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\begin{document}

\title{The problem of initial conditions in cosmology}

\classification{04.60.Gw, 04.62.+v, 98.80.Bp, 98.80. Qc} \keywords
{cosmology, quantum field theory, density matrix}

\author{A.O. Barvinsky}{
  address={Theory Department, Lebedev Physics Institute,
  Leninsky Prospect 53, 119991 Moscow, Russia }
}

\author{A.Yu. Kamenshchik}{address={Dipartimento di Fisica
and INFN, via Irnerio 46, 40126 Bologna, Italy\\
L.D. Landau Institute for Theoretical Physics, Kosygin street 2,
119334 Moscow, Russia}
}




\begin{abstract}
The creation of a quantum Universe is described by a density matrix
which yields an ensemble of universes with the cosmological constant
limited to a bounded range $\Lambda_{\rm min}\leq \Lambda \leq
\Lambda_{\rm max}$. The domain $\Lambda<\Lambda_{\rm min}$ is ruled
out by a cosmological bootstrap requirement (the self-consistent
back reaction of hot matter). The upper cutoff results from the
quantum effects of vacuum energy and the conformal anomaly mediated
by a special ghost-avoidance renormalization. The cutoff
$\Lambda_{\rm max}$ establishes a new quantum scale -- the
accumulation point of an infinite sequence of garland-type
instantons. The Euclidean path integral formalism used for the
construction of the fundamental density matrix for a mixed state of
the Universe is justified by proving its correspondence to the
microcanonical ensemble in quantum cosmology. The cosmological
evolution starting with these initial conditions also have some new
features: the stage of cosmic acceleration can be followed by a big
boost singularity -- a rapid growth up to infinity of the scale
factor acceleration parameter. From the developed approach it
follows that the notion of the density matrix plays a more
fundamental role than that was traditionally prescribed to it.
\end{abstract}

\maketitle



\section{Introduction}
Many years ago when speaking to his students L.D. Landau used to
say, according to I.M. Khalatnikov's reminiscences, that the future
physical theory should incorporate not only equations of motion but
also initial conditions for them \cite{Land-priv}. It is difficult
to imagine how preferred initial conditions could be prescribed to
an equation of motion in the framework of classical physics. The
same could be said about the non-relativistic quantum mechanics or
about quantum field theory in the Minkowski spacetime. The only
theory where the idea of natural initial conditions was realized is
quantum cosmology, the branch of quantum theory treating the
Universe as a unique quantum object described by its quantum state.
The basic equation of quantum cosmology - the Wheeler-DeWitt
equation was formulated in the sixties \cite{DeWitt}. However,
certain prescriptions for the wave function of the universe,
satisfying this equation were suggested only in the early eighties
in papers \cite{HH,Vil,Linde,Star,Rub}. In papers mentioned above
the two related approaches were used: the analogy with the tunneling
processes in quantum mechanics \cite{Vil} and the apparatus of the
Euclidean field theory \cite{HH}. In both cases the phenomenon of
the so called ``quantum birth of the universe from nothing'' was
employed. Both approaches used the instanton solutions of the
Euclidean Einstein equations, however their physical predictions
were different because the Euclidean action entered with different
signs the exponential of the wave function of the universe,
calculated in the semiclassical approximation. Namely, the
Hartle-Hawking or ``no-boundary'' wave function of the universe
\cite{HH} which behaves in the lowest order of the WKB approximation
as $\psi_{NB} \sim \exp(-\Gamma)$, where $\Gamma$ is the Euclidean
action on the underlying instanton, predicts the quantum birth of a
universe with a very large (infinite) initial radius, which looks
quite counter-intuitive. The tunneling or Vilenkin wave function of
the universe \cite{Vil} behaves as $\psi_T \sim \exp(+\Gamma)$ and
predicts the birth of a universe with an infinitely small radius.
Besides, both of these functions are non-normalizable and it is
hardly possible to prescribe to them the traditional
quantum-mechanical probabilistic interpretation.

Considering solutions of the Wheeler-DeWitte equation in the
one-loop approximation, one can achieve (imposing some constraints
on the particle content of the theory) the normalizability of the
wave function of the universe in both the tunneling and no-boundary
prescriptions \cite{Bar-Kam-norm}. Moreover, for the tunneling wave
function of the universe one can predict a peak of the probability
of the quantum birth of the universe with reasonable initial
parameters \cite{Bar-Kam-scale}.

