# Observational Imprints of Our Lost Twin Anti-Universe

Samuel Barroso Bellido  Main author<EMAIL_ADDRESS>Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451
Szczecin, Poland  Mariusz P. Da̧browski<EMAIL_ADDRESS>Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451
Szczecin, Poland  National Centre for Nuclear Research, Andrzeja Sołtana 7,
05-400 Otwock, Poland  Copernicus Center for Interdisciplinary Studies,
Szczepańska 1/5, 31-011 Kraków, Poland

###### Abstract

We consider observational consequences of the entanglement between our
universe and a hypothetical twin anti-universe in the third quantization
scheme of the canonical quantum gravity. Based on our previous investigations
we select some special form of the interuniversal interaction which allows the
entanglement entropy of the pair of universes to diverge at some critical
points of their classical evolution. We find that the modification of the
cosmic microwave background (CMB) power spectrum due to the entanglement with
our twin anti-universe is enlarged for small modes $k$ and small multipole
numbers $l$ with the Planck constraint onto the interaction coupling constant
$\lambda_{o}\lesssim\mathcal{O}(10^{-56})$. Some other coupling functions
which allow more critical points are also briefly commented on in the context
of their observational effect on CMB and other observations.

##  I Introduction

Year by year, the multiverse scenario reaches more and more important status
in theoretical physics Langhoff  _et al._ (2021); Aguilar-Gutierrez  _et al._
(2021); Alonso-Serrano and Jannes (2019); Da̧browski (2019); Linde and
Vanchurin (2010); Hartle (2021); Garriga  _et al._ (2016); Mersini-Houghton
(2017); Vilenkin (2014); Balcerzak and Lisaj (2021). The question whether the
multiverse exists or not is still open and under debate. In Ref. Bouhmadi-
López  _et al._ (2019), a level III multiverse scenario Tegmark (2004) is put
to the test for which the imprints on the Cosmic Microwave Background (CMB)
spectrum are found. It seems then appealing to apply its strategy as viable
and proceed to find the observable signals from the CMB for other scenarios.

The method relies on the third quantization scheme Caderni and Martellini
(1984); McGuigan (1988); Giddings and Strominger (1989); Robles-Pérez (2021)
of canonical quantum gravity DeWitt (1967); Kiefer (2007) which now is widely
considered the field theory of universes over the multiverse, that, assuming
they are Friedmann-Lemaitre-Robertson-Walker (FLRW) universes, each universe
is described by its wave function $\Psi(a,\\{\phi\\})$, where $a$ is the scale
factor and $\\{\phi\\}$ the matter content. Such wave function fulfils its own
renowned Wheeler-DeWitt (WDW) equation $\mathcal{H}\Psi=0$. In that kind of
the multiverse, the individual universes interact as particles of any other
field theory. In particular, they follow the most natural process of quantum
field theory which is the pair creation of universes Robles-Pérez and
González-Díaz (2010); Robles-Pérez (2020) ”from nothing” as an analogy with
particle-antiparticle pair creation Schwinger (1951). Unlike in Ref. Bouhmadi-
López  _et al._ (2019), where the number of interacting universes is arbitrary
and large, we will just consider entangled and interacting pair of universes
with no other interactions or environment.

Besides, a detail arose when the entanglement entropy of such pair of
universes was calculated Robles-Pérez  _et al._ (2017); Balcerzak  _et al._
(2021); Bellido (2021). It was found to be divergent close to any critical
point along the classical evolution of the universes, or equivalently, when
the Hubble parameter $H$ vanishes (turning points of possible cyclic evolution
- cf. Ref. Kiefer and Zeh (1995); Da̧browski and Larsen (1995)). As we know
from quantum mechanics, the Hamiltonian description of such a bipartite system
with a certain interaction between them can be written as

$\mathcal{H}=\mathcal{H}_{1}+\mathcal{H}_{2}+\mathcal{H}_{\text{int}},$ (1)

where $\mathcal{H}_{i}$ is the free Hamiltonian of each subsystem, and
$\mathcal{H}_{\text{int}}$ is the contribution to the Hamiltonian by their
interaction. Since the entanglement entropy diverges when the Hubble parameter
vanishes, the interaction must also diverge. One is then compelled to write
the Hamiltonian of interaction as the function of the Hubble parameter and not
solely as the function of the scale factor only $a$, as it was considered in
Ref. Bouhmadi-López  _et al._ (2019).

