#  A universe born in a metastable
false vacuum state needs not die

K. Urbanowski111e–mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
University of Zielona Góra, Institute of Physics,
ul. Prof. Z. Szafrana 4a, 65–516 Zielona Góra, Poland

###### Abstract

We try to find conditions, the fulfillment of which allows a universe born in
a metastable false vacuum state to survive and not to collapse. The conditions
found are in the form of inequalities linking the depending on time $t$
instantaneous decay rate ${\it\Gamma}(t)$ of the false vacuum state and the
Hubble parameter $H(t)$. Properties of the decay rate of a quantum metastable
states are discussed and then the possible solutions of the conditions found
are analyzed and discussed. Within the model considered it is shown that a
universe born in the metastable vacuum state has a very high chance of
surviving until very late times if the lifetime, $\tau_{0}^{F}$, of the
metastable false vacuum state is comparable to (or even shorter), than time
when the inflation process begins. Our analysis shows that the instability of
the electroweak vacuum does not have to result in the tragic fate of our
Universe leading to its death.

##  1 Introduction

The problem of whether the Universe can live forever or whether sooner or
later will have to decay (i. e. to die) was considered in mathematical
cosmology based on solutions to Einstein’s equations practically from the very
beginning. This discussion gained a new face and intensified after the
publication of a seminal series of papers by Coleman et al. [1, 2, 3], where
the instability of a physical system, which is not at an absolute energy
minimum, and which is separated from the absolute minimum by an effective
potential barrier was discussed. They showed that if the early Universe is too
cold to activate the energy transition to the minimum energy state then a
quantum decay, from the false vacuum to the true vacuum, is still possible
through a barrier penetration via the macroscopic quantum tunneling. In other
words they showed that quantum decay processes can play an important role in
the early Universe and their result means that all early universes, in which
the the state describing their lowest energy was the quantum metastable vacuum
state should decay and die sooner or later. Such a possibility of a tragic
fate of the Universe has been discussed in many cosmological models (see e. g.
[4, 5, 6, 8, 7, 10, 11, 12, 13, 14, 15, 16, 17] and also [18, 19, 20, 21, 22,
23] and others). Some authors focus on the search for models in which the
lifetime, $\tau^{FV}_{0}$, of the false vacuum is much greater than the age of
our Universe. Unfortunately, such models do not solve the problem. In the
quantum theory of unstable states, long lifetime of a unstable particle does
not mean that there are no decays before that time passes, so the very long
lifetime, $\tau_{0}^{FV}$, of the metastable false vacuum state does not
guarantee that the universe with such a vacuum will survive at least until
$t\sim\tau_{0}^{FV}$ and that such a universe considered as the quantum
unstable particle will certainly not collapse earlier (or even much earlier)
than the lifetime will pass. On the other hand, there are also cosmological
models under study in which the lifetime of a false vacuum is very short, and
even significantly shorter than the time at which the inflation phase begins.
In [12, 13], a hypothesis was formulated that regions of the space with such a
fale vacuum could survive well until very late times. The same conclusion
concerns universes with such a vacuum.

The study of cosmological models with the metastable false vacua gained
additional, even more important significance after the discovery of the Higgs
boson and the determination of its mass $m_{H}$ [24, 25]. Estimations of the
Higgs and top quark masses [26] $m_{H}\simeq 125.10\pm 0.14$ [GeV] and
$m_{t}\simeq 172.76\pm 0.30$ [GeV] based on the experimental results place the
Standard Model in the region of the metastable vacuum (see e. g. [27, 28, 29,
30, 31, 32, 33, 34, 36, 37]). A consequence of the metastability of the Higgs
vacuum is that it should induce the decay of the electroweak vacuum in the
early Universe with catastrophic consequences for the Universe (for a
discussion see e.g. [38]). Considering all of these, the question arises: Why
does our Universe still exist? Probably for this reason various mechanisms
inducing the electroweak decay process of false vacuum or slowing it down have
been discussed in many papers (see e. g. [38, 39] and references therein).
Unfortunately there is no satisfactory answer to the above formulated question
in the available literature. Here we try to clarify this issue.

Properties of a universe born in metastable false vacuum state having very
short lifetime have been analyzed in [40]. In the analysis performed therein
the model of the dark anergy close to that considered by Landim and Abdalla
[17], in which the observed vacuum energy is the value of the scalar potential
at the false vacuum was assumed. (Similar idea was used and discussed in many
papers — see eg. [10, 41, 42, 43]). In other words, it was assumed that the
current stage of accelerated expansion of the universe is described by a
canonical scalar field $\Phi$ such that its potential, $V(\Phi)$, has a local
and true minimums. So, the field at the false vacuum represents the dark
energy. In such a situation, the quantum state of the system in the local
minimum is described by a state vector corresponding to the metastable false
vacuum state whereas the quantum state of the system in the true minimum
corresponds to the state of the lowest energy of the system and it is a true
vacuum. This means that the density of the energy of the system in the false
vacuum state, $\rho_{vac}^{\,\text{F}}$, is identified with the density of the
dark energy, $\rho_{vac}^{\,\text{F}}\equiv\rho_{de}$, (or, equivalently, as
the cosmological term $\Lambda$) in the Einstein equations [44, 45].

This paper complements the considerations and analysis presented in [40] and
the same assumptions as in [40] were adopted herein wherever necessary. Here
we propose conditions those must be met by a universe born in a metastable
false vacuum state so that it does not collapse and die.

The paper is organized as follows: In Section 2 an effective volume of an
expanding universe born in a metastable fale vacuum state is discussed and a
possible criterion for the survival of such a universe is proposed. Properties
of the decay rate of a quantum unstable state and energy of the system in such
a state are analyzed in Sec. 3. In Sec. 4 we describe a relation between some
cosmological parameters and quantities discussed in Sec. 3. In Sec.5 an
attempt is made to find solutions to the "survival" criterion formulated in
Sec. 2. Section 6 contains final remarks.

##  2 An effective volume of an expanding Universe born in a metastable fale
vacuum state

Krauss’s and Dent’s statement in [12] that "It may occur even though some
regions of false vacua by assumption should decay exponentially, gravitational
effects force space in a region that has not decayed yet to grow exponentially
fast" may be generalized and expressed in the general case in a mathematical
form in terms of the rate, ${\it\Gamma}(t)$, of the decay process, and of the
rate of the volume expansion. So, let $V_{eff}(t)$ denote an effective volume
of a universe born in the metastable false vacuum state. According to Krauss’s
and Dent’s hypothesis this volume is the effect of two independent opposing
processes taking place simultaneously: One of them is the quantum process of a
decaying false vacuum characterized by the survival probability ${\cal P}(t)$,
the other process is the process of growing the volume, $V(t)$, of the
universe depending on the properties of the parameter $\Lambda(t)$ in
Einstein’s equations, which with the Robertson–-Walker metric in the standard
form of Friedmann equations [45, 46] look as follows: The first one,

$\frac{{\dot{a}}^{2}(t)}{a^{2}(t)}+\frac{kc^{2}}{R_{0}^{2}\,a^{2}(t)}=\frac{8\pi
G_{N}}{3}\,\rho+\frac{\Lambda\,c^{2}}{3},$ (1)

and the second one,

$\frac{{\ddot{a}}(t)}{a(t)}=\,-\,\frac{4\pi
G_{N}}{3}\,\left(\frac{3p}{c^{2}}+\rho\right)+\frac{\Lambda\,c^{2}}{3}.$ (2)

where "dot" denotes the derivative with respect to time $t$,
${\dot{a}}(t)=\frac{da(t)}{dt}$, $\rho$ and $p$ are mass density and pressure
respectively, $k$ denote the curvature signature, and $a(t)=R(t)/R_{0}$ is the
scale factor, $R(t)$ is the proper distance at epoch $t$, $R_{0}=R(t_{0})$ is
the distance at the reference time $t_{0}$, (it can be also interpreted as the
radius of the Universe now) and here $t_{0}$ denotes the present epoch. The
pressure $p$ and the density $\rho$ are are related to each other through the
equation of state, $p=w\rho\,c^{2}$, where $w$ is constant [45]. There is
$w=0$ for a dust (for a matter dominated era), $w=1/3$ for a radiation and
$w=-1$ for a vacuum energy.

The simplest possibility is to define the effective volume, $V_{eff}(t)$, as
follows

$V_{eff}(t)={\cal P}(t)\,V(t),$ (3)

where $V(t)=\frac{4\pi}{3}[R(t)]^{3}=\frac{4\pi}{3}[R_{0}\,a(t)]^{3}$. Then
the rate, $\nu_{V}(t)$, of the expansion of the volume, $V_{eff}(t)$, equals

$\nu_{V}(t)\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{\dot{V}_{eff}(t)}{V_{eff}(t)}=\frac{d\,[{\cal
P}(t)\,V(t)]}{dt},$ (4)

which gives

$\nu_{V}(t)=\frac{\dot{V}_{eff}(t)}{V_{eff}(t)}\equiv
3\frac{\dot{a}(t)}{a(t)}\,-\frac{{\it\Gamma}(t)}{\hbar},$ (5)

where,

$\frac{{\it\Gamma}(t)}{\hbar}\,=\,-\,\frac{1}{{\cal P}(t)}\,\frac{d{\cal
P}(t)}{dt},$ (6)

is the rate of the decay process. Now if $\nu_{V}(t)\geq 0$ then the process
of the expansion overcomes the process of the collapse of the universe and as
a result the universe does not have to decay. On the other hand if
$\nu_{V}(t)<0$ then the fate of such a universe is doomed, and sooner or later
it must collapse and die.

