# Probing Lorentz Invariance Violation with Absorption of Astrophysical Gamma-
rays by Solar Photons

Justin D. Finke<EMAIL_ADDRESS>Parshad Patel<EMAIL_ADDRESS>U.S. Naval Research Laboratory, Code 7653, 4555 Overlook Ave. SW, Washington,
DC 20375-5352, USA

###### Abstract

We compute in detail the absorption optical depth for astrophysical
$\gamma$-ray photons interacting with solar photons to produce electron
positron pairs. This effect is greatest for $\gamma$-ray sources at small
angular distances from the Sun, reaching optical depths as high as
$\tau_{\gamma\gamma}\sim 10^{-2}$. We also calculate this effect including
modifications to the absorption cross section threshold from subluminal
Lorentz invariance violation (LIV). We show for the first time that subluminal
LIV can lead to increases or decreases in $\tau_{\gamma\gamma}$ compared to
the non-LIV case. We show that, at least in principle, LIV can be probed with
this effect with observations of $\gamma$-ray sources near the Sun at $\gtrsim
20\ \mathrm{TeV}$ by HAWC or LHAASO, although a measurement will be extremely
difficult due to the small size of the effect.

##  1 Introduction

Lorentz invariance is one of the fundamental principles of special relativity.
However, the violation of Lorentz invariance has been explored in various
theories beyond the Standard Model, such as string theory, brane worlds, and
loop quantum gravity (e.g., Amelino-Camelia et al., 1998; Kifune, 1999;
Amelino-Camelia & Piran, 2001; Stecker & Glashow, 2001; Mattingly, 2005;
Christiansen et al., 2006; Jacobson et al., 2006; Ellis et al., 2008). In the
presence of Lorentz invariance violation (LIV), the normal relativistic
dispersion relation for photons is modified as

$\displaystyle E^{2}-p^{2}c^{2}=\pm E^{2}\left(\frac{E}{E_{\rm
LIV}}\right)^{n}.$ (1)

Here $E_{\rm LIV}$ is the energy scale, $n$ is an integer (the order of the
leading correction), and “$+$” represents superluminal LIV, and “$-$”
represents subluminal LIV. For LIV brought about by quantum gravity models, it
would be natural for $E_{\rm LIV}$ to be near the Planck Energy, $E_{\rm
Planck}=1.2\times 10^{28}\ \mathrm{eV}$. LIV can lead to a number of
potentially observable effects (for a review see (Mattingly, 2005)), including
a number of astrophysically interesting effects (e.g., Sarkar, 2002; Martínez-
Huerta et al., 2020; Desai, 2023). One is that the speed of light is no longer
constant, and is energy-dependent. Using this effect, time-of-flight
experiments can be used with astrophysical $\gamma$-ray transients (such as
gamma-ray bursts [GRBs] and blazars) to constrain LIV (e.g., Abdo et al.,
2009; Vasileiou et al., 2013; Ellis et al., 2019). The lack of photon decay in
Galactic $\gamma$-ray sources measured with HAWC has led to strong constraints
on superluminal LIV (Albert et al., 2020).

Another effect is the modification of the threshold for pair production from
photon-photon interactions, i.e., the process $\gamma+\gamma\rightarrow
e^{+}+e^{-}$. Subluminal LIV can be constrained with this process with
$\gamma$-rays from extragalactic sources (such as blazars and GRBs), which are
absorbed by ultraviolet through far-infrared photons from the extragalactic
background light (EBL; e.g., Nikishov, 1962; Gould & Schréder, 1967a; Fazio &
Stecker, 1970; Franceschini et al., 2008; Razzaque et al., 2009; Finke et al.,
2010; Kneiske & Dole, 2010; Domínguez et al., 2011; Helgason & Kashlinsky,
2012; Stecker et al., 2012; Scully et al., 2014; Khaire & Srianand, 2015;
Stecker et al., 2016; Franceschini & Rodighiero, 2017; Andrews et al., 2018;
Khaire & Srianand, 2019; Saldana-Lopez et al., 2021; Finke et al., 2022). The
EBL is the integrated background light from all the stars that have existed in
the observable universe, either through direct emission, or through absorption
and re-radiation by dust. Both the EBL and the $\gamma$-ray spectra of
extragalactic sources are rather uncertain. Nevertheless, after the discovery
of photons out to 20 TeV from Mrk 501 by HEGRA (Aharonian et al., 1999), a
number of authors suggested LIV was needed to explain how the $\gamma$-rays at
these high energies could avoid EBL absorption and reach Earth (e.g., Kifune,
1999; Protheroe & Meyer, 2000). Since then, studies of the
$\gamma+\gamma\rightarrow e^{+}+e^{-}$ process between extragalactic
$\gamma$-ray sources and the EBL have been used to constrain LIV (e.g., Biteau
& Williams, 2015; Tavecchio & Bonnoli, 2016; Abdalla et al., 2019; Dzhappuev
et al., 2022; Baktash et al., 2022; Li & Ma, 2023; Finke & Razzaque, 2023).
This will continue with the greatly improved sensitivity of the Cherenkov
Telescope Array (CTA; Abdalla et al., 2021). Constraints have also been made
from the upper limits on ultra-high energy $\gamma$-ray flux measured by the
Auger Observatory (e.g., Guedes Lang et al., 2018). In this manuscript, we
explore only the effect of subluminal LIV on $\gamma\gamma$ absorption.

There are (at least) two formulations of the effects of LIV on the photon-
photon pair production process commonly used in the literature. One is by
Jacob & Piran (2008). In their formulation, the standard model cross section
formula is used, modifying only the threshold energy and the cross-section’s
dependence on this threshold. The other is by Fairbairn et al. (2014). Those
authors define an effective mass that is related to the LIV energy scale, and
use that to define an invariant center of mass energy. The two formulations do
not give equivalent results. A comparison of the two is given by Tavecchio &
Bonnoli (2016).

Another possible application of the $\gamma+\gamma\rightarrow e^{+}+e^{-}$
process to LIV constraints could come from the extinction of astrophysical
$\gamma$-ray photons by photons from the Sun. Preliminary estimates of this
process were done by (Loeb, 2022; Balaji, 2023). Here we explore this effect
in much greater detail, including its application to constraining LIV. As we
will see below, subluminal LIV can make the $\gamma$-ray absorption optical
depth more transparent or more opaque than it would be without LIV, depending
on the photon distribution interacting with the $\gamma$-rays. This may allow
LIV to be constrained with measurements of the $\gamma$-ray sky at energies of
$\gtrsim 20$ TeV, although such measurements will be difficult to make.

