# Mean and variance of the cardinality of particles in polyanalytic Ginibre
processes via a quantization method

Zouhaïr Mouayn∗, Mohamed Mahboubi♭ and Othmane El Moize♭

###### Abstract

We discuss the mean and variance of the number “point-particles”
$\sharp_{D_{R}}$ inside a disk $D_{R}$ centered at the origin of the complex
plane $\mathbb{C}$ and of radius $R>0$ with respect to a Ginibre-type
(polyanalytic) process of index $m\in\mathbb{\mathbb{Z}}_{+}$ by quantizing
the phase space $\mathbb{C}$ via a set of generalized coherent states
$\left|z,m\right\rangle$ of the harmonic oscillator on
$L^{2}\left(\mathbb{R}\right)$. By this procedure, the spectrum of the quantum
observable representing the indicator function
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}$
of $D_{R}$ (viewed as a classical observable) allows to compute the mean value
of $\sharp_{D_{R}}$. The variance of $\sharp_{D_{R}}$ is obtained as a special
eigenvalue of a quantum observable involving to the auto-convolution of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}.$
By adopting a coherent states quantization approach, we seek to identify
classical observables on $\mathbb{C},$ whose quantum counterparts may encode
the first cumulants of $\sharp_{D_{R}}$ through spectral properties.

∗ Department of Mathematics, Faculty of Sciences and Technics (M’Ghila),
Sultan Moulay Slimane University, P.O. Box. 523, Béni Mellal, Morocco
♭ Department of Mathematics, Faculty of Sciences,
Ibn Tofaïl University, P.O. Box. 133, Kénitra, Morocco

##  1 Introduction

Random point processes, which may be defined as distributions over
configurations of points, have arose in many different areas (and still may
emerge in unexpected places), such as statistical mechanics, combinatories,
representation theory and probability theory. They have been used to model
fermions in quantum mechanics, zeros of Gaussian analytic functions, in
classical Ginibre and circular unitary ensembles from random matrix theory,
for examples arising from non-intersecting random walks and random spanning
trees see [2, 1]. They also have been used in machine learning [3], where the
state space is finite and in wireless communication to model the locations of
network nodes [4, 5].

Characterizing these processes may be viewed as studying systems of
interacting “point-particles”. In this respect Macchi [6] introduced a time
independent model (called the fermions point process) describing the
statistical distribution of a fermions system in thermal equilibrium, with the
property that the $k-$point correlation functions have the form of
determinants built from a correlation kernel, implying that particles obey the
Pauli exclusion principle. Such a point process with a kernel function
encoding repulsiveness, exhibit hyperuniformity or rigidity properties [7],
that is the fluctuations of the number of points in a given region is smaller
than compared to a Poisson point process with the same intensity. To predict
the large deviation asymptotics for the number of points in large balls
becomes a subject to intense investigating (see [8] as an example) keeping in
mind the basic electrostatic fact stating that the variance of the number of
points in a box should grow like the surface area instead of the volume.

The Ginibre point process, in particular, was first introduced in [9] and
specifically formed by the eigenvalues of a hermitian matrix with standard
complex Gaussian entries (also known as the Gaussian Unitary Ensemble GUE). It
models the positions of charges of a two-dimensional Coulomb gas in harmonic
potential at the temperature $\beta=2$. At the limit of these eigenvalues
appears the infinite analytic Ginibre process on $\mathbb{C}$, denoted
$\mu_{\exp},$ associated with the reproducing kernel $e^{z\overline{w}}$ of
the Fock-Bargmann space $\mathcal{A}_{0}\left(\mathbb{C}\right)$ of entire
complex-valued functions which are $e^{-\left|z\right|^{2}}d\nu-$square
integrable on $\mathbb{C}$ , $d\nu$ being the Lebesgue measure on
$\mathbb{C=}$ $\mathbb{R}^{2}$. $\mu_{\exp}$ arises from the two-dimensional
one-component plasma or jellium model at a special temperature and it is well
known to be exactly solvable [10].

By [11], it turns out that the space $\mathcal{A}_{0}\left(\mathbb{C}\right)$
coincides with the eigenspace of the magnetic Laplacian
$\tilde{\Delta}_{1}=-\partial\overline{\partial}+\overline{z}\overline{\partial}$
associated with the lowest Landau level $m=0$. The eigenspace
$\mathcal{A}_{m}(\mathbb{C}):=\\{\varphi\in
L^{2}(\mathbb{C},e^{-\left|z\right|^{2}}d\nu);\,\tilde{\Delta}_{1}\varphi=m\varphi\\}$
corresponding to higher Landau levels $m\in\mathbb{Z}_{+}$ also admit explicit
reproducing kernels. For each fixed $m\in\mathbb{Z}_{+}$, the associated
determinantal point process (Ginibre-type) has been considered by Shirai [12]
who was concerned by the random variable $\sharp_{D_{R}}$ counting the number
of points inside a disk $D_{R}$ centered at the origin of $\mathbb{C}$ and of
radius $R>0$. He proved that the variance $\mathbb{V}ar(\sharp_{D_{R}})$ can
be expressed as an integral involving the Laguerre polynomial (see Eq.(3.8)
below) and investigated its asymptotic behavior as $R\rightarrow\infty.$
Hyperuniformity of these Ginibre-types processes has been also discussed by
Abreu et al [13] in the framework of the so-called infinite Weyl-Heisenberg
Ensemble.

In this paper, our purpose is to recover the mean and variance of
$\sharp_{D_{R}}$ from a coherent states quantization view point. We precisely
derive the mean value of $\sharp_{D_{R}}$ by summing eigenvalues of the
quantized quantum counterpart (operator) form of the indicator function
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}$
of the disk $D_{R}$. To obtain the variance we proceed by
quantizing/dequantizing, essentially, the auto-convolution of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}.$
The coherent states with respect to which we are performing this quantization
belong to $L^{2}\left(\mathbb{R}\right)$, and are obtained by displacing “à la
Perelomov”, via the Schrödinger representation of the Weyl-Heisenberg group,
an eigenfunction of the harmonic oscillator, given by a Hermite function. By
regarding this determinantal point process from a coherent states viewpoint we
seek to identify classical observables on the phase space $\mathbb{C},$
attached to the geometry of the disk $D_{R}$, and whose quantum counterpart
operators may encode, through spectral properties, the first cumulants of the
random variable $\sharp_{D_{R}}$.

The organization of the paper is as follows. In section 2, we briefly review
the formalism of determinantal point processes. In section 3, we recall a
generalized Bargmann-Fock spaces with their reproducing kernels. We also
summarize the polyanalytic Ginibre process in $\mathbb{C}$ with some of its
properties. We give a brief review of the standard coherent states in section
4. Section 5 is devoted to an explicit quantization of the complex plane via
coherent states attached to generalized Bargmann-Fock spaces. In section 6, we
compute the mean value and the variance of the random variable
$\sharp_{D_{R}}$.

##  2 Determinantal point processes

We briefly recall some basic notions on DPP, for more details we refer to [2]
and references therein.

Let $\mathcal{E}$ be a one-particle space, generally a separable Hausdorff
space, and $B(\mathcal{E})$ the topological Borel $\sigma$-field. We fix a
radon measure $\lambda(dx)$ on $(\mathcal{E},\mathcal{B}(\mathcal{E}))$. Let
$X$ be the space of countable configurations of particles in $\mathcal{E}$,
usually $X$ is called the configuration space. We assume that each
configuration $\xi:=(x_{i})$, $x_{i}\in\mathcal{E}$, $i\in\mathbb{Z}$ or
$i\in\mathbb{Z}_{+}$ if $d>1$, is locally finite, that is, for every compact
set $K\subset\mathcal{E}$ the number of particles in $K$ is finite. Let
$B\subset\mathcal{E}$ be a Borel set and define $\mathcal{B}$ as the
$\sigma$-algebra generated by all cylinder sets $C_{\ell}^{B}=\\{\xi\in
X:\sharp_{B}(\xi)=\ell\\}$, $\ell\in\mathbb{Z}_{+}$. We call a random point
process (RPP) the triplet $(X,\mathcal{B},P_{r})$ where $P_{r}$ is a
probability measure on $(X,\mathcal{B})$, which may be described by its
correlation functions.

Denoting
$\mathcal{E}^{k}=\mathcal{E}\times\mathcal{E}\times...\times\mathcal{E}$, a
locally integrable function
$\rho_{k}:\mathcal{E}^{k}\rightarrow\mathbb{R}_{+}$ is called a $k-$point
correlation function for this RPP if for every bounded Borelian disjoint
subsets $A_{1},...,A_{n}$ of $\mathcal{E}$ and for every multi-index
$(k_{1},...,k_{n})\in(\mathbb{Z}_{+})^{n}$ with $k_{1}+...+k_{n}=k$ the
following formula holds true

$\mathbb{E}\left(\prod_{i=1}^{n}\frac{(\sharp A_{i})!}{(\sharp
A_{i}-k_{i})!}\right)=\int_{A_{1}^{k_{1}}\times...\times
A_{n}^{k_{n}}}\rho_{k}(x_{1},...,x_{k})\lambda^{\otimes}(dx_{1}...dx_{k})$
(2.1)

where $\mathbb{E}$ is the mathematical expectation with respect to $P_{r}$.
The problem of existence and uniqueness for a RPP defined by its correlation
functions was studied in [14, 15]. This is the case when the distribution of
the random variable $\sharp_{A}$ counting the number of particles inside a
bounded Borel set $A$ is uniquely determined by its moments.

A determinantal (or fermions) point process (DPP) with kernel $K(x,y)$ is a
RPP in $\mathcal{E}$ such that its $k$-points correlation function is given by

$\rho_{k}(x_{1},...,x_{k})dx_{1}...dx_{k}=\det(K(x_{i},x_{j}))_{1\leq i,j\leq
k}\lambda^{\otimes}(dx_{1}...dx_{k}).$ (2.2)

Now let $\mathfrak{K}$ be a self adjoint integral operator on
$L^{2}(\mathcal{E},\lambda(dx))$ with kernel $K(x,y)$ and suppose that
$Spec(\mathfrak{K})\subset[0,1]$ and $\mathfrak{K}$ is of locally trace class
then there exist a unique DPP associated with kernel $K$ and $\lambda(dx)$
such that its $k$-point correlation function is given by (2.2). For
${D}\subset\mathcal{E}$ a relatively compact, the random variable
$\sharp_{{D}}$, in this DPP, has the same distribution as a sum of independent
Bernoulli random variables $(\lambda_{i}^{{D}})_{i}$, where
$\lambda_{i}^{{D}}$ is an eigenvalue of the restriction to ${D}$ of the
operator $\mathfrak{K}$. The mean value of $\sharp_{{D}}$ is given by

$\mathbb{E}\left(\sharp_{{D}}\right)=\int\limits_{\mathcal{D}}K(x,x)\lambda(dx)$
(2.3)

and its variance

$\mathbb{V}ar\left(\sharp_{D}\right)=\int\limits_{\mathcal{D}}\lambda(dx)\int\limits_{\mathcal{E\setminus
D}}\lambda(dy)\left|K\left(x,y\right)\right|^{2}.$ (2.4)

##  3 Ginibre-type processes

###  3.1 Generalized Bargmann-Fock spaces $\mathcal{A}_{m}(\mathbb{C})$

The motion of a charged particle in a constant uniform magnetic field in
$\mathbb{R}^{2}$ with a strength proportional to $B>0$, is described (in
suitable units) by the Schrödinger operator

$H_{B}:=-\frac{1}{4}\left((\partial_{x}+iBy)^{2}+(\partial_{y}-iBx)^{2}\right)-\frac{1}{2}$
(3.1)

on $L^{2}(\mathbb{R}^{2},d\nu).$ By identifying $\mathbb{R}^{2}$ with
$\mathbb{C}$ in the usual way and setting
$d\lambda\left(z\right)=e^{-|z|^{2}}d\nu\left(z\right)$ then, with the help of
the unitary map $Q:L^{2}(\mathbb{C},d\nu)\longrightarrow
L^{2}(\mathbb{C},d\lambda),$ defined by
$Q[\phi](z):=e^{\frac{1}{2}B|z|^{2}}\phi(z)$, one can intertwine the operator
(3.1) as
$e^{\frac{1}{2}B|z|^{2}}H_{B}e^{-\frac{1}{2}B|z|^{2}}=\tilde{\Delta}_{B}$. We
take $B=1$ and consider the operator, called magnetic Laplacian