##  3 Quantum metastable states: decay rate
and related quantities

We assume that the decay of the metastable false vacuum is the quantum decay
process. In general, in such a situation the survival probability equals,

${\cal P}(t)=|{\cal A}(t)|^{2},$ (7)

where

${\cal
A}(t)=\langle\phi|\phi(t)\rangle\equiv\langle\phi|e^{\textstyle{-\frac{i}{\hbar}\mathfrak{H}t}}|\phi\rangle,$
(8)

is the survival amplitude, $|\phi\rangle$ is the initial metastable state and
$|\phi(t)\rangle$ is the solution of the time depended Schrödinger equation
with initial conditions $|\phi(t=t_{0}^{init}\equiv
0)\rangle\stackrel{{\scriptstyle\rm def}}{{=}}|\phi\rangle$. Here
$\mathfrak{H}$ denotes the complete (full), self-adjoint Hamiltonian of the
system acting in the Hilbert space ${\cal H}$ of states of this system,
$|\phi\rangle,|\phi(t)\rangle\in{\cal H}$,
$\langle\phi|\phi\rangle=\langle\phi(t)|\phi(t)\rangle=1$ and therefore ${\cal
A}(0)=1$.

Note that the important formula, equivalent to (6)), for the decay rate,
strictly speaking for the instantaneous decay rate ${\it\Gamma}(t)$, can be
found using the equation (7). Namely differentiating the product $|{\cal
A}(t)|^{2}={\cal A}(t)[{\cal A}(t)]^{\ast}$ with respect to time $t$ one finds
that (see e. g. [47, 48] or [40, 49] and references therein),

${\it\Gamma}(t)=-2\Re\,[h(t)],$ (9)

where $\Im\,(z)$ denotes the imaginary parts of a complex number $z$, and

$h(t)=i\hbar\frac{1}{{\cal A}(t)}\,\frac{\partial{\cal A}(t)}{\partial
t}\equiv\frac{\langle\phi|\mathfrak{H}|\phi(t)\rangle}{{\cal A}(t)},$ (10)

is the effective Hamiltonian. The effective Hamiltonian $h(t)$ governs the
time evolution in the subspace of unstable states ${\cal
H}_{\parallel}=\mathbb{P}{\cal H}$, where
$\mathbb{P}=|\phi\rangle\langle\phi|$ in the case considered (see [48] and
also [50, 51, 52] and references therein):

$i\hbar\frac{\partial{\cal A}(t)}{\partial t}=h(t){\cal A}(t).$ (11)

The subspace ${\cal H}\ominus{\cal H}_{\parallel}={\cal
H}_{\perp}\equiv\mathbb{Q}{\cal H}$ is the subspace of decay products. Here
$\mathbb{Q}=\mathbb{I}-\mathbb{P}$. One meets the effective Hamiltonian $h(t)$
when one starts from the Schrödinger equation for the total state space ${\cal
H}$ and looks for the rigorous evolution equation for a distinguished subspace
of states ${\cal H}_{||}\subset{\cal H}$ [48, 53, 54]. In general $h(t)$ is a
complex function of time and in the case of ${\cal H}_{\parallel}$ of two or
more dimensions the effective Hamiltonian governing the time evolution in such
a subspace is a non-hermitian matrix $H_{\parallel}$ or non-hermitian
operator. There is

$h(t)=E(t)-\frac{i}{2}{{\it\Gamma}}(t),$ (12)

where $E(t)=\Re\,[h(t)]$, ${\it\Gamma}(t)=-2\,\Im\,[h(t)]$, are the
instantaneous energy (mass), $E(t)$, and the instantaneous decay rate,
${\it\Gamma}(t)$. (Here $\Re\,(z)$ denotes the part of $z$). In the case of
the metastable false vacuum state the instantaneous energy $E(t)$ allows one
to analyze early as well as late properties of the energy density (and thus of
the cosmological "constant" $\Lambda(t)$) in the false vacuum state (see [40]
and references therein).

In some cases it is convenient to consider the survival amplitude ${\cal
A}(t)$ as a solution of the Schrödinger–like equation (11) with the initial
condition ${\cal A}(0)=1$. Then

${\cal
A}(t)=e^{\textstyle{-\frac{i}{\hbar}\int_{0}^{t}h(\varsigma)d\varsigma}}\equiv
e^{\textstyle{-\frac{i}{\hbar}\left[\overline{E(t)}-\frac{i}{2}\overline{{\it\Gamma}(t)}\right]t}},$
(13)

where $\overline{E(t)}=\frac{1}{t}\int_{0}^{t}E(\varsigma)d\varsigma$ is the
effective energy of the system in the metastable state $|\phi\rangle$ and
$\overline{{\it\Gamma}(t)}=\frac{1}{t}\int_{0}^{t}{\it\Gamma}(\varsigma)d\varsigma$
is the effective decay rate. Using (13) we get

${\cal P}(t)=e^{\textstyle{-\overline{\it\Gamma(t)}\,t}}.$ (14)

Important general properties of the amplitude ${\cal A}(t)$, and thus the
survival probability ${\cal P}(t)$, can be found using integral representation
of ${\cal A}(t)$ as the Fourier transform. Namely, using a basis in ${\cal H}$
build from normalized eigenvectors $|E\rangle,\,\
E\in\sigma_{c}(\mathfrak{H})=[E_{{min}},{\infty})$ (where
$\sigma_{c}(\mathfrak{H})$ is the continuous part of the spectrum of
$\mathfrak{H}$) of $\mathfrak{H}$ and using the expansion of $|\phi\rangle$ in
this basis one can express the amplitude ${\cal A}(t)$ as the following
Fourier integral

${\cal
A}(t)\equiv\int_{E_{min}}^{\infty}\omega(E)\,e^{\textstyle{-\,\frac{i}{\hbar}\,E\,(t)}}\,d{E},$
(15)

where $\omega(E)=\omega(E)^{\ast}$ and $\omega(E)>0$ is the probability to
find the energy of the system in the state $|\phi\rangle$ between $E$ and
$E\,+\,dE$, and $E_{{min}}$ is the minimal energy of the system. The last
relation (15) means that the survival amplitude ${\cal A}(t)$ is a Fourier
transform of an absolute integrable function $\omega(E)$. If we apply the
Riemann-Lebesgue lemma to the integral (15) then one concludes that there must
be ${\cal A}(t)\to 0$ as $t\to\infty$. This property and the relation (15) are
the essence of the Fock–Krylov theory of unstable states [55, 56]. Within this
approach the amplitude ${\cal A}(t)$, and thus the decay law ${\cal P}(t)$ of
the metastable state $|\phi\rangle$, are determined completely by the density
of the energy distribution $\omega(E)$ for the system in this state [55, 56]
(see also [57, 58], and so on). (This approach is also applicable to models in
quantum field theory [59, 60]).

Assuming that
$\langle\phi|\mathfrak{H}|\phi\rangle,\;\langle\phi|\mathfrak{H}^{2}|\phi\rangle$
exist and that $|\langle\phi|\mathfrak{H}|\phi\rangle|<\infty$ and
$|\langle\phi|\mathfrak{H}^{2}|\phi\rangle|<\infty$ one can find that (see
[48])

${\cal P}(t)\simeq
1-\left(\frac{\delta\mathfrak{H}}{\hbar}\right)^{2}\,t^{2},$ (16)

at early times $t\to 0$, where
$\delta\mathfrak{H}=\sqrt{\langle\phi|\mathfrak{H}^{2}|\phi\rangle\,-\,E_{\phi}^{2}}$
and $E_{\phi}=\langle\phi|\mathfrak{H}|\phi\rangle$. This formula can be
obtained also, e. g., from the formula (8) using first terms of the expansion
of $\exp\,[-\frac{i}{\hbar}\mathfrak{H}t]$ in series for $t\to 0$: ${\cal
A}(t)\underset{t\to\,0}{\simeq}1-\frac{i}{\hbar}\langle\phi|\mathfrak{H}|\phi\rangle-\frac{1}{2!}\left(\frac{t}{\hbar}\right)^{2}\langle\phi|\mathfrak{H}^{2}|\phi\rangle+\ldots$
.

Using (16) and (6) one finds that at early times

$\frac{{\it\Gamma}(t)}{\hbar}=2t\left(\frac{\delta\mathfrak{H}}{\hbar}\right)^{2},\;\;({\rm
for}\,\,t\to 0),$ (17)

which agrees with the inference from Equations (10). Namely, from the last
term of Eq (10) one concludes that

$h(t)\equiv
E_{\phi}+\frac{\langle\phi|\mathfrak{H}\mathbb{Q}|\phi(t)\rangle}{{\cal
A}(t)}\stackrel{{\scriptstyle\rm def}}{{=}}E_{\phi}+v(t).$ (18)

Taking into account that $\mathbb{Q}=\mathbb{I}-\mathbb{P}$ and
$\mathbb{P}=|\phi\rangle\langle\phi|$, and thus that
$\mathbb{Q}|\phi\rangle=0$, the conclusion follows:

$h(0)\equiv\Re\,[h(0)]=E(0)\equiv
E_{\phi},\;\;\;\;\Im\,[h(0)]=0\;\Rightarrow\;{\it\Gamma}(0)=0,$ (19)

(if the matrix element $\langle\phi|H|\phi\rangle$ exists). On the other hand
making use of the relation
$|\phi(t)\rangle=\exp\,[-\frac{i}{\hbar}\mathfrak{H}t]|\phi\rangle$ one can
find that at early times

$v(t)\underset{t\to\,0}{\simeq}-\frac{i}{\hbar}\langle\phi|\mathfrak{H}\mathbb{Q}\mathfrak{H}|\phi\rangle
t\equiv-\frac{i}{\hbar}\left(\delta\mathfrak{H}\right)^{2}t,$ (20)

which leads to the same early time formula for ${\it\Gamma}(t)$ as that
obtained directly from the definition (6) of ${\it\Gamma}(t)$. It results from
(12), (18) and (20) that at early times

$h(t)\underset{t\to\,0}{\simeq}E_{\phi}-\frac{i}{\hbar}\left(\delta\mathfrak{H}\right)^{2}t.$
(21)

Hence

$E(t)\underset{t\to\,0}{\simeq}E_{\phi},\;\;\;\;{\rm
and}\;\;\;\;{\it\Gamma}(t)\underset{t\to\,0}{\simeq}\frac{2}{\hbar}\left(\delta\mathfrak{H}\right)^{2}t,$
(22)

(compare Eq. (17)). Note that the last relations are quite general and do not
depend on the form of the function of the energy density distribution
$\omega(E)$ in the metastable state considered.