In Section 2, we describe our detailed calculations of absorption of
astrophysical $\gamma$-rays by solar photons, both with and without LIV
included. In Section 3 we describe the potential for observing this effect and
how it could be used to constrain LIV. Finally we conclude with a summary in
Section 4.

##  2 Formalism

###  2.1 Setup

Consider an astrophysical $\gamma$-ray photon coming towards the Earth from
outside the solar system. Let the angle at the Earth between the sun and the
$\gamma$-ray photon be $\theta_{E}\equiv\cos^{-1}(\mu_{E})$; the angle at the
Sun between the Earth and the $\gamma$-ray photon is
$\theta_{S}\equiv\cos^{-1}(\mu_{S})$. The distance between the Earth and the
Sun is $a$; the distance between the Sun and the incoming $\gamma$-ray photon
is $x_{S}$; and the distance between the Earth and the $\gamma$-ray photon is
$x_{E}$. See Fig. 1 for an illustration of the geometric setup.

The $\gamma$-ray photon will interact with solar photons, annihilating the
photons and producing electron-positron pairs (i.e., $\gamma+\gamma\rightarrow
e^{+}+e^{-}$). The $\gamma\gamma$ absorption cross section is (e.g., Gould &
Schréder, 1967b; Brown et al., 1973)

$\displaystyle\sigma_{\gamma\gamma}(s)$ $\displaystyle=\frac{1}{2}\pi
r_{e}^{2}(1-\beta_{\rm cm}^{2})\Biggr{[}(3-\beta_{\rm
cm}^{4})\ln\left(\frac{1+\beta_{\rm cm}}{1-\beta_{\rm cm}}\right)$
$\displaystyle-2\beta_{\rm cm}(2-\beta_{\rm cm}^{2})\Biggr{]}$ (2)

where

$\displaystyle\beta_{\rm cm}=\sqrt{1-s^{-1}}\ ;$ (3)

$\sqrt{s}$ is the Lorentz factor of the resulting electron and positron in
their center-of-momentum frame; and $r_{e}=2.817\times 10^{-13}$ cm is the
classical electron radius.

Figure 1:  Geometry of solar system and $\gamma$-ray heading towards Earth.
Distances and angles are labeled.

The $\gamma$-ray absorption optical depth per unit distance is given by (e.g.,
Dermer & Menon, 2009)

$\displaystyle\frac{d\tau_{\gamma\gamma}(\epsilon_{1})}{dx_{E}}$
$\displaystyle=\int^{2\pi}_{0}d\phi\int^{1}_{-1}d\mu(1-\cos\psi)$
$\displaystyle\times\int^{\infty}_{0}d\epsilon\
n(\epsilon,\Omega)\sigma_{\gamma\gamma}\left[\frac{\epsilon\epsilon_{1}(1-\cos\psi)}{2}\right]\
,$ (4)

where $\epsilon_{1}$ is the dimensionless energy of the $\gamma$-rays;
$\epsilon$ is the dimensionless energy of the soft photons interacting with
and absorbing the $\gamma$-rays; and $n(\epsilon,\Omega)$ is the photon
density of the photon field that interacts with and absorbs the $\gamma$-rays.
We use the notation where $\epsilon_{1}$ and $\epsilon$ represent
dimensionless photon energies, i.e. photon energies in units of the electron
rest energy, $m_{e}c^{2}=511\ \mathrm{keV}$. Here $\psi$ is the angle between
the incoming $\gamma$-ray photon and the photon from the Sun. In general, the
radiation field $n(\epsilon,\Omega)$ and the interaction angle $\psi$ can be
functions of both the polar angle ($\theta$) and the azimuthal angle ($\phi$).
However, for our geometry here they are independent of $\phi$. Some
trigonometric effort yields

$\displaystyle\cos\psi=\mu_{S}\mu_{E}-\sqrt{1-\mu_{S}^{2}}\sqrt{1-\mu_{E}^{2}}\
.$ (5)

Note that Balaji (2023) neglect the $1-\cos\psi$ term in Equation (2.1). This
is the primary difference in our calculations and theirs.

The $\gamma$-ray comes from a source with intrinsic (i.e., unabsorbed)
$\gamma$-ray flux $F_{\rm int}(\epsilon_{1})$. It will be observed by a
detector on or orbiting the Earth with a $\gamma$-ray flux $F_{\rm
obs}(\epsilon_{1})=\exp(-\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E}))F_{\rm
int}(\epsilon_{1})$.

###  2.2 Monochromatic Approximation for Sun

We follow the formalism of Boettcher & Dermer (1995); Dermer & Menon (2009);
Dermer et al. (2009); Finke (2016) to derive formulae for the photoabsorption
of astrophysical $\gamma$-rays by solar photons as a function of $\gamma$-ray
photon energy ($\epsilon_{1}$) and angular distance on the sky of the source
from the Sun ($\theta_{E}$). In spherical coordinates
$(R_{r},\theta_{r},\phi_{r})$ are the radial distance, polar angle, and
azimuthal angle, respectively, for the generic radiating medium, and
$\mu_{r}=\cos\theta_{r}$. In our case the radiating medium will be the Sun.

We approximate the Sun as a point source emitting monochromatic photons with
dimensionless energy $\epsilon_{S}=2.7\Theta$ and luminosity
$L_{S}=3.846\times 10^{33}$ ergs s-1. Here $\Theta=k_{B}T_{S}/(m_{e}c^{2})$
where $k_{B}=1.380\times 10^{-16}$ erg K-1 is the Boltzmann Constant and
$T_{S}=5780\ \mathrm{K}$ is the effective temperature of the Sun. The center
of the Sun is at the origin (see Fig. 1). In this case, the emissivity of the
photons emitted by the Sun is

$\displaystyle\dot{n}(\epsilon,\Omega)=\frac{L_{S}}{m_{e}c^{2}\epsilon_{S}}\frac{\delta(\epsilon-\epsilon_{S})\delta(R_{r}-R_{S})}{4\pi
R_{S}^{2}}\ ,$ (6)

where $R_{S}=6.98\times 10^{10}\ \mathrm{cm}$ is the radius of the Sun. The
photon density at a distance $x$ from the origin is (e.g., Boettcher & Dermer,
1995; Finke, 2016)

