# Distortion of extra dimensions
in the inflationary Multiverse

Sergey G. Rubin1,2,a Júlio C. Fabris1,3

(1 National Research Nuclear University MEPhI
(Moscow Engineering Physics Institute),
115409, Kashirskoe shosse 31, Moscow, Russia
2N.I. Lobachevsky Institute of Mathematics and Mechanics,
Kazan Federal University,
Kremlevskaya street 18, 420008 Kazan, Russia
a<EMAIL_ADDRESS>
3 Núcleo Cosmo-ufes & Departamento de Física,
Universidade Federal do Espírito Santo, Vitória,
ES, CEP 29075-910, Brazil)

###### Abstract

We discuss the effect of the quantum fluctuations at high energies on the
final shape of compact extra dimensions. The quantum fluctuations produce a
wide range of the initial extra metrics in causally disconnected regions
(pocket universes) of the Multiverse during the inflationary stage. This set
of initial extra metrics evolves to a set of inhomogeneous metrics at the
present time. The low energy physics appears to be different in different
pocket universes. The numerical estimate of the probability of finding a
specific metric is based on the model of the compact 2-dimensional extra
space.

##  1 Introduction

The inflationary paradigm has been developed decades ago, starting with the
papers [1, 2]. The existence of an inflationary period in the history of our
Universe seems inevitable since it explains the key problems of the Big Bang
theory: the horizon problem, the flatness problem, the galaxies formation etc.
[3, 4]. The first inflationary mechanisms were based on self-consistent
equations of the scalar and gravitational fields. The key point is that the
self gravitating scalar field leads to an exponential growth of initially
small spatial regions at high energies. The horizon size appears to be much
smaller then the expanding region. Therefore, the latter is separated into
quickly increasing number of the causally disconnected regions. The estimation
made by A. Linde [3] gives $\mathfrak{N}\sim 10^{10^{12}}$ such regions -
pocket universes (PU) [5] after the inflation is finished.

Immediate question arises - are the physical parameters equal in the different
PU? Positive answer open the way for solution of the fine tuning problem - one
of the most serious enigma of modern physics to our mind. To make physical
laws different in the different PU, we need a substantial variety of ground
states. Nevertheless, it seems that the PU have similar properties for simple
potentials. Even the idea of landscape does not improve the situation without
knowledge of the ground states distribution.

In this paper we discuss the way of endowing each PU by a specific ground
state. To this end, we employ the idea of the compact extra dimensions. The
extra dimensions are widely used tool for study deep problems of the modern
physics. In particular, the physical parameters $h_{i}$ of the Standard Model
depend on the extra space metric at low energies after the extra coordinates
have been integrated out.

It is usually assumed that an extra space metric should satisfy generalized
Einstein differential equations. Hence, there is a continuum set of solutions
that differs by arbitrary constants. The choice of their specific values is
usually made by comparison of the observations with the model predictions. As
was shown in [6, 7], the simplest metric is not necessarily favorite.

Let us mention several examples for illustration. In the paper [8], arbitrary
constants are fixed by energetic argument which is not very strong one if the
gravity is included, see discussion in [9]. Very often, like in the paper
[10], the constants used to restore the 4-dim Planck mass and the cosmological
constants. The classical solutions explicitly containing two arbitrary
constants in 5-dim brane world are considered in [11, 12] A flat, two-
dimensional toroidal compactification is discussed in [13]. where the relative
angle between the two internal directions can be continuously varied to
alternate the extra space volume.

We conclude that there is an enormous number $\mathfrak{N}$ of separate
universes - PU - on the one hand, and a continuum set of different extra
metrics, on the second hand. The idea is to relate different extra metric and
different PU.

The discussion above does not specify extra space metric and topology. Here,
we focus on particular set of the compact extra spaces characterized by
inhomogeneous extra metrics. The latter has been revealed in [6] and their
stability was discussed in [7]. Their application leads to variety of effects,
see [14, 15, 16, 17]. Such kind of metrics is used in the other research like
the Universal Extra Dimensions approach [18], the branes formation [6] the
justification of the 3 generation of leptons [19] and the baryon asymmetry of
the Universe [20].