As it was shown (see, e.g. [48, 61] and references therein), in a general case
at later time when the decay law (the survival probability) has an exponential
form to very good approximation, that is at canonical decay times, there are

${\it\Gamma}(t)\simeq{\it\Gamma}_{\phi}^{(1)}\stackrel{{\scriptstyle\rm
def}}{{=}}{\it\Gamma}_{0},\;\;\;\;{\rm and},\;\;\;E(t)\simeq
E_{0}\stackrel{{\scriptstyle\rm def}}{{=}}E_{\phi}-\Delta_{\phi}^{(1)}\equiv
E(0)-\Delta_{\phi}^{(1)},$ (23)

where

${\it\Gamma}^{(1)}_{\phi}=2\pi\langle\phi|\mathfrak{H}\mathbb{Q}\,\delta(\mathbb{Q}\mathfrak{H}\mathbb{Q}-E_{\phi})\,\mathbb{Q}\mathfrak{H}|\phi\rangle,$
(24)

is the decay width, and

$\Delta^{(1)}_{\phi}=\langle\phi|\mathfrak{H}\mathbb{Q}\;\,{\rm
P}_{v}\,\frac{1}{\mathbb{Q}\mathfrak{H}\mathbb{Q}-E_{\phi}}\;\,\mathbb{Q}\mathfrak{H}|\phi\rangle,$
(25)

(here ${\rm P}_{v}$ denotes principal value), and $\Delta^{(1)}_{\phi}$ is the
correction to the energy of the system in the metastable state,
$\Delta_{\phi}^{(1)}=-\Re\,[v(t)]$, ($v(t)\simeq const.$ at canonical decay
times) and $|\Delta_{\phi}^{(1)}|\ll|E_{\phi}|$. It should be noted that more
detailed considerations show that these approximate results describe behavior
of the system in the metastable state $|\phi\rangle$ accurate enough only for
canonical decay times (i.e. when the exponential decay law holds with
sufficient accuracy [62]).

The representation of the survival amplitude ${\cal A}(t)$ as the Fourier
transform (15) can be used to find the late time asymptotic form of ${\cal
A}(t)$, ${\cal P}(t)$, the decay rate ${\it\Gamma}(t)$ and the instantaneous
energy $E(t)$ (see [50, 51]). There are,

${{\it\Gamma}(t)\,\vline}_{\,t\rightarrow\infty}\stackrel{{\scriptstyle\rm
def}}{{=}}{\it\Gamma}_{lt}(t)=\alpha_{1}\frac{\hbar}{t}+\alpha_{3}\left(\frac{\hbar}{t}\right)^{3}+\ldots,$
(26)

and

${E(t)\,\vline}_{\,t\rightarrow\infty}\stackrel{{\scriptstyle\rm
def}}{{=}}E_{lt}(t)=E_{min}+\alpha_{2}\left(\frac{\hbar}{t}\right)^{2}+\alpha_{4}\left(\frac{\hbar}{t}\right)^{4}+\ldots,$
(27)

where $\alpha_{k}$, are real numbers for $k=1,2,\ldots$ and $\alpha_{1}>0$ and
the sign of $\alpha_{k}$ for $k>1$ depends on the model considered (see [51]).
Results (26), (27) are completely general. They do not depend on the form of
$\omega(E)$. The absolute integrability of $\omega(E)$ and its first few
derivatives are the only conditions that must be met to obtain the asymptotic
expansions (26) and (27) (for details see [51]). An example of such a general
function $\omega(E)$ is the function
$\omega(E)=\omega_{\lambda}(E)={\it\Theta}(E-E_{min})\,(E-{E}_{min})^{\lambda}\,\eta(E)$,
where $\eta({E})$ is an absolutely integrable function, ${\it\Theta}(E)$ is a
step function: ${\it\Theta}(E)=0\;\;{\rm for}\;\;E\leq 0$ and
${\it\Theta}(E)=1\;\;{\rm for}\;\;E>0$, and $0\leq\lambda<1$, (see [51, 61]).
In this case, in (26) we have $\alpha_{1}=2(\lambda+1)$, (see [61]).

The above analysis shows that the following phases can be distinguished in the
quantum decay process. The initial phase, $t\in[t_{0}^{init},T_{0})$, where
$t_{0}^{init}$ is the instant of a creation of the quantum metastable state,
and $T_{0}$ is the indicative maximal time such that ${\it\Gamma}(t)$ is
smaller than ${\it\Gamma}_{0}$ for $t<T_{0}$. (For simplicity we assume that
$t_{0}^{init}=0$). In other words, the initial phase comprises a time period
$t\in[0,T_{0})$ in which the approximations (17), (22) are sufficiently
accurate and ${\it\Gamma}(t)<{\it\Gamma}_{0}$. The times $t\in(T_{0},T_{1})$
form the next phase called the canonical decay phase. During this phase, to a
very high accuracy, the decay law has the form of an exponentially decreasing
function of time. The time $T_{1}$ denotes approximately the end of the
canonical decay phase and the beginning of the transition times phase. For
$t>T_{1}$ the decay law as a function of time still decreases but slower and
slower and begins to be oscillatory modulated up to the time $T_{2}$, when the
late time inverse power law phase begins to dominate. So times
$t\in(T_{1},T_{2})$ form the transition times phase and times
$t\in(T_{2},\infty)$ form the late time inverse power law  phase. During the
canonical decay phase, $t\in(T_{0},T_{1})$, the decay rate ${\it\Gamma}(t)$
and the instantaneous energy $E(t)$ are constant to a very good approximation
and they are given by Eqs (23), (24), (25). At times $t\in(T_{1},T_{2})$ the
decay rate, ${\it\Gamma}(t)$, and the instantaneous energy, $E(t)$, are
rapidly decreasing and oscillatory modulated functions of time. During the
inverse power law phase, $t\in(T_{2},\infty)$, the decay law behaves as a
function of powers of $1/t$ and the decay rate ${\it\Gamma}(t)$ and the energy
$E(t)$ take the form described by Eqs (26) and (27) respectively.

In order to calculate the survival amplitude ${\cal A}(t)$ within the
Fock–Krylov theory of unstable states we need the energy density distribution
function $\omega(E)$. The simplest choice is to take $\lambda=0$ in formula
for $\omega(E)\equiv\omega_{\lambda}(E)$ and to assume that $\eta(E)$ has a
Breit–Wigner (BW) form of the energy distribution density. It turns out that
the decay curves obtained in this simplest case are very similar in form to
the curves calculated for a more general $\omega(E)$, (see [63, 64] and
analysis in [57]). So to find the most typical properties of the decay process
it is sufficient to make the relevant calculations for $\omega(E)$ modeled by
the the Breit–Wigner distribution of the energy density:

$\begin{split}\omega(E)&\equiv\omega_{{BW}}(E)\\\
&\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{N}{2\pi}\,{\it\Theta}(E-E_{{min}})\
\frac{{\it\Gamma}_{0}}{(E-E_{0})^{2}+(\frac{{\it\Gamma}_{0}}{2})^{2}},\end{split}$
(28)

where $N$ is a normalization constant. The parameters $E_{0}$ and
${\it\Gamma}_{0}$ correspond to the energy of the system in the metastable
state and its decay rate at the exponential (or canonical) regime of the decay
process. $E_{{min}}$ is the minimal (the lowest) energy of the system. For
$\omega(E)=\omega_{BW}(E)$ one can find relatively easy an analytical form of
${\cal A}(t)$ (see e.g. [65, 66]). Simply, inserting $\omega_{{BW}}(E)$ into
formula (15) for the amplitude ${\cal A}(t)$ after some algebra one can find
${\cal A}(t)$. Then having the amplitude ${\cal A}(t)$ we can use it to
analyze properties of decay rate ${\it\Gamma}(t)$ and the instantaneous energy
$E(t)$. These quantities are defined using the effective Hamiltonian $h(t)$
which is build from ${\cal A}(t)$. (Details of calculations can be found e. g.
in [40, 49, 50, 51, 52, 66]). Then one can find and an asymptotic analytical
form of ${\cal A}(t),h(t)$, $E(t)$ and ${\it\Gamma}(t)$ at very late times
(see e.g . [50, 52, 66]) as well as to find numerically the form of the
survival probability ${\cal P}(t)$, the decay rate ${\it\Gamma}(t)$ and the
instantaneous energy $E(t)$ over a wide range of times $t$, starting from the
initial instant $t=t_{0}^{init}=0$, through canonical decay and then
intermediate times, up to asymptotically late times $t\to\infty$. So within
the model based on the Breit–Wigner distribution of the energy density one
finds that at late times, $t\to\infty$, there are (see [49, 66]):

${\it\Gamma}(t)=-2\Im\,[h(t)]\underset{t\to\,\infty}{\simeq}2\,\frac{\hbar}{t}+\ldots\,.$
(29)

and

$E(t)={\Re\,[h(t)]}\\\
\underset{t\to\,\infty}{\simeq}{E}_{{min}}\,-\,2\,\frac{\beta}{{\it\Gamma}_{0}\,(\beta^{2}+\frac{1}{4})}\,\left(\frac{\hbar}{t}\right)^{2}+\ldots,$
(30)

where $\beta=\frac{E_{0}-E_{min}}{{\it\Gamma}_{0}}>0$.