$\tilde{\Delta}_{1}:=-\frac{\partial^{2}}{\partial
z\partial\bar{z}}+\bar{z}\frac{\partial}{\partial\bar{z}}$ (3.2)

on $L^{2}(\mathbb{C},d\lambda)$. The spectrum of $\tilde{\Delta}_{1}$ consists
of eigenvalues (Euclidean Landau levels) of infinite multiplicity of the form
$\,m\in\mathbb{Z}_{+}.$ In [11] generalized Bargmann-Fock spaces have been
introduced as eigenspaces of $\tilde{\Delta}_{1}$ as

$\mathcal{A}_{m}(\mathbb{C}):=\\{\varphi\in
L^{2}(\mathbb{C},d\lambda);\,\tilde{\Delta}_{1}\varphi=m\varphi\\}.$ (3.3)

Actually, for $m=0$, $\mathcal{A}_{0}(\mathbb{C})$ turns out to be the
realization by harmonic functions with respect to $\tilde{\Delta}_{1}$ of the
Bargmann-Fock space whose elements are entire functions in
$L^{2}(\mathbb{C},d\lambda)$. For $m\geq 0,$ $\mathcal{A}_{m}(\mathbb{C})$
turns out to be the space of true-$m$-polyanalytic functions that is the
orthogonal difference
$\mathfrak{F}_{m}(\mathbb{C})\ominus\mathfrak{F}_{m-1}(\mathbb{C})$ between
two consecutive $m$-polyanalytic spaces

$\mathfrak{F}_{m}(\mathbb{C}):=\left\\{g\in L^{2}(\mathbb{C},d\lambda),\ \
\overline{\partial}^{m}g=0\right\\}.$ (3.4)

The eigenspaces (3.3) are pairwise orthogonal in $L^{2}(\mathbb{C},d\lambda)$
which decomposes as $\bigoplus_{m=0}^{+\infty}\mathcal{A}_{m}(\mathbb{C})$.
Furthermore, the reproducing kernel of the eigenspace (3.3) reads

$\widetilde{K}_{m}(z,w)=\pi^{-1}e^{z\bar{w}}L_{m}^{(0)}(|z-w|^{2})$ (3.5)

which may be transfered back into the Hilbert space $L^{2}(\mathbb{C},d\nu)$
in which the kernel (3.5) becomes

$K_{m}(z,w)=\pi^{-1}e^{-\frac{1}{2}z\overline{z}}\widetilde{K}_{m}(z,w)e^{-\frac{1}{2}w\overline{w}}=\left(e^{z\bar{w}-\frac{1}{2}z\overline{z}-\frac{1}{2}w\overline{w}}\right)L_{m}^{(0)}(|z-w|^{2}),\text{
}z,w\in\mathbb{C}$ (3.6)

where $L_{m}^{(\alpha)}(\cdot)$ is the Laguerre polynomial ([16], p.239).

###  3.2 The DPP $\mu_{K_{m},d\nu}$

According to [12], we recall the following facts. Let $m\in\mathbb{Z}_{+}$ and
$\mathfrak{K}_{m}$ be the projection operator defined on
$L^{2}(\mathbb{C},d\nu)$, whose integral kernel is given in (3.6). The DPP
associated with $\left(\mathfrak{K}_{m},d\nu\right)$ is the Ginibre-type point
process on $\mathbb{C}$ with index $m,$ denoted by $\mu_{K_{m},d\nu},$ which
may also be called polyanalytic Ginibre point process. Note that
$\mu_{K_{m},d\nu}$ is the same as $\mu_{\widetilde{K}_{m},d\lambda}$ and is
translation and rotation invariant. According to (2.3), the first intensity
function $\rho_{1}(z)=K_{m}\left(z,z\right)=\pi^{-1}$ leads to the expected
number of particles inside the disk $D_{R}$
$=\\{\xi\in\mathbb{C},\,|\xi|<R\\}$ as

$\mathbb{E}_{\mu_{K_{m},d\nu}}\left(\sharp_{D_{R}}\right)=\int\limits_{D_{R}}\pi^{-1}d\nu\left(z\right)=R^{2}.$
(3.7)

While, by (2.4), the variance reads

$\mathbb{V}ar_{\mu_{K_{m},d\nu}}(\sharp_{D_{R}})=\frac{R}{\pi}\int\limits_{0}^{+\infty}e^{-t}\left(L_{m}^{\left(0\right)}(t)\right)^{2}\left(\int_{0}^{t\wedge
4R^{2}}\left(1-\frac{x}{4R^{2}}\right)^{1/2}x^{-1/2}dx\right)dt$ (3.8)

which behaves as $C_{m}R$ as $R\rightarrow+\infty$. The constant

$C_{m}=\frac{2}{\pi
m!}\Gamma\left(m+\frac{3}{2}\right)._{3}F_{2}\left(-\frac{1}{2},-\frac{1}{2},m;1,-\frac{1}{2}-m;1\right)\sim\frac{8}{\pi^{2}}\sqrt{m}$
(3.9)

as $m\rightarrow+\infty$ (at very higher Landau levels).

In [17], Demni and Lazag added a geometrical view point to $\left(4.2\right)$
by rewriting it as

$Var_{\mu_{K_{m},d\nu}}(\xi(D_{R}))=\frac{1}{\pi^{2}}\int\limits_{\mathbb{C}}e^{-|z|^{2}}\left(L_{m}^{\left(0\right)}(|z|^{2})\right)^{2}\left[Area\left(D_{R}^{c}\cap
D_{R}\left(z\right)\right)\right]d\nu(z)$ (3.10)

where $D_{R}\left(z\right)$ is the disk centered at $z$ and of radius $R$ and
$D_{R}^{c}$ denotes the complementary in $\mathbb{C}$ of $D_{R}.$ If
$\left|z\right|$ $\geq 2R$, then $Area\left(D_{R}^{c}\cap
D_{R}\left(z\right)\right)=\pi R^{2}$ since $D_{R}\left(z\right)\subset
D_{R}^{c}.$ Otherwise, if $\left|z\right|<2R$, then $D_{R}^{c}\cap
D_{R}\left(z\right)$ is the complementary in $D_{R}\left(z\right)$ of the
overlapping of the disks $D_{R}$ and $D_{R}\left(z\right).$
Remark 3.1. For $\left|z\right|\leq 2R,$ we may also write $\
\left[Area\left(D_{R}^{c}\cap D_{R}\left(z\right)\right)\right]/\pi
R^{2}=1-\alpha_{R}\left(\left|z\right|\right)$ where

$\alpha_{R}\left(r\right):=\frac{4}{\pi}\int\limits_{0}^{\arccos\left(\frac{1}{2R}r\right)}\sin^{2}\theta
d\theta,\text{ \ }r=\left|z\right|,$ (3.10)

is the well known scaled intersection area for a circular window of diameter
$2R.$ It is the area of the intersection of two disks each with radius $R$ and
with distance $\left|z\right|$ between centers. It is known that for
$R=2\sqrt{\pi}$, the Fourier transform $\widehat{\alpha}_{2\sqrt{\pi}}$ of the
radial function $\alpha_{2\sqrt{\pi}}$ generates a translationally invariant
DPP, with the kernel given in terms of the Bessel function $J_{1}$, called a
Fermi-circle point process in $\mathbb{R}^{2}.$ For more details, see [7]
where the authors analytically provided an $\mathbb{R}^{n}$ generalization for
the unique point process on $\mathbb{R}$ to which one can map certain
properties of random matrices, fermionic gases and zeros of the Riemann zeta
function.

##  4 Coherent states

Coherent states (CS) were first introduced by E. Schrödinger [18] in order to
obtain quantum states in $L^{2}(\mathbb{R})$ that follow the classical flow
associated to the harmonic oscillator Hamiltonian

$\hat{H}=-\frac{\hslash}{2}\frac{d^{2}}{dx^{2}}+\frac{1}{2}x^{2}-\frac{1}{2}.$
(4.1)

Namely, they are states $\left|z\right\rangle\in L^{2}(\mathbb{R})$, labeled
by elements of $\,z\in$ $\mathbb{C}\simeq T^{\ast}\mathbb{R}$ (the phase space
of a particle moving on $\mathbb{R}$) given by

$\langle
x\left|z\right\rangle=\left(e^{z\overline{z}}\right)^{-\frac{1}{2}}\frac{1}{(\pi\hslash)^{4}}\left(-\frac{1}{2\hslash}(\overline{z}^{2}+x^{2}-2\sqrt{2}\overline{z}x)\right),\text{
\ \ }x\in\mathbb{R},$ (4.2)

$\hslash$ is the Planck parameter (take $\hslash=1$). Their most important
property is the resolution of the identity operator

$\mathbf{1}_{L^{2}(\mathbb{R})}=\frac{1}{\pi}\int\limits_{\mathbb{C}}d\nu(z)\left|z\right\rangle\left\langle
z\right|.$ (4.3)

The property (4.3) bridges between classical and quantum mechanics in the
sense that every operator acting on $L^{2}(\mathbb{R})$ or any vector lying
there may be decomposed over the phase space $\mathbb{C}$. The Klauder-
Berezin-Toeplitz (or ”anti-Wick”) quantization, here named coherent states
quantization, of the complex plane consists in associating to a function
$z\mapsto F\left(z,\overline{z}\right)$ (with specific properties) the
operator $P_{F}$ in $L^{2}(\mathbb{R})$ defined by

$F\mapsto
P_{F}:=\frac{1}{\pi}\int\limits_{\mathbb{C}}d\nu(z)F\left(z,\overline{z}\right)\left|z\right\rangle\left\langle
z\right|.$ (4.4)

By the operator-valued integral (4.4) we mean the sesquilinear form

$\mathcal{S}_{F}\left(\phi_{1},\phi_{2}\right)=\frac{1}{\pi}\int\limits_{\mathbb{C}}d\nu(z)F\left(z,\overline{z}\right)\langle\phi_{1}\left|z\right\rangle\left\langle
z\right|\phi_{2}\rangle$ (4.5)

The form $\mathcal{S}_{F}$ is assumed to be defined on a dense subspace of
$L^{2}(\mathbb{R})$. If $F$ is real-valued and at least semi-bounded, the
Friedrich’s extension ([19], Vol.II,Th.X.23) $\mathcal{S}_{F}$ allows to
define $P_{F}$ as a self-adjoint operator. The function is the upper (or
contravariant) symbol of $P_{F}$, and the mean value of the latter in the
state $\left|z\right\rangle$,

$z\mapsto\langle
z\left|P_{F}\right|z\rangle=\frac{1}{\pi}\int\limits_{\mathbb{C}}d\nu(w)F\left(w,\overline{w}\right)\left|\langle
z\left|w\right\rangle\right|^{2}=\mathcal{B}\left[F\right]\left(z\right)$
(4.6)

is the lower (or covariant) symbol of the operator $P_{F}.$ The map $F\mapsto$
$\mathcal{B}\left[F\right]$ generalizes the Berezin transform. Following [20],
a possible criteria for a function $D\ni z\mapsto$
$F\left(z,\overline{z}\right)$ defined on a certain domain
$D\subseteq\mathbb{C}$ to be considered as a “quantizable classical
observable” via the map (4.6) is that
$z\mapsto\mathcal{B}\left[F\right]\left(z\right)$ be a smooth
$\left(\text{i.e. }C^{\infty}\text{ on }D\right)$ with respect to
$\left(q,p\right)\equiv\frac{1}{2}\left(q+ip\right)=z.$ For more details on
the CS quantization, we refer to [21, 22, 23, 24].

