different orders. In short, our theoretical developments indicate that the
final expression for the spectral shape of the signal should reduce to a
regular function with may be expanded in $1/(\nu/\nu_{c})^{\kappa}$-dependent
series. In addition, the proposal defined in equation (119) contemplates the
condition that the maximum of the function corresponds to the value of the
characteristic frequency, $\nu/\nu_{c}=1$, as well as additionally makes it
possible to fit simulated data with the simple choice of the only formulation
parameter, $a$ (see for example [5], in which the choice of $a$ corresponds to
$2.8$ to fit simulated data to describe spectrum of the gravitational
radiation determined by numerical simulations using the ’envelope
approximation’ [78]).

##  VII Discussion of Results

There are numerous research works in the literature that address the topic of
quantum gravity in a non-commutative environment. However, with regard to the
description of the wave function of the Universe based on the formalism of
Hořava-Lifshitz quantum gravity and the Wheeler-DeWitt equation, or based on
other alternative approaches, the number of articles dealing with this topic
are not significant. Most of the published works deal with the temporal
evolution of the scale factor of the Universe, a topic of studies commonly
designated as “dynamical equations”. In some articles the authors deal with
both topics. We indicate a few references on these topics [79, 80, 81, 82, 83,
84, 85, 86, 87, 88, 89, 90]). We emphasize once again that concerning the wave
function of the Universe in a non-commutative environment, most authors, due
to the inherent computational difficulties imposed by the formalism, use
approximations that significantly limit their conclusions and direct their
studies towards formal aspects without a numerical approach or algebraic
results.

Regarding the dynamical equations, some studies involving non-commutative
formulations appear to be more improved but are still limited by significant
computational approximations. It is important to mention, although this is a
non-central topic in the present work, studies involving strings and non-
commutative gauge theories which have contributed significantly to a better
understanding of the influence of non-commutative algebra on the deformation
of geometric structures and the impacts of these studies in understanding the
accelerated evolution of our Universe in string theory. Edward Witten,
precursor of this line of investigation [91], was followed by many scientists
who have followed this line of investigation (see for instance [92, 93] and
references therein).

In our case, in describing the evolution of the Universe’s wave function, as
well as in the dynamic equations involving the Universe’s scale factor as well
as in relic gravitational waves, we focus on consistently numerically solving
the resulting equations without using, from the point from a computational
point of view, no approximation, through a numerical approach based on the
method known as Runge-Kutta-Fehlberg. This method proves to be quite powerful
for solving differential equations, without adopting any numerical
approximation. This computational approach made possible, despite the extreme
formal complexity of the theory and the resulting equations, a broad numerical
study of the evolutionary process focused on in the present work, through the
exploration of the theory’s parameter space. This study thus enabled a broad
study of the evolution of the functions $\Psi(\eta)$, $\Psi(\xi)$, $\eta(t)$,
and $\xi(t)$ and a wide class of solutions that for reasons of brevity we
limited their presentation to just a few figures. The most expressive results
involving both the contraction phase and the expansion phase of the branched
Universe indicate an accelerating cosmic expansion.

##  VIII Conclusions

The paradigm supporting the theory of renormalization groups -organizing
physical phenomena based on energy or distance scales- holds firm in
commutative quantum field theories. In their non-commutative counterparts,
however, one encounters an uncertain territory. One of the features of non-
commutative field theories is the mixing of short and long scales. A striking
illustration of this phenomenon can be found in UV/IR mixing [94]. In cases
where non-commutativity exists at a small scale, the UV/IR mixing effect is
anticipated to manifest at an earlier epoch in the Universe’s history, thereby
contributing to raising new questions about the hierarchy problem. Another
striking example pertains to the inhomogeneities within the distribution of
large-scale structures and anisotropies observed in the CMB radiation. These
anomalies bear traces of the non-commutative nature of the early Universe.
Specifically, the power spectrum of these structures becomes direction-
dependent in non-commutative spacetime [95].