Results of the numerical calculations depend on the value of the ratio $\beta$
(see [40] for details). A visualization of these results is presented below in
graphical form in Figs 1, 2, 3, 4, 5, 6, 7. There is

$\kappa(t)=\frac{E^{\,\text{F}}(t)-E_{{min}}}{E_{0}^{\,\text{F}}-E_{{min}}},$
(31)

in Figs 5, 6 and 7. In some of these Figures, the aforementioned times
$T_{0},T_{1}$ and $T_{2}$ are marked for illustration. And so, the time
$T_{0}$ denotes the maximal time such that ${\it\Gamma}(t)$ is smaller then
${\it\Gamma}_{0}$ for $t<T_{0}$: There is ${\it\Gamma}(t)<{\it\Gamma}_{0}$ for
$t\in[0,T_{0})$ (see Fig 4). The time $T_{1}$ is the time when the canonical
(exponential) decay law ceases to apply and the transition from the
exponential decay phase to the phase in which the decay law takes the form of
powers $1/t$ begins, simply $T_{1}$ denotes approximately the beginning of the
transition times (see Figs 1, 5, 6). The time $T_{2}$ denotes an indicative
end of the transition phase between exponential and inverse power like forms
of the decay law (it is also the indicative beginning of this last phase — see
Fig 6). In general the transition times are such times $t$ that
$t\in(T_{1},T_{2})$.

Figure 1: A typical form of the survival probability ${\cal P}(\tau)$. Here
$\tau=\frac{t}{\tau_{0}}$.$\tau_{0}=\frac{\hbar}{{\it\Gamma}_{0}}$ is the
lifetime, $t$ is the time. The case $\beta=10$. Figure 2: The instantaneous
decay rate ${\it\Gamma}(t)$ obtained for $\omega_{BW}(E)$ given by Eq. (28).
The case $\beta=10$. $\tau=\frac{t}{\tau_{0}}$,
$\tau_{0}=\frac{\hbar}{{\it\Gamma}_{0}}$ is the lifetime, $t$ is the time. The
solid line: The ratio ${\it\Gamma}(t)/{\it\Gamma}_{0}$. The dashed line:
${\it\Gamma}(t)={\it\Gamma}_{0}$. Figure 3: The effective decay rate
$\overline{{\it\Gamma}(t)}\equiv\frac{1}{t}\int_{0}^{t}{\it\Gamma}(\varsigma)\,d\varsigma$
obtained for $\omega_{BW}(E)$ given by Eq. (28). The case $\beta=10$.
$\tau=\frac{t}{\tau_{0}}$, $\tau_{0}=\frac{\hbar}{{\it\Gamma}_{0}}$ is the
lifetime, $t$ is the time. The solid line: The ratio
$\overline{{\it\Gamma}(t)}/{\it\Gamma}_{0}$. The dashed line:
$\overline{{\it\Gamma}(t)}={\it\Gamma}_{0}$.  Figure 4: The instantaneous
decay rate ${\it\Gamma}(t)$ obtained for $\omega_{BW}(E)$ given by Eq. (28).
The case $\beta=100$. $\tau=\frac{t}{\tau_{0}}$,
$\tau_{0}=\frac{\hbar}{{\it\Gamma}_{0}}$ is the lifetime, $t$ is the time. The
solid line: The ratio ${\it\Gamma}(t)/{\it\Gamma}_{0}$. The dashed line:
${\it\Gamma}(t)={\it\Gamma}_{0}$.

##  4 Metastable states: cosmological applications

Within the standard approach transitions from a metastable false vacuum to the
true vacuum state are described in cosmology using quantum field theory
methods (see eg. [1] — [10] and many other papers). If one wants to generalize
the results obtained in the previous Section on the basis of quantum mechanics
to quantum field theory one should take into account among others volume
factors so that survival probabilities per unit volume should be considered
and similarly the energies and the decay rate:
$E\mapsto\varepsilon(E)=\frac{E}{V_{0}}$,
${\it\Gamma}_{0}\mapsto\gamma=\frac{{\it\Gamma}_{0}}{V_{0}}$, where
$V_{0}=V(t_{0}^{init})$ is the volume of the considered system at the initial
instant $t_{0}^{init}$, when the time evolution starts. The volume $V_{0}$ is
used in these considerations because the initial unstable state
$|\phi\rangle\equiv|0\rangle^{\text{F}}$ at $t=t_{0}^{init}=0$ is expanded
into eigenvectors $|E\rangle$ of $\mathfrak{H}$, (where
$E\in\sigma_{c}(\mathfrak{H})$), and then this expansion is used to find the
density of the energy distribution $\omega(E)$ at this initial instant
$t_{0}^{init}$. Now, if we identify $\varepsilon_{{de}}(t_{0}^{init})$ with
the energy $E_{0}^{\;\text{F}}$ of the unstable system divided by the volume
$V_{0}$:
$\varepsilon_{{de}}(t_{0}^{init})\equiv\varepsilon_{0}^{\,\text{F}}\equiv\varepsilon_{0}^{{qft}}\stackrel{{\scriptstyle\rm
def}}{{=}}\varepsilon_{{de}}^{0}=\frac{E_{0}^{\,\text{F}}}{V_{0}}$ and
$\varepsilon_{{bare}}=\frac{E_{{min}}}{V_{0}}$, (where
$\varepsilon_{{0}}^{{qft}}$ is the vacuum energy density calculated using
quantum field theory methods) then it is easy to see that the mentioned
changes $E\mapsto\frac{E}{V_{0}}$ and
${\it\Gamma}_{0}\mapsto\frac{{\it\Gamma}_{0}}{V_{0}}$ do not changes the
parameter $\beta$:

$\beta=\frac{E_{0}^{\,\text{F}}-E_{{min}}}{{\it\Gamma}_{0}}\equiv\frac{\varepsilon_{de}^{0}-\varepsilon_{bare}}{{\gamma}_{0}}>0,$
(32)

(where $\gamma_{0}={\it\Gamma}_{0}/V_{0}$, or equivalently,
${\it\Gamma}_{0}/V_{0}\equiv\frac{\varepsilon_{de}^{0}-\varepsilon_{bare}}{\beta}$).
This means that the relations (27), (30), can be replaced by corresponding
relations for the densities $\varepsilon_{{de}}$ or $\Lambda$ (see, eg., [49,
61, 67, 68]). Simply, within this approach $E(t)=E^{\,\text{F}}(t)$
corresponds to the running cosmological constant $\Lambda(t)$ and $E_{{min}}$
to the $\Lambda_{bare}$. For example, we have

$\begin{split}\kappa(t)&=\frac{E^{\,\text{F}}(t)-E_{{min}}}{E_{0}^{\,\text{F}}-E_{{min}}}\equiv\frac{\frac{E^{\,\text{F}}(t)}{V_{0}}-\frac{E_{{min}}}{V_{0}}}{\frac{E_{0}^{\,\text{F}}}{V_{0}}-\frac{E_{{min}}}{V_{0}}}\\\
&=\frac{\varepsilon^{F}(t)-\varepsilon_{bare}}{\varepsilon^{F}_{0}-\varepsilon_{{bare}}}=\frac{\varepsilon_{{de}}(t)-\varepsilon_{{bare}}}{\varepsilon_{{de}}^{0}-\varepsilon_{{bare}}}\\\
&=\frac{\Lambda(t)-\Lambda_{{bare}}}{\Lambda_{0}-\Lambda_{{bare}}}.\end{split}$
(33)

Here
$\varepsilon^{F}(t)=\frac{E^{\,\text{F}}(t)}{V_{0}}\equiv\varepsilon_{{de}}(t)\equiv\rho_{de}c^{2}$,
$\Lambda(t)=\frac{8\pi G}{c^{4}}\,\varepsilon_{de}(t)\equiv\frac{8\pi
G}{c^{2}}\,\rho_{de}(t)$. Equivalently, $\rho_{de}(t)=\frac{c^{2}}{8\pi
G}\Lambda(t)$.