##  5 Quantization via CS $\left|z,m\right\rangle$

The Heisenberg group $\mathbb{H}_{1}$ (of degree $1$) is the Lie group whose
underlying manifold is $\mathbb{C\times R=R}^{3}$ with coordinates ($x,y,t)$
and whose group law is
$(x,y,t).(x^{\prime},y^{\prime},s)=(x+x^{\prime},y+y^{\prime},t+s+\frac{1}{2}(xy^{{}^{\prime}}-x^{\prime}y)).$
The continuous unitary irreducible representations (UIR) of $\mathbb{H}_{1}$
are well known ([25], p.37).

Here, we will be concerned with the UIR of $\mathbb{H}_{1}$ on the Hilbert
space $L^{2}(\mathbb{R})$ defined by shift and multiplication operators ([26],
§1.1) as :

$T_{\tau}(x,y,t)\left[\psi\right](\xi):=\exp i\left(\tau
t-\sqrt{\tau}y\xi+\frac{\tau}{2}xy\right)\psi\left(\xi-\sqrt{\tau}x\right),\xi\in\mathbb{R}$
(5.1)

for $(x,y,t)\in\mathbb{H}_{1}\mathbf{,}$ $\tau>0$ and $\psi\in
L^{2}(\mathbb{R})$, called the Schrödinger representation. This representation
is square integrable modulo the center $\mathbb{R}$ of $\mathbb{H}_{1}$ and
the Borel section $\sigma_{0}$ of $\mathbb{H}_{1}$ over $\mathbb{R\equiv
H}_{1}\diagup\mathbb{R}$, which is given by
$\sigma_{0}\left(x,y\right)=\left(x,y,0\right).$ Further, by [27] there exists
a self-adjoint, positive semi-invariant operator $\delta$ in
$L^{2}\left(\mathbb{R}\right)$ such that

$\int\limits_{\mathbb{R}^{2}}\left\langle\psi_{1},T_{\tau}(\sigma_{0}\left(x,y\right))\left[\phi_{1}\right]\right\rangle\left\langle
T_{\tau}(\sigma_{0}\left(x,y\right))\left[\phi_{2}\right],\psi_{2}\right\rangle
d\mu(x,y)=\left\langle\psi_{1},\psi_{2}\right\rangle\left\langle\delta^{\frac{1}{2}}\phi_{1},\delta^{\frac{1}{2}}\phi_{2}\right\rangle$
(5.2)

for all $\psi_{1},\psi_{2}\in L^{2}(\mathbb{R},d\xi)$ and
$\phi_{1},\phi_{2}\in Dom\left(\delta^{\frac{1}{2}}\right).$ The group
$\mathbb{H}_{1}$ here being unimodular, therefore $\delta$ must be the
identity operator [28]. The connection of the Schrödinger representation
$T_{\tau}$ with spin $\frac{1}{2}$ representations and magnetic field can be
found in [29].

Now, according to [30], CS are obtained “à la Perelomov”[31] as orbits of the
unitary operator $T_{\tau}$ acting on the eigenfunction $\phi_{m}\in$
$L^{2}(\mathbb{R)}$ of the harmonic oscillator $\hat{H}$ in (4.1) associated
with eigenvalue $m\in\mathbb{Z}_{+}$ and given by

$\phi_{m}(\xi):=\left(\sqrt{\pi}2^{m}m!\right)^{-\frac{1}{2}}e^{-\frac{1}{2}\xi^{2}}H_{m}(\xi),\qquad\xi\in\mathbb{R},$
(5.3)

where $H_{m}$ is the Hermite polynomial ([16], p.249), as

$\left|\left(x,y\right),\tau,m\right\rangle:=T_{\tau}(\sigma_{0}(x,y))[\phi_{m}].$
(5.4)

Setting $z\equiv x+iy$ and $\tau=2,$ the wavefunction of the state (5.4) is
given by

$\langle\xi\left|z,m\right\rangle=\left(-1\right)^{m}\left(\sqrt{\pi}2^{m}m!\right)^{-\frac{1}{2}}e^{-\frac{1}{2}\overline{z}^{2}+\sqrt{2}\xi\overline{z}-\frac{1}{2}\left|z\right|^{2}-\frac{1}{2}\xi^{2}}H_{m}\left(\xi-\frac{z+\overline{z}}{\sqrt{2}}\right).$
(5.5)

These coherent states are completely justified by the square integrability
property (5.2) of $T_{\tau}$ modulo the subgroup $\mathbb{R}$ and the section
$\sigma_{0},$ which leads to the resolution of the identity operator

$\mathbf{1}_{L^{2}(\mathbb{R})}=\frac{1}{\pi}\int\limits_{\mathbb{C}}d\nu(z)\left|z,m\right\rangle\left\langle
z,m\right|.$ (5.6)

As for the canonical CS $\left|z\right\rangle\equiv\left|z,0\right\rangle$,
Eq. (5.6) allows to implement a CS quantization of the set of parameters $z$
$\in\mathbb{C}$ by associating to a function $\mathbb{C\ni}z\mapsto
F\left(z,\overline{z}\right)\in\mathbb{R}$ the operator on
$L^{2}\left(\mathbb{R}\right):$

$F\longmapsto
P_{F}:=\frac{1}{\pi}\int\limits_{\mathbb{C}}d\nu(z)\left|z,m\right\rangle\left\langle
z,m\right|F\left(z,\overline{z}\right).$ (5.7)

We can prove (Appendix A) that $P_{F}$ admits the following discrete
resolution with respect to the orthonormal basis $|j\rangle\equiv\phi_{j}$
given by (5.3) as

$P_{F}=\sum_{j,k=0}^{+\infty}\mathfrak{a}_{j,k}^{F}\left|j\right\rangle\left\langle
k\right|$ (5.8)

with the matrix elements

$\mathfrak{a}_{j,k}^{F}=c_{j,k}^{\left(m\right)}\int\limits_{\mathbb{C}}e^{-|z|^{2}}|z|^{|k-m|+|j-m|}e^{i(j-k)\arg
z}L_{m\wedge j}^{(|j-m|)}(|z|^{2})L_{m\wedge
k}^{(|k-m|)}(|z|^{2})F(z)d\nu(z),$ (5.9)

where

$c_{j,k}^{\left(m\right)}:=\frac{1}{\pi m!\sqrt{j!k!}}(-1)^{m\wedge j+m\wedge
k}(m\wedge k)!(m\wedge j)!.$ (5.10)

In particular, when $F$ is a radial function, i.e., $F(z)=\phi(r),$
$r=\left|z\right|$, then, we may use polar coordinates to rewrite (5.9) as

$\mathfrak{a}_{j,k}^{F}=c_{j,k}^{\left(m\right)}\int\limits_{0}^{\infty}e^{-r^{2}}r^{|k-m|+|j-m|}L_{m\wedge
j}^{(|j-m|)}(r^{2})L_{m\wedge
k}^{(|k-m|)}(r^{2})\phi\left(r\right)rdr\int\limits_{0}^{2\pi}e^{i(j-k)\theta}d\theta.$
(5.11)

From (5.11), only the coefficient corresponding to $j=k$ is nonzero and this
amounts to the eigenvalues of $P_{F}$ with the form

$\lambda_{k}^{F}:=\mathfrak{a}_{k,k}^{\digamma}=\frac{(m\wedge k)!}{(m\vee
k)!}\int\limits_{0}^{\infty}e^{-\rho}\rho^{|k-m|}\left(L_{m\wedge
k}^{(|k-m|)}(\rho)\right)^{2}\phi(\sqrt{\rho})d\rho$ (5.12)

which provides the following reduced form of (5.8):

$P_{F}=\sum_{k=0}^{+\infty}\lambda_{k}^{F}\left|k\right\rangle\left\langle
k\right|.$ (5.13)

Moreover, it’s not difficult to see that
$P_{F}\left|k\right\rangle=\lambda_{k}^{F}\left|k\right\rangle,$ meaning that
the Hermite functions (5.3) are eigenfunctions of $P_{F}$. The latter one may
also be unitarly intertwined, via the coherent states transform
$\mathcal{W}_{m}:L^{2}(\mathbb{R})\longrightarrow\mathcal{A}_{m}(\mathbb{C})$
defined by [30] :

$\mathcal{W}_{m}[\phi](z)=(-1)^{m}(2^{m}m!\sqrt{\pi})^{-\frac{1}{2}}e^{-\frac{1}{2}\overline{z}^{2}}\int\limits_{\mathbb{R}}\phi(\xi)e^{\sqrt{2}\xi\overline{z}-\frac{1}{2}\xi^{2}}H_{m}\left(\xi-\frac{z+\overline{z}}{\sqrt{2}}\right)d\xi,$
(5.14)

to act as $\widetilde{P}_{F}=\mathcal{W}_{m}\circ
P_{F}\circ\mathcal{W}_{m}^{-1}$ on the $m$-true-polyanalytic space
$\mathcal{A}_{m}(\mathbb{C}).$ If
$F\equiv{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}$
the indicator function of the disk $D_{R}$, $\widetilde{P}_{F}$ turns out to
be the concentration operator. That is, the restriction to the disk $D_{R}$ of
the integral operator $\tilde{\mathfrak{K}}_{m}$ on
$L^{2}(\mathbb{C},d\lambda)$ with the kernel $\widetilde{K}_{m}(z,w)$ in
(3.5), which is the reproducing kernel of $\mathcal{A}_{m}(\mathbb{C})$.

##  6 Mean value and variance of $\sharp_{D_{R}}$

We here will be dealing with two functions: $\left(i\right)$ the indicator
function
$F_{R}\equiv{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}$
of the disk $D_{R}$ and $\left(ii\right)$ the function

$G_{R}\left(z,\overline{z}\right)\equiv Area\left(D_{R}^{c}\cap
D_{R}\left(z\right)\right)=\begin{cases}\pi R^{2}&,\left|z\right|>2R\\\ \pi
R^{2}-2R^{2}\arccos(\frac{1}{2R}\left|z\right|)+\frac{1}{2}\left|z\right|\sqrt{4R^{2}-\left|z\right|^{2}}&,\left|z\right|\leq
2R\end{cases}$ (6.1)

Each of these functions will be treated as quantizable classical observables
on the phase space $\mathbb{C}=\mathbb{R}^{2}$ in order to derive some
spectral properties for their  quantum counterpart observables which are
(essentially) self-adjoint operators on $L^{2}\left(\mathbb{R}\right)$,
resulting by applying the coherent states quantization mapping (5.7).

###  6.1 The number mean of $\sharp_{D_{R}}$

For $F_{R}$
$\equiv{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}},$
the eigenvalues (5.12) take the form (see Appendix B):

$\beta_{k}^{(m,R)}=\sum\limits_{j=0}^{2\left(m\wedge
k\right)}\mathfrak{a}_{j}^{\left(m,k\right)}\gamma\left(\left|k-m\right|+j+1,R^{2}\right)$
(6.2)

where

$\mathfrak{a}_{j}^{\left(m,k\right)}=m!k!\left(-1\right)^{j}\sum\limits_{\ell=0}^{j}\frac{1}{\ell!\left(j-\ell\right)!\left(m\wedge
k-j+\ell\right)!\left(m\wedge
k-\ell\right)!\left(\left|k-m\right|+j-\ell\right)!\left(\left|k-m\right|+\ell\right)!}$
(6.3)

and

$\gamma\left(\alpha,\tau\right)=\int\limits_{0}^{\tau}t^{\alpha-1}e^{-t}dt,\text{
\ \ }\mathop{\mathrm{R}e}\alpha>0$ (6.4)

denotes the incomplete gamma function ([16], p.337).

For the DPP $\mu_{K_{m},d\nu}$ the number of points $\sharp_{D_{R}},$ that
fall in the disk $D_{R}$ has the same distribution as a sum of independent
Bernoulli random variables $(\beta_{k}^{(m,R)})_{k\geq 0}$. Straightforward
calculations (see Appendix C) leads to the mean value

$\mathbb{E}\left(\sharp_{D_{R}}\right)=\sum\limits_{k=0}^{+\infty}\beta_{k}^{(m,R)}=R^{2}$
(6.5)

as expected.
Remark 6.1. Note that for $m=0,$ the eigenvalues (6.2) reduce to

$\beta_{k}^{(0,R)}=\frac{1}{k!}\gamma\left(k+1,R^{2}\right)$ (6.6)

from which we recover, up to a scaling $R\rightarrow\frac{1}{\sqrt{2}}R,$ the
Daubechies’s result ([32], p.610).