$\displaystyle n(\epsilon,\Omega)$ $\displaystyle=\frac{1}{4\pi
c}\int_{0}^{2\pi}d\phi_{r}\int^{1}_{-1}d\mu_{r}\int_{0}^{\infty}dR_{r}\left(\frac{R_{r}}{x}\right)^{2}$
$\displaystyle\times\dot{n}(\epsilon,\Omega)\delta(\phi_{r}-\phi_{S})\delta(\mu-\mu_{S})\
.$ (7)

The coordinate system is chosen so that the azimuthal angle of the sun is
$\phi_{S}=0$. Putting Equation (6) in Equation (2.2) gives

$\displaystyle
n(\epsilon,\Omega)=\frac{L_{S}}{m_{e}c^{2}\epsilon_{S}c}\frac{1}{4\pi
x^{2}}\frac{\delta(\mu-\mu_{S})}{2\pi}\delta(\epsilon-\epsilon_{S})\ .$ (8)

We are interested in the case where the $\gamma$-ray photon interacts with a
solar photon, which occurs at $x=x_{S}$. Substituting Equation (8) into
Equation (2.1) results in

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E})$
$\displaystyle=\frac{L_{S}}{4\pi\epsilon_{S}m_{e}c^{3}}\int_{0}^{\infty}dx_{E}\frac{1-\cos\psi}{x_{S}^{2}}$
$\displaystyle\times\sigma_{\gamma\gamma}\left[\frac{\epsilon_{S}\epsilon_{1}(1-\cos\psi)}{2}\right]\
,$ (9)

where, from the law of cosines,

$\displaystyle x_{S}^{2}=a^{2}+x_{E}^{2}-2ax_{E}\mu_{E}\ $ (10)

and

$\displaystyle\mu_{S}=\frac{a^{2}+x_{S}^{2}-x_{E}^{2}}{2ax_{S}}\ .$ (11)

If $\theta_{E}=\pi$ then $\mu_{E}=-1$, $\mu_{S}=1$, and the integral in
Equation (2.2) can be performed analytically, giving

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E}=-1)=\frac{L_{S}}{2\pi\epsilon_{S}m_{e}c^{3}a}\sigma_{\gamma\gamma}(\epsilon_{S}\epsilon_{1})\
.$ (12)

This is a factor of 2 larger than the preliminary calculation by Loeb (2022).

###  2.3 Blackbody Approximation for Sun

Now we make the more realistic assumption that, instead of emitting
monochromatically, the sun emits as a blackbody. We still assume the Sun is a
point source. In this case, the emissivity of photons emitted by the Sun is

$\displaystyle\dot{n}(\epsilon,\Omega)=\frac{L_{S}}{m_{e}c^{2}\epsilon}\frac{\delta(R_{r}-R_{S})}{4\pi
R_{S}^{2}}\frac{15}{\Theta^{4}\pi^{4}}\frac{\epsilon^{3}}{\exp(\epsilon/\Theta)-1}\
.$ (13)

Following the same procedure as in Section 2.2, we get

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E})$
$\displaystyle=\frac{L_{S}}{4m_{e}c^{3}}\frac{15}{\Theta^{4}\pi^{5}}\int_{0}^{\infty}d\epsilon\frac{\epsilon^{2}}{\exp(\epsilon/\Theta)-1}$
$\displaystyle\times\int_{0}^{\infty}dx_{E}\frac{1-\cos\psi}{x_{S}^{2}}$
$\displaystyle\times\sigma_{\gamma\gamma}\left[\frac{\epsilon\epsilon_{1}(1-\cos\psi)}{2}\right]\
.$ (14)

Again, for $\theta_{E}=\pi$, the integral over $x_{E}$ is analytic, so that

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E}=-1)$
$\displaystyle=\frac{L_{S}}{2m_{e}c^{3}a}\frac{15}{\Theta^{4}\pi^{5}}$
$\displaystyle\times\int_{0}^{\infty}d\epsilon\frac{\epsilon^{2}}{\exp(\epsilon/\Theta)-1}\sigma_{\gamma\gamma}(\epsilon\epsilon_{1})\
.$ (15)

Figure 2:  Absorption optical depth for astrophysical $\gamma$-rays
interacting with solar photons as a function of $\gamma$-ray energy, $E_{1}$,
at different angular distances from the Sun ($\theta_{E}$) as indicated by the
legend. We show the monochromatic approximation (Equation [2.2]; dashed
curves) and blackbody approximation (Equation [2.3]; solid curves).

In Figure 2 we plot $\tau_{\gamma\gamma}$ versus $\gamma$-ray energy
$E_{1}=m_{e}c^{2}\epsilon_{1}$ for different angles from the Sun
($\theta_{E}$) for the monochromatic approximation (Equation [2.2]) and the
blackbody approximation (Equation [2.3]). As one can see, the results for the
monochromatic and blackbody approximations are fairly similar, although the
monochromatic approximation has a hard cutoff at about 200 GeV. One cannot
observe $\gamma$-ray photons that come from behind the Sun. Since the Sun has
an angular diameter of $\approx$0.5°, and solar-blind $\gamma$-ray telescopes
typically have angular resolutions of $\sim 1$°, one could not reasonably
expect to detect an astrophysical $\gamma$-ray source at an angular distance
of $\theta_{E}\lesssim$1° from the center of the Sun.

###  2.4 Lorentz Invariance Violation

Subluminal LIV can modify the threshold for the $\gamma\gamma$ absorption
cross section, leading to an increase or decrease in $\tau_{\gamma\gamma}$
compared to the case without LIV. There are (at least) two ways of
implementing this used in the literature, that of Jacob & Piran (2008) and
Fairbairn et al. (2014). The two formulations do not give identical results.

