# Reply to “Masanes-Galley-Müller and the State-Update Postulate”

Thomas D. Galley<EMAIL_ADDRESS>Institute for Quantum Optics and
Quantum Information, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria Lluís Masanes  Department of Computer
Science, University College London, United Kingdom  London Centre for
Nanotechnology, University College London, United Kingdom  Markus P. Müller
Institute for Quantum Optics and Quantum Information, Austrian Academy of
Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria Perimeter Institute for Theoretical
Physics, 31 Caroline Street North, Waterloo ON N2L 2Y5, Canada  Vienna Center
for Quantum Science and Technology (VCQ), Faculty of Physics, University of
Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

(December 7, 2022)

###### Abstract

In a recent comment on the arXiv, Blake C. Stacey criticizes our derivation of
the quantum state update rule in Nat. Commun. 10, 1361 (2019). Here we argue
that the criticism is unfounded. In particular, and in contrast to Stacey’s
claims, our proof does not assume linearity.

##  1 Introduction

In a recent comment on the arXiv [1], Blake Stacey offers a critique of our
work [2]. The results of [2] demonstrate that the measurement postulates of
quantum theory are the only possible ones consistent with the structure of
pure states, unitary dynamics and the composition rule of quantum theory.
Stacey’s critique is not levelled at the main theorem (the Born rule), but
rather at its corollary (the post-measurement state-update rule.) Stacey
claims that our proof of the state-update rule is circular, allegedly assuming
the linearity that it intends to show.

In the following, we provide an overview of our paper [2] and outline Stacey’s
argument from [1] before offering our response.

##  2 Overview of [2]

The quantum measurement postulates state that each measurement is
mathematically characterised by a POVM, and that the probability of
measurement outcomes is given by the Born rule. The main theorem of [2] shows
that the quantum measurement postulates are the only operationally consistent
postulates given that one assumes the ‘rest of’ quantum theory; namely that
pure states are represented by rays in some complex projective space,
reversible dynamics by unitary transformations, and pure states compose using
the tensor product.

We stress that for single systems (i.e. without any constraints on the
composition of systems) this is not the case: there is an infinite family of
conceivable systems with the same pure states and reversible dynamics as
quantum theory but different measurement rules [3]. A key point to emphasise
is that these alternative systems have sets of mixed states which do not
correspond to the set of density operators, and sets of outcome probability
functions which do not correspond to the set of POVM elements.

The density operator already contains information about the measurements of
quantum theory since density operators tell us which ensembles of states are
indistinguishable. Here indistinguishability is relative to the possible
measurements. Mathematically, this is evident by the fact that if one assumes
that the states of a system are given by density operators (with statistical
mixing represented by convex combinations), it is immediate that the
measurements must be represented by POVMs and probabilities by the Born rule.
As Stacey points out, this type of argument goes back at least to Holevo.

Thus, in order to be non-trivial, any proof of the Born rule must not assume
that probabilities are linear in density operators (or in POVM elements for
similar reasons). The starting point of the theorem of [2] is that a
measurement outcome should be some function of the pure states, namely of the
form $\mathbf{f}(\psi)$. This function need not be linear in
$|\psi\rangle\\!\langle\psi|$ (in fact, it was proven in [3] that $F$ has to
be linear in $|\psi\rangle\\!\langle\psi|^{\otimes n}$, for some fixed integer
$n\geq 1$.) Thus the main theorem of [2] neither assumes that the outcome
probability functions are linear in $|\psi\rangle\\!\langle\psi|$ nor that
outcomes correspond to POVM elements but rather derives these facts.

In [2] it is also shown that, once the measurement postulates are known to be
the quantum ones, the only consistent state update rule is the quantum one.
The argument for this follows from standard arguments and can be found in the
proof of Lemma 22 in the supplementary materials of [2].

##  3 Stacey’s argument

The proof that the measurement update rule is given by the standard form
requires introducing a POVM $\\{H_{j,i}\\}_{j,i}$ which corresponds to the
sequence of two POVMs $\\{F_{i}\\}_{i}$ and $\\{G_{j}\\}_{j}$. Stacey argues
that, although the main theorem of [2] applies to single-step measurements, it
does not apply to two step transformations. He states:

> In other words, the linearity of the map $\Lambda_{F_{i}}(\rho)$ follows
> from the assumption that $H_{i,j}$ is a POVM element that can be used as any
> other. But $\\{H_{j,i}\\}$ is not a measurement that is performed at a
> single time. It is, by necessity, a sequence of two experimental
> interventions, with a state-change in between.

His main point of contention is that, while it may be justified in assigning
POVM elements to the individual measurements outcomes of $\\{F_{i}\\}_{i}$ and
$\\{G_{j}\\}_{j}$, it is not automatically justified to assert that the
outcomes $\\{(F_{i},G_{j})\\}_{j,i}$ corresponding to the sequential
measurement can be associated to a single POVM $\\{H_{j,i}\\}_{j,i}$. That is,
in order to associate a POVM elements to the outcomes of a sequence of
measurements we must have made some assumptions about sequential measurements:

> the assumption that two-stage experiments have their mathematical
> representations in exactly the same set as one-stage experiments is an
> assumption that state-change is undramatic. (We have already accustomed
> ourselves to the fact that the AND operation is not always meaningful for
> quantum experiments; on what deep basis do we say it should apply trivially
> here?)

To back up his argument he provides the following example where a sequence of
measurements $\\{F_{i}\\}$ and $\\{G_{j}\\}$ would not lead to a POVM elements
$\\{H_{i,j}\\}$ being associated to the sequence:

> We could have a first measurement $\\{F_{i}\\}$ and a second measurement
> $\\{G_{j}\\}$, but the transformation taking place in between would be
> nonlinear, and so the conclusion about the map $\Lambda_{F_{i}}$ would
> simply not follow. For instance, suppose the initial state $\rho$ is a
> mixture of the Pauli eigenstate $|+z\rangle\\!\langle+z|$ and the garbage
> state $\frac{{\mathbb{I}}}{2}$, and let the state-update map implement a
> logistic transformation of the Bloch-sphere radial coordinate: $r\to
> r^{\prime}=\lambda r(1-r)$ with $\lambda\in[0,4]$. If the first and second
> POVMs are both the computational basis measurement
> $\\{|+z\rangle\\!\langle+z|,|-z\rangle\\!\langle-z\\}|$, then the
> probability of either outcome in the second measurement is trivially a
> nonlinear function of the initial state $\rho$.

The measurement with outcomes $\\{(F_{i},G_{j})\\}_{i,j}$ would in this case
not be a linear function of $\rho$ and would therefore not correspond to a
POVM $\\{H_{j,i}\\}_{j,i}$ with probabilities given by the Born rule.
Therefore, Stacey concludes that we have therefore implicitly assumed
linearity of the state update rule in the proof of the linearity of the state
update rule.

##  4 Response

The key point of our response is that any experiment that takes a quantum
system, described by a Hilbert space $\mathbb{C}^{d}$, as its input and
generates one of several possible outcomes is a measurement. Our derivation of
the Born rule is completely agnostic as to how the outcome comes about or how
many steps it takes.

More specifically, our result is a mathematical theorem about the form of what
we call “outcome probability functions” (OPFs)
$\mathbf{f}:\mathrm{P}\mathbb{C}^{d}\to[0,1]$, $\psi\mapsto\mathbf{f}(\psi)$.
Every $\mathbf{f}$ that satisfies a list of assumed mathematical properties
(given in our paper) is then proven to be of the form
$\psi\mapsto\langle\psi|Q|\psi\rangle$, where $Q$ is some POVM element.
Physically, our result will thus apply to all experiments in which the
mathematical properties are enforced by the scenario at hand. Such experiments
can be thought of as consisting of two stages:

  * •

The preparation P of a quantum state $\psi$. Note that our result is agnostic
as to what the state means — it could be understood as the real state of
affairs of the physical system that exits a well-characterized preparation
device; or an agent’s formulation of their beliefs about future experiences
involving the quantum system (as QBists might prefer), or something completely
different.

  * •

The rest R of the experiment, generating an outcome to which we want to assign
probabilities.

Our assumed mathematical properties correspond to physical assumptions on how
our experiment relates to other experiments that we could possibly perform
instead (or in addition), given the experiment that we have. For example,
Property 2 of [2] assumes that if $\mathbf{f}$ is an OPF, and if $U$ is a
unitary, then $\mathbf{f}\circ U$ is also a valid OPF. Operationally, this
means the following: after performing preparation P, we could also (if we
wanted) implement transformation $U$ on the system, before we complete the
experiment by performing R. If we know how to assign probabilities to the
possible outcomes of our original experiment (via $\mathbf{f}$), then we also
know how to assign probabilities to the possible outcomes of this new
experiment (namely via $\mathbf{f}\circ U$). Hence $\mathbf{f}\circ U$ must be
a valid OPF (note that this argumentation does not involve any assumptions
about the mathematical structure of $\mathbf{f}$, like being linear on
$|\psi\rangle\\!\langle\psi|$, etc.).