Our analysis shows that the properties of the instantaneous energy $E(t)$
change with time. The same is true for $\Lambda(t)$ (see Figs 5, 6). Taking
into account relations connecting $E(t)$ and $\Lambda(t)$ and using (22), (23)
and (27) one can conclude that there should be,

$\Lambda(t)\simeq\Lambda_{0}\simeq\frac{8\pi
G}{c^{4}}\,\varepsilon_{0}^{F}\equiv\frac{8\pi
G}{c^{4}}\,\frac{{}^{\text{F}}\langle
0|\mathfrak{H}|0\rangle^{\text{F}}}{V_{0}},\;\;\;t\in[0,T_{0})\cup(T_{0},T_{1}),$
(34)

at canonical decay times $t<T_{1}$, were
$\varepsilon_{0}^{F}=\varepsilon_{0}^{qft}=\frac{{}^{\text{F}}\langle
0|\mathfrak{H}|0\rangle^{\text{F}}}{V_{0}}.$ (Here we used relations (22),
(23) and the property that $|\Delta_{\phi}^{(1)}|\ll|E_{\phi}|$ from which it
follows that in our analysis it is enough to assume that
$E_{0}^{\,\text{F}}\simeq E_{\phi}$, i.e., that
$E_{0}^{\,\text{F}}\simeq\;\,^{\text{F}}\langle
0|\mathfrak{H}|0\rangle^{\text{F}}$). In other words there should be
$\Lambda(t)\simeq\Lambda_{0}\equiv\Lambda_{qft}=\frac{8\pi
G}{c^{4}}\,\varepsilon_{0}^{qft}$ at times $t<T_{1}$. Then latter, when time
$t$ runs from $t=T_{1}$ to $t=T_{2}$ the quantum effect discussed above forces
this $\Lambda(t)\simeq\Lambda_{0}$ to reduce its value for $(t>T_{2})$ to the
following one:

$\Lambda(t)\simeq\Lambda_{eff}(t)=\Lambda_{\text{bare}}+\frac{\alpha_{2}^{\Lambda}}{t^{2}}+\frac{\alpha_{4}^{\Lambda}}{t^{4}}+\ldots\ll\Lambda_{0},\;\;\;(t>T_{2}).$
(35)

(A detailed discussion of this case is given in [49, 67, 68, 69]). At times
$t\in(T_{1},T_{2})$ the energy $E(t)$ is oscillatory modulated and decreases
from the value $E(t)=E_{0}$ to the late time asymptotic value $E_{lt}(t)$ (see
(27). An analogous behavior takes place for $\Lambda(t)$ at this time
interval. The typical behavior of $\Lambda(t)$ as a function of time is shown
below in Figs 5, 6, 7.

Figure 5:  An illustration of a behavior of energy $E(\tau)$ (or cosmological
"constant" $\Lambda(t)$). Results obtained for $\omega_{BW}(E)$ given by Eq.
(28). The case $\beta=100$. The solid line:
$\kappa(\tau)=\left(E(\tau)-E_{min}\right)/\left(E_{0}-E_{min}\right)\equiv\left(\Lambda(t)-\Lambda_{bare}\right)/\left(\Lambda_{0}-\Lambda_{bare}\right)$.
The dashed line: $E(\tau)=E_{0}=\text{const}$, or,
$\Lambda(\tau)=\Lambda_{0}=\text{const}$, ($\kappa(\tau)=1$). The time, $t$,
is measured in lifetimes $\tau_{0}$: $\tau=t/\tau_{0}$ and
$\tau_{0}=\hbar/{\it\Gamma}_{0}$ is the lifetime.  Figure 6:  The modulus of
the results presented in Fig (5) in the logarithmic scale. The solid line:
$\left|\kappa(\tau)\right|=\left|\left(E(\tau)-E_{min}\right)/\left(E_{0}-E_{min}\right)\right|\equiv\left|\left(\Lambda(t)-\Lambda_{bare}\right)/\left(\Lambda_{0}-\Lambda_{bare}\right)\right|$.
The dashed line: $E(\tau)=E_{0}=\text{const}$, or,
$\Lambda(\tau)=\Lambda_{0}=\text{const}$, ($\kappa(\tau)=1$). The time, $t$,
is measured in lifetimes $\tau_{0}$: $\tau=t/\tau_{0}$ and
$\tau_{0}=\hbar/{\it\Gamma}_{0}$ is the lifetime.  Figure 7:  The modulus of
$\kappa(\tau)$ in the logarithmic scale. The case $\beta=10000$. The solid
line:
$\left|\kappa(\tau)\right|=\left|\left(E(\tau)-E_{min}\right)/\left(E_{0}-E_{min}\right)\right|\equiv\left|\left(\Lambda(t)-\Lambda_{bare}\right)/\left(\Lambda_{0}-\Lambda_{bare}\right)\right|$.
The dashed line: $E(\tau)=E_{0}=\text{const}$, or,
$\Lambda(\tau)=\Lambda_{0}=\text{const}$, ($\kappa(\tau)=1$). The time, $t$,
is measured in lifetimes $\tau_{0}$: $\tau=t/\tau_{0}$ and
$\tau_{0}=\hbar/{\it\Gamma}_{0}$ is the lifetime.

##  5 Analysis and discussion

According to conclusion drawn in Sec. 2 the universe born in a false vacuum
state will survive well up to late times if the rate $\nu_{V}(t)$ of the
expansion of the effective volume $V_{eff}(t)$ is nonnegative. Using the
formula (5) for $\nu_{V}(t)$ this conclusion can be written as follows

$\nu_{V}(t)=3\frac{\dot{a}(t)}{a(t)}\,-\frac{{\it\Gamma}(t)}{\hbar}\geq 0,$
(36)

or, equivalently,

$3H(t)\,-\frac{{\it\Gamma}(t)}{\hbar}\geq 0,$ (37)

where $H(t)=\frac{\dot{a}(t)}{a(t)}$ is the Hubble parameter and
${\it\Gamma}(t)$ is the decay rate of the metastable false vacuum state. Using
the first Einstein equation (1) one can replace the condition (36) by the
following one:

$\left(\frac{{\it\Gamma}(t)}{\hbar}\right)^{2}\leq
9\left[-\frac{kc^{2}}{R_{0}^{2}\,a^{2}(t)}+\frac{8\pi
G_{N}}{3}\,\rho+\frac{\Lambda\,c^{2}}{3}\right].$ (38)

Now if the system is in the false vacuum state, $|0\rangle^{F}$, then at times
$t\in[0,T_{0})$, (where $t_{0}^{init}=0$ is the initial instant), and at
canonical decay times, $T_{0}<t<T_{1}$, the energy of the system in this state
equals
$E^{\,\text{F}}(t)=E_{0}^{\,\text{F}}\approx\,{{}^{F}\langle}0|\mathfrak{H}|0\rangle^{F}$
to a very good approximation and then $\Lambda(t)\simeq\Lambda_{0}$ is given
by the formula (34) and it can be very large.

At very early times, $t\in[0,T_{0})$, there is no matter and the vacuum energy
dominates. Therefore the behavior of expansion rate $\dot{a}(t)$ at these
times is such that the curvature signature in Eq (1) can be approximated as
$k\approx 0$ (see, e.g. [45]) and then Eq (1) simplifies to

$H^{2}(t)=\frac{{\dot{a}}^{2}(t)}{a^{2}(t)}\simeq\frac{\Lambda_{0}\,c^{2}}{3}.$
(39)

So in this case the universe considered will survive well at these times if
the following inequality replacing (38) will be satisfied:

$\left(\frac{{\it\Gamma}(t)}{\hbar}\right)^{2}\leq 3\Lambda_{0}\,c^{2}\simeq
24\pi G_{N}\,\frac{{}^{\text{F}}\langle
0|\mathfrak{H}|0\rangle^{\text{F}}}{c^{2}\,V_{0}}\equiv 24\pi
G\,\frac{\varepsilon^{F}_{0}}{c^{2}}.$ (40)

Taking into account the results (17), (22), i.e. that,
$\frac{{\it\Gamma}(t)}{\hbar}\underset{t\to\,0}{\sim}t$, (see also Figs 2, 4)
one can conclude that at very early times, $t\in[0,T_{0})$, the left hand side
of the inequality (40) is always smaller then the right hand side of this
inequality. This means that at these times the universe can not decay.

Recent studies based on the results of astronomical observations suggest that
our Universe is flat [70, 71, 72]. Therefore if one wants to apply Eqs. (37),
(38) to our Universe one should insert $k=0$ for the curvature $k$ in Eq. (1).
Then the first Friedmann Equation simplifies to the following one:

$H^{2}(t)=\frac{8\pi G_{N}}{3}\,\rho+\frac{\Lambda\,c^{2}}{3},$ (41)

which allows to rewrite the condition (38) as follows

$\left(\frac{{\it\Gamma}(t)}{\hbar}\right)^{2}\leq 3\left[8\pi
G_{N}\,\rho\,+\,\Lambda\,c^{2}\right],$ (42)

and only this case will be analyzed in the following.

Let us assume now that the lifetime, $\tau_{0}^{F}$, of the false vacuum state
is much shorter then duration of the inflationary epoch and that the time
$T_{1}$ is comparable with the duration of this epoch (such a scenario was
analyzed in [40]). Then let us consider times $t\in(T_{0},T_{1})$, i. e. the
epoch of the canonical decay times. In such a case, similarly to the case of
very early times $t\in[0,T_{0})$, we can ignore the matter density $\rho$ in
Eqs (1), (41) and the curvature $k$ in Eq (1) can be approximated as $k\approx
0$. So in this case the universe considered will survive well at times
$T_{0}<t<T_{1}$ if again the inequality (40) will be satisfied.

