# Dispersive effects in two- and three-dimensional peridynamics

A. Coclite<EMAIL_ADDRESS>G. M. Coclite
<EMAIL_ADDRESS>G. Fanizza<EMAIL_ADDRESS>F. Maddalena
<EMAIL_ADDRESS>Dipartimento di Ingegneria Elettrica e
dell’Informazione (DEI), Politecnico di Bari,
Via Re David 200 – 70125 Bari, Italy  Dipartimento di Meccanica, Matematica e
Management, Politecnico di Bari (DMMM),
Via Re David 200 – 70125 Bari, Italy  Instituto de Astrofisíca e Ciências do
Espaço, Faculdade de Ciências, Universidade de Lisboa,
Edificio C8, Campo Grande, P-1749-016, Lisbon, Portugal

###### Abstract

In this paper we study the dispersive properties related to a model of
peridynamic evolution, governed by a non local initial value problem, in the
cases of two and three spatial dimensions. The features of the wave
propagation characterized by the nontrivial interactions between nonlocality
and the regimes of low and high frequencies are studied and suitable numerical
investigations are exposed.

###### keywords:

Peridynamics , nonlocal Continuum Mechanics , Elasticity , Dispersion

###### MSC:

[2020] 74A70 , 74B10 , 70G70 , 35Q70

## Introduction

As it is well known, an evolution partial differential equation is dispersive
if, when no boundary conditions are imposed, its wave solutions spread out in
space as they evolve in time. This phenomenon is ruled by the so called
dispersion relation which states that plane waves travel at different speeds
according to their wave number, hence the spreading of a wave packet occurs as
the time evolves [1, 16, 17]. The dispersion relation is intrinsically
connected to the differential operator characterizing the evolution equation,
then when the evolution problem is governed by nonlocal operators, as in the
case of peridynamic theory, it is a natural question to investigate the
dispersive properties exhibited in these situations. Peridynamics, initiated
by S.A. Silling (see [11, 12, 13, 14, 15]), constitutes a nonlocal mechanical
theory in which the evolution problem is governed by an integral operator
which takes into account a scale length characterizing the dynamical
interactions inside the material. In some previous works ([5, 6]) we have
focused on the analysis of the dispersive properties of the nonlocal evolution
equation related to a model of peridynamics, studied in detail in [3, 8, 9,
10], by limiting the analysis to the scalar case modeling a one-dimensional
infinite material body. In that case the study of the interplay between
nonlocality and dispersion has revealed interesting features related to the
asymptotic at low and high frequencies, suggesting new pictures in the
framework of wave propagation in continua where nonlocal characteristics are
taken into account. The numerical aspects of the problem have been studied in
[2, 4, 7].

In this paper we extend the analysis of these phenomena to the vectorial case,
namely we study the dispersive properties of the peridynamic evolution
equation in the physical relevant cases of two and three spatial dimensions.
The results obtained show that the two- and three-dimensional cases share the
same scaling with the uni-dimensional solutions since the dispersive relations
exhibit the same dependence on the nonlocal interaction length $\delta$ and
this is due to the scaling of the elastic parameter $\kappa$ vs the
interaction length $\delta$, hence this dependence has universal character,
regardless of the dimension of the system. Moreover, the dispersive relations
depends solely on the modulus of the Fourier variable $\xi$ and the two
scenario related to $\xi\delta\ll 1$ and $\xi\delta\gg 1$ are analyzed in
detail.

The paper is organized as follows. In Section 1 the general equations
governing the peridynamic model studied here are exposed and the results
pertaining the linear scalar problem are recalled. In Section 2 and Section 3
the dispersion relations and the relative asymptotic properties for the two
and three dimensional problems are obtained. In Section 4 the results deduced
through the previous analysis are carefully discussed and suitable numerical
investigations are deepen by exploiting the evolution of a class of initial
data.

##  1 The peridynamic model

In [3] a general model for nonlocal continuum mechanics was proposed and
studied by exploiting the analytic aspects of global solutions in energy space
in the framework of nonlinear hyperelastic constitutive assumptions.
Specifically, the momentum balance equation of motion of an infinite material
body takes the form of the following initial-value problem:

$\begin{cases}\partial_{tt}\mathbf{u}(t,\mathbf{x})=(K\mathbf{u}(t,\cdot))(\mathbf{x}),&\quad\mathbf{x}\in\mathbb{R}^{N},\>t>0,\\\
\mathbf{u}(0,\mathbf{x})=\mathbf{u}_{0}(\mathbf{x}),\>\partial_{t}\mathbf{u}(0,\mathbf{x})=\mathbf{v}_{0}(\mathbf{x}),&\quad\mathbf{x}\in\mathbb{R}^{N},\end{cases}$
(1.1)

where

${(K\mathbf{u}(t,\cdot))(\mathbf{x}):=\int_{B_{\delta}(\mathbf{x})}\mathbf{f}(\mathbf{x}^{\prime}-\mathbf{x},\mathbf{u}(\mathbf{x}^{\prime})-\mathbf{u}(\mathbf{x}))\,d\mathbf{x}^{\prime},\quad{\mbox{
for every }}\,\mathbf{x}\in\mathbb{R}^{N}}\,.$ (1.2)

$N=1,2,3$ corresponding to physically meaningful cases;
$\mathbf{u}\in\mathbb{R}^{N}$ denotes the displacement field and $\delta>0$
characterizes the non local interaction range governed by the integral kernel
$K$. The internal force $\mathbf{f}:\Omega\rightarrow\mathbb{R}^{N}$, with
$\Omega:=(\mathbb{R}^{N}\setminus\\{\mathbf{0}\\})\times\mathbb{R}^{N}$, is
supposed to satisfy the following general constitutive assumptions:

  1. (H.1)

$\mathbf{f}\in C^{1}(\Omega;\mathbb{R}^{N})$;

  2. (H.2)

$\mathbf{f}(-\mathbf{y},-\mathbf{u})=-\mathbf{f}(\mathbf{y},\mathbf{u}),$ for
every $(\mathbf{y},\,\mathbf{u})\in\Omega\times\mathbb{R}^{N}$;

  3. (H.3)

there exists a function $\Phi\in C^{2}(\Omega)$ such that

$\mathbf{f}=\nabla_{\mathbf{u}}\Phi,\qquad\Phi(\mathbf{y},\mathbf{u})=\kappa\frac{|\mathbf{u}|^{p}}{|\mathbf{y}|^{N+\alpha
p}}+\Psi(\mathbf{y},\mathbf{u}),\quad{\mbox{ for every
}}\,(\mathbf{y},\,\mathbf{u})\in\Omega,$

where $\kappa,\,p,\,\alpha$ are constants such that

$\kappa>0,\qquad 0<\alpha<1,\qquad p\geq 2,$

and

$\displaystyle\Psi(\mathbf{y},\mathbf{0})=0\leq\Psi(\mathbf{y},\mathbf{u}),$
$\displaystyle|\nabla_{\mathbf{u}}\Psi(\mathbf{y},\mathbf{u})|,|D^{2}_{\mathbf{u}}\Psi(\mathbf{y},\mathbf{u})|\leq
g(\mathbf{y}),\quad{\mbox{ for every }}\,(\mathbf{y},\,\mathbf{u})\in\Omega,$

for some non-negative function $g\in L^{2}_{\rm{loc}}(\mathbb{R}^{N})$.

Assumption (H.2) can be seen as a counterpart of Newton’s Third Law of Motion
(the Action-Reaction Law). Also, assumption (H.3) states that the material is
hyperelastic (the linear elastic case corresponding to $p=2$ and $\Psi=0$). In
this regard, we will first recall the main results obtained for the dispersive
propagation in the uni-dimensional case (see [6]), where the Cauchy problem
(1.1) takes the following form.

$\begin{cases}\rho\,u_{tt}(t,x)=-2\,\kappa\displaystyle\int_{-\delta}^{\delta}\frac{u(t,x)-u(t,x-y)}{|y|^{1+2\,\alpha}}dy=:\left(Ku(t,\cdot)\right)(x),&\quad
t>0,\,x\in\mathbb{R},\\\\[10.0pt] u(0,x)=v_{0}(x),&\quad
x\in\mathbb{R},\\\\[5.0pt] u_{t}(0,x)=v_{1}(x),&\quad
x\in\mathbb{R}\,,\end{cases}$ (1.3)

where $\delta$, $\kappa$ and $\rho$ are positive real constants and
$0<\alpha<1$. Let us clarify the adopted notation for vectors: bold
quantities, such as ${\bm{\xi}},\,\mathbf{x},\,...$ refer to the entire
vector, whereas their modulus is simply denoted by $\xi,\,x,\,...$

In [6], the following results are proved.

######  Theorem 1.1.

Let $v_{0},\,v_{1}\in\mathcal{S}(\mathbb{R})$ and $0<\alpha<1$. Then problem
(1.3) has the unique solution
$u:\mathbb{R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}$ given by

$u(t,x)=\int_{\mathbb{R}}e^{-i\xi
x}\left[\widehat{v_{0}}(\xi)\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{v_{1}}(\xi)}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d\xi\,,$
(1.4)

where $\widehat{v_{0}}(\xi)$ and $\widehat{v_{1}}(\xi)$ represent the Fourier
transform111We use here the non-unitary convention that  $\widehat{\bf
v}({\bm{\xi}}):=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}{\bf v}({\bf
x})\,e^{i\mathbf{x}\cdot{\bm{\xi}}}\,d\mathbf{x}.$ In this way, the inversion
formula reads  ${\bf v}(\mathbf{x})=\int_{\mathbb{R}^{N}}\widehat{\bf
v}({\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot\mathbf{x}}\,d{\bm{\xi}}.$ of $v_{0}(x)$
and $v_{1}(x)$, and $\omega:\mathbb{R}\rightarrow\mathbb{R}^{+}$ is the
dispersion relation defined by

$\omega(\xi)=\left(\frac{2\kappa}{\rho\,\delta^{2\alpha}}\,\int_{-1}^{1}\frac{1-\cos(\xi\delta
z)}{|z|^{1+2\alpha}}dz\right)^{1/2}\,.$ (1.5)

######  Theorem 1.2.

For $\delta>0$ and $0<\alpha<1$, the function $\xi\mapsto\omega(\xi)$ given by
(1.5) satisfies the following relations.

$\lim_{\xi\rightarrow
0}\xi^{-2}\omega^{2}(\xi)=\frac{\kappa\,\delta^{2(1-\alpha)}}{(1-\alpha)\,\rho},$
(1.6)

$\lim_{\xi\rightarrow\pm\infty}|\xi|^{-2\alpha}\omega^{2}(\xi)=\frac{4\kappa}{\rho}\,\int_{0}^{\infty}\frac{1-\cos\tau}{\tau^{1+2\alpha}}d\tau.$
(1.7)

This paper is devoted to the analysis of the two-dimensional and three-
dimensional counterparts of the dispersion relation (1.5). Analogous results
to the uni-dimensional case will be discussed here and the main differences
concerning the dispersive behavior emerging from the increased dimension of
the system will be analyzed in detail.

##  2 The two-dimensional case

######  Theorem 2.1.

Let ${\bf v}_{0},\,{\bf v}_{1}\in\mathcal{S}(\mathbb{R}^{2})$ and
$0<\alpha<1$. Then problem (1.1) with $N=2$, $p=2$ and $\Psi=0$ has the unique
solution
$\mathbf{u}:\mathbb{R}^{+}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ given
by

$\mathbf{u}(t,\mathbf{x})=\int_{\mathbb{R}^{2}}e^{-i{\bm{\xi}}\cdot\mathbf{x}}\left[\widehat{{\bf
v}_{0}}({\bm{\xi}})\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}({\bm{\xi}})}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d{\bm{\xi}}\,,$
(2.1)

where $\widehat{{\bf v}_{0}}({\bm{\xi}})$ and $\widehat{{\bf
v}_{1}}({\bm{\xi}})$ represent the Fourier transforms of ${\bf
v}_{0}(\mathbf{x})$ and ${\bf v}_{1}(\mathbf{x})$, and
$\omega:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is the dispersion relation
defined by

$\omega(\xi)=\left(\frac{4\pi\kappa}{\rho\,\delta^{2\alpha}}\,\int_{0}^{1}\frac{1-J_{0}(\xi\delta
z)}{z^{1+2\alpha}}dz\right)^{1/2},$ (2.2)

where $J_{0}(\cdot)$ is the $0$-th order Bessel function of the first kind.

