###### Abstract

Applying the renormalization group, we derive the stochastic equations as the
effective theory at the horizon. We focus on the conformal zero mode to
respect local Lorentz symmetry. Under Gaussian approximation, we derive the
fundamental equation for the Universe (EqU). We also derive the identical
equation from the first law of thermodynamics in a dual geometric picture.
There is a convincing evidence for de Sitter duality between quantum
stochastic physics on the boundary and classical thermodynamics in the bulk.
The equation for the Universe (EqU) possesses the solution with the
ultraviolet fixed point. It also contains the inflationary universe with the
power potentials. We discuss possible scenarios for the very early universe
with decreasing $\epsilon$. We argue inflationary universe subsequently
dominates to maximize the entropy and $\epsilon$ problem is naturally solved.

###### keywords:

Style file; LaTeX; Proceedings; World Scientific Publishing.

de Sitter Duality and Holographic Renormalization

KEK-TH-2382

Yoshihisa Kitazawa1),2), 111E-mail address<EMAIL_ADDRESS>
1) KEK Theory Center, Tsukuba, Ibaraki 305-0801, Japan
2) Department of Particle and Nuclear Physics
The Graduate University for Advanced Studies (Sokendai)
Tsukuba, Ibaraki 305-0801, Japan

##  1 Introduction

The importance of stochastic effect has been long recognized in inflation
theory[1, 2]. We have recognized that inflaton may very well be conformal zero
mode and Einstein gravity and inflation theory are dual to each other.[3]. In
this work we demonstrate the consistency and consequences )of de Sitter
duality.

dS space may be decomposed into the bulk and the boundary, i.e., the sub-
horizon and horizon. From a holographic perspective, we consider the conformal
zero mode dependence of the Einstein-Hilbert action:

$\displaystyle\frac{1}{16\pi
G_{N}}\int\sqrt{g}d^{4}x(Re^{2\omega}-6H^{2}e^{4\omega})\simeq\frac{\pi}{G_{N}H^{2}}(1-4\omega^{2}).$
(1)

Our gauge fixing procedure and the propagators are explained in [4]. It does
not contribute to the interaction vertices. The semi-classical dS entropy was
obtained by rotating the background spacetime $dS^{4}$ to $S^{4}$ in (1) [5].

The quadratic part of $\omega$ constitutes a Gaussian distribution function
for the conformal zero mode,

$\displaystyle\rho(\omega)=\sqrt{\frac{4}{\pi
g}}\exp\big{(}-\frac{4}{g}\omega^{2}\big{)}.$ (2)

It may represent an initial state of the Universe when the dS expansion
begins. In order to describe the time evolution of the conformal factor of the
Universe, we introduce a new parameter $\xi(t)$.

$\displaystyle\rho(\xi(t),\omega)=\sqrt{\frac{4\xi(t)}{\pi
g}}\exp\big{(}-\frac{4\xi(t)}{g}\omega^{2}\big{)}.$ (3)

$\xi$ is the only parameter in the Gaussian approximation. We work within the
Gaussian approximation since it is an excellent approximation for gravity with
the small coupling $g$. The von Neumann entropy
$S=-\text{tr}(\rho\log\rho)\sim{1/2\log(g/\xi)}$ becomes larger as $\xi$
becomes smaller. Thus the diffusion triggers an instability in de Sitter
space.

In terms of the distribution function, the $n$-point functions are defined as
follows

$\displaystyle\langle\omega^{n}(t)\rangle_{\text{boundary}}=\int
d\omega\rho(\xi(t),\omega)\omega^{n}.$ (4)

In particular, the two-point function of the conformal mode is given by

$\displaystyle\langle\omega^{2}(t)\rangle_{\text{boundary}}=\frac{g}{8\xi(t)}.$
(5)

$1/\xi$ is the enhancement factor of the conformal (scalar) perturbation over
tensor mode.

The negative norm of the bulk conformal mode indicates that the $\rho$
diffuses toward the future. In fact the perturbative quantum expectation of
(1) gives $g(t)\sim g(1-2\gamma Ht)\sim g(1-3gHt)$. The duality between the
inflation and quantum gravity is also based on this one loop effect [6, 7].
However such an estimate is reliable only locally $Ht\ll 1$. In cosmology it
is essential to resum all powers of IR logarithms $(Ht)^{n}$ to understand the
global picture. For such a purpose, we find our holographic approach is up to
the task. We investigate the dynamics of conformal mode after integrating the
bulk mode. The two point function gives rise to IR logarithms.

$\displaystyle<\omega^{2}>_{\text{bulk}}$
$\displaystyle=-\frac{3g}{4}\int_{Ha(t)}^{\Lambda}\frac{dk}{k}$
$\displaystyle=$ $\displaystyle\frac{3g}{4}Ht=\frac{3g}{4}N(t).$ (6)

We assume there is no time dependent UV contributions. We focus on the Hubble
scale physics where $a(t)=1/-\tau H=\exp(Ht)$. We recall that $-\tau\mu\sim 1$
at the Horizon. While the wave functions of the bulk mode oscillates, those of
the boundary mode do not. That is why we call it the zero mode. Our strategy
is to integrate out oscillating mode first. The finite bare distribution
function is given by subtracting the bulk mode contribution above the Hubble
scale. We thus construct low energy effective theory around the Hubble scale.
Such a theory is holographic and finite after the subtraction.

$\displaystyle\rho_{B}=\exp\big{(}-\frac{3g}{4}\frac{Ht}{2}\frac{\partial^{2}}{\partial\omega^{2}}\big{)}\rho.$
(7)

The renormalization scale of the low energy effective action is the Hubble
scale. As $\rho_{B}$ is independent of the renormalization scale, the
renormalized distribution function obeys the following renormalization group
equation.

$\displaystyle\dot{\rho}-\frac{3g}{4}\cdot\frac{H}{2}\frac{\partial^{2}}{\partial\omega^{2}}\rho=0,$
(8)

where $\dot{O}$ denotes a derivative of $O$ with respect to the cosmic time
$t$. The factor $3g/4$ in the diffusion term is the projection factor to the
IR region. The conformal mode $\omega$ consists of a minimally coupled field
$X$. $\omega=\sqrt{3}X/4+Y/4$ [8, 4]. We neglect $Y$ because it has the
effective mass $m^{2}=2H^{2}$. There is no drift term in the reduced space
consisting only of $X$.

The gravitational FP equation (8) is obtained by integrating the quantum bulk
modes inside the horizon. It turns out to be a diffusion equation due to the
lack of the drift term. The solution is the Brownian motion as it is jolted by
the horizon exiting modes. The FP equation is a dynamical renormalization
group equation. We can sum up the IR logarithms $\log^{n}a=(Ht)^{n}$ by this
equation to find a running coupling $g(t)$. The FP (diffusion) equation shows
that the solution is the Gaussian distribution with the standard deviation
increasing linearly with the e-folding number $N(t)$ [9].

$\displaystyle{3\over 4}gHt={3g\over 4}N(t)={g\over 8\xi}.$ (9)

The standard deviation is related to $N$ as $1/\xi=6N$ in (9). It is
consistent with a standard Brownian motion prediction. The von Neumann entropy
thus increases logarithmically,

$\displaystyle\delta S=\frac{1}{2}\log{1\over\xi}=\frac{1}{2}\log 6N(t).$ (10)

Identifying the von Neumann entropy of conformal zero mode with the quantum
correction to dS entropy, we obtain the bare action with the counter term

$\displaystyle\frac{1}{g_{B}}=\frac{1}{g(N)}-\frac{1}{2}\log(6N).$ (11)

By requiring the bare action is independent of the renormalization scale:
namely $N$ , we obtain the one loop $\beta$ function.

$\displaystyle\beta=\frac{\partial}{\partial\log(N)}g(N)=-\frac{1}{2}g(N)^{2}.$
(12)

We find the running gravitational coupling as 222The coefficient in front of
$N$ can be put to the identity.

$\displaystyle g(N)={2\over\log(N)}.$ (13)

The holographic investigation at the boundary shows that $g$ is asymptotically
free toward the future [10, 11]. The renormalization group trajectory must
reach Einstein gravity in the weak coupling limit for the consistency with
general covariance [12]. We find that it approaches a flat spacetime in
agreement with this requirement.