Similarly, our Property 1 (closedness under statistical mixing) formalizes the
physical possibility that if we have a finite set of experiments (labelled by
$x$) with identical preparation procedures, but different outcome-generating
stages $R_{x}$, then we obtain a new valid experiment by performing experiment
$x$ with some probability $p_{x}$.

Note that a very large class of experiments satisfies the above desiderata,
i.e. is in the realm of applicability of our derivation of the Born rule.

######  Example 1.

Consider an experiment where the preparation $P$ amounts to generating an
entangled state $\psi$ of the spin degrees of freedom of two electrons. The
outcome-generating stage $R$ of the experiment works as follows. We measure
whether electron $1$ has spin-up in our favorite $z$-direction. If so, then
the outcome of our experiment is by definition the string of five zeroes
$00000$. Otherwise, we repeatedly measure electron $2$ in the
x-,z-,x-,z-,x-basis, denoting ‘0’ for ‘up’ and ‘1’ for ‘down’. If the
resulting string $s$ contains less than four zeroes, then this is the outcome
string. Otherwise, we wait until the next day and repeat the procedure. And so
on, ad infinitum, until we terminate.

This experiment falls into the regime of applicability of our derivation of
the Born rule. The probabilities of the outcomes must therefore be given by
some POVM.

However, there are certainly situations to which our mathematical theorem does
not apply:

######  Example 2.

Imagine an experiment on a quantum system $S$ which is constrained by a
fundamental superselection rule, forbidding superpositions between some charge
sectors. In this case, for some unitaries $U$, it will be fundamentally
impossible to implement them on $S$. This eradicates the motivation for
Property 2 as explained above: the physical scenario does not motivate the
mathematical assumption that $\mathbf{f}\circ U$ is a valid OPF for every
unitary $U$ and every OPF $\mathbf{f}$. Alternatively, we might say that the
description of $S$ as a Hilbert space $\mathbb{C}^{d}$ has been incomplete in
the first place. Thus, our theorem is inapplicable and cannot be used to
derive the Born rule for $S$.

Similarly, we may think of Wigner’s-friend-type situations with a sequence of
outcomes that, intuitively, do not necessarily “co-exist” [4] so that it is
unclear whether, and how, it makes sense to assign (joint) probabilities to
these sequences of outcomes. The question of applicability of our theorem may
then depend on details of the metaphysical interpretation of such scenarios.

The main theorem of [2] shows that outcomes of measurements that satisfy the
above desiderata must correspond to POVM elements with probabilities given by
the Born rule (which is linear in $|\psi\rangle\\!\langle\psi|$). The theorem
does not assume linearity at the level of density operators or POVM elements,
but instead, it derives it.

In particular, a sequence of measurements $\\{F_{i}\\}_{i}$ and
$\\{G_{j}\\}_{j}$ can be interpreted as an outcome-generating procedure which
inputs a state $\rho$ and outputs an outcome pair $(i,j)$. The fact that this
outcome set has a direct product structure or that the parts of the outcome
pair are obtained sequentially (and potentially in some exotic manner) has no
bearing on the fact that an outcome $(i,j)$ is still the outcome of a
measurement that falls into the regime of physical applicability of our
mathematical theorem, in every case in which it makes sense to talk about the
probability $p(i,j)$ of the pair of outcomes $(i,j)$. Similarly two
measurements $\\{F_{i}^{\sf A}\\}_{i}$ and $\\{G_{j}^{\sf B}\\}_{j}$ performed
on subsystems ${\sf A}$ and ${\sf B}$ respectively, also form a single
measurement of the composite system ${\sf A}{\sf B}$.

Let us return to Stacey’s example and show how it can be ruled out by
appealing to our operational characterisation of measurements as any procedure
which inputs a system and outputs some probability distribution over classical
variables.

It is clear from this that if a non-linear transformation $\mathcal{T}$ were
to occur in between step 1 and step 2 of a sequential measurement as above,
the operation $\\{G_{j}\circ\mathcal{T}\\}_{j}$ would correspond to a
measurement operationally. Indeed it is a single step measurement (in the
language of Stacey), and since the outcome probabilities are non-linear in
$|\psi\rangle\\!\langle\psi|$, it is inconsistent with the other postulates of
quantum theory by the main theorem.

> The existence of the $H_{i,j}$ on which MGM’s lemma depends follows from the
> linearity they wish to prove. This linearity can be derived from other
> premises, such as assumptions about context-independence [4] as in the
> proofs of the POVM-Gleason theorem [5, 6]. However, these assumptions do not
> appear to be entailed in the postulates that MGM put forth. Indeed, MGM
> write, “We stress that our results, unlike previous contributions [ . . . ],
> do not assume” such premises.

We stress that the linearity of $H_{i,j}$ follows from the main theorem of [2]
which does not make any assumption of linearity.

We also want to mention that, independently of its tension with the main
theorem of [2], the consistency of Stacey’s example is not clear to us. The
example is formulated in terms of density matrices, as if these fully
characterise a state, or the outcome probabilities when a state is measured.
However, the non-linear dynamics of his example-theory would allow for
discriminating two ensembles of the same mixed state. This implies that the
density matrix does not provide a full characterisation the outcome
probabilities when a state is measured.

## References

  * [1] B. C. Stacey, Masanes-Galley-Müller and the State-Update Postulate, https://arxiv.org/abs/2211.03299v1.
  * [2] Ll. Masanes, T. D. Galley, and M. P. Müller, The measurement postulates of quantum mechanics are operationally redundant, Nat. Commun. 10, 1361 (2019).
  * [3] T. D. Galley and Ll. Masanes, Classification of all alternatives to the Born rule in terms of informational properties, Quantum 1, 15 (2017).
  * [4] K.-W. Bong, A. Utreras-Alcarcón, F. Ghafari, Y.-C. Liang, N. Tischler, E. G. Cavalcanti, G. J. Pryde, and H. M. Wiseman, A strong no-go theorem on the Wigner’s friend paradox, Nat. Phys. 16, 1199 (2020).


# The Low Energy Module (LEM): development of a CubeSat spectrometer for sub-
MeV particles and Gamma Ray Burst detection

Riccardo Nicolaidis<EMAIL_ADDRESS>Francesco Nozzoli  Roberto
Iuppa  Francesco Maria Follega  Veronica Vilona  Physics Department,
University of Trento, 38123 Trento, Italy.
INFN-Trento Institute of Fundamental Physics and Application, 38123 Trento,
Italy.  Giancarlo Pepponi  Pierluigi Bellutti  Evgeny Demenev  Fondazione
Bruno Kessler, 38123 Trento, Italy

###### Abstract

Accurate flux measurement of low energy charged particles, trapped in the
magnetosphere, is necessary for Space Weather characterization and to study
the coupling between the lithosphere and magnetosphere, allowing the
investigation of the correlations between seismic events and particle
precipitation from Van Allen Belts. In this work, the project of a CubeSat
space spectrometer, the Low Energy Module (LEM), is shown. The detector will
be able to perform an event-based measurement of energy, arrival direction,
and composition of low-energy charged particles down to 0.1 MeV. Moreover,
thanks to a CdZnTe mini-calorimeter, the LEM spectrometer also allows photon
detection in the sub-MeV range, joining the quest for the investigation of the
nature of Gamma Ray Bursts. The particle identification of the LEM relies on
the $\Delta E-E$ technique performed by thin silicon detectors. This
multipurpose spectrometer will fit within a 10x10x10 $\text{cm}^{3}$ CubeSat
frame and it will be constructed as a joining project between the University
of Trento, FBK, and INFN-TIFPA. To fulfil the size and mass requirements an
innovative approach, based on active particle collimation, was designed for
the LEM, this avoids heavy/bulky passive collimators of previous space
detectors. In this paper, we will present the LEM geometry, its detection
concept, and the results from the developed GEANT4 simulation.

Low Energy Module; LEM; Low-Energy Particles; Gamma Ray Bursts; Space Weather;
CubeSat; $\Delta E-E$ technique;

##  I Introduction

The Low Energy Module (LEM) will be a compact spectrometer able to perform an
event-based measurement of energy, direction and composition of low energy
charged particles, in particular down to 0.1 MeV for electrons. The physics
goal of this detector is the monitoring of the magnetosphere and ionosphere
environment. It is known that the measurements of the fluxes of low energetic
particles could allow the characterisation of the coupling between the
lithosphere, atmosphere, ionosphere, and the magnetosphere. In particular,
earthquakes are dynamical processes caused by a continuous and slow strain
accumulation. From studies on fault rupture mechanics, seismic wave
propagation, and geophysical parameters measured in the ionosphere and the low
magnetosphere, some anomalies correlated with catastrophic events were
discovered. Moreover, also a statistical evidence of temporal correlation
between particle precipitations from Van Allen Belts and strong seismic events
has been pointed out [1]. These observations, motivate the interest in
further, detailed, measurements of electron fluxes in the energy window
$0.1-7\ \text{MeV}$ that could be a promising channel to identify possible
seismic precursors.

Another interesting case study for the LEM instrument is the application to
space weather. Severe space weather storms could cause power outages and
telecommunication alterations. For this reason, the construction of new
instruments to monitor and (possibly) to predict the effects of solar activity
on Earth is crucial.