######  Theorem 2.2.

For $\delta>0$ and $0<\alpha<1$, the following asymptotic relations hold true.

$\lim_{\xi\rightarrow
0}\xi^{-2}\omega^{2}(\xi)=\frac{\pi\kappa\,\delta^{2(1-\alpha)}}{2(1-\alpha)\,\rho}$
(2.3)

and

$\lim_{\xi\rightarrow\infty}\xi^{-2\alpha}\omega^{2}(\xi)=\frac{4\pi\kappa}{\rho}\,\int_{0}^{\infty}\frac{1-J_{0}(\tau)}{\tau^{1+2\alpha}}d\tau.$
(2.4)

######  Corollary 2.3 (Radial solutions).

Let ${\bf v}_{0},\,{\bf v}_{1}\in\mathcal{S}(\mathbb{R}^{2})$ and
$0<\alpha<1$. Assume ${\bf v}_{0}$, ${\bf v}_{1}$ be such that

${\bf v}_{0}(\mathbf{x})={\bf
v_{0}}\left(\sqrt{x^{2}_{1}+x^{2}_{2}}\right)\qquad\text{and}\qquad{\bf
v}_{1}(\mathbf{x})={\bf v}_{1}\left(\sqrt{x^{2}_{1}+x^{2}_{2}}\right)\,.$
(2.5)

Then, the unique solution of the Cauchy problem (1.1) is

$\mathbf{u}(t,x)=2\pi\int_{0}^{\infty}\xi\,J_{0}(\xi x)\left[\widehat{{\bf
v}_{0}}(\xi)\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}(\xi)}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d\xi\,.$ (2.6)

######  Remark 2.4.

Notice that the relation (2.2) only depends on the modulus $\xi$ of the
Fourier variable ${\bm{\xi}}$ and the solution (2.6) is radial since it only
depends on the modulus $x$ of the vector $\mathbf{x}$.

As a consequence of Remark 2.4, the only way to have spatial anisotropic
effects in the dynamic evolution is to deal with anisotropic initial
conditions. We now proceed in proving of the aforementioned theorems.

###### Proof of Theorem 2.1.

The existence and uniqueness of the solution for the Cauchy problem (1.1) has
been proven in [3]. To prove that this solution is given by Eq. (2.1), we
proceed by applying the Fourier transform to (1.1). Indeed, we first have that

$\partial_{tt}\mathbf{u}(t,\mathbf{x})=\int_{\mathbb{R}^{2}}e^{-i{\bm{\xi}}\cdot\mathbf{x}}\partial_{tt}\widehat{\bf
u}(t,{\bm{\xi}})d{\bm{\xi}}\,.$ (2.7)

In the same way,

$\displaystyle(K{\bf u}$ $\displaystyle(t,\cdot))({\bf x})\equiv$ (2.8)
$\displaystyle\equiv$ $\displaystyle-2\kappa\int_{B_{\delta}(0)}\frac{{\bf
u}(t,{\bf x})-{\bf u}(t,{\bf x}-{\bf y})}{|{\bf y}|^{2+2\alpha}}d{\bf y}$
$\displaystyle=$
$\displaystyle-2\kappa\int_{\mathbb{R}^{2}}{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\int_{B_{\delta}(0)}\frac{1-e^{i{\bm{\xi}}\cdot{\bf y}}}{|{\bf
y}|^{2+2\alpha}}d{\bf y}d{\bm{\xi}}$ $\displaystyle=$
$\displaystyle-2\kappa\int_{\mathbb{R}^{2}}{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\int_{0}^{\delta}\int_{0}^{2\pi}\frac{1-e^{i\xi
y\cos\left(\theta-\theta_{\xi}\right)}}{y^{1+2\alpha}}d\theta dyd{\bm{\xi}}$
$\displaystyle=$
$\displaystyle-2\kappa\int_{\mathbb{R}^{2}}{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\int_{0}^{\delta}\left[\int_{0}^{2\pi}\frac{1-\cos(\xi
y\cos\left(\theta-\theta_{\xi}\right))}{y^{1+2\alpha}}d\theta-i\underbrace{\int_{0}^{2\pi}\frac{\sin(\xi
y\cos\left(\theta-\theta_{\xi}\right))}{y^{1+2\alpha}}d\theta}_{=0}\right]dyd{\bm{\xi}}$
$\displaystyle=$
$\displaystyle-4\pi\kappa\int_{\mathbb{R}^{2}}{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\int_{0}^{\delta}\frac{1-J_{0}(\xi y)}{y^{1+2\alpha}}dyd{\bm{\xi}}$
$\displaystyle=$
$\displaystyle-4\pi\kappa\delta^{-2\alpha}\int_{\mathbb{R}^{2}}{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\int_{0}^{1}\frac{1-J_{0}(\xi\delta z)}{z^{1+2\alpha}}dzd{\bm{\xi}}$
$\displaystyle=$
$\displaystyle-\rho\int_{\mathbb{R}^{2}}{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\omega^{2}(\xi)d{\bm{\xi}}\,.$ (2.9)

In the third line, we expressed the integral over $B_{\delta}(0)$ in polar
coordinates, namely
${\bm{\xi}}=\xi\left(\cos\theta_{\xi},\sin\theta_{\xi}\right)$ and
$\mathbf{y}=y\left(\cos\theta,\sin\theta\right)$ with $\xi,y\in\mathbb{R}^{+}$
and $\theta,\theta_{\xi}\in[0,2\pi)$, and we have defined $z:=y/\delta$ in the
second last line. Moreover, the under-braced integral is null thanks to the
symmetry of the integrand under the shift
$\theta\rightarrow\theta+\theta_{\xi}$ and the invariance of the integral for
any integration over an entire period of the integrand.

Now, we make use of Eqs. (2.7) and (2.9) to formulate our two-dimensional
Cauchy problem in Fourier space. We obtain

$\begin{cases}\partial_{tt}{\bf\widehat{u}}(t,{\bm{\xi}})+\omega^{2}(\xi){\bf\widehat{u}}(t,{\bm{\xi}})=0,&\quad{\bm{\xi}}\in\mathbb{R}^{2},\>t>0,\\\
\widehat{\mathbf{u}}(0,{\bm{\xi}})=\widehat{\mathbf{u}_{0}}({\bm{\xi}}),\>\partial_{t}\widehat{\mathbf{u}}(0,{\bm{\xi}})=\widehat{\mathbf{v}}_{0}({\bm{\xi}}),&\quad{\bm{\xi}}\in\mathbb{R}^{2},\end{cases}$
(2.10)

which is solved by

$\widehat{\mathbf{u}}(t,{\bm{\xi}})=\widehat{{\bf
v}_{0}}({\bm{\xi}})\cos(\omega(\xi)t)+\frac{\widehat{{\bf
v}_{1}}({\bm{\xi}})}{\omega(\xi)}\sin(\omega(\xi)t)\,,$ (2.11)

hence, the claim of theorem follows. ∎

###### Proof of Theorem 2.2.

To prove Eq. (2.3), let us first recall that the Bessel function of first kind
is given by

$J_{0}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!\,\Gamma(m+1)}\left(\frac{x}{2}\right)^{2m}\,,$
(2.12)

where $\Gamma(x)$ is Euler Gamma function. Hence, we have that the dispersion
relation becomes

$\displaystyle\omega^{2}(\xi)=$
$\displaystyle-\frac{4\pi\kappa}{\rho\,\delta^{2\alpha}}\,\int_{0}^{1}\frac{1}{z^{1+2\alpha}}\sum_{m=1}^{\infty}\frac{(-1)^{m}(\xi\delta)^{2m}}{m!\,\Gamma(m+1)}\left(\frac{z}{2}\right)^{2m}dz$
$\displaystyle=$
$\displaystyle\xi^{2}\frac{\pi\kappa\delta^{2(1-\alpha)}}{\rho}\,\int_{0}^{1}z^{1-2\alpha}dz-\frac{4\pi\kappa}{\rho\,\delta^{2\alpha}}\,\int_{0}^{1}\frac{1}{z^{1+2\alpha}}\sum_{m=2}^{\infty}\frac{(-1)^{m}(\xi\delta)^{2m}}{m!\,\Gamma(m+1)}\left(\frac{z}{2}\right)^{2m}dz$
$\displaystyle=$
$\displaystyle\xi^{2}\frac{\pi\kappa\delta^{2(1-\alpha)}}{2(1-\alpha)\rho}-\frac{4\pi\kappa}{\rho\,\delta^{2\alpha}}\,\sum_{m=2}^{\infty}\frac{(-1)^{m}\xi^{2m}\delta^{2m}}{m!\,\Gamma(m+1)}\int_{0}^{1}\frac{z^{2m-1-2\alpha}}{2^{2m}}dz\,.$
(2.13)

Therefore,

$\displaystyle\lim_{\xi\rightarrow 0}\xi^{-2}\omega^{2}(\xi)=$
$\displaystyle\frac{\pi\kappa\delta^{2(1-\alpha)}}{2(1-\alpha)\rho}-\lim_{\xi\rightarrow
0}\frac{4\pi\kappa}{\rho\,\delta^{2\alpha}}\,\sum_{m=2}^{\infty}\frac{(-1)^{m}\xi^{2m-2}\delta^{2m}}{m!\,\Gamma(m+1)}\int_{0}^{1}\frac{z^{2m-1-2\alpha}}{2^{2m}}dz\,$
$\displaystyle=$
$\displaystyle\frac{\pi\kappa\delta^{2(1-\alpha)}}{2(1-\alpha)\rho}\,,$ (2.14)

which proves Eq. (2.3).

To prove Eq. (2.4), firstly we define $\tau:=\xi\delta z$ such that the
dispersive relation becomes

$\omega^{2}(\xi)=\frac{4\pi\kappa}{\rho}\xi^{2\alpha}\,\int_{0}^{\xi\delta}\frac{1-J_{0}(\tau)}{\tau^{1+2\alpha}}d\tau\,.$
(2.15)

The desired limit then is

$\lim_{\xi\rightarrow\infty}\xi^{-2\alpha}\omega^{2}(\xi)=\frac{4\pi\kappa}{\rho}\,\int_{0}^{\infty}\frac{1-J_{0}(\tau)}{\tau^{1+2\alpha}}d\tau\,,$
(2.16)

eventually proving the theorem. ∎

###### Proof of Corollary 2.3.

Firstly, we evaluate the Fourier transform of ${\bf v}_{0}(\mathbf{x})$

$\displaystyle\widehat{{\bf v}_{0}}({\bm{\xi}})=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}{\bf
v}_{0}\left(\sqrt{x^{2}_{1}+x^{2}_{2}}\right)e^{i\mathbf{x}\cdot{\bm{\xi}}}d\mathbf{x}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{0}^{\infty}\int_{0}^{2\pi}x\,{\bf
v}_{0}\left(x\right)e^{ix\xi\cos\left(\theta-\theta_{\xi}\right)}d\theta dx$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int_{0}^{\infty}x\,{\bf
v}_{0}\left(x\right)J_{0}(\xi x)dx\,,$ (2.17)

where we have adopted polar coordinates
${\bm{\xi}}=\xi\left(\cos\theta_{\xi},\sin\theta_{\xi}\right)$ and
$\mathbf{x}=x\left(\cos\theta,\sin\theta\right)$. We obtained then that
$\widehat{{\bf v}_{0}}({\bm{\xi}})$ is solely function of $\xi$ rather than
${\bm{\xi}}$. The same occurs for $\widehat{{\bf v}_{1}}({\bm{\xi}})$. Hence,
thanks to the Theorem 2.1, the unique solution of the Cauchy problem (1.1) is

