Limits on the Weak Equivalence Principle and Photon Mass with FRB 121102 Subpulses

Fast radio bursts (FRBs) are short-duration radio transients with anomalously high dispersion measure in excess of the Galactic value (DM ≳ 200 pc cm⁻³; Lorimer et al. 2007; Keane et al. 2012; Thornton et al. 2013; Petroff et al. 2016). The first repeating burst FRB 121102 was localized in a star-forming dwarf galaxy at z = 0.193, which has confirmed the cosmological origin of FRBs (Scholz et al. 2016; Spitler et al. 2016; Chatterjee et al. 2017; Marcote et al. 2017; Tendulkar et al. 2017). Recently, two nonrepeating FRBs were precisely localized. The first one is FRB 180924, which was localized in a luminous and massive galaxy at redshift 0.3214 by the Australia Square Kilometre Array Pathfinder (Bannister et al. 2019). The other one is FRB 190523, which was localized with the Deep Synoptic Array 10-antenna prototype to a massive galaxy at redshift 0.66 (Ravi et al. 2019).

Although the progenitors and radiation mechanism are still debated, FRBs have been proposed to be promising tools for cosmological and astrophysical studies, such as locating the "missing" baryons (Mcquinn 2014), constraining the cosmological parameters (Gao et al. 2014; Zhou et al. 2014; Yang & Zhang 2016a; Walters et al. 2018), directly measuring Ω_b of the universe (Deng & Zhang 2014; Keane et al. 2016), probing the reionization history of the universe (Deng & Zhang 2014; Zheng et al. 2014; Caleb et al. 2019; Li et al. 2019), probing compact dark matter or precisely measuring the Lemaître–Hubble–Humason constant and the cosmic curvature through gravitationally lensed FRBs (Muñoz et al. 2016; Laha 2018a; Li et al. 2018; Wang & Wang 2018), measuring cosmic proper distances (Yu & Wang 2017), testing Einstein's weak equivalence principle (WEP; Wei et al. 2015; Nusser 2016; Tingay & Kaplan 2016; Wu et al. 2017; Yu et al. 2018), and constraining the rest mass of the photon (Bonetti et al. 2016, 2017; Wu et al. 2016b; Shao & Zhang 2017).

FRB emission arrives later at lower radio frequencies. In principle, the observed time delay for a cosmic transient between two different energy bands should include various terms (Gao et al. 2015; Wei et al. 2015), such as the intrinsic (astrophysical) time delay Δt_int, which means photons with different energies were emitted sequentially due to the dynamical process or radiation mechanism, the time delay contribution from the dispersion by the line-of-sight free electron content Δt_DM, the potential time delay caused by special relativistic effects (Δt_spe) in the case where the photons have a rest mass that is nonzero, and the potential time delay caused by the violation of Einstein's WEP (Δt_gra) where photons with different energies follow different trajectories while traveling in the same gravitational potential.⁹ In FRB observations, the arrival time delay Δt_obs is around 1 s and at a given frequency ν follows a ν⁻² law (Lorimer et al. 2007; Keane et al. 2012; Thornton et al. 2013; Petroff et al. 2016). This is compatible with both plasma and massive photon dispersion (de Broglie 1940). The time delay is routinely attributed to plasma, but there is not an independent, that is, by other means, confirmation that such an attribution is entirely correct.

In this case, a conservative constraint on the WEP can be obtained under the assumption that Δt_obs is mainly contributed by Δt_gra. Using FRB 110220 and two possible FRB/gamma-ray burst (GRB) association systems (FRB/GRB 101011A and FRB/GRB 100704A), Wei et al. (2015) obtained a strict upper limit on the differences of the parameterized post-Newtonian (PPN) parameter γ values, e.g., $\left|\gamma (1.23\ \mathrm{GHz})-\gamma (1.45\ \mathrm{GHz})\right|\lt 4.36\times {10}^{-9}$. Keane et al. (2016) reported the connection between a fading radio transient with FRB 150418, so that a putative host galaxy with redshift 0.492 ± 0.008 was identified (see counter opinions in Williams & Berger 2016, where the counterpart radio transient is claimed to be active galactic nucleus variability instead of an afterglow of FRB 150418). Assuming that 0.492 is the redshift of FRB 150418, Tingay & Kaplan (2016) and Nusser (2016) obtained more stringent upper limits on the differences of γ values as (1–2) × 10⁻⁹ and 10⁻¹²–10⁻¹³ by considering the Milky Way (MW) and the large-scale structure gravitational potential, respectively.