However, the traditional approach to quantum cosmology limited to
the consideration of only pure quantum states and associated with
them instantons looks too restrictive. It appears that relaxing the
requirement of the ``purity'' of possible quantum states of the
universe and taking into account the possibility of existence of the
gravitational instantons with more complicated geometries than those
considered in the above works on quantum cosmology, one can obtain
some, at first glance, unexpected results. In our papers
\cite{Bar-Kam-den,Bar-Kam-den1} we have generalized the traditional
scheme of quantum cosmology. The main goals of our approach were the
following:
\begin{enumerate}
\item Description of the birth of the universe from nothing in a mixed state
and the use of the density matrix instead of the wave function of
the universe.
\item Predicition of initial conditions for the cosmological evolution, which
we call ``cosmological landscape'' in analogy with a very popular
string landscape \cite{string}.
\item Elimination of ``infrared catastrophe'' (an infinitely large
probablity of the birth of the universe of an infinitely large size)
in the Hartle-Hawking prescription.
\item Establishing connections with string theory.
\end{enumerate}
The tools which we have used were
\begin{enumerate}
\item  Quantum theory of tunneling : Euclidean quantum gravity.
\item  Quantum  field theory: renormalization in curved spacetime;
effective action formalism.
\item  The account of non-local effects due to back reaction of
the conformal anomaly of quantum fields and their radiation.
\end{enumerate}
Our main results can be formulated as follows:
\begin{enumerate}
\item The closed system of equations describing the
quantum birth of the universe is derived: the generalized Friedmann
equation with the quantum radiation source and the ``bootstrap''
equation for the latter.
\item The solution of these equations gives the families of acceptable
parameters, characterizing initial conditions for cosmological
evolution - ``cosmological landscape''.
\item The problem of ``infrared catastrophe" in the
Hartle-Hawking prescription is resolved.
\item The scenarios of the future evolution of the Universe are studied.
\end{enumerate}

\section{Density matrix, radiation and instantons}
The idea to consider the density matrix of the universe instead of
the wave function of the universe was put forward in \cite{Page}
where it was also noticed that such a density matrix is based on an
instanton with two disjoint boundaries (see Fig.1). The density
matrix describes a mixed state which might account for the presence
of radiation \cite{radiation}.
\begin{figure}
  \includegraphics[height=.1\textheight]{hh5.eps}
  \caption{Picture of instanton representing the density matrix. Dashed lines
depict the Lorentzian Universe nucleating from the instanton at the
minimal surfaces $\Sigma$ and $\Sigma'$.}
\end{figure}
For the pure quantum state \cite{HH}  the instanton bridge between
$\Sigma$ and $\Sigma'$ breaks down (see Fig.2). However, the
radiation stress tensor prevents these half instantons from closure.
\begin{figure}
\includegraphics[height=.1\textheight]{hh1.eps}
\caption{Density matrix of the pure Hartle-Hawking state represented by the
union of two vacuum instantons.}
\end{figure}
Indeed, the Euclidean Friedmann equation for a closed universe with
the metric
\begin{equation}
ds^2 = N^2(\tau)\,d\tau^2 +
    a^2(\tau)\,d^2\Omega^{(3)}
\end{equation}
in the presence of a cosmological constant $\Lambda=3H^2$ and
radiation characterized by a constant $C$
\begin{equation}
\frac{\dot{a}^2}{a^2} =
    \frac{1}{a^2} - H^2 -\frac{C}{a^4}
\label{Fried}
\end{equation}
has the solution
%\begin{equation}
$a = \frac{1}{\sqrt{2}H}
    \sqrt{1-(1-4CH^2)^{1/2}\cos
    2H\tau}$
%\label{solution}
%\end{equation}
with two turning points, neither of them vanishing,
%\begin{equation}
$a_\pm = \frac{1}{\sqrt{2}H}
    \sqrt{1\pm(1-4CH^2)^{1/2}},\ \ 4H^2C \leq 1$.
%\label{turn}
%\end{equation}

The relevant density matrix is the path integral
\begin{equation}
\rho[\,\varphi,\varphi'\,]=\mbox{$e$}^{
    \Gamma}\!\!\!\!\!\!\!\!\!\!\!\!\!\!
    \int\limits_{\,\,\,\,\,\,\,\,g,\,
    \phi\,|_{\,\Sigma,\Sigma'}\,=\,(\,\varphi,\varphi')}
    \!\!\!\!\!\!\!\!\!D[\,g,\phi\,]\,
    \exp\big(-S_{\rm E}[\,g,\phi\,]\big).
\end{equation}
with the partition function $e^{-\Gamma}$ which follows from
integrating out the field $\varphi$ in the coincidence
$\varphi'=\varphi$ corresponding to the identification of $\Sigma'$
and $\Sigma$, the underlying instanton acquiring the toroidal
topology (see Fig.3).
\begin{figure}
\includegraphics[height=.1\textheight]{hh6.eps}
\caption{Calculation of the partition function represented by
compactification of the instanton to a torus with periodically
identified Euclidean time.}
\end{figure}