$\displaystyle\mathbf{u}(t,\mathbf{x})=$
$\displaystyle\int_{\mathbb{R}^{2}}e^{-i{\bm{\xi}}\cdot\mathbf{x}}\left[\widehat{{\bf
v}_{0}}\left(\xi\right)\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}\left(\xi\right)}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d{\bm{\xi}}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\int_{0}^{2\pi}\xi e^{-i\xi
x\cos\left(\theta-\theta_{\xi}\right)}\left[\widehat{{\bf
v}_{0}}\left(\xi\right)\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}\left(\xi\right)}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d\theta_{\xi}d\xi$
$\displaystyle=$ $\displaystyle 2\pi\int_{0}^{\infty}\xi J_{0}(\xi
x)\left[\widehat{{\bf
v}_{0}}\left(\xi\right)\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}\left(\xi\right)}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d\xi\,,$
(2.18)

which indeed only depends on the scalar variable $x$. ∎

##  3 The three-dimensional case

######  Theorem 3.1.

Let ${\bf v}_{0},\,{\bf v}_{1}\in\mathcal{S}(\mathbb{R}^{3})$ and
$0<\alpha<1$. Then problem (1.1) with $N=3$, $p=2$ and $\Psi=0$ has the unique
solution
$\mathbf{u}:\mathbb{R}^{+}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ given
by

$\mathbf{u}(t,\mathbf{x})=\int_{\mathbb{R}^{3}}e^{-i{\bm{\xi}}\cdot\mathbf{x}}\left[\widehat{{\bf
v}_{0}}({\bm{\xi}})\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}({\bm{\xi}})}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d{\bm{\xi}}\,,$
(3.1)

where $\widehat{{\bf v}_{0}}({\bm{\xi}})$ and $\widehat{{\bf
v}_{1}}({\bm{\xi}})$ represent the Fourier transforms of ${\bf
v}_{0}(\mathbf{x})$ and ${\bf v}_{1}(\mathbf{x})$, and
$\omega:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is the dispersion relation
defined by

$\omega(\xi)=\left(\frac{8\pi\kappa}{\rho\,\delta^{2\alpha}}\,\int_{0}^{1}\frac{1-j_{0}(\xi\delta
z)}{z^{1+2\alpha}}dz\right)^{1/2},$ (3.2)

where $j_{0}(x):=\frac{\sin x}{x}$ is the $0$-th order spherical Bessel
function of the first kind.

######  Theorem 3.2.

For $\delta>0$ and $0<\alpha<1$, the following asymptotic relations hold true.

$\lim_{\xi\rightarrow
0}\xi^{-2}\omega^{2}(\xi)=\frac{2\pi\kappa\,\delta^{2(1-\alpha)}}{3(1-\alpha)\,\rho}$
(3.3)

and

$\lim_{\xi\rightarrow\infty}\xi^{-2\alpha}\omega^{2}(\xi)=\frac{8\pi\kappa}{\rho}\,\int_{0}^{\infty}\frac{1-j_{0}(\tau)}{\tau^{1+2\alpha}}d\tau.$
(3.4)

######  Corollary 3.3 (Radial solution).

Let ${\bf v}_{0},\,{\bf v}_{1}\in\mathcal{S}(\mathbb{R}^{3})$ and
$0<\alpha<1$. Assume ${\bf v}_{0}$, ${\bf v}_{1}$ be such that

${\bf v}_{0}(\mathbf{x})={\bf
v}_{0}\left(\sqrt{x^{2}_{1}+x^{2}_{2}+x^{2}_{3}}\right)\qquad\text{and}\qquad{\bf
v}_{1}(\mathbf{x})={\bf
v}_{1}\left(\sqrt{x^{2}_{1}+x^{2}_{2}+x^{2}_{3}}\right)\,.$ (3.5)

Then, the unique solution of the Cauchy problem (1.1) is

$\mathbf{u}(t,x)=4\pi\int_{0}^{\infty}\xi^{2}\,j_{0}(\xi x)\left[\widehat{{\bf
v}_{0}}(\xi)\cos\left(\omega(\xi)\,t\right)+\frac{\widehat{{\bf
v}_{1}}(\xi)}{\omega(\xi)}\sin\left(\omega(\xi)\,t\right)\right]d\xi\,.$ (3.6)

######  Remark 3.4.

Analogously to the two-dimensional case, the dispersive relation (3.2) only
depends on the modulus $\xi$ of the Fourier vector ${\bm{\xi}}$. Moreover, the
solution (3.6) is radial on the sphere since it only depends on the modulus
$x$ of the vector $\mathbf{x}$.

###### Proof of Theorem 3.1.

This proof follows the one already presented for the two-dimensional case. To
this end, we need to apply the 3D Fourier transform within the operator $K$

$\displaystyle(K{\bf u}(t,\cdot))({\bf x})\equiv$
$\displaystyle-2\int_{B_{\delta}(0)}\frac{{\bf\widehat{u}}(t,{\bf
x})-{\bf\widehat{u}}(t,{\bf x}-{\bf y})}{|{\bf y}|^{3+2\alpha}}d{\bf y}$
$\displaystyle=$
$\displaystyle-2\int_{\mathbb{R}^{3}}\,{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\underbrace{\int_{\mathcal{S}^{3}(\delta)}\frac{1-e^{i{\bm{\xi}}\cdot{\bf
y}}}{|{\bf y}|^{3+2\alpha}}d{\bf y}}d{\bm{\xi}}$ $\displaystyle=$
$\displaystyle-2\int_{\mathbb{R}^{3}}\,{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\underbrace{\int_{0}^{\delta}\frac{4\pi}{y^{1+2\alpha}}\left(1-\frac{\sin\xi
y}{\xi y}\right)dy}d{\bm{\xi}}$ $\displaystyle=$
$\displaystyle-8\pi\,\delta^{-2\alpha}\int_{\mathbb{R}^{3}}\,{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\int_{0}^{1}\frac{1-j_{0}(\xi\delta z)}{z^{1+2\alpha}}dzd{\bm{\xi}}\,,$
(3.7)

where we have defined $z:=z\delta$ in the last line. The equality between the
under-braced terms in the second and third line is given as follows. Since the
integral over the sphere is invariant for rotation of the coordinate system,
we first align the component $y_{3}$ with the direction of ${\bm{\xi}}$. In
this way, the integral can be written in polar coordinates as

$\displaystyle\int_{\mathcal{S}^{3}(\delta)}\frac{1-e^{i{\bm{\xi}}\cdot{\bf
y}}}{|{\bf y}|^{3+2\alpha}}d{\bf y}=$
$\displaystyle\int_{0}^{\infty}\int_{-1}^{1}\int_{0}^{2\pi}\frac{1-e^{i\xi
y\nu}}{|{\bf y}|^{3+2\alpha}}y^{2}d\phi d\nu dy$ $\displaystyle=$
$\displaystyle 2\pi\int_{0}^{\infty}\int_{-1}^{1}\frac{1-e^{i\xi
y\nu}}{y^{1+2\alpha}}d\nu dy\,,$ (3.8)

where $\nu:=\frac{{\bm{\xi}}\cdot{\bf y}}{\xi y}$. At this point, the
integration over $\nu$ in the last equation can be written as

$\int_{-1}^{1}(1-e^{i\xi y\nu})d\nu=\int_{-1}^{1}(1-\cos\xi
y\nu)d\nu-i\underbrace{\int_{-1}^{1}\sin\xi y\nu
d\nu}_{=0}=2\left(1-\frac{\sin\xi y}{\xi y}\right)\,,$ (3.9)

leading then to

$(K{\bf u}(t,\cdot))({\bf
x})=-\rho\int_{\mathbb{R}^{3}}\,{\bf\widehat{u}}(t,{\bm{\xi}})\,e^{-i{\bm{\xi}}\cdot{\bf
x}}\omega^{2}(\xi)d{\bm{\xi}}\,,$ (3.10)

with

$\omega^{2}(\xi)=\frac{8\pi\kappa}{\rho\,\delta^{2\alpha}}\int_{0}^{1}\frac{dz}{z^{1+2\alpha}}\left(1-\frac{\sin\xi\delta
z}{\xi\delta z}\right)\,.$ (3.11)

Hence, the proof follows in the sam way as the one of Theorem 2.1. Indeed, the
three-dimensional Cauchy problem in Fourier space reads

$\begin{cases}\partial_{tt}{\bf\widehat{u}}(t,{\bm{\xi}})+\omega^{2}(\xi){\bf\widehat{u}}(t,{\bm{\xi}})=0,&\quad{\bm{\xi}}\in\mathbb{R}^{3},\>t>0,\\\
\widehat{\mathbf{u}}(0,{\bm{\xi}})=\widehat{\mathbf{u}}_{0}({\bm{\xi}}),\>\partial_{t}\widehat{\mathbf{u}}(0,{\bm{\xi}})=\widehat{\mathbf{v}}_{0}({\bm{\xi}}),&\quad{\bm{\xi}}\in\mathbb{R}^{3},\end{cases}$
(3.12)

which is solved by

$\widehat{\mathbf{u}}(t,{\bm{\xi}})=\widehat{{\bf
v}_{0}}({\bm{\xi}})\cos(\omega(\xi)t)+\frac{\widehat{{\bf
v}_{1}}({\bm{\xi}})}{\omega(\xi)}\sin(\omega(\xi)t)\,,$ (3.13)

hence, the proof of the theorem is concluded. ∎

###### Proof of Theorem 3.2.

To prove Eq. (3.3), let us expand the Bessel function $j_{0}(x)$ as

$j_{0}(x):=\frac{\sin
x}{x}=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(2m+1)!}x^{2m}\,.$ (3.14)

Hence, the dispersion relation becomes

$\displaystyle\omega^{2}(\xi)=$
$\displaystyle-\frac{8\pi\kappa}{\rho\,\delta^{2\alpha}}\int_{0}^{1}\frac{dz}{z^{1+2\alpha}}\sum_{m=1}^{\infty}\frac{(-1)^{m}}{(2m+1)!}(\xi\delta
z)^{2m}$ $\displaystyle=$
$\displaystyle\xi^{2}\frac{4\pi\kappa\delta^{2(1-\alpha)}}{3\rho}\int_{0}^{1}z^{1-2\alpha}dz-\frac{8\pi\kappa}{\rho\,\delta^{2\alpha}}\int_{0}^{1}\frac{dz}{z^{1+2\alpha}}\sum_{m=2}^{\infty}\frac{(-1)^{m}}{(2m+1)!}(\xi\delta
z)^{2m}$ $\displaystyle=$
$\displaystyle\xi^{2}\frac{2\pi\kappa\delta^{2(1-\alpha)}}{3(1-\alpha)\rho}-\frac{8\pi\kappa}{\rho\,\delta^{2\alpha}}\int_{0}^{1}\frac{dz}{z^{1+2\alpha}}\sum_{m=2}^{\infty}\frac{(-1)^{m}}{(2m+1)!}(\xi\delta
z)^{2m}\,,$ (3.15)

and then

$\lim_{\xi\rightarrow
0}\xi^{-2}\omega^{2}(\xi)=\frac{2\pi\kappa\delta^{2(1-\alpha)}}{3(1-\alpha)\rho}\,,$
(3.16)

which proves Eq. (3.3).

For the second part of the proof, we first define $\tau:=\xi\delta z$ such
that the dispersive relation becomes

$\omega^{2}(\xi)=\frac{8\pi\kappa}{\rho}\xi^{2\alpha}\,\int_{0}^{\xi\delta}\frac{1-j_{0}(\tau)}{\tau^{1+2\alpha}}d\tau\,.$
(3.17)

The desired limit then is

$\lim_{\xi\rightarrow\infty}\xi^{-2\alpha}\omega^{2}(\xi)=\frac{8\pi\kappa}{\rho}\,\int_{0}^{\infty}\frac{1-j_{0}(\tau)}{\tau^{1+2\alpha}}d\tau\,,$
(3.18)

eventually proving the theorem. ∎

###### Proof of Corollary 3.3.

Just as for the proof of Corollary 2.3, it is enough to prove that
$\widehat{{\bf v}_{0}}$ and $\widehat{{\bf v}_{1}}$ only depends on the
modulus $\xi$. Indeed, we have

$\displaystyle\widehat{{\bf v}_{0}}({\bm{\xi}})=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{2}}{\bf
v}_{0}\left(\sqrt{x^{2}_{1}+x^{2}_{2}+x^{2}_{3}}\right)e^{i\mathbf{x}\cdot{\bm{\xi}}}d\mathbf{x}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int_{-1}^{1}\int_{0}^{\infty}\int_{0}^{2\pi}x^{2}\,{\bf
v}_{0}\left(x\right)e^{ix\xi\nu}d\phi d\nu dx$ $\displaystyle=$
$\displaystyle\frac{2}{(2\pi)^{2}}\int_{0}^{\infty}x^{2}\,{\bf
v}_{0}\left(x\right)j_{0}(\xi x)dx\,,$ (3.19)

where we have adopted polar coordinates and aligned the $z$-axis of the
reference system with the direction of $\mathbf{x}$ and defined
$\nu:=\frac{{\bm{\xi}}\cdot\mathbf{x}}{\xi x}$. We obtained then that
$\widehat{{\bf v}_{0}}({\bm{\xi}})$ is solely function of $\xi$ rather than
${\bm{\xi}}$. The same occurs for $\widehat{{\bf v}_{1}}({\bm{\xi}})$. ∎

##  4 Results and discussion

###  4.1 Analytical findings

A direct comparison with the one-dimensional case shows that the two- and
three-dimensional cases share the same scaling with the uni-dimensional
solutions. As a matter of fact, the dispersive relations (2.2) and (3.2) have
the same dependence on $\delta^{-2\alpha}$ occurring in (1.5) as
$\xi\rightarrow 0$ and $\xi\rightarrow\infty$. Since this dependence is
intimately related to the scaling of $\kappa$ vs $\delta$, this feature is a
hint of the fact that this dependence is universal regardless of the dimension
of the system.