##  2 Equation for the Universe and de Sitter Duality

The Gaussian distribution of the conformal zero mode is characterized by the
standard deviation $1/\xi$. Although there is no inflaton in Einstein gravity,
we propose to identify the inflaton $f^{2}$ as $f^{2}\propto 1/\xi$. In our
interpretation, the inflaton is not a fundamental field but a stochastic
variale. The two point function at an equal time grows due to the Brownian
motion: IR logarithmic fluctuations $1/\xi\sim N$. While the inflation theory
is specified by the inflaton potential, the dynamics of quantum gravity is
determined by the FP equation which describes the stochastic process at the
horizon. We thus argue the classical solution of the inflation theory
satisfies the FP equation as well.

It is likely that there are multiple elements in the universality class of
quantum gravity/inflation theory. The inflation era of the early Universe may
be one of them. As we discuss shortly we find a pre-inflation era which is
indispensable to launch inflation era which in turn necessary to trigger the
big bang. We evaluated the time evolution of entropy to the leading log order
in (10). In order to take account of the higher loop corrections in $g$, the
FP equation should be generalized. It turns out to be just necessary to make
the dimensionless gravitational coupling time dependent $g(t)$:

$\displaystyle{\partial\over\partial
t}{\rho}-\frac{3g(t)}{4}\cdot\frac{H}{2}\frac{\partial^{2}}{\partial\omega^{2}}\rho=0.$
(14)

The Gaussian ansatz (3) is an exact solution if $g$ is a constant.

In what follows, we consider such a solutions.

$\displaystyle\rho=\sqrt{\frac{4\xi(t)}{\pi
g(t)}}\exp\big{(}-\frac{4\xi(t)}{g(t)}\omega^{2}\big{)}.$ (15)

We next put the ansatz into the FP equation and find the condition for the
background to satisfy. In the small $\omega$ limit, the background must
satisfy

$\displaystyle{\partial\over\partial t}{1\over 2}\log({\xi\over
g(t)})+3H\xi=0.$ (16)

We obtain a remarkably simple equation in terms of $N=Ht$.

$\displaystyle\frac{\partial}{\partial N}\log\frac{g(t)}{\xi}=6\xi.$ (17)

(17) determines the evolution of von Neumann entropy $S={1\over
2}\log{g\over\xi}$ with respect to $N$. This formula shows the validity of our
postulate that von Neumann entropy of conformal zero mode constitutes the
quantum correction to de Sitter entropy. We call it the equation for the
Universe (EqU).

The equation (17) has inflationary solutions with power potentials.

$\displaystyle g=c\tilde{N}^{\frac{m}{2}},\hskip
10.00002pt\xi=\frac{m+2}{12\tilde{N}}.$ (18)

Here $c$ is an integration constant. $m$ denotes the power of the potential:
$f^{m}$. It is convenient to replace $N$ by $\tilde{N}$ where
$\tilde{N}=N_{e}-N$. $N_{e}$ denotes the e-foldings at the end of inflation.
(17) becomes as follows after the change of the variable,

$\displaystyle-\frac{\partial}{\partial
N}\log\frac{g(\tilde{N})}{\xi(\tilde{N})}=6\xi(\tilde{N}).$ (19)

Although the dS entropy can be explained by quantum effects alone for the
weakly coupled inflaton solution, the strongly coupled inflaton solution is a
dual object in the sense that geometrical description is reliable. The
increase of the entropy $S=1/g$ can be evaluated by the first law $T\Delta
S=\Delta E$ where $\Delta E$ is the incoming energy flux of the inflaton [23].
In this way, the one of the Einstein’s equation is obtained:

$\displaystyle\dot{H}(t)=-4\pi G_{N}\dot{f}^{2}.$ (20)

From this formula, we obtain

$\displaystyle 2\epsilon=-{\partial\over\partial{N}}\log g(t).$ (21)

We add the same quantity ${1/\tilde{N}}$ to the both sides of the equation.

$\displaystyle
2\epsilon+{1\over\tilde{N}}=-{\partial\over\partial{N}}\log(g\tilde{N}).$ (22)

For power potential inflationary universe, (22) is rewritten as

$\displaystyle 6\xi={\partial\over\partial{N}}\log{g\over\xi}.$ (23)

It is precisely our EqU (17). We have derived it in a dual geometric picture
here in place of the original quantum stochastic picture. It proves the
consistency of the EqU and the confirmation of de Sitter duality. Since (17)
and (21) are equivalent, the solutions of the former satisfy the latter.

##  3 Inflation and UV completion

In the literature, $\delta N$ formalism is widely used to investigate the
curvature perturbation. It underscores the validity of the stochastic picture
of the inflation. [13, 14, 15, 16, 17] Let us consider the fluctuation of the
curvature perturbation $\zeta$.

$\displaystyle{\zeta}={\delta N}={H\over\dot{f}}\delta f,$
$\displaystyle<\delta f(t)\delta f(t^{\prime})>=({H^{2}\over
4\pi^{2}})H\delta(t-t^{\prime}).$ (24)

We obtain in the super-horizon regime:

$\displaystyle<\zeta^{2}(t)>=<({H\over\dot{f}})^{2}\delta f^{2}>$
$\displaystyle={1\over 2\epsilon M^{2}_{P}}<\delta f^{2}>$
$\displaystyle={H^{2}N(t)\over 8\pi^{2}\epsilon
M^{2}_{P}}={g\over\epsilon(t)}=P.$ (25)

We recall the following identity holds at the horizon exit $t=t_{*}$ 333We
follow the notations of [19].

$\displaystyle\dot{\rho}(t_{*})e^{\rho(t_{*})}=k.$ (26)

It is nothing but choosing our renormalization scale as $\log k=Ht$. Let
$P\sim k^{1-n_{s}}$. The scaling dimension of $P$ can be estimate as

$\displaystyle k{d\over dk}(4\log\dot{\rho}(t_{*})-2\log\dot{\phi}(t_{*}))$
$\displaystyle=$ $\displaystyle{d\over
dN_{*}}(4\log\dot{\rho}(t_{*})-2\log\dot{\phi}(t_{*}))$ $\displaystyle=$
$\displaystyle 2(\eta-3\epsilon),$ (27)

where

$\displaystyle\epsilon={\dot{\phi}^{2}\over
2H^{2}},~{}~{}\eta={\ddot{\phi}\over\dot{\rho}\dot{\phi}}+\epsilon.$ (28)

In power inflation potential models $V\sim f^{m}$, $\xi$ and $\epsilon$ scale
as $1/N$ due to the universality of the random walk. They belong to the same
universality class as the following scalar two point functions scale as

$\displaystyle<\zeta^{2}>={g\over\epsilon}=4gN(t),~{}~{}\epsilon={m\over
4N(t)}.$ (29)

The scaling exponents agree with our FP equation and the $\delta$ N formalism.

$\displaystyle 1-n_{s}={\partial\over\partial N}\log(gN(t))=2\epsilon+{1\over
N(t)}={m+2\over 2N(t)}.$ (30)

The conformal mode is indistinguishable from the curvature perturbation as far
as the scaling property is concerned. Our identification of the conformal mode
and the inflaton therefore works in generic quantum gravity with conformal
mode or inflation theory. The advantage of conformal mode is being Lorentz
scalar, well understood in covariant field theory. The precise normalization
of $P=g/\epsilon$ needs to be fit with the data: - normalization. It is still
a challenge to separate $g$ and $\epsilon$.

There is a UV fixed point in our renormalization group. We study it next. FP
equation (17) enables us to evaluate higher order corrections to the $\beta$
function. The expansion parameter is $1/\log N$. We can confirm that the
following $g_{f}$ and $\xi_{f}$ satisfies (17),

$\displaystyle g_{f}=\frac{2}{\log N}\big{(}1-\frac{1}{\log N}\big{)},\hskip
10.00002pt\xi_{f}=\frac{1}{6N}\big{(}1-\frac{1}{\log N}\big{)}.$ (31)

Thus, the $\beta$ function, $\epsilon$ and the semi-classical entropy
generation rate are given by

$\displaystyle\beta=\frac{\partial}{\partial\log
N}g_{f}=-{2\over\log^{2}N}+{4\over\log(N)^{3}}=-2({1-2\frac{1}{\log
N}}){1\over\log^{2}N}.$ (32)

$\displaystyle\epsilon_{f}=-{1\over 2}{\partial\over\partial
N}\log(g_{f})=-{1\over 2g_{f}N}\beta_{f}.$ (33)

$\displaystyle\frac{\partial}{\partial N}S_{sc}=\frac{\partial}{\partial
N}\frac{1}{g_{f}}=-\frac{1}{Ng_{f}^{2}}\beta_{f}.$ (34)

Before concluding this section, we explicitly construct the logarithmic
solution of (17). We start with the following basic structure and decorate on
it.

