Bounding the photon mass with gravitationally lensed fast radio bursts

As one of the fundamental postulates of Maxwell’s electromagnetism and Einstein’s special relativity, the principle of invariance of light speed implies that the rest mass of the photon should be exactly zero. Nevertheless, there exist some theories involving a finite photon rest mass, such as the famous de Broglie-Proca theory (De Broglie, 1922; Proca, 1936), the model of the nonvanishing photon mass as an explanation of dark energy (Kouwn et al., 2016), and other new ideas in the Standard-Model Extension with effectively massive photons (Spallicci et al., 2021). Despite the great success of the postulate of the constancy of light speed, those new theories with massive photons are interesting and worthy to explore, whereas the ultimate word on the photon mass ($m_{\gamma}$) stems from empirical facts.

Over the last few decades, various kinds of experimental approaches have been performed to push the empirical boundary on the masslessness of photons (see Goldhaber and Nieto (1971); Tu et al. (2005); Zhang (1997); Okun (2006); Goldhaber and Nieto (2010); Spavieri et al. (2011); Wei and Wu (2021) for reviews). These experiments include measurements of the frequency dependence of the speed of light ($m_{\gamma}\leq 3.8\times 10^{-51}\,\mathrm{kg}$; Lovell et al. (1964); Warner and Nather (1969); Schaefer (1999); Wu et al. (2016); Bonetti et al. (2016, 2017); Zhang et al. (2016); Shao and Zhang (2017); Wei et al. (2017); Wei and Wu (2018, 2020); Xing et al. (2019); Wang et al. (2021); Chang et al. (2023); Lin et al. (2023); Wang et al. (2023); Ran et al. (2024)), tests of Coulomb’s inverse square law ($m_{\gamma}\leq 1.6\times 10^{-50}\,\mathrm{kg}$; Williams et al. (1971)), measurement of Jupiter’s magnetic field ($m_{\gamma}\leq 8\times 10^{-52}\,\mathrm{kg}$; Davis et al. (1975)), analysis of the mechanical stability of magnetized gas in galaxies ($m_{\gamma}\leq 3\times 10^{-63}\,\mathrm{kg}$; Chibisov (1976)), tests of Ampère’s law ($m_{\gamma}\leq 8.4\times 10^{-48}\,\mathrm{kg}$; Chernikov et al. (1992)), magnetohydrodynamics of the solar wind ($m_{\gamma}\leq 1.4\times 10^{-49}-3.4\times 10^{-51}\,\mathrm{kg}$; Ryutov (1997, 2007); Retinò et al. (2016)), Cavendish torsion balance methods ($m_{\gamma}\leq 1.2\times 10^{-54}\,\mathrm{kg}$; Lakes (1998); Luo et al. (2003)), estimates of suppermassive black-hole spin ($m_{\gamma}\leq 7\times 10^{-56}\,\mathrm{kg}$; Pani et al. (2012)), analysis of pulsar spindown ($m_{\gamma}\leq 6.3\times 10^{-53}\,\mathrm{kg}$; Yang and Zhang (2017)), gravitational deflection of massive photons ($m_{\gamma}\leq 1.7\times 10^{-40}-4.1\times 10^{-45}\,\mathrm{kg}$; Lowenthal (1973); Accioly and Paszko (2004); Qian (2012); Egorov et al. (2014); Glicenstein (2017)), and so on. Among these experiments on photon mass, the resulting constraints obtained from the gravitational deflection of light are not the tightest ones; however, in view of model dependence of many experimental methods (see e.g. Tu et al. (2005); Goldhaber and Nieto (2010)), tests of the photon mass using different independent methods (such as gravitational deflection) are always interesting and important.