On the other hand, if one assumes that Δt_obs of an FRB is mainly contributed by Δt_spe, a conservative limit on the rest mass of the photon could be placed. For instance, taking z = 0.492 as the redshift of FRB 150418, a conservative upper limit for the rest mass of the photon was placed as m_ph ≤ 5.2 × 10⁻⁴⁷ g, which is 10²⁰ times smaller than the rest mass of the electron (Bonetti et al. 2016; Wu et al. 2016b). Later, Bonetti et al. (2017) applied a similar method to FRB 121102, and they obtained a similar result as m_ph ≤ 3.6 × 10⁻⁴⁷ g.

Most recently, Hessels et al. (2019) found that some bursts in FRB 121102 have complex time–frequency structures that include subbursts with finite bandwidths. After correcting for dispersive delay, the subbursts still show an interesting subpulse time–frequency downward drifting pattern, namely, the characteristic frequencies for subpulses drift lower at later times in the total burst envelope. The same features are also found in the second discovered repeating FRB source, FRB 180814.J0422+73 (CHIME/FRB Collaboration et al. 2019). Such features could be intrinsic (e.g., related to the burst emission mechanism; Wang et al. 2019), or they could also be imparted by propagation effects (Cordes et al. 2017; Hessels et al. 2019). Plasma lensing may cause upward and downward subpulse drifts, but only downward drifting is observed so far in the repeating FRBs. In the 1.1–1.7 GHz band, the typical time spans for subpulses are ∼0.5–1 ms, with a characteristic drift rate of ∼200 MHz ms⁻¹ toward lower frequencies.¹⁰ Considering that FRB 121102 is the only FRB with confirmed redshift observations, and the time delay between subpulses is almost 10⁴ times smaller than the dispersive delay, it is of great interest to place limits on the WEP and the photon mass with the FRB 121102 subpulses.

The Einstein WEP is an important foundation of many metric theories of gravity, including general relativity. One statement of the WEP is that test particles traveling in the same gravitational potential will follow the same trajectory, regardless of their internal structure and composition (Will 2006). Therefore, it has long been proposed that the accuracy of the WEP can be constrained with the time delay for different types of messenger particles (e.g., photons, neutrinos, or gravitational waves), or the same types of particles but with different energies or different polarization states, which are simultaneously radiated from the same astronomical sources.

According to the Shapiro time delay effect (Shapiro 1964), the time interval required for test particles to traverse a given distance would be longer by

$$\begin{eqnarray}&&{t}_{\mathrm{gra}}=-\displaystyle \frac{1+\gamma }{{c}^{3}}{\int }_{{r}_{e}}^{{r}_{o}}\,U(r){dr},\end{eqnarray} \tag{ 1 }$$

in the presence of a gravitational potential U(r), where the test particles are emitted at r_e and received at r_o. Here γ is one of the PPN parameters, which reflects how much space curvature is produced by unit rest mass. When the WEP is invalid, different particles might correspond to different γ values. In this case, two particles emitted simultaneously from the source will arrive at the Earth with a time delay difference