\section{Conformal anomaly and ghosts}
The metric of the instanton introduced above
\begin{equation}
ds^2 = a^2(\eta)(d\eta^2 + d^2\Omega^{(3)}),
\label{Ein-stat}
\end{equation}
is conformally equivalent to the metric of the Einstein static
universe:
\begin{equation}
d\bar{s}^2 = d\eta^2 + d^2\Omega^{(3)},
\label{Ein-stat1}
\end{equation}
where $\eta$ is the conformal time parameter. We shall consider
conformally invariant fields. As is well known, the quantum
effective action for such fields has a conformal anomaly first
studied in cosmology in \cite{Starob,Hu}. It has the form
\begin{equation}
g_{\mu\nu}\frac{\delta
    \Gamma_{\rm 1-loop}}{\delta g_{\mu\nu}} =
    \frac{1}{4(4\pi)^2}g^{1/2}
    \left(\alpha \Box R +
    \beta E +
    \gamma C_{\mu\nu\alpha\beta}^2\right),        \label{anomaly}
\end{equation}
where $E = R_{\mu\nu\alpha\gamma}^2 -4R_{\mu\nu}^2 + R^2$ and $\Box$
is the four-dimensional Laplacian. This anomaly, when integrated
functionally along the orbit of the conformal group, gives the
relation between the actions on conformally related backgrounds
\cite{BMZ}.
\begin{eqnarray}
    &&\Gamma_{\rm 1-loop}[\,g\,]=
    \Gamma_{\rm 1-loop}[\,\bar g\,]+\delta\Gamma[\,g,\bar
    g\,],\,\,\,
\\&&g_{\mu\nu}(x)=e^{\sigma(x)}\bar g_{\mu\nu}(x),
    \end{eqnarray}
where
\begin{eqnarray}
    &&\delta\Gamma[\,g,\bar g\,]
    =
    \frac{1}{2(4\pi)^2}\int d^4x \bar g^{1/2} \left\{\,\frac{1}{2}\,
    \Big[\,\gamma\, \bar C_{\mu\nu\alpha\beta}^2\right.\nonumber \\
        &&
    +\beta\,\Big(\bar E-\frac{2}{3}\,\bar{\Box} \bar R\Big)\Big]\,
    \sigma                                  \nonumber\\
    &&
    \left.+\,\frac{\beta}{2}\,\Big[\,(\bar\Box\sigma)^2
    +\frac{2}{3}\,\bar R\,(\bar\nabla_{\mu}\sigma)^2\,
    \Big]\,\right\}\nonumber\\
    &&-
    \,\frac{1}{2(4\pi)^2}\Big(\frac{\alpha}{12}
    +\frac{\beta}{18}\Big)\,\nonumber \\
    &&\times\int d^4x\,\Big(g^{1/2}R^2(g)-
    \bar{g}^{1/2}R^2(\bar{g})\Big).              \label{deltaW}
    \end{eqnarray}

One can show that the higher-derivative in $\sigma$ terms are all
proportional to the coefficient $\alpha$. The $\alpha$-term can be
arbitrarily changed by adding a local counterterm $\sim g^{1/2}R^2$.
We fix this local renormalization ambiguity by an additional
criterion of the absence of ghosts. The conformal contribution to
the renormalized action on the minisuperspace background equals
    \begin{eqnarray}
    &&\delta \Gamma[\,g,\bar g\,]\equiv
    \Gamma_{R}[\,g\,]
    -\Gamma_{R}[\,\bar g\,]\nonumber \\
&&=
    m_P^2\,B\!\int d\tau
    \left(\frac{\dot{a}^2}{a}
    -\frac16\,\frac{\dot{a}^4}a\right),   \label{correction}\\
    &&m_P^2\,B=\frac34\,\beta,       \label{Bm_P^2}
    \end{eqnarray}
with the constant $m_P^2\,B$ which for scalars, two-component
spinors and vectors equals respectively $1/240$, $11/480$ and
$31/120$.