###  6.2 The variance of $\sharp_{D_{R}}$

On one hand, we may apply the above quantization scheme, via the CS
$|z,m\rangle$, to the function $G_{R}$ taken as a radial weight function to
construct the discrete spectral resolution

$P_{G}=\sum_{k=0}^{+\infty}\lambda_{k}^{(m,R)}|k\rangle\langle k|$ (6.7)

for the corresponding operator $P_{G}$. Indeed, direct calculations (see
Appendix D) gives us the eigenvalues

$\lambda_{k}^{(m,R)}=\left(\pi
R\right)^{2}-\frac{m!\pi^{3/2}(2R)^{2(k-m)+4}\Gamma((k-m)+\frac{3}{2})}{4\Gamma(2(k-m)+1)\Gamma((k-m)+2)k!(k-m+2)(k-m+1)}$
$\times\sum_{s=0}^{2m}A_{s}^{(m,k)}~{}~{}_{3}F_{3}\left(\begin{matrix}2(k-m)+s+1,(k-m)+\frac{3}{2},(k-m)+1\\\
2(k-m)+1,(k-m)+2,(k-m)+3\end{matrix};-4R^{2}\right)$ (6.8)

with

$A_{s}^{(m,k)}:=\frac{(-1)^{s}(2(k-m)+s)!}{s!}\sum\limits_{r=0}^{s}\left(\begin{array}[]{c}s\\\
r\end{array}\right)\left(\begin{array}[]{c}k\\\
m-s+r\end{array}\right)\left(\begin{array}[]{c}k\\\
m-r\end{array}\right),\qquad k\geq m.$

Next, the variance of $\sharp_{D_{R}}$ counting the number of points inside
the disk $D_{R}$ can be expressed as a “dequantization” of $P_{R}$ with
respect to the set of CS $\left|w,m\right\rangle$ as

$\mathbb{V}ar(\sharp_{D_{R}})=\langle w,m\left|P_{R}\right|w,m\rangle\text{
}\mid_{w=0}$ (6.9)

that is the Berezin transform of $G_{R}$ evaluated at the point $w=0$. So
inserting (6.7) into (6.9), we

$\mathbb{V}ar(\sharp_{D_{R}})=\langle
0,m\left|\sum_{k=0}^{+\infty}\lambda_{k}^{\left(m,R\right)}\left|k\right\rangle\left\langle
k\right|\right|0,m\rangle=\sum_{k=0}^{+\infty}\lambda_{k}^{\left(m,R\right)}\left(R\right)\langle
m\left|k\right\rangle\left\langle k\right|m\rangle=\lambda_{m}^{(m,R)}$ (6.10)

Therefore, putting $k=m$ in (6.8), we obtain

$\mathbb{V}ar(\sharp_{D_{R}})=R^{2}\left[1-R^{2}\sum_{s=0}^{m}(-1)^{s}\left(\begin{array}[]{c}m\\\
s\end{array}\right){}_{3}F_{2}\left(\begin{matrix}-m,-s,-s\\\
1,m-s+1\end{matrix};-1\right)~{}~{}_{2}F_{2}\left(\begin{matrix}s+1,\frac{3}{2}\\\
3,2\end{matrix};-4R^{2}\right)\right].$ (6.11)

For the first Landau level $m=0$, Eq. (6.11) reduces to

$\mathbb{V}ar(\sharp_{D_{R}})=R^{2}\left[1-R^{2}~{}_{2}F_{2}\left(\begin{matrix}1,\frac{3}{2}\\\
3,2\end{matrix};-4R^{2}\right)\right].$ (6.12)

By applying the identity ([33], p.589):

$~{}_{2}F_{2}\left(\begin{matrix}1,\frac{3}{2}\\\
3,2\end{matrix};-4R^{2}\right)=R^{-2}\left[1-e^{-2R^{2}}\left(I_{0}(-2R^{2})-I_{1}(-2R^{2})\right)\right]$
(6.13)

in terms of modified Bessel functions $I_{0}$ and $I_{1}$, Eq. (6.12) also
reads

$\mathbb{V}ar(\sharp_{D_{R}})=R^{2}e^{-2R^{2}}\left(I_{0}(2R^{2})+I_{1}(2R^{2})\right)$
(6.14)

which corresponds to the (infinite) Ginibre process in agreement with the
result of Osada and Shirai ([34], p.2).

## Appendix A

In order to prove (5.8), we need to show that the operator $P_{F}$ defined by
(5.7) satisfies

$\langle P_{F}[\phi_{j}],\phi_{k}\rangle=\mathfrak{a}_{j,k}^{F}$ (A1)

where $\mathfrak{a}_{j,k}^{F}$ are the coefficients given by (5.9). Here
$\\{\phi_{j}\\}$ is the basis vector of Hermite functions defined in (5.3).
For this, we may use the number states expansion of the CS $|z,m\rangle$
([30], p.4) :

$|z,m\rangle=\left(e^{|z|^{2}}\right)^{-\frac{1}{2}}\sum_{j\geq
0}\frac{(-1)^{m}}{\sqrt{\pi m!j!}}m!|z|^{(j-m)}e^{i(m-j)\arg
z}L_{m}^{j-m}\left(|z|^{2}\right)\phi_{j}$ (A2)

to write the above coefficients as

$\displaystyle\mathfrak{a}_{j,k}^{\digamma}$ $\displaystyle=$
$\displaystyle\left\langle\frac{1}{\pi}\int_{\mathbb{C}}|z,m\rangle\langle
z,m|F(z,\bar{z})d\nu(z)[\phi_{j}],\phi_{k}\right\rangle_{L^{2}(\mathbb{R})}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{\mathbb{C}}\left\langle\phi_{j}|z,m\right\rangle\left\langle
z,m|\phi_{k}\right\rangle F(z,\bar{z})d\nu(z)$
$=\frac{1}{\pi^{2}}\frac{m!}{\sqrt{j!k!}}\int_{\mathbb{C}}|z|^{(j+k-2m)}e^{-|z|^{2}}e^{i(j-k)\arg
z}L_{m}^{(j-m)}\left(|z|^{2}\right)L_{m}^{(k-m)}\left(|z|^{2}\right)F(z,\bar{z})d\nu(z)$
(A3)

which completes the proof of Appendix A. $\square$

## Appendix B

For
$F={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{D_{R}}$
the indicator function of the disk $D_{R}$, the eigenvalues (5.12) take the
form

$\beta_{k}^{(m,R)}=\frac{(m\wedge k)!}{(m\vee
k)!}\int\limits_{0}^{R^{2}}e^{-\rho}\rho^{|k-m|}\left(L_{m\wedge
k}^{(|k-m|)}(\rho)\right)^{2}d\rho.$ (B1)

Making use of the Feldheim’s formula ([35], Eq.1.13) :

$L_{q}^{\left(\alpha\right)}\left(x\right)L_{p}^{\left(\alpha\right)}\left(x\right)=\sum\limits_{j=0}^{q+p}\left(-1\right)^{j}A_{j}^{\left(q,p,\alpha\right)}\frac{x^{j}}{j!}$
(B2)

where

$A_{j}^{\left(q,p,\alpha\right)}=\sum\limits_{\ell=0}^{j}\left(\begin{array}[]{c}j\\\
\ell\end{array}\right)\left(\begin{array}[]{c}q+\alpha\\\
q-j+\ell\end{array}\right)\left(\begin{array}[]{c}p+\alpha\\\
p-\ell\end{array}\right)$ (B3)

for parameters $p=q=m\wedge k,$ $\alpha=\left|k-m\right|$ and $x=\rho$, then
we may write

$\frac{(m\wedge k)!}{(m\vee k)!}\left(L_{m\wedge
k}^{(|k-m|)}(\rho)\right)^{2}=\sum\limits_{j=0}^{2\left(m\wedge
k\right)}\mathfrak{a}_{j}^{\left(m,k\right)}\rho^{j}.$ (B4)

By inserting (B4) into (B1), we obtain

$\beta_{k}^{(m,R)}=\sum\limits_{j=0}^{2\left(m\wedge
k\right)}\mathfrak{a}_{j}^{\left(m,k\right)}\int\limits_{0}^{R^{2}}e^{-\rho}\rho^{|k-m|+j}d\rho=\sum\limits_{j=0}^{2\left(m\wedge
k\right)}\mathfrak{a}_{j}^{\left(m,k\right)}\gamma\left(|k-m|+j-1,R^{2}\right)$
(B5)

as announced in (6.2). $\square$

## Appendix C

The random variable $\sharp_{D_{R}}$ has the same distribution as a sum of
independent Bernoulli random variables of parameters $0<\beta_{k}^{(m,R)}<1$,
$k\geq 0$. Therefore, the expectation of $\sharp_{D_{R}}$ is given by the sum

$\mathbb{E}\left(\sharp_{D_{R}}\right)=\sum_{k=0}^{+\infty}\beta_{k}^{(m,R)}=\sum_{k\geqslant
0}\frac{(m\wedge k)!}{(m\vee
k)!}\int_{0}^{R^{2}}e^{-\rho}\rho^{|k-m|}\left(L_{m\wedge
k}^{|k-m|}(\rho)\right)^{2}d\rho.$ (C1)

The integer $m$ being fixed, we may write the series

$\mathfrak{S}_{m}(\rho)=\sum_{k=0}^{+\infty}\frac{(m\wedge k)!}{(m\vee
k)!}\rho^{|k-m|}\left(L_{m\wedge k}^{|k-m|}(\rho)\right)^{2}$ (C2)

as $S_{<\infty}(m,\rho)+S_{\infty}(m,\rho)$, where

$S_{<\infty}(m,\rho)=\sum_{k=0}^{m-1}\frac{k!}{m!}\rho^{m-k}\left(L_{k}^{m-k}(\rho)\right)^{2}-\sum_{k=0}^{m-1}\frac{m!}{k!}\rho^{k-m}\left(L_{m}^{k-m}(\rho)\right)^{2}$
(C3)

and

$S_{\infty}(m,\rho)=\sum_{k=0}^{+\infty}\frac{m!}{k!}\rho^{k-m}\left(L_{m}^{k-m}(\rho)\right)^{2}.$
(C4)

By making use of the identity ([36], p. 98):

$L_{m}^{(k-m)}\left(\rho\right)=\left(-\rho\right)^{m-k}\frac{k!}{m!}L_{k}^{(m-k)}\left(\rho\right),$
(C5)

one can check that $S_{<\infty}(m,\rho)=0$. For the infinite sum in (C4), we
may apply the following formula of Bateman ([37], p.457):

$\sum_{l=0}^{+\infty}\frac{n!}{l!}\left(\sqrt{xy}e^{i\varphi}\right)^{l-n}L_{n}^{(l-n)}\left(x\right)L_{n}^{(l-n)}\left(y\right)=\exp\left(\sqrt{xy}e^{i\varphi}\right)L_{n}^{(0)}\left(x+y-2\sqrt{xy}\cos\varphi\right)$
(C6)

for $n=m,\,x=y=\rho$ and $\varphi=0$. This leads to
$S_{\infty}(m,p)=e^{\rho}$. Summarizing the above calculations, Eq. (C1) reads

$\mathbb{E}\left(\sharp_{D_{R}}\right)=\int_{0}^{R^{2}}e^{-\rho}\mathfrak{S}_{m}(\rho)d\rho=\int_{0}^{R^{2}}d\rho=R^{2}.\qquad$
(C7)

This completes the proof of Appendix C.

