# Bounding the Photon Mass with Ultrawide Bandwidth Pulsar Timing Data and
Dedispersed Pulses of Fast Radio Bursts

Yu-Bin Wang School of Physics and Electronic Engineering, Sichuan University
of Science & Engineering, Zigong 643000, China.
Xia Zhou Xinjiang Astronomical Observatory, Chinese Academy of Sciences,
Urumqi 830011, China.  Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, Nanjing 210008, China.  Xinjiang Key Laboratory of Radio
Astrophysics, 150 Science1-Street, Urumqi 830011, China.  Abdusattar Kurban
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011,
China.  Key Laboratory of Radio Astronomy, Chinese Academy of Sciences,
Nanjing 210008, China.  Xinjiang Key Laboratory of Radio Astrophysics, 150
Science1-Street, Urumqi 830011, China.  Fa-Yin Wang School of Astronomy and
Space Science, Nanjing University, Nanjing 210093, China.  Key Laboratory of
Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China. Xia Zhou<EMAIL_ADDRESS>
###### Abstract

Exploring the concept of a massive photon has been an important area in
astronomy and physics. If photons have mass, their propagation in nonvacuum
space would be affected by both the nonzero mass $m_{\gamma}$ and the presence
of a plasma medium. This would lead to a delay time proportional to
$m_{\gamma}^{2}\nu^{-4}$, which deviates from the classical dispersion
relation (proportional to $\nu^{-2}$). For the first time, we have derived the
dispersion relation of a photon with a nonzero mass propagating in plasma. To
reduce the impact of variations in the dispersion measure (DM), we employed
the high-precision timing data to constrain the upper bound of the photon
mass. Specifically, the DM/time of arrival (TOA) uncertainties derived from
ultrawide bandwidth (UWB) observations conducted by the Parkes Pulsar Timing
Array (PPTA) are used. The dedispersed pulses from fast radio bursts (FRBs)
with minimal scattering effects are also used to constrain the upper bound of
photon mass. The stringent limit on the photon mass is determined by
uncertainties of the TOA of pulsars, with an optimum value of $9.52\times
10^{-46}\,\rm kg\,\,(5.34\times 10^{-10}\,\rm eV/c^{2}$). In the future, it is
essential to investigate the photon mass, as pulsar timing data are collected
by PTA and UWB receivers, or FRBs with wideband spectra are detected by UWB
receivers.

Radio pulsars (1353); Pulsar timing method (1305); Radio transient sources
(2008); Intergalactic medium (813)

##  1 Introduction

In accordance with Maxwell’s electromagnetism and Einstein’s special
relativity, it is assumed that all electromagnetic waves travel at a constant
speed, $c$, in a vacuum (Einstein, 1905). This constant velocity implies that
the photons do not have mass (Weinberg, 1995). If photons were to be found to
have a nonzero mass, this would contradict the prediction and could lead to
new physics. Therefore, testing the mass of photons ($m_{\gamma}$) using
various methods across different scales, from particle physics to cosmology,
is an interesting topic in modern physics. Examples of such tests include the
null tests of Coulomb’s inverse square law (Williams et al., 1971), measuring
the Schumann resonance in a cavity (Kroll, 1971), examining Jupiter’s magnetic
field (Davis et al., 1975), surveying magnetohydrodynamic phenomena of the
solar wind (Ryutov, 1997, 2007; Retinò et al., 2016), observing the spin down
of a white dwarf pulsar (Yang & Zhang, 2017), studying the spin of
supermassive black holes (Pani et al., 2012), and measuring the gravitational
deflection of massive photons (Lowenthal, 1973).

The photon mass can be evaluated through the dispersion method, which involves
analyzing the signals of cosmological transients at different frequencies to
detect any potential frequency-dependent (chromatic) velocity caused by
photons with a nonzero mass. This velocity variation would lead to frequency-
dependent arrival times during observations (see review by Wei & Wu (2021) and
references therein). The dispersion method is used to identify frequency-
dependent time delays from cosmological distances at different frequencies or
energy bands, thus providing bounds on the photon mass. Radio pulse emissions
of millisecond durations from sources such as radio pulsars and fast radio
bursts (FRBs) are used to establish the upper limit of the photon mass, as
shown in Table 1.

Table 1: The stringent constraints on the photon mass have been determined through observations of both Galactic and Extragalactic sources. Reference | Source (s) | Bandwidth |  $m_{\gamma}$ (kg)
---|---|---|---
Galactic source |  |  |
Lovell et al. (1964) | flare stars | 0.54 $\mu$m$-$1.2 m | $1.6\times 10^{-45}$
Warner & Nather (1969) | Crab Nebula pulsar | 0.35$-$0.55 $\mu$m | $5.2\times 10^{-44}$
Chang et al. (2023) | Crab pulsar |  $9.25$ GHz | $5.7\times 10^{-46}$
This work | 22 millisecond pulsars in Milky Way | 0.7$\sim$4.0 GHz | $9.52\times 10^{-46}$
Extragalactic source |  |  |
Schaefer (1999) | GRB 980703 |  $5.0\times 10^{9}-1.2\times 10^{20}$ Hz | $4.2\times 10^{-47}$
Zhang et al. (2016) | GRB 050416A |  $8.46$ GHz$-$15 keV | $1.1\times 10^{-47}$
Wei et al. (2017) | Extragalactic radio pulsar (PSR J0451-67) |  $\sim 1.4$ GHz | $2.0\times 10^{-48}$
Wei et al. (2017) | Extragalactic radio pulsar (PSR J0045-7042) |  $\sim 1.4$ GHz | $2.3\times 10^{-48}$
Wei & Wu (2018) | 22 radio pulsars in the LMC |  $\sim 1.4$ GHz | $1.5\times 10^{-48}$
Wei & Wu (2018) | 5 radio pulsars in the SMC |  $\sim 1.4$ GHz | $1.6\times 10^{-48}$
Wu et al. (2016) | FRB 150418 |  $1.2-1.5$ GHz | $5.2\times 10^{-50}$
Bonetti et al. (2016) | FRB 150418 |  $1.2-1.5$ GHz | $3.2\times 10^{-50}$
Bonetti et al. (2017) | FRB 121102 |  $1.1-1.7$ GHz | $3.9\times 10^{-50}$
Shao & Zhang (2017) | 21 FRBs (20 of them without redshift test) |  $\sim$ GHz | $8.7\times 10^{-51}$
Xing et al. (2019) | FRB 121102 subbursts |  $1.34-1.37$ GHz | $5.1\times 10^{-51}$
Wei & Wu (2020) | 9 localized FRBs |  $\sim$ GHz | $7.1\times 10^{-51}$
Wang et al. (2021) | 129 FRBs |  $\sim$ GHz | $3.1\times 10^{-51}$
Lin et al. (2023) | 17 localized FRBs |  $\sim$ GHz | $7.1\times 10^{-51}$
Wang et al. (2023c) | 23 localized FRBs |  $\sim$ GHz | $3.8\times 10^{-51}$
Chang et al. (2023) | FRB 180916.J0158+65 |  $110-188$ MHz | $6.0\times 10^{-47}$
This work | 3 localized nonrepeaters (NE2001) | 1.10$\sim$1.44 GHz | $(4.52-4.72)\times 10^{-44}$
This work | 3 localized nonrepeaters (YMW16) | 1.10$\sim$1.44 GHz | $(4.70-4.93)\times 10^{-44}$