Note that at the canonical decay times (see (24) and Figs 2, 3, 4) there is
${\it\Gamma}(t)\simeq{\it\Gamma}^{(1)}\simeq{\it\Gamma}_{0}$ to a very good
approximation. So considering the false vacuum we can replace ${\it\Gamma}(t)$
in (40) by ${\it\Gamma}_{0}$ and use ${\it\Gamma}_{0}\equiv{\it\Gamma}_{F}$ to
obtain

$\left(\frac{{\it\Gamma}_{F}}{\hbar}\right)^{2}\equiv\left(\frac{1}{\tau_{0}^{F}}\right)^{2}\leq
24\pi G_{N}\,\frac{\varepsilon^{F}_{0}}{c^{2}},$ (43)

for the epoch considered, $t\in(T_{0},T_{1})$. Here
$\tau_{0}^{F}=\frac{\hbar}{{\it\Gamma}_{F}}$ and ${\it\Gamma}_{F}$ are the
lifetime and decay rate (decay width) of the system in the metastable false
vacuum state $|0\rangle^{F}$ respectively. There is $\frac{24\pi
G_{N}}{c^{2}}\simeq 55,9\times 10^{-27}$ [m/kg] and hence $\frac{c^{2}}{24\pi
G_{N}}\simeq 1.8\times 10^{25}$ [kg/m]. So, for the epoch $t\in(T_{0},T_{1})$
the condition (43) can be rewritten as follows:

$\varepsilon^{F}_{0}\geq 1.8\times
10^{25}{\,\rm[kg/m]}\times\left(\frac{1}{\tau_{0}^{F}}\right)^{2}.$ (44)

As it has been stated in Sec. 1, this paper complements analysis presented in
[40], where the possibility that vacuum energy could drive the inflation
process (or contribute significantly to it) was discussed. Within such a
scenario, as an example, it was analyzed the case that
$T_{1}\sim(50-100)\tau_{0}^{F}$ and $\tau_{0}^{F}\sim(10^{-38}-10^{-36})$[s].
We will begin our analysis of the possibility that the Universe will survive
with this scenario. So, let us consider the case $\tau_{0}^{F}\sim 10^{-36}$
[s]: From (44) if follows that in such a case the Universe can survive at the
epoch $t\in(T_{0},T_{1})$ only if $\varepsilon_{0}^{F}>1.8\times 10^{97}$
$[\frac{{\rm J}}{{\rm m}^{3}}]$. Similarly, if to assume that
$\tau_{0}^{F}\sim 10^{-38}$ [s], then inequality (44) implies that the
Universe has a chance of the survival only when $\varepsilon_{0}^{F}>1.8\times
10^{101}$ $[\frac{{\rm J}}{{\rm m}^{3}}]$. The surprise in this scenario is
that the minimum values of the false vacuum energy density,
$\varepsilon_{0}^{F}$, thus obtained fit well with the energy density
resulting from the estimates made on the assumption that quantum field theory
can be applied up to the Planck scale, $M_{Pl}\sim 10^{19}$ [GeV/$c^{2}$],
(see e.g. Sec. 7 in [46], Eq. (19) in [73], or the end of Sec. 3.1 in [74]).

In order to analyze inequalities (37) or (42) for the epoch
$t\in(T_{1},T_{2})$ within this scenario one should know the form of the
potential $V(\Phi)$ and thus the Lagrangian and the Hamiltonian $\mathfrak{H}$
allowing to find $|0\rangle^{F}$ and then to calculate ${\it\Gamma}(t)$ and
$E(t)$ for $t\in(T_{1},T_{2})$. It is because ${\it\Gamma}(t)$ is scillatory
modulated in this epoch and these modulations depend on $\mathfrak{H}$. The
same is true for $E(t)$ and therefore for $\Lambda(t)$ (see Figs 5, 6 and 7).

Within the scenario analyzed the picture is simpler at the epoch
$t\in(T_{2},\infty)$. At this epoch ${\it\Gamma}(t)$ is given by the formulae
(26), (29) and (35). (Note that within the scenario analyzed we are living now
at the epoch $t>T_{2}$, i.e. $t\in(T_{2},\infty)$). This means that now we
should use Eq. (29). Inserting
${\it\Gamma}(t)\underset{t\to\,\infty}{\simeq}2\,\frac{\hbar}{t}$ into (37) we
obtain the following inequality

$\frac{2}{t}\leq 3H(t),$ (45)

or

$\frac{3}{2}\,t\,H(t)\geq 1,$ (46)

for $t\in(T_{2},\infty)$. First, let us consider for simplicity the special
case, assuming for a moment that that $t\sim t_{0}$, where $t_{0}$ is the age
of the Universe today. Then using the present–day values of
$H(t)=H(t_{0})=H_{0}$ and $t_{0}$ (see [26]) one obtains that
$\frac{3}{2}\,t_{0}\,H_{0}\simeq 1.4$. So the condition (46) holds for the
today Universe and today we are safe. In general, if the universe has
successfully survived the epoch $(T_{1},T_{2})$ and has been fortunate enough
to survive until $t>T_{2}$, then the simplest choice is to use solutions of
the Friedmann Eqs. (2), (41),

$H(t)=\frac{2}{3t},$ (47)

for the matter dominated universe (see, e.g. [45]). Inserting (47) into the
weak inequality (46) one gets the result $\frac{3}{2}\,t\,H(t)=1$ for
$t\in(T_{2},\infty)$. So in such a case the universe should not decay.

Note that the above analysis concerning the epoch $t\in(T_{2},\infty)$ bases
on the assumption that a metastable state is modeled by the Breit–Wigner
energy density distribution which leads, among others, to the result (29) used
in Eqs. (45), (46). In the general case considering the epoch
$t\in(T_{2},\infty)$ one should use Eq. (26) instead of (29), which leads to
the following inequality,

$\frac{\alpha_{1}}{t}\leq 3H(t),$ (48)

replacing (45), where $\alpha_{1}$ depends on the model considered. So,
depending on $\alpha_{1}$ and $H(t)$, (or on the right hand side of (42)), the
inequality can be satisfied or not.

Let us consider now another scenario. Let $\tau_{0}^{F}\sim t_{0}$. Note that
if this assumption is true, then ${\it\Gamma}(t)\simeq{\it\Gamma}_{F}$ and
$T_{1}>t_{0}$ and are living in the canonical decay times epoch
$t\in(T_{0},T_{1})$. As a result the term $\frac{{\it\Gamma}(t)}{\hbar}$ in
(37) can be replaced by
$\frac{{\it\Gamma}(t)}{\hbar}=\frac{{\it\Gamma}_{F}}{\hbar}=\frac{1}{\tau_{0}^{F}}\simeq\frac{1}{t_{0}}$.
Hence the inequality (37) can be rewritten as follows

$3\,\tau_{0}^{F}\,H(t)\equiv 3\,t_{0}\,H(t_{0})\geq 1.$ (49)

Taking $t\sim t_{0}$ and using the present–day values of $H(t_{0})=H_{0}$ and
$t_{0}$ one finds that $3\,t_{0}\,H_{0}\simeq 2,798$. This means that when
$\tau_{0}^{F}\sim t_{0}$, then the condition (49) is satisfied and in this
case our Universe should survive well up to times $t>T_{1}>t_{0}$.
Unfortunately another problem arises. Namely if the lifetime is of order of
the age of the Universe and $t<T_{1}$ then, according to Eg. (34),
$\Lambda(t)=\Lambda_{0}\simeq\frac{8\pi G}{c^{4}}\,\varepsilon_{0}^{F}\simeq
const.$, where
$\varepsilon_{0}^{F}=\varepsilon_{0}^{qft}=\frac{{}^{\text{F}}\langle
0|\mathfrak{H}|0\rangle^{\text{F}}}{V_{0}}$, that is, it is given by
expectation value of the Hamiltonian $\mathfrak{H}$ in the false vacuum state
$|0\rangle^{F}$. On the other hand Eq. (41) says that if $H(t)$ is small then
$\frac{\Lambda(t)c^{2}}{3}=\frac{\Lambda_{0}c^{2}}{3}$ must be even much
smaller. Note that until now, no one has proposed a quantum field theory model
predicting a false vacuum state $|0\rangle^{F}$ with the lifetime,
$\tau_{0}^{F}$, of the order of the age of the Universe today, $t_{0}$, (or
longer) such that the expected value of the operator $\mathfrak{H}$ in this
state is correspondingly small. So, in the light of this analysis we should
rather forget about the possibility that $\tau_{0}^{F}\sim t_{0}$ or
$\tau_{0}^{F}>t_{0}$.

##  6 Final remarks

An analysis and estimations presented in the previous Section show that a
universe born in the metastable false vacuum state has a very high chance of
surviving until very late times if the lifetime, $\tau_{0}^{F}$, of the
metastable vacuum is comparable to, or even shorter, than the moment when the
inflation process begins. In such a case, the universe should survive at least
the epoch $t\in(T_{0},T_{1})$ and survive at least until $t>T_{1}$. Using
inequalities (36), (37), (42) the fate of the universe in the epoch
$t\in(T_{1},T_{2})$ is difficult to determine because at times
$t\in(T_{1},T_{2})$ the decay rate ${\it\Gamma}(t)$ and $\Lambda(t)$, and
hence the parameter $H(t)$, fluctuate and the amplitude of these fluctuations,
as well as the time periods during which they reach their local maxima and
minima depend on the potential $V(\Phi)$. It is because the Hamiltonian
$\mathfrak{H}$ depends on $V(\Phi)$ and $\mathfrak{H}$ determines the form and
properties of the energy density distribution $\omega(E)$ in the false vacuum
state $|0\rangle^{F}$. In turn knowing $\omega(E)$ and using the equation (15)
we can find the survival amplitude ${\cal A}(t)$ and then ${\it\Gamma}(t)$,
$E(t)$ and thus $\Lambda(t)$ too. Within this scenario, if the universe
survived the epoch $t\in(T_{1},T_{2})$ and still exists in the epoch
$t>T_{2}$, the picture is simpler. In this case the decay rate of the false
vacuum state has the form given by Eqs (26) and (29) which allows to rewrite
inequalities (36), (37) in a simple form (45), (46) leading to the conclusion
that within the considered scenario the universe born in metastable false
vacuum state should be safe. All these conclusions are important because they
also apply to our Universe, as according to the Standard Model (there is no
replacement model today), the Higgs boson and the top quark masses suggest
that our Universe was created in the state of metatable vacuum. Another
important conclusion from our analysis is that cosmological models in which
the lifetime, $\tau_{0}^{F}$, of a metastable false vacuum is comparable to
the age, $t_{0}$, of the Universe or is much longer are useless and do not
solve the problems discussed in this paper.