####  2.4.1 Jacob & Piran Formulation

Following Jacob & Piran (2008); Biteau & Williams (2015), in the presence of
subluminal LIV, Equation (2.2) for the monochromatic approximation becomes

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E})$
$\displaystyle=\frac{L_{S}}{4\pi\epsilon_{S}m_{e}c^{3}}\int_{0}^{\infty}dx_{E}\frac{1-\cos\psi}{x_{S}^{2}}$
$\displaystyle\times\sigma_{\gamma\gamma}\left[\frac{\epsilon_{S}\epsilon_{1}(1-\cos\psi)}{2(1+0.25(\epsilon_{1}/\epsilon_{\rm
LIV})^{n}\epsilon_{1}^{2})}\right]\ ,$ (16)

where $\epsilon_{\rm LIV}=E_{\rm LIV}/(m_{e}c^{2})$. Similarly for the
blackbody approximation, Equation (2.3) becomes

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E})$
$\displaystyle=\frac{L_{S}}{4m_{e}c^{3}}\frac{15}{\Theta^{4}\pi^{5}}\int_{0}^{\infty}d\epsilon\frac{\epsilon^{2}}{\exp(\epsilon/\Theta)-1}$
$\displaystyle\times\int_{0}^{\infty}dx_{E}\frac{1-\cos\psi}{x_{S}^{2}}$
$\displaystyle\times\sigma_{\gamma\gamma}\left[\frac{\epsilon\epsilon_{1}(1-\cos\psi)}{2(1+0.25(\epsilon_{1}/\epsilon_{\rm
LIV})^{n}\epsilon_{1}^{2})}\right]\ .$ (17)

We parameterize LIV with the parameter $\xi\equiv E_{\rm LIV}/E_{\rm Planck}$.
In Figure 3 we show $\tau_{\gamma\gamma}$ plotted versus $E_{1}$ for
$\theta_{E}=1$°, $\theta_{E}=10$° and $\theta_{E}=90$° for various values of
$\xi$ in the linear ($n=1$) LIV case. As the plot demonstrates, there is an
increase in the gamma-ray absorption optical depth above 10 TeV, depending on
the value of $\xi$. This is because of the LIV effect modifying the argument
for $\sigma_{\gamma\gamma}$. With this modification, when sampling the cross
section at energies where $0.25(\epsilon_{1}/\epsilon_{\rm
LIV})^{n}\epsilon_{1}^{2}\gg 1$, one gets a cross section equivalent to the
cross section at a lower energy. As described by Jacob & Piran (2008): “[A]t
any energy above $E^{*}$ [there is] an optical depth identical to the optical
depth at some energy below $E^{*}$ for which the pair-production threshold is
the same.” In the notation of Jacob & Piran (2008), $E^{*}$ is the energy
above which LIV effects become important for $\gamma\gamma$ absorption. Thus,
if the optical depth without LIV is increasing with energy, at energies where
LIV is important, one will be sampling the optical depth at lower energies
where $\tau_{\gamma\gamma}$ is lower, and thus one will find a lower
$\tau_{\gamma\gamma}$ than without LIV. However, if $\tau_{\gamma\gamma}$ is
decreasing with energy, the LIV effect allows one to sample
$\tau_{\gamma\gamma}$ at a lower energy where it is higher, and thus one will
find a higher $\tau_{\gamma\gamma}$ than without LIV.

Figure 3:  Absorption optical depth for astrophysical $\gamma$-rays
interacting with solar photons as a function of $\gamma$-ray energy, $E_{1}$,
at different angular distances from the Sun ($\theta_{E}$) as indicated by the
legend, using the blackbody approximation. We show the result without LIV
(solid curves; Equation [2.3]) and with $n=1$ LIV using the Jacob & Piran
(2008) formulation (Equation [17]) for $\xi=1$ (dashed curves), $\xi=10$
(dotted curves), and $\xi=100$ (dashed-dotted curves).

This is demonstrated in Fig. 4 for EBL absorption. Here we plot the EBL
absorption optical depth from the model of Finke et al. (2022) with no LIV,
and with LIV for various values of $\xi$. Here the LIV effect is computed as
described by Finke & Razzaque (2023), following the formulation of Jacob &
Piran (2008). The dashed lines with arrows indicate the lower energies where
$\tau_{\gamma\gamma}$ is being sampled for higher energies with LIV. The plot
demonstrates that when $\tau_{\gamma\gamma}$ is increasing, the LIV effect
causes $\tau_{\gamma\gamma}$ to be lower than it otherwise would be. But where
$\tau_{\gamma\gamma}$ is decreasing with energy, as it is here above a few
hundred TeV, the LIV can cause an increase in $\tau_{\gamma\gamma}$.

Figure 4:  EBL absorption optical depth versus $\gamma$-ray energy for the EBL
model of Finke et al. (2022) at $z=0.15$. This is shown without LIV (solid
curve), and with $n=1$ LIV for the Jacob & Piran (2008) formulation (dotted
curves) and Fairbairn et al. (2014) formulation (dashed curves) for various
values of $\xi$, as shown in the plot. For the LIV curves from the Jacob &
Piran (2008) formulation, the dashed lines with arrows indicate the
$\tau_{\gamma\gamma}$ at lower energies where the $\tau_{\gamma\gamma}$ at
higher energies are sampling, due to the LIV effect. Absorption by cosmic
microwave background photons is not included in this plot.

In Figures 5 and 6 we plot $\tau_{\gamma\gamma}$ versus $\theta_{E}$ for
various $\gamma$-ray energies including effects from LIV. The absorption
optical depth decreases rapidly with increasing anglular distance from the
Sun. In most cases, the decrease in $\tau_{\gamma\gamma}$ is monotonic with
angle. This is because at large $x_{E}$ as $\theta_{E}$ increases, the cross
section argument $\epsilon_{1}\epsilon(1-\cos\psi)/2$ increases, and the cross
section generally decreases with increasing energy, as long as the invariant
argument is above threshold. However, the optical depth for 100 GeV decreases
with increasing $\theta_{E}$ until it reaches a minimum at $\theta_{E}\approx
75$°, then increases until $180$°. In this case, the quantity
$\epsilon_{1}\epsilon(1-\cos\psi)/2$ will go below threshold for parts of the
integral over $x_{E}$ for smaller $\theta_{E}$. The exact results thus depend
on the detailed kinematics.

Figure 5:  Absorption optical depth for astrophysical $\gamma$-rays
interacting with solar photons as a function of angular distances from the Sun
($\theta_{E}$), for different $\gamma$-ray energies as indicated by the
legend, using the blackbody approximation. We show the result without LIV
(solid curves; Equation [2.3]) and with $n=1$ LIV using the Jacob & Piran
(2008) formulation (Equation [17]) for $\xi=1$ (dashed curves), $\xi=10$
(dotted curves), and $\xi=100$ (dashed-dotted curves). The 1 TeV curve has
been scaled up by a factor of 2 and the 100 TeV $\xi=1$ curve has been scaled
up by a factor of 1.5 for clarity. Figure 6:  The same as Fig. 5, but with
$\theta_{E}$ from 0° to 20°.