$\displaystyle{\partial\over\partial N}\log({2N})={1\over N}.$ (35)

Let us subtract $({1}/{2})({\partial\over\partial N}\log(\log N))={1}/{2(N\log
N)}$ on the both sides of (35).

$\displaystyle{\partial\over\partial N}\log(\frac{2N}{\log N})={1\over
2N}-{1\over 2N\log N}.$ (36)

This equation is equivalent to (17).

A remarkable feature is that the coupling has the maximum value $g=1/2$ at the
beginning. It steadily decreases toward the future as the $\beta$ function is
negative in the whole region of time flow. It has two fixed points at the
beginning and at the future of the Universe. The existence of the UV fixed
point may indicate the consistency of quantum gravity. The single stone solves
the $\epsilon$ problem [18] as well since it vanishes at the fixed point. The
$\beta$ function describes a scenario that our Universe started the dS
expansion with a minimal entropy $S=2$ while it has $S=10^{120}$ now. It
corresponds to $N=e^{2}\sim 7.4,a=e^{N}\sim 1.6\times 10^{3}$. The just born
Universe is rather large which reflects the critical coupling $g=1/2$ is
rather small. In terms of the reduced Planck mass,
$H^{2}/{4\pi}^{2}M_{P}^{2}=1$.

Since we work with the Gaussian approximation, our results on the UV fixed
points are not water tight as the coupling is not weak. Nevertheless we find
it remarkable that they support the idea that quantum gravity has a UV fixed
point with a finite coupling. In fact 4 dimensional de Sitter space is
constructed in the target space at the UV fixed point of $2+\epsilon$
dimensional quantum gravity [12]. 4 dimensional de Sitter space also appears
at the UV fixed point of the exact renormalization group [20][21].

Such a theory might be a strongly interacting conformal field theory. However,
it is not an ordinary field theory as the Hubble scale is Planck scale. Our
dynamical $\beta$ function is closely related to the cosmological horizon and
physics around it. The existence of the UV fixed point could solve the trans-
Planckian physics problem. A consistent quantum gravity theory can be
constructed under the assumption that there are no degrees of freedom at
trans-Planckian physics [22]. In this sense, it is consistent with string
theory and matrix models. The Universe might be governed by (31) in the
beginning as it might be indispensable to construct the UV finite solutions of
the FP equation.

##  4 The Inflationary Universe and Pre-History

The inflaton may be identified with the stochastic variable $f$ whose
correlators show characteristic features of Brownian motion.
$<f^{2}>=\tilde{N}$ and $g\propto\tilde{N}^{m/2}=<f^{m}>$. $6\xi=1-n_{s}$
measures the extra tilt of the scalar two point function $k^{1-n_{s}}$ with
respect to $k$.

We thus conclude:

$\displaystyle g=c\tilde{N}^{\frac{m}{2}},~{}~{}\epsilon=-{1\over
2}{\partial\over\partial N}log(g)=\frac{m}{4\tilde{N}},$ $\displaystyle
1-n_{s}=6\xi=\frac{m+2}{2\tilde{N}}.$ (37)

In this case, the concave power solutions can be obtained by formally
replacing $m$ by $1/n$. We admit it is not entirely clear why concave
potentials are relevant. It is possible that the convex potentials are already
excluded by observations.

In contrast, there is more room for concave potentials to accommodate the
observational information with the judicious choice of $n$.

$\displaystyle\epsilon=-{1\over 2}{\partial\over\partial
N}log(g)=\frac{1}{4n\tilde{N}},$ $\displaystyle
1-n_{s}=\frac{\frac{1}{n}+2}{2\tilde{N}}.$ (38)

In our view the inflaton behaves as an particle in the Brownian motion. Its
trajectory is made of infra-red fluctuations. Nevertheless the convex and the
concave potentials in the inflation models seems to form two distinct groups.

The comparison (31) and (38) indicates the effective couplings are $1/\log{N}$
in the former

$\displaystyle\epsilon={1\over\log N\cdot 2N}{(1-{2\over\log
N})\over(1-{1\over\log N})}.$ (39)

and $1/2n$ in the latter .

$\displaystyle\epsilon={1\over 2n\cdot 2\tilde{N}}.$ (40)

So concave potentials are in the weak coupling regime while convex potentials
belong to the strong coupling regime. We believe this corrependence should be
taken seriously. At the UV fixed point $\log N=2$ suggests $n=1$. Although the
$1/\log N$ correction becomes small only when $n\sim 3$, it is still close to
the critical point. The concave potentials are promising avenue to explore
right now [24].

|  | $n=1$ | $n=2$ | $n=3$
---|---|---|---|---
| $\epsilon$ | ${1\over 4\tilde{N}}$ | ${1\over 8\tilde{N}}$ | ${1\over 12\tilde{N}}$
| $1-n_{s}$ | $\frac{3}{2\tilde{N}}$ | $\frac{5}{4\tilde{N}}$ | $\frac{9}{8\tilde{N}}$

)

In Table 4, we list expected $\epsilon$ and $1-n_{s}$ in the power potential
model with $f^{1/n},n=1,2,3$. We note $1-n_{s}$ is bounded from below by
$1/\tilde{N}$ while $r=16\epsilon$ is not. It is consistent with the current
observations. It is important to establish the bound on $n$. We argue the
order of magnitude estimate can be trusted for the concave potentials as they
are in the weak coupling regime. The semi-classical de Sitter entropy favors
smaller $n$ as the corresponding entropy grows faster:
$S\sim(1/\tilde{N})^{1\over 2n}$.

Although it is still highly speculative, we argue that our Universe is likely
to be located close to the point $m=n=1$. Since the convex potentials are
excluded by observation, it must be a concave potential with a small $n$. The
convex potentials correspond to strongly coupled systems. Here the naturalness
and the anthropic principle may come in. It is hard to believe that our
Universe comes out of a strongly coupled system. It is much easier to
accommodate hierarchies in the weakly coupled system. Our Universe might to be
located near the boundary between the stable and unstable universes like the
standard model of particle physics. Since the Universe can stay at the fixed
point forever, it is likely to be not far from it.

We recall here the curvature perturbation:

$\displaystyle P\sim{H^{2}\over(2\pi)^{2}2M_{P}^{2}\epsilon}\sim 2.2\times
10^{-9}.$ (41)

$P=<\zeta^{2}>=g/\epsilon$ must be generated during inflation. It is bounded
from below $g>10^{-11}$ if $\epsilon>1/200$. In order to explain this
magnitude, we must assume the inflation lasted much beyond what CMB isotropy
generation requires as we examine it in the last part of this section.

The left-hand side of (17) can be identified with $1-n_{s}$ where $1-n_{s}$ is
the scalar spectral index. Let us recall that $1-n_{s}$ is expressed by the
slow-roll parameters $\epsilon$ and $\eta$,

$\displaystyle 1-n_{s}=6\epsilon-2\eta.$ (42)

With the identification (38), the equation (17) is equivalent to (42) for the
$f^{m}$ inflaton potential where $\eta=(m-1)/(2\tilde{N})$. Our FP equation is
expected to go beyond the resummation of leading IR logarithms which results
in the logarithmic decay of $g$. It is still a great surprise to find
inflation theory as its solutions. Fortunately EqU can be derived in a dual
geometrical picture. This fact gives a non-perturbative evidence for the dS
duality.

The solution (31) is UV complete. $g$ is attracted to the fixed point as we
roll back the history of the Universe. However, it cannot terminate the
eternal inflation as $\epsilon=-(1/2)\partial\log g/\partial N\sim 1/(2N\log
N)$ decreases with time. On the other hand, the solution (18) is not UV
complete but it can end the inflation as $\epsilon\sim m/4\tilde{N}$ increases
with time. These solutions generate the entropy in different ways: We consider
the leading de Sitter entropy $1/g(t)$. The $t$ dependence of $g(t)$ depends
on the quantum corrections through the EqU equation. $1/g\sim\log N$ for the
former and $1/g\sim 1/\tilde{N}^{\frac{1}{2n}}$ for the latter. From the
perspective of the dominant entropy principle, (31) is chosen initially and
(18) is chosen after $\log N\sim 1/\tilde{N}^{1\over 2n}$. That is to say,
$g_{1}$ in (31) describes the newly born Universe and $g_{2}$ in (18)
describes the inflation era.