Moreover, the dispersive relation depends solely on the modulus $\xi$ rather
than the entire Fourier vector ${\bm{\xi}}$, no matter what the dimension of
the system is. This behavior is somehow expected for small value of $\xi$,
i.e. $\xi\delta\ll 1$. Indeed, in this regime the spatial scale of the Fourier
mode is much greater than the peridynamics radius $\delta$: here the dynamics
of the standard elastic theory must be recovered and then the dispersive
relation must depend only on $\xi$. On the other hand, having recovered this
isotropy also on small scales Fourier modes, where $\xi\delta\gg 1$, is less
trivial. We address this interesting feature to the fact that the integral
peridynamic kernel $K$ averages with the same weight all the possible
directions of the shift vector
$\mathbf{u}(\mathbf{x})-\mathbf{u}(\mathbf{x}-\mathbf{y})$.

| $\xi\delta\ll 1$ | $\xi\delta\gg 1$
---|---|---
$N=1$ | $\frac{\kappa\,\delta^{2(1-\alpha)}}{(1-\alpha)\,\rho}\xi^{2}$ | $-\frac{4\kappa}{\rho}\cos(\pi\alpha)\Gamma(-2\alpha)\,\xi^{2\alpha}$
$N=2$ | $\frac{\pi\kappa\,\delta^{2(1-\alpha)}}{2(1-\alpha)\,\rho}\xi^{2}$ | $-\frac{2^{1-2\alpha}\pi\kappa}{\rho}\frac{\Gamma(-\alpha)}{\Gamma(1+\alpha)}\xi^{2\alpha}$
$N=3$ | $\frac{2\pi\kappa\,\delta^{2(1-\alpha)}}{3(1-\alpha)\,\rho}\xi^{2}$ | $\frac{8\pi\kappa}{\rho}\cos(\pi\alpha)\Gamma(-1-2\alpha)\,\xi^{2\alpha}$
Table 1: Dispersive relations in the limit regimes of large ($\xi\delta\ll 1$)
and small ($\xi\delta\ll 1$) scales in terms of the dimensionality of the
system.

To thoroughly characterize the dispersive relation, we resume in Table 1 the
asymptotic limits of $\omega^{2}(\xi)$ for $\xi\delta\ll 1$ and $\xi\delta\gg
1$ for different values of the dimension $N$. As a further step with respect
to the claims of Theorems 1.7, 2.4 and 3.4, the small scales limits have been
formally written in terms of the Euler Gamma function $\Gamma(x)$ and these
are shown in Fig.1.

Figure 1: Small scales ($\xi\delta\gg 1$) amplitude of the dispersive relation
versus the value of $\alpha$ for different dimensions, with $\kappa=\rho=1$.

Two different features concerning small and large scales regimes clearly
emerge:

  * 1.

for $\xi\delta\ll 1$, the limits for different values of $N$ have different
amplitudes. However, they shares an universal scaling with the constitutive
parameters of the peridynamics $\delta$ and $\alpha$;

  * 2.

on the contrary, when $\xi\delta\gg 1$, the overall amplitudes do not depend
on $\delta$. Nevertheless, the dependence on $\alpha$ changes with $N$. From
Fig. 1, it is evident that the qualitative behavior is the same, even though
the lower the dimension, the smaller the amplitude. This feature is much
evident for small values of $\alpha$.

###  4.2 Numerical experiments

To conclude this work, we provide a numerical study for the propagation of
elastic deformations in the two and three dimensional case. The analysis that
we are going to present is analogous to the one for the one dimensional case
in [6, 5]. For the purposes of this paper, we will show numerical evidences
that the speed of propagation for an elastic deformation is weakly dependent
on the dimensionality of the system, whereas the constitutive micro-scale
parameters $\alpha$ and $\delta$ are the main ingredients to estimate such
speed of propagation. To this end, we study numerical solutions in
correspondence of the following initial conditions

$v_{0}(r)=\left(2\pi\right)^{N/2}\exp\left(-\frac{r^{2}}{2\,\sigma^{2}}\right)\qquad\text{and}\qquad
v_{1}(r)=0\,,$ (4.1)

in complete analogy with the one dimensional system. We also remark that we
are limiting our case study by taking $v_{0}$ and $v_{1}$ as function in
$\mathbb{R}$ rather than $\mathbb{R}^{N}$. This choice has been done for the
sake of simplicity since our conclusion can be immediately extended to the
general case.

According to our convention, the Fourier transforms of the initial conditions
(4.1) then become

$\widehat{v_{0}}(\xi)=\sigma^{N}\exp\left(-\frac{\xi^{2}\sigma^{2}}{2}\right)\qquad\text{and}\qquad\widehat{v_{1}}(\xi)=0\,,$
(4.2)

which satisfy the assumptions of Corollaries 2.3 and 3.3. The role of $\sigma$
in the initial conditions is crucial to put in evidence the dispersive
features due to peridynamic. Indeed, Eq. (4.2) undisputedly privileges the
Fourier modes such that $\xi\sigma\lesssim 1$ in the imprinted deformations.
The peculiar scales $\xi$ and $\sigma$ then must be compared with the unique
constitutive scale $\delta$, since the latter marks the change in the regime
for the dispersive relation.

In details, based on the results obtained for the one dimensional case [6, 5],
we expect also for the multi-dimensional scenarios that:

  * i)

When $\sigma\gtrsim\delta$, the dispersion due to the peridynamic kernel is
hidden since only large scale modes are switched on, given the hierarchy
$\xi\lesssim\sigma^{-1}\lesssim\delta^{-1}$. With reference to Table 1, these
modes can only be dispersed in qualitative agreement with the standard theory
of elasticity and the peridynamics parameters may play a role only in the
effective group velocity.

  * ii)

Conversely, when $\sigma\ll\delta$, the above-mentioned hierarchy among the
scales translates as $\xi\lesssim\sigma^{-1}$ and $\delta^{-1}\ll\sigma^{-1}$:
in principle, both modes $\xi>\delta^{-1}$ and $\xi<\delta^{-1}$ can propagate
throughout the system, enabling then the peridynamics evolution to be
investigated. Moreover, as a matter of fact, smaller scale modes will travel
slower than the large ones, due to the sub-linearity of the dispersive
relation.

The understanding described so far is based on the results shown in the
previous sections and the detailed study of the one-dimensional case. In the
following, we will provide numerical evidences for our claimed phenomenology.
To this end, for both the two and three dimensional propagation, we will keep
fixed $\kappa=\rho=\delta=1$ and consider the elucidative cases of
$\alpha=\\{1/10,1/2,9/10\\}$ and $\sigma=1$ or $\sigma=1/10$, respectively
case i) and ii) in the item list above. The choice for $\delta$ to be equal to
$1$ might look peculiar. However, it will simplify our discussion concerning
the speed of propagation of the elastic deformations, since it will hide all
the factors $\delta^{\alpha-1}$ in Table 1. We remark that this will not
provide any loss of generality if our numerical investigations, since what
physically matters to enlighten the dispersive phenomena is the ratio
$\delta/\sigma$ rather than their individual amplitudes.

We begin by focusing on the two-dimensional case for the case i) in Fig. 2 and
case ii) in Fig. 3. For $\sigma=1$, Fig. 2 exhibits a really mild dispersion
as time goes on, regardless on the value of $\alpha$. What changes instead is
the speed of propagation of the deformation imprinted by the initial
conditions. An estimate of this speed of propagation can be actually obtained
from Table 1. In fact, for the science case $\sigma=\delta=1$, a non-rigorous
but nonetheless accurate estimate for the speed of propagation can be inferred
by the group velocity222A detailed Proof of equivalence between the velocity
of the energy flux and the group velocity for to one-dimensional linear
peridynamics has been provided in [5]. For the two and three dimensional case,
we limit our analysis to the numerical evidences.
$v_{g}:=\omega^{\prime}(\xi)$. When $\xi\delta\lesssim 1$, we have that
$v_{g}\approx\delta^{1-\alpha}\sqrt{\frac{\pi\kappa\,}{2(1-\alpha)\,\rho}}=\sqrt{\frac{\pi\,}{2(1-\alpha)}}$:
the higher $\alpha$, the faster the propagation. In details,

$\alpha$ | $v_{g}$
---|---
$9/10$ | $3.96$
$1/2$ | $1.77$
$1/10$ | $1.32$
Table 2: Estimation of the large scale group velocity for the two-dimensional
case with $\kappa=\rho=\delta=1$ for the chosen values of $\alpha$ in Fig. 2.

despite its roughness, our estimation for the speed of propagation of the
signals is in quite good agreement with what emerges from Fig. 2.

Figure 2: Isotropic 2D solution with $\kappa=\rho=\delta=1$ and $\sigma=1$ for
$t=0,1,2$ (from right to left panels) and for $\alpha=9/10,1/2,1/10$ (from top
to bottom panels).

The same numerical investigation has been repeated for the case ii) with
$\sigma=1/10$ and reported in Fig. 3. Two main differences are evident between
the two cases: when $\sigma=1/10$, the dispersion is evident and the looks
slower than the case i).

Figure 3: Isotropic 2D solution with $\kappa=\rho=\delta=1$ and $\sigma=1/10$
for $t=0,1,2$ (from right to left panels) and for $\alpha=9/10,1/2,1/10$ (from
top to bottom panels).

These features can be explained again by looking at the Table 1. Indeed, by
invoking the role of the velocity group also for the case ii), all the scales
such that $\xi\gg\delta$ travel with group velocity which is suppressed by a
factor $\xi^{\alpha-1}$ and this suppression is undisputedly more severe for
smaller values of $\alpha$.

The phenomenology of the three dimensional cases is qualitatively the same as
the the two dimensional ones and this can be appreciated in Figs. 4 and 5
respectively for $\sigma=1$ and $\sigma=1/10$.

Figure 4: Isotropic 3D solution with $\kappa=\rho=\delta=1$ and $\sigma=1$ for
$t=0,1,2$ (from right to left panels) and for $\alpha=9/10,1/2,1/10$ (from top
to bottom panels).

The main quantitative disclaimer w.r.t. to the two dimensional scenario can be
provided in regard of the group velocity for the large scale modes (i.e.
$\xi\delta\lesssim 1$). From Table 1, we get that the large scale group
velocity for the three dimensional case is
$v_{g}\approx\delta^{1-\alpha}\sqrt{\frac{2\pi\kappa}{3(1-\alpha)\,\rho}}=\sqrt{\frac{2\pi}{3(1-\alpha)}}$.

$\alpha$ | $v_{g}$
---|---
$9/10$ | $4.58$
$1/2$ | $2.05$
$1/10$ | $1.53$
Table 3: Estimation of the large scale group velocity for the three-
dimensional case with $\kappa=\rho=\delta=1$ for the chosen values of $\alpha$
in Fig. 4. Figure 5: Isotropic 3D solution with $\kappa=\rho=\delta=1$ and
$\sigma=1/10$ for $t=0,1,2$ (from right to left panels) and for
$\alpha=9/10,1/2,1/10$ (from top to bottom panels).