In 1947, Hartland Sweet Snyder introduced the concept of quantizing spacetime
in a seminal paper [96]. While this paper has received relatively little
citation, it sparked one of the most remarkable inquiries in the realm of
physics, namely the possibility of discretizing spacetime. In alignment with
Snyder’s proposal, the uncertainty principle of Werner Heisenberg suggests
that spacetime possesses a non-commutative structure, which can be represented
as

$[x^{\mu},x^{\nu}]=i\theta^{\mu\nu}\,.$ (120)

This non-commutative property implies a minimum scale of approximately
$\sim\sqrt{\theta}$. From a cosmological perspective, to assess the
implications of this concept on the dynamics of the Universe, it is most
appealing to investigate how the non-commutativity of coordinates affects the
deformation of spacetime algebra. We assume that this equation holds within a
comoving frame, a coordinate system in which galaxies are freely falling.

In this work we adopt an alternative approach by considering a non-commutative
quantum cosmological scenario based on the deformation of a mini-superspace of
variables obeying Poisson algebra. We aim to examine whether such a
perspective can help identify the mechanism that drives the acceleration of
the Universe [97]. While the results presented here are preliminary, they are
promising in suggesting that non-commutative branch-cut gravity offers an
algebraic framework that impacts both the statistical distribution and
gravitational dynamics of the matter constituting the primordial Universe. As
previously emphasized, the implications of non-commutative algebra are evident
in the solutions presented in this study. These implications manifest in
various aspects, including the wave function of the early Universe, the
dynamical equations involving the cosmic scale factor and its dual
counterpart, as well as relic gravitational waves. The results point to a
dynamic acceleration driven by a force that in our view, arises from the
reconfiguration of matter in the early Universe due to the algebraic structure
of non-commutative geometry. This structure captures the intrinsic properties
of spacetime at short distances, with significant implications for the
dynamical symmetries and conventional duality symmetries of quantum spacetime
geometry.

A seminal aspect of our study involves the propagation of relic gravitational
waves in the stages before and after the transition involving our Universe and
its mirror counterpart. In our study, we have identified a similar behavior
involving relic gravitational waves and the wave function of the Universe,
with respect to a topological quantum leap in the branch-cut transition
region, addressed in other studies [16, 8, 9]. Based on Bekenstein’s universal
upper bound of a localized quantum system that establishes that

$S\leq\frac{2\pi k_{B}RE}{\hbar c}\,,.$ (121)

with $E$ representing the total energy of a system enclosed in a circumference
with surface area $A$ and radius $R$, we thus developed a Generalized
Heisenberg Uncertainty Principle (GHUP) version for the Bekenstein’s criterion
[9], establishing a point of contact with Snider’s predictions

$\Delta S\Delta E\lesssim\frac{\hbar c}{2}\frac{\pi
k_{B}\bigl{(}1+\theta\bigr{)}}{\sqrt{\theta}\ell_{P}}\,,$ (122)

in which $\sqrt{\theta}$ represents the minimum scale in the non-commutative
space-time algebra, so the minimum observable GHUP length is assumed as
$\Delta x\sim\sqrt{\theta}\ell_{P}$.

The relic cosmological stochastic background, as expected, is characterized by
stability, isotropy, absence of polarization, whose origin are fundamental
processes that supposedly occurred in the early Universe [5]. Among these
processes, quantum vacuum fluctuations, cosmic phase transitions and cosmic
strings stand out [73]. And insofar, as pointed out earlier, beyond redshift
$z\sim 1100$, the early Universe is opaque to electromagnetic radiation, these
relic gravitational waves are the only capable messengers of the prevailing
physical conditions of the early Universe.

In this contribution, the results obtained for the description of
gravitational waves bring a new perspective, in line with the implications of
equation (122) which indicate an uncertainty around the previous boundary
conditions, as a result of a minimum scale. And also in line with the
implications of the Bekenstein criterion, indicating a quantum leap in the
boundary of the branch-cut transition region. These results also indicate, in
view of the mixture of intensities of the original sources of relic
gravitational waves identified by the distribution of colors associated with
the quantities that describe the corresponding power spectra and density
parameters, two fundamental aspects. The first concerns the primordial vision
of a smooth and isotropic Universe, symmetrical to ours. In this description,
the stress-energy must take the form of a perfect fluid fully described by an
energy density $\rho_{\epsilon}$ and pressure p, in which the variations of
the internal equation of state are so small that they make no qualitative
difference in the cosmic evolutionary process. This premise seems to collapse
when we examine the results obtained in this contribution, which indicate
asymmetric Universes, whether in the distribution of intensities of
gravitational wave sources, or in the implications of non-symmetric algebra in
the short-distance structure of spacetime. The second aspect concerns the
transition region between the different phases of the Universe, bringing to
light the discussion about the possibility of observing signatures not only
corresponding to the era marked by the dissociation of matter and radiation in
our Universe, but also the phases associated with the cosmic seeds, sources of
gravitational waves, existing in the mirror Universe.