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{gra}}=\displaystyle \frac{{\gamma }_{1}-{\gamma }_{2}}{{c}^{3}}{\int }_{{r}_{o}}^{{r}_{e}}\,U(r){dr},\end{eqnarray} \tag{ 2 }$$

where γ₁ and γ₂ correspond to two different test particles. For a cosmic source, in principle, U(r) has contributions from the host galaxy potential U_host(r), the intergalactic potential U_IG(r), and the local gravitational potential U_local(r). Since the potential models for U_IG(r) and U_host(r) are poorly known, for the purposes of obtaining a lower limit, it is reasonable to extend the local potential out to cosmic scales to bracket from below the potential function of U_IG(r) and U_host(r). In the previous works, the gravitational potential of the MW or the Laniakea supercluster (Tully et al. 2014) has been used as the local potential, which could be expressed as a Keplerian potential¹¹ U(r) = −GM/r. In this case, we have

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}{t}_{\mathrm{gra}}=\left({\gamma }_{1}-{\gamma }_{2}\right)\displaystyle \frac{{GM}}{{c}^{3}}\times \\ \mathrm{ln}\left\{\displaystyle \frac{\left[d+{\left({d}^{2}-{b}^{2}\right)}^{1/2}\right]\left[{r}_{L}+{s}_{{\rm{n}}}{\left({r}_{L}^{2}-{b}^{2}\right)}^{1/2}\right]}{{b}^{2}}\right\},\end{array}\end{eqnarray} \tag{ 3 }$$

where d is the distance from the transient to the MW/Laniakea center and b is the impact parameter of the light rays relative to the center. Here we use s_n = +1 or s_n = −1 to denote the cases where the transient is located along the direction of the MW/Laniakea or anti-MW/Laniakea center. For a cosmic source, d is approximated as the distance from the source to the Earth. The impact parameter can be estimated as

$$\begin{eqnarray}&&b={r}_{L}\sqrt{1-{(\sin {\delta }_{s}\sin {\delta }_{L}+\cos {\delta }_{s}\cos {\delta }_{L}\cos ({\beta }_{s}-{\beta }_{L}))}^{2}},\end{eqnarray} \tag{ 4 }$$

where β_s and δ_s are the source coordinates, β_L and δ_L represent the coordinates of the local (MW/Laniakea) center, and r_L is the distance from the Earth to the center.

In the literature, many investigations have been done to achieve stringent limits on γ differences between particles emitted from the same astrophysical sources, such as supernovae 1987A (Krauss & Tremaine 1988; Longo 1988), GRBs (Gao et al. 2015; Wei et al. 2016b; Wu et al. 2017; Yang et al. 2017; Wei & Wu 2019), FRBs (Wei et al. 2015; Nusser 2016; Tingay & Kaplan 2016; Wu et al. 2017), blazars (Wang et al. 2016; Wei et al. 2016a, 2019), the Crab pulsar (Yang & Zhang 2016b; Zhang & Gong 2017), and gravitational-wave (GW) sources (Kahya & Desai 2016; Wu et al. 2016a; Abbott et al. 2017; Wei et al. 2017a; Shoemaker & Murase 2018). The previous results are summarized in Table 1. When the test particles are of different species, the best constraint is $\left|{\gamma }_{1}-{\gamma }_{2}\right|\lt 1.3\times {10}^{-13}$ for keV photons and the TeV neutrino from GRB 110521B (Wei et al. 2016b). When the test particles are the same species but with different energies, the best constraint is $\left|{\gamma }_{1}-{\gamma }_{2}\right|\lt (0.6-1.8)\times {10}^{-15}$ for 8.15–10.35 GHz photons from the Crab pulsar (Yang & Zhang 2016b). When the test particles are of the same species but with different polarization states, the best constraint is $\left|{\gamma }_{1}-{\gamma }_{2}\right|\lt 0.8\times {10}^{-33}$ for polarized gamma-ray photons from GRB 061122 (Wei & Wu 2019).