The UV fixed point endows with a special initial condition for the inflation
theory. $\epsilon$ vanishes at the UV fixed point while it grows subsequently
(39) up to $1/200$ at $N\sim 10$. It then decays again like $1/(2N\log N)$.
Let us investigate pre-inflation scenario. The entropy grows logarithmicaly
$1/g=(\log N)/2$ as $N$ grows after the Universe emerges out of the UV fixed
point. In the inflationary phase, the entropy grows power like $1/g\sim
1/\tilde{N}^{1/2n}$. The entropy increases slowly in the pre-inflation era
while it changes much faster in the inflation era. Inflation phase inevitably
takes over the pre-inflation phase as the entropy of the former dominates in
time.

We may follow the evolution of $\epsilon$. It decreases as $1/(2N\log N)$ in
the pre-inflation era and it grows as $1/4\tilde{N}$ in the inflationary era .
To the leading order of $N$, $\epsilon$ decay is canceled out by the
subsequent growth during inflation. More precise scenario is
$N=30,~{}\epsilon=1/200$ in the first stage. This reduction of $\epsilon$ will
be compensated by the growth in the second stage with $\tilde{N}=50$. This is
the minimum consistent scenario. However $g$ is reduced by only the factor
$20$ in this process and we need to stretch the length of the both stages
enormously: $10^{16}N\sim 10^{17}$ and $\tilde{N}=50\rightarrow
10^{16}\tilde{N}$ by the common factor $10^{17}$ to explain $g\sim 10^{-10}$.
$\epsilon$ decreases as small as $10^{-19}$ before bouncing back to $0(1)$.

During the inflation, the gravitational coupling decreases $\tilde{N}^{1\over
2n}$. While $\epsilon$ decreases in the pre-inflation, it increases during the
inflation. It is in the same ball park with the curvature perturbation $P\sim
10^{-9}$. It is because $P=g/\epsilon$ and the inflation terminates at
$\epsilon\sim 1$. We believe this coincidence is not an accident but an
evidence for the existence of pre-inflation era prior to the conventional
inflation era.

The history of the Universe is likely to be much more versatile than the
minimal inflation which CMB anisotropy requires. Inflation requires fine
tuning such as minuscule $\epsilon$. We believe we have found a natural
mechanism to accomplish it. So far in this section, we investigated pre-
inflation to inflation transition as if it is the second order phase
transition. There maybe a different picture. It may be possible to construct
the UV finite inflationary Universe as composite models [25]. Although it is
beyond the scope of the paper, we have initiated some research along these
ideas [26].

## acknowledgments

This work is supported by Grant-in-Aid for Scientific Research (C) No.
16K05336. I thank Chong-Sun Chu, Masashi Hazumi, Satoshi Iso, Hikaru Kawai,
Kozunori Kohri, Takahiko Matsubara, Jun Nishimura, Hirotaka Sugawara, and
Takao Suyama for discussions. I also thank the organizers of EAJS especially
Hiroshi Itoyama for let me present this work at Osaka.

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Some references include a sum over $q'' = q+q'$ which is omitted here because of properties of Clebsh-Gordan coefficients about projections of angular momentum.

CHAPTER: RACAH FORM OF QUADRUPOLE AND ANGULAR MOMENTUM OPERATORS

The Racah form of a tensor operator $T^{k}_q$ is obtained as

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
&\hspace{20mm}T^{k}_q = \mathbb{I}T^{k}_q\mathbb{I} = \sum_{l,m,l',m'}\left|lm\right>\left< lm \right|T^{k}_q\left|l'm'\right>\left< l'm'\right| = \sum_{l,m,l',m'}\left< lm \right|T^{k}_q\left|l'm'\right>b^{\dagger}_{lm}b_{l'm'}\\
& =  \sum_{l,l'}\frac{\left< l||T^{k}||l'\right>}{\sqrt{2l+1}}\sum_{m,m'}\left<l'm'kq|lm\right>b^{\dagger}_{lm}b_{l'm'} =  \sum_{l,l'}\frac{\left< l||T^{k}||l'\right>}{\sqrt{2l+1}}\sum_{m,m'}(-1)^{l'-m'}\left<lml'-m'|kq\right>\sqrt{\frac{2l+1}{2k+1}}b^{\dagger}_{lm}b_{l'm'}\\& = \frac{1}{\sqrt{2k+1}}\sum_{l,l'}\langle l||T^{k}||l'\rangle\sum_{m,m'}(-1)^{l'+m'}\left<lml'm'|kq\right>b^{\dagger}_{lm}b_{l'-m'} = \frac{1}{\sqrt{2k+1}}\sum_{l,l'}\langle l||T^{k}||l'\rangle\left(b^{\dagger}_{l}\widetilde{b}_{l'}\right)^k_q
\label{TRacah}
\end{aligned}
\end{equation}

Thus for $L^1_q$

\begin{equation}
\begin{aligned}
&L^{1}_q = \frac{1}{\sqrt{3}}\sum_{l,l'}\langle l||L^{1}||l'\rangle\left(b^{\dagger}_{l}\widetilde{b}_{l'}\right)^1_q,
\label{LRacah}
\end{aligned}
\end{equation}

where the reduced matrix element is given by

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
&\hspace{15mm}\langle l||L^{1}||l'\rangle = \frac{\sqrt{2l+1}}{\left<l'm'10|lm\right>}\langle lm|J^1_0 | l'm' \rangle = \frac{\sqrt{2l+1}m}{m\sqrt{2l+1}\sqrt{\frac{1}{l(l+1)(2l+1)}}} = \sqrt{l(l+1)(2l+1)},
\label{LRacah2q}
\end{aligned}
\end{equation}

which implies that

\begin{equation}
\begin{aligned}\hspace{5mm}L^{1}_q = \sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{3}}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^1_q.
\label{LRacah2}
\end{aligned}
\end{equation}

For $Q^2_q$ as defined in <ref>

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
&Q^{2}_q = \frac{1}{\sqrt{5}}\sum_{l,l'}\langle\eta l||Q^{2}||\eta l'\rangle\left(b^{\dagger}_{l}\widetilde{b}_{l'}\right)^2_q =  \frac{\sqrt{4\pi}}{5}\sum_{l,l'}\left(\langle\eta l||r^2Y^2(\Omega)||\eta l'\rangle+ \langle\eta l||p^2Y^2(\Omega_p)||\eta l'\rangle\right)\left(b^{\dagger}_{l}\widetilde{b}_{l'}\right)^2_q,
\label{QRacah}
\end{aligned}
\end{equation}

where the states require the label of the major shell $\eta$ involved because of $r$ and $p$ coordinates appearance. Both reduced matrix elements have the same value

\begin{equation}
\begin{aligned}
&\langle\eta l||r^2Y^2(\Omega)||\eta l'\rangle = \frac{\sqrt{2l+1}}{\left< l'm'20 | lm \right>}\langle \eta l |r^2  | \eta l' \rangle \langle \eta l m |Y^2_0(\Omega) | \eta l' m' \rangle\\
& = \sqrt{\frac{5}{4\pi}}\frac{2l+1}{\sqrt{2l'+1}}\langle \eta l |r^2  | \eta l' \rangle \frac{\left< l-m20 | l'-m \right>\left< l020 | l'0 \right>}{\left< l'm20 | lm \right>},
\label{QRaca2}
\end{aligned}
\end{equation}

where the spherical harmonic addition theorem [36] was used. The quotient of Clebsh-Gordan coefficients has the allowed values

\begin{equation}
\begin{aligned}
& \frac{\left< l-m20 | l'-m \right>\left< l020 | l'0 \right>}{\left< l'm20 | lm \right>} = \begin{cases}
&-\sqrt{\frac{l(l+1)}{(2l-1)(2l+3)}}\hspace{3mm}\text{if}\hspace{3mm} l'=l,\\
&\sqrt{\frac{3}{2}}\frac{1}{2l+1}\sqrt{\frac{(l+1)(l+2)(2l+5)}{2l+3}}\hspace{3mm}\text{if}\hspace{3mm} l'=l\pm 2.
\end{cases}
\label{QRaca3}
\end{aligned}
\end{equation}