In conclusion, the results suggesting a primordial dynamic acceleration of
spacetime demonstrate that non-commutative quantum branch-cut gravity provides
a viable theoretical alternative to models such as inflation [6, 7] and
bouncing [35, 36, 37]. This exploration aligns with a fundamental
characteristic of non-commutative algebra, namely the interplay between small
and large scales. As a result, if the effects of non-commutative algebra were
indeed present in the primordial Universe, it is reasonable to anticipate
their persistence in the present day.

A fundamental question then emerges: Can inhomogeneities in the distribution
of large-scale structures and anisotropies in the stochastic gravitational
wave background (SGWB), if they indeed exist, carry traces of the non-
commutative nature of the early Universe? Our results indicate a scenario
during the early stages of the Universe, characterized by an SGWB distribution
that deviates significantly from the homogeneity expected to be observed
today.

As a consequence, inhomogeneities in the SGWB distribution, if influenced by
traces of non-commutativity, could serve as crucial windows into the initial
phases of the early Universe, preceding the recombination era. However,
answering this question necessitates further observations, and this theme will
remain the primary focus of our ongoing investigations.

A final word regarding the results obtained to describe relic gravitational
waves in a non-commutative formulation: the regular function that describes
the spectral shape of the signal is the result, among many other functional
possibilities, of the Lagrangian structure of the branched quantum gravity
formulation. proposal. There are still aspects to be revealed in the future,
in order to introduce a functional formulation, albeit regular, that
highlights the behavior of the spectral form of the signal parametrically
dependent on the non-commutative structure. This aspect, although in our
formulation it is implicitly contained in the functional structure of the
expression (119), deserves a consistent specific approach.

##  IX Acknowledgements

P.O.H. acknowledges financial support from PAPIIT-DGAPA (IN116824). F.W. is
supported by the U.S. National Science Foundation under Grant PHY-2012152.

## Appendix A

In what follows, we will elaborate on how to find a transformation from non-
commuting coordinates to commuting ones. We assume that $u$ and $v$ still
commute, i.e., $\sigma=0$.

Let us define the commuting and non-commuting coordinates respectively as

$({\tilde{x}}_{i})=({\tilde{u}},\tilde{p}_{u},{\tilde{v}},{\tilde{p}}_{v})\quad\mbox{and}\quad(x_{i})=(u,p_{u},v,p_{v})~{}.$
(123)

The commuting and non-commuting coordinates satisfy both the respective
Poisson-brackets

$\left\\{{\tilde{x}}_{i},{\tilde{x}}_{j}\right\\}={\tilde{g}}_{ij}\quad\mbox{and}\quad\left\\{x_{i},x_{j}\right\\}=g_{ij}\,,$
(124)

where on the right side of these equations we have the symplectic metrics,
satisfying respectively

${\tilde{g}}_{ji}=-{\tilde{g}}_{ij}\quad\mbox{and}\quad g_{ji}=-g_{ij}~{}.$
(125)

This symmetry property is the reason why we can call it a symplectic space.
The matrix structure of these metrics is

$\displaystyle\left({\tilde{g}}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right)\,;$ (130) $\displaystyle\left(g\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&1&0&\gamma\\\
-1&0&-\chi&\alpha\\\ 0&\chi&0&1\\\ -\gamma&-\alpha&-1&0\end{array}\right)~{}.$
(135)

Using (124), this leads to the non-zero Poisson brackets (only the Poisson
brackets are listed which will be non-zero in the non-commuting case)