For $E_{1}\lesssim 10$ TeV, there is no effect of LIV on $\gamma$-ray
absorption, as shown in Figures 3 and 5. For larger $E_{1}$, the absorption
optical depth will “mimic” the absorption at lower energy when LIV is
included, as described above. Thus at 1 PeV with $\xi=10$, the result mimics
the behavior of the 100 GeV curve, including the minimum at $\approx 75$°. At
100 TeV for $\xi=1$ and 1 PeV for $\xi=10$ the results are identical to the 1
TeV curve.

####  2.4.2 Fairbairn et al. Formulation

Fairbairn et al. (2014) follow Protheroe & Meyer (2000) in their approach to
LIV. In their formulation, the absorption optical depth for astrophysical
$\gamma$-rays due to interactions with solar photons is given by

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E})$
$\displaystyle=\frac{L_{S}}{4\pi\epsilon_{S}m_{e}c^{3}}\int_{0}^{\infty}dx_{E}\frac{1-\cos\psi}{x_{S}^{2}}$
$\displaystyle\times\sigma_{\gamma\gamma}\left[\frac{\epsilon_{S}\epsilon_{1}(1-\cos\psi)}{2}-\frac{(\epsilon_{1}/\epsilon_{LIV})^{n}\epsilon_{1}^{2}}{4}\right]\
$ (18)

with the monochromatic approximation, and

$\displaystyle\tau_{\gamma\gamma}(\epsilon_{1},\mu_{E})$
$\displaystyle=\frac{L_{S}}{4m_{e}c^{3}}\frac{15}{\Theta^{4}\pi^{5}}\int_{0}^{\infty}d\epsilon\frac{\epsilon^{2}}{\exp(\epsilon/\Theta)-1}$
$\displaystyle\times\int_{0}^{\infty}dx_{E}\frac{1-\cos\psi}{x_{S}^{2}}$
$\displaystyle\times\sigma_{\gamma\gamma}\left[\frac{\epsilon\epsilon_{1}(1-\cos\psi)}{2}-\frac{(\epsilon_{1}/\epsilon_{LIV})^{n}\epsilon_{1}^{2}}{4}\right]\
$ (19)

with the blackbody approximation.

The absorption optical depth of astrophysical $\gamma$-rays with solar photons
with LIV using the Fairbairn et al. (2014) formulation is shown in Fig. 7;
this can be compared with the Jacob & Piran (2008) formulation plotted in Fig.
3. The Figure demonstrates that the calculation using the Fairbairn et al.
(2014) formulation also can lead to an increase in the absorption optical
depth relative to the calculation with no LIV, although the increase is much
lower in the Fairbairn et al. (2014) formulation. This is also true for
absorption of $\gamma$-rays by EBL photons, as demonstrated in Fig. 4; here
again, on the region of the plot where $\tau_{\gamma\gamma}$ is decreasing
with $E_{1}$ in the non-LIV case, the LIV can lead to an increase in
$\tau_{\gamma\gamma}$ relative to the non-LIV case. In general, the Fairbairn
et al. (2014) formulation predicts lower absorption optical depths than the
Jacob & Piran (2008) formulation, as discussed by Tavecchio & Bonnoli (2016).

Figure 7:  Same as Figure 3, but using the Fairbairn et al. (2014)
formulation.

##  3 An Experiment for LIV

In the preceding section, we have computed the absorption optical depth of
$\gamma$-ray photons from the Sun, both with and without LIV effects included.
These calculations make potentially observable predictions. The absorption is
generally greatest at small angular distances from the Sun. This implies an
experiment that could potentially detect or constrain LIV. One would need to
observe a bright $\gamma$-ray source when it is at small angular distances
from the Sun, where the effect is strongest; and when it is quite far from the
Sun, where the absorption is minimal. The flux near the Sun would be

$\displaystyle F({\rm near})$ $\displaystyle=F({\rm
far})\exp[-\tau_{\gamma\gamma}({\rm near})]$ $\displaystyle\approx F({\rm
far})[1-\tau_{\gamma\gamma}({\rm near})]\ ,$ (20)

where $F({\rm far})$ is the flux far from the Sun. The combination of these
observations would give $\tau_{\gamma\gamma}({\rm near})$, the absorption
optical depth near the Sun. A comparison of this with our model predictions
could then constrain $\xi$. Assuming $F({\rm far})\approx F({\rm near})\approx
F$ for the purposes of estimating uncertainties, and $F({\rm far})$ and
$F({\rm near})$ have the same uncertainty, $\sigma_{F}$, standard Gaussian
error propagation gives the uncertainty in $\tau_{\gamma\gamma}({\rm near})$
as

$\displaystyle\sigma_{\tau_{\gamma\gamma}}=\sqrt{2}\left(\frac{\sigma_{F}}{F}\right)\
.$ (21)

In Fig. 8 the absorption optical depth is plotted versus $\xi$ for
$\theta_{E}=2$° for a variety of photon energies. In general, observations of
photons with higher energies have the greatest ability to constrain $\xi$.
Although note there is some ambiguity; a measurement of
$\tau_{\gamma\gamma}=10^{-2}$ at 500 TeV could indicate $\xi\approx 5$ or
$\xi\approx 100$. However, there are other ways to resolve these ambiguities;
for instance, according to time-of-flight constraints from GRBs, $\xi\gtrsim
10$ (Vasileiou et al., 2013). To constrain LIV one must be able to observe a
$\gamma$-ray source near the Sun at these very high energies. Imaging
Atmospheric Cherenkov Telescopes (IACTs) such as VERITAS, H.E.S.S., MAGIC, and
the upcoming CTA are not able to observe during the day and thus could not
observe $\gamma$-ray sources near the Sun, but could potentially observe this
effect at large angles. These leaves water Cherenkov detectors, such as HAWC
and LHAASO. Fermi-LAT could potentially measure the attenuation of
$\gamma$-ray photons near the Sun, but it is not sensitive at energies
relevant for constraining LIV.

Figure 8: Absorption optical depth for astrophysical $\gamma$-rays interacting
with solar photons as a function of $\xi$ for $n=1$ and $\theta_{E}=2$°, for
different $\gamma$-ray energies, $E_{1}$, as indicated by the legend, using
the blackbody approximation and the Jacob & Piran (2008) formulation of LIV.

Compared with constraints on LIV from the EBL, the main advantage to this
method is the higher degree of certainty. With EBL absorption, neither the
absorbing photon source (the EBL) nor the $\gamma$-ray source spectrum is
completely known (e.g., Mattingly, 2005). With the experiment outlined here,
the absorbing photon source (the Sun) has a known spectrum that is
approximated well by a blackbody. The $\gamma$-ray source spectrum can be
known from observations when the source is not near the Sun, when absorption
by this effect is negligible. The disadvantage of using the Sun is that the
absorption is much smaller than with the EBL; for solar photons,
$\tau_{\gamma\gamma}\sim 10^{-2}$ at most.