## Appendix D

By choosing the radial weight function (6.1), the eigenvalues in Eq.(5.12) can
be decomposed into three integrals as

$\lambda_{k}^{(m,R)}=\sigma_{1}-\sigma_{2}+\sigma_{3}$ (D1)

where

$\sigma_{1}:=\pi
R^{2}\frac{m!}{k!}\int\limits_{0}^{\infty}e^{-\rho}\rho^{k-m}\left(L_{m}^{(k-m)}(\rho)\right)^{2}d\rho,$
(D2)
$\sigma_{2}:=2R^{2}\int_{0}^{2R}\rho^{2(k-m)+1}\arccos\left(\frac{\rho}{2R}\right)e^{-\rho^{2}}\left(L_{m\wedge
k}^{(k-m)}\left(\rho^{2}\right)\right)^{2}d\rho,$ (D3)

and

$\sigma_{3}:=\frac{1}{2}\int_{0}^{2R}\sqrt{4R^{2}-\rho^{2}}e^{-\rho^{2}}\rho^{2(k-m)+2}\left(L_{m}^{(k-m)}\left(\rho^{2}\right)\right)^{2}d\rho.$
(D4)

To calculate $\sigma_{1}$, we make use of the orthogonality relation of
Laguerre polynomials ([38], p.809), the integral (D2) reduces to
$\sigma_{1}=\pi R^{2}.$

For $\sigma_{2}$, we set $a=2R$ and $\alpha=k-m$, then

$\sigma_{2}=\frac{a^{2}}{2}\int_{0}^{a}\rho^{2\alpha+1}\arccos\left(\frac{\rho}{a}\right)e^{-\rho^{2}}\left(L_{m}^{(\alpha)}\left(\rho^{2}\right)\right)^{2}d\rho.$
(D5)

Next, we apply the Feldheim’s formula ([35], Eq.1.14):

$L_{m}^{(\alpha)}(x)L_{m}^{(\alpha)}(x)=\sum_{s=0}^{2m}C_{s}(m,\alpha)L_{s}^{(2\alpha)}(x)$
(D6)

with

$C_{s}(m,\alpha)=(-1)^{s}\sum_{r=0}^{s}\left(\begin{array}[]{l}s\\\
r\end{array}\right)\left(\begin{array}[]{c}m+\alpha\\\
m-s+r\end{array}\right)\left(\begin{array}[]{c}m\\\ m-r\end{array}\right).$
(D7)

This gives

$\sigma_{2}=\frac{a^{2}}{2}\sum_{s=0}^{2m}C_{s}(m,\alpha)\int_{0}^{a}\rho^{2\alpha+1}\arccos\left(\frac{\rho}{a}\right)e^{-\rho^{2}}L_{s}^{(2\alpha)}\left(\rho^{2}\right)d\rho.$
(D8)

By using the expression of the Laguerre polynomials
$L_{s}^{(2\alpha)}\left(u^{2}\right)=\sum_{l=0}^{s}\frac{(-1)^{l}}{l!}\left(\begin{array}[]{c}s+2\alpha\\\
s-l\end{array}\right)u^{2l}$ and setting $\rho=ta$, Eq. (D8) becomes

$\sigma_{2}=\frac{a^{2(\alpha+2)}}{2}\sum_{s=0}^{2m}C_{s}(m,\alpha)\sum_{l=0}^{s}\frac{(-1)^{l}a^{2l}}{l!}\left(\begin{array}[]{c}s+2\alpha\\\
s-l\end{array}\right)\sigma_{2}^{\prime}$ (D9)

where

$\sigma_{2}^{\prime}:=\int_{0}^{1}t^{2(\alpha+l)+1}\arccos\left(t\right)e^{-a^{2}t^{2}}dt.$
(D10)

In order to calculate $\sigma_{2}^{\prime}$, we set $u=\arccos(t)$ ,
$dv=t^{2(\alpha+l)+1}e^{-a^{2}t^{2}}$ and we can easily check that

$v=\frac{a^{-2(\alpha+l)-2}}{2}\gamma\left(\alpha+l+1,a^{2}t^{2}\right)$ (D11)

where $\gamma$ is the incomplete Gamma function. Next, we apply the
integration by parts to get that

$\sigma_{2}^{\prime}=\frac{a^{-2(\alpha+l)-2}}{2}\int_{0}^{1}\frac{\gamma\left(\alpha+l+1,a^{2}t^{2}\right)}{\sqrt{1-t^{2}}}dt.$
(D12)

Performing again the change of variables $t^{2}=x$, we may rewrite (D12) as

$\sigma_{2}^{\prime}=\frac{a^{-2(\alpha+l)-2}}{8}\int_{0}^{1}x^{-\frac{1}{2}}(1-x)^{-\frac{1}{2}}\gamma\left(\alpha+l+1,a^{2}x\right)dx$
(D13)

By applying the following formula ([33], p.143) :

$\int_{0}^{\zeta}x^{n-1}\left(\zeta^{r}-x^{r}\right)^{\beta-1}\gamma(\nu,cx)dx=\frac{\zeta^{n-r\beta-r+\nu}c^{\nu}}{r}\Gamma(\beta)\sum_{s=0}^{\infty}\frac{(-\zeta
c)^{s}}{s!(\nu+s)}\frac{\Gamma((n+\nu+s)/r)}{\Gamma((n+\nu+s)/r+\beta)}$ (D14)

for the parameters $n=1/2$, $\zeta=1$, $\beta=1/2$, $\nu=\alpha+l+1$, $r=1$,
$c=a^{2}$, to get that

$\sigma_{2}^{\prime}=\frac{\Gamma(1/2)}{8}\sum_{s=0}^{\infty}\frac{\left(-a^{2}\right)^{s}}{s!(\alpha+l+1+s)}\frac{\Gamma(s+\alpha+l+3/2)}{\Gamma(s+\alpha+l+2)}$
(D15)

which can be expressed as

$\sigma_{2}^{\prime}=\frac{\sqrt{\pi}}{8}\frac{\Gamma(\alpha+l+3/2)}{(\alpha+l+1)^{2}\Gamma(\alpha+l+1)}{}_{2}F_{2}\left(\begin{array}[]{c}\alpha+l+1,\alpha+l+3/2\\\
\alpha+l+2,\alpha+l+2;-a^{2}\end{array}\right).$ (D16)

By inserting (D16) into (D9), we obtain

$\sigma_{2}=\frac{a^{2}\sqrt{\pi}}{2}\sum_{s=0}^{2m}C_{s}(m,\alpha)K_{l,s}(\alpha)$
(D17)

where

$K_{l,s}(\alpha)=a^{2\alpha+2}\frac{\Gamma(\alpha+1)\Gamma\left(\alpha+\frac{3}{2}\right)}{\Gamma(\alpha+2)\Gamma(\alpha+2)}\frac{(s+2\alpha)!}{s!\Gamma(2\alpha+1)}$
$\times\sum_{l=0}^{s}\left(\begin{array}[]{l}s\\\
l\end{array}\right)\frac{\left(-a^{2}\right)^{l}}{(2\alpha+1)_{l}}\frac{(\alpha+1)_{l}\left(\alpha+\frac{3}{2}\right)_{l}}{(\alpha+2)_{l}(\alpha+2)_{l}}{}_{2}F_{2}\left(\begin{array}[]{c}\alpha+l+1,\alpha+l+3/2\\\
\alpha+l+2,\alpha+l+2;-a^{2}\end{array}\right).$ (D18)

By appealing the identity ([33], p.391) :

$\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)\frac{x^{k}}{(\beta)_{k}}\frac{\prod\left(a_{p}\right)_{k}}{\prod\left(b_{q}\right)_{k}}{}_{p}F_{q}\left(\begin{array}[]{c}\left(a_{p}\right)+k;x\\\
\left(b_{q}\right)+k\end{array}\right)={}_{p+1}F_{q+1}\left(\begin{array}[]{c}\left(a_{p}\right),\beta+n;x\\\
\left(b_{q}\right),\beta\end{array}\right),$ (D19)

the sum in (D18) takes the form

${}_{3}F_{3}\left(\begin{array}[]{c}\alpha+1,\alpha+\frac{3}{2},2\alpha+1+s;-a^{2}\\\
\alpha+2,\alpha+2,2\alpha+1\end{array}\right).$ (D20)

Therefore,

$K_{l,s}(\alpha)=a^{2\alpha+2}\frac{\Gamma(\alpha+1)\Gamma\left(\alpha+\frac{3}{2}\right)}{\Gamma(\alpha+2)\Gamma(\alpha+2)}\frac{(s+2\alpha)!}{s!\Gamma(2\alpha+1)}{}_{3}F_{3}\left(\begin{array}[]{c}\alpha+1,\alpha+\frac{3}{2},2\alpha+1+s;-a^{2}\\\
\alpha+2,\alpha+2,2\alpha+1\end{array}\right),$ (D21)

and our integral $\sigma_{2}$ can be expressed as

$\sigma_{2}=\frac{(2R)^{2}\sqrt{\pi}}{8}\frac{(2R)^{2(k-m)+2}\Gamma(k-m+1)\Gamma\left(k-m+\frac{3}{2}\right)}{\Gamma(2(k-m)+1)\Gamma(k-m+2)\Gamma(k-m+2)}$
$\times\sum_{s=0}^{2m}C_{s}(m,k-m)\frac{(s+2(k-m))!}{s!}{}_{3}F_{3}\left(\begin{array}[]{c}k-m+1,k-m+\frac{3}{2},2(k-m)+1+s;-(2R)^{2}\\\
k-m+2,k-m+2,2(k-m)+1\end{array}\right).$ (D22)

For the last integral, we again use Feldheim’s formula (D6), then Eq.(D4)
becomes

$\sigma_{3}=\frac{1}{2}\sum_{s=0}^{2m}C_{s}(m,k-m)\int_{0}^{2R}\sqrt{4R^{2}-\rho^{2}}e^{-\rho^{2}}\rho^{2(k-m)+2}L_{s}^{(2(k-m))}\left(\rho^{2}\right)d\rho$
(D23)

where $C_{s}(m,k-m)$ is given by (D7). Setting $\alpha=k-m$,
$\rho^{2}=(2R)^{2}t,$ we may rewrite (D23) as

$\sigma_{3}=\frac{(2R)^{2\alpha+4}}{4}\sum_{s=0}^{2m}C_{s}(m,\alpha)\int_{0}^{1}t^{\alpha+\frac{1}{2}}e^{-(2R)^{2}t}(1-t)^{\frac{1}{2}}L_{s}^{(2\alpha)}\left((2R)^{2}t\right)dt.$
(D24)

Now, we use the formula ([38], p.810):

$\int_{0}^{1}(1-x)^{\mu-1}x^{\lambda-1}e^{-\beta x}L_{s}^{(\gamma)}(\beta
x)dx=\frac{\Gamma(\gamma+s+1)}{s!\Gamma(\gamma+1)}\frac{\Gamma(\lambda)\Gamma(\mu)}{\Gamma(\lambda+\mu)}{}_{2}F_{2}\left(\begin{array}[]{c}\gamma+s+1,\lambda\\\
\gamma+1,\lambda+\mu;-\beta\end{array}\right)$ (D25)

for the parameters $\mu=\frac{3}{2}$, $\lambda=\alpha+\frac{3}{2}$,
$\beta=(2R)^{2}$, $\gamma=2\alpha$, to get that

$\sigma_{3}=\frac{(2R)^{2\alpha+4}\Gamma(\alpha+\frac{3}{2})\Gamma(\frac{3}{2})}{4\Gamma(2\alpha+1)\Gamma(\alpha+3)}\sum_{s=0}^{2m}C_{s}(m,\alpha)\frac{\Gamma(2\alpha+s+1)}{s!}{}_{2}F_{2}\left(\begin{array}[]{c}2\alpha+s+1,\alpha+\frac{3}{2}\\\
2\alpha+1,\alpha+3;-(2R)^{2}\end{array}\right).$ (D26)

Next, to compute the difference $\sigma_{3}-\sigma_{2}$ we may apply the
following identity