$\displaystyle\left\\{{\tilde{u}},{\tilde{p}}_{u}\right\\}$ $\displaystyle=$
$\displaystyle
1\,;\quad\left\\{{\tilde{v}},{\tilde{p}}_{v}\right\\}=1\,;\quad\mbox{and}\quad\left\\{{\tilde{u}},{\tilde{p}}_{v}\right\\}=0\,;$
$\displaystyle\left\\{{\tilde{v}},{\tilde{p}}_{u}\right\\}$ $\displaystyle=$
$\displaystyle
0\,;\quad\mbox{and}\quad\left\\{{\tilde{p}}_{u},{\tilde{p}}_{v}\right\\}=0\,;$
(136)

and for the non-commuting case

$\displaystyle\left\\{u,p_{u}\right\\}$ $\displaystyle=$ $\displaystyle
1\,;\quad\left\\{v,p_{v}\right\\}=1,;\quad\mbox{and}\quad\left\\{u,p_{v}\right\\}=\gamma\,;$
$\displaystyle\left\\{v,p_{u}\right\\}$ $\displaystyle=$
$\displaystyle\chi\,;\quad\mbox{and}\quad\left\\{p_{u},p_{v}\right\\}=\alpha~{}.$
(137)

Now, we look for a transformation

$\displaystyle{\tilde{x}}_{i}$ $\displaystyle=$
$\displaystyle\sum_{j}M_{ij}x_{j}\,,$ (138)

such that the above Poisson brackets are satisfied. This gives us conditions
for the matrix elements $M_{ij}$. Equation (138) can be cast into the form

$\displaystyle{\tilde{u}}$ $\displaystyle=$ $\displaystyle
M_{11}u+M_{12}p_{u}+M_{13}v+M_{14}p_{v}\,;$ $\displaystyle{\tilde{p}}_{u}$
$\displaystyle=$ $\displaystyle M_{21}u+M_{22}p_{u}+M_{23}v+M_{24}p_{v}\,;$
$\displaystyle{\tilde{v}}$ $\displaystyle=$ $\displaystyle
M_{31}u+M_{32}p_{u}+M_{33}v+M_{34}p_{v}\,;$ $\displaystyle{\tilde{p}}_{v}$
$\displaystyle=$ $\displaystyle
M_{41}u+M_{42}p_{u}+M_{43}v+M_{44}p_{v}\,;~{}.$ (139)

Using that $\tilde{u}=u$ and $\tilde{v}=v$, we can reduce the matrix $M$ to
the expression

$\displaystyle\left(M\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\
M_{21}&M_{22}&M_{23}&M_{24}\\\ 0&0&1&0\\\
M_{41}&M_{42}&M_{43}&M_{44}\end{array}\right)\,.$ (144)

Using (137), and the transformation (138) with (144), we obtain

$\displaystyle\left\\{{\tilde{u}},{\tilde{p}}_{u}\right\\}$ $\displaystyle=$
$\displaystyle 1=M_{22}+M_{24}\gamma\,;$
$\displaystyle\left\\{{\tilde{v}},{\tilde{p}}_{v}\right\\}$ $\displaystyle=$
$\displaystyle 1=M_{42}\chi+M_{44}\,;$
$\displaystyle\left\\{{\tilde{u}},{\tilde{p}}_{v}\right\\}$ $\displaystyle=$
$\displaystyle 0=M_{42}+M_{44}\gamma\,;$
$\displaystyle\left\\{{\tilde{v}},{\tilde{p}}_{u}\right\\}$ $\displaystyle=$
$\displaystyle 0=M_{22}\chi+M_{24}~{},$ (145)

where the Poisson-bracket $\left\\{\tilde{p}_{u},\tilde{p}_{v}\right\\}=0$
will be calculated later. First of all we resolve the set of equations (145),
which leads to

$\displaystyle M_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\gamma\chi}\,;\quad\quad
M_{24}=-\frac{\chi}{1-\gamma\chi}\,;$ $\displaystyle M_{42}$ $\displaystyle=$
$\displaystyle-\frac{\gamma}{1-\gamma\chi}\,;\quad
M_{44}=\frac{1}{1-\gamma\chi}~{},$ (146)

which leads consequently to the matrix

$\displaystyle\left(M\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\
M_{21}&\frac{1}{1-\gamma\chi}&M_{23}&-\frac{\chi}{1-\gamma\chi}\\\ 0&0&1&0\\\
M_{41}&-\frac{\gamma}{1-\gamma\chi}&M_{43}&\frac{1}{1-\gamma\chi}\end{array}\right)~{}.$
(151)

An additional information we get when calculating the missing commutator:

$\left\\{{\tilde{p}}_{u},{\tilde{p}}_{v}\right\\}=0=-M_{41}+M_{23}+\frac{\alpha}{a-\gamma\chi}~{},$
(152)

which leads to

$\displaystyle M_{41}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{1-\gamma\chi}+M_{23}~{}~{}~{}.$ (153)

So, finally we obtain the structure of the transformation matrix, with still
some liberty because $M_{21}$, $M_{23}$ and $M_{43}$ are free to choose! That
the momenta now commute is important, because this allows us to write these
momenta proportional to derivatives of the conjugate variable.