Here we considered the time–frequency structure of FRB 121102. As shown in Hessels et al. (2019), after correcting for dispersive delay, several subpulses still exist in some of the FRB 121102 repeated bursts. Each subpulse corresponds to its own characteristic frequency, and subpulses with higher frequencies arrive earlier. Assuming that these subpulses are emitted simultaneously and the observed time delay is mainly caused by the WEP effect, a stringent limit on the WEP can be placed by using the time delay between any two subpulses. In order to get more stringent constraints, here we consider the closest neighboring subpulses in AO-05, where the time delay between f₁ = 1374.16 MHz and f₂ = 1343.69 MHz is 0.4 ms. With the inferred coordinates and redshifts for FRB 121102 (here we adopt R.A. = β_s = 05^h32^m, decl. = δ_s = +33°08' and z = 0.193; Spitler et al. 2016), we obtain

$$\begin{eqnarray}&&\left|{\gamma }_{1}-{\gamma }_{2}\right|\lt 2.5\times {10}^{-16},\end{eqnarray} \tag{ 5 }$$

where we consider the gravitational potential of the Laniakea supercluster as the local potential, the Great Attractor (β_L = 10^h32^m, δ_L = −46°00') is adopted as the gravitational center of Laniakea (Lynden-Bell et al. 1988), M_L ≃ 10¹⁷M_⊙ is the Laniakea mass, and r_L = 79 Mpc is the distance from the Earth to the center of Laniakea (Tully et al. 2014). The result is 1000 times better than previous constraints from FRBs and 4 times better than previous best constraints for the case when the test particles are of the same species but with different energies. It is worth noticing that the uncertainty of the adopted gravitational potential would affect the constraints. Conservatively, we can consider the Keplerian potential for the MW, where the impact parameter can be estimated as $b\,={r}_{G}\sqrt{1-{(\sin {\delta }_{s}\sin {\delta }_{G}+\cos {\delta }_{s}\cos {\delta }_{G}\cos ({\beta }_{s}-{\beta }_{G}))}^{2}}$, where r_G = 8.3 kpc is the distance from the Earth to the Galaxy center, and (β_G = 17^h45^m 4004, δ_G = −29°00' 281) are the coordinates of the Galaxy center in the equatorial coordinate system (J2000; Gillessen et al. 2009). In this case, we have

$$\begin{eqnarray}&&\left|{\gamma }_{1}-{\gamma }_{2}\right|\lt 9.6\times {10}^{-12},\end{eqnarray} \tag{ 6 }$$

where the Galaxy mass is adopted as M = 6 × 10¹¹M_☉ (Kafle et al. 2012).

The postulate that all electromagnetic radiation propagates in vacuum at the constant speed c, namely, that the photons should have a zero rest mass, is one of the most important foundations of Einstein's theory of special relativity. If the photon mass is nonzero, a mass term should be added to the Lagrangian density for the electromagnetic field to describe the effective range of the electromagnetic interaction (de Broglie 1923, 1940; Proca 1936). In this case, some abnormal phenomena will appear for the electromagnetic potentials and their derivatives, for instance, the speed of light is no longer constant but depends on the frequency of the photons, magnetic dipole fields would decrease with distance very rapidly due to the addition of a Yukawa component, longitudinal electromagnetic waves could exist, and so on. Such effects could be applied to make restrictive constraints on the photon rest mass (Goldhaber & Nieto 1971, 2010; Tu et al. 2005; Pani et al. 2012). The current limit on the photon mass accepted by the Particle Data Group (PDG) is from the solar wind observation (Tanabashi et al. 2018), which is set to m_γ ≤ 1.5 × 10⁻⁵¹ g (but see Retinò et al. 2016 for comments and an experiment in the solar wind referring to the PDG upper limit ).