Generally, the pulse arrival times of radio sources follow the classical
dispersion relation ($\Delta t_{\rm d}\propto\nu^{-2}$) for the propagation of
radio signals through a cold plasma. A constraint on the photon mass
$m_{\gamma}$ is usually derived from the delay time with the accepted first-
order Taylor expansion term ($\Delta t_{m_{\gamma},1}\propto
m_{\gamma}^{2}\nu^{-2}$; Wu et al. 2016; Chang et al. 2023). However, the
second-order delay time term ($\Delta t_{m_{\gamma},2}\propto
m_{\gamma}^{4}\nu^{-4}$) has been used to limit the photon mass (Chang et al.,
2023). This method produces upper limits that are 4 orders of magnitude higher
(or worse) than those obtained from the first-order term (Chang et al., 2023).
As the classical dispersion relation is calculated based on the speed of light
for zero photon mass, it is hard to differentiate between the two dispersions,
which leads to anomalous results when considering the second-order term. In
this study, our aim is to address this problem by considering the new first-
order effects of delay time ($\Delta t_{m_{\gamma}}^{\prime}\propto
m_{\gamma}^{2}\nu^{-4}$) that are caused by the combination of a nonzero
photon mass and the plasma medium.

The delay time caused by the nonzero photon mass is an intrinsic
characteristic of light that should be preserved in the dedispersed pulse of a
radio source across high and low frequencies. Additionally, the geometric
contribution to the delay time ($\Delta t_{\rm g}$), which is induced by the
frequency-dependent propagation path, is proportional to $\nu^{-4}$ (Er et
al., 2020; Wang et al., 2022) and changes over time (Cordes et al., 2017; Wang
et al., 2022), can significantly affect the dedispersed pulse, thus masking
the delay time caused by the non-zero photon mass and leading to an increased
upper limit of the photon mass. To reduce the geometric effects, two types of
radio sources are observed to set an upper limit on the photon mass. Radio
sources, such as Galactic pulsars and extragalactic FRBs, can be used to
detect the delay time caused by the nonzero photon mass. The delay time of
each dedispersed pulse between high and low frequencies in Galactic pulsars
may be preserved in the time-independent white noise, which can be identified
through the use of high-precision pulsar timing data and reflected in the time
of arrival (TOA) uncertainties of pulsars. This delay time is mainly affected
by the properties of photons with a nonzero mass and is significantly
increased when radio pulses experience minimal scattering effects and the
propagation path of the cosmological distance. Therefore, FRBs are also a
suitable source for testing the photon mass.

Agazie et al. (2023) reported the results of pulsar timing using data from
various radio telescopes and receivers. However, the bandwidths at the center
frequency were found to be nonuniform, which could lead to larger TOA and
dispersion measure (DM) uncertainties. To address this issue, Curyło et al.
(2023) proposed wideband timing measurements to simultaneously estimate TOAs
and DMs, along with their uncertainties and initial noise analysis. This
method can provide comprehensive information on dispersive delay and different
sources of timing noise, as well as system-independent white noise.
Furthermore, Macquart et al. (2020) highlighted the importance of studying
localized FRBs to differentiate the DM contributions from the intergalactic
medium and the host galaxy.

This paper will explore the upper limit of the photon mass by using data from
millisecond pulsars in the Milky Way, which were observed with the Ultrawide
Bandwidth receiver by the Parkes Pulsar Timing Array (PPTA; Curyło et al.
2023. Additionally, the observed results of dedispersed radio pulses from
certain localized FRBs, as provided by Day et al. (2020), will be
incorporated. The paper is structured as follows. Section 2 will present the
theoretical framework that takes into account the nonzero photon mass and
plasma effects, from which the limits of $m_{\gamma}$ can be determined.
Section 3 will analyze the pulsar timing data of millisecond pulsars and
dedispersed radio pulses from certain localized FRBs to set an upper limit for
the photon mass. Finally, Section 4 will discuss and conclude the findings.

##  2 Theoretical Framework

###  2.1 Dispersion relation combined with the nonzero photon mass and plasma
effects

Previous research has studied the delay time caused by a nonzero photon mass
as a separate factor (Wu et al., 2016; Shao & Zhang, 2017; Wei & Wu, 2021).
The classical dispersion relation is derived from the alteration in the speed
of light due to plasma, assuming that the photon has zeromass (Wu et al.,
2016; Shao & Zhang, 2017). Therefore, the total delay time is usually
calculated by adding the delay times caused by the two independent
dispersions. This implies that it is difficult to differentiate between the
two effects and to obtain the maximum limit of the photon mass through direct
observations (Shao & Zhang, 2017). We looked into how radio signals with a
nonzero photon mass are affected by the plasma in space. This will be
discussed in the following.

The speed of light depends on the frequency, expressed as $\upsilon_{\rm
g}^{\gamma}=c\sqrt{1-A\nu^{-2}}$, where $A=m_{\gamma}^{2}c^{4}/h^{2}$ and $h$
is the Planck constant. When comparing the group velocities of radio signals
at different frequencies with the speed of light in a vacuum, it is essential
to take into account the combined effects of plasma and the nonzero photon
mass, rather than treating them as distinct entities. Therefore, the group
velocity can be expressed as

$\displaystyle\upsilon_{\rm g}^{\rm new}=\upsilon_{\rm g}^{\gamma}n_{\rm
ref},$ (1)

where $n_{\rm ref}$ is the index of refraction and given as (Stix, 1992)

$\displaystyle n_{\rm ref}=\left(1-\frac{\nu_{\rm
p}^{2}}{\nu^{2}}\right)^{1/2},$ (2)

where $\nu_{\rm p}=(n_{\rm e}e^{2}/\pi m_{\rm e})^{1/2}$ is the plasma
frequency, $n_{\rm e}$ represents the electron number density of the plasma
medium along the line of sight, $e$ and $m_{\rm e}$ denote the charge and mass
of an electron, respectively. Therefore, the observed delay time should be
$\Delta t_{\rm obs}=\Delta t_{\rm d}+\Delta t_{m_{\gamma}}^{\prime}$ instead
of $\Delta t_{\rm obs}=\Delta t_{\rm d}+\Delta t_{{m_{\gamma}}}$ ($\Delta
t_{{m_{\gamma}}}\propto
a_{1}m_{\gamma}^{2}\nu^{-2}+a_{2}m_{\gamma}^{4}\nu^{-4}$).