In this letter, we pay attention to the way of such metrics formation. We show
here that the quantum fluctuations are responsible for the final shape of
metric. It is known that the scalar field tends to a potential minimum if the
gravity is not involved into consideration. It seems that nothing changed if
we add maximally symmetric extra dimensions. Any scalar field excitation
acting in the extra dimensions (like KK-modes) also is attenuated sooner or
later. The picture is changed drastically if the gravity is taken into
account. Indeed, the energy density of fluctuations in the FRW universe
evolves into dense local objects forming the large scale structure of the
Universe. In this letter, we show that the gravity influences the scalar field
fluctuations within the compact extra dimensions in the same manner. The
quantum fluctuations produced at the inflationary stage deform the extra space
metric that tends to stationary state after the end of inflation.

In Section II we describe the basis of the following discussion - the
$D$-dimensional scalar-tensor model with arbitrary function $f(R)$. 4-dim
space is endowed by the de Sitter metric. Section III is devoted to stationary
states in extra dimensions mainly based on paper [7]. The classical states are
defined up to the quantum fluctuation which could be large at high energies.
In Section IV we elaborate the way to work with the scalar field quantum
fluctuations in $4+n$ dimensional space. In Section V we estimate the
formation probability of specific pocket universes.

##  2 Outline

Consider the manifold with the topology $M_{1}\times M_{3}\times M_{n}$. The
space $M_{1}\times M_{3}$ \- is described by the de-Sitter metric, $M_{n}$ \-
represents the ideal $n$-dimensional sphere, its deformation is described
below. Our model is based on the action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{D}x\sqrt{|g_{D}|}\bigg{[}\frac{m_{D}^{D-2}}{2}f(R)+\frac{1}{2}g^{AB}\phi_{,A}\phi_{,B}-V(\phi)\bigg{]},$
$\displaystyle A,B=0,\ldots,5.$

where $g_{D}=\det(g_{AB})$, $D=4+n$, $n$ is a number of extra dimensions.
$f(R)$ and $V(\phi)$ are some functions of the D-dim scalar curvature $R$ and
the scalar field $\phi$, respectively. Requirements for the potential are
quite easy and we choose the simplest form $V(\phi)=m^{2}\phi^{2}/2$.
Variation of action $S$ with respect to $\phi$ and $g^{AB}$ leads to the field
equations

$\displaystyle\square\phi+V_{\phi}=0,\quad{\rm where}\
\square\phi=\nabla_{A}\nabla^{A}\phi,\quad V_{\phi}=dV/d\phi,$ (2)

$\displaystyle-\frac{1}{2}\delta_{A}^{B}f(R)+\big{[}R^{B}_{A}+\nabla_{A}\nabla^{B}-\delta^{B}_{A}\square\big{]}f_{R}=-\frac{1}{m_{D}^{2}}T^{B}_{A},$
(3) $\displaystyle\hskip 8.53581ptf_{R}=df/dR,$

Let it be pure de-Sitter 4-dim space. Then the stress-energy tensor of the
scalar field $\phi(t,u)$ reads

$\displaystyle
T^{B}_{A}[\phi]=\phi_{,A}\phi^{,B}-\frac{1}{2}\delta^{B}_{A}\phi_{,C}\phi^{,C}+\delta^{B}_{A}V.$
(4)

The interval

$ds^{2}=dt^{2}-e^{2Ht}\delta_{ij}dx^{i}dx^{j}-du^{2}-r(u)^{2}d\varphi^{2}$ (5)

for the 2-dim compact extra space $(n=2)$ with the sphere topology and
$H=const$. Here $i,j=1,2,3$ and the extra space coordinates $y=(u,\varphi)$.