In order to find the Hamiltonian of the third quantized theory, we need to
find the action for which each universe fulfils its WDW equation and interacts
with the other. In order to do so, we find the individual actions, sum them
up, and include the yet unknown interaction term. Our proposal is to include
rather general form for the interaction Hamiltonian based on the nearest
neighbour interaction process

$\mathcal{H}_{\text{int}}(a,H)=-a\lambda^{2}(a,H)(\Psi_{2}-\Psi_{1})^{\ast}(\Psi_{2}-\Psi_{1}),$
(2)

where $\lambda^{2}(a,H)$ is a coupling function whose constraints are to be
based on two conditions. Firstly, this function must depend on the Hubble
parameter in such a way that the coupling diverges when the Hubble parameter
vanishes, i.e.

$\lim_{H\to 0}\lambda(a,H)\to\infty,$ (3)

and secondly (cf. Ref. Robles-Pérez  _et al._ (2016)), the probability of the
false vacuum to decay which is computed for such kind of interaction

$\mathcal{P}\sim\exp\left\\{-\left[\frac{a^{4}}{\lambda^{2}(a,H)}\right]^{3}\right\\},$
(4)

is suppressed at large values of the scale factor $a\to\infty$.

The development to find the Wheeler-DeWitt equation and the semiclassical
Friedmann equation which corresponds to that model is the purpose of Section
II. Using the new dynamics, we find the spectra of the CMB in Section III. A
comment of the choice of a certain coupling function
$\lambda^{2}(\alpha,\phi)$ is given in Section IV. The conclusion follows in
Section V.

##  II Third Quantized Wheeler-DeWitt Equation and the Semiclassical Friedmann
Equation

Considering a pair of flat universes containing a single homogeneous scalar
field $\phi$, the mode expansion of its perturbations $v_{k}$, and the
perturbations of the metric, the Lagrangian of each universe of the pair can
be expressed, in terms of the conformal time $\text{d}\eta=\text{d}t/a(t)$,
like Bouhmadi-López  _et al._ (2019); Morais  _et al._ (2018)

$\mathcal{L}=\frac{1}{2}\left[-(a^{\prime})^{2}+a^{2}(\phi^{\prime})^{2}-2a^{4}V(\phi)+\sum_{k}(v_{k}^{\prime}{v_{k}^{\ast}}^{\prime}+\omega^{2}_{k}v_{k}v^{\ast}_{k})\right],$
(5)

where the prime denotes the derivative with respect to $\eta$, we have assumed
$3/4\pi G=1$, $G$ is the gravitational constant, $V(\phi)$ is the potential of
the scalar field, and the frequency is given by

$\omega^{2}_{k}=k^{2}-\frac{z^{\prime\prime}}{z},\qquad\qquad
z=a\frac{\phi^{\prime}}{\mathfrak{H}},\qquad\qquad\mathfrak{H}=\frac{a^{\prime}}{a},$
(6)

where $z$ is called the Mukhanov-Sasaki variable, and the perturbations
$v_{k}$ are the solutions to the Mukhanov-Sasaki equation

$v^{\prime\prime}_{k}+\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0.$
(7)

The conjugated momenta are found from Eq. (5) to be

$p_{a}=-a^{\prime},\qquad\qquad p_{\phi}=a^{2}\phi^{\prime},\qquad\qquad
p_{v_{k}}={v_{k}^{\ast}}^{\prime},$ (8)

and a Legendre transformation yields the Hamiltonian

$\mathcal{H}=\frac{1}{2}\left[-p_{a}^{2}+2a^{4}\rho(\phi)+\sum_{k}\left(p_{v_{k}}p_{v^{\ast}_{k}}-\omega^{2}_{k}v_{k}v_{k}^{\ast}\right)\right],$
(9)

where we kept the scalar field as classical through the relation Weinberg
(2008)

$\rho(\phi)=\frac{1}{2}\frac{(\phi^{\prime})^{2}}{a^{2}}+V(\phi).$ (10)

In order to find the Wheeler-DeWitt equation $\mathcal{H}\Psi=0$ of a single
universe, where $\Psi(a,\\{v_{k}\\})$ is the wave function of each one, we now
quantize the momenta, for a certain factor ordering, as

$p_{a}^{2}=-\frac{1}{a}\frac{\partial}{\partial
a}\left(a\frac{\partial}{\partial a}\right),\qquad\qquad
p_{v_{k}}=-\text{i}\frac{\partial}{\partial v_{k}},$ (11)

and replace them into the Hamiltonian (9), so that the kinetic part could be
expressed covariantly as the Laplace-Beltrami operator. Thus, one finds the
Wheeler-DeWitt equation

$\left[\frac{1}{a}\frac{\partial}{\partial a}\left(a\frac{\partial}{\partial
a}\right)+2a^{4}\rho(\phi)+\sum_{k}\left(-\frac{\partial^{2}}{\partial
v_{k}^{2}}+\omega^{2}_{k}v_{k}^{2}\right)\right]\Psi(a,\\{v_{k}\\})=0,$ (12)

where $\Psi(a,\\{v_{k}\\})$ is the wave function of the universe, which is a
function on the scale factor and the modes of the perturbations.