It has long been proposed that the photon rest mass could be constrained by using the frequency-dependent time delays of multiwavelength emissions from astrophysical sources (Lovell et al. 1964; Warner & Nather 1969; Schaefer 1999; Bonetti et al. 2016, 2017; Wu et al. 2016b; Shao & Zhang 2017; Wei et al. 2017b; Wei & Wu 2018). According to Einstein's special relativity, if the photon has a rest mass m_γ, the photon energy can be written as

$$\begin{eqnarray}&&E=h\nu =\sqrt{{p}^{2}{c}^{2}+{m}_{\gamma }^{2}{c}^{4}},\end{eqnarray} \tag{ 7 }$$

where h is the Planck constant. In vacuum, the speed of photons with energy E can be derived as

$$\begin{eqnarray}&&\upsilon =\displaystyle \frac{\partial E}{\partial p}.\end{eqnarray} \tag{ 8 }$$

When m_γ = 0, we have υ = c. If ${m}_{\gamma }\ne 0$, we have

$$\begin{eqnarray}&&\upsilon =\displaystyle \frac{\partial E}{\partial p}=c\sqrt{1-\displaystyle \frac{{m}_{\gamma }^{2}{c}^{4}}{{E}^{2}}}\approx c\left(1-\displaystyle \frac{1}{2}\displaystyle \frac{{m}_{\gamma }^{2}{c}^{4}}{{h}^{2}{\nu }^{2}}\right),\end{eqnarray} \tag{ 9 }$$

where the last approximation is applicable when ${m}_{\gamma }\,\ll h\nu /{c}^{2}\simeq 7\times {10}^{-39}\left(\tfrac{\nu }{\mathrm{GHz}}\right)\ {\rm{g}}$. In this case, two photons with different frequencies, which are emitted simultaneously from the same source, would arrive on the Earth at different times with a time–frequency downward drifting pattern. For a cosmic source, the arrival time difference is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{{m}_{\gamma }}=\displaystyle \frac{{m}_{\gamma }^{2}{c}^{4}}{2{h}^{2}{H}_{0}}\left({\nu }_{l}^{-2}-{\nu }_{h}^{-2}\right){\int }_{0}^{z}\displaystyle \frac{{(1+z^{\prime} )}^{-2}{dz}^{\prime} }{\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}{(1+z^{\prime} )}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}},\end{eqnarray} \tag{ 10 }$$

where H₀ is the Lemaître–Hubble–Humason constant (Lemaître 1927; Hubble 1929; Hubble & Humason 1931). Thus, the photon mass can be constrained as (Wu et al. 2016b)

$$\begin{eqnarray}&&{m}_{\gamma }=\left(1.56\times {10}^{-47}{\rm{g}}\right){\left[\displaystyle \frac{{H}_{70}{\rm{\Delta }}{t}_{{m}_{\gamma }}}{\left({\nu }_{l,9}^{-2}-{\nu }_{h,9}^{-2}\right){\int }_{0}^{z}\tfrac{{(1+z^{\prime} )}^{-2}{dz}^{\prime} }{\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}{(1+z^{\prime} )}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}}\right]}^{1/2},\end{eqnarray} \tag{ 11 }$$

where ν₉ is the radio frequency in units of 10⁹ Hz and H₇₀ is the Lemaître–Hubble–Humason constant in units of 70 km s⁻¹ Mpc⁻¹.

In the literature, many attempts have been made to obtain constraints on the photon rest mass by considering various astrophysical sources, including flare stars (Lovell et al. 1964), the Crab Nebula pulsar (Warner & Nather 1969), FRBs (Bonetti et al. 2016, 2017; Wu et al. 2016b; Shao & Zhang 2017), GRBs (Schaefer 1999), and pulsars in the Large and Small Magellanic Clouds (Wei et al. 2017b; Wei & Wu 2018). The constraint results are shown in Figure 1. The current best constraint on the photon mass through the velocity dispersion method is made by using the radio emissions from FRB 121102, m_γ ≤ 3.6 × 10⁻⁴⁷ g (Bonetti et al. 2017), where the time delay between the whole observational bandwidth is considered, and Δt_obs is on the order of 1 s.