As was shown in [6, 7], system (3) is reduced to simpler system

$\displaystyle\phi^{\prime\prime}+\phi^{\prime}\frac{r^{\prime}}{r}=V^{\prime}_{\phi},$
(6)
$\displaystyle-\frac{1}{2}f(R)+3H^{2}f_{R}+f_{R}^{\prime\prime}+\frac{r^{\prime}}{r}f_{R}^{\prime}=m_{D}^{-2}\left(-\frac{\phi^{\prime
2}}{2}-V\right),$ (7)
$\displaystyle-\frac{1}{2}f(R)-\frac{r^{\prime\prime}}{r}f_{R}+\frac{r^{\prime}}{r}f_{R}^{\prime}=m_{D}^{-2}\left(\frac{\phi^{\prime
2}}{2}-V\right),$ (8)
$\displaystyle-\frac{1}{2}f(R)-\frac{r^{\prime\prime}}{r}f_{R}+f_{R}^{\prime\prime}=m_{D}^{-2}\left(-\frac{{\phi^{\prime}}^{2}}{2}-V\right),$
(9)

for metric (5) and the stationary scalar field depending on the internal
coordinate $u$, $\phi=\phi(u)$. Here $f^{\prime}_{R}=df_{R}/du$, etc. The
system described above is the basis for the following discussion.

The Ricci scalar is expressed in terms of the metric functions as

$R=12H^{2}-\frac{2r^{\prime\prime}}{r}$ (10)

where ${}^{\prime}\equiv d/du$.

Throughout this letter, we use the conventions for the curvature tensor
$R_{\mu\nu\alpha}^{\beta}=\partial_{\alpha}\Gamma_{\mu\nu}^{\beta}-\partial_{\nu}\Gamma_{\mu\alpha}^{\beta}+\Gamma_{\sigma\alpha}^{\beta}\Gamma_{\nu\mu}^{\sigma}-\Gamma_{\sigma\nu}^{\beta}\Gamma_{\mu\alpha}^{\sigma}$
and the Ricci tensor is defined as $R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}\,$.

##  3 Static inhomogeneous extra space

###  3.1 f(R) gravity without a matter

In this section, we follow the analysis of system (3) made in [7]. More
definitely, the difference (7)-(8) leads to equation

$\displaystyle 3H^{2}+\frac{r^{\prime\prime}}{r}=0$ (11)

for the de Sitter space with the Hubble parameter $H$. Here we have kept in
mind that $\phi=0$, $f_{R}\neq 0,\,R=const$ and hence $R^{\prime}=0$. The
regular center is characterized by conditions $r(0)=0,r^{\prime}(0)=1$.
Together with equation (11), these conditions give the solution

$r(u)=r_{0}\sin(u/r_{0}).$

with the constant radius $r_{0}$ of extra dimensions depending on $H$,

$r_{0}=\frac{1}{\sqrt{3}H}$ (12)

Expressions (10) and (11) give the 6-dim Ricci scalar

$R=18H^{2}$ (13)

The 2-dim metric has the standard form

$ds_{2}=r_{0}^{2}[dv^{2}+\sin^{2}(v)d\varphi^{2}]$ (14)

in terms of new coordinate $v=u/l_{0}$. This final metric does not contradict
our assumption $R=const$ and gives the relation (12) between the Hubble
parameter and the extra dimensional radius $r_{0}$. At the present time
$H\simeq 0$ and hence $r_{0}\to\infty$ which strongly contradicts
observations. As has been shown in [6, 7], the inhomogeneous compact extra
dimensions are free from this defect. Here we study one of the way of such
metrics formation.

At high energies, the maximally symmetrical extra space does exist. Its Ricci
scalar is proportional to the Hubble parameter square. Below we consider
influence of the scalar field distribution to these basic metric.

###  3.2 Continuous set of the extra dimensional metrics. The scalar field in
the D-dim space.

Let us decompose the scalar field in action (2) in the standard manner

$\phi(x,y)=\sum_{a}\Phi_{a}(x)Y_{a}(y)$ (15)

where the orthonormal functions $Y_{a}(y)$ satisfy the equations

$\square_{n}Y_{a}=\lambda_{a}Y_{a},\quad
2\pi\int_{0}^{u_{max}}|Y_{a}|^{2}\sqrt{|g_{n}|}du=1.$ (16)