\section{Effective action on a static Einstein instanton}
For a conformal scalar field
\begin{equation}
    S[\,\bar g,\phi\,] = \frac12\,
    \sum_{\omega}\int_0^{\eta}
    d\eta' \left(\Big(\frac{d\phi_\omega}{d\eta'}\Big)^2+
    \omega^2\,\phi^2_\omega\right),                   \label{scal-action}
    \end{equation}
where $\omega=n$, $n=0,1,2,...$, labels a set of eigenmodes and
eigenvalues of the Laplacian on a unit 3-sphere. Thus
\begin{eqnarray}
    &&\mbox{$e$}^{\textstyle
    -\Gamma_{\rm 1-loop}[\,\bar g\,]}\nonumber \\&&=\int \prod_\omega
    d\varphi_\omega
    \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
    \int\limits_{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
    \,\,\,\,\,\,\,\,\,\,\,\,
    \phi_\omega(\eta)=\phi_\omega(0)=\varphi_\omega}
    \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
    \!\!\!\!\!\!
    D[\,\phi\,]\,
    \exp\big(-S[\,\bar
    g,\phi\,]\big)
    \nonumber\\
    &&
    ={\rm const}\,\prod_\omega\left(\sinh
    \frac{\omega\eta}2\right)^{-1},
    \end{eqnarray}
and the effective action equals the sum of contributions of the
vacuum energy $E_0$ and free energy $F(\eta)$ with the inverse
temperature played by $\eta$ --- the circumference of the toroidal
instanton in units of a conformal time,
    \begin{eqnarray}
    &&\Gamma_{\rm 1-loop}[\,\bar g\,]
    =\sum_{\omega}\left[\,\eta\,
    \frac{\omega}{2}
    +\ln\big(1-e^{-\omega\eta}\big)\,\right]\nonumber \\&&=
    m_P^2\,E_0\,\eta+F(\eta),            \label{1000}\\
    &&m_P^2\,E_0=\sum_{\omega}
    \frac{\omega}{2}=\sum_{n=1}^\infty
    \frac{n^3}{2},                           \label{E_0}\\
    &&F(\eta)=\sum_{\omega}
    \ln\big(1-e^{-\omega\eta}\big)\\
    &&=\sum_{n=1}^\infty n^2\,
    \ln\big(1-e^{-n\eta}\big).
    \end{eqnarray}

Similar expressions hold for other conformally invariant fields of
higher spins. In particular, the vacuum energy (an analog of the
Casimir energy) on Einstein static spacetime is
\begin{eqnarray}
    m_P^2\,E_0=\frac1{960}\times\left\{\begin{array}{c} 4 \\
    17\\
    88\end{array}\right.
    \end{eqnarray}
respectively for scalar, spinor and vector fields.

We should take into account the effect of the finite ghost-avoidance
renormalization denoted below by a subscript $R$, which results in
the replacement of $E_0$ above by a new parameter $C_0$:
\begin{eqnarray}
    &&\Gamma_{R}[\,\bar g\,]
    =m_P^2\,C_0\,\eta_0+F(\eta),          \label{GammaRR}\\
    &&m_P^2\,C_0=m_P^2\,E_0+\frac3{16}\,\alpha.  \label{GammaR}
    \end{eqnarray}

A direct observation indicates the following universality relation
for all conformal fields of low spins
\begin{eqnarray}
    m_P^2\,C_0=\frac12\,m_P^2\,B.         \label{universality}
    \end{eqnarray}

\section{Effective Friedmann and bootstrap equations}
Now we can write down the effective Friedmann equation governing the
Euclidean evolution of the universe. First of all, the full
conformal time on the instanton is
\begin{equation}
    \eta = 2\int_{\tau_-}^{\tau_+}
    \frac{d\tau\,N(\tau)}{a(\tau)},                \label{fulltime}
    \end{equation}
where $\tau_\pm$ label the turning points for $a(\tau)$ -- its
minimal and maximal values.

The effective action is ($m_P^2\equiv 3/4\pi G$)
\begin{eqnarray}
    &&\Gamma[\,a(\tau),N(\tau)\,]\nonumber \\&&=
    2 m_P^2\int_{\tau_-}^{\tau_+} d\tau\left(-\frac{a\dot{a}^2}N
    - Na + N H^2 a^3\right)\nonumber\\
    &&
    +2B m_P^2\int_{\tau_-}^{\tau_+}
    d\tau \left(\frac{\dot{a}^2}{Na}
    -\frac16\,\frac{\dot{a}^4}{N^3 a}\right)\nonumber\\
    &&
    +F\left(2\int_{\tau_-}^{\tau_+}
    \frac{d\tau\,N}{a}\right)+ B m_P^2
    \int_{\tau_-}^{\tau_+}
    \frac{d\tau\,N}{a}\,,
    \end{eqnarray}
and the effective Friedmann equation reads
\begin{eqnarray}
    &&\frac{\delta\Gamma}{\delta N}=
    2m_P^2\left(\frac{a\dot{a}^2}{N^2}
    - a + H^2 a^3\right)
    \nonumber\\
    &&
    +2Bm_P^2\left(-\frac{\dot{a}^2}{N^2 a}
    +\frac12\,\frac{\dot{a}^4}{N^4 a}\right)\nonumber \\&&
    +\frac2a \left(\frac{dF(\eta)}{d\eta}+\frac{B}2
    m_P^2\right)=0.
    \end{eqnarray}
In the gauge $N = 1$ this equation takes form
\begin{eqnarray}
    \frac{\dot{a}^2}{a^2}
    +B\,\left(\frac12\,\frac{\dot{a}^4}{a^4}
    -\frac{\dot{a}^2}{a^4}\right) =
    \frac{1}{a^2} - H^2 -\frac{C}{ a^4},     \label{Friedmann}
    \end{eqnarray}
where the amount of radiation constant $C$ is given by the bootstrap
equation
   \begin{equation}
    m_P^2 C = m_P^2\frac{B}2 +\frac{dF(\eta)}{d\eta}
    \equiv \frac{B}2 m_P^2+
    \sum_{\omega}\frac{\omega}{e^{\omega\eta}-1}. \label{bootstrap}
    \end{equation}