The LEM will be a particle telescope performing an event-based measurement of
energy, direction and composition of low energy charged particles, in
particular electrons down to 0.1 MeV. This capability is not possible with the
existing detectors that fails the possibility of an event-based PID or fails
the possibility of monitoring the particle flux from different directions at
the same time, or are not able to measure directions of low energy particles
because of the multiple scattering occurred in the first layer of a particle-
tracking configuration.

##  II The current landscape of space-based particle detectors

There is an extensive literature about particle detectors in space using
silicon technologies. The instruments under examination, are the following:
The Instrument for the Detection of Particles (IDP) on the DEMETER
microsatellite [2, 3, 4], The High Energy Particle Package (HEPP-H and HEPP-L)
on CSES [5, 6, 7, 8], the Mars Energetic Particle Analyser (Mars-EPA) on the
Tianwen-1 mission [9, 10, 11], and the Radiation Assessment Detector on the
Curiosity rover [12, 13, 14, 15, 16].

Table 1: Summary of some features of the detectors studied in this section. The references from which I extracted the information are quoted within the text. Instrument | Size | Directions | Angular | Energy | PID | Detector
---|---|---|---|---|---|---
| Weight |  | Resolution | range |  | Elements
IDP DEMETER | 525 g | 1 | FOV 32 deg. | e: [0.07, 0.8] MeV | No | Silicon Diode
RAD Curiosity |  $\sim$10x10x10 cm3 | Complex | FOV 36.7 deg. | e: [0.1, 20] MeV | Yes | PIPS (3 segments)
|  | segmentation |  | p: [5, 200] MeV |  | CsI(Tl)
|  |  |  |  $\alpha$: [5, 200] MeV |  | Plast. Scint.
|  |  |  | l.Z: [10, 300] MeV |  |
HEPP-L | Large | \- 5 Narrow | \- FOV 6.5 deg. | e: [0.1, 3] MeV | Yes | Si det. (2 layers)
| Collimators | \- 4 Wide | \- FOV 15 deg. | p: [2, 20] MeV |  | Plast. Scint.
Mars-EPA | 270x180x148 cm3 | 1 | FOV 60 deg. | e: [0.1, 12] MeV | Yes | PIPS (2 layers)
|  |  |  | p: [2, 100] MeV |  | CsI(Tl)
|  |  |  |  $\alpha$: [25, 400] MeV |  |
|  |  |  | l.Z: [25, 400] MeV |  |

Some of the most important features of these instruments are listed in table
1. Even though all of these experiments have different scientific purposes and
goals, their detection concept and scheme are very similar, allowing a
comparison between their structure, size, components, and performances. In
table 1, the reader can see a summary of the features characterising the
previously mentioned experiments. By comparing the six detectors studied, we
can conclude that the larger is the number of layers inserted into the design,
the better are the performances in detecting energetic particles. Furthermore,
by adding additional layers at the bottom of the instrument, as we can see in
the RAD, on the Curiosity mission, or in the Mars-EPA, the maximum energy that
can be detected with particle identification is increased. On the other hand,
to minimise the low energy threshold, one has to minimise the thickness of the
$\Delta E$ layer. As an example, the Mars-EPA, can detect electrons in the
energy range of $0.1-2\ \text{MeV}$ by adopting a $\Delta E$ layer made of
Passivated Implanted Planar Silicon (PIPS) detectors with a thickness of $15\
\mu\text{m}$. Finally, the use of an inorganic scintillator as a calorimeter
could be problematic. In particular, many scintillator crystals like Sodium-
Iodide or Caesium-Iodide, are very fragile and hygroscopic. These aspects will
unavoidably result in the introduction of mechanical supports or metallic
wrapping providing additional dead layers in which particles could deposit
part of their energy.

Furthermore , it is required that the LEM is compact (within 10x10x10cm3) and
that it can monitor the particle flux in a large field of view from different
directions at the same time. These capabilities are not simultaneously
fulfilled by the past detectors. Therefore, a different and innovative design
is required for the LEM.

##  III The LEM concept: the active collimation technique

The idea that allows the reduction of weight and size of the LEM detector
relies upon the active collimation technique. More precisely, a drilled
plastic scintillator is acting as a veto. Only particles with directions
aligned with one of the 16 channels are detected by one of the 16 silicon
sensor pairs. Thus, direction information is obtained. Particles with wrong
direction are stopped in the Aluminium shield or will release a signal in the
drilled plastic scintillator veto. This technique is an alternative to the
tracking one affected by the multiple scattering problem. On the other hand,
the low density of the plastic scintillator veto avoids the significant weight
required by a totally passive metallic collimator. However, the price to pay
is a relatively high veto rate. This high veto rate will unavoidably result in
a enhancement of the dead-time of the detector. For this reason, a small
drilled aluminium shield is still necessary to suppress very low-energy
particles.

In figure 1, the detection concept and a schematised cross section of the
instrument are shown. From the top, we can see the drilled aluminium mask
suppressing the flux of very low energetic particles. Below the aluminium
shield, the active anti-coincidence is obtained by using a drilled plastic
scintillator (polyvinyl toluene). The aluminium drilled mask and the drilled
Anti-Coincidence Detector (ACD) define the so-called active collimator.

Figure 1: Detection concept embedded within the detector’s geometry. The
legend allow to distinguish a partial energy deposition from a total kinetic
energy deposition.

For a LEM operating in Low Earth Orbit (LEO), an Aluminium thickness larger
than $0.5\ \text{cm}$ is necessary to reduce the veto rate from several MHz to
the affordable rate of $\sim\text{kHz}$. Below the active collimation system,
we place the 16 independent $\Delta E-E$ modules. These $\Delta E-E$ modules
will measure the angular flux of particles crossing the veto channels
($\oslash 1$cm $\times$ 1.3 cm), encoding one specific solid angle in the sky
with a resolution of $6\degree-7\degree$. The size of commercially available
PIPS detectors manufactured by Ametek (50mm2 \- $\oslash$ 8mm each) have been
considered to define a realistic geometry in the detector simulation. The
$\Delta E$ detector consists of a $100\ \mu\text{m}$ thick PIPS detector
while, the $E$ detector is a CdZnTe detector with $1\ \text{mm}$ thickness.
These two $\Delta E$ \- $E$ layers, allow a good particle identification in
the energy ranges: about $0.1-10\ \text{MeV}$ for electrons, $3-30\
\text{MeV}$ for protons, and $10-100\ \text{MeV}$ for alpha particles. A
bottom plastic scintillator (ACD) is added at the very end of the LEM to
ensure the energy release is confined within the above layers. In particular,
Particle Identification (PID) is not possible both for energetic particles
crossing the ACD as well as for slow particles stopped in the front PIPS.
Events with undefined direction are rejected thanks to a signal released in
the active veto/collimator. Finally, events which are fully contained within
the LEM, are selected. In this very last case, the direction is well defined
and it is also possible to perform an accurate PID. Thanks to the high density
and high averaged atomic number of CZT [17, 18], the LEM can identify low
energy $\gamma$-rays converting in the CZT mini-calorimeter and using all the
surrounding low-Z sensors as anticoincidence. The ability of observing
energetic photons will allow to use this compact particle spectrometer also as
a Gamma-Ray Burst (GRB) monitor [19].

##  IV Performance characterisation with GEANT 4 simulation

The detection concept adopted in the LEM is a consolidated technique
denominated $\Delta E-E$ [20, 21, 22]. Basically, a $\Delta E-E$ particle
spectrometer is composed of a thin detection layer and a thicker one behind.
When a particle impinges on the spectrometer, if the kinetic energy is enough,
the particle could cross the first layer, releasing a part of its kinetic
energy $\Delta E$. Then, the residual kinetic energy $E$ could be entirely
deposited within a second, thicker, layer. This experimental layout allows
particle identification by measuring the energy deposited in the thinner
layer, the $\Delta E$ energy, and the energy deposited in the thick sensor,
the $E$ energy. If a sub-MIP particle passes through a thin detector layer,
the energy deposited $\Delta E$ will be velocity dependent:

$\Delta E\approx\frac{Z^{2}}{\beta^{2}}$ (1)

where $Z$ is the projectile’s charge and $\beta$ is its velocity in natural
units. On the other hand, also the residual kinetic energy, $E$, of a particle
stopping in a subsequent thick detector is velocity dependent:

$E=mc^{2}(\gamma-1)\approx\frac{1}{2}m(\beta c)^{2}$ (2)

where the non-relativistic approximation holds for sub-MIP particles.
Therefore, in a $\Delta E-E$ spectrometer, a useful PID classifier could be
defined in the following way:

$\begin{split}\text{PID}_{\text{classifier}}&=\log_{10}\left[\frac{\Delta
E}{1\ \text{MeV}}\frac{E}{1\ \text{MeV}}\right]\approx\\\
&\approx\text{constant}+\log_{10}Z^{2}\left(\frac{mc^{2}}{1\
\text{MeV}}\right)\\\ \end{split}$ (3)

Figure 2: PID classifier vs kinetic energy. Left: $100\ \mu\text{m}-500\
\mu\text{m}$ PIPS detectors. Right: $100\ \mu\text{m}$ PIPS and $1\ \text{mm}$
CZT detectors. The three different clusters in each plot represent (from the
top to the bottom) : alpha particles, protons, electrons. Figure 3:
Improvement of detection yield for $\gamma$-rays thanks to CZT sensors. The
legend refers to the material and the thickness considered for the $E$
detector.