Interestingly, for the three-dimensional case the intrinsic velocity of the
deformation is slightly higher than the two dimensional case and this is
consistent with Fig. 4. From a more qualitative point of view, this speed of
propagation is suppressed for smaller scales perturbations exactly as expected
by the $\xi^{\alpha-1}$ suppression already discussed for the two dimensional
case.

##  5 Anisotropic initial conditions

Figure 6: Anisotropic 3D solution with $\kappa=\rho=\delta=1$ and
$\sigma=1/10$ for $t=0,2,4$ (from right to left panels) and for
$\alpha=9/10,1/2,1/10$ (from top to bottom panels).

The machinery developed so far can be applied to investigate the propagation
of anisotropic initial conditions even though the peridynamic kernel is
isotropic. Indeed, according to the hypothesis of Theorem 3.1, an anisotropic
3D Cauchy problem is given by two initial conditions which are
functions333Here $\mathbf{n}$ indicates the direction subtended by the angles
$\left(\theta,\phi\right)$. To lighten the notation, in the remaining part of
the work, we will use $\mathbf{n}$ to indicate

$\displaystyle v_{0}=v_{0}(r,\mathbf{n})\qquad\text{and}\qquad
v_{1}=v_{1}(r,\mathbf{n})\,,$ (5.1)

where, in polar coordinates, $v_{0},v_{1}\in\mathcal{S}(\mathbb{R})\times
L^{2}\left(\left[0,\pi\right]\times\left[0,2\pi\right]\right)$. Therefore,
they can be written in terms of the spherical harmonic $Y_{\ell
m}(\mathbf{n})$ basis, namely

$\displaystyle
v_{0}(r,\mathbf{n})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}v^{0}_{\ell
m}(r)Y_{\ell m}(\mathbf{n})\,,$ $\displaystyle
v_{1}(r,\mathbf{n})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}v^{1}_{\ell
m}(r)Y_{\ell m}(\mathbf{n})\,,$ (5.2)

where

$\displaystyle v^{0}_{\ell
m}(r):=\int_{B^{2}(1)}v_{0}(r,\mathbf{n})Y^{*}_{\ell
m}(\mathbf{n})d\mathbf{n}\,,$ $\displaystyle v^{1}_{\ell
m}(r):=\int_{B^{2}(1)}v_{1}(r,\mathbf{n})Y^{*}_{\ell
m}(\mathbf{n})d\mathbf{n}\,.$ (5.3)

Here $v^{0}_{\ell m},v^{1}_{\ell m}(r):\mathbb{R}^{+}\rightarrow\mathbb{C}$,
$v^{0}_{\ell m},v^{1}_{\ell m}(r)\in\mathcal{S}(\mathbb{R})$ and $B_{1}(0)$ is
the unitary 2D sphere. In this case, we have the explicit solution (3.1) given
by

$\displaystyle u\left(t,r,\mathbf{n}\right)=$
$\displaystyle\frac{2}{\pi}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\,Y_{LM}\left(\mathbf{n}\right)\int_{0}^{\infty}\xi^{2}j_{L}\left(\xi
r\right)\,\cos\left(\omega(\xi)\,t\right)\int_{0}^{\infty}v^{0}_{LM}(s)s^{2}\,j_{L}\left(\xi
s\right)dsd\xi$
$\displaystyle+\frac{2}{\pi}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\,Y_{LM}\left(\mathbf{n}\right)\int_{0}^{\infty}\xi^{2}j_{L}\left(\xi
r\right)\,\frac{\sin\left(\omega(\xi)\,t\right)}{\omega(\xi)}\int_{0}^{\infty}v^{1}_{LM}(s)s^{2}\,j_{L}\left(\xi
s\right)dsd\xi\,.$ (5.4)

The detailed proof of (5.4) for a generic angular dependence is reported in A.
The remarkable technical point is that the factorisation occurring between the
radial and the angular dependence in (5.4) is general. Moreover, the numerical
advantage of this chosen form stands in the fact that the needed numerical
scheme to study the anisotropic initial conditions is the same as the
isotropic one, namely a countable series two-dimensional integrals. This
property is a general feature of the above-mentioned factorisation between the
angular and the radial dependences and follows from the explicit expansion in
the spherical harmonics basis.

As an informative science case study, we consider the 3D solution (5.4) with
initial conditions given by

$\displaystyle v_{0}(r,\theta)=$
$\displaystyle\left(2\pi\right)^{3/2}\exp\left(-\frac{r^{2}}{2\,\sigma^{2}}\right)r\cos\theta$
$\displaystyle=$ $\displaystyle
4\pi^{2}\sqrt{\frac{2}{3}}\exp\left(-\frac{r^{2}}{2\,\sigma^{2}}\right)r\,Y_{10}\left(\mathbf{n}\right)\,,$
$\displaystyle v_{1}(r,\theta)=$ $\displaystyle 0\,.$ (5.5)

To our ends, it is then enough to evaluate the multipoles of the initial
conditions in Eqs. (5.3) to make use of the factorised solution (5.4). We
straightforwardly have that

$\displaystyle v^{0}_{\ell m}(r)=$ $\displaystyle
4\pi^{2}\sqrt{\frac{2}{3}}\exp\left(-\frac{r^{2}}{2\,\sigma^{2}}\right)r\int_{B^{2}(1)}Y_{10}\left(\mathbf{n}\right)Y^{*}_{\ell
m}(\mathbf{n})d\mathbf{n}$ $\displaystyle=$ $\displaystyle
4\pi^{2}\sqrt{\frac{2}{3}}\exp\left(-\frac{r^{2}}{2\,\sigma^{2}}\right)r\delta_{\ell
1}\,\delta_{m0}\,,$ $\displaystyle v^{1}_{\ell m}(r)=$ $\displaystyle 0\,.$

The Kronecker delta for $v^{0}_{\ell m}$ follows from the orthonormality of
the spherical harmonics. From a physical viewpoint, the fact that only
$\ell=1$ multipoles survive means that the we are dealing with a dipolar
anisotropy, namely a our initial conditions have a front-back asymmetry. The
numerical solutions for this dipolar anisotropy and different values of
$\alpha$ are shown in Fig. 6.

Here we have considered just the case $\sigma/\delta=10^{-1}\ll 1$ where the
dispersive behavior is more evident. It is evident from the numerical
solutions that the “shape” of the anisotropy is preserved as time passes by.
This is a consequence of the factorization already discussed. All the other
features regarding the dependence of the speed of propagation and the
attenuation of the overall amplitude are manifest in accordance with what
already pointed out in the radial case.

## Conclusions and future works

The analysis of the dispersive relation for the peridynamics evolution of a
two- and three-dimensional system has been deepened in this work through
suitable numerical and analytical findings. We show that cases with different
dimensionalities exhibit the dependence on the nonlocal interaction length
$\delta$ in their dispersive relation, thus enlightening the universal
characteristic of this dependence. Moreover, the dispersive relations are
found to depend solely on the modulus of the Fourier variable $\xi$ and the
two scenario related to small ($\xi\delta\ll 1$) and large scales
($\xi\delta\gg 1$) are analyzed in details.

## Acknowledgments

AC is a member of Gruppo Nazionale per il Calcolo Scientifico (GNCS), GF of
Gruppo Nazionale per la Fisica Matematica (GNFM), GMC and FM of Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work was
partially supported by the project “Research for Innovation” (REFIN) - POR
Puglia FESR FSE 2014-2020 - Asse X - Azione 10.4 (Grant No. CUP -
D94120001410008); Fundacao para a Ciencia e a Tecnologia (FCT) under the
program “Stimulus” with grant no. CEECIND/04399/2017/CP1387/CT0026 and through
the research project with ref. number PTDC/FIS-AST/0054/2021; Italian Ministry
of Education, University and Research under the Programme Department of
Excellence Legge 232/2016 (Grant No. CUP - D94I18000260001). GMC expresses its
gratitude to HIAS - Hamburg Institute for Advanced Study for their warm
hospitality.

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##  Appendix A Proof of Eq. (5.4)

To prove that the initial conditions (5.1) with the multipolar decompositions
(5.2) actually admit Eqs. (5.4) for the proposed anisotropic Cauchy problem,
let us first evaluate the Fourier transform of $v_{0}(r,\mathbf{n})$. Without
any loss of generality, we write a generic anisotropy in the form

$v_{0}(r,\mathbf{n})=\sum_{L=0}^{\infty}\sum_{M=-L}^{L}v^{0}_{LM}(r)Y_{LM}(\mathbf{n})\,,$
(A.1)

where
$\mathbf{n}:=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\phi\right)$ is
the direction subtended by the position $\mathbf{x}$ and $Y_{LM}$ are the
spherical harmonics. The advantage of using $Y_{LM}$ stands in the fact that
they provide a basis for all space of the functions
$f\left(\mathbf{n}\right)$, i.e. we can express any anisotropy in the 3D space
as a linear combination of spherical harmonics. Moreover, the dependences on
$\theta$ and $\phi$ are respectively discretised by the indices $\ell$ and
$m$.

Hence, according to our conventions for the Fourier transform, we have

$\displaystyle\widehat{v_{0}}(\xi,\mathbf{n}_{\xi})=$
$\displaystyle\frac{1}{\left(2\pi\right)^{3}}\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}v_{0}(r,\mathbf{n})\,e^{i\,r\xi\,\mathbf{n}\cdot\mathbf{n}_{\xi}}\,r^{2}\sin\theta\,drd\theta
d\phi$ $\displaystyle=$
$\displaystyle\frac{1}{\left(2\pi\right)^{3}}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}v^{0}_{LM}(r)Y_{LM}\left(\mathbf{n}\right)\,e^{i\,r\xi\,\mathbf{n}\cdot\mathbf{n}_{\xi}}\,r^{2}\sin\theta\,drd\theta
d\phi\,,$ (A.2)

where $\mathbf{n}$ and $\mathbf{n}_{\xi}$ are respectively the direction on
the unitary sphere of $\mathbf{x}$ and ${\bm{\xi}}$. To our ends, we use the
spherical harmonics decomposition for the exponential

$e^{i\,r\xi\,\mathbf{n}\cdot\mathbf{n}_{\xi}}=4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}i^{\ell}\,j_{\ell}\left(\xi
r\right)\,Y_{\ell m}\left(\mathbf{n}\right)Y^{*}_{\ell
m}\left(\mathbf{n}_{\xi}\right)\,,$ (A.3)

where $j_{\ell}(x)$ are the spherical Bessel functions of $\ell$-th order and
$*$ denotes the complex conjugation. In this way, we can write Eq. (A.2) as

$\displaystyle\widehat{v_{0}}(\xi,\mathbf{n}_{\xi})=$
$\displaystyle\left(-1\right)^{M}\frac{4\pi}{\left(2\pi\right)^{3}}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}i^{\ell}\,Y^{*}_{\ell
m}\left(\mathbf{n}_{\xi}\right)\int_{0}^{\infty}v^{0}_{LM}(r)r^{2}\,j_{\ell}\left(\xi
r\right)dr$
$\displaystyle\times\underbrace{\int_{0}^{\pi}\int_{0}^{2\pi}Y^{*}_{L-M}\left(\mathbf{n}\right)\,Y_{\ell
m}\left(\mathbf{n}\right)\,\sin\theta d\theta d\phi}_{=\delta_{\ell
L}\delta_{m-M}}$ $\displaystyle=$
$\displaystyle\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\left(-1\right)^{M}\frac{4\pi}{\left(2\pi\right)^{3}}i^{L}\,Y^{*}_{L-M}\left(\mathbf{n}_{\xi}\right)\int_{0}^{\infty}v^{0}_{LM}(r)r^{2}\,j_{L}\left(\xi
r\right)dr\,,$ (A.4)

where in the second line we have used the orthonormal condition of the
spherical harmonics

$\int_{0}^{\pi}\int_{0}^{2\pi}Y_{\ell
m}\left(\mathbf{n}\right)Y^{*}_{\ell^{\prime}m^{\prime}}\left(\mathbf{n}\right)\,\sin\theta
d\theta d\phi=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$