The Friedmann equation can be rewritten as
\begin{eqnarray}
    &&\dot{a}^2 = \sqrt{\frac{(a^2-B)^2}{B^2}
    +\frac{2H^2}{B}\,(a_+^2-a^2)(a^2-a_-^2)}\nonumber \\
    &&-\frac{(a^2-B)}{B}          \label{time-der1}
    \end{eqnarray}
and has the same two turning points $a_\pm$ as in the classical case
provided
    \begin{equation}
    a_-^2 \geq B.                \label{require}
    \end{equation}
This requirement is equivalent to
\begin{equation}
    C \geq B-B^2 H^2,\,\,\,\,B H^2\leq\frac12. \label{restriction1}
    \end{equation}
Together with
\[CH^2 \leq \frac{1}{4},\]
the admissible domain for instantons reduces to the curvilinear
wedge below the hyperbola and above the straight line to the left of
the critical point (see Figure 4)
\[C = \frac{B}{2},\ \ H^2 = \frac{1}{2B}.\]

\begin{figure}
\includegraphics[height=.4\textheight]{hh10.eps}
\caption{The instanton domain in the $(H^2,C)$-plane is located
between bold segments of the upper hyperbolic boundary and lower
straight line boundary. The first one-parameter family of instantons
is labeled by $k=1$. Families of garlands are qualitatively shown
for $k=2,3,4$. $(1/2B,B/2)$ is the critical point of accumulation of
the infinite sequence of garland families.}
\end{figure}

For a scalar field the numerical analysis of the Friedmann and
bootstrap equations shows that the one-parameter family of
instantons interpolates between the point on the lower line boundary
with the parameters
\begin{eqnarray}
    H^2\approx2.997 \,m_P^2,\,\,
    C\approx0.004\, m_P^{-2},\,\,\,
    \Gamma_0\approx-0.1559,    \label{lower-point}
    \end{eqnarray}
and the point on the upper hyperbolic boundary
    \begin{eqnarray}
    H^2\approx12.968 \,m_P^2,\,\,
    C\approx0.0193 \,m_P^{-2},\,\,\,
    \Gamma_0\approx-0.0883.   \label{upper-point}
    \end{eqnarray}
The last instanton  describes the creation of a static Einstein
Universe of the constant size \[a=a_+=a_-=1/(\sqrt{2 H})\] with the
hot gas of a conformally-invariant scalar field particles in the
equilibrium state with the temperature
    \begin{eqnarray}
    T=\frac1{a\eta}=\frac{H}{\pi\sqrt{1-2BH^2}}.   \label{T}
    \end{eqnarray}

\section{Infrared catastrophe is eliminated.}
The suggested approach allows to resolve the problem of the
so-called infrared catastrophe for the no-boundary state of the
Universe based on the Hartle-Hawking instanton. This problem is
related to the fact that the Euclidean action on this instanton is
negative and inverse proportional to the value of the effective
cosmological constant. This means that the probability of the
universe creation with an infinitely big size is infinitely high. We
shall show now that the conformal anomaly effect allows one to avoid
this counter-intuitive conclusion.




Indeed, outside of the admissible domain for the instantons with two
turning points, obtained above, one can also construct instantons
with one turning point which smoothly close at $a_- = 0$ with $\dot
a(\tau_-)=1$. Such instantons correspond to the Hartle-Hawking pure
quantum state. However, in this case the on-shell effective action,
which reads for the set of solutions obtained above as
    \begin{eqnarray}
    &&\Gamma_0= F(\eta)-\eta\frac{dF(\eta)}{d\eta}
    \nonumber \\
    &&
    +4m_P^2\int_{a_-}^{a_+}
    \frac{da \dot{a}}{a}\left(B-a^2
    -\frac{B\dot{a}^2}{3}\right),              \label{action-instanton}
    \end{eqnarray}
diverges to plus infinity. Indeed, for $a_-=0$ and $\dot a_{-}=1$
\begin{eqnarray}
&&\eta = \int_0^{a_+}\frac{da}{\dot a
a}=\infty,\,\,\,F(\infty)=F'(\infty)=0,
\end{eqnarray}
and hence the effective Euclidean action diverges at the lower limit
to $+\infty$. Thus,
\[\Gamma_0 = +\infty,\ \ \exp(-\Gamma_0) = 0,\]
and this fact completely rules out all pure-state instantons,
and only mixed quantum states of the universe, described by the cosmological
density matrix appear to be admissible.