This $\text{PID}_{\text{classifier}}$ allows to remove the main dependence on
the particle’s velocity for sub-MIP particles, therefore it is mainly
dependent only from the particle’s nature. For the LEM detector, this
approximation is very good for protons and other nuclei but for electrons in
the LEM kinetic energy range the sub-MIP approximation fails and PID
classifier for electrons will grow roughly like $\log_{10}\frac{E}{1\
\text{MeV}}$. However a good identification of electrons from protons based on
this classifier is still achieved thanks to the fact that proton mass is 2000
times larger than electron mass. In figure 2, the results from a GEANT4 [23]
simulation are shown. In particular, the PID vs Energy identification
capability for the case of a mini-calorimeter made of $500\ \mu\text{m}$
Silicon (left plot) is compared with the case of a mini-calorimeter composed
by 1 mm thick CZT.

Finally, to quantify the advantage of the use of CZT sensors in the LEM as a
monitor for GRBs, a comparison of the relative photon detection efficiency for
the two mini-calorimeter configurations is shown in Figure 3.

##  V Conclusion and Outlooks

In this work we described the Low Energy Module (LEM): a compact particle
spectrometer, suitable for a CubeSat , for measurement of the differential
flux of low-energy particles in the lower magnetosphere. Here, it is worth
summarising the structure of the LEM. To avoid a bulky and heavy detector, we
design an active collimator based on a thin Aluminium shield followed by an
anti-coincidence detector. The drilled aluminium shield protects the drilled
ACD, made of a plastic scintillator, from the large flux of very low energetic
electrons in LEO. The holes in the aluminium and in the ACD are used to select
a known direction of the particles with an angular resolution of
$6\degree-7\degree$. The LEM Field of View is $60\degree\times 60\degree$
monitoring 16 directions in the sky at the same time. The particle
identification relies in a series of 16 $\Delta E-E$ modules based on PIPS and
CdZnTe detectors placed below each collimator channel. An additional layer of
plastic scintillator at the bottom is added as a veto to identify non-
contained, particles. Test on sensor prototypes are ongoing at INFN-TIFPA
laboratory.

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††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks<EMAIL_ADDRESS>
# Cosmology-independent Photon Mass Limits from Localized Fast Radio Bursts by
using Artificial Neural Networks

Jing-Yu Ran  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, China  School of Astronomy and Space Sciences, University of Science
and Technology of China, Hefei 230026, China  Bao Wang  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, China  School of
Astronomy and Space Sciences, University of Science and Technology of China,
Hefei 230026, China  Jun-Jie Wei  Purple Mountain Observatory, Chinese Academy
of Sciences, Nanjing 210023, China  School of Astronomy and Space Sciences,
University of Science and Technology of China, Hefei 230026, China

###### Abstract

A hypothetical photon mass, $m_{\gamma}$, can produce a frequency-dependent
vacuum dispersion of light, which leads to an additional time delay between
photons with different frequencies when they propagate through a fixed
distance. The dispersion measure–redshift measurements of fast radio bursts
(FRBs) have been widely used to constrain the rest mass of the photon.
However, all current studies analyzed the effect of the frequency-dependent
dispersion for massive photons in the standard $\Lambda$CDM cosmological
context. In order to alleviate the circularity problem induced by the
presumption of a specific cosmological model based on the fundamental
postulate of the masslessness of photons, here we employ a new model-
independent smoothing technique, Artificial Neural Network (ANN), to
reconstruct the Hubble parameter $H(z)$ function from 34 cosmic-chronometer
measurements. By combining observations of 32 well-localized FRBs and the
$H(z)$ function reconstructed by ANN, we obtain an upper limit of
$m_{\gamma}\leq 3.5\times 10^{-51}\;\rm{kg}$, or equivalently $m_{\gamma}\leq
2.0\times 10^{-15}\;\rm{eV/c^{2}}$ ($m_{\gamma}\leq 6.5\times
10^{-51}\;\rm{kg}$, or equivalently $m_{\gamma}\leq 3.6\times
10^{-15}\;\rm{eV/c^{2}}$) at the $1\sigma$ ($2\sigma$) confidence level. This
is the first cosmology-independent photon mass limit derived from
extragalactic sources.

###### pacs:

14.70.Bh, 41.20.Jb, 52.25.Os, 95.85.Bh

##  I Introduction

As a fundamental postulate of Maxwell’s electromagnetism and Einstein’s
special relativity, the principle of the constancy of the speed of light
implies the masslessness of photons, as elucidated by the particle-wave
duality. This principle is also embraced within the framework of general
relativity (GR). However, even a very minute photon mass, if existent, would
necessitate new physical theories, such as the renowned de Broglie-Proca
theory (De Broglie, 1922; Proca, 1936), the model of massive photons as an
explanation for dark energy (Kouwn  _et al._ , 2016), and other new ideas in
the standard-model extension with massive photons (Spallicci  _et al._ ,
2021). Consequently, achieving precise constraints on the rest mass of photon,
denoted as $m_{\gamma}$, remains an imperative.

Numerous experimental and observational constraints on $m_{\gamma}$ have been
derived from various effects resulting from the hypothetical nonzero photon
mass. These include tests of Coulomb’s inverse square law (Williams  _et al._
, 1971), the Cavendish torsion balance (Lakes, 1998; Luo  _et al._ , 2003),
gravitational deflection of electromagnetic waves (Lowenthal, 1973),
mechanical stability of magnetized gas in galaxies (Chibisov, 1976), magneto-
hydrodynamic phenomena of the solar wind (Ryutov, 1997, 2007; Liu and Shao,
2012; Retinò  _et al._ , 2016), Jupiter’s magnetic field (Davis  _et al._ ,
1975), the spindown of a pulsar (Yang and Zhang, 2017), among others. However,
these constraints are contingent upon specific theories of massive photons and
are thus dynamic tests in nature.

In contrast, a kinematic test based on the dispersion effect of the speed of
light in vacuum is more inherently pure (Tu  _et al._ , 2005a, b). The
dispersion relation for massive photons is governed by

$E^{2}=p^{2}c^{2}+m_{\gamma}^{2}c^{4}\;,$ (1)

where $p$ represents the momentum. The group velocity of a photon with energy
$E=h\nu$ can be expressed as

$v=\frac{\partial E}{\partial
p}=c\sqrt{1-\frac{m_{\gamma}^{2}c^{4}}{E^{2}}}\approx
c\left(1-\frac{1}{2}\frac{m_{\gamma}^{2}c^{4}}{h^{2}\nu^{2}}\right)\;,$ (2)

where $h$ is the Planck constant and the last term is valid when
$m_{\gamma}\ll h\nu/c^{2}\simeq 7\times 10^{-42}\mathrm{(\nu/GHz)\;kg}$. Eq.
(2) suggests that if two photons with different frequencies are emitted
simultaneously from the same source, they will be observed at different times
due to their different velocities. Fast radio bursts (FRBs) provide an
excellent celestial laboratory for detecting this dispersion effect, owing to
their characteristics of (i) short time durations, (ii) long propagation
distances, and (iii) low-frequency emissions.

Based on the vacuum dispersion method, several studies have used FRBs to put
stringent upper limits on the photon mass (Wu  _et al._ , 2016; Bonetti  _et
al._ , 2016, 2017; Shao and Zhang, 2017; Xing  _et al._ , 2019; Wei and Wu,
2020; Wang  _et al._ , 2021; Chang  _et al._ , 2023; Lin  _et al._ , 2023;
Wang  _et al._ , 2023, 2024). Since FRBs originate at cosmological distances,
the cosmic expansion rate $H(z)$ has to be considered in constraining the
photon mass $m_{\gamma}$. In all previous studies, the required $H(z)$
information is calculated within the standard $\Lambda$CDM cosmological model.
It should, however, be emphasized that $\Lambda$CDM is rooted in the framework
of GR, which also embraces the postulate of the constancy of light speed.
Thus, there is a circularity problem in constraining the photon mass. To
address this problem, one has to determine the Hubble parameter $H(z)$ in a
cosmology-independent way.

In this work, we propose a novel nonparametric approach, utilizing the
Artificial Neural Network (ANN) technology, to reconstruct a smooth $H(z)$
function that best approximates the discrete $H(z)$ measurements. By combining
observations of well-localized FRBs and the $H(z)$ function reconstructed by
ANN, we aim to establish a cosmology-independent constraint on the rest mass
of the photon. The paper is structured as follows. Section II introduces the
theoretical framework and the observational data utilized for constraining the
photon mass. Section III presents the cosmology-independent constraints on the
photon mass and other relevant parameters. Finally, a brief summary and
discussions are given in Section IV.

##  II Analysis Method and Data

###  II.1 Theoretical framework

In our analysis, the observed time delay between different frequencies for
FRBs is attributed to two main factors: (i) the propagation of photons through
the plasma distributed between the source of the FRB and the observer, and
(ii) the non-zero rest mass of the photon.