In the third quantization picture, Eq. (12) is the equation of motion for a
certain field $\Psi^{\ast}$, meaning that the universe described by $\Psi$ in
Eq. (12) is just an excitation of a field of universes. The action from where
we recover it, reads

$S_{3Q}=\int\text{d}a\prod_{k}\text{d}v_{k}a\left[-\frac{\partial\Psi^{\ast}}{\partial
a}\frac{\partial\Psi}{\partial
a}+2a^{4}\rho(\phi)\Psi^{\ast}\Psi+\sum_{k}\left(\frac{\partial\Psi^{\ast}}{\partial
v_{k}}\frac{\partial\Psi}{\partial
v_{k}}+\omega^{2}_{k}v_{k}^{2}\Psi^{\ast}\Psi\right)\right].$ (13)

Identifying the momentum of the universe as

$P_{\Psi}=-a\frac{\partial\Psi}{\partial a},$ (14)

we find the third quantized Hamiltonian

$\mathcal{H}_{3Q}=a\left[-\frac{1}{a^{2}}P_{\Psi}P_{\Psi^{\ast}}-2a^{4}\rho(\phi)\Psi^{\ast}\Psi-\sum_{k}\left(\frac{\partial\Psi^{\ast}}{\partial
v_{k}}\frac{\partial\Psi}{\partial
v_{k}}+\omega^{2}_{k}v_{k}^{2}\Psi^{\ast}\Psi\right)\right],$ (15)

which accounts for one of the universes of the bipartite system of two
interacting universes we want to focus on. The whole system is described by a
Hamiltonian like the one in Eq. (1)

$\mathcal{H}_{\text{pair}}=\mathcal{H}_{3Q}^{1}+\mathcal{H}_{3Q}^{2}-a\lambda^{2}(a,H)(\Psi_{2}-\Psi_{1})^{\ast}(\Psi_{2}-\Psi_{1}),$
(16)

where the interaction term has been taken as in Eq. (2), controlled by a
coupling function $\lambda(a,H)$ which here is the clue to justify the
divergence of the entanglement entropy at the critical points of the classical
evolution as it was recently found in Ref. Balcerzak  _et al._ (2021), and to
maintain the vacuum stability.

To simplify a bit the calculations, we expand the wave functions in its
frequency modes like

$\Psi_{1,2}=\frac{1}{\sqrt{2}}\left(\widetilde{\Psi}_{2}\mp\widetilde{\Psi}_{1}\right),\qquad\qquad
P_{\Psi_{1,2}}=\frac{1}{\sqrt{2}}\left(\widetilde{P}_{\widetilde{\Psi}_{2}}\mp\widetilde{P}_{\widetilde{\Psi}_{1}}\right),$
(17)

since there are only two universes. Substituting (17) into (16), we get the
third quantized Hamiltonian of the system, which is

$\widetilde{\mathcal{H}}_{3Q}^{\text{pair}}=\widetilde{\mathcal{H}}_{3Q}^{1}+\widetilde{\mathcal{H}}_{3Q}^{2}-2a\lambda^{2}(a,H)\widetilde{\Psi}^{\ast}_{1}\widetilde{\Psi}_{1},$
(18)

where

$\widetilde{\mathcal{H}}_{3Q}^{l}=a\left[-\frac{1}{a^{2}}\widetilde{P}_{\widetilde{\Psi}_{l}}\widetilde{P}^{\ast}_{\widetilde{\Psi}_{l}}-2a^{4}\rho(\phi)\widetilde{\Psi}_{l}^{\ast}\widetilde{\Psi}_{l}-\sum_{k}\left(\frac{\partial\widetilde{\Psi}_{l}^{\ast}}{\partial
v_{k}}\frac{\partial\widetilde{\Psi}_{l}}{\partial
v_{k}}+\omega^{2}_{k}v_{k}^{2}\widetilde{\Psi}_{l}^{\ast}\widetilde{\Psi}_{l}\right)\right],\qquad
l=\\{1,2\\}.$ (19)

The equation of motion for $\widetilde{\Psi}_{l}^{\ast}$, which is the
Wheeler-DeWitt equation for $\widetilde{\Psi}_{l}$, after a parameterization
of the scale factor as $\alpha=\ln(a)$, is