The semi-classical gravity predicts that the deflection of massive photons in an external gravitational field would be energy-dependent (Lowenthal, 1973; Accioly et al., 2000; Accioly and Ragusa, 2002; Accioly and Paszko, 2004). Therefore, an upper bound on the photon mass can be obtained by comparing the difference between the measured deflection angle and the calculated deflection angle for massless photons (Lowenthal, 1973). Exploiting the gravitational deflection of radio waves by the Sun, Accioly and Paszko (2004) obtained an upper limit of $m_{\gamma}\leq{10}^{-43}\,\mathrm{kg}$. Based on the astrometry of several strong gravitational lensing systems, Qian (2012) investigated the photon mass limit at a cosmological scale, yielding $m_{\gamma}\leq 8.7\times{10}^{-42}\,\mathrm{kg}$. With the precise astrometry of the gravitationally lensed quasar MG J2016$+$112, Egorov et al. (2014) further improved the limit to be $m_{\gamma}\leq 4.1\times{10}^{-45}\,\rm kg$. However, these astrometric limits do not use lens models and simply assume that the angular separation of lensed images is equivalent to the deflection angle of light. Glicenstein (2017) argued that this is a strong assumption. Modeling lens galaxies with central suppermassive black holes by a singular isothermal model, Glicenstein (2017) used the time delays between compact images from three lensed active galactic nuclei (AGNs) to derive a photon mass limit of $m_{\gamma}\leq 1.7\times{10}^{-40}\,\rm kg$.

Fast radio bursts (FRBs) are bright millisecond-long radio flashes originating at cosmological distances (Lorimer et al., 2007; Cordes and Chatterjee, 2019; Petroff et al., 2019, 2022; Zhang, 2023). Their cosmological origin, energetic nature, and high all-sky rate make them ideal for probing cosmology (e.g., Deng and Zhang (2014)). With tens of thousands of signals that will be guaranteed in the future, FRBs have gained attention as potential targets for lensing studies (Muñoz et al., 2016; Wang and Wang, 2018; Liao et al., 2020; Sammons et al., 2020; Krochek and Kovetz, 2022; Leung et al., 2022; Zhou et al., 2022). Very recently, Chang et al. (2024) employed the autocorrelation algorithm to search for potential lensed FRBs in the first Canadian Hydrogen Intensity Mapping Experiment (CHIME) FRB catalogue, and identified FRB 20190308C as a lensed candidate with a significance of $3.4\sigma$. The information about the time delay and flux ratio between the two substructures of FRB 20190308C can be easily extracted. Inspired by Glicenstein (2017), a natural question arises: is it possible to improve the photon mass lensing limits by using the gravitational time delays of lensed FRBs?

In this work, we propose a new method to place an upper limit on the photon mass by applying the time delay information from lensed FRBs. The rest of this paper is arranged as follows. In Section II, we discuss the photon-mass dependence of the time delay in the Chang-Refsdal lens model. The constraints on the photon mass from a lensed FRB candidate are presented in Section III. Finally, a brief summary and discussion are provided in Section IV.

The Chang-Refsdal lens model describes the lensing effect of a star, which can be considered as a point-mass lens under the gravitational perturbation of a background galaxy. The lens equation of the Chang-Refsdal lens model is given by (Chang and Refsdal, 1979, 1984; An and Evans, 2006; Chen et al., 2021; Gao et al., 2022)

where $\beta$ and $\theta$ represent the positions in the source and deflector planes, respectively, $\theta_{E}$ is the Einstein angle, and $\gamma$ is the external shear strength. Note that $\alpha=\beta-\theta$ stands for the deflection angle of light.

Assuming a weak gravitational field, the basic formulas for the time delay and position of lensed images of a massive photon source were derived by Lowenthal (1973) and Glicenstein (2017). These studies show that the deflection angle for massive photons is similar as that of massless photons, except for the $(1+\frac{1}{2}\mu^{2})$ multiplicative factor. Here $\mu^{2}=\frac{{m_{\gamma}}^{2}c^{2}}{P_{0}^{2}}$, where $m_{\gamma}$ is the rest mass of the photon and $P_{0}$ is the time component of the four-momentum. Therefore, for the scenario of massive photons, the lens equation (Equation 1) can be simply rewritten by replacing $\gamma$ and $\theta_{E}^{2}$ with $\gamma^{\prime}=(1+\frac{1}{2}\mu^{2})\gamma$ and ${\theta_{E}^{\prime}}^{2}=(1+\frac{1}{2}\mu^{2})\theta_{E}^{2}$. Scaling the angular coordinates with $\theta_{E}^{\prime}$: $y^{\prime}=\beta/\theta_{E}^{\prime}$ and $x^{\prime}=\theta/\theta_{E}^{\prime}$, the dimensionless lens equation reads as