Here $n$ is the dimensionality of the extra space and we assume the
homogeneity in $x$ directions. The extra dimensions are described by $u,\phi$
coordinates. The upper limit of the integral can be found in the capture to
Fig.1. Following the Kaluza-Klein approach, one can substitute decomposition
(15) into (2) to obtain an action as a sum of scalar modes. The destiny of
these scalar modes acting in the 4-dim de Sitter universe is well known - they
are attenuated according to the equation

$\ddot{\Phi}_{a}+3H\dot{\Phi}_{a}+\mu^{2}_{a}\Phi_{a}=0;\quad\mu_{a}^{2}=m^{2}+\lambda_{a}$
(17)

for the quadratic potential $V(\phi)={m^{2}}\phi^{2}/2$. The final state is
common stationary state $\Phi_{a}=0$ for all modes. Therefore, we are back to
the situation described above in subsection 3.1. The scalar field equals zero
and the only asymtote of the 2-dim metric is the maximally symmetric one with
positive curvature.

The picture changes drastically when we keep in mind a deformation of the
extra space metric under the scalar field influence. In this case, we have to
solve the system of equations (3), rather than single equation (17). This is a
difficult task, and we will restrict ourselves to discuss the asymptotes for
$t\to\infty$.

There is infinite number of stationary solutions to system (6), (7), (8), (9)
depending on the additional conditions [6, 7]. It was revealed there that the
continuum set of nontrivial stationary solutions does exist. Each solution is
characterized by a specific additional condition. For illustration, we
represent one of the stationary solution of the system, see Fig.1.

Figure 1: The extra space metric function $r(u)$ (solid line, upper figure)
and the scalar field $\phi(u)$ (lower figure) for $f(R)=aR^{2}+bR+c$ and
$V(\phi)=(m^{2}/2)\phi^{2}$ (units $m_{D}=1$). Dashed line relates to the
maximally symmetric 2-dim sphere. The parameter values are $m=0.1,\ b=1,\
a=-10.9,\ c=-0.0021$. Additional conditions are: $\phi_{0}=0.9,\ H=0.1$ and
$R(0)\simeq 0.182$. The variation range of the parameter $u$ is $(0\div
u_{max}),\,u_{max}=17.1.$

As was shown in [6, 7], each stationary metric relates to a specific value of
the cosmological constant. The maximally symmetric 2-dim extra space is one of
the solutions. The latter is unacceptable because of the Hubble parameter
smallness at the present time. Indeed, the observable value $H_{0}\sim
10^{-42}GeV$ that gives too big size (12) of extra dimensions comparable to
the Universe size.

Fortunately, there is infinite set of the stationary states. As one can see
from the Figure 1, upper panel, that the size of distorted extra space is
quite small, $\sim 5$ in $m_{D}$ units. As was revealed in [6, 16, 7], there
exists a solution to the system (6) - (9) for which the extra space size
remains small even if $H=0$. So, the observable value of the cosmological
constant and small size of the extra space may not contradict each other.

Each area under the horizon - PU - can be equipped with the specific extra
space metrics. It means that different universes are characterized by
different cosmological constants.

Before moving further, an important remark is necessary. The quantum
fluctuations $\delta\phi$ play important role at the inflationary stage when
the Hubble parameter is large. It is assumed that the scalar field consists of
classical and quantum parts

$\phi(t,u)=\phi_{cl}(u)+\delta\phi(t,u).$ (18)

The classical part $\phi_{cl}$ is good approximation if the quantum
fluctuations are small,

$\sqrt{<\delta\phi^{2}>}\ll\phi_{cl}(u).$ (19)

This condition is considered true when the stationary solutions of the system
(6), (7), (8), (9) are discussed.

##  4 Scalar field fluctuations in D-dim space

The previous section is devoted to the discussion on the inhomogeneous extra
dimensional metrics. They form continuous set of stable metrics. Now we will
consider the way they are formed.

Analogy to the 4-dim collapse of dust to the black hole is good support for
our intuition. Indeed, the specific black hole mass is the result of initial
dust distribution which collapses to this black hole. In its turn, the
collapsing energy density distribution is the result of the quantum
fluctuations at the inflationary stage. Evidently, the resulting black hole
masses form a continuous set. The quantum fluctuations of scalar field are
also a reason of the continuum set of inhomogeneous extra metrics.