With the last form of the transformation matrix and an appropriate choice of
the remaining matrix elements, we obtain for the transformation of the
momenta:

$\displaystyle{\tilde{p}}_{u}$ $\displaystyle=$ $\displaystyle
M_{21}u+\frac{1}{1-\gamma\chi}p_{u}+M_{23}v-\frac{\chi}{1-\gamma\chi}p_{v}$
$\displaystyle{\tilde{p}}_{v}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha}{1-\gamma\chi}+M_{23}\right)u-\frac{\gamma}{1-\gamma\chi}p_{u}$
(154) $\displaystyle+$ $\displaystyle
M_{43}v+\frac{1}{1-\gamma\chi}p_{v}~{}~{}~{}.$

We also define $\Gamma=(\gamma\chi-1)$.

There is still an ambiguity, which we call gauge transformation, to choose the
remaining matrix elements. There are two possible paths: The first is to
select a particular choice for ${\tilde{p}}_{u}$ and ${\tilde{p}}_{v}$ in
terms of the non-commuting variables. The second one is to invert the matrix
and select a particular choice for the non-commuting momenta $p_{u}$ and
$p_{v}$ in terms of the commuting ones. We chose the second path and the
particular choice is listed in the main text. The choice then gives conditions
to the remaining matrix elements, which can be resolved.

The present deduction also implies that there are several options in choosing
${\tilde{p}}_{u}$ and ${\tilde{p}}_{v}$, depending on the particular choice of
$M_{21}$, $M_{23}$ and $M_{43}$. Still, for each alternative one has to verify
the relations of the Poisson brackets.

## Appendix B

In our calculations for the generation of gravitational waves as a result of
bubble collisions, we use $\alpha_{\infty}\simeq 2.71\times 10^{-2}$, and
assume $u_{\omega}\to 1$ (runaway regime) [5]. The parameter $\kappa_{\infty}$
holds a crucial role in our analysis, as it quantifies the efficiency of
converting latent heat into bulk motion, a pivotal factor in defining the
amplitude of GW signals. In reference to [98], $\kappa_{\infty}$ is
approximately given as $\kappa_{\infty}\approx 3.516\times 10^{-2}$. When
considering the parameter $\alpha_{T}$, from [5] we use $\alpha_{T}\approx
3.68\times 10^{-2}(100/g_{*}).$ Using the above values for $\kappa_{\infty}$
and $\alpha_{T}$, we obtain

$\Biggl{(}\frac{\alpha_{T}\,\kappa_{\infty}}{1+\alpha_{T}}\Biggr{)}^{2}\approx
1.557411\times 10^{-6}\,,$ (155)

with $g_{*}$ is given as [99]

$g_{*}=g_{*}^{\rm{[SM]}}=100,\quad\mbox{and
thus}\quad\Biggl{(}\frac{100}{g_{*}}\Biggr{)}^{1/3}=1\,.$ (156)

Assuming the runaway regime ($u_{\omega}\to 1$), we get

$\Bigl{(}\frac{0.11u^{3}_{\omega}}{0.42+u^{2}_{\omega}}\Bigr{)}\approx
7.746479\times 10^{-2}\,,$ (157)

and

$\Bigl{(}\frac{0.62}{1.8+u^{2}_{\omega}-0.1n_{\omega}}\Bigr{)}\approx
2.29629\times 10^{-1}\,.$ (158)

In a first order phase transition, the bubble nucleation process is fixed by
the tunnelling probability between the two vacua states of the effective
potential (for the details see [5])

$V(T,\phi)=\frac{\gamma}{2}\Bigl{(}T^{2}-T_{0}^{2}\Bigr{)}\phi^{2}-\frac{\sigma}{3}T\phi^{3}+\frac{\lambda}{4}\phi^{4}\,.$
(159)