From the classical dispersion relation, the dispersive delay is derived by the
comparison between the time of monochrome light in the propagation of plasma
medium and vacuum (Lorimer & Kramer, 2012), so it is reasonable for the zero
mass photon that the dispersive delay difference is caused due to different
group velocities of light between the low and high frequencies in the plasma
medium. However, if the photon had mass, the group velocity of the photon in
vacuum would be $\upsilon_{\rm g}^{\gamma}$ rather than a constant $c$. Thus,
assuming that two massive photons with different frequencies ($\nu_{\rm
low},\nu_{\rm high}$; $\nu_{\rm low}<\nu_{\rm high}$) are emitted at the same
time from a radio source in the Milky Way and travel a distance of $d$ to
reach the observer, the dispersive delay between the two photons (or two radio
signals) can be expressed as

$\displaystyle\Delta t_{\rm d}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{d(z_{\rm d})}\left(\frac{1}{\upsilon_{\rm g,low}^{\rm
new}}-\frac{1}{\upsilon_{\rm g,low}^{\gamma}}\right){\rm
d}l-\int_{0}^{d(z_{\rm d})}\left(\frac{1}{\upsilon_{\rm g,high}^{\rm
new}}-\frac{1}{\upsilon_{\rm g,high}^{\gamma}}\right){\rm d}l$
$\displaystyle\simeq$ $\displaystyle\frac{e^{2}}{2\pi m_{\rm
e}c}\left(\frac{1}{\nu_{\rm low}^{2}}-\frac{1}{\nu_{\rm high}^{2}}\right){\rm
DM}+\frac{Ae^{2}}{4\pi m_{\rm e}c}\left(\frac{1}{\nu_{\rm
low}^{4}}-\frac{1}{\nu_{\rm high}^{4}}\right){\rm DM}+O(\nu)$ $\displaystyle=$
$\displaystyle\Delta t_{\rm d}+\Delta t_{m_{\gamma}}^{\prime}+O(\nu),$

where $O(\nu)$ is the high-order delay time, which is not considered in this
case; $\Delta t_{m_{\gamma}}^{\prime}$ is the delay time arises from the
combined effects of plasma and nonzero photon mass and is written as

$\displaystyle\Delta t_{m_{\gamma}}^{\prime}=\frac{Ae^{2}}{4\pi m_{\rm
e}c}\left(\frac{1}{\nu_{\rm low}^{4}}-\frac{1}{\nu_{\rm high}^{4}}\right){\rm
DM}.$ (4)

For the radio pulses from the extragalactic source at redshift $z_{\rm d}$,
the dispersive delay, taking into account the results given by Deng & Zhang
(2014), can be expressed as

$\displaystyle\Delta t_{\rm d}^{\prime}$ $\displaystyle\simeq$
$\displaystyle\frac{e^{2}}{2\pi m_{\rm e}c}\left(\frac{1}{\nu_{\rm
low}^{2}}-\frac{1}{\nu_{\rm low}^{4}}\right){\rm DM}+\frac{Ae^{2}}{4\pi m_{\rm
e}c}\left(\frac{1}{\nu_{\rm high}^{2}}-\frac{1}{\nu_{\rm high}^{4}}\right)$
$\displaystyle\times\left[{\rm DM_{\rm MW}}+{\rm DM_{\rm
halo}}+\int_{0}^{d(z_{\rm d})}\frac{n_{\rm e}(z^{\prime}){\rm
d}l}{(1+z^{\prime})^{3}}+\frac{\rm DM_{\rm host}}{(1+z_{\rm
d})^{3}}\right]+O(\nu)$ $\displaystyle=$ $\displaystyle\Delta t_{\rm d}+\Delta
t_{m_{\gamma}}^{\prime}+O(\nu),$

where $z^{\prime}$ is the redshift and

$\displaystyle\Delta t_{m_{\gamma}}^{\prime}=\frac{Ae^{2}}{4\pi m_{\rm
e}c}\left(\frac{1}{\nu_{\rm low}^{4}}-\frac{1}{\nu_{\rm high}^{4}}\right){\rm
DM_{\gamma}},$ (6)

where $\rm DM_{\rm MW}$, $\rm DM_{\rm halo}$, and $\rm DM_{\rm host}$ are DMs
contributed by the Milky Way, the Galactic halo, and the host galaxy,
respectively. $n_{\rm e}(z^{\prime})$ represents the electron density from the
Galaxy to the source (or in the intergalactic space) and $\rm DM_{\gamma}$ is
an equivalent DM

$\displaystyle{\rm DM_{\gamma}}={\rm DM_{\rm MW}}+{\rm DM_{\rm
halo}}+\int_{0}^{d(z_{\rm d})}\frac{n_{\rm e}(z^{\prime}){\rm
d}l}{(1+z^{\prime})^{3}}+\frac{\rm DM_{\rm host}}{(1+z_{\rm d})^{3}}.$ (7)

The delay time caused by a nonzero photon mass is proportional to
$m_{\gamma}^{2}\nu^{-4}$, which is different from the results of earlier
studies (Wu et al., 2016; Shao & Zhang, 2017, $\propto
m_{\gamma}^{2}\nu^{-2}$). Compared to Equation (4), Equation (6) implies that
the photon mass constrained by extragalactic transient radio sources may be
more uncertain, due to factors such as ${\rm DM_{\rm halo}}$, $n_{\rm
e}(z^{\prime})$, and $\rm DM_{\rm host}$ (Prochaska & Zheng, 2019; Macquart et
al., 2020; Cook et al., 2023). The Milky Way’s ${\rm DM_{\rm MW}}$ can be
estimated using the NE2001 (Cordes & Lazio, 2002) or YMW16 model (Yao et al.,
2017), though some uncertainty is introduced due to the uncertain electron
density models within the Milky Way. Therefore, the photon mass would be more
accurately determined by radio sources in the Galaxy than by radio pulses from
extragalactic distances unless the uncertainties, such as $\rm DM_{\rm halo}$,
$n_{\rm e}(z^{\prime})$, and $\rm DM_{\rm host}$, are taken into account.