$\frac{1}{(d+2)}{}_{p}F_{q}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{p-1},a_{p}\\\
b_{1},b_{2},\ldots,b_{q-1},d+3\end{array};z\right)-\frac{1}{(d+1)}{}_{p+1}F_{q+1}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{p-1},a_{p},d+1\\\
b_{1},b_{2},\ldots,b_{q-1},d+2,d+2\end{array};z\right)$
$=-\frac{1}{(d+2)(d+1)}{}_{p+1}F_{q+1}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{p-1},a_{p},d+1\\\
b_{1},b_{2},\ldots,b_{q-1},d+2,d+3\end{array};z\right),\qquad
p,q\in\mathbb{N},\;z\in\mathbb{C}.$ (D27)

which is easy to check from the definition of the hypergeometric series
${}_{p}F_{q}$. Finally, we obtain, after summarizing all the calculations in
(D1), the expression of the eigenvalues given by (6.8). $\square$

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# Sibling Rivalry: SNeIa Diversity and the Hubble Tension

Richard S. Miller Johns Hopkins University Applied Physics Laboratory
11100 Johns Hopkins Road
Laurel, MD 20723, USA

###### Abstract

Thermonuclear supernovae, or Type-Ia supernovae (SNeIa), are an essential tool
of cosmology. Precise cosmological constraints are extracted from a Hubble
diagram defined by homogeneous distance indicators, but supernova homogeneity
is not guaranteed. The degree of heterogeneity within the SNeIa parent
population is unknown. In addition, event selections and standardization
procedures are based on empirical, optically-measured observables rather than
fundamental thermonuclear properties. Systematics are a natural consequence of
event selection from a diverse parent population. Quantifying the impact of
diversity-driven systematics is crucial to optimizing SNeIa as cosmic probes.
In this work, the empirical observables are used to calibrate previously
unidentified diversity-driven systematic uncertainties. The foundation of this
approach is the concept of “supernova siblings”, two or more supernovae hosted
by the same parent galaxy. Sibling-based calibrations isolate intrinsic
differences between supernovae; they control for source distance and host
galaxy dependencies that can conceal systematics or lead to their
underestimation. Newly calibrated distance modulus uncertainties are
approximately an order of magnitude larger than previously reported. The
physical origin of these uncertainties is plausibly attributed to the diverse
thermonuclear scenarios responsible for SNeIa and the inhomogeneous apparent
magnitudes induced by this diversity. Systematics mitigation strategies are
discussed. Cosmological parameter constraints extracted from a re-analysis of
the Pantheon+ SNeIa dataset are weaker than previously reported. Agreement
with early-Universe parameter estimates is achieved for a $\Lambda$CDM
cosmology, including a reduction of the Hubble Tension from $\sim$5$\sigma$ to
$<$1$\sigma$.

††software:  CosmoSIS (Zuntz et al., 2015), matplotlib (Hunter, 2007),
Mathematica (Wolfram Research, Inc., 2022), Pantheon+ Data Release
(https://github.com/PantheonPlusSH0ES/DataRelease)

##  1 Introduction

Thermonuclear, or Type-Ia, supernovae (SNeIa) are deeply connected to topics
throughout astrophysics and cosmology. They have synthesized the majority of
iron in the universe and have supplied much of the energy input to the
interstellar medium. Thermonuclear supernovae are also effective distance
indicators, making them one of the premiere probes of cosmology. Their
treatment as standardizable candles and visibility out to high redshifts has
had sweeping ramifications, including direct evidence for an accelerating
universal expansion and the existence of dark energy (Riess et al., 1998;
Perlmutter et al., 1999).

Supernovae continue to be used to constrain cosmological models. The most
precise constraints are those extracted from a systematics-free Hubble diagram
defined by a homogeneous sample of standard (or standardizable) candles
(Goobar & Leibundgut, 2011). A homogeneous sample of supernovae is not
guaranteed; event selections and optically-calibrated standardization
procedures (Phillips, 1993; Guy et al., 2007) that homogenize these events are
based on secondary observables, rather than thermonuclear scenarios that
govern these cosmic beacons (Hillebrandt & Niemeyer, 2000; Maoz et al., 2014;
Seitenzahl & Townsley, 2017; Clark et al., 2021). Rapid thermonuclear burning
of a white dwarf is central to the SNeIa paradigm (e.g., Nomoto et al.
(1984)), but the nature of progenitor(s) and the method(s) of ignition remain
a topic of intense debate (Hillebrandt & Niemeyer, 2000; Maoz et al., 2014;
Seitenzahl & Townsley, 2017; Clark et al., 2021).

Systematics are a natural consequence of event selection from a heterogenous
(i.e., diverse) parent SNeIa population. Quantifying diversity-driven
systematics is important. For example, over-constrained cosmological
parameters are a direct result of a Hubble diagram defined by distance moduli
with underestimated uncertainties. The imperative to understand systematics
has taken on new relevance since there now exists an unresolved inconsistency
between measurements of the Hubble-Lemaître parameter ($H_{\circ}$), the
fundamental metric of cosmic expansion (Freedman, 2021; Kamionkowski & Riess,
2022; Abdalla et al., 2022).

A comprehensive new analysis, and calibration, of SNeIa systematic
uncertainties is presented here. The foundation of this work is the concept of
“supernova siblings”, two or more supernovae hosted by the same parent galaxy.
Galaxies that host multiple supernovae are unique cosmic laboratories; they
isolate sibling-to-sibling differences by controlling for source distance and
global host galaxy properties. These scientific controls mitigate the
confounding effects that conceal diversity-driven systematics to reveal newly
identified distance modulus uncertainties.

Previous sibling-based studies were motivated by the goal to characterize the
distance modulus uncertainties that define Hubble diagram quality; these
qualitative analyses identified evidence for correlations between sibling
observables, but did not quantitatively account for them (Scolnic et al.,
2020, 2022). Doing so is critical; incognito intrinsic correlations conceal
the presence of systematics by artificially biasing statistical measures to
low values. The sibling-based calibration procedures derived below fully
account for intrinsic correlations to give a rigorous accounting of systematic
uncertainties for SNeIa distance moduli and associated observables.

This paper is organized as follows: Supernova sibling datasets are described
in Section 2. Sibling-based calibration procedures are presented in Section 3.
The impact of newly calibrated distance modulus uncertainties is discussed in
Section 4, with a focus on how the revised uncertainties resolve the tension
between early- and late-time estimates of $H_{\circ}$. Finally, the physical
origin of the newly calibrated systematic distance modulus uncertainties is
discussed in Section 5, along with recommendations for their mitigation.

Figure 1: Parameter ($m_{\mathrm{B}},c,x_{1},\mu-M$) differences between
supernova siblings. Differences for DES (open circles) and Pantheon+ (filled
circles) siblings are shown, ordered by increasing host galaxy redshift
(bottom to top); measurement errors of each sibling pair were added in
quadrature. Differences should be zero if sibling supernovae are of the same
subclass and originate from the same host galaxy.

##  2 Sibling Datasets

The primary challenge to compiling a statistically meaningful sample of
supernova siblings is a low event rate ($\ll$1 SNeIa galaxy-1 century-1;
Horiuchi & Beacom (2010)). Large cosmic volumes must be surveyed over extended
periods of time to compensate. This work focuses on previously identified
siblings within the Dark Energy Survey (DES) and Pantheon+ datasets (Scolnic
et al., 2020, 2022). The former carried out a deep, wide-area survey of
millions of galaxies over six years (2013-2019); the latter includes 18
different samples, each defined as a single survey of supernovae over a
discrete period of time. A total of thirty-one (31) sibling pairs were
identified in twenty (20) host galaxies: eight (8) within the DES sample
($0.228\leq z\leq 0.648$) and twenty-three (23) in Pantheon+ ($0.004\leq z\leq
0.025$). This accounting includes all pairwise combinations within a host;
multiple observations of the same supernova are considered unique events, but
sibling pairs consisting of the same object are excluded.

Additional sibling compilations exist; they are not included here for a
variety of reasons. Individual sibling pairs identified within other surveys
are insufficient to support a robust independent statistical analysis (Burns
et al., 2020; Biswas et al., 2021). Spectroscopic twins are excluded because
they do not meet the definition of siblings; their relationships are defined
solely on the basis of their spectral properties (Boone et al., 2021). Lastly,
siblings identified using combined historical datasets are excluded because
the individual measurements require recalibration to a homogenous standard
(e.g., Scolnic et al. (2015)).

All DES and Pantheon+ siblings were processed using the same analysis
pipelines as their parent datasets; each was also spectroscopically confirmed
as Type-Ia. The light curves of each supernova were fit using the SALT2
(Spectral Adaptive Light-curve Template) model (Guy et al., 2007) as part of
event processing; this well-established empirical approach uses the emergent
spectral energy distributions of SNeIa to standardize their apparent
magnitudes. The fits return three parameters: an overall amplitude ($x_{0}$),
which is then converted to a brightness
($m_{\mathrm{B}}=-2.5\log_{10}{x_{0}}$), a light-curve stretch ($x_{1}$), and
a light-curve color ($c$). A distance modulus is computed for each supernova
using the Tripp relation (Tripp, 1998),

$\mu=m_{\mathrm{B}}+\alpha x_{1}-\beta c-M.$ (1)

where $\alpha,\beta$ are correlation coefficients of luminosity with $x_{1}$
and $c$, respectively, and $M$ is the absolute magnitude of SNeIa; here,
$\alpha=0.14$ and $\beta=3.112$ (Brout et al., 2019). The Tripp estimate is
intended to be a standardized measure of supernova distance in units of an
apparent magnitude. Additional details associated with event selections, data
processing, as well as tabulated fit parameters, distance moduli, their
uncertainties, and host galaxy redshifts for all sibling supernovae can be
found elsewhere (Scolnic et al., 2020, 2022).

Sibling relationships are defined by the degree of similarity among their
observables ($m_{\mathrm{B}},c,x_{1},\mu-M$). Sibling-to-sibling differences
show a diversity that belies their fundamental relationship (Fig. 1). Intra-
sibling observables are (imperfectly) correlated; inter-sibling connections
are uncorrelated (Tab. 1). These links imply that siblings are
distinguishable, with pair-to-pair differences governed by stochastic
processes. In short, the supernova siblings are merely relatives, rather than
twins.

Table 1: Supernova Siblings Analysis Summary.

Observable, $x$ | $\chi^{2}_{\mathrm{D}}$ | $P(\chi^{2}\geq\chi^{2}_{\mathrm{D}})$ | $R_{x}$ | KS $p$-value | $\sigma_{x,m}$ | $S_{x}$ | $S_{x}/\sigma_{x,m}$ | $\sigma_{x,\mathrm{tot}}$
---|---|---|---|---|---|---|---|---
DES ($N_{\mathrm{p}}=8$) |  |
$\mu-M$ | 29.8 | $2.3\times 10^{-4}$ | 0.942 | 0.474 | 0.149 | $1.31\pm 0.34$ | 8.8 | 1.32
$m_{\mathrm{B}}$ | 430.3 | $<10^{-16}$ | 0.464 | 0.137 | 0.063 | $0.70\pm 0.18$ | 11.1 | 0.70
$c$ | 34.4 | $3.4\times 10^{-5}$ | 0.237 | 0.011 | 0.045 | $0.11\pm 0.03$ | 2.4 | 0.12
$x_{1}$ | 24.1 | $2.2\times 10^{-3}$ | 0.187 | 0.813 | 0.761 | $1.41\pm 0.47$ | 1.9 | 1.60
Pantheon+ ($N_{\mathrm{p}}=23$) |  |
$\mu-M$ | 114.2 | $4.5\times 10^{-14}$ | 0.989 | 0.277 | 0.086 | $1.89\pm 0.28$ | 21.9 | 1.89
$m_{\mathrm{B}}$ | 1104.2 | $<10^{-16}$ | 0.985 | 0.022 | 0.056 | $3.26\pm 0.48$ | 58.2 | 3.26
$c$ | 67.8 | $2.6\times 10^{-6}$ | 0.552 | 0.187 | 0.037 | $0.09\pm 0.01$ | 2.4 | 0.10
$x_{1}$ | 1220.0 | $<10^{-16}$ | 0.563 | 0.168 | 0.181 | $2.05\pm 0.30$ | 11.3 | 2.1

Note. — (1) Tripp/SALT2 parameter, $m_{\mathrm{B}}$ and $\mu-M$ in units of
magnitude; (2) from Eqn. 2; (3) probability of $\chi^{2}$ exceeding
$\chi^{2}_{\mathrm{D}}$ under baseline i.i.d. assumption; (4) intra-sibling
(sibling-to-sibling) correlation; (5) Kolmogorov-Smirnov $p$-value, inter-
sibling (pair-to-pair) comparison to uniform distribution; (6) measurement
error, pooled from measurement uncertainties reported by Scolnic et al. (2020,
2022); (7) sibling-derived systematic uncertainty; (8) systematic uncertainty-
to-measurement error ratio; (9) total uncertainty,
$(S^{2}_{x}+\sigma_{x,m}^{2})^{1/2}$.