\section{Instanton garlands}
One should consider also the multiple instanton configurations,
which could be called ``Instanton garlands'' (see Figure 5). The
total conformal time for such an instanton garland is
\begin{equation}
    \eta_0^{(k)} =
    2k \int_{\tau_-}^{\tau_+} \frac{d\tau}{a}
    = 2k\int_{a_-}^{a_+}
    \frac{da}{a\dot{a}},           \label{conf-time-k}
    \end{equation}
where $k$ is the number of simple instanton folds in a garland.

Numerical analysis for $k=2$ shows the existence of the
one-parameter family of instantons similar to the case of $k=1$. It
interpolates between the point on the lower boundary of
$(C,H^2)$-plane
\begin{equation}
    H^2_{(2)}\approx45.89 \,m_P^2,\,\,
    C_{(2)}\approx0.0034\, m_P^{-2},\,\,\,
    \Gamma_0^{(2)}\approx-0.0113,    \label{lower-point2}
\end{equation}
and the point on the upper (hyperbolic) boundary
    \begin{equation}
    H^2_{(2)}\approx61.12 \,m_P^2,\,\,
    C_{(2)}\approx0.0041\, m_P^{-2},\,\,\,
    \Gamma_0^{(2)}\approx-0.0145.    \label{upper-point2}
    \end{equation}

Such families exist for all $k, 1 \leq k \leq \infty,$ and their
infinite sequence is saturated at the critical point,
\begin{eqnarray}
&&\eta_0^{(k)} \simeq \ln k^2\\
&&H^2_{(k)}\simeq \frac1{2B}
    \left(1 - \frac{\ln^2k^2}{2k^2\pi^2}
    \right),                            \label{h-tilde}\\
    &&C_{(k)}\simeq \frac{B}{2}
    \left(1+\frac{\ln^2k^2}{2k^2\pi^2}
    \right),                        \label{c-tilde}\\
    &&\Gamma_0^{(k)}\simeq
    -m_P^2B\,\frac{\ln^3 k^2}{4k^2\pi^2}.         \label{action-k}
    \end{eqnarray}
The length of instanton families decreases as $1/k^4$. Infinite
garlands ($k \rightarrow \infty$) do not dominate the instanton
distribution because their action grows with $k$ rather than
decreases to $-\infty$.
\begin{figure}
\includegraphics[height=.05\textheight]{hh11.eps}
\caption{Segment of the garland consisting of three folds of a
simple instanton glued at surfaces of a maximal scale factor.}
\end{figure}

A growing spin of a conformal particle decreases the instanton size
and makes its probability weight higher. For $N$ fields
\begin{eqnarray}
&&C \rightarrow NC,\\
&&B \rightarrow NB,\\
&&\eta_0 \rightarrow \eta_0,\\
&&F(\eta_0) \rightarrow NF(\eta_0),\\
&&H^2 \rightarrow \frac{H^2}{N}.
\end{eqnarray}
The initial size of the universe grows with the growing spin and
number of fields.

\section{Where Euclidean Quantum Gravity and Cosmology comes from ?}
In the preceding sections we have described a new approach to the
problem of initial conditions in cosmology based on the use of the
combination of two ideas: the density matrix formalism and Euclidean
quantum gravity. A natural question arises: where Euclidean quantum
gravity comes from? The answer can be formulated briefly as follows:
from the Lorentzian quantum gravity (LQG) \cite{Bar-den}. Namely,
the density matrix of the Universe for the microcanonical ensemble
in Lorentzian quantum cosmology of spatially closed universes
describes an equipartition in the physical phase space of the
theory, but in terms of the observable spacetime geometry this
ensemble is peaked about a set of cosmological instantons (solutions
of the Euclidean quantum cosmology) limited to a bounded range of
the cosmological constant. These instantons obtained above as
fundamental in Euclidean quantum gravity framework, in fact, turn
out to be the saddle points of the LQG path integral, belonging to
the imaginary axis in the complex plane of the Lorentzian signature
lapse function \cite{Bar-den}.