In the same way, we also have that

$\displaystyle\widehat{v_{1}}(\xi,\mathbf{n}_{\xi})=$
$\displaystyle\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\left(-1\right)^{M}\frac{4\pi}{\left(2\pi\right)^{3}}i^{L}\,Y^{*}_{L-M}\left(\mathbf{n}_{\xi}\right)\int_{0}^{\infty}v^{1}_{LM}(r)r^{2}\,j_{L}\left(\xi
r\right)dr\,,$ (A.5)

Now we can finally prove our solution directly from the theorem 3.1 with the
aim of Eqs. (A.4). We get

$\displaystyle u(t,r,\mathbf{n})=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}e^{-i\,\xi
r\,\mathbf{n}\cdot\mathbf{n}_{\xi}}\widehat{v_{0}}(\xi,\mathbf{n}_{\xi})\cos\left(\omega(\xi)\,t\right)\xi^{2}\sin\theta_{\xi}d\xi
d\theta_{\xi}d\phi_{\xi}$
$\displaystyle+\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}e^{-i\,\xi
r\,\mathbf{n}\cdot\mathbf{n}_{\xi}}\widehat{v_{1}}(\xi,\mathbf{n}_{\xi})\frac{\sin\left(\omega(\xi)\,t\right)}{\omega(\xi)}\xi^{2}\sin\theta_{\xi}d\xi
d\theta_{\xi}d\phi_{\xi}$ $\displaystyle=$
$\displaystyle\frac{\left(4\pi\right)^{2}}{\left(2\pi\right)^{3}}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\left(-i\right)^{\ell}\,i^{L}\,Y_{\ell
m}\left(\mathbf{n}\right)\underbrace{\int_{0}^{\pi}\int_{0}^{2\pi}\,\left(-1\right)^{M}Y^{*}_{L-M}\left(\mathbf{n}_{\xi}\right)Y^{*}_{\ell
m}\left(\mathbf{n}_{\xi}\right)\sin\theta_{\xi}d\theta_{\xi}d\phi_{\xi}}_{=\delta_{L\ell}\delta_{Mm}}$
$\displaystyle\times\int_{0}^{\infty}\xi^{2}j_{\ell}\left(\xi
r\right)\,\cos\left(\omega(\xi)\,t\right)\int_{0}^{\infty}v^{0}_{LM}(s)s^{2}\,j_{L}\left(\xi
s\right)dsd\xi$
$\displaystyle+\frac{\left(4\pi\right)^{2}}{\left(2\pi\right)^{3}}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\left(-i\right)^{\ell}\,i^{L}\,Y_{\ell
m}\left(\mathbf{n}\right)\underbrace{\int_{0}^{\pi}\int_{0}^{2\pi}\,\left(-1\right)^{M}Y^{*}_{L-M}\left(\mathbf{n}_{\xi}\right)Y^{*}_{\ell
m}\left(\mathbf{n}_{\xi}\right)\sin\theta_{\xi}d\theta_{\xi}d\phi_{\xi}}_{=\delta_{L\ell}\delta_{Mm}}$
$\displaystyle\times\int_{0}^{\infty}\xi^{2}j_{\ell}\left(\xi
r\right)\,\frac{\sin\left(\omega(\xi)\,t\right)}{\omega(\xi)}\int_{0}^{\infty}v^{1}_{LM}(s)s^{2}\,j_{L}\left(\xi
s\right)dsd\xi$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\,Y_{LM}\left(\mathbf{n}\right)\int_{0}^{\infty}\xi^{2}j_{L}\left(\xi
r\right)\,\cos\left(\omega(\xi)\,t\right)\int_{0}^{\infty}v^{0}_{LM}(s)s^{2}\,j_{L}\left(\xi
s\right)dsd\xi$
$\displaystyle+\frac{2}{\pi}\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\,Y_{LM}\left(\mathbf{n}\right)\int_{0}^{\infty}\xi^{2}j_{L}\left(\xi
r\right)\,\frac{\sin\left(\omega(\xi)\,t\right)}{\omega(\xi)}\int_{0}^{\infty}v^{1}_{LM}(s)s^{2}\,j_{L}\left(\xi
s\right)dsd\xi\,,$ (A.6)

where we have made use again of the decomposition (A.3) and of the properties
of the spherical harmonics.


# Unitary quantum gravitational physics and the CMB parity asymmetry

Enrique Gaztañaga  Institute of Cosmology & Gravitation, University of
Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, United
Kingdom  Institute of Space Sciences (ICE, CSIC), 08193 Barcelona, Spain
Institut d´Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain  K.
Sravan Kumar  Institute of Cosmology & Gravitation, University of Portsmouth,
Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, United Kingdom

###### Abstract

The proposal of Direct-Sum Quantum Field Theory (DQFT) offers a new
perspective for quantum fields by combining parity and time reversal
operations, blurring the distinction between quantum past and future while
preserving causality. This approach provides a unitary QFT in curved
spacetime, resolving the information-loss paradox. When applied to
inflationary quantum fluctuations, DQFT predicts variations in CMB
measurements, explaining longstanding anomalies as a result of parity
asymmetry. The data strongly supports the unitary treatment of quantum
fluctuations, with a probability over 650 times greater than the standard
prediction. This significant discrepancy underscores the validity and
importance of the DQFT approach in understanding the intricate relationship
between gravity and quantum mechanics.

Introduction - Understanding and testing the interplay between gravity and
quantum mechanics (QM) is the most important key to unraveling the new physics
beyond the standard model of particle physics and Einstein’s general
relativity (GR). The merge of GR and QM is an essential physics to be unlocked
at the length scales of gravitational horizons. For this, we perhaps need an
in-depth understanding of how one can resolve an age-old conundrum in the
concept of time between the GR and QM. The success of Quantum field theory
(QFT) in Minkowski spacetime Coleman (2018) already teaches us several lessons
in this regard. Indeed QFT is the culmination of special relativity (SR) and
QM where the concept of time already enters into the conflict. In SR, the
space and time are on equal footing but in QM spatial coordinates are elevated
to operators while time is just a parameter due to its anti-unitary nature
under discrete transformation ($\mathcal{T}:t\to-t$). The SR and QM are merged
by imposing the commutativity of operators corresponding to space-like
distances. The underlying reason is that SR forbids any sort of communication
happening in space-like distances and we respect it and achieve it in
formulating the rules of second quantization. This is exactly the lesson one
must keep in mind for understanding quantum fields in curved spacetime. That
is we take lessons from classical physics and apply them according to the
rules of quantum physics. The quantum field theory in curved spacetime (QFTCS)
is the first step in seeing GR and QM together besides the renormalizable
ultra-violet complete quantum gravity that may be required at the Planck
scales. Furthermore, QFTCS is a necessity in the context of understanding
inflationary quantum fluctuations Mukhanov and Chibisov (1981); Sasaki (1986).

In this letter, we address crucial questions related to the meaning of time
reversal and parity inversions in the context of inflationary background
spacetime to place quantum fields on it and uncover their true nature in the
cosmic microwave background (CMB). We explain how the direct-sum quantum field
theory (DQFT) is a necessity to resolve the unitarity and information-loss
problem of QFTCS. The question of unitarity in QFTCS precedes the question of
a completely renormalizable quantum gravity theory of Planck scales. In
contrast to the widespread schools of thought that argue unitarity in QFTCS
needs to be given up within GR and QM, we came up with a fundamental
understanding of it and an observational test of it by the remarkable
revelation of parity asymmetry in the CMB. We find that the theory of
inflationary quantum fluctuations with DQFT (which we call shortly DSI which
stands for ”Direct-Sum inflation”) fits up to 650 times better than the
Standard Inflation (SI). This letter ultimately presents the picture of how a
new theory of QFTCS emerged from fundamental questions of quantum
gravitational physics that can potentially explain the observed CMB large-
scale anomalies. The new results in this letter are further complemented with
our companion paper Gaztañaga and Sravan Kumar (2024).

DQFT, in a nutshell, ($\hbar=c=1$)- DQFT formulation emerges from a simple
demand that a quantum state or a quantum field operator respects the discrete
symmetries (Parity $\mathcal{P}$ and time reversal $\mathcal{T}$) of the
background spacetime. Here we summarize the conceptual understanding of DQFT
and direct the reader to Gaztañaga and Sravan Kumar (2024); Sravan Kumar and
Marto (2023a, b) for more details. We start with the direct-sum Schrödinger
equation

$i\frac{\partial|\Psi\rangle}{\partial
t_{p}}=\begin{pmatrix}\hat{\mathbb{H}}&0\\\
0&-\hat{\mathbb{H}}\end{pmatrix}|\Psi\rangle$ (1)

where $\mathbb{H}$ is the time-independent Hamiltonian (which is Hermitian)
that is $\mathcal{P}\mathcal{T}$ symmetric. The quantum state $|\Psi\rangle$
is the direct-sum of two components

$\displaystyle|\Psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\Psi_{+}\rangle\oplus|\Psi_{-}\rangle\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}|\Psi_{+}\rangle\\\
|\Psi_{-}\rangle\end{pmatrix}$ (2)
$\displaystyle\int_{-\infty}^{\infty}\langle\Psi|\Psi\rangle dx$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\Big{[}\langle\Psi_{+}|\Psi_{+}\rangle+\langle\Psi_{-}|\Psi_{-}\rangle\Big{]}dx=1\,.$

where $|\Psi_{+}\rangle,\,|\Psi_{-}\rangle$ are the two components of the same
state at parity conjugate points. The component $|\Psi_{+}\rangle$ is a
positive energy state that evolves forward in time as
$|\Psi_{+}\rangle_{t_{p}}=e^{-i\mathbb{H}t_{p}}|\Psi_{+}\rangle_{0}$ where the
parametric time arrow $t_{p}:-\infty\to\infty$ while the $|\Psi_{-}\rangle$
can also be seen as positive energy state that evolves back in time as
$|\Psi_{-}\rangle_{t_{p}}=e^{i\mathbb{H}t_{p}}|\Psi_{-}\rangle_{0}$ with the
parametric time convention $t_{p}:+\infty\to-\infty$. One can notice here the
purely anti-unitary character of time associated with the replacement $i\to-i$
Donoghue and Menezes (2019). The direct-sum Schrödinger equation does not have
any arrow of time associated with it in comparison to the conventional
Schrödinger equation. The direct-sum splitting of the quantum state (2)
implies the Hilbert space splitting into a direct-sum of the two corresponding
to parity conjugate points in position space:
$\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$. The position and momentum
commutation relations now become double associated with parity conjugate
position and momentum space operators denoted by subscripts $+,\,-$ and their
relations are

$\displaystyle\hat{p}_{+}=-i\frac{\partial}{\partial
x_{+}}\quad,\quad\Big{[}\hat{x}_{+},\,\hat{p}_{+}\Big{]}$
$\displaystyle=i,\quad\Big{[}\hat{x}_{-},\,\hat{p}_{-}\Big{]}=-i\,$ (3)
$\displaystyle\hat{p}_{-}=+i\frac{\partial}{\partial
x_{-}}\quad,\quad[\hat{x}_{+},\,\hat{x}_{-}]$
$\displaystyle=[\hat{p}_{+},\,\hat{p}_{-}]=0\,,$

This framework only makes the wavefunction of the quantum system explicitly
$\mathcal{P}\mathcal{T}$ symmetric that resonates with the assumed
$\mathcal{P}\mathcal{T}$ symmetry of the physical system i.e.,
$[\mathbb{H},\,\mathcal{P}\mathcal{T}]=0$ and the results of standard quantum
mechanics remains the same as shown with the example of harmonic oscillator
worked out in Gaztañaga and Sravan Kumar (2024). The DQFT in Minkowski
spacetime ($ds^{2}=-dt_{m}^{2}+d\textbf{x}^{2}$ which is
$\mathcal{P}\mathcal{T}$ symmetric) is just built on the direct-sum
Schrödinger equation, following how standard QFT is built. See Appendix A for
more details and Fig.1 for an illustration.