$\left[\text{e}^{-2\alpha}\frac{\partial^{2}}{\partial\alpha^{2}}+2\text{e}^{4\alpha}\rho(\phi)-\frac{\partial^{2}}{\partial
v_{k}^{2}}+\omega^{2}_{k}v_{k}^{2}+2\lambda^{2}(a,H)\delta(l-1)\right]\widetilde{\Psi}_{l}(\alpha,\\{v_{k}\\})=0,$
(20)

where $\delta(l-1)$ is a Dirac delta function. Here, $\widetilde{\Psi}_{l}$ is
the wave function with mode $l$ of the bipartite system. From (20), we can see
that the mode $l=2$ does not feel the interaction at all, hence the dynamics
of the classical universe related to it is equivalent to a non-interacting
one. The question is in what combination of them is our universe, and this is
impossible to answer. We will, therefore, focus our attention over the mode
$l=1$ for which the dynamics differs, and from now on, no index will be used.

Using the semiclassical ansatz

$\widetilde{\Psi}(\alpha,\\{v_{k}\\})=\text{e}^{\text{i}S_{0}(\alpha)}\prod_{k}\psi_{k}(\alpha,\\{v_{k}\\}),$
(21)

we find the approximate relation111For the explicit calculation, see the
Appendix A of the analogous case in Ref. Bouhmadi-López  _et al._ (2019).

$-\text{e}^{-2\alpha}\left(\frac{\partial
S_{0}}{\partial\alpha}\right)^{2}=-2\text{e}^{4\alpha}\rho(\phi)-2\lambda^{2}(a,H).$
(22)

Recognizing (22) it as the Hamilton-Jacobi equation for the slowly moving
background, we use it together with

$\frac{\partial S_{0}}{\partial a}=p_{a}=-a\frac{\text{d}a}{\text{d}t},$ (23)

in order to obtain the Friedmann equation

$H^{2}=2\rho(\phi)+2\frac{\lambda^{2}(a,H)}{a^{4}},$ (24)

where we have recovered the scale factor and the cosmological time $t$. For
the fluctuations $v_{k}$, it is found that they obey the Schrödinger-like
equation Bouhmadi-López  _et al._ (2019), as expected.

##  III Observational Consequences: CMB Angular Power Spectrum

The new dynamics ruled by the Friedmann equation (24) which takes into account
a small interaction between our universe and its twin by means of the coupling
function $\lambda(a,H)$, may have an impact on the anisotropies of the CMB.
The intention of this section is to find out how it makes a difference from
the best fit found by Planck Aghanim  _et al._ (2020a), where $\Lambda$CDM
model is assumed. We will consider pretty general coupling function fulfilling
the condition (3), which reads

$\lambda^{2}(a,H)=\frac{\lambda_{o}}{2}\frac{a^{q}}{H^{n}}=\frac{\lambda_{o}}{2}\frac{a^{q+n}}{\dot{a}^{n}},$
(25)

where $n>0$, and $q$ are some real constants, and $\lambda_{o}$ is a constant
small enough to see the interaction between the pair as a perturbation for
large values of the scale factor, whose units are $[T^{-2-n}]$. However, a
priori it is not valid to perform any approximation around $\lambda_{o}$ since
such a term contributes mainly to the dynamics close to the initial
singularity.

In order to also adopt other types of extreme points of the universe discussed
in Ref. Balcerzak  _et al._ (2021) such as exotic singularities Da̧browski
(2014); Da̧browski and Marosek (2018) which may have regular energy density
but irregular the pressure and its derivatives, one may also postulate some
more general ansatz

$\lambda^{2}(a,H,\bar{q})=\frac{\lambda_{o}}{2}\frac{a^{q}}{H^{n}\bar{q}^{m}}=(-1)^{m}\frac{\lambda_{o}}{2}\frac{a^{q+n-m}}{\dot{a}^{n-2m}\ddot{a}^{m}},$
(26)

where $\bar{q}\equiv-(\ddot{a}a)/\dot{a}^{2}$ is the deceleration parameter,
and the constant $m>0$. For example, in a sudden future singularity (SFS)
Barrow (2004a) the scale factor $a=a_{s}=$ const., and its first derivative
(proportional to the energy density) $\dot{a}=\dot{a}_{s}=$ const., while the
second derivative (proportional to the pressure) $\ddot{a}\to-\infty$. This
procedure may further be extended into the singularities in higher derivatives
of pressure (for example generalized sudden future singularities (GSFS) Barrow
(2004b)) by inserting higher-order kinematic quantities in the denominator
such as jerk, snap, and pop Caldwell and Kamionkowski (2004); Dunajski and
Gibbons (2008); Da̧browski (2005) which would involve higher order derivatives
of the scale factor into (26).