\section{Cosmological evolution and Big Boost singularity}
Now let us consider the cosmological evolution of the unverse
starting from the initial conditions described above. Making the
transition from the Euclidean time to the Lorentzian one, $\tau=it$,
we can write the modified Lorentzian Friedmann equation as
\cite{Bar-Kam-Def}
    \begin{eqnarray}
    &&\frac{\dot{a}^2}{a^2} + \frac{1}{a^2} =
    \frac{1}{B}\left\{1 - \sqrt{1-\frac{16\pi G}3\,B\,
    \varepsilon}\,\right\},\\
    &&\varepsilon=\frac3{8\pi G}\left(H^2+\frac{\cal
    C}{a^4}\right),\\
    &&{\cal C} \equiv C -\frac{B}{2}, \label{Lor-ev}
    \end{eqnarray}
where $\varepsilon$ is a total gravitating matter density in the
model (including at later stages also the contribution of particles
created during inflationary expansion and thermalized at the
inflation exit). A remarkable feature of this equation is that the
Casimir energy is totally screened here and only the thermal
radiation characterized by $\cal C$ weighs.

If one wants to compare the evolution described by Eq.
(\ref{Lor-ev}) with the real evoltuion of the universe, first of all
it is necessary to have a realistic value for an effective
cosmological constant $\Lambda=3H^2$. The only way to achieve this
goal is to increase the number of conformal fields and the
corresponding parameter $B$, (\ref{Bm_P^2}), of the conformal
anomaly (\ref{anomaly}). The mechanisms for growing number of the
conformal fields exist in some string inspired cosmological models
with extra dimensions \cite{Bar-den}. If some of these mechanisms
work we can encounter an interesting phenomenon: if the $B$ grows
with $a$ faster than the rate of decrease of the energy density
$\varepsilon$ one encounters a new type of the cosmological
singularity - Big Boost. This singularity is characterized by finite
values of the cosmological radius $a_{BB}$ and of its time
derivative $\dot{a}_{BB}$, while the second time variable $\ddot{a}$
has an infinite positive value. The universe reaches this
singularity at some finite moment of cosmic time $t_{BB}$:
\begin{eqnarray}
&&a(t_{BB}) = a_{BB} < \infty,\\
&&\dot{a}(t_{BB}) = \dot{a}_{BB} < \infty,\\
&&\lim_{t \rightarrow t_{BB}} \ddot{a}(t) = \infty.
\label{BigBoost}
\end{eqnarray}

It is interesting to compare this singularity with other types of
cosmological singularities arising in isotropic and homogeneous
Friedmann cosmological models. The most known and well studied one
is the Big Bang (or Big Crunch) singularity which is characterized
by a vanishing scale factor at the initial or final moments of the
cosmological evolution:

\begin{equation}
a(t_{In,Fin}) = 0.
\end{equation}
Such a singularity arises, for example, in Friedmann universes
filled by dust, radiation and other types of ``standard'' matter
with the equation of state parameter $w = p/\rho>-1/3$.

During the last decade another type of singularity has acquired some
popularity. This is the so called Big Rip singularity
\cite{Rip,Rip1}, when the radius of the universe, its first time
derivative and the Hubble variable tend to infinity at some finite
moment of time:
\begin{eqnarray}
&&\lim_{t \rightarrow t_{BR}}a(t) = \infty,\\
&&\lim_{t \rightarrow t_{BR}}\dot{a}(t) = \infty,\\
&&\lim_{t \rightarrow t_{BR}}\frac{\dot{a}(t)}{a(t)} = \infty.
\label{Rip}
\end{eqnarray}
Such a singularity arises in the models where the phantom dark
energy (i.e. dark energy for which the equation of state parameter
$w < -1$) is present. Some modern cosmological observations give
certain indications in favour of  models, including phantom dark
energy.

Another type of cosmological singularity was found in some
cosmological models based on tachyon (Born-Infeld-type) field
\cite{Brake}. Such a singularity is a result of  a decelerating
evolution of the universe which culminates at some finite moment of
time $t_{BBr}$  when the cosmological radius has some finite value,
its first time derivative and the Hubble variable are equal to zero,
while the second time derivative of the cosmological radius tends to
$-\infty$:
\begin{eqnarray}
&&a(t_{BBr}) = a_{BBr} < \infty,\\
&&\dot{a}(t_{BBr}) = 0,\\
&&\lim_{t \rightarrow t_{BBr}} \ddot{a}(t) = -\infty.
\label{BigBrake}
\end{eqnarray}
This type of  cosmological singularity arises, for example in a
simple model, representing a flat Friedmann universe filled with the
``anti-Chaplygin gas'' \cite{Brake,Brake1}, whose equation of state
is $p = A/\rho$ (where $A$ ia a positive constant) 
in analogy with the Chaplygin gas cosmological model
\cite{Chap}, whose equation of state is $p = -A/\rho$ and which
describes a unified model of dark energy and dark matter. However,
when one considers a richer tachyon model \cite{Brake} the situation
looks more interesting. Indeed, the cosmic deceleration era in this
model can follow a long period of accelerated expansion. Thus, one
cannot exclude, that after the period of the accelerated quasi-de
Sitter expansion which we experience now, a quite new phase of
cosmological evolution will come.