Figure 1: The left-hand side depicts the picture of a light cone where a
quantum field in standard QFT evolves from the past to the future while the
field operators commute for spacelike distances
$\left[\hat{\phi}\left(x\right),\
\hat{\phi}\left(y\right)\right]=0,\,\left(x-y\right)^{2}>0$ which is often
called the causality condition. The right-hand side light cone represents how
a quantum field in DQFT has two direct-sum components with opposite time
evolutions at parity conjugate regions satisfying an additional new causality
condition
$\left[\hat{\phi}_{+}\left(x\right),\,\hat{\phi}_{-}\left(-x\right)\right]=0$.

DQFT in curved spacetime, gravitational horizons unitarity and information-
loss paradox ($G=\hbar=c=1$)- The de Sitter (dS) spacetime is a perfect
testing ground for exploring the theory of QFTCS because of its maximally
symmetric aspect and it is also relevant for unlocking the nature of
inflationary quantum fluctuations. Furthermore, dS spacetime has the closest
resemblance with black holes where the most important conundrums in quantum
physics emerge such as the loss of unitarity and information-loss paradox.

We start with the dS metric in the flat Friedman-Lemaître-Robertson-Walker
(FLRW) coordinates

$ds^{2}=-dt^{2}+e^{2Ht}d\textbf{x}^{2}=\frac{1}{H^{2}\tau^{2}}\left(-d\tau^{2}+d\textbf{x}^{2}\right)\,,$
(4)

where the scale factor $a=e^{Ht}$ is the clock that determines the expansion
of the Universe, $H=\frac{\dot{a}}{a}$ is the Hubble parameter and
$\tau=\int\frac{dt}{a}=-\frac{1}{aH}$ is the conformal time. If one has to
place quantum theory in dS spacetime (4) the first and foremost things to be
understood are the parity and time reversal operations. The usual conception
of understanding the expansion of the Universe is to restrict to $\tau<0$. But
the metric carries a symmetry

$\mathcal{T}:\tau\to-\tau,\quad\mathcal{P}:\textbf{x}\to-\textbf{x}$ (5)

and restricting to $\tau<0$ before quantization of a classical field (as it is
usually followed in standard QFTCS) means throwing away the symmetry by hand
(5). The expanding Universe can be realized in two ways:

$\displaystyle\implies\begin{cases}t:-\infty\to+\infty,\quad
H>0&(\tau:-\infty\to 0)\\\ t:+\infty\to-\infty,\quad H<0&(\tau:\infty\to
0)\end{cases}$ (6)

This would explain why in the Wheeler-de Witt equation the cosmic time does
not explicitly appear, rather the scale factor is the clock that runs with
expanding Universe Kiefer (2007). Note that $H\to-H$ does not change at all
any properties of dS space. For example, curvature invariants such as Ricci
scalar $R=12H^{2}$ remain the same under discrete sign flip of H. The equation
(6) illustrates two possible time realizations for an expanding universe. By
convention, choosing $\tau>0$ or $\tau<0$ results in information loss beyond
the horizon. Schrödinger in his famous monograph of 1956 Erwin Schrodinger
(1956) rejected the idea of two Universes because it would lead to a conundrum
of ignorance of information beyond the horizon which in modern language pure
states evolving into mixed states which leads the violation of unitarity and
information-loss Gibbons and Hawking (1977); Parikh  _et al._ (2003).

Similar to the Minkowski space (see Appendix A), the quantum fields in the dS
space are created in a direct-sum vacuum $|0_{T}\rangle_{\rm
dS}=|0_{+}\rangle_{\rm dS}\oplus|0_{-}\rangle_{\rm dS}$ corresponding to the
direct-sum Fock space $\mathcal{F}_{\rm dS}=\mathcal{F}_{\rm
dS+}\oplus\mathcal{F}_{\rm dS-}$. This implies, that everywhere in dS space a
quantum field is tied by time forward and backward evolutions at the parity
conjugate points, see Appendix B. At any moment of dS expansion, the parity-
conjugate points on the comoving horizon $r_{H}=|\frac{1}{aH}|$ are space-like
separated for an imaginary classical observer as shown in Fig. 2.

Figure 2: This figure depicts the concept of unitarity that can be novelly
achieved in de Sitter spacetime (in flat FLRW coordinates) with DQFT. Given
the particle horizon ($r_{H}$) at any moment of de Sitter expansion the above
picture depicts how DQFT formulation projects back all information of the
states outside the horizon back inside at the antipodal point of the physical
space. This implies all the information that any imaginary classical observer
perceives is within the horizon and pure states cannot evolve into mixed
states.  Figure 3: This picture depicts the conformal (not) Penrose diagram of
static dS spacetime (8) where we depict direct-sum quantum states
$|\phi_{+}\rangle\oplus|\phi_{-}\rangle$ entering and exiting the horizon with
opposite time evolutions. This is not a Penrose diagram because each point
represents half of the $S_{2}$ sphere in contrast to the usual Penrose
diagram. This diagram is to represent quantum physics in dS space rather than
classical physics for which the usual Penrose diagram is used.

Note that by DQFT construction, the direct-sum quantum states
$|\phi_{\pm}\rangle=\hat{d}_{(\pm)\,\textbf{k}}^{\dagger}|0_{\pm}\rangle_{\rm
dS}$ that follow from (20) are the opposite time evolving positive energy
states at the parity conjugate points and this holds for any imaginary
classical observer. Furthermore, DQFT preserves the causality and locality
too. A consequence of this is that any state that disappears beyond the
horizon happens to be a direct-sum state that reappears at the antipodal
point. This is because all the imaginary classical observers including those
on the horizons of each other should observe the same direct-sum quantum
states $|\phi_{+}\rangle\oplus|\phi_{-}\rangle$.

This means the horizon acts as a $\mathcal{P}\mathcal{T}$ mirror which maps
every information outside the horizon to the states inside the horizon. Fig. 2
gives the pictorial representation of this. This feature of DQFT implies any
maximally entangled state (a pure state)

$|\phi\rangle\otimes|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\phi_{+}\rangle\otimes|\psi_{+}\rangle\right)\oplus\frac{1}{\sqrt{2}}\left(|\phi_{-}\rangle\otimes|\psi_{-}\rangle\right)$
(7)

remains within the horizon because no information is lost beyond the horizon.
Every information of each of the entanglement pairs is mapped to within the
horizon, thus the pure state will remain a pure state. In other words, DQFT
forbids any entanglement beyond the gravitational horizons. DQFT constructs a
direct-sum Hilbert space with the interior and exterior being separated by a
superselection rule Wick  _et al._ (1952); nLab authors (2023). Thus, there
would be an observer complementarity. We can also understand the same from the
point of view of dS space in static coordinates which reads as

$\displaystyle ds^{2}$
$\displaystyle=-\left(1-H^{2}r^{2}\right)dt_{s}^{2}+\frac{1}{\left(1-H^{2}r^{2}\right)}dr^{2}+r^{2}d\Omega^{2}$
(8)
$\displaystyle=\frac{1}{H^{2}\left(1-UV\right)^{2}}\left(-4dUdV+\left(1+UV\right)^{2}d\Omega^{2}\right)$

where $r=\big{|}\frac{1}{H}\big{|}\frac{1+UV}{1-UV}$ and
$\frac{V}{U}=-e^{2Ht_{s}}$. The relation between the flat FLRW (4) and (8) is
just a simple coordinate change Lanczos and Hoenselaers (1997); Hartman
(2017). A very practiced view of flat dS spacetime (4) is that it covers only
half of the dS space. But we must keep in mind the symmetry (6) which implies
that (4) can cover the entire dS space. Thus the maximally symmetric nature of
dS spacetime sets no difference whether it is expressed in flat, closed, open
FLRW coordinates or the static coordinates Mukhanov and Winitzki (2007). The
use of mapping flat coordinates to static coordinates here is that we can
realize how unitarity can be achieved with DQFT whereas it is lost in the
standard QFT as it was shown by Gibbons and Hawking in 1977 Gibbons and
Hawking (1977). In Fig. 3, we show the unitary quantum physics in the static
dS space through the conformal (not) Penrose diagram, where we see the three-
dimensional space split by parity for the interior (the white region) and
exterior (the dark region) of the cosmological event horizon. We can picture
here how none of the observers lose any information beyond the horizon. If any
state leaves the horizon it is only the direct-sum component of what is inside
the horizon. This means any observer can perfectly reconstruct information
about what is beyond the horizon. Any state beyond the horizon is not
entangled with what is inside. This means the Hilbert spaces of the interior
and exterior states cannot be connected by direct-product ($\otimes$) but
rather should be related by direct-sum ($\oplus$). This has to be like this
because beyond-the-horizon time is not the same, so one cannot do the same
quantum mechanics everywhere. In other words, the gravitational horizons are
special, they separate different spacetimes by a boundary. Thus, they cannot
be treated as a standard null surface where quantum states can be entangled
across. Furthermore, we stress that no entanglement across the horizon does
not lead to firewalls which happens in standard QFTCS but not in direct-sum
QFTCS. We suggest Ref. Sravan Kumar and Marto (2023b) to the reader for
further analogous discussion on unitarity and no information loss in the
context of the Schwarzschild black hole. In a nutshell, DQFT offers a new
understanding of dS space where any observer’s physics of the universe is
complete, and no information is lost outside any observer’s Universe. This is
exactly what Schrödinger envisioned a quantum theory to achieve in 1956 Erwin
Schrodinger (1956). Furthermore, our achievement of DQFT echoes well with the
recent classical understanding of dS through the Black Hole Universe (BHU)
proposal Gaztanaga (2022); Gaztañaga (2023).

Direct-sum Inflation (DSI) and CMB parity asymmetry - Inflationary quantum
fluctuations are the initial seeds for large-scale structure formation. It is
important to understand their quantum nature to derive consistent CMB
predictions. Inflation is by definition a quasi-dS expansion Starobinsky
(1980) and this means that $\mathcal{P}\mathcal{T}$ (6) is spontaneously
broken by the presence of a non-perturbative scalar field in addition to the
GR’s tensor degree of freedom. Our focus here is on single-field inflation.
The metric and scalar field matter quantum fluctuations are generated during
inflation. The quantity which is a gauge invariant combination of them is
called the curvature perturbation ($\zeta$), and it is an effective (scalar)
degree of freedom. It connects the quantum physics of inflation to CMB
observations. The action for curvature perturbation at the linear order is

$\delta^{(2)}S_{s}=\frac{1}{2}\int d\tau
d^{3}xa^{2}\frac{\dot{\phi}^{2}}{H^{2}}\Bigg{[}\zeta^{\prime
2}-\left(\partial\zeta\right)^{2}\Bigg{]}\,\text{,}$ (9)

where $H=\frac{\dot{a}}{a}$ which is nearly constant during inflation
characterized by the smallness of slow-roll parameters
$\epsilon=-\frac{\dot{H}}{H^{2}},\,\eta=\frac{\dot{\epsilon}}{H\epsilon}$ for
$N=50-60$ number of e-foldings. The canonical variable which is a redefinition
of curvature perturbation that we eventually quantize is

$v=a\zeta\dot{\phi}/H$ (10)

In the framework of DSI, the canonical variable is promoted to field operator
that is split into direct-sum of two components

$\displaystyle\hat{v}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{v}_{(+)}\left(\tau,\,\textbf{x}\right)&0\\\
0&\hat{v}_{-}\left(-\tau,\,-\textbf{x}\right)\end{pmatrix}$ (11)

which generate quantum fields in a direct-sum vacuum

$|0\rangle_{\rm qdS}=|0\rangle_{\rm qdS_{I}}\oplus|0\rangle_{\rm
qdS_{II}}=\begin{pmatrix}|0\rangle_{\rm qdS_{I}}\\\ |0\rangle_{\rm
qdS_{II}}\end{pmatrix}$ (12)

in which they evolve forward and backward in time at the parity conjugate
points in physical space. In analogy with (6), the time reversal operation
(quantum mechanically) in an expanding quasi-dS Universe is

$\tau\to-\tau\implies(t,H,\epsilon,\eta)\to(-t,-H,-\epsilon\,-\eta).$ (13)

Figure 4:  The top panels show the anti-symmetric,
$\mathcal{T}^{-}(\hat{n})=\left[\mathcal{T}(\hat{n})-\mathcal{T}(-\hat{n})\right]/2$,
component of the Planck 2018 CMB temperature map Akrami  _et al._ (2020a) in
North and South polar galactic caps. The visualization is such that the
Galactic plane, shown in Fig. 5, remains obscured along the edges. Parity
conjugate positions can be identified by horizontally flipping into the
opposite pole, where both caps have the same information with opposing signs:
$\mathcal{T}^{-}(-\hat{n})=-\mathcal{T}^{-}(\hat{n})$. The bottom panel
presents the observed $\mathcal{T}=\mathcal{T}^{+}+\mathcal{T}^{-}$ maps,
including both the symmetric
$\mathcal{T}^{+}(\hat{n})=\left[\mathcal{T}(\hat{n})+\mathcal{T}(-\hat{n})\right]/2$
and anti-symmetric components. The evident resemblance between the top and
bottom panels indicates the prevalence of the anti-symmetric component, in
perfect agreement with DSI predictions (21).