Replacing in <ref> and recalling that radial functions are real implying that $\langle \eta l |r^2  | \eta l' \rangle = \langle \eta l' |r^2  | \eta l \rangle$ it is obtained that

\begin{equation}
\begin{aligned}
Q^{2}_q = &-\sum_{l}\langle \eta l |r^2  | \eta l \rangle\sqrt{\frac{l(l+1)(2l+1)}{5(2l-1)(2l+3)}}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^2_q \\&+ \sum_{l}\langle \eta l |r^2  | \eta l +2 \rangle\sqrt{\frac{6(l+1)(l+2)}{5(2l+3)}}\left( \left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^2_q  + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^2_q  \right),
\label{QRaca4}
\end{aligned}
\end{equation}

the expected values of $r^2$ are given by [94]

\begin{equation}
\begin{aligned}
&\langle \eta l |r^2  | \eta l \rangle = \eta+\frac{3}{2},\\
&\langle \eta l |r^2  | \eta l+2 \rangle = \sqrt{(\eta-l)(\eta+l+3)},
\label{lawsonr2}
\end{aligned}
\end{equation}

thus it is obtained

\begin{equation}
\begin{aligned}
Q^{2}_q = &-\left(\eta+\frac{3}{2}\right)\sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{5(2l-1)(2l+3)}}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^2_q \\&+ \sum_{l}\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{5(2l+3)}}\left(\left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^2_q  + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^2_q  \right),
\label{QRaca4}
\end{aligned}
\end{equation}

With these definitions and the result of appendix C, the commutators of the $SU(3)$ algebra can be computed. Two of them are shown now and the third one is left as an exercise for the reader

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
&\hspace{20mm}\left[L^1_q, L^1_{q'}\right] = \sum_{l,l'}\sqrt{\frac{l(l+1)(2l+1)}{3}}\sqrt{\frac{l'(l'+1)(2l'+1)}{3}}\left[\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^1_q, \left(b^{\dagger}_{l'}\widetilde{b}_{l'}\right)^1_{q'}\right]\\
1 & 1 & k'' \\
l' & l & l \end{Bmatrix}\left(b^{\dagger}_{l}\widetilde{b}_{l'}\right)^{k''}_{q+q'}\delta_{ll'}-(-1)^{l+l'}\begin{Bmatrix}
1 & 1 & k'' \\
l' & l & l \end{Bmatrix}\left(b^{\dagger}_{l'}\widetilde{b}_{l}\right)^{k''}_{q+q'}\delta_{ll'}\right]\\&=\sum_{l,k''}l(l+1)(2l+1)\left<
1 & 1 & k'' \\
l & l & l \end{Bmatrix}\left[(-1)^{-k''}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{k''}_{q+q'}- \left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{k''}_{q+q'}\right]\\&=\sum_{l}l(l+1)(2l+1)\left<
1 & 1 & 1 \\
l & l & l \end{Bmatrix}(-2)\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{1}_{q+q'}\\&=\sum_{l}l(l+1)(2l+1)\left<
1q1q'|1q+q'\right>\sqrt{\frac{2}{3}}\sqrt{\frac{(2l-1)!}{(2l+2)!}}(-2)\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{1}_{q+q'}\\& = \sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{3}}\left<
1q1q'|1q+q'\right>(-\sqrt{2})\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{1}_{q+q'}\\& = -\sqrt{2}\left<
\label{COMMSU3}
\end{aligned}
\end{equation}

The 6-j symbols were computed using Mathematica.

\begin{equation}
\begin{aligned}
&\left[L^1_q, Q^2_{q'}\right] =  -\left(\eta+\frac{3}{2}\right)\sum_{l,l'}\sqrt{\frac{l(l+1)(2l+1)}{3}}\sqrt{\frac{l'(l'+1)(2l'+1)}{5(2l'-1)(2l'+3)}}\left[\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^1_q, \left(b^{\dagger}_{l'}\widetilde{b}_{l'}\right)^2_{q'}\right]\\&+ \sum_{l,l'}\sqrt{\frac{l(l+1)(2l+1)}{3}}\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{5(2l+3)}}\\&\hspace{50mm} \times\left(\left[\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^1_q, \left(b^{\dagger}_{l'}\widetilde{b}_{l'+2}\right)^2_{q'}\right]  + \left[\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^1_q, \left(b^{\dagger}_{l'+2}\widetilde{b}_{l'}\right)^2_{q'}\right] \right),
\label{COMMSU31}
\end{aligned}
\end{equation}

both sums will be computed separately. The first is analog to the commutator for angular momentum components and has the form

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
& -\left(\eta+\frac{3}{2}\right)\sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{3}}\sqrt{\frac{l(l+1)(2l+1)}{5(2l-1)(2l+3)}}\sum_{k''}\sqrt{15}\left<
1q2q' |k''q+q' \right>\begin{Bmatrix}
1 & 2 & k'' \\
l & l & l \end{Bmatrix}\\&\hspace{80mm}\times\left[ (-1)^{-k''}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{k''}_{q+q'}+\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{k''}_{q+q'} \right],
\label{COMMSU32}
\end{aligned}
\end{equation}

where $k''= 1,2,3$ of which only the value of 2 contributes

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
& -\left(\eta+\frac{3}{2}\right)\sum_{l}\frac{l(l+1)(2l+1)}{\sqrt{(2l-1)(2l+3)}}\left<
1q2q' |2q+q' \right>\begin{Bmatrix}
1 & 2 & 2 \\
l & l & l \end{Bmatrix}\times2\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{2}_{q+q'}\\& =  -\left(\eta+\frac{3}{2}\right)\sum_{l}\frac{l(l+1)(2l+1)}{\sqrt{(2l-1)(2l+3)}}\left<
1q2q' |2q+q' \right>\left(-\sqrt{\frac{6}{5}}\frac{1}{2\sqrt{l(l+1)(2l+1)}}\right)\times2\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{2}_{q+q'}\\& =  \sqrt{6}\left<
1q2q' |2q+q' \right>\left(\eta+\frac{3}{2}\right)\sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{5(2l-1)(2l+3)}}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^{2}_{q+q'},
\end{aligned}
\label{COMMSU32}
\end{equation}

for the second term a few steps will be skipped but can be easily followed. It will be equal to

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
\\& \sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{3}}\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{5(2l+3)}}\sum_{k''}\sqrt{15}\left< 1q2q'| k'' q+q'\right>\\&\hspace{20mm}\times \begin{Bmatrix}
1 & 2 & k'' \\
l+2 & l & l \end{Bmatrix}\left(
(-1)^{k''}\left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^{k''}_{q+q'} + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^{k''}_{q+q'}\right)
\\& +\sum_{l'}\sqrt{\frac{(l'+2)(l'+3)(2l'+5)}{3}}\sqrt{\frac{6(l'+1)(l'+2)(\eta-l')(\eta+l'+3)}{5(2l'+3)}}\sum_{k''}\sqrt{15}\left< 1q2q'| k'' q+q'\right>\\&\hspace{20mm}\times
\begin{Bmatrix}
1 & 2 & k'' \\
l' & l'+2 & l'+2 \end{Bmatrix}\left(
(-1)^{k''}\left(b^{\dagger}_{l'+2}\widetilde{b}_{l'}\right)^{k''}_{q+q'} + \left(b^{\dagger}_{l'}\widetilde{b}_{l'+2}\right)^{k''}_{q+q'}\right)\\&
= \sum_{l}\sqrt{l(l+1)(2l+1)}\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{(2l+3)}}\left< 1q2q'| 2 q+q'\right>\\&\hspace{20mm}\times \sqrt{\frac{2}{15}}\sqrt{\frac{l}{(l+1)(2l+1)}}\left(
\left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^{2}_{q+q'} + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^{2}_{q+q'}\right)
\\& +\sum_{l'}\sqrt{(l'+2)(l'+3)(2l'+5)}\sqrt{\frac{6(l'+1)(l'+2)(\eta-l')(\eta+l'+3)}{(2l'+3)}}\left< 1q2q'| 2 q+q'\right>\\&\hspace{20mm}\times
\left( -\sqrt{\frac{2}{15}}\sqrt{\frac{l'+3}{(2l'+5)(l'+2)}} \right)\left( \left(b^{\dagger}_{l'+2}\widetilde{b}_{l'}\right)^{2}_{q+q'} + \left(b^{\dagger}_{l'}\widetilde{b}_{l'+2}\right)^{2}_{q+q'}\right)\\&
= \sum_{l}\frac{\sqrt{6}}{3}l\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{5(2l+3)}}\left< 1q2q'| 2 q+q'\right>\left(
\left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^{2}_{q+q'} + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^{2}_{q+q'}\right)
\\& -\sum_{l'}\frac{\sqrt{6}}{3}(l'+3)\sqrt{\frac{6(l'+1)(l'+2)(\eta-l')(\eta+l'+3)}{5(2l'+3)}}\left< 1q2q'| 2 q+q'\right>\left( \left(b^{\dagger}_{l'+2}\widetilde{b}_{l'}\right)^{2}_{q+q'} + \left(b^{\dagger}_{l'}\widetilde{b}_{l'+2}\right)^{2}_{q+q'}\right)\\&=-\sqrt{6}\left< 1q2q'| 2 q+q'\right>\sum_{l}\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{5(2l+3)}}\left(
\left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^{2}_{q+q'} + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^{2}_{q+q'}\right) ,
\label{COMMSU33}
\end{aligned}
\end{equation}

where in the last equality it was considered the fact that $\sum_l=\sum_{l'}$. Adding <ref> and <ref> it is seen that

\begin{equation}
\begin{aligned}
&\left[L^1_q, Q^2_{q'}\right] = -\sqrt{6}\left< 1q2q'| 2 q+q'\right>Q^2_{q+q'}.
\label{COMMSU35}
\end{aligned}
\end{equation}