However, just focusing our attention on the coupling function (25), in order
to recover the spectrum, we should use a scalar field whose equation of state
is $p=\omega\rho$, where $\omega$ is expected to be very close to
$\omega_{\Lambda}=-1$, since the expansion must be almost de Sitter Bassett
_et al._ (2006). Let us then write

$\omega=-1+\frac{2\alpha}{3},\qquad\qquad\alpha\gtrsim 0.$ (27)

For a universe with a scalar field fulfilling the above condition, the density
Weinberg (2008) scales like
$\rho(\phi)=(1/2)H_{\text{dS}}^{2}(a_{d}/a)^{2\alpha}$, and thus Eq. (24) is
written as

$H^{2}=H_{\text{dS}}^{2}\left[\left(\frac{a_{d}}{a}\right)^{2\alpha}+\frac{\lambda_{o}}{H_{\text{dS}}^{2}H^{n}a^{4-q}}\right],$
(28)

where $a_{d}$ is a constant to be determined.

The condition for the probability (4) can be fulfilled, when $4-q-n>0$. Hence,
at the very beginning, the universe is dominated by a term which is
essentially different from the one of Ref. Bouhmadi-López  _et al._ (2019),
since it depends on the Hubble parameter in the denominator.

The analysis of an arbitrary $q$ and $n$ is left for future research. Here we
will only consider the values $q=1$ and $n=1$, which is the extreme case since
the Hubble parameter goes like $1/H$ in the interaction term, and it is
positive during inflation. Therefore, any other value of $n$ will decrease the
interaction and the effects would be less noticeable. This case is one of the
family of cases for which the analysis is simplified fulfilling the condition
$2-q-n=0$, so that the Friedmann equation can be written as

$\accentset{\mbox{\large.}}{a}^{n+2}=H_{\text{dS}}^{2}\accentset{\mbox{\large.}}{a}^{n}a_{d}^{2\alpha}a^{2(1-\alpha)}+\lambda_{o}.$
(29)

The asymptotic behaviour one can obtain from Eq. (29) of the scale factor
close to the singularity is

$a(t)\approx\sqrt[3]{\lambda_{o}}t,$ (30)

which is an extremely slow motion of the (Milne) universe which appears on the
edge of inflationary condition $\ddot{a}\geq 0$. Using Eq. (30) as feedback
for Eq. (28), we infer that a more precise expansion is still slightly
accelerated.

Even if there is a short period before inflation, it should have been in a
constant thermal equilibrium and no much physics is expected to happen.
Indeed, one finds that $(aH)\sim\sqrt[3]{\lambda_{o}}$ there, which is
constant throughout the initial stage. Furthermore, the dynamics of the very
beginning of the universe is not perfectly obtained from canonical quantum
gravity due to the problem of the factor ordering Kiefer (2007). For these
reasons, we will not repeat the method used in Ref. Bouhmadi-López  _et al._
(2019), and we will neglect the effects at the very beginning of the universe
assuming that $\lambda_{o}$ can be taken as a perturbation at any point of the
evolution.

The equation (28) can be expanded at first order around $\lambda_{o}$ as

$H\approx
H_{\text{dS}}\left(\frac{a_{d}}{a}\right)^{\alpha}+\frac{\lambda_{o}}{2H_{\text{dS}}Ha^{3-\alpha}a_{d}^{\alpha}}\approx
H_{\text{dS}}\left(\frac{a_{d}}{a}\right)^{\alpha}+\frac{\lambda_{o}}{2H_{\text{dS}}^{2}a^{3}}.$
(31)

Trying the ansatz

$a(t)=a_{o}(t)\left[1+\xi(t,\lambda_{o})\right],$ (32)

where $a_{o}(t)$ is the solution for Eq. (31) setting $\lambda_{o}$ to zero,
and $\xi(t,\lambda_{o})$ is a function such that $\xi\ll 1$, after cancelling
second order derivatives of $\xi$, we find

$a(t)\approx a_{d}\left(\alpha
H_{\text{dS}}t\right)^{1/\alpha}\left[1-\frac{\lambda_{o}}{6a_{d}^{3}H_{\text{dS}}^{2+3/\alpha}(\alpha
t)^{-1+3/\alpha}}\right].$ (33)

Using the slow-roll parameter Bassett  _et al._ (2006)