###  2.2 Residual Time delay caused by nonzero photon mass

The total time delay of two photons with different frequencies can be
attributed to several components, such as the geometric delay $\Delta t_{\rm
g}\propto\nu^{-4}$ caused by the geometry and radiation processes of the radio
source itself (Wang et al., 2019; Tuntsov et al., 2021), the effects of plasma
lensing in the propagation path (Cordes et al., 2016; Er et al., 2020; Wang et
al., 2022, 2023b), and an unknown associated delay time $\Delta t_{\rm sys}$
with the observing systems of pulsars (Shannon et al., 2014) or FRBs (Gajjar
et al., 2018). This can be expressed as

$\Delta t_{\rm obs}=\Delta t_{\rm g}+\Delta t_{\rm d}+\Delta
t_{m_{\gamma}}^{\prime}+\Delta t_{\rm sys}$ (8)

For a pulsar located in the Milky Way, the uncertainties in TOA ($\sigma_{\rm
TOA}$) due to the white noise from the observing system and analysis may
include the delay time caused by nonzero photon mass and $\Delta t_{\rm sys}$.
The geometric delay is possibly variable with the time due to the variations
of DM and hidden in the red noise of the pulsar timing data. Therefore, the
rest delay time is calculated as

$\Delta t_{\rm rest}^{\rm Gal}=\Delta t_{\rm obs}-\Delta t_{\rm g}-\Delta
t_{\rm d}=\Delta t_{m_{\gamma}}^{\prime}+\Delta t_{\rm sys}$ (9)

Assuming $\Delta t_{\rm sys}>0$, then $\Delta t_{m_{\gamma}}^{\prime}=\Delta
t_{\rm rest}^{\rm Gal}-\Delta t_{\rm sys}\leq\Delta t_{\rm rest}^{\rm Gal}$,
which sets an upper boundary on the photon mass

$\displaystyle m_{\gamma}\leq hc^{-2}\left[\frac{4\pi m_{\rm
e}c}{e^{2}}\frac{\Delta t_{\rm rest}^{\rm Gal}}{\left(\nu_{\rm
low}^{-4}-\nu_{\rm high}^{-4}\right){\rm DM}}\right]^{1/2},$ (10)

which can be simplified as

$\displaystyle m_{\gamma}\leq(1.62\times 10^{-43}\,{\rm kg})\left[\frac{\Delta
t_{\rm rest,\mu s}^{\rm Gal}}{\left(\nu_{\rm low,GHz}^{-4}-\nu_{\rm
high,GHz}^{-4}\right){\rm DM}_{\rm pc\,cm^{-3}}}\right]^{1/2},$ (11)

where $\Delta t_{\rm rest,\mu s}^{\rm Gal}$ is normalized with $1\rm\,\mu s$,
$\nu_{\rm low,GHz}$ and $\nu_{\rm high,GHz}$ are normalized with $\rm{1GHz}$,
${\rm DM}_{\rm pc\,cm^{-3}}$ is the normalized DM with ${1\,\rm pc\,cm^{-3}}$.

For a radio transient source from a cosmic distance with redshift $z_{\rm d}$,
the residual delay time of the signal between high and low frequencies after
the single pulse has been dedispersed is represented by both $\Delta t_{\rm
g}$ and $\Delta t_{m_{\gamma}}^{\prime}$. The rest delay time should be

$\displaystyle\Delta t_{\rm rest}^{\rm extr}$ $\displaystyle=$
$\displaystyle\Delta t_{\rm obs}-\Delta t_{\rm d}$ (12) $\displaystyle=$
$\displaystyle\Delta t_{m_{\gamma}}^{\prime}+\Delta t_{\rm g}+\Delta t_{\rm
sys}.$

If $\Delta t_{\rm g}+\Delta t_{\rm sys}>0$ and the source has a relatively
clean line-of-sight path, then $\Delta t_{m_{\gamma}}^{\prime}=\Delta t_{\rm
rest}^{\rm extr}-\Delta t_{\rm g}-\Delta t_{\rm sys}\leq\Delta t_{\rm
rest}^{\rm extr}$. We can simplify calculations by approximating the effective
DM from Equation (7) as follows:

$\displaystyle{\rm DM_{\gamma}}\gtrsim{\rm DM_{\rm MW}}+{\rm DM_{\rm
halo}}+\frac{{\rm DM_{\rm IGM}+DM_{\rm host}/(1+z_{\rm d})}}{(1+z_{\rm
d})^{2}},$ (13)

where ${\rm DM_{\rm IGM}}$ is the DM of intergalactic medium. Thus, the photon
mass is constrained by considering the observed results of the extragalactic
source

$\displaystyle m_{\gamma}\leq hc^{-2}\left\\{\frac{4\pi m_{\rm
e}c}{e^{2}}\frac{\Delta t_{\rm rest}^{\rm extr}}{\left(\nu_{\rm
low}^{-4}-\nu_{\rm high}^{-4}\right)}\left[{\rm DM_{\rm MW}}+{\rm DM_{\rm
halo}}+\frac{{\rm DM_{\rm IGM}+DM_{\rm host}/(1+z_{\rm d})}}{(1+z_{\rm
d})^{2}}\right]^{-1}\right\\}^{1/2},$ (14)

which can be further reduced to

$\displaystyle m_{\gamma}\leq(1.62\times 10^{-43}\,{\rm
kg})\left\\{\frac{\Delta t_{\rm rest,\mu s}^{\rm extr}}{\left(\nu_{\rm
low,GHz}^{-4}-\nu_{\rm high,GHz}^{-4}\right)}\left[{\rm DM_{\rm MW}}+{\rm
DM_{\rm halo}}+\frac{{\rm DM_{\rm IGM}+DM_{\rm host}/(1+z_{\rm d})}}{(1+z_{\rm
d})^{2}}\right]^{-1}_{\rm pc\,cm^{-3}}\right\\}^{1/2},$ (15)

where $\Delta t_{\rm rest,\mu s}^{\rm extr}$ is normalized with $1\,\rm\mu s$,
and ${\rm DM_{\rm MW}},{\rm DM_{\rm halo}},{\rm DM_{\rm IGM}},{\rm DM_{\rm
host}}$ are normalized with ${\rm 1pc\,cm^{-3}}$. Since the numerator of the
third term on the right side of Equation (13) can be easily estimated by $\rm
DM_{\rm tot}-DM_{\rm MW}-DM_{\rm halo}$, where $\rm DM_{\rm tot}$ is the total
DM of a FRB. This method by comparing to the previous works is not affected by
the uncertainties of $\rm DM_{\rm host}$ and $\rm DM_{\rm IGM}$ (Zhang et al.,
2020, 2021).

##  3 Photon mass limits

Generally, the discrepancies between the timing model and the observations of
a pulsar’s TOA can be seen as two types of timing noise: red and white noise
(Lorimer & Kramer, 2012). The former may include time-dependent effects, such
as pulsar intrinsic spin noise, variations in DM, gravitational waves, and so
on. On the other hand, the latter does not depend on the observed time (Curyło
et al., 2023). The delay time caused by nonzero photon mass may be present in
the white noise of pulsar timing data, which is reflected in the TOA
uncertainties.