We estimate the possibility of detecting absorption of astrophysical photons
by the Sun at 1 TeV by HAWC and LHAASO, without regard to constraining LIV. At
$E_{1}=1$ TeV and $\theta_{E}=2^{\circ}$, $\tau_{\gamma\gamma}\approx 3\times
10^{-3}$. Assuming Poisson uncertainties and a precisely known background,
$\sigma_{F}/F\approx\sqrt{S+B}/S$, where $S$ and $B$ are the source and
background count rates, respectively, collected by the detector. For HAWC and
LHAASO we expect $B\gg S$, so that $\sqrt{S+B}/S\approx\sqrt{B}/S$. Using
Equation (21),

$\displaystyle\frac{S}{\sqrt{B}}=\frac{\sqrt{2}}{\tau_{\gamma\gamma}}\left(\frac{\tau_{\gamma\gamma}}{\sigma_{\tau_{\gamma\gamma}}}\right)\
.$ (22)

For a $3\sigma$ detection,
$\tau_{\gamma\gamma}/\sigma_{\tau_{\gamma\gamma}}=3$.

One can search for the absorption of individual astrophysical $\gamma$-ray
sources, or the $\gamma$-ray background. The background has the advantage of
always being observable near the Sun. Its disadvantage is it is extremely
faint. Indeed, the extragalactic $\gamma$-ray background has not yet been
detected directly (as far as the authors know) but estimates have been made by
Inoue & Tanaka (2016); Qu et al. (2019); Qu & Zeng (2022).

Table 1: VHE Sources Within $10^{\circ}$ of the Sun.  Name | RA | Dec | Ecliptic Longitude [deg] | Ecliptic Latitude [deg] | Flux [Crab]
---|---|---|---|---|---
Sources from TevCat
RGB J0152+017 | 1h52m33s | 1°46′40′′ | 26.78 | -9.15 | 0.02
1ES 0229+200 | 2h32m53s | 20°16′21′′ | 42.29 | 4.99 | 0.018
RBS 0413 | 3h19m47s | 18°45′42′′ | 52.46 | 0.39 | 0.01
VER J0521+211 | 5h21m45s | 21°12′51′′ | 81.09 | -1.93 | 0.092
Crab | 5h34m30s | 22°0′44′′ | 84.09 | -1.30 | 1
HAWC J0543+233 | 5h43m7.2s | 23°24′0.0′′ | 86.13 | 0.02 | -
IC 443 | 6h16m51s | 22°30′11′′ | 93.89 | -0.88 | 0.03
Geminga | 6h32m28s | 17°22′0.0′′ | 97.79 | -5.85 | 0.23
RX J0648.7+1516 | 6h48m45s | 15°16′12′′ | 101.86 | -7.67 | 0.033
1ES 0647+250 | 6h50m46s | 25°3′0.0′′ | 101.49 | 2.12 | 0.03
2HWC J0700+143 | 7h0m28s | 14°19′12′′ | 104.80 | -8.35 | -
2HWC J0819+157 | 8h19m55s | 15°47′24′′ | 123.56 | -3.67 | -
RBS 0723 | 8h47m12s | 11°33′50′′ | 131.06 | -6.12 | 0.025
OJ 287 | 8h54m49s | 20°5′58′′ | 130.51 | 2.59 | 0.013
3C 279 | 12h56m11s | -5°47′22′′ | 195.17 | 0.20 | -
2HWC J1309-054 | 13h9m14s | -5°29′24′′ | 198.06 | 1.72 | -
PKS 1510-089 | 15h12m52s | -9°6′21′′ | 228.30 | 8.49 | 0.03
AP Librae | 15h17m41s | -24°22′19′′ | 233.44 | -5.94 | 0.02
TXS 1515-273 | 15h18m3.6s | -27°31′34′′ | 234.35 | -8.96 | 0.06
HESS J1741-302 | 17h41m15s | -30°22′37′′ | 265.93 | -7.00 | 0.01
HESS J1745-303 | 17h45m2.1s | -30°22′14′′ | 266.75 | -6.97 | 0.05
Galactic Centre | 17h45m39s | -29°0′22′′ | 266.85 | -5.61 | 0.05
HESS J1746-308 | 17h46m17s | -30°50′28′′ | 267.03 | -7.44 | 0.03
VER J1746-289 | 17h46m19s | -28°57′58′′ | 267.00 | -5.56 | -
HESS J1746-285 | 17h46m23s | -28°52′33′′ | 267.01 | -5.47 | -
MAGIC J1746.4-2853 | 17h46m25s | -28°52′55′′ | 267.01 | -5.48 | -
SNR G000.9+00.1 | 17h47m23s | -28°9′6.0′′ | 267.21 | -4.74 | 0.02
Terzan 5 | 17h47m49s | -24°48′30′′ | 267.23 | -1.40 | 0.015
HESS J1800-240B | 18h0m26s | -24°2′20′′ | 270.10 | -0.60 | -
W 28 | 18h1m42s | -23°20′6.0′′ | 270.39 | 0.10 | -
HESS J1800-240A | 18h1m57s | -23°57′43′′ | 270.45 | -0.52 | -
HESS J1804-216 | 18h4m31s | -21°42′0.0′′ | 271.05 | 1.74 | 0.25
HESS J1808-204 | 18h8m37s | -20°25′36′′ | 272.02 | 3.00 | -
HESS J1809-193 | 18h10m31s | -19°18′0.0′′ | 272.49 | 4.12 | 0.14
HESS J1813-178 | 18h13m36s | -17°50′24′′ | 273.25 | 5.56 | 0.06
2HWC J1814-173 | 18h14m4.8s | -17°18′36′′ | 273.38 | 6.09 | -
SNR G015.4+00.1 | 18h18m4.8s | -15°28′1.0′′ | 274.40 | 7.90 | 0.018
2HWC J1819-150* | 18h19m19s | -15°3′36′′ | 274.71 | 8.30 | -
LHAASO J1825-1326 | 18h25m48s | -13°27′0.0′′ | 276.37 | 9.85 | 3.57
HESS J1825-137 | 18h25m49s | -13°46′35′′ | 276.36 | 9.52 | 0.54
2HWC J1825-134 | 18h25m50s | -13°24′0.0′′ | 276.38 | 9.90 | -
LS 5039 | 18h26m15s | -14°49′30′′ | 276.41 | 8.47 | 0.03
1RXS J195815.6-301119 | 19h58m14s | -30°11′11′′ | 295.61 | -9.34 | -
Additional Sources from LHAASO Catalog
1LHAASO J0534+2200 | 5h34m26s | 22°3${}^{\prime}36^{\prime\prime}$ | 84.08 | -1.25 | -
1LHAASO J0542+2311 | 5h42m50s | 23°12${}^{\prime}0.0^{\prime\prime}$ | 86.06 | -0.18 | -
1LHAASO J0617+2234 | 6h17m24s | 22°35${}^{\prime}48^{\prime\prime}$ | 94.02 | -0.78 | -
1LHAASO J0634+1741 | 6h34m16s | 17°42${}^{\prime}36^{\prime\prime}$ | 98.20 | -5.49 | -
1LHAASO J0703+1405 | 7h3m19s | 14°6${}^{\prime}0.0^{\prime\prime}$ | 105.52 | -8.49 | -
1LHAASO J1809-1918 | 18h9m31s | -20°42’$0.0^{\prime\prime}$ | 272.23 | 2.72 | -
1LHAASO J1814-1719 | 18h13m4.0s | -18°7${}^{\prime}24^{\prime\prime}$ | 273.12 | 5.28 | -
1LHAASO J1814-1636 | 18h14m52s | -17°23${}^{\prime}12^{\prime\prime}$ | 273.57 | 6.01 | -
1LHAASO J1825-1418 | 18h25m9.0s | -15°41${}^{\prime}12^{\prime\prime}$ | 276.11 | 7.62 | -
1LHAASO J1825-1337 | 18h25m48s | -14°23${}^{\prime}48^{\prime\prime}$ | 276.32 | 8.90 | -