CHAPTER: QUADRUPOLE AND ANGULAR MOMENTUM OPERATORS AS $SU(3)$ TENSORS

In this appendix the expansion of operators $L^1_q$ and $Q^2_q$ as $SU(3)$ tensors is done. Consider the creation operator in the spherical basis $b^\dagger_{lm}$ which creates a state

\begin{equation}
\begin{aligned}
&b^\dagger_{lm}|0\rangle = |l,m\rangle,
\label{bdaggercreation}
\end{aligned}
\end{equation}

corresponding to an irrep $[1]$ of $U((\eta+1)(\eta+2)/2)$ whose branching to $SU(3)\supset SO(3)\supset SO(2)$ corresponds to $(\eta,0)$, with $K = 0$, $l = \eta, \eta-2,\eta-4,...0$ or 1 and $m = -l,...,l$. Thus $b^\dagger_{lm}$ is a $(\eta,0)$ tensor in $SU(3)$ and

\begin{equation}
\begin{aligned}
b^{\dagger}_{(\eta,0),K=0,l,m}|0\rangle = |[1],(\eta,0),K=0,l,m\rangle.
\label{bdecompinsu3}
\end{aligned}
\end{equation}

The $SU(3)$ labels are required for the adjoint destruction operator $\widetilde{b}_{lm} = (-1)^{l+m}b_{l-m}$ as well. To obtain them consider the number operator which must be a tensor of kind

\begin{equation}
\begin{aligned}
n_{(0,0),K=0,l=0,m=0} = \sum_{l,m}b^{\dagger}_{(\eta,0),K=0,l,m}b_{(\lambda,\mu),K,l,m},
\label{numberoperator}
\end{aligned}
\end{equation}

which imposes $b_{(0,\eta),K=0,l,m}$ and for the adjoint $\widetilde{b}_{(0,\eta),K=0,l,m} = (-1)^{\phi+l+m}b_{(0,\eta),K=0,l,-m}$ where $\phi$ is a phase to be determined by coupling the operators as follows

\begin{equation}
\begin{aligned}
n_{(0,0),K=0,l=0,m=0} &= \sum_{l,m}b^{\dagger}_{(\eta,0),K=0,l,m}b_{(0,\eta),K =0,l,m} \\
&= \sum_{l,m}(-1)^{\phi+l+m}b^{\dagger}_{(\eta,0),K=0,l,m}\widetilde{b}_{(0,\eta),K=0,l,-m}\\
&= \sum_{l,m}(-1)^{\phi+l+m} \langle(\eta,0),K=0,l;(0,\eta),K=0,l||(0,0),K=0,0\rangle \\ &\hspace{30mm}\times
\langle l m l -m|00\rangle
\left[b^{\dagger}_{(\eta,0),l}\widetilde{b}_{(0,\eta),l}\right]_{(0,0),K=0,l=0,m=0},
\label{phasecalculation}
\end{aligned}
\end{equation}

where no additional sumations where added because of the coupling to $(0,0)K=0,l=0, m=0$. The values of the Clebsch-Gordan coefficients can be determined to be [35]

\begin{equation}
\begin{aligned}
\langle(\eta,0),K=0,l;(0,\eta),&K=0,l||(0,0),K=0,0\rangle = (-1)^\eta\sqrt{\frac{2l+1}{\text{dim}(\eta,0)}},\\
&\langle l m l -m|00\rangle = (-1)^{l-m}\sqrt{\frac{1}{2l+1}},
\label{Clebschbothalgebras}
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
n_{(0,0),K=0,l=0,m=0}  = \sum_{l,m} (-1)^{\phi+\eta}\sqrt{\frac{1}{\text{dim}(\eta,0)}}
\left[b^{\dagger}_{(\eta,0),l}\widetilde{b}_{(0,\eta),l}\right]_{(0,0),K=0,l=0,m=0},
\label{numbercontinuation}
\end{aligned}
\end{equation}

where $\phi=\eta$ and $\sqrt{1/\text{dim}(\eta,0)}$ is a normalization factor.

In order to obtain the Racah form of the coupling between creation and destruction operators consider

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
&\left(b^{\dagger}_{l}\widetilde{b}_{l'}\right)_{(\lambda,\lambda),K,L,q}= \sum_{\lambda,K}(-1)^{\eta}\langle(\eta,0),K=0,l;(0,\eta),K=0,l'||(\lambda,\lambda),K,L\rangle \left[b^{\dagger}_{(\eta,0),l}\widetilde{b}_{(0,\eta),l'}\right]_{(\lambda,\lambda),K,L,q},
\label{Racahexpansiontermsll}
\end{aligned}
\end{equation}

where the coupling must be of the form $(\lambda,\lambda)$ because of the product

\begin{equation}
\begin{aligned}
&(\eta,0)\otimes(0,\eta) = (0,0)+(1,1)+(2,2)+...+(\eta-1,\eta-1)+(\eta,\eta).
\label{prdetazero}
\end{aligned}
\end{equation}

Notice that the $(1,1),K=1$ representation generates $l=1,2$ corresponding to the labels of the operators

\begin{equation}
\begin{aligned}\hspace{5mm}L^{1}_q = \sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{3}}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^1_q,
\label{LRacah2appendix5}
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
Q^{2}_q = &-\left(\eta+\frac{3}{2}\right)\sum_{l}\sqrt{\frac{l(l+1)(2l+1)}{5(2l-1)(2l+3)}}\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)^2_q \\&+ \sum_{l}\sqrt{\frac{6(l+1)(l+2)(\eta-l)(\eta+l+3)}{5(2l+3)}}\left(\left(b^{\dagger}_{l}\widetilde{b}_{l+2}\right)^2_q  + \left(b^{\dagger}_{l+2}\widetilde{b}_{l}\right)^2_q  \right),
\label{QRaca4appendix5}
\end{aligned}
\end{equation}

where the following couplings must be computed

\begin{equation}
\resizebox{0.9\hsize}{!}{$
\begin{aligned}
&\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)_{(1,1),K=1,l=1,q}= \sum_{\lambda,K}(-1)^{\eta}\langle(\eta,0),K=0,l;(0,\eta),K=0,l||(1,1),K=1,1\rangle \left[b^{\dagger}_{(\eta,0),l}\widetilde{b}_{(0,\eta),l}\right]_{(1,1),K=1,l=1,q},\\
&\left(b^{\dagger}_{l}\widetilde{b}_{l}\right)_{(1,1),K=1,l=2,q}= \sum_{\lambda,K}(-1)^{\eta}\langle(\eta,0),K=0,l;(0,\eta),K=0,l||(1,1),K=1,2\rangle \left[b^{\dagger}_{(\eta,0),l}\widetilde{b}_{(0,\eta),l}\right]_{(1,1),K=1,l=2,q},\\
\label{racahcoupledterms11}
\end{aligned}$}
\end{equation}

using analytic expressions found in [95] and [35]. This shows that angular momentum and quadrupole operators are $SU(3)$ tensors of kind $L_{(1,1),K=1,l=1,q}$ and $Q_{(1,1),K=1,l=2,q}$.