The usual standard model potential does not generates a strong transition
required to produce a significant background. In general, the potential must
be modified and a minimum change implies in additional gauge bosons (at least
two new ones), what in practice means to modify the $\phi T^{3}$ term. The
solution for the vacuum states permits the evaluation of the nucleation
temperature, that is of the order of $T_{*}=166$ GeV [5]. During the EW phase
transition, a fraction of the latent heat is used to excite sound waves,
turbulence, and the bulk motion of bubbles, which are able to generate
gravitational waves. Thus, the physical conditions of transition must be used
for all mechanisms, which are not independent. Once the nucleation temperature
$T_{*}$ is computed (or fixed), the duration of the transition can be
estimated from [5]

$\Biggl{(}\frac{\beta}{H_{*}}\Biggr{)}\simeq
4\,\ln\Biggl{(}\frac{M_{P}}{T_{*}}\Biggr{)}\,.$ (160)

However, fixing the nucleation temperature to be $T_{*}=166\,GeV$, the
relation above implies $(\beta/H_{*})\approx 155.$

## Appendix C

In this appendix we briefly describe the steps to obtain the gravitational
waves spectra described by the spectral shape of the signal, considering the
sub-horizon condition with a matter-energy source given by equation (115). To
solve this equation, due to the technical difficulties of its resolution, as
we stressed before, we used powerful computational algorithms, based on the
Runge-Kutta-Fehlberg method, that made it possible to obtain algebraic
solutions without the need to implement simplifications or any computational
approximation. The corresponding solution of the field equation that describes
the evolution over time of the metric disturbance having as its source the
potential defined in equation (93), that configures the dynamic composition of
matter in the primordial cosmic period, is

$\displaystyle{h}(k,\tau)$ $\displaystyle=$ $\displaystyle
c_{2}(\eta)\sin(k\tau)+c_{1}(\eta)\cos(k\tau)+\frac{1}{k^{4}\eta^{3}}\Bigl{[}k^{2}\Bigl{(}6.28319\tau^{2}\eta^{2}+\tau\bigl{(}-4.18879\eta^{5/2}+16.7384\eta^{6}-23.4572\eta^{5}$
(161) $\displaystyle-$ $\displaystyle
16.7552\eta^{4}-6.70206\eta^{2}-0.150796\bigr{)}+50.2655\eta^{7/2}\Bigr{)}-12.5664\eta^{2}\Bigr{]}\,.$

$S_{b}(\nu/\nu_{c})$, the spectral shape of the signal, is determined in terms
of the solution (161) of equation (115), after integrating out the time
dependence of $S_{b}(\nu/\nu_{c},\tau)$, taking the limits $0$ to
$\beta/H_{*}$, with $k\to\nu/\nu_{c}$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{b}(\nu/\nu_{c})$
$\displaystyle=$ $\displaystyle{\cal
A}(\nu/\nu_{c})\int_{0}^{\beta/H_{*}}\\!\\!\\!\\!\\!S_{b}(\nu/\nu_{c},\tau)d\tau$
(162) $\displaystyle=$ $\displaystyle{\cal
A}(\nu/\nu_{c})\int_{0}^{\beta/H_{*}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{1}{4\pi(\nu/\nu_{c})}\frac{h^{2}(\nu/\nu_{c},\tau)}{\eta^{2}(\tau)}d\tau\,,$

where the regulatory function ${\cal A}(\nu/\nu_{c})$ guarantees the
normalization of the spectral shape, the boundary condition
$S_{n}(\nu/\nu_{c})\to 0$, as well as the position of the maximum of the
function at the point $\nu/\nu_{c}=1$. The final expression for the spectral
shape of the signal is quite intricate and difficult to manipulate. As a
preliminary study, in the following we insert a generic scale power law
behavior for the branch-cut scale factor, $\eta(\tau)=a_{n}\tau^{n}$, which
covers the cases of radiation ($n=1$) and matter ($n=2$) domination, as well
as the de Sitter inflation ($n=-1$) to determine the main ingredients of the
time-independent spectral shape of the signal, expressed as
$S_{b}(\nu/\nu_{c})$. Taking for simplicity the case $n=1$, we show below a
series expansion of the resulting integral, limiting for convergence reasons
the final expression to the $(\nu/\nu_{c})^{6}$\- and
$(\beta/H_{*})^{6}$-dependent terms. The mean ingredients of this calculation,
except for determining the regulatory function ${\cal A}(\nu/\nu_{c})$, are
synthesized in the following expression:

$\displaystyle S_{n}(\nu/\nu_{c})$ $\displaystyle\approx$
$\displaystyle\Bigl{(}\sin\Bigl{(}k\frac{\beta}{H_{*}}\Bigr{)}+\cos\Bigl{(}k\frac{\beta}{H_{*}}\Bigr{)}\Bigr{)}\Biggl{(}0.0795775\Bigl{(}\frac{\nu_{c}}{\nu}\frac{H_{*}}{\beta}\Bigr{)}-0.159155\Bigl{(}\frac{\nu_{c}}{\nu}\frac{H_{*}}{\beta}\Bigr{)}^{\\!\\!3}+1.90986\Bigl{(}\frac{\nu_{c}}{\nu}\frac{H_{*}}{\beta}\Bigr{)}^{\\!\\!5}\\!\Biggr{)}+{\cal
O}\Bigl{(}\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}^{6}\Bigr{)}$ (163)
$\displaystyle+$
$\displaystyle\Bigl{(}\frac{\nu_{c}}{\nu}\Bigr{)}^{5}\Biggl{(}3.15843\Bigl{(}\frac{\beta}{H_{*}}\Bigr{)}+92.4467\Bigl{(}\frac{\beta}{H_{*}}\Bigr{)}^{1/2}+169.119log\Bigl{(}\frac{\beta}{H_{*}}\Bigr{)}+0.150401\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}+0.733704\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}^{3/2}$
$\displaystyle-$ $\displaystyle
0.000596831\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}^{3}\Biggr{)}+0.0795775log\Bigl{(}\frac{\nu_{c}}{\nu}\Bigr{)}-0.0397887log\Bigl{(}\Bigl{(}\frac{\nu_{c}}{\nu}\Bigr{)}^{2}\Bigr{)}+0.125+{\cal
O}\Bigl{(}\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}^{11/2}\Bigr{)}$ $\displaystyle+$
$\displaystyle\Bigl{(}\sin\Bigl{(}k\frac{\beta}{H_{*}}\Bigr{)}-\cos\Bigl{(}k\frac{\beta}{H_{*}}\Bigr{)}\Bigr{)}\Biggl{(}0.0795775\Bigl{(}\frac{\nu_{c}}{\nu}\frac{H_{*}}{\beta}\Bigr{)}^{2}-0.477465\Bigl{(}\frac{\nu_{c}}{\nu}\frac{H_{*}}{\beta}\Bigr{)}^{4}+{\cal
O}\Bigl{(}\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}^{6}\Biggr{)}$ $\displaystyle+$
$\displaystyle\Bigl{(}\sin\Bigl{(}k\frac{\beta}{H_{*}}\Bigr{)}+\cos\Bigl{(}k\frac{\beta}{H_{*}}\Bigr{)}\Bigr{)}\Biggl{(}-0.0795775\Bigl{(}\frac{\nu_{c}}{\nu}\frac{H_{*}}{\beta}\Bigr{)}+{\cal
O}\Bigl{(}\Bigl{(}\frac{H_{*}}{\beta}\Bigr{)}^{6}\Biggr{)}\Biggr{)}\,.$

## Author Contributions

Conceptualization, C.A.Z.V.; methodology, C.A.Z.V. and B.A.L.B. and P.O.H and
J.A.deF.P. and D.H. and F.W. and M.M.; software, C.A.Z.V. and B.A.L.B. and
M.R. and M.M.; validation, C.A.Z.V. and B.A.L.B. and D.H. and P.O.H. and
J.A.deF.P. and F.W.; formal analysis, C.A.Z.V. and B.A.L.B. and P.O.H. and
J.A.deF.P. and D.H. and F.W.; investigation, C.A.Z.V. and B.A.L.B. and P.O.H.
and J.A.deF.P. and M.R. and M.M. and F.W.; resources, C.A.Z.V.; data curation,
C.A.Z.V. and B.A.L.B.; writing—original draft preparation, C.A.Z.V.;
writing—review and editing, C.A.Z.V. and B.A.L.B. and P.O.H. and J.A.deF.P.
and D.H. and M.R. and M.M. and F.W.; visualization, C.A.Z.V. and B.A.L.B.;
supervision, C.A.Z.V.; project administration, C.A.Z.V.; funding acquisition
(no funding acquisition). All authors have read and agreed to the published
version of the manuscript.

## References

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