The advantage of individual sources is they should have many more photons than
the background. The disadvantage is they do not stay near the Sun for long,
since the Sun moves across the sky by approximately 1° day-1. This means a
point source will only be observable with $10^{\circ}$ of the Sun for at most
20 days, so that for point sources it will be observed a fraction of the time
$f_{t}=0.5\times(20\ {\rm days})/(365\ {\rm days})\approx 0.03$. We searched
the TeVCat (http://tevcat.uchicago.edu; Wakely & Horan (2008)) for
astrophysical TeV sources that get near the Sun. In Table 1 we list the TeV
sources with ecliptic latitude $<10$°. As of this writing, the TeVCat has not
included sources from the first LHAASO catalog (Cao et al., 2023). We included
these sources in our table as well. The ecliptic latitude will be the sources’
minimum distance on the sky from the Sun. Variable sources such as blazars
would not be ideal for these purposes, since one would want to compare their
flux when they are near and far from the Sun. Two sources stand out as the
brightest: the Crab (Aharonian et al., 2021; Cao et al., 2021a) and LHAASO
J1825$-$1326 (Cao et al., 2021b), seen by LHAASO up to $\approx 1$ PeV and 500
TeV, respectively. For point sources with photon flux $\Phi_{S}\equiv E\
dN/dE$ the signal count rate can be approximated as

$\displaystyle S\approx\Phi_{S}A_{\rm eff}f_{t}\Delta t.$ (23)

where $\Delta t$ is the length of time needed to observe the source and
$A_{\rm eff}$ is the effective area of the detector. For HAWC, we take the
effective area from DeYoung & HAWC Collaboration (2012); for LHAASO we take it
from Cao et al. (2019). Cosmic rays are a significant background for HAWC and
LHAASO. The analysis with HAWC is able to reject all but a fraction
$f_{r}\approx 6\times 10^{-3}$ of these cosmic rays (DeYoung & HAWC
Collaboration, 2012), and LHAASO can reject all but $f_{r}\approx 2\times
10^{-3}$ of them (Aharonian et al., 2021). The cosmic ray background count
rate is thus

$\displaystyle B\approx\Phi_{\rm CR}f_{r}A_{\rm eff}f_{t}\Delta t\Omega\ ,$
(24)

where $\Phi_{\rm CR}$ is the $E\ dN/dE$ cosmic ray intensity and $\Omega$ is
the solid angle around the source. We take $\Phi_{\rm CR}$ from Chapter 29 of
Tanabashi et al. (2018). For HAWC and LHAASO, with angular resolution of about
$\theta_{\rm psf}\approx 1\arcdeg$, $\Omega\approx\pi\theta_{\rm
psf}^{2}\approx 10^{-3}\ \mathrm{srad}$. For a precisely known background, the
number of counts above background is

$\displaystyle\frac{S}{\sqrt{S+B}}\approx\frac{S}{\sqrt{B}}\approx\frac{\Phi_{S}}{\sqrt{\Phi_{\rm
CR}f_{r}}}\sqrt{\frac{A_{\rm eff}f_{t}\Delta t}{\Omega}}\ .$ (25)

Table 2: Detection of $\tau_{\gamma\gamma}$ at 1 TeV by VHE $\gamma$-ray instruments.  | Crab | LHAASO J1825$-$1326
---|---|---
$\Phi_{S}$ [$\mathrm{cm}^{-2}\ \mathrm{s}^{-1}$] | $4\times 10^{-11}$ | $1.5\times 10^{-11}$
$\Phi_{\rm CR}$ [$\mathrm{cm}^{-2}\ \mathrm{s}^{-1}\ \mathrm{srad}^{-1}$] | $1.4\times 10^{-5}$ | $1.4\times 10^{-5}$
$\tau_{\gamma\gamma}(\theta_{E}=2^{\circ})$ | $3\times 10^{-3}$ | $3\times 10^{-3}$
HAWC $f_{r}$ | $6\times 10^{-3}$ | $6\times 10^{-3}$
HAWC $A_{\rm eff}$ [km2] | 0.02 | 0.02
HAWC $\Delta t$ [year] | $560$ | $4000$
LHAASO $f_{r}$ | $2\times 10^{-3}$ | $2\times 10^{-3}$
LHAASO $A_{\rm eff}$ [km2] | 0.1 | 0.1
LHAASO $\Delta t$ [year] | $37$ | $260$

Combining Equations (22) and (25) gives the time needed for a detection,

$\displaystyle\Delta
t\approx\frac{2}{\tau_{\gamma\gamma}^{2}}\left(\frac{\tau_{\gamma\gamma}}{\sigma_{\tau_{\gamma\gamma}}}\right)^{2}\frac{\Phi_{\rm
CR}f_{r}}{\Phi_{S}^{2}}\frac{\Omega}{A_{\rm eff}f_{t}}\ .$ (26)

Here we estimate the time for detection for the Crab and LHAASO J1825$-$1326
at 1 TeV. The results are in Table 2. In all cases, it would take $>30$ years
to make a detection. Note that although LHAASO J1825$-$1326 is listed as
brighter than the Crab in Table 1, the energy ranges here are different, so
they are not directly comparable. For the LHAASO J1825$-$1326 at 1 TeV listed
in Table 2, we have extrapolated the power-law spectrum from Cao et al.
(2021b) to lower energies.