Important question is the probability of appropriate quantum fluctuations. The
subsequent mechanism has been substantially studied in the inflationary
scenario [4] and we use it below.

###  4.1 4-dim fluctuations

This subsection is needed to remind results for the standard 4-dim inflation
which will be used later.

Let us start with the scalar field of mass $m$ and the standard action (2) in
4 dimensions. It is known that the quantum fluctuations of scalar field
quickly acquires classical behavior described by equation (2). The latter
depends on metric but not on the same function $f(R)$. Hence, we may use known
results for the scalar field fluctuations acting in the de Sitter metric. We
are interested in the super horizon scales for which the fluctuations do not
depend on the space coordinates. The probability to find the field value
$\phi_{2}$ at the instant $t_{2}=t_{1}+t$ in a space region of the horizon
size $H^{-1}$ is, see [4]

$dP(\phi_{2},t_{1}+t;\phi_{1},t_{1}|m)=d\phi_{2}\cdot\sqrt{q/\pi}\exp\left[-q(\phi_{2}-\phi_{1}e^{-\mu
t})^{2}\right]$ (20)

where

$\displaystyle q=\frac{\mu}{\sigma^{2}}\left(1-e^{-2\mu t}\right)^{-1},$ (21)
$\displaystyle\mu=\frac{m^{2}}{3H}=const,$ (22)
$\displaystyle\sigma=\frac{H^{3/2}}{2\pi}=const.$ (23)

The average field value can be derived from (20)

$\displaystyle<\phi(t)^{2}>=\int_{-\infty}^{\infty}\phi^{2}dP(\phi,t_{1}+t;\phi_{1},t_{1}|m)$
(24) $\displaystyle=\phi_{1}^{2}e^{-2\mu
t}+\frac{3H^{4}}{8\pi^{2}m^{2}}\left(1-e^{-2\mu t}\right).$

This expression coincides with that known in the literature [21]. The first
term relates to the classical motion in the 4-dim space while the second one
is responsible for the quantum fluctuations.

###  4.2 D-dim fluctuations

In this subsection, we extend the results discussed above to the $D$-dim model
with metric (5) and action (2). In the spirit of the Kaluza-Klein approach,
the scalar field can be decomposed in the series (15) of the orthogonal
normalized functions $Y_{a}$. The discrete set of eigen-values
$\lambda_{a}\,a=0,\pm 1,\pm 2...$ depends on the extra space metric. Here
remark is necessary. The analytical manipulations are much more clear if the
extra metric is maximally symmetric. Fortunately, the metric satisfying the
system of equations is almost maximally symmetric indeed as can be seen from
Fig.1, upper panel, at least for some additional conditions.

The scalar part of the action acquires the form

$S=\sum_{a}\frac{1}{2}\int
d^{4}x\sqrt{g_{4}}[(\partial\Phi_{a})^{2}-\mu_{a}^{2}\Phi_{a}^{2}],\quad\mu_{a}^{2}=m^{2}+\lambda_{a}.$
(25)

where decomposition (15) has been used.

Now, formulas (20)-(24) can be applied to each mode $\Phi_{a}$. In particular,

$\displaystyle<\phi(t)^{2}>=\sum_{a=-\infty}^{\infty}<\Phi_{a}(t)^{2}>=\sum_{a=-\infty}^{\infty}\frac{3H^{4}}{8\pi^{2}\mu_{a}^{2}},\quad
t\to\infty$ (26)

according to (24). This value characterises the amplitude of fluctuations.

The fluctuation probability for first $A$ modes $\Phi_{a},\,(a=0,\pm 1,\pm
2,..,\pm A)$ can be found as (see (20) for $t\to\infty$)

$\displaystyle
dP=\prod_{a=-A}^{A}dP_{a}=\prod_{a=-A}^{A}d\Phi_{a}\cdot\sqrt{q_{a}/\pi}\exp\left[-q_{a}\Phi_{a}^{2}\right],\quad$
(27)