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Here we propose to use the observed time delay between subpulses in FRB 121102 to obtain more stringent constraints on the photon mass, by assuming that the time delay between subpulses is mainly due to the massive photon effect, which is reasonable since the subpulses with lower frequencies arrive later. We adopt the closest neighboring subpulses in AO-05 (Δt_obs = 0.4 ms between f₁ = 1374.16 MHz and f₂ = 1343.69 MHz) and obtain a stringent limit on the photon mass as m_γ < 5.1 × 10⁻⁴⁸ g, where z = 0.193 is adopted for FRB 121102, and the Planck results are adopted for cosmological parameters, e.g., H₀ = 67.8 km s⁻¹ Mpc⁻¹, Ω_m = 0.308, and Ω_Λ = 0.692 (Planck Collaboration et al. 2018).¹² Note that since the redshift of FRB 121102 is relatively small, the uncertainty of cosmological parameters would not affect the constraints too much. For instance, when H₀ varies from 67.8 to 73.2 km s⁻¹ Mpc⁻¹ (Riess et al. 2016), the limit on the photon mass varies from m_γ < 5.1 × 10⁻⁴⁸ g to m_γ < 5.3 × 10⁻⁴⁸ g. When the value of Ω_m varies from 0.2 to 0.4 (Ω_Λ thus varies from 0.8 to 0.6), the limit on the photon mass varies from m_γ < 5.0 × 10⁻⁴⁸ g to m_γ < 5.1 × 10⁻⁴⁸g.

As shown in Figure 1, our result is 10 times better than that obtained using other FRB sources, and ∼10⁴ times better than that obtained by GRBs, ∼10³ times better than that obtained by pulsars in the Large and Small Magellanic Clouds, ∼10⁶ times better than flare stars, and ∼10⁷ times better than the Crab Nebula pulsar.

Using the time–frequency structure of subpulses in some bursts of FRB 121102, here we have obtained a stringent limit on the γ differences between photons with different energies of $\left|{\gamma }_{1}-{\gamma }_{2}\right|\lt 2.5\times {10}^{-16}$, which is 1000 times better than previous constraints from FRBs through similar methods. In addition, we also obtained a stringent constraint on the photon mass of m_γ < 5.1 × 10⁻⁴⁸ g, which is 10 times better than the previous best limits on the photon mass using the velocity dispersion method.

It is worth stressing the advantages of the method for placing limits on the WEP and the photon mass using the time–frequency structure of the subpulses of, e.g., FRB 121102. In previous works, the time delay between the whole observational bandwidth of FRBs (on the order of 1 s) were applied to make constraints on the WEP or the photon mass. Such a time delay at a given frequency ν follows a ν⁻² law, which is compatible with both plasma and massive photon dispersion. Considering that the column density of free electrons inferred from the time delay is roughly consistent with the theoretical predictions (accumulated contributions from MK, IGM, and host galaxy; Chatterjee et al. 2017), the time delay is routinely attributed to plasma, although without an independent confirmation that such attribution is entirely correct. The time–frequency structure of the FRB 121102 subpulses, however, emerges after correcting for dispersive delays. Therefore, the time delay between subpulses is largely reduced to the order of milliseconds or even submilliseconds, which is very advantageous for further improving the accuracy of a basic physical analysis. Moreover, it has been proposed that the observed downward drifting of the subpulse frequency is more likely intrinsic, namely, a generic geometrical effect within the framework of coherent curvature radiation by bunches of electron–positron pairs in the magnetosphere of a neutron star (Wang et al. 2019). If this is the case, the constraints on the WEP and the photon mass would become even tighter.

This work is supported by the National Natural Science Foundation of China (NSFC) under grant Nos. 11690024, 11722324, 11603003, 11633001, 11725314, 11603076, U1831122; the Strategic Priority Research Program of the Chinese Academy of Sciences, grant No. XDB23040100; and the Fundamental Research Funds for the Central Universities. W.Y.W. acknowledges the support from MoST grant 2016YFE0100300; NSFC under grant Nos. 11633004, 11473044, 11653003; and the CAS grant QYZDJ-SSW-SLH017. P.M. acknowledges the Eberly Foundation.