  * •

Due to the interaction between the plasma and electromagnetic waves, photons
with higher frequency $\nu_{h}$ travel faster than those with lower frequency
$\nu_{l}$, resulting in a dispersion effect. The time delay $\Delta
t_{\mathrm{DM}}$ between photons of different frequencies caused by this
dispersion effect can be described as (Lorimer and Kramer, 2012; Bentum  _et
al._ , 2017)

$\displaystyle\Delta t_{\mathrm{DM}}$
$\displaystyle=\int\frac{\nu_{p}^{2}}{2c}\left(\nu_{l}^{-2}-\nu_{h}^{-2}\right)\mathrm{d}l$
(3)
$\displaystyle=\frac{e^{2}}{8\pi^{2}m_{e}\epsilon_{0}c}\left(\nu_{l}^{-2}-\nu_{h}^{-2}\right)\mathrm{DM_{astro}}\;,$

where $\nu_{p}\equiv\sqrt{\frac{n_{e}e^{2}}{4\pi^{2}m_{e}\epsilon_{0}}}$ is
the plasma frequency with the number density of electrons $n_{e}$, the charge
of an electron $e$, the mass of an electron $m_{e}$, and the permittivity of
vacuum $\epsilon_{0}$. Here $\mathrm{DM_{astro}}$ is the dispersion measure
(DM) contributed by the plasma, which is defined as the integral of the number
density of electrons $n_{e}$ along the line of propagation, given by
$\mathrm{DM_{astro}}\equiv\int n_{e}\mathrm{d}l$.

For an extragalactic source, $\mathrm{DM_{astro}}$ can be primarily divided
into four components: the contributions from the Milky Way’s interstellar
medium ($\mathrm{DM_{ISM}^{MW}}$), the Galactic halo
($\mathrm{DM_{halo}^{MW}}$), the intergalactic medium (IGM;
$\mathrm{DM_{IGM}}$), and the host galaxy ($\mathrm{DM_{host}}$). Therefore,
$\mathrm{DM_{astro}}$ is given by

$\displaystyle\mathrm{DM_{astro}}=$
$\displaystyle\mathrm{DM_{ISM}^{MW}}+\mathrm{DM_{halo}^{MW}}$ (4)
$\displaystyle+\mathrm{DM_{IGM}}+\frac{\mathrm{DM_{host,0}}}{1+z}\;,$

where the factor $(1+z)$ converts the DM component of the host galaxy in the
rest frame, denoted as $\mathrm{DM_{host,0}}$, to the observed value
$\mathrm{DM_{host}}$.

  * •

According to Eq. (2), the time delay $\Delta t_{m_{\gamma}}$ between massive
photons of high and low frequencies can be expressed as:

$\Delta
t_{m_{\gamma}}=\frac{1}{2}\left(\frac{m_{\gamma}c^{2}}{h}\right)^{2}\left(\nu_{l}^{-2}-\nu_{h}^{-2}\right)H_{\gamma}(z)\;,$
(5)

where $H_{\gamma}(z)$ is a newly defined function related to redshift,

$H_{\gamma}(z)=\int_{0}^{z}\frac{\left(1+{z}^{\prime}\right)^{-2}}{H(z^{\prime})}\mathrm{d}z^{\prime}\;,$
(6)

where $H(z)$ is the Hubble parameter at redshift $z$. Note that here
$H_{\gamma}(z)$ differs from the dimensionless redshift function that defined
in previous works, and it is in units of $\left[{\rm
km\;s^{-1}\;Mpc^{-1}}\right]^{-1}$.

In our analysis, the total observed time delay is

$\Delta t_{\rm obs}=\Delta t_{\mathrm{DM}}+\Delta t_{m_{\gamma}}\;.$ (7)

Observationally, the time delay of all FRBs exhibits a frequency dependence of
$\nu^{-2}$, while both $\Delta t_{\mathrm{DM}}$ and $\Delta t_{m_{\gamma}}$
follow a $\nu^{-2}$ behavior. Hence, it is natural for us to analogously
define the equivalent DM arising from the massive photons as (Shao and Zhang,
2017)

$\mathrm{DM}_{\gamma}(z)\equiv\frac{4\pi^{2}m_{e}\epsilon_{0}c^{5}}{h^{2}e^{2}}H_{\gamma}(z)m_{\gamma}^{2}\;.$
(8)

Thus, the observed DM obtained from fitting the $\nu^{-2}$ behavior in the
total frequency-dependent time delay, can be written as:

$\mathrm{DM_{obs}}=\mathrm{DM_{astro}}+\mathrm{DM}_{\gamma}\;.$ (9)

Once we are able to properly estimate the value of each DM term in Eq. (4),
$\mathrm{DM}_{\gamma}$ can then be effectively extracted from
$\mathrm{DM_{obs}}$, thereby providing an upper limit on the photon mass
$m_{\gamma}$.

The $\mathrm{DM_{ISM}^{MW}}$ term arising from the ionizing medium around our
galaxy is well modeled by some galactic electron distribution models. Here we
adopt the NE2001 model (Cordes and Lazio, 2002) for its wide application. The
value of $\mathrm{DM_{halo}^{MW}}$ is challenging to estimate accurately, but
it has been expected to lie within the range of $\mathrm{50-80\;pc\;cm^{-3}}$
(Prochaska and Zheng, 2019; Keating and Pen, 2020). Here, we conservatively
adopt a Gaussian prior of $\mathrm{DM_{halo}^{MW}=65\pm 15\;pc\;cm^{-3}}$ (Wu
_et al._ , 2022; Wang  _et al._ , 2023).

The DM due to the IGM, $\mathrm{DM_{IGM}}$, depends on the cosmological model
and is largely influenced by the number of halos intersected along the
propagation path. Due to density perturbations on large-scale structures, it
is challenging to calculate the precise value of $\mathrm{DM_{IGM}}$ for
individual FRBs. Instead, we typically calculate the average value of
$\mathrm{DM_{IGM}}$ using (Deng and Zhang, 2014)

$\left\langle\mathrm{DM_{IGM}}(z)\right\rangle=\frac{21c\Omega_{b}H_{0}^{2}f_{\mathrm{IGM}}}{64\pi
Gm_{p}}H_{e}(z)\;,$ (10)

where $\Omega_{b}$ is the baryon density parameter at the present day, $H_{0}$
is the Hubble constant, $f_{\mathrm{IGM}}\simeq 0.83$ is the fraction of
baryon in the IGM (Fukugita  _et al._ , 1998), $G$ is the gravitational
constant, and $m_{p}$ is the mass of proton. The redshift-dependent function
$H_{e}(z)$ (in units of $[{\rm km\;s^{-1}\;Mpc^{-1}}]^{-1}$) is defined by

$H_{e}(z)\equiv\int_{0}^{z}\frac{(1+z^{\prime})}{H(z^{\prime})}\mathrm{d}z^{\prime}\;.$
(11)

However, the actual value of $\mathrm{DM_{IGM}}$ may deviate from the average
due to the inhomogeneity of IGM. To account for this variability, we employ a
one-parameter model to simulate the probability distribution of
$\mathrm{DM_{IGM}}$ (Miralda-Escudé  _et al._ , 2000; Macquart  _et al._ ,
2020; Wang  _et al._ , 2021), which is given by

$P_{\mathrm{IGM}}(\mathrm{DM_{IGM}}|z)=A\Delta^{-\beta}\exp\left[-\frac{\left(\Delta^{-\alpha}-C_{0}\right)^{2}}{2\alpha^{2}s^{2}}\right]\;,$
(12)

where
$\Delta\equiv\mathrm{DM_{IGM}}/\left\langle\mathrm{DM_{IGM}}\right\rangle>0$,
$A$ is a normalization constant to ensure the integral of
$P_{\mathrm{IGM}}(\mathrm{DM_{IGM}}|z)$ to be unity, $C_{0}$ is chosen to
satisfy $\left\langle\Delta\right\rangle=1$, and $\alpha=\beta=3$ (Macquart
_et al._ , 2020). The fractional standard deviation of $\mathrm{DM_{IGM}}$ is
approximated as $s=Fz^{-0.5}$, where $F$ is the free parameter that quantifies
the strength of baryon feedback (Macquart  _et al._ , 2020). Both semi-
analytic models and numerical simulations of the IGM and galaxy halos showed
that the probability distribution of $\mathrm{DM_{IGM}}$ can be well fitted by
the quasi-Gaussian form (i.e., Eq. 12) (McQuinn, 2014; Macquart  _et al._ ,
2020). Therefore, this analytic form is adopted in our analysis.

The $\mathrm{DM_{host}}$ term is contributed by the source environment and the
interstellar medium of the host galaxy, implying that the value of
$\mathrm{DM_{host}}$ may vary significantly across different sources. To model
this variability, we adopt a log-normal distribution for the probability
density function of $\mathrm{DM_{host}}$ with the expression as (Macquart  _et
al._ , 2020)

$\displaystyle P_{\mathrm{host}}\mathrm{(DM_{host}|\mu,\sigma_{host})}$
$\displaystyle=\frac{1}{\sqrt{2\pi}\mathrm{DM_{host}\sigma_{host}}}$ (13)
$\displaystyle\times\exp\left[-\frac{\left(\ln{\mathrm{DM_{host}}}-\mu\right)^{2}}{2\sigma_{\mathrm{host}}^{2}}\right]\;,$

where $\mu$ and $\sigma_{\mathrm{host}}$ are free parameters which are used to
estimate the mean and standard deviation of $\ln{\mathrm{DM_{host}}}$,
respectively.