CHAPTER: LIE ALGEBRAS SOFTWARE IMPLEMENTATION

The softwares used in this thesis were Schur: An interactive programme for calculating properties of Lie groups and symmetric functions [54], UNTOU3 [51], UNtoU3 [96] and SU3lib [97], all of them of public access. The UNTOU3 and UNtoU3 compute the irreps of $U(3)$ contained in an irrep of $U(\mathcal{N})$ exclusively for $\mathcal{N} = (\eta+1)(\eta+2)/2$ where $\eta = $1,2,3,4,5,6,7, that is, the branching problem $U((\eta+1)(\eta+2)/2)\supset SU(3)$. UNTOU3 code written in FORTRAN is more general than UNtoU3 since it allows irreps $[f_1,f_2,f_3,...,f_{\mathcal{N}}]$ for any value of $f_i$ and additionally prints the Gel’fand patterns [49], Casimir values and dimensions as well, while the UNtoU3 written in C++ only allows irreps of the form $[2^a,1^b,0^c]$ but has the advantage of a more robust architecture with the possibility of parallel computations which makes it a lot faster. A new software that combines the advantages of these two might be a necessity for nuclear physics. During the writing of this thesis an extension of UNtoU3 was developed by P. Fasano which admits irreps $[4^a, 3^b, 2^c,1^d,0^e]$ and can be found in [98].

Above: Partial output of UNTOU3 software performing the irreps of $U(3)$ included in $[2^4]$ of $U(21)$. Below: Partial output of the same calculation of UNtoU3.

An example of their outputs is seen partially in figure <ref>. Notice that UNTOU3 prints the $SU(3)$ irreps $(\lambda,\mu)$ obtained form $[f_1,f_2,f_3]$ of $U(3)$ as $(f_1-f_2,f_2-f_3)$ and requires to enter the $\mathcal{N}$ numbers separated by spaces while for UNtoU3 it must be specified in a .cpp file of which some examples are provided in the source of the code along with details on the parallelization of the computations. In order to organize the irreps obtained from UNtoU3 by decreasing value of Casimir operator a proper code must be written by the user.

Since only the leading irreps are considered for each proton $SU_\nu(3)$ and neutron $SU_\pi(3)$ spaces, to obtain the coupled irreps of $SU_{\pi+\nu}(3)$ it was used the Schur software in the current version 6.11. Among several many functionalities it provides, one of them is the possibility to calculate tensor (Kronecker) products between irreps of a same algebra. Schur has four modes of operation, by default it starts in direct product mode (DP) but the one of interest is the representation mode (REP) where operations on a single Lie algebra can be performed. Schur is case insensitive so in order to change to REP mode one can type the command "REP" or "rep" once the program starts. Then it requires the group to be specified which for $SU(3)$ can be done by the command "gr1 su3". The command to calculate tensor products is "prod". For the group $SU(3)$, the irreps must be entered between curly brackets and in $U(3)$ notation. Integers of the irreps must be separated by spaces and if it has two digits one must type it with a previous "!" character. As an example, for $(26,4)\otimes(34,4)$ it must be entered "prod !30 4,!38 4" as shown in figure <ref>. The resulting irreps of $SU(3)$ are obtained by subtracting the second digit from the first one. The reader is encouraged to obtain and read the Schur program manual where its many uses are explained in detail.

Partial output of calculation $(26,4)\otimes(34,4)$ in software Schur.

As an illustration of how these three software can be applied, the 0$\hbar\omega$, 1$\hbar\omega$ and 2$\hbar\omega$ model space are computed for the isotope $^{220}$Ra. The core is $^{208}$Pb and possesses three quartets. These are described by the irrep $[2^3]_\pi\otimes[2^3]_\nu$ in $U(21)\otimes U(28)$ for the 0$\hbar\omega$ ground state. Using UNTOU3 or UNtoU3 it is found that the leader irrep of $U(21)\supset SU(3)_\pi$ contained in $[2^3]_\pi$ is $(24,0)_\pi$. Similarly, for $U(28)\supset SU(3)_\nu$ the leader irrep contained in $[2^3]_\nu$ is $(30,0)_\nu$. Now, coupling $SU(3)_\pi\otimes SU(3)_\nu$ irreps to $SU(3)_{\pi+\nu}$, the total model space is obtained. This is, $(24,0)_\pi\otimes(30,0)_\nu = ( 54 , 0 ) + ( 52 , 1 )+( 50 , 2 )+( 48 , 3 )+( 46 , 4 )+( 44 , 5 )+( 42 , 6 )+...$ The last product is performed by the Schur software. For 1$\hbar\omega$, the excitation of one proton it corresponds to $\{[2^2,1]\otimes[1]\}_\pi\otimes[2^3]_\nu$ in $\{U(21)\otimes U(28)\}\otimes U(28)$ and taking its leader irreps under decomposition will result in $\{(20,1)\otimes(6,0)\}_\pi\otimes(30,0)_\nu$. Coupling the proton part and considering its leader only results in $(26,1)_\pi\otimes (30,0)_\nu$. For the excitation of one neutron it corresponds to $[2^3]_\pi\otimes\{[2^2,1]\otimes[1]\}_\nu$ in $U(21)\otimes \{U(28)\otimes U(36)\}$. Similarly, it results in $(24,0)_\pi\otimes\{(25,1)\otimes(7,0)\}_\nu$ and under coupling to  $(24,0)_\pi\otimes (32,1)_\nu$. The total model space for 1$\hbar\omega$ will be $(26,1)_\pi\otimes (30,0)_\nu + (24,0)_\pi\otimes (32,1)_\nu$ which after deleting spurious states it is obtained $( 56 , 1 )^2 + ( 54 , 2 )^2 + ( 55 , 0 )+( 52 , 3 )^2+( 50 , 4 )^2+( 48 , 5 )^2+( 46 , 6 )^2+...$ Last, for 2$\hbar\omega$ the possibilities are excitation of two protons corresponding to $\{[2^2]\otimes[2]\}_\pi\otimes[2^3]_\nu$, excitation of two neutrons corresponding to  $[2^3]_\pi\otimes\{[2^2]\otimes[2]\}_\nu$ and excitation of one proton and one neutron corresponding to $\{[2^2,1]\otimes[1]\}_\pi\otimes \{[2^2,1]\otimes[1]\}_\nu$. Repeating the same procedure results in $\{(16,2)\otimes(12,2)\}_\pi\otimes(30,0)_\nu$, $(24,0)_\pi\otimes\{(20,2)\otimes(14,0)\}_\nu$ and $\{(20,1)\otimes(6,0)\}_\pi\otimes\{(25,1)\otimes(7,0)\}_\nu$ respectively, whose total model space will be  $(28,2)_\pi\otimes (30,0)_\nu + (24,0)_\pi\otimes (24,2)_\nu + (26,1)_\pi\otimes (32,1)_\nu$. After deletion of spurious states the real model space will be $( 58 , 2 )^3+( 59 , 0 )+( 56 , 3 )^3+( 57 , 1 )^2+( 54 , 4 )^3+( 52 , 5 )^3+( 50 , 6 )^3+...$ This completes the model space for the ground and first excited states of $^{220}$Ra.

The other computational necessity was the calculation of $SU(3)\supset SO(3)$ reduced Wigner coefficients required for calculating $B(E2)$ transitions in equation <ref> which was solved by the $SU(3)$ library SU3lib in particular the  module SU3_SO3_WignerCoeffs.cpp programmed in C++. The use of the program is very straightforward, once it starts it asks for the labels $(\lambda,\mu)$ and $L$ of the coefficient which after being entered fills the allowed labels $K$ and shows the values of the coefficients for each multiplicity coupling $\rho$. An example of the calculation of coefficient $\langle(60,8),1,0; (11),1,2||(60,8),1,2\rangle_{\rho}$ used in the electromagnetic transition of $^{224}$Th is shown in figure <ref>. The reader is again encouraged to obtain these free software and deepen his knowledge in their  many more uses than those exposed in this appendix.

Calculation of coefficient $\langle(60,8),1,0; (11),1,2||(60,8),1,2\rangle_{\rho}$ in software SU3lib.

CHAPTER: $B(E2)$ IN LIQUID DROP AND $SU(3)$ MODELS

In this section are presented the expressions for the reduced transition probability $B(E2)$ in the liquid drop and $SU(3)$ which are used in calculations on the main text. Both expressions must be coherent in their predictions and can be complemented in order to expand the understanding of electromagnetic interactions in atomic nuclei.