##  3 Calibration of Distance Modulus Uncertainties

Sibling-based experimental controls guarantee that the distance moduli of
identical twins are equal, by definition. Any observed differences between
them should be consistent with measurement errors; otherwise, the siblings
must be considered members of an inhomogeneous sample. An ensemble measure of
sibling-to-sibling differences in $\mu-M$ is given by

$\chi^{2}_{\mathrm{D}}=\sum_{i}^{N_{\mathrm{p}}}(\mu_{1,i}-\mu_{2,i})^{2}/(\sigma_{\mu_{1},i}^{2}+\sigma_{\mu_{2},i}^{2}),$
(2)

where $\mu_{1,i}$ and $\mu_{2,i}$ are the distance moduli of the $i$-th
sibling pair, $\sigma_{\mu_{1},i}$ and $\sigma_{\mu_{2},i}$ are the
corresponding measurement uncertainties, and $N_{\mathrm{p}}$ is the number of
pairwise comparisons within the sibling ensemble under study. This test
statistic follows a $\chi^{2}$ distribution ($N_{\mathrm{p}}$ degrees of
freedom) if the observables are independent and identically distributed
(i.i.d.), and measurement errors adequately characterize their uncertainties.
These assumptions are inadequate, as evidence by the small probability of
obtaining a value of $\chi^{2}$ that exceeds $\chi^{2}_{\mathrm{D}}$
($P(\chi^{2}\geq\chi^{2}_{\mathrm{D}})$) for the sibling datasets considered
here (Tab. 1). Alternative descriptions are required.

Sibling distance moduli are known to be highly correlated (Fig. 2).
Correlations violate the premise of i.i.d.-based observables; they skew sample
statistics (e.g., $\chi^{2}$, root mean squared, standard error) to lower
values and, as a result, may mask the presence of an unidentified systematic
or lead to its underestimation. It is imperative to account for this effect in
order to unmask the presence of, and quantify, a systematic uncertainty in
$\mu-M$.

A diverse collection of artificial datasets was generated, each a set of
$N_{\mathrm{p}}$ random variate pairs with a specified systematic uncertainty
and degree of correlation imposed on them. Consistency with the data was
determined by comparing $\chi^{2}_{\mathrm{D}}$ to a bootstrapped $\chi^{2}$
distribution associated with each artificial dataset; this approach is model-
independent and agnostic to the existence and origin of any systematic.

Variates for each evaluation scenario were drawn from a zero-mean normal
distribution $N(0,\sigma_{\mu,\mathrm{tot}})$, where
$\sigma_{\mu,tot}\,(=(S_{\mu}^{2}+\sigma_{\mu,m}^{2})^{1/2})$, $S_{\mu}$, and
$\sigma_{\mu,m}$ are the total uncertainty, hypothesized systematic
uncertainty, and known measurement error for the observable $\mu-M$.
Measurement errors were computed by pooling the measurement uncertainties
reported for each observable (Tab. 1), i.e.,

$\sigma_{\mu,m}^{2}=\frac{1}{N_{\mathrm{p}}}\sum_{i}^{N_{\mathrm{p}}}(\sigma_{\mu_{1},i}^{2}+\sigma_{\mu_{2},i}^{2}).$
(3)

Correlation between each normally distributed random variate pair $(x,y)$ was
imposed using

$\begin{pmatrix}x^{\prime}\\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}1&0\\\
R_{\mu}&\sqrt{1-R_{\mu}^{2}}\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix},$
(4)

where $R_{\mu}$ is the hypothesized degree of correlation among sibling
distance moduli. A bootstrapped chi-squared distribution
($\chi^{2}_{\mathrm{B}}$) was constructed from $10,000$ trials of each
evaluation scenario, where

$\chi^{2}_{\mathrm{trial}}=\sum_{i}^{N_{\mathrm{p}}}(\mu_{1,i}-\mu_{2,i})^{2}/(2\sigma_{\mu,tot}^{2}),$
(5)

a modification of Equation 2, defined the test statistic for each trial. In
toto, five hundred thousand ($5\times 10^{5}$) different
($S_{\mu},R_{\mu}$)-evaluation scenarios were considered. Allowed scenarios
were those that satisfy $(1-\alpha)/2\leq
P(\chi^{2}_{\mathrm{B}}\geq\chi^{2}_{\mathrm{D}})\leq(1+\alpha)/2$, where
$\alpha$ is the desired confidence level (C.L.). The allowed
($S_{\mu},R_{\mu}$)-parameter space is shown in Figure 3; scenarios consisting
of measurement errors alone ($S_{\mu}=0$; Eqn. 2) are excluded at $>$$95\%$
C.L.

Figure 2: Comparison of supernova sibling distance moduli, $\mu-M$. The
distributions for the DES (top) and Pantheon+ (bottom) ensembles are shown.
The choice of supernova for the $x$-$y$ values is arbitrary and can be
exchanged. Equal-valued sibling moduli sit on the diagonal line. The
$\chi_{D}^{2}$ relative to that line is given along with the correlation
coefficient $R_{\mu}$ for sibling pairs. Contours show the 68% (yellow) and
95% C.L. (green) regions of linear fits to the data.  Figure 3: Allowed
regions of ($S_{\mu},R_{\mu}$)-parameter space associated with supernova
sibling distance moduli, $\mu-M$. The regions for the DES (top) and Pantheon+
(bottom) sibling ensembles are shown, along with the measured intra-sibling
correlations $R_{\mu}$ (dashed). Shading corresponds to 68% (dark) and 95%
C.L. (light).

The required non-zero systematic distance modulus uncertainties are
calibration measurements; they were extracted using measured intra-sibling
correlations as an additional parameter space constraint (Tab. 1):
$S_{\mu}=1.31\pm 0.34$ mag and $S_{\mu}=1.89\pm 0.28$ mag for the DES and
Pantheon+ sibling ensembles, respectively. A constant $S_{\mu}$ is excluded at
$\sim$$90\%$ C.L.; the calibration measurements decrease as parent galaxy
distance increases. Redshift dependence was characterized with a fit to the
local ($\bar{z}=0.011$) and deep ($\bar{z}=0.48$) calibrations; $z$-dependence
was assumed to mimic distance moduli, i.e., $S_{\mu}(z)\propto\log_{10}z$
(Fig. 4). Extrapolation to low-$z$ gives
$S_{\mu}^{\mathrm{max}}=S_{\mu}(0.001)=2.24$ mag, the intrinsic distance
modulus uncertainty in the absence of redshift effects.

Figure 4: Calibrated distance modulus uncertainties. (Top) Calibration
measurements of SNeIa distance modulus uncertainties derived from the DES and
Pantheon+ sibling ensembles. Redshift dependence was characterized with a fit
of the form $S_{\mu}(z)\propto\log_{10}z$ (blue); extrapolation to low-$z$
gives $S_{\mu}^{\mathrm{max}}=S_{\mu}(0.001)=2.24$ mag, the intrinsic distance
modulus uncertainty in the absence of redshift effects. (Bottom) Systematic
distance modulus uncertainties $S_{\mu}(z_{i})$ for each of the 1701 SNeIa of
the Pantheon+ dataset (Scolnic et al., 2022; Brout et al., 2022), where
$z_{i}$ is the tabulated redshift of the $i$-th supernova.

###  3.1 Phenomenological Origin of Non-Zero $S_{\mu}$

The origin of sibling-calibrated distance modulus uncertainties ($S_{\mu}$)
was isolated with supplemental analyses of Tripp parameters. Each parameter
($m_{\mathrm{B}}$, $c$, $x_{1}$) was analyzed independently, using procedures
analogous to those described in Section 3. Scenarios consisting of measurement
errors alone (i.e., $S_{m_{\mathrm{B}}}=S_{c}=S_{x_{1}}=0$) were excluded at
$>$$95\%$ C.L. Key findings also include (Tab. 1): a redshift dependent
$S_{m_{\mathrm{B}}}$ (local and deep values of $S_{m_{\mathrm{B}}}$ differ by
$5\sigma$), absence of a $z$-dependence for $S_{c}$ or $S_{x_{1}}$, and an
uncertainty-to-measurement error ratio
($S_{m_{\mathrm{B}}}/\sigma_{m_{\mathrm{B}},m}$) that is at least $5\times$
larger than the equivalent color or stretch ratios.

Evidence derived from the supplementary analyses supports the hypothesis that
$m_{\mathrm{B}}$ diversity is primarily responsible for sibling-to-sibling
differences in $\mu-M$ and, ultimately, non-zero $S_{\mu}$; the observed
redshift dependence is reasonably attributed to a degraded ability to
differentiate sibling $m_{\mathrm{B}}$ as a function of distance (likely due
to instrument sensitivity limitations). Isolating apparent magnitudes as the
principal driver of distance modulus uncertainties was uniquely enabled by
sibling-based experimental controls that remove known dependencies between
light curve parameters ($c$, $x_{1}$) and host galaxy properties (e.g. Hamuy
et al. (1995, 1996, 2000); Riess et al. (1999); Sullivan et al. (2003);
Johansson et al. (2013); Henne et al. (2017); Kim et al. (2019); Pruzhinskaya
et al. (2020)). The physical origin of $m_{\mathrm{B}}$ diversity is discussed
in Section 5.

###  3.2 Comparison with Previous Results

An earlier investigation of Pantheon+ siblings found no evidence that sibling
distance moduli were correlated (Scolnic et al., 2022). This is inconsistent
with the notion of siblings and in conflict with the findings presented here
(e.g., Fig. 2). That work compared the standard deviation of sibling
differences to random supernova pairs using a model-dependent bootstrap
approach. Also noted in that work was weak evidence for sibling $x_{1}$
correlations, a result consistent with an earlier DES-sibling study (Scolnic
et al., 2020); this is also inconsistent with the direct measures of
correlation presented here (Tab. 1). The origin of these inconsistencies is
unclear, but may stem from the use of inter-sibling statistics to make
inferences about intra-sibling relationships. Analyses that do not account for
intra-sibling correlations are susceptible to a numerical conspiracy: skewed
statistics that influence data interpretation and mask systematics (Sec. 3).

##  4 Impact

Sibling-derived $S_{\mu}$ are direct, model-independent calibrations of SNeIa
distance modulus uncertainties; they are different from, and exacerbate, the
systematic effects known to dominate supernova-based cosmology studies (e.g.,
Scolnic et al. (2019)). Their impact is profound: larger distance modulus
uncertainties dilute the relationship between (light curve-derived) distance
moduli and independent measures of host-galaxy redshift, weakening
cosmological model constraints derived from the SNeIa-defined Hubble diagram.

###  4.1 Hubble Tension

The so-called Hubble Tension refers to an $\sim$$5\sigma$ inconsistency
between early- and late-time determinations of $H_{\circ}$. Early-time ($z\geq
1100$) estimates are extracted from model-dependent analyses of the cosmic
microwave background (CMB). In contrast, local (late-time) estimates use a
model-independent ladder of cross-calibrated distance metrics; SNeIa are an
integral component of this approach.

The Hubble Tension is mitigated by the calibrated distance modulus
uncertainties (Sec. 3). To quantify impact, cosmological parameter constraints
were obtained by minimizing $\chi^{2}$ likelihoods, an approach commonly used
in supernova cosmology analyses. In general, likelihoods are constructed from
a vector of distance modulus residuals and a covariance matrix that includes
statistical and systematic uncertainties; the residuals are differences
between the distance modulus of each supernovae (Eqn. 1) and predicted
redshift-dependent distances (Conley et al., 2011).