With the analyses stated above, we formulate the joint likelihood function for
a sample of localized FRBs as

$\mathcal{L}=\prod_{i=1}^{N}P_{i}\left(\mathrm{DM}_{\mathrm{E},i}|z_{i}\right)\;,$
(14)

where $N$ is the number of FRBs and
$P_{i}\left(\mathrm{DM}_{\mathrm{E},i}|z_{i}\right)$ denotes the likelihood
for the $i$-th FRB with the corrected observable DM as

$\displaystyle\mathrm{DM_{E}}$
$\displaystyle\equiv\mathrm{DM_{obs}}-\mathrm{DM_{halo}^{MW}}-\mathrm{DM_{ISM}^{MW}}$
(15)
$\displaystyle=\mathrm{DM_{IGM}}+\mathrm{DM_{host}}+\mathrm{DM_{\gamma}}\;.$

For a certain FRB at the redshift of $z_{i}$, we can estimate the probability
distribution of $\mathrm{DM_{E}}$ with the combination of Eqs. (12), (13), and
(15), which gives the expression as

$\displaystyle P($
$\displaystyle\mathrm{DM_{E}}|z_{i})=\int_{0}^{\mathrm{DM_{E}}-\mathrm{DM_{\gamma}}}P_{\rm
host}\left(\mathrm{DM_{host}}|\mu,\sigma_{\mathrm{host}}\right)$ (16)
$\displaystyle\times
P_{\mathrm{IGM}}\left(\mathrm{DM_{E}}-\mathrm{DM_{host}}-\mathrm{DM_{\gamma}}|z_{i}\right)\mathrm{dDM_{host}}\;.$

Note that by Eq. (8), $\mathrm{DM}_{\gamma}$ is a constant at a given $z_{i}$.
Denoting $P_{X}(X)$, $P_{Y}(Y)$, and $P_{Z}(Z)$ as the probability density
functions of random variables $X$, $Y$, and $Z$, respectively. The equality
$P_{Z}(Z)=\int_{0}^{Z}P_{X}(X)P_{Y}(Z-X)dX$ holds for every non-negative
independent random variables $X$, $Y$, and $Z$, such that $Z=X+Y$. As
$\mathrm{DM}_{\mathrm{IGM}}$ and $\mathrm{DM}_{\mathrm{host}}$ are non-
negative independent random variables, by setting
$Z=\mathrm{DM}_{\mathrm{E}}-\mathrm{DM}_{\gamma}$,
$X=\mathrm{DM}_{\mathrm{host}}$, and $Y=\mathrm{DM}_{\mathrm{IGM}}$, it is
easy to derive Eq. (16).

By maximizing the likelihood function $\mathcal{L}$, we can place constraints
on the parameters in our model, including the rest mass of the photon
$m_{\gamma}$. Nevertheless, there are two redshift-dependent functions,
$H_{\gamma}(z)$ and $H_{e}(z)$ (see Eqs. 6 and 11), to be determined. In
previous works (Wu  _et al._ , 2016; Bonetti  _et al._ , 2016, 2017; Shao and
Zhang, 2017; Xing  _et al._ , 2019; Wei and Wu, 2020; Wang  _et al._ , 2021;
Chang  _et al._ , 2023; Lin  _et al._ , 2023; Wang  _et al._ , 2023), the
conventional approach is to adopt the flat $\Lambda$CDM cosmological model and
replace the denominator $H(z)$ with
$H_{0}[\Omega_{m}\left(1+z\right)^{3}+\Omega_{\Lambda}]^{1/2}$. Subsequently,
$H_{\gamma}(z)$ and $H_{e}(z)$ are calculated by treating the cosmological
parameters $\Omega_{m}$ and $\Omega_{\Lambda}$ as fixed values. However, it is
crucial to note that the $\Lambda$CDM model is established within the
framework of Einstein’s theory of GR, which is derived from one of the
fundamental postulates–the masslessness of photons. Consequently, adopting
this model introduces a logical circularity problem in constraining the rest
mass of the photon. To circumvent this issue, it is necessary to employ a
cosmology-independent method to describe the relation between the Hubble
parameter $H(z)$ and redshift $z$.

###  II.2 Artificial Neural Network

Gaussian Process (GP) is a widely-used approach for studying cosmology in a
model-independent manner. This method assumes that the reconstructed value at
a given point follows a Gaussian distribution, and the relationship between
function values at different points is characterized by a selected covariance
function (Seikel  _et al._ , 2012). Besides, the reconstructed function of
$H(z)$ from GP tends to underestimate errors and is significantly influenced
by the prior of the Hubble constant $H_{0}$ (Wei and Wu, 2017; Wang  _et al._
, 2017). In contrast, ANN, inspired by biological neural networks, is a
mathematical model primarily used to discover intricate relationships between
input and output data. The reconstructed function based on ANN makes no
assumptions about the observational data and is entirely data-driven without
parameterization of the function. Recently, Ref. Wang  _et al._ (2020)
developed a public code called ReFANN111https://github.com/Guo-Jian-
Wang/refann for reconstructing functions from data using ANN. Their
reconstructed $H(z)$ function showed no sensitivity to the setting of $H_{0}$,
indicating that reconstructing $H(z)$ from observational data using ANN may be
more reliable than using GP. To make a cosmology-independent determination of
$H(z)$, in this work we will reconstruct a smooth curve of $H(z)$ from 34
model-independent measurements of $H(z)$ using ReFANN.

The first step in using ReFANN to reconstruct a function from data is to find
the best parameter configuration for the model used to train the data.
Following the approach of Ref. Wang  _et al._ (2020), we determine the optimal
network model by minimizing the risk (Wasserman  _et al._ , 2001):

$\displaystyle\mathrm{risk}$ $\displaystyle=\sum_{i=1}^{N}{\rm
Bias}_{i}^{2}+\sum_{i=1}^{N}{\rm Variance}_{i}$ (17)
$\displaystyle=\sum_{i=1}^{N}\left[H(z_{i})-\bar{H}(z_{i})\right]^{2}+\sum_{i=1}^{N}\sigma^{2}_{H(z_{i})}\;,$

where $N$ is the number of sample $H(z)$ and $\bar{H}(z_{i})$ is the fiducial
value of $H(z)$ at redshift $z_{i}$. For our purposes, we only consider the
number of hidden layers and neurons in the hidden layer. Through
experimentation, we determined that the optimal model configuration involves a
single hidden layer with 4096 neurons. Additionally, to reduce sensitivity to
initialization and stabilize the distribution among variables, we employ batch
normalization (Ioffe and Szegedy, 2015) in our network model, with the batch
size set to half of the number of $H(z)$ data points. We also use the
Exponential Linear Unit (Clevert  _et al._ , 2015) as the activation function,
with the hyperparameter $\alpha$ set to be 1. Furthermore, we utilize the Adam
optimizer (Kingma and Ba, 2014) to update the network parameters in each
iteration of the training process.

Subsequently, we train this optimal network model with the observational
$H(z)$ data. This training process provides us with a predicted value of
$H(z_{i})$ and its associated error $\sigma_{H(z_{i})}$ at a given redshift
$z_{i}$. This predicted function represents an approximate reconstruction of
$H(z)$ based on the ANN model trained with observational data.

Table 1: 34 $H(z)$ measurements obtained with the CC method.  $z$ | $H(z)$ | $\sigma$ | Refs.
---|---|---|---
| ($\rm km\;s^{-1}\;Mpc^{-1}$) | ($\rm km\;s^{-1}\;Mpc^{-1}$) |
0.09 | 69 | 12 | Jimenez  _et al._ (2003)
0.17 | 83 | 8 | Simon  _et al._ (2005)
0.27 | 77 | 14
0.4 | 95 | 17
0.9 | 117 | 23
1.3 | 168 | 17
1.43 | 177 | 18
1.53 | 140 | 14
1.75 | 202 | 40
0.48 | 97 | 62 | Stern  _et al._ (2010)
0.88 | 90 | 40
0.1791 | 75 | 4 | Moresco  _et al._ (2012)
0.1993 | 75 | 5
0.3519 | 83 | 14
0.5929 | 104 | 13
0.6797 | 92 | 8
0.7812 | 105 | 12
0.8754 | 125 | 17
1.037 | 154 | 20
0.07 | 69 | 19.6 | Zhang  _et al._ (2014)
0.12 | 68.2 | 26.2
0.2 | 72.9 | 29.6
0.28 | 88.8 | 36.6
1.363 | 160 | 33.6 | Moresco (2015)
1.965 | 186.5 | 50.4
0.3802 | 83 | 13.5 | Moresco  _et al._ (2016)
0.4004 | 77 | 10.2
0.4247 | 87.1 | 11.2
0.4497 | 92.8 | 12.9
0.4783 | 80.9 | 9
0.47 | 89 | 49.6 | Ratsimbazafy  _et al._ (2017)
0.75 | 98.8 | 33.6 | Borghi  _et al._ (2022)
0.8 | 113.1 | 25.22 | Jiao  _et al._ (2023)
1.26 | 135 | 65 | Tomasetti  _et al._ (2023)