$\epsilon=-\frac{\dot{H}}{H^{2}}\approx\alpha-\lambda_{o}\frac{(3-\alpha)(3-2\alpha)(1-\alpha)}{6H_{\text{dS}}^{2}a^{2}(\eta)}\eta,$
(34)

the Mukhanov-Sasaki equation (7), at first order in $\lambda_{o}$, is just

$v^{\prime\prime}_{k}(\eta)+\left(k^{2}-\frac{2}{\eta^{2}}-\frac{3\alpha}{\eta^{2}}+\lambda_{o}\frac{\left[a_{d}^{\alpha}H_{\text{dS}}(1-\alpha)\right]^{\frac{2}{1-\alpha}}}{6H_{\text{dS}}^{2}}|\eta|^{\frac{1+\alpha}{1-\alpha}}\right)v_{k}(\eta)=0,$
(35)

where we have included the relations

$\eta\approx-\frac{t\alpha}{(1-\alpha)a(t)}\approx-\frac{1}{(1-\alpha)aH},\qquad\qquad\frac{z^{\prime\prime}}{z}=(aH)^{2}(2-\epsilon).$
(36)

As expected, the Bunch-Davies vacuum Mukhanov (2005)

$v_{k}(\eta)=\frac{\text{e}^{-\text{i}k\eta}}{\sqrt{2k}},$ (37)

cannot be recovered at $\eta\to-\infty$, since the term due to the interaction
between universes (the third one into the parenthesis in Eq. (35)), dominates,
and hence it cannot be used as a good boundary condition. However, since
$(aH)\sim\sqrt[3]{\lambda_{o}}$ during the initial phase, such a size
corresponds to the conformal time

$\eta_{\text{knee}}\approx-\frac{1}{(1-\alpha)\sqrt[3]{\lambda_{o}}}\approx-\frac{1}{\sqrt[3]{\lambda_{o}}},$
(38)

for the late time evolution. We will use a solution to Eq. (35) such that it
behaves like (37) at $\eta_{\text{knee}}$. This approximation is thought to be
very good since $\absolutevalue{\eta_{\text{knee}}}$ must be very large.

Trying a solution of the form

$v_{k}(\eta)=v_{k}^{(0)}(\eta)\text{e}^{f_{k}(\eta,\lambda_{o})},$ (39)

where $v_{k}^{0}(\eta)$ is the solution to Eq. (35) when the third term in the
parenthesis is vanishing, and $f_{k}(\eta,\lambda_{o})$ is a slow varying
function whose value is expected to be small enough in its whole domain.
Introducing (39) into (35), and cancelling the terms containing
$f^{\prime\prime}$ and $(f^{\prime})^{2}$, we find the solution going like

$v_{k}(\eta)=v_{k}^{(0)}(\eta)\exp\left[-\lambda_{o}\frac{\left[a_{d}^{\alpha}H_{\text{dS}}(1-\alpha)\right]^{\frac{2}{1-\alpha}}}{12H_{\text{dS}}^{2}}\int^{\eta}|\tilde{\eta}|^{\frac{1+\alpha}{1-\alpha}}\frac{v_{k}^{(0)}(\tilde{\eta})}{{v^{\prime}_{k}}^{(0)}(\tilde{\eta})}\text{d}\tilde{\eta}+\Delta_{k}\right],$
(40)

where $\Delta_{k}$ is a constant of integration. This constant $\Delta_{k}$ is
very close to be purely imaginary, and for the future calculation of the power
spectrum the phase to which it contributes is not relevant. The solution
$v^{(0)}_{k}(\eta)$ is well-known as Bassett  _et al._ (2006)

$v^{(0)}_{k}(\eta)=\frac{\sqrt{\pi|\eta|}}{2}H^{(1)}_{\mu}\left(k|\eta|\right),\qquad\qquad\mu\approx\frac{3}{2}+\frac{5}{9}\alpha,$
(41)

where $H_{n}^{(1)}(z)$ is the Hankel function of the first kind (for special
functions, see Olver  _et al._ (2010)), since it fulfils the condition (37) at
$\eta\to-\infty$.

The anisotropies of the CMB are expressed via the power spectrum of the
curvature perturbations Mukhanov (2005)

$P_{\mathcal{R}}(k)=\frac{k^{3}}{2\pi^{2}}\frac{|v_{k}|^{2}}{z^{2}}.$ (42)

One expects to adjust the spectrum to a power law like

$P_{\mathcal{R}}(k)=A_{s}\left(\frac{k}{k_{\ast}}\right)^{n_{s}-1},$ (43)

when $aH\sim k$, or equivalently, $k|\eta|\sim 1$, where $A_{s}$, and $n_{s}$
depend on the given value $k_{\ast}$ set by the experiment. From the Planck
analysis Aghanim  _et al._ (2020b), where they considered $k_{\ast}=0.05$
Mpc-1, it was obtained $A_{s}=(2.105\pm 0.030)\times 10^{-9}$, and
$n_{s}=(0.9665\pm 0.0038)$. That way, the power spectrum is found to have the
known form Bouhmadi-López  _et al._ (2019); Morais  _et al._ (2018); Bassett
_et al._ (2006)