This lens equation can have up to four solutions, resulting in multiple imaging scenarios. It is difficult to obtain the general analytical solutions to the lens equation, so do the expressions for the time delay $\Delta t$ and flux ratio $R$ between lensed images. Fortunately, the “permitted region” in the $R$–$\Delta t$ space can be determined with the three boundary conditions that the source is on the symmetry axis of the lensing system (i.e., $y_{1}^{\prime}=0$ or $y_{2}^{\prime}=0$) or at the tips of the inner caustics, thereby proving bounds on the photon mass.

Chen et al. (2021) focused on two-image configurations with $\gamma\ll 1$. For the case of $\gamma\ll 1$, the size of caustic is much less than $\theta_{E}^{\prime}$ and the cross section of four-image configurations can be ignored. Their analysis can be extended to the case of $\gamma<1$ if we only consider two-image configurations.

For $y_{1}^{\prime}=0$, the lower boundary of the permitted region in the $R$–$\Delta t$ space can be determined. The solutions for Equation (2) are

and

Using Equation (3) and the magnification of each image (see Chen et al. (2021) for the detailed derivation), we obtain the time delay

and the flux ratio

where $M$ and $z_{l}$ are the point mass and redshift of the lens, respectively, and $s_{2}^{\prime}=\sqrt{{y_{2}^{\prime}}^{2}+4(1-\gamma^{\prime})}$.

For $y_{2}^{\prime}=0$, the solutions for Equation (2) are

and

When $y_{2}^{\prime}=0$, the corresponding formulas for the time delay and flux ratio can be treated as the upper boundary of the permitted region, i.e.,

and

where $s_{1}^{\prime}=\sqrt{{y_{1}^{\prime}}^{2}+4(1+\gamma^{\prime})}$.

When the source is located at the tips of the inner caustics, i.e., $y_{1}^{\prime}=0$ and $y_{2}^{\prime}=\pm 2\gamma^{\prime}/\sqrt{1+\gamma^{\prime}}$ (or $y_{1}^{\prime}=\pm 2\gamma^{\prime}/\sqrt{1-\gamma^{\prime}}$ and $y_{2}^{\prime}=0$), the left boundary of the permitted region in the $R$–$\Delta t$ space can be determined. The lower limit on $\Delta t$ can be written as

The corresponding leading-to-trailing flux ratio $R$ is $0$ or $+\infty$.

These boundary conditions can still provide some information about the mass of the lens and the photon-mass-dependent time delay between lensed images in the absence of the general analytical solutions. Therefore, they can be used for further study on photon mass limits.

The permitted region of all possible $R$–$\Delta t$ pairs for a point-mass $+$ external shear lens model with the lens mass $M(1+z_{l})=4277\,{\rm M}_{\odot}$ and the equivalent external shear strength $\gamma^{\prime}=0.01$ is shown in Figure 1. One can see from this plot that all possible $R$–$\Delta t$ pairs between two lensed images are actually bracketed by three boundary lines. The lower boundary of the permitted region (solid curve on bottom) corresponds to the $R$–$\Delta t$ relation along the $y_{2}^{\prime}$-axis (i.e., $y_{1}^{\prime}=0$), which is determined by Equations (5) and (6). The upper boundary (solid curve on top) corresponds to the $R$–$\Delta t$ relation along the $y_{1}^{\prime}$-axis (i.e., $y_{2}^{\prime}=0$), which is determined by Equations (9) and (10). This $y_{2}^{\prime}=0$ curve reaches its minimum at $y_{1}^{\prime}=\sqrt{\frac{2\gamma^{\prime}(1+2\gamma^{\prime})}{1-\gamma^{\prime}}}$ with

where $\Delta^{\prime}=\sqrt{8{\gamma^{\prime}}^{3}+20{\gamma^{\prime}}^{2}+8\gamma^{\prime}}$. It is obvious that $R_{\rm min}$ is larger than $1$ when $\gamma^{\prime}>0$. The left boundary (vertical dashed line) corresponds to the lower limit of $\Delta t$, which is determined by Equation (11). As shown in Figure 1, with the fixed $R$, the observed time delay $\Delta t_{\rm obs}$ between the lensed images should always be larger than the lower limit $\Delta t_{\rm min}$ (vertical dashed line). With Equation (11), it is thus easy to obtain