FRBs are bright, extragalactic, transient radio pulses with a duration of a
few milliseconds. The TOA of bursts at different frequencies may show a slight
deviation from the classical dispersion relation (Thornton et al., 2013). This
discrepancy could potentially be caused by the nonzero photon mass.
Consequently, FRBs can be employed to limit the photon mass.

###  3.1 Timing data of 22 millisecond pulsars

Recently, Curyło et al. (2023) reported the timing results of 35 millisecond
pulsars that were observed using a UWB low-frequency receiver installed on the
64 m Parkes radio telescope. This receiver provided instantaneous frequency
coverage in the range of 0.704-4.032$\rm\,GHz$. However, some pulsars that had
either no or only one eigenprofile had a low flux measured above
2.5$\rm\,GHz$, and their bands above $3-3.5\,\rm GHz$ were excluded (Curyło et
al., 2023). Therefore, we chose data of 22 millisecond pulsars with more than
two high signal-to-noise ratio (S/N) eigenprofiles as reported by Curyło et
al. (2023). The data in Table 2 include the DM and the median uncertainties of
the TOA ($\sigma_{\rm TOA}$) which can be thought of as the results of white
noise. It is clear that the PPTA observations are highly precise, with an
uncertainty of TOA as low as $10\,\rm ns$ for PSR J1939$+$2134\. With the
timing results of all millisecond pulsars in Table 2 and $\Delta t_{\rm
rest}^{\rm Gal}=\sigma_{\rm TOA}$, the upper limits of the photon mass, as
given by Equation (11), are provided in the last column of Table 2. The limits
of the photon mass range from $2.48\times 10^{-44}\,\rm kg$ (or $1.39\times
10^{-8}\,\rm eV/c^{2}$) to $9.52\times 10^{-46}\,\rm kg$ (or $5.34\times
10^{-10}\,\rm eV/c^{2}$). Since the nonzero photon mass is an inherent
property, there exists a strict upper limit for its mass

$\displaystyle m_{\gamma}\leq 9.52\times 10^{-46}\,\rm kg\simeq 5.34\times
10^{-10}\,\rm eV/c^{2}.$ (16)

The observing strategy and precision of the radio telescope, the frequency-
dependent delay caused by the nonzero photon mass and the cold plasma medium,
and the fluctuations of the interstellar medium can all have an effect on the
DM perturbations of radio pulses (Cordes & Shannon, 2010; Curyło et al.,
2023). These factors can lead to an uncertainty in the DM of a radio pulse. We
assume that the DM uncertainty is mainly due to the nonzero photon mass, and
thus Equation (11) can be used to calculate the upper limit of the photon
mass. Curyło et al. 2023 reported the median uncertainties of the DM
($\sigma_{\rm DM}$) of millisecond pulsars at the epoch length of the timing
observations, which are presented in Table 2. It was found that the limits on
the photon mass are slightly higher when the median uncertainties of the DM of
pulsars are taken into account, compared to those calculated using the median
uncertainties of the TOA. The best result from the example of PSR J1939$+$2134
yielded an upper limit of $m_{\gamma}\leq 1.92\times 10^{-45}\rm\,kg\simeq
1.08\times 10^{-9}\,eV/{c^{2}}$. It is clear that the uncertainties of the DM
have a moderate effect on the limits of the photon mass.

Table 2: The data obtained from PPTA (Curyło et al., 2023) and the calculated
photon mass in this work (last two column). The PPTA, using a UWB receiver
($0.704-4.032\rm\,GHz$), obtained timing results (${\rm DM,median\ \sigma_{\rm
TOA},median\ \sigma_{\rm DM}}$) for 22 millisecond pulsars with more than two
high-S/N edge profiles (Curyło et al., 2023).

No. | Pulsars | DM | Med. $\sigma_{\rm TOA}$ | Med. $\sigma_{\rm DM}$ | $m_{\gamma}^{\rm TOA}$ | $m_{\gamma}^{\rm DM}$
---|---|---|---|---|---|---
|  | ($\rm pc\,cm^{-3}$) | ($\mu$s) | ($\times 10^{-4}\,\rm pc\,cm^{-3}$) | (kg) | (kg)
01 | PSR J0613$-$0200 | 38.78 | 0.158 | 0.807 | $5.12\times 10^{-45}$ | $1.04\times 10^{-44}$
02 | PSR J0711$-$6830 | 18.41 | 0.550 | 3.211 | $1.39\times 10^{-44}$ | $3.02\times 10^{-44}$
03 | PSR J0900$-$3144 | 75.61 | 0.612 | 3.421 | $7.22\times 10^{-45}$ | $1.54\times 10^{-44}$
04 | PSR J1017$-$7156 | 94.22 | 0.120 | 0.684 | $2.86\times 10^{-45}$ | $6.16\times 10^{-45}$
05 | PSR J1024$-$0719 | 6.48 | 0.619 | 4.105 | $2.48\times 10^{-44}$ | $5.76\times 10^{-44}$
06 | PSR J1045$-$4509 | 58.15 | 0.781 | 4.173 | $9.30\times 10^{-45}$ | $1.94\times 10^{-44}$
07 | PSR J1125$-$6014 | 52.93 | 0.119 | 0.744 | $3.81\times 10^{-45}$ | $8.57\times 10^{-45}$
08 | PSR J1545$-$4550 | 68.39 | 0.212 | 2.144 | $4.47\times 10^{-45}$ | $1.28\times 10^{-44}$
09 | PSR J1600$-$3053 | 52.33 | 0.113 | 0.832 | $3.73\times 10^{-45}$ | $9.12\times 10^{-45}$
10 | PSR J1603$-$7202 | 38.04 | 0.361 | 2.129 | $7.82\times 10^{-45}$ | $1.71\times 10^{-44}$
11 | PSR J1643$-$1224 | 62.41 | 0.286 | 1.459 | $5.43\times 10^{-45}$ | $1.11\times 10^{-44}$
12 | PSR J1713$+$0747 | 15.92 | 0.048 | 0.384 | $4.41\times 10^{-45}$ | $1.12\times 10^{-44}$
13 | PSR J1730$-$2304 | 9.62 | 0.287 | 1.677 | $1.39\times 10^{-44}$ | $3.02\times 10^{-44}$
14 | PSR J1744$-$1134 | 3.14 | 0.080 | 0.524 | $1.28\times 10^{-44}$ | $2.95\times 10^{-44}$
15 | PSR J1857$+$0943 | 13.30 | 0.257 | 2.189 | $1.12\times 10^{-44}$ | $2.93\times 10^{-44}$
16 | PSR J1939$+$2134 | 71.01 | 0.010 | 0.050 | $9.52\times 10^{-46}$ | $1.92\times 10^{-45}$
17 | PSR J2051$-$0827 | 20.73 | 0.680 | 5.657 | $1.45\times 10^{-45}$ | $3.78\times 10^{-43}$
18 | PSR J2145$-$0750 | 9.00 | 0.186 | 1.179 | $1.15\times 10^{-44}$ | $2.62\times 10^{-44}$
19 | PSR J2241$-$5236 | 11.41 | 0.070 | 0.465 | $6.29\times 10^{-45}$ | $1.46\times 10^{-44}$
20 | PSR J0125$-$2327 | 9.60 | 0.129 | 0.974 | $9.30\times 10^{-45}$ | $2.30\times 10^{-44}$
21 | PSR J1022$+$1001 | 10.25 | 0.254 | 1.517 | $1.26\times 10^{-44}$ | $2.78\times 10^{-44}$
22 | PSR J0437$-$4715 | 2.64 | 0.005 | 0.043 | $3.49\times 10^{-45}$ | $9.23\times 10^{-45}$