\section{Conclusion: initial conditions, singularities, density
matrix and Landau's legacy}

In conclusion we would like to make some remarks concerning
connections between topics touched in the reported series of works
\cite{Bar-Kam-den,Bar-Kam-den1,Bar-den,Bar-Kam-Def} and the
scientific legacy of L.D. Landau.

First, let us remember that the main goal of works in quantum
cosmology is the construction of the quantum state of the universe,
which can predict initial conditions for its subsequent classical
evolution. As we have already noticed in the Introduction, it was
L.D. Landau who pointed out that a consistent physical theory should
not only present the equations of motion for the system under
consideration, but also be able to predict initial conditions for
these equations \cite{Land-priv}.

Second, considering some string-inspired cosmological models, we
have found a new type of the cosmological singularity - the Big
Boost singularity, characterized by an infinite value of the cosmic
acceleration \cite{Bar-Kam-Def}. The importance of the problem of
cosmological  singularity was also underlined by Landau in the
fifties on equal footing with such topics as the theory of phase
transitions and superconductivity \cite{Land-priv}. Development of
both the theoretical and observational cosmology has confirmed the
correctness of his opinion. Indeed, the theoretical study of the
anisotropic universe in the vicinity of the Big Bang (Crunch) moment
has resulted  in the discovery of the phenomenon of the oscillatory
approach to the cosmological singularity \cite{KL,BKL,Misner}, while
the recent observations of the cosmic acceleration phenomenon have
stimulated study of other types of cosmological singularities (for a
recent review see e.g. \cite{Khal-Kam}).

Finally, the results of the reported works open a new insight into
the role of a density matrix in quantum theory. Let us recollect the
main features of the density matrix of the universe, which we
advocate:
\begin{enumerate}
\item It was shown that the density matrix, corresponding to mixed
quantum states is a fundamental object, not less fundamental than
the wave function, describing pure quantum states.
\item Appearance of the density matrix is not necessarily a result
of our ignorance or an artifact of tracing out a part of the degrees
of freedom.
\item The universe in the framework of quantum cosmology can be born
in a mixed and not in a pure quantum state.
\end{enumerate}
This means that the density matrix plays a more fundamental role
than that which was prescribed to it until now.

Here we would like to say that it  was L.D. Landau who has introduced
the notion of the density matrix in 1927 in the paper ``{\it Das
D\"{a}mpfungsproblem in der Wellenmechanics}'', Z. Physik, 45 (1927)
430 in parallel with F. Bloch and J. von Neumann \cite{vonNeumann}.
This happened at the dawn of quantum mechanics and all three of them
were very young. Landau was the youngest of them -- he was 19 year
old.

The existence of mixed quantum states  described by the density
matrix was and is considered as a result of lack of information,
which is not connected with the basic laws of quantum theory. The
probability weights present in a  density matrix are the numbers
reflecting traditional statistic probabilities which exist already
in the framework of the classical theory. These probability weights
which are given by eigenvalues of the density matrix are the
relative weights of different pure quantum states constituting a
mixed quantum state. Thus, these probability weights play the role
similar to one played by the probability distribution function on
the phase space in the classical statistical mechanics. Considering
pure classical states, where all the coordinates and momenta have
determined values, one does not encounter the probabilities. In the
case of quantum theory considering pure quantum states, one
eliminates the statistical probabilities, while quantum mechanical
probabilities are always present. Hence, the statistical
probabilities are usually considered to be less fundamental than the
quantum mechanical ones. Moreover, one normally thinks that the
universe is born and, hence, always exists in the pure quantum
state, while its subsystems are usually represented by density
matrices due to quantum entanglement effects. The results of the
presented series of works show that, at least in some cosmological
models, the universe as a whole always exists in the mixed quantum
state represented by the cosmological density matrix, while the
probability of its birth in the pure quantum state (Hartle-Hawking
no-boundary state) is equal to zero. That means that the statistical
or ``thermodynamical'' probability is in a way no less fundamental
than the quantum mechanical probability, because, even in principle,
we cannot get rid of it, choosing to work only with pure states.
Thus, the density matrix introduced by L.D. Landau, F. Bloch and J.
von Neumann in 1927, is a more fundamental object than these authors
could have imagined at that time.


\begin{theacknowledgments}
The work of A.B. was supported by the RFBR grant 08-02-00725 and the
grant LSS-1615.2008.2. The work of A.K. was partially supported by
RFBR grant No 08-02-00923 and by the grant LSS-4899.2008.2.

\end{theacknowledgments}


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\end{document}