So the photon mass can be constrained as

Very recently, Chang et al. (2024) searched for potential lensed FRBs within the first CHIME/FRB catalogue using the autocorrelation algorithm and verification through signal simulations. Only FRB 20190308C was identified as a plausible candidate for gravitational lensing. The observed time delay and flux ratio between the two substructures of FRB 20190308C are $\Delta t_{\rm obs}=8.85\,\rm ms$ and $R_{\rm obs}=0.5$. As an example, we now use the time delay information from FRB 20190308C to demonstrate how to obtain the constraints on the lens mass $M(1+z_{l})$, thereby placing constraints on the photon mass $m_{\gamma}$. For the doubly lensed FRB 20190308C with $R_{\rm obs}=0.5$ and $\Delta t_{\rm obs}=8.85\,\rm ms$, the upper boundary does not provide any useful information because $0.5$ will always be smaller than $R_{\rm min}$ for any $\gamma^{\prime}$. Therefore, only two $M(1+z_{l})$–$\gamma^{\prime}$ relations derived from the lower and left boundaries are seen in Figure 2. The $M(1+z_{l})$–$\gamma^{\prime}$ relation corresponding to the lower boundary (blue line) is derived from Equations (5) and (6). The $M(1+z_{l})$–$\gamma^{\prime}$ relation corresponding to the left boundary (orange line) is derived from Equation (11). For a moderate shear of $\gamma^{\prime}=0.01$, the lower limit on the lens mass is about $M(1+z_{l})=4277\,{\rm M}_{\odot}$. Since the photon mass term $\mu\ll 1$, it is reasonable to assume that $\gamma\simeq\gamma^{\prime}=0.01$. With the observed time delay $\Delta t_{\rm obs}=8.85\,\rm ms$, the lowest observed frequency $\nu=\frac{P_{0}c}{h}=400\,\rm MHz$, and the lower lens mass limit $M(1+z_{l})=4277\,{\rm M}_{\odot}$ corresponding to $\gamma^{\prime}=0.01$, a stringent upper limit on the photon mass from Equation (14) is

for FRB 20190308C.

In our above analysis, the external shear strength is set to be $\gamma^{\prime}=0.01$. To explore the effect of different $\gamma^{\prime}$ values, we estimate the sensitivity as we vary $\gamma^{\prime}$. For $\gamma^{\prime}<1$, the solutions (i.e., Equation 3) for the dimensionless lens equation always exist when $y_{1}^{\prime}=0$. Therefore, $\left|y_{2}^{\prime}\right|$ needs to be greater than $\frac{2\gamma^{\prime}}{\sqrt{1+\gamma^{\prime}}}$ to satisfy the two-image condition and ensure that the boundary conditions discussed in Section II remain applicable. As shown in Figure 2, with a fixed $\gamma^{\prime}$, a lower limit on the lens mass limit $M(1+z_{l})$ can be obtained, leading to the establishment of an upper limit on the photon mass $m_{\gamma}$. Figure 3 shows that as $\gamma^{\prime}$ increases, $m_{\gamma}$ decreases first and then increases. The upper limit of the photon mass has a minimum value of $m_{\gamma}=2.4\times 10^{-42}\,\text{kg}$, corresponding to $\gamma^{\prime}=0.38$ and a minimum lens mass of $M(1+z_{l})=221.7\,{\rm M}_{\odot}$. The maximum value of the upper photon mass limit is difficult to determine because, as $\gamma^{\prime}$ approaches 1, the wave properties of light become significant, making the geometric approximation of the lensing equation invalid. Therefore, we calculate the upper limit of the photon mass to be $m_{\gamma}<2.1\times 10^{-41}\,\text{kg}$ when $\gamma^{\prime}=0.99$, corresponding to a minimum lens mass of $M(1+z_{l})=4.3\,{\rm M}_{\odot}$. That is, within the range of $0<\gamma^{\prime}<1$, the photon mass can be constrained to be

for FRB 20190308C, which is almost 10-100 times tighter than the constraints from the time delays of lensed AGNs (Glicenstein, 2017).