  1. Note. The upper limit of the photon mass, $m_{\gamma}^{\rm TOA}$ and $m_{\gamma}^{\rm DM}$, was calculated from the median $\sigma_{\rm TOA}$ and $\sigma_{\rm DM}$, respectively.

###  3.2 Dedispersed Pulses of three localized FRBs

FRBs are divided into two categories: one-off events (nonrepeaters) and
repeating bursts (repeaters). More than 20 FRBs have been identified, along
with their locations and host galaxies (Macquart et al., 2020; Niu et al.,
2022; Caleb et al., 2023; Driessen et al., 2023). Repeaters have much higher
DM in their host galaxy than in intergalactic space (Niu et al., 2022; Wu et
al., 2024). The degree of linear polarization for repeaters is significantly
dependent on frequency (Feng et al., 2022; Wang et al., 2022a). These studies
suggest that bursts from a repeater at different frequencies may come from
different propagation paths, which can be attributed to the presence of an
inhomogeneous plasma medium (Er et al., 2020; Feng et al., 2022; Wang et al.,
2022, 2023b; Yang, 2023). This phenomenon leads to a considerable geometric
delay in the dedispersion of bursts between low and high frequencies (Er et
al., 2020; Wang et al., 2022). Furthermore, there is no obvious evidence that
nonrepeaters have spectral depolarizations (Uttarkar et al., 2023). Therefore,
we select nonrepeaters with a scattering timescale lower than $1\,\rm ms$ at
$1\,\rm GHz$ to reduce the geometric effects in the propagation path (Er et
al., 2020), and neglect the geometric delay.

FRBs can be dedispersed in two ways: by maximizing the average S/N across the
band to reach its peak value, or by optimizing the alignment of the frequency-
time structure (Gajjar et al., 2018). The difference in the DM between the two
methods ($\Delta\rm DM$) is due to the delay time of
$t_{m_{\gamma}}^{\prime}\propto\nu^{-4}$ caused by the existence of a nonzero
photon mass. So the rest delay time is taken as $\Delta t^{\rm extr}_{\rm
rest}\approx 4.15\,{\rm ms}\times\Delta\rm DM(\nu^{-2}_{\rm
low,GHz}-\nu^{-2}_{\rm high,GHz})$. Table 3 presents the data of three non-
repeaters detected by the Australian Square Kilometer Array Pathfinder (ASKAP;
$1.1045-1.4405\,\rm GHz$), including the names of FRBs, the DMs from the de-
dispersed method of the optimized alignment of frequency-time structure,
$\Delta\rm DM$, and $\rm DM_{\rm MW}$ from the electron density model of
NE2001 (Cordes & Lazio, 2002) or YMW16 (Yao et al., 2017). The spectrum of
three FRBs appears in all bands of ASKAP. Furthermore, the observation of FRB
in M81 shows $\rm DM_{\rm halo}\lesssim 32-42\,\rm pc\,cm^{-3}$ (Kirsten et
al., 2022). CHIME/FRB detected a nearby FRB with DM of no more than 250 pc
cm-3, leading to the $\rm DM_{\rm halo}$ range of 52-111 pc cm-3 (Cook et al.,
2023). X-ray observations give $\rm DM_{\rm halo}\lesssim 10\,\rm pc\,cm^{-3}$
(Keating & Pen, 2020). Thus, we take the possible $\rm DM_{\rm halo}$ in the
range of 0-111 pc cm-3. By combining the parameters in Table 3, $\rm DM_{\rm
halo}$, and Equation (15), the upper limits of the photon mass can be
calculated

$\displaystyle m_{\gamma}\leq(4.52-4.92)\times 10^{-44}\,\rm
kg\simeq(2.54-2.76)\times 10^{-8}\,\rm eV/c^{2}$ (17)

for the NE2001 model, and

$\displaystyle m_{\gamma}\leq(4.70-5.16)\times 10^{-44}\,\rm
kg\simeq(2.64-2.90)\times 10^{-8}\,\rm eV/c^{2}$ (18)

for the YMW16 model. The photon mass can only be estimated with the reliable
electron density models from the Milky Way and the potential contributions of
different types of DM from the Galactic halo. This makes it difficult to
accurately determine the photon mass using extragalactic sources.

Table 3: The data of three FRBs observed by ASKAP (Day et al., 2020) and the
calculated photon mass in this work (last two column). The parameters
$\rm{z_{\rm d},DM,\Delta DM}$ of three FRBs were obtained from observations
made by ASKAP (Day et al., 2020).

Source | $z_{\rm d}$ | DM | $\Delta\rm DM$ | $\rm DM_{\rm MW}^{N}$ | $\rm DM_{\rm MW}^{Y}$ | $m_{\gamma}^{\rm N}$ | $m_{\gamma}^{\rm Y}$
---|---|---|---|---|---|---|---
|  | ($\rm pc\,cm^{-3}$) | ($\rm pc\,cm^{-3}$) | ($\rm pc\,cm^{-3}$) | ($\rm pc\,cm^{-3}$) | ($\times 10^{-44}\,$kg) | ($\times 10^{-44}\,$kg)
FRB 180924 | 0.3212 | 362.16 | 0.04 | 40.5 | 27.65 | $10.08-12.19$ | $11.19-12.34$
FRB 190102 | 0.2912 | 364.545 | 0.007 | 57.3 | 3.48 | $4.52-4.92$ | $4.70-5.16$
FRB 190611 | 0.3778 | 332.63 | 0.03 | 57.83 | 3.07 | $9.91-11.12$ | $10.46-11.91$

  1. Note. The upper limits of the photon mass, denoted by the superscripts ‘N’ and ‘Y’, were calculated by the NE2001 (Cordes & Lazio, 2002) and YMW16 (Yao et al., 2017) models, respectively.