$P_{\mathcal{R}}(k\lesssim
aH)\approx\left[(1-\alpha)\frac{\Gamma(\nu)}{\Gamma(3/2)}\frac{H}{2\pi}\right]^{2}\left(\frac{k|\eta|}{2}\right)^{-2\alpha}\left|\text{e}^{f_{k}(\eta,\lambda_{o})}\right|^{2},$
(44)

which is found, in our case, at first order in $\epsilon\sim\alpha$, and at
the moment when the modes cross the horizon $k_{\ast}\approx
a_{\ast}H_{\text{dS}}$, to be like

$P_{\mathcal{R}}(k\lesssim
aH)\approx\frac{1}{2^{-2\alpha}}\left\\{\left[1-\alpha+\Psi\left(\frac{3}{2}\right)\alpha\right]\frac{H_{\text{dS}}}{2\pi}\right\\}^{2}\left(\frac{k}{a_{\ast}H_{\text{dS}}}\right)^{-2\alpha}\left|\text{e}^{f_{k}(\eta,\lambda_{o})}\right|^{2},$
(45)

where $\Psi(z):=\Gamma^{\prime}(z)/\Gamma(z)$ is the digamma function. The
absolute value of the exponential can be considered very close to unity such
that we can derive, comparing with the fit (43), the values

$\displaystyle\alpha=0.0168\pm 0.0019,\qquad\qquad a_{d}\equiv
a_{\ast}=(9.052\pm 0.067)\cdot 10^{-56},$ (46) $\displaystyle
H_{\text{dS}}=(2.896\pm 0.021)\cdot 10^{-4}=(5.372\pm 0.039)\cdot
10^{39}\,\text{s}^{-1},$ (47)

which we can compare with the actual values

$a_{0}=1,\qquad\qquad H_{0}\approx 1.2\cdot 10^{-61}.$ (48)

The power spectrum (42) is used then to find the angular power spectrum, given
in terms of the coefficients

$C_{l}=2T_{0}^{2}l(l+1)\int_{0}^{\infty}\frac{\text{d}k}{k}P_{\mathcal{R}}(k)\Delta_{l}^{2}(k),$
(49)

where $T_{0}=(2.72548\pm 0.00057)$ K is the temperature of the CMB Fixsen
(2009), and camb has been used to obtain the transfer functions
$\Delta_{l}(k)$. For us it depends on $\lambda_{o}$. The power spectrum (42)
and the angular power spectrum are shown in Fig. 1.

Figure 1: Left panel: The power spectrum for different values of
$\lambda_{o}$. The dashed line represents the spectrum (43) given by Planck
data. The smaller the value of $\lambda_{o}$, the spectrum is more similar to
the standard, as expected. The effect of the multiverse interaction is visible
only for the smaller modes $k$. Right panel: The normalized angular power
spectrum for different values of $\lambda_{o}$. The solid line represents the
standard spectrum. The data points and the error bars come from the Planck
measurement. The dashed lines are the spectrum found for different
$\lambda_{o}$. For $\lambda_{o}=10^{-56}$, the dashed line is overlapped by
the solid line. Whence, the smaller the $\lambda_{o}$, the lesser the affect
onto the multipoles. For both figures, the considered values for the coupling
constant $\lambda_{o}$ are: $10^{-46}$, $10^{-50}$, $10^{-54}$, $10^{-55}$,
$10^{-56}$.

As expected, the interaction with our hypothetical partner universe affects
only the smaller values of $k$ and the multipoles. The lighter the
interaction, the more suppressed the effects. Since the spectra differ too
fast from the standard one, we can easily constraint the value of the coupling
constant $\lambda_{o}$, for a kind of coupling function as (25), to be

$\lambda_{o}\lesssim\mathcal{O}(10^{-56}).$ (50)

Coming back to a more general coupling function like in (26), the spectrum for
a certain family is expected to be very similar as the one in Fig. 1. Such
family is analogous to the one we analysed, but the condition is that
$4-q+m-n>0$ in order to keep the vacuum stability assuming $\dot{a}>0$ and
taking $m$ even, and $2-q+m-n=0$, which automatically satisfies the first
condition. Within the family we find the choice $m=2$, $n=4$ and $q=0$, for
which the Friedmann equation (24) is

$H^{2}=H_{\text{dS}}^{2}\left[\left(\frac{a_{d}}{a}\right)^{2\alpha}+\frac{\lambda_{o}}{H_{\text{dS}}^{2}a^{2}\ddot{a}^{2}}\right].$
(51)

The scale factor goes like $t^{3/2}$ close to the initial singularity, and
taking $\lambda_{o}$ as a perturbation for late times, the scale factor is
then evolving like $t^{1/\alpha}$. The transition between both states is
expected to be smooth, like if a single scalar field whose barotropic
parameter varies from $\omega=-5/9$ to $\omega\gtrsim-1$ monotonically fills
the universe. It implies that the power spectrum for the modes crossing the
horizon $k\sim aH$, during the early states of the universe, when the
interaction dominates, is just enlarged. Those modes are the smallest ones. It
will not change significantly the results in Fig. (1) since the slopes are
quite steep already for small $k$.

