementary particle physics
(the numerical value of the crossover temperature is
estimated~\cite{Csikor-etal1999}
as $T_\text{ew} \sim 300\;\text{GeV}$ for $m_\text{Higgs}\sim 150\;\text{GeV}$).
The new physics at the $\text{TeV}$ scale mentioned in the previous
paragraph and Sec.~\ref{sec:Electroweak-kick}
is assumed not to affect the nature of the electroweak crossover.
But the massive relic particles of the new physics can make a significant
contribution to the EOS parameter $\kappa_{M}(t)$ and can also
increase the numerical value of the effective energy scale $E_\text{ew}$,
thereby augmenting the magnitude of the estimated dark energy
\eqref{eq:classical_value}--\eqref{eq:general-solution-rhoV}
and bringing the theoretical value \eqref{eq:vac_energy_at_present}
closer to the observed value~\cite{Weinberg2008,Komatsu2008} of approximately
$(2\;\text{meV})^4$.

The electroweak scenario of Ref.~\cite{ArkaniHamed-etal2000} may
solve part of the triple cosmic coincidence
puzzle, as the same order of magnitude follows naturally for
the cold-dark-matter density and the radiation density in the present epoch.
Combined with the argument for the effective cosmological
constant \eqref{eq:vac_energy_at_present} of the present article,
this suggests that $\text{TeV}$--scale physics may be responsible for the
triple coincidence of vacuum, matter, and radiation energy densities
in the present Universe (perhaps even a quintuple coincidence if also
the baryon and neutrino energy densities are considered~\cite{ArkaniHamed-etal2000}).

For the present epoch, the vacuum energy density would be essentially
time-independent according to \eqref{eq:general-solution-rhoV}
and, observationally, the corresponding universe would be indistinguishable
from the one of the $\Lambda$CDM model (cf. Footnote~\ref{ftn:LambdaCDM}).
But, theoretically, we would have gained in understanding the
magnitude of the cosmological ``constant'' $\Lambda$
as given by \eqref{eq:vac_energy_at_present}, in addition
to explaining the triple or quintuple cosmic coincidence mentioned above.

\section*{\hspace*{-4.5mm}ACKNOWLEDGMENTS}
\noindent
It is a pleasure to thank A.A. Starobinsky and A.R. Zhitnitsky
for useful discussions on basic physics issues
and the referee for helpful comments on an earlier version of this article.
GEV is supported in part by the Academy of Finland,
Centers of Excellence Program 2006--2011,
the Russian Foundation for Basic Research (Grant No. 06--02--16002--a),
and the Khalatnikov--Starobinsky leading scientific school (Grant No. 4899.2008.2).

redpajama_set_name: RedPajamaArXiv