##  4 Conclusions and discussion

In this paper, we present a revised dispersion relation for photons with
nonzero mass as they travel through a plasma medium. This updated relation,
denoted as $t_{m_{\gamma}}^{\prime}\propto m_{\gamma}^{2}\nu^{-4}$, is
different from the classical dispersion relation ($t_{\rm d}\propto\nu^{-2}$)
and the previously established delay time proportional to $\propto
m_{\gamma}^{2}\nu^{-2}$. The intrinsic delay time of light,
$t_{m_{\gamma}}^{\prime}$, is independent of the observed time and can be
observed as DM/TOA uncertainties in pulsar timing data or as dedispersed radio
pulses across a range of frequencies. In this study, we used high-precision
pulsar timing data from the UWB receiver of PPTA and the dedispersed pulse of
the FRB with minimal scattering effects to determine the upper limit of photon
mass. The results of this work are listed below.

  * •

From Pulsar timing. The median uncertainties of the TOA from 22 millisecond
pulsars were used to calculate the upper limit of the photon mass, which is
$m_{\gamma}\leq 9.52\times 10^{-46}\,\rm kg$ ($m_{\gamma}\leq 5.34\times
10^{-10}\,\rm eV/c^{2}$).

  * •

From FRB. The upper limit of the photon mass was determined by considering the
time delay between high and low frequencies for the dedispersed radio pulses
of three localized FRBs. The calculated upper limit is
$m_{\gamma}\leq(4.52-5.16)\times 10^{-44}\,\rm kg$
($m_{\gamma}\leq(2.54-2.90)\times 10^{-8}\,\rm eV/c^{2}$).

It should be noted that the pulsar timing data collected in the frequency
range of 0.704$-$4.032$\rm\,GHz$ provide a more stringent constraint on the
value of $m_{\gamma}$ than the delay time difference of the FRBs observed
between 1.1045 and 1.4405$\,\rm GHz$. This increased precision is due to the
fact that the dispersion relation caused by the presence of a photon mass is a
higher-order term that depends on frequency. Thus, even after applying the
classical dispersion relation to remove dispersion effects, the delay time
associated with the photon mass can still be preserved in a radio pulse. This
delay time is especially prominent at frequencies below $1\,\rm{GHz}$.
Consequently, it is plausible to suggest that the upper limit of the photon
mass can be more accurately determined by analyzing data obtained from
observations conducted at frequencies below $1\,\rm{GHz}$.

We have given an upper bound for the photon mass in our investigation through
a newly updated formula. This is the first time that the interaction between a
nonzero photon mass and the plasma medium has been taken into account and
calculated as the photon propagates through the plasma medium. Our estimated
limit exceeds the limits set in prior studies ($m_{\gamma}\leq 10^{-47}\,\rm
kg$ or $10^{-51}\,\rm kg$), which used the second-order dispersion relation or
the first-order dispersion relation, by approximately 2 or 5 orders of
magnitude, respectively. It is worth noting that these earlier limits were
determined under the assumption of a photon with zero mass traveling through
the plasma medium. Therefore, it may be necessary to revisit these limits once
more by incorporating the factors we have developed in this work.

The upper limit of the photon mass given in our work is not much lower than
the one given by Chang et al. 2023. This could be due to recent studies that
provide only the median TOA uncertainties from long-term pulsar timing data,
without considering the minimum white noise level of the pulsar (Agazie et
al., 2023; Curyło et al., 2023). Additionally, the use of UWB receivers has
been shown to increase the S/N ratio in recorded pulsar profiles, thus
reducing the uncertainty in measuring pulse TOA (Liu et al., 2014). Comparison
of noise models from narrowband and wideband datasets can help distinguish
between different sources of noise (Alam et al., 2021; Agazie et al., 2023;
Curyło et al., 2023). Taking advantage of highly sensitive radio facilities
such as the Five-hundred-meter Aperture Spherical Radio Telescope (Jiang et
al., 2019) and the Qitai Radio Telescope (Wang et al., 2023a), along with the
implementation of UWB receivers, there is potential for further improvement in
determining the upper limit of photon mass in the near future.

Although we take these nonrepeaters with short scattering timescales in order
to reduce the geometric effects acting on the TOA of bursts, the chosen FRBs
still provide an upper limit for the photon mass that is about ten times
higher than the values obtained from ultrawideband pulsar timing data. This
suggests that some physical mechanisms around the source or in the propagation
pathway may be retained in the dedispersed pulses of bursts yet. According to
the observed results, the spectra of many nonrepeaters have a broader range
than the full CHIME/FRB band ($400-800\rm\,MHz$), implying the improved TOA
accuracy of nonrepeater detections by using multiple facilities simultaneously
or UWB receivers (Pleunis et al., 2021). In addition, the effecitive DM ($\rm
DM_{\gamma}$) included the term of photon mass is an approximate value here.
In future studies, the upper limit of the photon mass may be revised by the
dedispersed pulses of nonrepeaters if the contributions of these physical
mechanisms on the TOA of bursts is eliminated and the estimated method of $\rm
DM_{\gamma}$ is improved.

##  5 Acknowledgments

We thank the anonymous referee for value suggestions that can allow us to
improve our manuscript significantly. This work is supported in part by the
National Key R&D Program of China (No. 2022YFC2205202), the Natural Science
Foundation of Xinjiang Uygur Autonomous Region (No. 2023D01E20, 2022D01A363),
the National Natural Science Foundation of China (No. 12033001, 12288102,
12273028, 12273009), the Sichuan Science and Technology Program (No.
24QYCX0261, 2023YFG0024, 4GJHZ0046), the Sichuan University of Science &
Engineering Program (No. 2022RC07), the Opening Foundation of Xinjiang Key
Laboratory (No. 2021D04016), the Major Science and Technology Program of
Xinjiang Uygur Autonomous Region (No. 2022A03013-1, 2022A03013-4), and the
Urumqi Nanshan Astronomy and Deep Space Exploration Observation and Research
Station of Xinjiang (XJYWZ2303). XZ would like to thank Qing-min Li for her
fruitful suggestions.