Two likelihoods were used here: the publicly available likelihood for the full
Pantheon+ sample (1701 SNeIa, $0.001\leq z\leq 2.26$) and a modified version
of this likelihood that accommodates updated distance modulus uncertainties.
Following Brout et al. (2022), likelihoods were evaluated using the PolyChord
sampler (Handley et al., 2015) in the CosmoSIS package (Zuntz et al., 2015)
with 250 live points, 30 repeats, and an evidence tolerance of 0.1. The
$\chi^{2}$ minimization procedure was validated using the unmodified Pantheon+
likelihood; the resulting estimate of $H_{\circ}$ differs from the one
reported by Brout et al. (2022) by only $\sim 0.04\sigma$.

The modified likelihood was evaluated using the same procedure. Diagonal
elements of the original Pantheon+ (statistical) covariance matrix,
$C_{\mathrm{stat}}$, give the square of the reported full distance error for
each of the 1701 supernovae (Brout et al., 2022). These elements were modified
to account for the newly identified uncertainties, i.e.,
$C_{\mathrm{stat}}(i,i)+S_{\mu}(z_{i})^{2}$, where $z_{i}$ is the tabulated
redshift of the $i$-th supernova; the $S_{\mu}(z_{i})$ are shown in Figure 4.
No other changes were made to the likelihood. Modified full distance errors
are approximately an order of magnitude larger than previously reported; they
are also larger than the dispersions reported for SH0ES (Supernova and
$H_{\circ}$ for the Equation of State of dark energy) calibrator SNeIa and
Cepheids (0.130 mag) (Riess et al., 2022), the SH0ES Hubble flow sample (0.135
mag), and the RMS scatter of Hubble-diagram residuals for the full Pantheon+
dataset (0.17 mag) (Brout et al., 2022).

Minimization of the modified likelihood gives $H_{\circ}=72.26\pm 7.40$ (68%
C.L.). The central value of the Hubble-Lemaître parameter is essentially
unchanged relative to SH0ES, but its uncertainty is $\sim$$6.3\times$ larger.
The tension between early- and late-universe estimates of $H_{\circ}$ is then

$\frac{H_{\circ}({\mathrm{late}})-H_{\circ}({\mathrm{early}})}{\sqrt{\sigma^{2}_{H_{\circ}}({\mathrm{late}})+\sigma^{2}_{H_{\circ}}({\mathrm{early}})}}=\begin{cases}0.7\sigma,\,\mathrm{this\,work}\\\
4.9\sigma,\,\mathrm{SH0ES}\\\ 5.2\sigma,\,\mathrm{Pantheon+}\end{cases}$ (6)

where previously reported late-time estimates of the Hubble Tension are
provided for context (Fig. 5); the Planck constraint for a baseline
$\Lambda$CDM cosmology was used for the early-time estimate (Planck
Collaboration et al., 2020).

Figure 5: Cosmological parameter constraint comparisons for a $\Lambda$CDM
cosmology. (Top) Hubble-Lemaître parameter ($H_{\circ}$) constraints,
including those obtained by SH0ES (Supernova and $H_{\circ}$ for the Equation
of State of dark energy) (Riess et al., 2022), Pantheon+ (Brout et al., 2022),
and Planck (Planck Collaboration et al., 2020). (Bottom) Confidence contours
of $\Omega_{\mathrm{M}}$ and $\Omega_{\Lambda}$. The 68% and 95% confidence
levels obtained using the original (red) and modified (blue) Pantheon+
likelihoods are shown, along with confidence region for combined BAO
likelihoods (purple); the Planck constraint (black) for a baseline
$\Lambda$CDM cosmology is also shown (Planck Collaboration et al., 2020). The
modified likelihood accounts for newly calibrated systematic distance modulus
uncertainties. The Hubble-Lemaitre parameter was free to vary during the
evaluation all likelihoods. Two lines are shown for reference: a flat universe
where $\Omega_{\mathrm{M}}+\Omega_{\Lambda}=1$ and another that indicates an
accelerating universe.

###  4.2 Cosmological Model Constraints

Constraints on additional selected cosmological parameters were also obtained
as part of the likelihood minimization procedure. For a $\Lambda$CDM ($w=-1$)
cosmological model, the modified Pantheon+ likelihood gives
$\Omega_{M}=0.41\pm 0.18$, $\Omega_{\Lambda}=0.63\pm 0.34$; a flat universe is
within the 68% confidence region. Confidence regions are less restrictive than
those obtained using the unmodified Pantheon+ likelihood (Brout et al., 2022),
a direct reflection of the larger calibrated distance modulus uncertainties
identified here (Fig. 4). Parameter constraints do not change substantially if
$S_{\mu}^{\mathrm{max}}$ is reduced from its extrapolated value (2.24 mag;
Sec. 3) to the measured maximum (1.89 mag; Tab. 1). Finally, it is notable
that the three cosmological parameters considered here ($H_{\circ}$,
$\Omega_{M}$, $\Omega_{\Lambda}$) are in concordance with early-Universe
values after accounting for the newly calibrated systematic uncertainties.

##  5 Discussion

The phenomenological origin of the newly-calibrated systematic distance
modulus uncertainties is $m_{\mathrm{B}}$ diversity (Sec. 3.1). This inference
is not surprising; diversity is known to exist at the UV-optical-infrared
(UVOIR) wavelengths traditionally of interest to supernova cosmology
investigations (Benetti et al., 2005; Parrent et al., 2014; Kromer et al.,
2017). Event selections and empirical standardization procedures (like SALT2)
mitigate the impact of SNeIa diversity, but it persists for a fundamental
reason: UVOIR-based selectivity is not the same as selectivity based on a
specific thermonuclear scenario. As noted above, sample inhomogeneity is a
natural consequence of events selected from a diverse parent population.

Ideally, standard (or standardizable) candles are members of a homogeneous
population; supernovae from a single progenitor class with a common explosion
mechanism meet this criterion. Unfortunately, SNeIa selections based on
specific thermonuclear scenarios are not currently possible, in part because
of the physics responsible for UVOIR radiation and its light curves.
Thermonuclear supernovae are fundamentally nuclear objects, governed by
nuclear physics processes. During a supernova, a large fraction of the
progenitor’s mass is fused into a single radioactive isotope (56Ni) in just a
few seconds; the ensuing
${}^{56}\mathrm{Ni}\rightarrow$${}^{56}\mathrm{Co}\rightarrow$${}^{56}\mathrm{Fe}$
decay chain powers the UVOIR light curves. Emergent UVOIR radiation is
reprocessed nuclear radiation.

Information loss is a natural consequence of the nuclear radiation
reprocessing responsible for the UVOIR signatures (Colgate & McKee, 1969). A
diverse SNeIa population is expected to produce diverse nuclear light curves
(Fig. 6) and, by extension, diverse low-energy light curves (e.g., Summa et
al. (2013)). But the degree of information loss, the relation between peak
luminosity and $m_{\mathrm{B}}$ (“Arnett’s rule”), the effects of complex
interstellar and circumstellar environments on UVOIR emission, and the impact
of dependencies associated with distinct thermonuclear scenarios are currently
unknown (Arnett, 1979, 1982; Sukhbold, 2019; Khatami & Kasen, 2019). As a
result, the mapping between non-nuclear signatures and specific thermonuclear
scenarios (if any) is unknown; the low-energy light curves alone are limited
in their ability to differentiate thermonuclear scenarios.

Figure 6: Modeled nuclear light curves of selected SNeIa models (0.1-10 MeV).
The 15 models shown here include both single and double-degenerate
progenitors, deflagration and detonation scenarios, as well as hybrid models
(The & Burrows (2014), and references therein). Each light curve has been
normalized to the peak flux of the W7 supernova model (black).

Selectivity is an issue because SNeIa are believed to derive from multiple
evolutionary pathways. Candidate progenitors include single and double-
degenerate scenarios, corresponding to binaries with one or two white dwarf
components; these are typically considered to be carbon/oxygen white dwarfs
(e.g., Nomoto et al. (1984)), but hybrid systems have also been studied (e.g.,
Denissenkov et al. (2015); Augustine et al. (2019)). Nuclear burning ignition
mechanisms include compression-heating in the core of a white dwarf (initially
driving a deflagration), to detonations in a helium layer of an accreting
white dwarf, and detonations produced through white dwarf collisions (Maoz et
al., 2014). The shape and intensity of emergent nuclear light curves are
dictated by the explosion process, progenitor central densities, the
distribution of radioactive elements, internal structure and composition, as
well as the temporal evolution of ejecta opacities.

Although the demographics of SNeIa thermonuclear scenarios are currently
unknown, they are knowable. Emergent nuclear radiation is encoded with
critical diagnostic evidence of the underlying thermonuclear scenario(s) and
can travel great distances unscathed. Only measurements of this radioactivity
– the ashes of nuclear burning – will unmask the true identities of
thermonuclear supernovae to mitigate diversity-driven systematics. The
tremendous potential of supernova radioactivity measurements is currently
unrealized for the simple reason that instrument sensitivity has been
inadequate. To date, gamma rays from just one SNeIa have been observed
(Churazov et al., 2015; Diehl et al., 2015; Isern et al., 2016); this sample
of one is inadequate to quantify demographic trends. An all-sky observing
capability with sufficient sensitivity to characterize the temporal- and
spectral-evolution signatures of emergent nuclear radiation from a
representative sample of SNeIa is required. One such mission concept, the
Lunar Occultation eXplorer (LOX) (e.g., Miller et al. (2019); Miller &
Peplowski, in preparation) was recently proposed to fill this knowledge gap.

##  6 Conclusion

Newly calibrated distance modulus uncertainties are approximately an order of
magnitude larger than previously reported by other supernova cosmology
investigations. A reanalysis of the Pantheon+ SNeIa sample, updated with
calibrated uncertainties, weakens constraints on cosmological parameters
relative to the original Pantheon+ results; estimates of the Hubble-Lemaître
parameter, matter- and dark energy-densities are in agreement for a
$\Lambda$CDM cosmology. The Hubble Tension was relieved after rigorous
calibration of SNeIa distance modulus uncertainties. Calibration procedures
leveraged the power of supernova siblings (i.e., multiple supernovae from the
same host galaxy); their unique experimental controls mitigate the confounding
effects that have concealed diversity-driven systematics and over-constrained
cosmological parameters.

Refinements to the sibling-based calibrations are expected. Substantial
resources have been committed to measure the UVOIR emission from supernovae; a
key objective is to mitigate the systematics for cosmology and dark-energy
studies. Future surveys by the Vera C. Rubin Observatory (Ivezic et al.,
2009), the Nancy Grace Roman space telescope (Akeson et al., 2019), and other
assets (both space- and ground-based) will increase the sample size of
supernova and supernova siblings. The quality of sibling-based calibrations
will naturally improve because of better sample statistics; as a result, the
characterization of systematic distance modulus uncertainties and their
redshift dependence ($S_{\mu}(z)$) will be more precise.

Limitations will continue to persist due to a reliance on secondary, low-
energy emission. Identifying the physical origin of $m_{\mathrm{B}}$
diversity, the primary source of newly identified systematic uncertainties,
must be a high-priority objective. The next-level precision required to
address open questions in cosmology (e.g., cosmic expansion history, dark
energy) requires an understanding of the fundamental nature of SNeIa
progenitors and their engines, including the level of heterogeneity within
their population. Formal connections between thermonuclear scenarios
(identified using nuclear gamma-rays) and UVOIR-derived metrics (like the
Tripp parameters) are required to inform new SNeIa event selections, mitigate
the impact of intrinsic population diversity, reduce systematic uncertainties,
and ultimately tighten cosmological model constraints. Intrinsic population
diversity will continue to be a limiting factor without a detailed census of
the thermonuclear scenarios that govern these cosmic beacons.

We thank Drs. Patrick Peplowski, Jack Wilson, Robert Schaefer, John Beacom,
and Chris Fryer for useful feedback and conversations. The author also
acknowledges the support of Dr. Scott Murchie.

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