###  II.3 Hubble parameter $H(z)$ Data

The Hubble parameter $H(z)$, which quantifies the expansion rate of the
universe as a function of time (or equivalently redshift), has been
extensively used for exploring the nature of dark energy and testing modified
theories of gravity. $H(z)$ can be measured using two main methods. One
approach is based on the detection of the radial baryon acoustic oscillation
features (Gaztañaga  _et al._ , 2009; Blake  _et al._ , 2012; Samushia  _et
al._ , 2013). However, this method relies on assumptions about the underlying
cosmological model, typically the $\Lambda$CDM model. Therefore, in our
analysis, we focus exclusively on $H(z)$ measurements obtained using another
method, namely the cosmic chronometers (CC) method (Jimenez and Loeb, 2002).
This method is model-independent as it relies on minimal assumptions,
primarily the use of a Friedmann-Lemaître-Robertson-Walker metric. In this
method, by combining the definition of $H(z)\equiv\dot{a}/a$ and the relation
between the scale factor $a(t)$ and redshift, $1+z=1/a(t)$, $H(z)$ can be
rewritten as

$H(z)=-\frac{1}{1+z}\frac{\mathrm{d}z}{\mathrm{d}t}\;.$ (18)

This means that one can achieve a model-independent determination of $H(z)$ by
calculating the differential age of the universe using passively evolving
galaxies at various redshifts. We compile the latest 34 CC $H(z)$ measurements
in Table 1, covering the redshift range of $0.07<z<1.965$.

Figure 1: Reconstruction of the Hubble parameter $H(z)$ from 34 CC $H(z)$
measurements using ANN. The shaded area corresponds to the 1$\sigma$
confidence region of the reconstruction. The blue dots with error bars depict
the observational $H(z)$ data. Figure 2:  The redshift dependence of the
reconstructed $H_{\gamma}(z)$ and $H_{e}(z)$ functions (solid lines). The
corresponding theoretical curves (dashed lines) derived from the flat
$\Lambda$CDM model with $\Omega_{m}=0.315$ and $H_{0}=67.4\;\rm
km\;s^{-1}\;Mpc^{-1}$ are also shown.

Having obtained the dataset of $H(z)$, we adopt ANN to reconstruct the $H(z)$
function within the redshift range of $0<z<2$, and the results are shown in
Fig. 1. The black line represents the best fit, and the blue shaded area
indicates the 1$\sigma$ confidence region of the reconstructed function. The
redshift-dependent $H_{\gamma}(z)$ and $H_{e}(z)$ functions can then be
derived by integrating the reconstructed $H(z)$ function with respect to
redshift. As shown in Fig. 2, the reconstructed $H_{\gamma}(z)$ and $H_{e}(z)$
functions exhibit significant differences in behavior as a function of
redshift $z$. Such distinctions are crucial for breaking parameter degeneracy
and enhancing sensitivity in testing the photon mass when a few redshift
measurements of FRBs are available (Bonetti  _et al._ , 2016, 2017; Shao and
Zhang, 2017; Wei and Wu, 2020; Bentum  _et al._ , 2017). For comparison, we
also plot $H_{\gamma}(z)$ and $H_{e}(z)$ functions within the framework of the
flat $\Lambda$CDM model using Planck 2018 parameters ($\Omega_{m}=0.315$ and
$H_{0}=67.4\;\rm km\;s^{-1}\;Mpc^{-1}$) (Planck Collaboration  _et al._ ,
2020). The corresponding theoretical curves are presented in Fig. 2 as dashed
lines. It is evident that the reconstructions of $H_{\gamma}(z)$ and
$H_{e}(z)$ are roughly consistent with the predictions of the standard
$\Lambda$CDM model, implying that the ANN method can offer a reliable
reconstructed function from the observational data.

###  II.4 Fast Radio Burst Data

Expanding upon the sample of 23 localized FRBs utilized in Ref. Wang  _et al._
(2023), we incorporate 9 new localized FRBs recently detected by the
110-antenna Deep Synoptic Array Law  _et al._ (2023). Note that another two
new FRBs with redshift measurements, FRB 20220319D and FRB 20220914A, are not
included in our sample. FRB 20220319D exhibits an observed $\mathrm{DM_{obs}}$
value of $110.98\;\mathrm{pc\;cm^{-3}}$, while the $\mathrm{DM_{ISM}^{MW}}$
value calculated using the NE2001 model stands at
$133.3\;\mathrm{pc\;cm^{-3}}$, leading to its exclusion from our analysis.
Furthermore, FRB 20220914A is omitted due to its notable deviation from the
expected $\mathrm{DM_{IGM}}$–$z$ relation outlined in Eq. (10). The total 32
localized FRBs, including their respective redshifts, $\mathrm{DM_{obs}}$, and
$\mathrm{DM_{ISM}^{MW}}$, are provided in Table 2.

Table 2: Properties of 32 localized FRBs.  Name | $z$ | $\mathrm{DM_{obs}}$ | $\mathrm{DM_{ISM}^{MW}}$ | Refs.
---|---|---|---|---
|  | $(\mathrm{pc\,cm^{-3}})$ | $(\mathrm{pc\,cm^{-3}})$ |
FRB 20121102 | 0.19273 | 557 | 188.0 | Chatterjee  _et al._ (2017)
FRB 20180301 | 0.3304 | 536 | 152.0 | Bhandari  _et al._ (2022)
FRB 20180916 | 0.0337 | 348.76 | 200.0 | Marcote  _et al._ (2020)
FRB 20180924 | 0.3214 | 361.42 | 40.5 | Bannister  _et al._ (2019)
FRB 20181112 | 0.4755 | 589.27 | 102.0 | Prochaska  _et al._ (2019)
FRB 20190102 | 0.291 | 363.6 | 57.3 | Bhandari  _et al._ (2020)
FRB 20190523 | 0.66 | 760.8 | 37.0 | Ravi  _et al._ (2019)
FRB 20190608 | 0.1178 | 338.7 | 37.2 | Chittidi  _et al._ (2021)
FRB 20190611 | 0.378 | 321.4 | 57.83 | Heintz  _et al._ (2020)
FRB 20190614 | 0.6 | 959.2 | 83.5 | Law  _et al._ (2020)
FRB 20190711 | 0.522 | 593.1 | 56.4 | Heintz  _et al._ (2020)
FRB 20190714 | 0.2365 | 504 | 38.0 | Heintz  _et al._ (2020)
FRB 20191001 | 0.234 | 506.92 | 44.7 | Heintz  _et al._ (2020)
FRB 20191228 | 0.2432 | 297.5 | 33.0 | Bhandari  _et al._ (2022)
FRB 20200430 | 0.16 | 380.1 | 27.0 | Heintz  _et al._ (2020)
FRB 20200906 | 0.3688 | 577.8 | 36.0 | Bhandari  _et al._ (2022)
FRB 20201124 | 0.098 | 413.52 | 123.2 | Ravi  _et al._ (2022)
FRB 20210117 | 0.2145 | 730 | 34.4 | James  _et al._ (2022)
FRB 20210320 | 0.27970 | 384.8 | 42 | James  _et al._ (2022)
FRB 20210807 | 0.12927 | 251.9 | 121.2 | James  _et al._ (2022)
FRB 20211127 | 0.0469 | 234.83 | 42.5 | James  _et al._ (2022)
FRB 20211212 | 0.0715 | 206 | 27.1 | James  _et al._ (2022)
FRB 20220207C | 0.043040 | 262.38 | 79.3 | Law  _et al._ (2023)
FRB 20220307B | 0.28123 | 499.27 | 135.7 | Law  _et al._ (2023)
FRB 20220310F | 0.477958 | 462.24 | 45.4 | Law  _et al._ (2023)
FRB 20220418A | 0.622000 | 623.25 | 37.6 | Law  _et al._ (2023)
FRB 20220506D | 0.30039 | 396.97 | 89.1 | Law  _et al._ (2023)
FRB 20220509G | 0.089400 | 269.53 | 55.2 | Law  _et al._ (2023)
FRB 20220610A | 1.016 | 1457.624 | 31 | Ryder  _et al._ (2022)
FRB 20220825A | 0.241397 | 651.24 | 79.7 | Law  _et al._ (2023)
FRB 20220920A | 0.158239 | 314.99 | 40.3 | Law  _et al._ (2023)
FRB 20221012A | 0.284669 | 441.08 | 54.4 | Law  _et al._ (2023)