In general, the above briefly summarized results and conclusions obtained for
times $t<T_{2}$ are quite general and do not depend on the form of the energy
density distribution function, $\omega(E)$, and thus on $V(\Phi)$ and
$\mathfrak{H}$, except that the values of $T_{1}$ and $T_{2}$ depend on the
parameters of the model under consideration, i.e. on $\omega(E)$. Only the
results obtained and conclusions drawn for times $t>T_{2}$ are based on the
assumption that $\omega(E)$ is given by the Breit–Wigner energy density
distribution, $\omega(E)\equiv\omega_{BW}(E)$. We should be aware that for
other $\omega(E)\neq\omega_{BW}(E)$ the inequality (48) replaces inequalities
inequalities (45), (46) which allows us to study other scenarios for the epoch
$t>T_{2}$ than the scenario described above.

The author contribution statement: The author declares that there are no
conflicts of interest regarding the publication of this article and that all
results presented in this article are the author’s own results.

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$\displaystyle=\mathrm{sym}\left[\sum_{\gamma|\ell\in\gamma}\frac{a^{\gamma}_{i}}{2}\mathrm{tr}^{d^{\gamma}_{<\ell}}_{d^{\gamma}_{>\ell}}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-2}\right]\right]_{jk}$

###### Proof.

First, observe the following:

$\displaystyle\frac{\partial}{\partial
t_{i}}\log\mathrm{det}\left[\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right]$
$\displaystyle=\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\frac{\partial}{\partial
t_{i}}\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)\right]$
$\displaystyle=\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}a_{i}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}\right]$
$\displaystyle=a_{i}^{\gamma}\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\right]$

Furthermore, note that:

$\displaystyle\frac{\partial}{\partial
t_{i}}\sum_{\ell\in\gamma}\frac{\mathbf{t}^{T}\mathbf{a}^{\gamma}}{L^{\gamma}}\mathrm{tr}\left[\mathbf{\mathbf{S}^{\gamma}_{\ell}}\right]$
$\displaystyle=\sum_{\ell\in\gamma}\frac{a^{\gamma}_{i}}{L^{\gamma}}\mathrm{tr}\left[\mathbf{S}^{\gamma}_{\ell}\right]$
$\displaystyle=\frac{a^{\gamma}_{i}}{L^{\gamma}}\mathrm{tr}\left[\mathbf{S}^{\gamma}\right]$

Putting these together yields the first claim:

$\displaystyle\frac{\partial}{\partial t_{i}}\mathrm{NLL}$
$\displaystyle=\sum_{\gamma}-\frac{a^{\gamma}_{i}}{2}\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\right]+\frac{a^{\gamma}_{i}}{2}\mathrm{tr}\left[\mathbf{S}^{\gamma}\right]$

Differentiating a second time results in the following:

$\displaystyle\frac{\partial}{\partial t_{i}\partial t_{j}}\mathrm{NLL}$
$\displaystyle=\sum_{\gamma}\frac{a^{\gamma}_{i}}{2}\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\left(a^{\gamma}_{i}\mathbf{I}_{d^{\gamma}_{\forall}}\right)\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\right]$
$\displaystyle=\sum_{\gamma}\frac{a^{\gamma}_{i}a^{\gamma}_{j}}{2}\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-2}\right]$

Likewise, differentiating w.r.t. $\psi_{\ell_{jk}}$ gives us the final part of
our result. We’ll delay the symmetrization step until the end to prevent the
equation running off the right margin of the page.

$\displaystyle\frac{\partial^{\mathrm{nosym}}}{\partial
t_{i}\partial\psi_{\ell_{jk}}}\mathrm{NLL}$
$\displaystyle=\sum_{\gamma|\ell\in\gamma}\frac{a^{\gamma}_{i}}{2}\mathrm{tr}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\left(\mathbf{I}_{d^{\gamma}_{<\ell}}\otimes\mathbf{J}^{jk}\otimes\mathbf{I}_{d^{\gamma}_{>\ell}}\right)\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-1}\right]$
$\displaystyle=\sum_{\gamma|\ell\in\gamma}\frac{a^{\gamma}_{i}}{2}\mathrm{tr}^{d^{\gamma}_{<\ell}}_{d^{\gamma}_{>\ell}}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-2}\right]_{jk}$
$\displaystyle\frac{\partial}{\partial
t_{i}\partial\psi_{\ell_{jk}}}\mathrm{NLL}$
$\displaystyle=\mathrm{sym}\left[\sum_{\gamma|\ell\in\gamma}\frac{a^{\gamma}_{i}}{2}\mathrm{tr}^{d^{\gamma}_{<\ell}}_{d^{\gamma}_{>\ell}}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I}_{d^{\gamma}_{\forall}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell}\right)^{-2}\right]\right]_{jk}$

This completes the proof. ∎

###### Lemma 23

$\displaystyle\frac{\partial^{\mathrm{nosym}}}{\partial\psi_{\ell^{1}_{i^{1}j^{1}}}\partial\psi_{\ell^{2}_{i^{2}j^{2}}}}\mathrm{NLL}$
$\displaystyle=\frac{1}{2}\sum_{\gamma|\ell^{1},\ell^{2}\in\gamma}\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\right]_{i^{1}j^{1}}$

###### Proof.

This follows from Lemma 21. First we will compute the unsymmetrized
derivative.

$\displaystyle\frac{\partial^{\mathrm{nosym}}}{\partial\psi_{\ell^{1}_{i^{1}j^{1}}}\partial\psi_{\ell^{2}_{i^{2}j^{2}}}}\mathrm{NLL}$
$\displaystyle=-\frac{1}{2}\frac{\partial}{\partial\psi_{\ell^{2}_{i^{2}j^{2}}}}\sum_{\gamma|\ell^{1}\in\gamma}\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{t}^{T}\mathbf{a}^{\gamma}\mathbf{I_{d^{\gamma}_{\forall}}}+\bigoplus_{\ell\in\gamma}\mathbf{\Psi}_{\ell^{1}}\right)^{-1}\right]_{i^{1}j^{1}}$
$\displaystyle=-\frac{1}{2}\frac{\partial}{\partial\psi_{\ell^{2}_{i^{2}j^{2}}}}\sum_{\gamma|\ell^{1}\in\gamma}\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\right]_{i^{1}j^{1}}$
$\displaystyle=\frac{1}{2}\sum_{\gamma|\ell^{1},\ell^{2}\in\gamma}\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\right]_{i^{1}j^{1}}$

If we let the unsymmetrized derivative be
$\mathbf{X}_{\ell^{1}_{i^{1}j^{1}}\ell^{2}_{i^{2}j^{2}}}$, the symmetrized
version is
$\frac{\mathbf{X}_{\ell^{1}_{i^{1}j^{1}}\ell^{2}_{i^{2}j^{2}}}+\mathbf{X}_{\ell^{1}_{j^{1}i^{1}}\ell^{2}_{i^{2}j^{2}}}+\mathbf{X}_{\ell^{1}_{i^{1}j^{1}}\ell^{2}_{j^{2}i^{2}}}+\mathbf{X}_{\ell^{1}_{j^{1}i^{1}}\ell^{2}_{j^{2}i^{2}}}}{4}$.
∎

With these, we can derive $\mathbf{F}$. As the derivation of inter-axis
connections in $\mathbf{F}$ is complicated, and will be aided by the following
lemma.

###### Lemma 24

Let $\left\\{\mathbf{M}_{i}\right\\}_{i}$ be a subset of matrices where some
subset $X$ has the property that, for all $i\in X$,
$\mathbf{M}_{i}=\mathbf{I}_{d_{i}\times d_{i}}$, where $d_{i}$ represents the
size of the $i$th element.

Then $\mathrm{tr}\left[\bigotimes_{i}\mathbf{M}_{i}\right]=\left(\prod_{i\in
X}d_{i}\right)\mathrm{tr}\left[\bigotimes_{i\notin X}\mathbf{M}_{i}\right]$.

###### Proof.

This follows directly from the definition of the Kronecker product; taking the
Kronecker product of the identity matrix with some other matrix creates a
block-diagonal matrix that repeats the second matrix along the diagonal. While
the Kronecker product is not symmetric, the set of diagonal elements of a
Kronecker product is preserved when flipping the ordering. ∎

###### Corollary 25

Let $\left\\{\mathbf{M}_{i}\right\\}_{i}$ be a subset of matrices where some
subset $X$ has the property that, for all $i\in X$,
$\mathbf{M}_{i}=\mathbf{I}_{d_{i}\times d_{i}}$, where $d_{i}$ represents the
size of the $i$th element. Further, when $i\notin X$ we have that
$\mathbf{M}_{i}=\mathbf{J}^{j_{i}k_{i}}$.