For $\gamma^{\prime}>1$, there are two caustics, each with one cusp on the $y_{2}^{\prime}$ axis and two cusps off-axis. The boundary conditions for the two-image scenario are not clear. When $y_{1}^{\prime}=0$, the two-image scenario has two possibilities: either $\left|y_{2}^{\prime}\right|\geq\frac{2\gamma^{\prime}}{\sqrt{1+\gamma^{\prime}}}$ or $\left|y_{2}^{\prime}\right|\leq 2\sqrt{\gamma^{\prime}-1}$. For $y_{1}^{\prime}=0$ and $\left|y_{2}^{\prime}\right|\geq\frac{2\gamma^{\prime}}{\sqrt{1+\gamma^{\prime}}}$, the time delay and flux ratio become:

and

For $y_{1}^{\prime}=0$ and $\left|y_{2}^{\prime}\right|\leq 2\sqrt{\gamma^{\prime}-1}$, $\Delta t=0$ and $R=1$. Numerical calculations suggest that it may still be possible to impose certain constraints on the combined lens mass $M(1+z_{l})$ using Equations (17) and (18) by adding some restrictions, such as limiting the leading-to-trailing flux ratio $R<1$. However, the information obtained from the dynamic spectrum may not be sufficient to constrain the photon mass $m_{\gamma}$ due to the lack of an analytical relationship between $M(1+z_{l})$ and $m_{\gamma}$.

It has been suggested that the strong lensing effect of a point mass $+$ external shear lens model on a single-peak FRB can produce double peaks (i.e., lensed images). Based on this lens model, here we proposed a method of using the two observables of the time delay $\Delta t$ and the leading-to-trailing flux ratio $R$ from lensed FRBs to set a stringent upper limit on the photon mass. In particular, we showed the process of constraining photon mass using the observed values of $\Delta t$ and $R$ from a lensed FRB candidate, i.e., FRB 20190308C, as a reference.

For a point mass $+$ external shear lens model, there is no one-to-one correspondence between the upper photon mass limit $m_{\gamma}$ and $\Delta t$ and $R$ due to the extra freedom of the external shear. Nevertheless, we showed that an upper limit on $m_{\gamma}$ can still be derived from $\Delta t$ and $R$ for a given external shear strength of $\gamma^{\prime}$ (Section III). For FRB 20190308C with $\Delta t=8.85\,\rm ms$ and $R=0.5$, we obtained a strict constraint on the photon mass $m_{\gamma}<5.3\times{10}^{-42}\,\rm kg$ for a fixed external shear strength of $\gamma^{\prime}=0.01$. We also inspected the influences of different $\gamma^{\prime}$ values, finding that this effect has a modest impact on the photon mass limits. That is, within the range of $0<\gamma^{\prime}<1$, one can derive $m_{\gamma}<2.1\times 10^{-41}-2.4\times 10^{-42}\,\text{kg}$.

Previously, by analyzing the gravitational time delays from lensed AGNs, Glicenstein (2017) set a severe limit on the photon mass of $m_{\gamma}\leq 1.7\times{10}^{-40}\,\rm kg$. In the present Letter, using the sharp features of the lensed FRB signals, we have obtained the most stringent limit to date on the photon mass through gravitational lensing time delays, namely $\sim 2.1\times 10^{-41}-2.4\times 10^{-42}\,\text{kg}$, which represents an improvement of 1 to 2 orders of magnitude over the results previously obtained from lensed AGNs.

So far, only a lensed FRB candidate with a significance of $3.4\sigma$ has been identified (Chang et al., 2024). Nevertheless, given the high all-sky event rate and sustained efforts in FRB searches, more FRB signals lensed by point-mass lenses with higher significance are expected to be identified in the near future. Due to the short-lived nature and unpredictability of FRBs, it may be hard to perform a full lens modeling with the observed data. The method presented in this work offers an alternative, straightforward way of constraining the photon mass from easily obtained observables of $\Delta t$ and $R$.