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prd}^{1,\rm{op}}\big{\\}}$ is bounded in probability according to [9, Th.
2.1]. Suppose, aiming for a contradiction, that it is not the case. Then there
exists an $\epsilon>0$ and a sequence
$(\Gamma_{n})_{n\in\mathbb{N}}\subseteq\mathcal{S}_{\rm prd}^{1,\rm{op}}$
satisfying for all $n\in\mathbb{N}$ that

$\displaystyle P\left(\left\lVert\int_{0}^{T}\Gamma_{n}\,{\rm
d}I(\Psi)\right\rVert>n\right)\geq\epsilon.$ (7.18)

For each $\Psi\in\mathcal{S}_{\rm prd}^{\rm HS}$ and
$\Gamma\in\mathcal{S}_{\rm prd}^{1,\rm{op}}$, the very definitions of
stochastic integrals show

$\int_{0}^{T}\Gamma\,{\rm d}I(\Psi)=\int_{0}^{T}\Gamma\Psi\,{\rm d}L.$

This equality can be generalised to arbitrary $\Psi\in\mathcal{I}_{{\rm
prd},L}^{\rm HS}$ and $\Gamma\in\mathcal{S}_{\rm prd}^{1,\rm{op}}$ by a
standard approximation argument. Using this to rewrite Equation (7.18), we
obtain for all $n\in\mathbb{N}$ that

$\displaystyle\epsilon\leq P\left(\left\lVert\int_{0}^{T}\Gamma_{n}\,{\rm
d}I(\Psi)\right\rVert>n\right)=P\left(\left\lVert\int_{0}^{T}\frac{1}{n}\Gamma_{n}\Psi\,{\rm
d}L\right\rVert>1\right).$ (7.19)

On the other hand, since
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\frac{1}{n}\Gamma_{n}\Psi\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L}\rightarrow
0$ as $n\rightarrow\infty$, Corollary 7.3 implies

$\displaystyle\lim_{n\rightarrow\infty}E\left[\left\lVert\int_{0}^{T}\frac{1}{n}\Gamma_{n}\Psi\,{\rm
d}L\right\rVert\wedge 1\right]=0,$

which contradicts (7.19) because of the equivalent characterisation of the
topology in $L^{0}_{P}(\Omega,H)$. ∎

We finish this section with a stochastic dominated convergence theorem.

######  Theorem 7.5.

Let $(\Psi_{n})_{n\in\mathbb{N}}$ be a sequence of processes in
$\mathcal{I}_{{\rm prd},L}^{\rm HS}$ such that

  1. (1)

$(\Psi_{n})_{n\in\mathbb{N}}$ converges $P_{T}$-a.e. to an $L_{2}(G,H)$-valued
predictable process $\Psi$;

  2. (2)

there exists a process $\Upsilon\in\mathcal{I}_{{\rm prd},L}^{\rm HS}$
satisfying for all $n\in\mathbb{N}$ that

$(k_{L}+l_{L})(\Psi_{n}(\omega,t))\leq(k_{L}+l_{L})(\Upsilon(\omega,t))\quad\text{for
$P_{T}$-a.a.\ $(\omega,t)\in\Omega\times[0,T]$.}$

Then it follows that $\Psi\in\mathcal{I}_{{\rm prd},L}^{\rm HS}$ and

$\displaystyle\lim_{n\to\infty}\displaystyle\sup_{\Gamma\in\mathcal{S}_{{\rm
prd}}^{1,{\rm op}}}E\Bigg{[}\left\lVert\int_{0}^{T}\Gamma(\Psi_{n}-\Psi)\;{\rm
d}L\right\rVert\wedge 1\Bigg{]}=0.$

###### Proof.

By assumption, there exists a set $N\subseteq\Omega\times[0,T]$ with
$P_{T}(N)=0$ such that
$\lim_{n\rightarrow\infty}\Psi_{n}(\omega,t)=\Psi(\omega,t)$ and
$(k_{L}+l_{L})(\Psi_{n}(\omega,t))\leq(k_{L}+l_{L})(\Upsilon(\omega,t))$ for
all $(\omega,t)\in N^{c}$ and $n\in\mathbb{N}$. Fubini’s theorem yields that

$0=P_{T}(N)=\int_{\Omega}{\rm Leb}|_{[0,T]}(N_{\omega})\,P({\rm d}\omega),$

where

$\displaystyle
N_{\omega}\\!:=\\!\left\\{t\in[0,T]:\lim_{n\rightarrow\infty}\Psi_{n}(\omega,t)\neq\Psi(\omega,t)\,\text{or}\,(k_{L}+l_{L})(\Psi_{n}(\omega,t))>(k_{L}+l_{L})(\Upsilon(\omega,t))\right\\}.$

It follows that there exists an $\Omega_{1}\subseteq\Omega$ with
$P(\Omega_{1})=1$ such that ${\rm Leb}|_{[0,T]}(N_{\omega})=0$ for all
$\omega\in\Omega_{1}$. Consequently, for each $\omega\in\Omega_{1}$ we have
$(k_{L}+l_{L})(\Psi_{n}(\omega,t))\leq(k_{L}+l_{L})(\Upsilon(\omega,t))$ and
$\lim_{n\rightarrow\infty}\Psi_{n}(\omega,t)=\Psi(\omega,t)$ for Lebesgue
almost every $t\in[0,T]$. Theorem 7.1 guarantees that there exists
$\Omega_{2}\subseteq\Omega$ with $P(\Omega_{2})=1$ such that
$m_{L}(\Upsilon(\omega,\cdot))<\infty$ for all $\omega\in\Omega_{2}$.
Continuity of $k_{L}$ and $l_{L}$ at $0$, see Lemma 3.12, and the classical
version of Lebesgue’s dominated convergence theorem implies that for all
$\omega\in\Omega_{1}\cap\Omega_{2}$ we have

$\displaystyle\lim_{m,n\rightarrow\infty}m_{L}(\Psi_{m}(\omega,\cdot)-\Psi_{n}(\omega,\cdot))$
$\displaystyle\qquad=\lim_{m,n\rightarrow\infty}\Bigg{(}\int_{0}^{T}k_{L}(\Psi_{m}(\omega,t)-\Psi_{n}(\omega,t))+l_{L}(\Psi_{m}(\omega,t)-\Psi_{n}(\omega,t))\,{\rm
d}t$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{0}^{T}\left\lVert\Psi_{m}(\omega,t)-\Psi_{n}(\omega,t)\right\rVert_{L_{2}(G,H)}^{2}\wedge
1\,{\rm d}t\Bigg{)}=0.$

Hence, for each $\omega\in\Omega_{1}\cap\Omega_{2}$ the sequence
$(\Psi_{n})_{n\in\mathbb{N}}$ is Cauchy in the modular topology, which by
Lemma 3.20 and the fact that $\Psi_{n}(\omega)\rightarrow\Psi(\omega)$ for
Lebesgue a.a. $t\in[0,T]$ allows us to conclude that
$\Psi(\omega)\in\mathcal{M}_{{\rm det},L}^{\rm HS}$. Since
$P(\Omega_{1}\cap\Omega_{2})=1$, Theorem 7.1 shows $\Psi\in\mathcal{I}_{{\rm
prd},L}^{\rm HS}$. Another application of Lebesgue’s dominated convergence
theorem establishes that
$\lim_{n\rightarrow\infty}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\Psi_{n}-\Psi\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L}=0$,
which completes the proof by an application of Corollary 7.3. ∎

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