The CMB provides temperature fluctuations
$\mathcal{T}\left(\hat{n}\right)\equiv\Delta T\left(\hat{n}\right)/T_{0}$ all
over the sky $\hat{n}$:

$\mathcal{T}\left(\hat{n}\right)=\sum a_{\ell m}Y_{\ell
m}\left(\hat{n}\right)\quad;\quad C_{\ell}=\frac{1}{2\ell+1}|a_{\ell m}^{2}|$
(14)

the Fourier coefficients $a_{\ell m}$ are characterized by the angular power
spectrum $C_{\ell}$. Because $\mathcal{T}$ (and $\mathcal{P}\mathcal{T})$ is
broken by the slow-roll parameters (13), we expect the quantum field during
inflation to evolve time asymmetrically at parity conjugate points in physical
space. This leads to an angular power spectrum $C_{\ell}$ of the CMB
exhibiting an excess of power in the odd multipoles compared to the even ones,
as described by (21) in Appendix C.

In Fig. 4 we can see visually the parity asymmetry by the close resemblance
between odd parity temperature maps and the original CMB maps from Planck data
Akrami  _et al._ (2020a). This reflects that the Planck temperature maps
contain an additional pure odd parity component. The DSI framework exactly
generates this additional odd symmetry as we can see in (21). It is worth
noticing here that if we take the average of even and odd power spectra we
recover the well-known near-scale invariant power spectrum of Standard
Inflation (SI). The benefit of DSI is that there are no additional free
parameters. The parity asymmetry just depends on the spectral index
$n_{s}\approx 0.9634\pm 0.0048$ at the scale $k_{\ast}=0.05\,{\rm Mpc}^{-1}$
($\ell\approx 200$ or $\theta\lesssim 1^{\circ}$) as measured by the Planck
data Akrami  _et al._ (2020b).

In Fig. 6 we present how the observational data prefers significantly DSI over
SI. Observed quantities include the observed low quadrupolar amplitude
$C_{2}=C_{\ell=2}$, the even-to-odd multipole ratio of angular power spectra:

$R^{TT}=\frac{\sum_{\ell=even}^{\ell
max=20}\ell(\ell+1)C_{\ell}}{\sum_{\ell=odd}^{\ell
max=20}\ell(\ell+1)C_{\ell}}$ (15)

and the mean $Z^{1}=<Z(\hat{n})>$ of the parity map
$Z(\hat{n})\equiv\mathcal{T}(\hat{n})\mathcal{T}(-\hat{n})$ displayed in
Fig.5, where $-\hat{n}=(\pi-\theta,\pi+\varphi)$ represents the parity
conjugate or antipode of the sky position $\hat{n}=(\theta,\varphi)$. The
likelihood probability associated with DSI in these plots is up to 650 times
larger than that of SI. DSI favors odd parity, leading to a lower $C_{2}$.
This, in turn, diminishes the significance of other CMB anomalies Schwarz  _et
al._ (2016), such as the Hemispherical Power Asymmetry, the axis of Evil, and
the anomalous 2-point correlation function (See Gaztañaga and Sravan Kumar
(2024) for further details).

Figure 5: Parity map: $Z(\hat{n})=\mathcal{T}(\hat{n})\mathcal{T}(-\hat{n})$
for Fig. 4 with galactic mask. This is a $\mathcal{P}$ conjugate image of
itself. Positive values (red) correspond to the symmetric (even parity)
component of $\mathcal{T}$, while negative values (blue) give the
antisymmetric (odd parity) component. There is a clear excess of negative
values, indicating a preference for odd parity. This together with Fig. 4
serves as a visual testament to the unitary QFTCS in nature. Figure 6: Parity
measurements derived from different mask (component separation) CMB
temperature maps $\mathcal{T}(\hat{n})$ (dashed lines) against data
simulations (grey areas represent 1, 2, and 3 sigma contours), alongside
predictions from standard inflationary (SI) models (in red) and Direct-Sum
inflation (DSI) predictions derived from DQFT (in blue).

Conclusions- In this letter, we underscore the pivotal role of unitary Quantum
Field Theory in Curved Spacetime (QFTCS) in elucidating the intricacies of
early Universe physics. Contrary to prevailing notions, we contend that the
issue of unitarity in QFTCS is not inherently tied to quantum gravity at
Planck scales, but rather stems from our incomplete comprehension of quantum
fields in curved spacetime.

With the $\mathcal{P}\mathcal{T}$-symmetric formulation of Schrödinger
equation (1), the DQFT framework reinstates unitarity, which was previously
thought to be lost in curved spacetime. Fig. 2-3 visually represent this
concept, illustrating how a hypothetical classical observer consistently
experiences pure states without encountering information loss beyond the
horizon.

Furthermore, the striking correlation illustrated in Fig. 4-6 between the
observed hot and cold mirrored structures of the CMB map at antipodal points
strongly supports the unitary treatment of inflationary quantum fluctuations
within the framework of DQFT. Our findings indicate that superhorizon
fluctuations at antipodes manifest as $\mathcal{P}\mathcal{T}$ conjugates, a
prediction derived from direct-sum QFTCS. This prediction is substantially
favored with a probability over 650 times greater than that of the standard
QTFCS.

###### Acknowledgements.

EG acknowledges grants from Spain Plan Nacional (PGC2018-102021-B-100) and
Maria de Maeztu (CEX2020-001058-M). KSK acknowledges the Royal Society for the
Newton International Fellowship.

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##  Appendix A Appendix A: Klein-Gordon field operator in DQFT

The Klein-Gordon field operator for DQFT in Minkowski spacetime split into
direct-sum

$\hat{\phi}=\frac{1}{2}\hat{\phi}_{+}\left(t_{m},\,\textbf{x}\right)\oplus\hat{\phi}_{-}\left(-t_{m},\,-\textbf{x}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{\phi}_{+}&0\\\
0&\hat{\phi}_{-}\end{pmatrix}$ (16)

where

$\hat{\phi}_{\pm}=\int\frac{d^{3}k}{\left(2\pi\right)^{3/2}}\frac{1}{\sqrt{2|k_{0}|}}\Bigg{[}\hat{a}_{(\pm)\,\textbf{k}}e^{\pm
ik\cdot x}+\hat{a}^{\dagger}_{(\pm)\,\textbf{k}}e^{\mp ik\cdot x}\Bigg{]}$
(17)

where $k\cdot x=-k_{0}t_{m}+\textbf{k}\cdot\textbf{x}$. The creation and
annihilation operators satisfy the canonical relations

$\displaystyle=\delta^{(3)}\left(\textbf{k}-\textbf{k}^{\prime}\right),\quad[\hat{a}_{(\pm)\,\textbf{k}},\,\hat{a}_{(\mp)\,\textbf{k}^{\prime}}]=0$
(18)

which defines the positive norm states in the direct-sum vacuums
$|0_{T}\rangle=|0_{+}\rangle\oplus|0_{-}\rangle,\,a_{(\pm)}|0_{\pm}\rangle=0$.
The DQFT splits the Fock space as the direct-sum of two Fock spaces
corresponding to the field’s components
$\mathcal{F}_{T}=\mathcal{F}_{+}\oplus\mathcal{F}_{-}$ where states of forward
in time and backward in time at parity conjugate regions of position space can
be created and annihilated by the pair of operators (18). Since the Minkowski
spacetime is $\mathcal{P}\mathcal{T}$ symmetric the scalar field operator in
DQFT respects the same symmetry. For further details on DQFT and
$\mathcal{C}\mathcal{P}\mathcal{T}$ invariance see Sravan Kumar and Marto
(2023a).

##  Appendix B Appendix B: DQFT in dS

Applying DQFT in dS Gaztañaga and Sravan Kumar (2024); Sravan Kumar and Marto
(2023a), by construction the scalar field operator is the direct-sum of two
components

$\displaystyle\hat{\phi}$
$\displaystyle=\frac{1}{\sqrt{2}}\hat{\phi}_{+}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{\phi}_{-}\left(-\tau\,-\textbf{x}\right)\,,$
(19)

where

$\displaystyle\hat{\phi}_{\pm}$
$\displaystyle=\int\frac{d^{3}k}{\left(2\pi\right)^{3/2}}\Bigg{[}\hat{d}_{(\pm)\,\textbf{k}}\phi_{\pm\,k}e^{\pm
i\textbf{k}\cdot\textbf{x}}+\hat{d}^{\dagger}_{(\pm)\,\textbf{k}}\phi_{\pm\,k}^{\ast}e^{\mp
i\textbf{k}\cdot\textbf{x}}\Bigg{]}$ (20)

Here the operators $d_{\pm\,\textbf{k}},\,d^{\dagger}_{\pm\,\textbf{k}}$
satisfy the similar relations like in (18). The dS mode functions
$\phi_{\pm\,k}=\frac{1}{\sqrt{2k}}\left(1\mp\frac{i}{k\tau}\right)e^{\mp
ik\tau}$ corresponding to the standard Bunch-Davies vacuum of the DQFT.

##  Appendix C Appendix C: DSI angular power spectrum

Following Gaztañaga and Sravan Kumar (2024), the CMB power spectrum $C_{\ell}$
in DSI is:

$\displaystyle
C_{\ell}^{odd}=\frac{2}{9\pi}\int_{0}^{k_{c}}\frac{dk}{k}j_{\ell}^{2}\left(\frac{k}{k_{s}}\right)\mathcal{P}_{\zeta}(k)\left(1+\Delta\mathcal{P}_{v}\right)$
(21) $\displaystyle
C_{\ell}^{even}=\frac{2}{9\pi}\int_{0}^{k_{c}}\frac{dk}{k}j_{\ell}^{2}\left(\frac{k}{k_{s}}\right)\mathcal{P}_{\zeta}(k)\left(1-\Delta\mathcal{P}_{v}\right)$

where $\mathcal{P}_{\zeta}=A_{s}\left(\frac{k}{k_{\ast}}\right)^{n_{s}-1}$,
$A_{s}=2.2\times 10^{-9}$ and

$\Delta\mathcal{P}_{v}=\left(1-n_{s}\right)\operatorname{Re}\left[\frac{1}{H_{3/2}^{(1)}\left(\frac{k}{k_{\ast}}\right)}\frac{\partial
H^{(1)}_{\nu_{s}}\left(\frac{k}{k_{\ast}}\right)}{\partial\nu_{s}}\Bigg{|}_{\nu_{s}=\frac{3}{2}}\right]$
(22)

Here, $H^{(1)}_{\nu_{s}}\left(z\right),j_{\ell}\left(z\right)$ are the Hankel
and Bessel functions of the first kind, $k_{s}=7\times 10^{-5}\,{\rm
Mpc}^{-1}$ and $k_{c}=0.02k_{\ast}$ is the cut-scale we impose as (22) is
accurate enough for low-$\ell$ or large angular scales. This cut-off scale is
also related to the coarse gaining scale of stochastic inflation Gaztañaga and
Sravan Kumar (2024) which determines the large wavelength modes that have
already become classical on the onset of inflation while the mode $k_{\ast}$
exiting the horizon. In DSI, these large wavelength modes create parity
asymmetric inhomogeneities that correct the FLRW spacetime on large scales.
The parity asymmetry ceases to exist towards the small scales because the
$\mathcal{P}\mathcal{T}$ symmetry recovered towards the short distance scales.
This can be seen in Fig. 11 of Gaztañaga and Sravan Kumar (2024). An intuitive
explanation for this is that large wavelength modes experience quantum
gravitational effects more than those of short wavelength modes.