##  5 Formation of extra space metrics by quantum fluctuations

Let we have compact $n$-dim maximally symmetric extra space (14) of the radius
$r_{0}$. The classical form of metric is defined by system (6) - (9)
approximately, up to the quantum fluctuations, see (19). Some fluctuations
could serve as the initial conditions of system (6) - (9) if their amplitudes
are large with respect to the average value $<\phi>$. There is a nonzero
probability (27) of such large amplitude scalar field fluctuation $\Phi_{a}$,

$\displaystyle\Phi_{a}\gg\sqrt{<\phi^{2}>},\quad a=0,\pm 1,\pm 2,..,\pm A$
(28)

Being created, it participates in the classical dynamics, influences both the
metric and the field evolvution to a stationary state, the solution to general
system (6) - (9). Stationary $n$-dim extra space metrics form a set of the
cardinality of the continuum, as discussed in the Introduction and studied in
[6, 16, 7]. The realisation of a particular stationary metric depends on the
initial conditions - an accidental set of fluctuations $\\{\Phi_{a}\\}$. The
latter forms an initial scalar field distribution

$\displaystyle\phi_{in}(u)\equiv\sum_{a=-A}^{A}\Phi_{a}Y_{a}(u)$ (29)

Here $A$ is finite number. The equality $\phi_{-A}=\phi_{A}^{*}$ is assumed
because the scalar field is real. The probability of such fluctuation contains
the interval $d\Phi_{a}$ which also can not be smaller than the average
quantum fluctuation $\sqrt{<\phi^{2}>}$. Otherwise, the fluctuations quickly
push the field out from the desired interval. For estimations, we suppose that

$d\Phi_{a}\simeq\Delta\Phi_{a}\sim\sqrt{<\phi(t)^{2}>}\simeq\sqrt{\sum_{a=-A}^{A}\frac{3H^{4}}{8\pi^{2}\mu_{a}^{2}}}$
(30)

The probability of first $A$ modes fluctuation can be found as (see (27))

$\displaystyle\Delta
P\simeq\prod_{a=-A}^{A}\sqrt{<\phi^{2}>}\cdot\sqrt{q_{a}/\pi}\cdot\exp\left[-q_{a}\Phi_{a}^{2}\right];$
(31) $\displaystyle q_{a}=4\pi^{2}\frac{\mu_{a}}{H^{3}}.$

Now we can estimate the probability to find the scalar field distribution
$\phi_{in}(u)$ in a specific point of our 3-dim space. It will help us to
evaluate a number of pocket universes containing the particular deformed extra
space. It is known that the masses of the Kaluza-Klein excitations are
proportional to the inverse size of extra dimensions, $\lambda_{a}\sim 1/r$.
Hence, the approximation $\mu_{a}\sim 1/r$ looks reasonable and we come to the
following approximate equality, see (30):

$\sqrt{<\phi^{2}>}\sim\sqrt{\frac{3A}{4\pi^{2}}}H^{2}r$
$q_{a}\sim\frac{4\pi^{2}}{H^{3}r}$

We have to consider modes with large amplitudes, see (28), so that

$\Phi_{a}=\xi\cdot\sqrt{<\phi^{2}>}\sim\xi\cdot\sqrt{\frac{3A}{4\pi^{2}}}\cdot{H^{2}}{r},\quad\xi\gg
1$ (32)

The parameter $\xi$ characterises the ratio of first modes as compared to the
averaged fluctuation and should be big in our case. Suppose that the modes
number $A=5$ which looks enough for suitable approximation. The choice
$\xi=10,H=0.1,r=10$ is discus below. Substitution these values into (31) gives
an estimation for a fraction of universes

$\frac{\Delta N}{N}=\Delta P\sim e^{-6A^{2}\xi^{2}Hr}\sim e^{-12500}\sim
10^{-5428}$ (33)

containing such fluctuations. Here pre-exponent is neglected. This fraction is
extremely small, but it does not mean that a number $\Delta N$ of such pocket
universes is also small. Indeed, a total number $N$ of pocket universes - the
causally disconnected domains under the present horizon of the Multiverse -
can be estimated on the basis of the chaotic inflation [3] as

$\mathfrak{N}\sim 10^{10^{12}}$ (34)

for the quadratic potential $V(\phi)={m^{2}}\phi^{2}/2$. This number is much
greater than the number $10^{500}$ of the pocket universes usually mentioned
in connection to the string theory. Therefore, the number of such pocket
universes

$\Delta N\sim\Delta P\cdot\mathfrak{N}\sim 10^{10^{12}}$ (35)

is really huge if one takes into account the total number of pocket universes
(34).