Then $\mathrm{tr}\left[\bigotimes_{i}\mathbf{M}_{i}\right]=\left(\prod_{i\in
X}d_{i}\right)\left(\prod_{i\in X}\delta_{j_{i}k_{i}}\right)$.

We are now equipped with all the tools we need to prove Lemma 16, the values
of $\mathbf{F}$ under the null hypothesis.

######  Lemma (Lemma 16).

Under the null hypothesis where
$\mathbf{\Omega}^{\gamma}=\frac{1}{\left(\sigma^{\gamma}\right)^{2}}\mathbf{I}_{d^{\gamma}_{\forall}}$,
we have the following:

$\displaystyle\mathbf{F}_{t_{i}t_{j}}$
$\displaystyle=\sum_{\gamma}\frac{d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2}\left(\mathbf{a}^{\gamma}\mathbf{a}^{\gamma,T}\right)_{ij}$
$\displaystyle\mathbf{F}_{t_{i}\psi_{\ell_{jk}}}$
$\displaystyle=\sum_{\gamma|\ell\in\gamma}\frac{a^{\gamma}_{i}d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2d_{\ell}}\delta_{jk}$
$\displaystyle\mathbf{F}_{\psi_{\ell^{1}_{i^{1}j^{1}}}\psi_{\ell^{2}_{i^{2}j^{2}}}}$
$\displaystyle=\frac{1}{2}\sum_{\gamma|\ell^{1},\ell^{2}\in\gamma}\left(\sigma^{\gamma}\right)^{4}\frac{d_{\forall}^{\gamma}}{d_{\ell^{1}}d_{\ell^{2}}}\delta_{i^{2}j^{2}}\delta_{i^{1}j^{1}}$
($\ell^{1}\neq\ell^{2}$)
$\displaystyle\mathbf{F}_{\psi_{\ell^{1}_{i^{1}j^{1}}}\psi_{\ell^{2}_{i^{2}j^{2}}}}$
$\displaystyle=\frac{1}{2}\sum_{\gamma|\ell^{1},\ell^{2}\in\gamma}\left(\sigma^{\gamma}\right)^{4}\frac{d^{\gamma}_{\forall}}{d_{\ell}}\left(\delta_{i^{1}j^{1}i^{2}j^{2}}+(1-\delta_{i^{1}j^{1}i^{2}j^{2}})\frac{\delta_{i^{1}i^{2}}\delta_{j^{1}j^{2}}}{2}\right)$
($\ell^{1}=\ell^{2}$)

###### Proof.

$\displaystyle\frac{\partial}{\partial t_{i}\partial t_{j}}\mathrm{NLL}$
$\displaystyle=\sum_{\gamma}\frac{a^{\gamma}_{i}a^{\gamma}_{j}}{2}\mathrm{tr}\left[\left(\mathbf{\Omega}^{\gamma}\right)^{-2}\right]$
$\displaystyle=\sum_{\gamma}\frac{a^{\gamma}_{i}a^{\gamma}_{j}d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2}$
$\displaystyle=\sum_{\gamma}\frac{d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2}\left(\mathbf{a}^{\gamma}\mathbf{a}^{\gamma,T}\right)_{ij}$
$\displaystyle\frac{\partial}{\partial
t_{i}\partial\psi_{\ell_{jk}}}\mathrm{NLL}$
$\displaystyle=\mathrm{sym}\left[\sum_{\gamma}\frac{a^{\gamma}_{i}}{2}\mathrm{tr}^{d^{\gamma}_{<\ell}}_{d^{\gamma}_{>\ell}}\left[\left(\mathbf{\Omega}^{\gamma}\right)^{-2}\right]_{jk}\right]$
$\displaystyle=\mathrm{sym}\left[\sum_{\gamma}\frac{a^{\gamma}_{i}d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2d_{\ell}}\left[\mathbf{I}_{d^{\gamma}_{\backslash\ell}}\right]_{jk}\right]$
$\displaystyle=\frac{1}{2}\sum_{\gamma}\frac{a^{\gamma}_{i}d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2d_{\ell}}\delta_{jk}+\frac{a^{\gamma}_{i}d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2d_{\ell}}\delta_{kj}$
$\displaystyle=\sum_{\gamma}\frac{a^{\gamma}_{i}d^{\gamma}_{\forall}\left(\sigma^{\gamma}\right)^{4}}{2d_{\ell}}\delta_{jk}$

$\displaystyle\frac{\partial^{\mathrm{nosym}}}{\partial\psi_{\ell^{1}_{i^{1}j^{1}}}\partial\psi_{\ell^{2}_{i^{2}j^{2}}}}\mathrm{NLL}$
$\displaystyle=\frac{1}{2}\sum_{\gamma|\ell^{1}\in\gamma}\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\left(\mathbf{\Omega}^{\gamma}\right)^{-1}\right]_{i^{1}j^{1}}$
$\displaystyle=\frac{1}{2}\sum_{\gamma|\ell^{1}\in\gamma}\left(\sigma^{\gamma}\right)^{4}\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\right]_{i^{1}j^{1}}$

Let us focus on what happens to the stridewise-blockwise trace term under
algebraic manipulations.

$\displaystyle\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}+\mathbf{J}^{j^{2}i^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\right]_{i^{1}j^{1}}$
$\displaystyle=\mathrm{tr}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\left(\mathbf{I}_{d^{\gamma}_{<\ell^{1}}}\otimes\mathbf{J}^{i^{1}j^{1}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{1}}}\right)\right]$

We now split the $(\psi_{\ell^{1}_{i^{1}j^{1}}},\psi_{\ell^{2}_{i^{2}j^{2}}})$
computation into two cases, one in which $\ell=\ell^{1}=\ell^{2}$ and another
where $\ell^{1}\neq\ell^{2}$. In both cases we use Lemma 24.

Case 1:  $\ell^{1}\neq\ell^{2}$

$\displaystyle\mathrm{tr}^{d^{\gamma}_{<\ell^{1}}}_{d^{\gamma}_{>\ell^{1}}}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell^{2}}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell^{2}}}\right)\right]_{i^{1}j^{1}}$
$\displaystyle=\frac{d_{\forall}^{\gamma}}{d_{\ell^{1}}d_{\ell^{2}}}\delta_{i^{2}j^{2}}\delta_{i^{1}j^{1}}$

Case 2:  $\ell=\ell^{1}=\ell^{2}$

$\displaystyle\mathrm{tr}^{d^{\gamma}_{<\ell}}_{d^{\gamma}_{>\ell}}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell}}\right)\right]_{i^{1}j^{1}}$
$\displaystyle=\mathrm{tr}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell}}\otimes\mathbf{J}^{i^{2}j^{2}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell}}\right)\left(\mathbf{I}_{d^{\gamma}_{<\ell}}\otimes\mathbf{J}^{i^{1}j^{1}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell}}\right)\right]$
$\displaystyle=\mathrm{tr}\left[\left(\mathbf{I}_{d^{\gamma}_{<\ell}}\otimes\mathbf{J}^{i^{2}j^{2}}\mathbf{J}^{i^{1}j^{1}}\otimes\mathbf{I}_{d^{\gamma}_{>\ell}}\right)\right]$
$\displaystyle=\frac{d^{\gamma}_{\forall}}{d_{\ell}}\left(\mathrm{tr}\left[\mathbf{j}^{i^{2}}\mathbf{j}^{j^{2},T}\mathbf{j}^{i^{1}}\mathbf{j}^{j^{1},T}\right]\right)$
$\displaystyle=\frac{d^{\gamma}_{\forall}}{d_{\ell}}\left(\mathrm{tr}\left[\mathbf{j}^{j^{2},T}\mathbf{j}^{i^{1}}\mathbf{j}^{j^{1},T}\mathbf{j}^{i^{2}}\right]\right)$
$\displaystyle=\frac{d^{\gamma}_{\forall}}{d_{\ell}}\left(\delta_{j^{2}i^{1}}\delta_{j^{1}i^{2}}\right)$

This is the nonsymmetric derivative. Symmetricizing requires replacing
$\delta_{j^{2}i^{1}}\delta_{j^{1}i^{2}}$ with the average over all symmetries,
swapping $i^{1}$ and $j^{1}$ and swapping $i^{2}$ and $j^{2}$. These
symmetries do not affect the different axis case, but do affect the same-axis
case.

One of the symmetries of $\delta_{j^{2}i^{1}}\delta_{j^{1}i^{2}}$ is
$\delta_{i^{1}i^{2}}\delta_{j^{1}j^{2}}$, which we will rewrite as
$\delta_{(i^{1},j^{1})(i^{2},j^{2})}$ to make it more clear what is being
compared.

Suppose $i^{1}=j^{1}$ is diagonal. Then this is only true for
$i^{1}=i^{2}=j^{1}=j^{2}$, which if true under one symmetry then it is true
under all symmetries.

Suppose now that neither are diagonal, and recall we required $i\leq j$. Then,
any symmetries that swap the ordering of $i^{1}$ and $j^{1}$ will never be
satisfied, unless they also swap $i^{2}$ and $j^{2}$. This lets us rewrite the
same-axis case as:

$\displaystyle\delta_{i^{1}j^{1}i^{2}j^{2}}+(1-\delta_{i^{1}j^{1}i^{2}j^{2}})\frac{\delta_{i^{1}i^{2}}\delta_{j^{1}j^{2}}}{2}$

This completes the proof.

∎

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