Next we look at whether Fermi-LAT could detect this effect. This is
complicated by the fact that for the LAT the Sun has a $\gamma$-ray halo
around it extending for $\approx 20$° caused by the Compton scattering of
solar photons by cosmic ray electrons (Moskalenko et al., 2006; Orlando &
Strong, 2007, 2008; Abdo et al., 2011; Linden et al., 2022). The extragalactic
gamma-ray background (EGB; Ackermann et al., 2015) becomes brighter than this
halo at about 5°; we will assume the $\tau_{\gamma\gamma}$ at
$\theta_{E}=10^{\circ}$, although the exact details will likely not effect our
conclusions. We use the LAT’s Pass 8 on-axis instrument response
function111https://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm.
The LAT is a sky survey instrument that sees the entire sky every 3 hours with
a field of view of approximately 20% of the sky. So we take $f_{t}=0.2\times
20/365=0.01$ for point sources. The LAT has an anti-coincidence detector to
veto cosmic rays, so that at high energies it is signal dominated ($S\gg B$),
and $\sigma_{F}/F\approx\sqrt{S+B}/S\approx\sqrt{S}/S=1/\sqrt{S}$. This leads
to

$\displaystyle\Delta t\approx\frac{2}{\Phi_{S}A_{\rm
eff}f_{t}\tau_{\gamma\gamma}^{2}}\left(\frac{\tau_{\gamma\gamma}}{\sigma_{\tau_{\gamma\gamma}}}\right)^{2}\
.$ (27)

Table 3: Detection of $\tau_{\gamma\gamma}$ by Fermi-LAT 3FHL sources.  | 50-150 GeV | 150-500 GeV | 500-2000 GeV
---|---|---|---
$\Phi_{S}$ | $3.9\times 10^{-9}$ | $9.9\times 10^{-10}$ | $2.3\times 10^{-10}$
$\tau_{\gamma\gamma}(\theta_{E}=10^{\circ})$ | $2.4\times 10^{-5}$ | $3.8\times 10^{-4}$ | $6.3\times 10^{-4}$
$A_{\rm eff}$ [cm2] | 9000 | 9000 | 9000
$\Delta t$ [year] | $3\times 10^{9}$ | $4\times 10^{7}$ | $7\times 10^{7}$

For individual sources, we searched the Fermi-LAT Third Hard Source Catalog
(Ajello et al., 2017) for sources with ecliptic latitude $<10$°, i.e., sources
that get within 10 degrees of the Sun, and found 253 sources. Using Equation
(27), we determine the total count rate for all of these sources for three
energy bins given in the catalog: the 50-150 GeV, 150-500 GeV, and 500-2000
GeV bins. These results are in Table 3. Even a stacking analysis of all hard
LAT sources will not detect this effect in the likely lifetime of the Fermi
spacecraft. Further, most of these sources are blazars, which are often highly
variable in $\gamma$-rays, and so would not be appropriate for this sort of
measurement.

If HAWC or LHAASO could reject cosmic rays like the LAT, we could follow the
same procedure for those experiments as we did for the LAT. In that case we
estimate HAWC and LHAASO would be able to detect this effect in the Crab at 1
TeV in 6.5 and 1.3 years, respectively. Distinguishing between cosmic rays and
photons is the strongest limitation on these experiments.

The prospects for detecting the absorption of $\gamma$-rays by solar photons
at 1 TeV are poor. Higher energies are more interesting from the point of view
of constraining LIV, but prospects are even worse here, since fewer photons
are detected. We note that our approximation in this section is rather
imprecise; we neglect all but Poisson uncertainties, and our approximations
for source and background counts (Equations (23) and (24)) do not properly
integrate over energy. However they are good enough for the order of magnitude
level estimates provided here.

##  4 Summary

In this paper:

  * •

We compute in detail the effect of absorption of astrophysical $\gamma$-ray
sources by solar photons, a process for which preliminary calculations were
made by Loeb (2022); Balaji (2023).

  * •

We show the absorption effect is greatest for $\gamma$-ray sources at small
angular distances from sun, reaching as high as $\tau_{\gamma\gamma}\sim
10^{-2}$. As pointed out by Loeb (2022), this effect causes the $\gamma$-ray
background at $\gtrsim 100\ \mathrm{GeV}$ to have a greater anisotropy than
the cosmic microwave background from the solar system’s motion relative to the
cosmic frame. Unfortunately, the EGB is $\sim 10^{6}$ fainter than the CMB
(e.g., Hauser & Dwek, 2001), making the anisotropy much more difficult to
detect.

  * •

We make calculations for this effect that include subluminal LIV. For the
first time, we show that subluminal LIV can lead to a decrease or increase in
the absorption optical depth from the pair production process compared to the
non-LIV case, depending on the spectrum of the absorbing photon source. This
is true for both the Jacob & Piran (2008) and Fairbairn et al. (2014)
formulation of LIV. he Jacob & Piran (2008) formulation predicts higher values
of $\tau_{\gamma\gamma}$ than the Fairbairn et al. (2014) one.

  * •

We show that this effect is unlikely to be observed with water Cherenkov
observatories like HAWC or LHAASO. These can observe near the Sun, unlike
IACTs, but the difficulties in separating $\gamma$-rays and hadrons means the
needed precision will not be reached in a reasonable amount of time. However,
a clever stacking analysis may lead to a possible detection of the effect in a
more reasonable amount of time.

We are grateful to the referee for comments that have improved this paper. We
would like to thank Teddy Cheung for pointing out there are sources in the
LHAASO catalog that are not in the TeVCat at the time of this writing; and
Brian Humensky for useful discussions on the observation of the effect
described in this manuscript. The authors were supported by NASA through
contract S-15633Y. J.D.F. was also supported by the Office of Naval Research.

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