##  IV A Short Comment About an Interuniversal Contribution to the Dark Energy
of the Universe

Here we shortly present a way for the multiverse interaction to contribute to
the dark energy of the universe via its entanglement. The Friedmann equation
(24) rules the dynamics of the universe with mode $K=1$ for a given
interaction $\lambda(a,H)$ as we have reasoned previously. Now, let us
consider the special form of the coupling function

$\lambda^{2}(a,H)=\frac{\lambda_{o}}{2}a^{4}f(H)H^{2},$ (52)

where $f(H)$ is an explicit and well-behaved function of $H$, and
$\lambda_{o}$ is a positive constant. In order to be sure that the interaction
term diverges when $H$ vanishes, we require $f(H)H^{2}$ to diverge. The
Friedmann equation (24), without any field, yields the trivial result

$H=f^{-1}(\lambda_{o}^{-1})=\text{constant}.$ (53)

The dynamics is then equivalent to the one of the de Sitter space

$a(t)=a_{o}\exp\left(Ht\right).$ (54)

Even if the Friedmann equation (24) has been obtained without assuming that
such coupling function was small, one needs it to be so small since the
probability condition (4), which is of the form
$\mathcal{P}\sim\exp\\{-[\lambda_{o}H^{2}f(H)]^{-3}\\}$, prevents
$\lambda_{o}$ to be very large in order to keep a small initial probability.
Besides, the probability condition tells us that not considering any kind of
matter produces a constant probability along the entire evolution of the
universe. There is no other chance than having some kind of matter
contributing to the dynamics, so that the probability decreases leading to the
recovery of the anisotropies of the CMB.

##  V Conclusions

The multiverse scenario derived from the third quantization in quantum
cosmology has been considered. In this scenario, the universes behave like
particles of the standard quantum field theory. We have considered the
interaction of a pair of universes (our universe and our twin anti-universe),
whose entanglement entropy was found to diverge at the critical points of the
classical evolution Balcerzak  _et al._ (2021). It has motivated us to
consider the coupling function like (25) or a more general one like (26),
which both depend on an arbitrary coupling constant $\lambda_{o}$ which is to
be determined and on the Hubble parameter. We have assumed that the functions
must fulfil the condition (3) (coupling function diverges when the Hubble
parameter vanishes) and the condition (4) (the probability is supressed at
large values of the scale factor). For the function (25), we have found the
modified Friedmann equation (28) while taking into account the interaction of
the pair of universes.

For a certain family of solutions, we have found the modifications of the
imprints on the CMB power spectrum. The plots have been given in Fig. 1 in
which one can see that the interaction with our hypothetical partner enlarges
the spectra for small $k$ and the small monopoles $l$. The constraint on the
interaction coupling constant which is in agreement with Planck observations
Aghanim  _et al._ (2020a) has been found to be such as
$\lambda_{o}\lesssim\mathcal{O}(10^{-56})$.

It is the future problem to fix the parameters of the coupling function and
its specific form. However, for the set of solutions fulfilling the condition
$2-q-n=0$, where $n>0$, the results presented in the paper are approximately
equivalent. Once those parameters are known, then one can extrapolate our
conclusions and say that the CMB angular power spectrum is insensitive to some
small interaction term. The need for more observational experiments digging
into the very early stages of our universe, like the search for primordial
gravitational waves or the cosmic neutrino background, would be of interest in
order to falsify the existence of a twin anti-universe or some other entangled
universes. From what we have found, we can only constrain the strength of the
interaction, which is considerably small.

A short comment has also been given for two empty universes whose interaction
was defined by a very special coupling function (52). The dynamics of such
universes is similar to the de Sitter universe. It could mimic the effect of a
cosmological constant at early times. Nevertheless, in order to recover the
spectra, one needs some kind of matter to be introduced to such a simple
scenario.

###### Acknowledgements.

SBB would like to thank Michele Liguori and Eleonora Di Valentino for
providing useful hints to calculate the CMB angular power spectrum and Mar
Bastero Gil and Fabian Wagner for some useful discussions and coffees. The
work of SBB was supported by the Polish National Research and Development
Center (NCBR) project UNIWERSYTET 2.0. STREFA KARIERY,
POWR.03.05.00-00-Z064/17-00.

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