The transition probability per unit time (also called transition rate) from an initial nuclear state $i$ to a final state $f$ is described by Fermi's golden rule and reads as [10, 4]

\begin{equation}
\begin{aligned}
T_{i\xrightarrow{}f}^{\sigma\lambda\mu} = \frac{2}{\epsilon_0\hbar}\frac{\lambda+1}{\lambda((2\lambda+1)!!)^2}\left(\frac{E_{\gamma}}{\hbar c}\right)^{2\lambda+1}|\langle \Xi_f L_f M_f|\mathcal{M}_\mu^{\sigma\lambda}|\Xi_i L_i M_i\rangle|^2,
\end{aligned}
\label{Tfermigoldenrule}
\end{equation}

where $\sigma = E,M$ to identify electric or magnetic transitions,
$\lambda,\mu$ represent the multipole component, $\mathcal{M}_\mu^{\sigma\lambda}$ the electromagnetic interaction operator, $E_{\gamma}$ the energy of the gamma ray emitted and the $\Xi $ represent any additional labels for the quantum state. No spin (or isospin) will be considered in this section so it will be assumed that $L = J$. By averaging over initial states, summing over final states and components $\mu$ it is obtained the expression

\begin{equation}
\begin{aligned}
&T_{i\xrightarrow{}f}^{\sigma\lambda} = \frac{1}{2L_i+1}\sum_{M_f,\mu,M_i}T_{i\xrightarrow{}f}^{\sigma\lambda\mu} \\ &= \frac{2}{\epsilon_0\hbar}\frac{\lambda+1}{\lambda((2\lambda+1)!!)^2}\left(\frac{E_{\gamma}}{\hbar c}\right)^{2\lambda+1} \frac{|\langle \Xi_f L_f||\mathcal{M}^{\sigma\lambda}||\Xi_i L_i\rangle|^2}{2L_i+1} \sum_{M_f,\mu,M_i} |\langle L_i M_i\lambda\mu | L_f M_f
\rangle|^2 \\
& = \frac{2}{\epsilon_0\hbar}\frac{\lambda+1}{\lambda((2\lambda+1)!!)^2}\left(\frac{E_{\gamma}}{\hbar c}\right)^{2\lambda+1} B(\sigma\lambda; \Xi_i L_i\xrightarrow{}\Xi_f L_f),
\end{aligned}
\label{Tsums}
\end{equation}

where Wigner-Eckart theorem without the factor $1/\sqrt{2L_f+1}$ convention was used and the reduced transition probability is defined as

\begin{equation}
\begin{aligned}
B(\sigma\lambda; \Xi_i L_i\xrightarrow{}\Xi_f L_f) =
\frac{2L_f+1}{2L_i+1} |\langle \Xi_f L_f||\mathcal{M}^{\sigma\lambda}||\Xi_i L_i\rangle|^2.
\end{aligned}
\label{Redtransitionprobability}
\end{equation}

For the particular case of interest, the $B(E2)$ transition will involve the quadrupole operator $\mathcal{M}_\mu^{E2} = Q_\mu^{2}$ as

\begin{equation}
\begin{aligned}
B(E2; \Xi_i L_i\xrightarrow{}\Xi_f L_f) =
\frac{2L_f+1}{2L_i+1} |\langle \Xi_f L_f||Q^{2}||\Xi_i L_i\rangle|^2,
\end{aligned}
\label{BE2transitionprobability}
\end{equation}

where the mathematical form of $Q_\mu^2$ and its reduced matrix element will depend on the theoretical framework used. For the liquid drop model it will come from classical electrodynamics expression

\begin{equation}
\begin{aligned}
Q_\mu^2 = \int d^3\mathbf{r}\hspace{1.5mm}r^2 Y_{2\mu}(\Omega) = \int \hspace{1.5mm}d\Omega Y_{2\mu}(\Omega)\int_0^{R(\Omega)} dr\hspace{1.5mm}r^4,
\end{aligned}
\label{Qliquiddrop}
\end{equation}

where $R(\Omega)$ will be given by equation <ref>. After performing the integral considering terms up to order two in deformation parameters $\alpha_{2\mu}$ it is obtained

\begin{equation}
\begin{aligned}
Q_\mu^2 = \frac{3ZR_0^2}{4\pi}\left(\alpha_{2\mu}-\frac{10}{\sqrt{70\pi}}[\alpha_2\times\alpha_2]_{2\mu}\right).
\end{aligned}
\label{QliquiddropExpanded}
\end{equation}

The calculation of $\langle L_f||Q^{2}|| L_i\rangle$ in this framework [4] for the ground state (g.s.) band results in the expression

\begin{equation}
\begin{aligned}
B(E2; L_i\hspace{1.5mm} \text{g.s.}\xrightarrow{} L_f\hspace{1.5mm}\text{g.s.}) =
\left( \frac{3ZR_0^2}{4\pi} \right)^2\left(
\beta+\frac{2}{7}\sqrt{\frac{5}{\pi}}\beta^2\right)^2\langle L_i020|L_f0\rangle^2,
\end{aligned}
\label{BE2transition}
\end{equation}

where $R_0 = 1.2A^{1/3} \hspace{0.5mm}\text{fm}$, only matrix elements with respect to $SO(3)\supset SO(2)$ were considered and $\beta$ is the deformation parameter introduced in chapter 2.

In the $SU(3)$ scheme the considered algebra chain is $SU(3)\supset SO(3)\supset SO(2)$ and its corresponding basis is $|(\lambda\mu)KLM\rangle$. The quadrupole operator is a $(11)K=1L=2$ tensor as shown in appendix E and $B(E2)$ for intraband transitions will be

\begin{equation}
\begin{aligned}
B\big(E2; (\lambda\mu)KL_i\xrightarrow\ (\lambda\mu)KL_f\big) =
\frac{2L_f+1}{2L_i+1} |\langle (\lambda\mu)KL_f||Q^{(11)1,2}||(\lambda\mu)KL_i\rangle|^2,
\end{aligned}
\label{BE2transitionprobabilitySU3}
\end{equation}

the reduced matrix element is given by [58]

\begin{equation}
\begin{aligned}
\langle (\lambda\mu)KL_f||Q^{(11)1,2}||(\lambda\mu)KL_i\rangle = (-1)^\phi\sqrt{4\mathcal{C}_2[SU(3)]}\langle(\lambda\mu)KL_i;(11)2||(\lambda\mu)KL_f\rangle_{\rho = 1},
\end{aligned}
\label{reducedQSU3}
\end{equation}

where $\phi = 0$ for $\mu=0$ and $\phi = 1$ for $\mu\neq 0$, the reduced Wigner coefficients can be computed with the software [56]. Finally

\begin{equation}
\begin{aligned}
B\big(E2; (\lambda\mu)KL_i\xrightarrow\ (\lambda\mu)KL_f\big) =
4\hspace{0.8mm}\frac{2L_f+1}{2L_i+1}\hspace{0.8mm}\mathcal{C}_2[SU(3)]\hspace{0.8mm}|\langle(\lambda\mu)KL_i;(11)2||(\lambda\mu)KL_f\rangle_{\rho = 1}|^2.
\end{aligned}
\label{BE2SU3afterreduced}
\end{equation}

Since the chain $SU_{\pi}(3)\otimes SU_{\nu}(3)\supset SU_{\pi+\nu}(3)$ is considered for the modelling in this thesis, the quadrupole operator will be a coupling between proton and neutron quadrupole operators so a parameter $\alpha$ of the effective transition must be introduced [99] which is fitted to the the experimental value of the $2^+_1\xrightarrow{} 0^+_1$ transition [7]. The resulting expression is

\begin{equation}
\begin{aligned}
B\big(E2; (\lambda\mu)KL_i\xrightarrow\ (\lambda\mu)KL_f\big) =
\hspace{0.8mm}\frac{2L_f+1}{2L_i+1}\hspace{0.8mm}\mathcal{C}_2[SU(3)]\hspace{0.8mm}\alpha^2|\langle(\lambda\mu)KL_i;(11)2||(\lambda\mu)KL_f\rangle_{\rho = 1}|^2.
\end{aligned}
\label{YSUNBE2}
\end{equation}

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Structure of Heavy Nuclei Based on
Nucleon Quartet Condensates

Algebraic Approach in the Proxy-SU(3) SchemeAlejandro Restrepo GiraldoJanuary20231

Structure of Heavy Nuclei Based on
Nucleon Quartet Condensates

Algebraic Approach in the Proxy-SU(3) SchemeAlejandro Restrepo GiraldoJanuary20231