##  III Results

We explore the posterior probability distributions of the free parameters by
maximizing the joint likelihood function $\mathcal{L}$ (Eq. 14) using emcee,
an affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler
implemented in Python (Foreman-Mackey  _et al._ , 2013). The free parameters
of our model now include the baryon density parameter $\Omega_{b}h_{0}^{2}$
(where $h_{0}\equiv H_{0}/{\rm(100\,km\,s^{-1}\,Mpc^{-1})}$), the DM
contribution from the Milky Way’s halo $\mathrm{DM_{halo}^{MW}}$, the
parameters related to the probability distributions for $\mathrm{DM_{IGM}}$
and $\mathrm{DM_{host}}$ (i.e., $F$, $\mu$, and $\sigma_{\mathrm{host}}$), and
the photon mass $m_{\gamma}$. In our MCMC analysis, we set uniform priors on
$\Omega_{b}h_{0}^{2}\in[0.01,\;1]$, $F\in[0.01,\;0.5]$, $e^{\mu}\in[20,\;200]$
$\mathrm{pc\;cm^{-3}}$, $\sigma_{\mathrm{host}}\in[0.2,\;2]$, and
$m_{\gamma}\in[0,\;10^{-49}]$ $\mathrm{kg}$. For $\mathrm{DM_{halo}^{MW}}$, we
set a Gaussian prior, $\mathrm{DM_{halo}^{MW}=65\pm 15\;pc\;cm^{-3}}$, within
the wide $3\sigma$ range of $[20,\;110]$ $\mathrm{pc\;cm^{-3}}$. To
incorporate the error of the ANN reconstruction into our analysis, at each
MCMC step, we sample the Hubble rate $\tilde{H}(z)$ function according to the
Gaussian distribution $\tilde{H}(z)=\mathcal{N}(H(z),\,\sigma_{H(z)})$ with
mean $H(z)$ and standard deviation $\sigma_{H(z)}$. Here $H(z)$ is the best-
fit function reconstructed by ANN and $\sigma_{H(z)}$ is the $1\sigma$ error
of the reconstructed $H(z)$ function.

The $1\sigma$ constraint results for these six parameters are summarized in
Table 3. For the photon mass $m_{\gamma}$, both the $1\sigma$ and $2\sigma$
upper limits are presented. Posterior distributions and $1-2\sigma$ confidence
regions for these parameters are displayed in Fig. 3. Notably, the constraints
on $m_{\gamma}$ are determined as

$m_{\gamma}\leq 3.5\times 10^{-51}\,\rm{kg}\simeq 2.0\times
10^{-15}\,\rm{eV/c^{2}}$ (19)

and

$m_{\gamma}\leq 6.5\times 10^{-51}\,\rm{kg}\simeq 3.6\times
10^{-15}\,\rm{eV/c^{2}}$ (20)

at the $1\sigma$ and $2\sigma$ confidence level, respectively. Meanwhile, we
find that the baryon density parameter is optimized to be
$\Omega_{b}h_{0}^{2}=0.030_{-0.007}^{+0.006}$, which is compatible with the
value inferred from Planck 2018 ($\Omega_{b}h_{0}^{2}=0.0224\pm 0.0001$) at
the $1.2\sigma$ confidence level (Planck Collaboration  _et al._ , 2020).

To investigate the impact of the prior assumption of $\mathrm{DM_{halo}^{MW}}$
on our results, we also perform a parallel comparative analysis of the FRB
data using a flat prior on $\mathrm{DM_{halo}^{MW}}\in[20,\;110]$
$\mathrm{pc\;cm^{-3}}$. The corresponding resulting constraints on all
parameters are also reported in Table 3. Comparing these inferred parameters
with those obtained from the Gaussian prior (see line 1 in Table 3), it is
clear that, except for the best-fit value of $\mathrm{DM_{halo}^{MW}}$, the
prior assumption of $\mathrm{DM_{halo}^{MW}}$ only has a minimal influence on
our results.

Table 3: The $1\sigma$ constraints for all parameters with different priors on $\mathrm{DM_{halo}^{MW}}$. Both the $1\sigma$ and $2\sigma$ upper limits for the photon mass $m_{\gamma}$ are provided. Priors of $\mathrm{DM_{halo}^{MW}}$ | $\Omega_{b}h_{0}^{2}$ | $F$ | $\mathrm{DM_{halo}^{MW}}$ | $\mu$ | $\sigma_{\mathrm{host}}$ | $m_{\gamma}$
---|---|---|---|---|---|---
|  |  | $(\mathrm{pc\;cm^{-3}})$ |  |  | $(10^{-51}\,{\rm kg})$
Gaussian prior | $0.030_{-0.007}^{+0.006}$ | $\geq 0.449$ | $39_{-10}^{+10}$ | $4.02_{-0.29}^{+0.46}$ | $1.01_{-0.37}^{+0.22}$ | $\leq 3.5\;(\leq 6.5)$
Flat prior | $0.031_{-0.006}^{+0.006}$ | $\geq 0.451$ | $\leq 32$ | $4.07_{-0.30}^{+0.48}$ | $0.99_{-0.37}^{+0.21}$ | $\leq 3.8\;(\leq 7.2)$
Figure 3: Plots of 1D marginalized probability distributions and 2D
$1-2\sigma$ confidence contours for the parameters $\Omega_{b}h_{0}^{2}$, $F$,
$\mathrm{DM_{halo}^{MW}}$, $\mu$, $\sigma_{\mathrm{host}}$, and $m_{\gamma}$,
constrained by 32 localized FRBs.

##  IV Discussion and conclusions

As a matter of fact, it is impossible to prove experimentally that the rest
mass of a photon is strictly zero. According to the uncertainty principle of
quantum mechanics, the ultimately measurable order of magnitude for the photon
mass is $m_{\gamma}\approx\hbar/(\Delta tc^{2})\approx 10^{-69}\,\rm{kg}$,
where $\hbar$ is the reduced Planck constant and $\Delta t\approx
10^{10}\,\rm{years}$ is the age of our universe. However, due to the immense
importance of Maxwell’s theory and Einstein’s theory of relativity, it is
still intellectually stimulating and scientifically significant to approach
this ultimate limit by various experiments.

FRBs provide the current best celestial laboratory to test the photon mass
$m_{\gamma}$ via the dispersion method. Constraining $m_{\gamma}$ with
cosmological FRBs, however, one has to know the cosmic expansion rate $H(z)$.
In all previous studies, the required $H(z)$ information is estimated within
the standard $\Lambda$CDM cosmological model. Such $H(z)$ estimations would
involve a circularity problem in constraining $m_{\gamma}$, since $\Lambda$CDM
itself is built on the framework of GR and GR embraces the postulate of the
constancy of light speed. In this work, aiming to overcome the circularity
problem, we have employed an ANN technology to reconstruct a cosmology-
independent $H(z)$ function from the discrete CC $H(z)$ data.

By combining the DM–$z$ measurements of 32 localized FRBs with the
reconstructed $H(z)$ function from 34 CC $H(z)$ data, we have placed the first
cosmology-independent photon mass limit. Our results show that the $1\sigma$
and $2\sigma$ confidence-level upper limits on the photon mass are
$m_{\gamma}\leq 3.5\times 10^{-51}\,\rm{kg}$ (or equivalently $m_{\gamma}\leq
2.0\times 10^{-15}\,\rm{eV/c^{2}}$) and $m_{\gamma}\leq 6.5\times
10^{-51}\,\rm{kg}$ (or equivalently $m_{\gamma}\leq 3.6\times
10^{-15}\,\rm{eV/c^{2}}$), respectively. Previously, under the assumption of
fiducial $\Lambda$CDM cosmology, Ref. (Wang  _et al._ , 2021) obtained an
upper limit of $m_{\gamma}\leq 3.1\times 10^{-51}\,\rm{kg}$ at the $1\sigma$
confidence level by using a catalog of 129 FRBs (most of them without redshift
measurement, and the observed $\mathrm{DM_{obs}}$ values were used to estimate
the pseudo redshifts). Ref. (Lin  _et al._ , 2023) obtained $m_{\gamma}\leq
4.8\times 10^{-51}\,\rm{kg}$ ($1\sigma$) by analyzing a sample of 17 localized
FRBs in the flat $\Lambda$CDM model. Ref. (Wang  _et al._ , 2023) obtained
$m_{\gamma}\leq 3.8\times 10^{-51}\,\rm{kg}$ ($1\sigma$) for flat $\Lambda$CDM
using a sample of 23 localized FRBs. Despite not assuming a specific
cosmological model, the precision of our constraint from 32 localized FRBs is
comparable to these previous results. Most importantly, this highlights the
validity of our approach and suggests that as the number of CC $H(z)$
measurements increases, we can expect even more reliable model-independent
tests of the photon mass.

###### Acknowledgements.

We are grateful to the anonymous referees for their helpful comments. This
work is partially supported by the National SKA Program of China
(2022SKA0130100), the National Natural Science Foundation of China (grant Nos.
12373053, 12321003, and 12041306), the Key Research Program of Frontier
Sciences (grant No. ZDBS-LY-7014) of Chinese Academy of Sciences,
International Partnership Program of Chinese Academy of Sciences for Grand
Challenges (114332KYSB20210018), the CAS Project for Young Scientists in Basic
Research (grant No. YSBR-063), the CAS Organizational Scientific Research
Platform for National Major Scientific and Technological Infrastructure:
Cosmic Transients with FAST, and the Natural Science Foundation of Jiangsu
Province (grant No. BK20221562).

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