It is also of interest to find such a truncated set $\tilde{\Phi}_{a}$, the
evolution of which is finished by a particular stationary solution
$\phi_{cl}(u)$ of equations (6)-(9). The quantum fluctuations prevent the
exact solution of this task. Nevertheless, approximate way is evident. To this
end, one has to approximate the required distribution $\phi_{cl}(u)$ by the
truncated set $\tilde{\Phi}_{a},\,a=1,2,..,A$.

$\displaystyle\phi_{cl}(u)\simeq\tilde{\phi}_{in}(u)=\sum_{a=-A}^{A}\tilde{\Phi}_{a}Y_{a}(u)$
(36)

to obtain required modes

$\displaystyle\tilde{\Phi}_{a}=2\pi\int_{0}^{u_{max}}\tilde{\phi}_{in}(u)Y_{a}(u)\sqrt{|g_{n}|}du,\quad|a|\leq
A$ (37)

and find the probability according to (31). Let us roughly estimate it for the
stationary state represented in Fig. 1 where necessary numerical values can
also be found. In fact, it has been done just above, see formula (33)-(35).
The only what remains is to explain the estimation $\xi=10$ for the multiplier
$\xi$ defined in (30). This estimation looks reasonable because
$\sqrt{<\phi^{2}>}\sim 0.1$ (see last equality in (30)) and $\Phi_{a}\sim 1$
for the first several modes of the scalar field distribution in Fig.1. As the
result, we have more than enough amount of appropriate pocket universes with
chosen truncated set of modes $\Phi_{a}$. This set eventually goes to a
stationary state, similar to that shown in the Fig. 1.

We would like to stress once again that the initial field distribution
$\tilde{\phi}_{in}$ can evolve to a variety of slightly different stationary
metrics $\phi_{cl}(u)$ under the influence of the quantum fluctuations. All of
them have similar probability to be formed.

We can conclude that there are regions of the horizon size where the
amplitudes of specific modes are quite large as compared to the quantum
fluctuations. These modes are stretched in 3 dimensions. Simultaneously, such
fluctuations influence the extra space metric and both of them evolve
classically to stationary states.

##  6 Conclusion

The extra space paradigm is a powerful tool in the modern theoretical physics.
They can influence low energy physics by determining the parameters $h_{i}$ of
the Lagrangian. The latter are fixed or limited by observations, which gives
an indirect way of choosing a specific metric of extra space. As has been
shown in [7], there exists a continuous set of the extra metrics. The measure
of maximally symmetrical metrics appears to be extremely small.

In this letter, we discuss the mechanism of various extra space metrics
formation. It is shown here that the quantum fluctuations of scalar field
during the inflation are the reason of its nontrivial distribution over the
compact extra dimensions. In its turn, this inhomogeneous distribution acts on
the extra space metric leading to formation the brane-like structures studied
in [7]. The way to estimate the probability of finding the specific extra
metric $g_{n}$ in a space under the present horizon is developed. Each volume
under the present horizon (the pocket universe) is characterised by a specific
value of the extra metric.

The physical parameters $h_{i}$ depend on the extra space metric $g_{n}$ after
the extra coordinates are integrated out. We come to conclusion that each
pocket universe is endowed by a specific set of the physical parameters
$h_{i}(g_{n})$. A subset of such pocket universes could contain the observable
parameters. The measure of such universes is estimated to be small but
nonzero, which is enough to an intelligent life nucleated.

##  7 Acknowledgments

The work of SGR has been supported by the Ministry of Science and Higher
Education of the Russian Federation, Project “Fundamental properties of
elementary particles and cosmology” N 0723-2020-0041 and the Kazan Federal
University Strategic Academic Leadership Program. JCF thanks CNPq (Brazil) and
FAPES (Brazil) for partial financial support.

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