Dark photon dark matter and fast radio bursts

The nature of dark matter is still a mystery that may indicate the necessity for extensions of the Standard Model (SM). The lack of positive signatures for well-known dark matter candidates, such as Weakly Interacting Massive Particles, opens new avenues of exploration. Among them, a new abelian gauge field (dark photon), which kinetically mixes with the SM hypercharge field, is a possible mediator of the interaction between a dark matter candidate and the SM particles. Light dark photons may also comprise partially or entirely the observed dark matter density and existing limits for the dark photon dark matter parameter space arise from several cosmological and astrophysical sources. In the present work we investigate dark photon dark matter using cosmic transients, specifically fast radio bursts (FRBs). The observed time delay of radio photons with different energies have been used to constrain the photon mass or the Weak Equivalence Principle (WEP), for example. Due to the mixing between the visible and the dark photon, the time delay of photons from these cosmic transients, caused by free electrons in the intergalactic medium, can change and impact those constraints from FRBs. We use five detected FRBs (FRB 110220, FRB 121102, FRB 150418, FRB180924 and FRB 190523) and two associations of FRBs with gamma-ray bursts (FRB/GRB 101011A and FRB/GRB 100704A) to investigate the correspondent variation on the time delay caused by the presence of dark photon dark matter. The result is virtually independent of the FRB used and this variation is very small to be detectable in the near future, considering the still allowed dark photon dark matter parameter space. The existing bounds on the photon mass or the WEP arising from FRBs are, therefore, not jeopardized.

The nature of dark matter is still a mystery that may indicate the necessity for extensions of the Standard Model (SM). The lack of positive signatures for well-known dark matter candidates, such as Weakly Interacting Massive Particles, opens new avenues of exploration. Among them, a new abelian gauge field (dark photon), which kinetically mixes with the SM hypercharge field, is a possible mediator of the interaction between a dark matter candidate and the SM particles. Light dark photons may also comprise partially or entirely the observed dark matter density and existing limits for the dark photon dark matter parameter space arise from several cosmological and astrophysical sources. In the present work we investigate dark photon dark matter using cosmic transients, specifically fast radio bursts (FRBs). The observed time delay of radio photons with different energies have been used to constrain the photon mass or the Weak Equivalence Principle (WEP), for example. Due to the mixing between the visible and the dark photon, the time delay of photons from these cosmic transients, caused by free electrons in the intergalactic medium, can change and impact those constraints from FRBs. We use five detected FRBs (FRB 110220, FRB 121102, FRB 150418, FRB180924 and FRB 190523) and two associations of FRBs with gamma-ray bursts (FRB/GRB 101011A and FRB/GRB 100704A) to investigate the correspondent variation on the time delay caused by the presence of dark photon dark matter. The result is virtually independent of the FRB used and this variation is very small to be detectable in the near future, considering the still allowed dark photon dark matter parameter space. The existing bounds on the photon mass or the WEP arising from FRBs are, therefore, not jeopardized.

I. INTRODUCTION
Dark matter (DM) is one the biggest puzzles in cosmology, and lately in particles physics, whose existence is a hint for physics beyond the standard model (SM). Weakly Interacting Massive Particles (WIMPs) have been the most well-known DM candidates (see Ref. [1] for a review), but the lack of positive signatures opens new avenues of exploration. Among several extensions of the SM, a new U (1) gauge field was proposed as mediator of the interaction between DM and SM [2][3][4][5][6][7][8][9]. The so-called 'dark photon' (DP) interacts with the visible sector through the kinetic-mixing with the hypercharge field 1 [11][12][13][14][15][16]. Its parameter space has been constrained through plenty of observations and experiments [5,, and has led to theoretical explanations for the (expected) smallness of the kinetic-mixing parameter [13,[39][40][41][42][43][44].
Since the visible photon mixes with DP, the dispersion velocity of the former changes when compared with the absence of the mixing. This fact is reflected in the photon frequency, when it travels through the intergalactic medium (IGM), and as a result the time delay caused by the dispersion in the IGM ∆t IGM may change. In turn, this change could influence current bounds on the photon mass or WEP, for instance.
In this paper we apply five detected FRBs (FRB 110220, FRB 121102, FRB 150418, FRB 180924 and FRB 190523) and two association of FRBs with gammaray bursts (GRB) (FRB/GRB 101011A and FRB/GRB 100704A) to investigate the contribution of DP DM to the time delay caused by IGM effects. The seven FRBs have their source measured or inferred in a redshift range of 0.15 < z < 1, whose time delays of photons with differ-ent frequencies (mostly between 1.2 GHz and 1.5 GHz) are in the range 0.15 s ∆t obs 1 s. The variation of the time delay caused by IGM effects is very small for the still allowed DP DM parameter space, not jeopardizing current bounds on other contributions of the observed time delay.
We organize the paper in the following manner. Sect. II reviews the necessary expressions for DP DM in a charged plasma. In Sect. III we present the detected FRBs and the resulting variation in the IGM time delay. Sect. IV is reserved for conclusions.

II. DARK PHOTON IN A PLASMA
After diagonalizing the photon and DP kinetic terms, the DP Lagrangian takes the form (1) where J µ is the SM electric current.
The IGM has a plasma frequency given by 2 where α is the fine-structure constant, m e is the electron mass and n e (z) = n e,0 (1 + z) 3 is the free electron number density, where the electron number density today is n e,0 ∼ 10 −7 cm −3 [64]. The mixing between visible and hidden photon changes the photon's dispersion relation. In order to reach the appropriate expression, the Proca and Maxwell equations are solved along with the equations for a non-relativistic plasma [10]. In the case of a non-relativistic DP, which is assumed as a DM candidate (k ω), the longitudinal and transverse components of the gauge fields obey the same (mixed) equation. The diagonalization of this equation gives [10] where ν is the frequency of electron-ion collision The electron temperature is about T e ∼ 10 4 − 10 7 K in the IGM [99], and as we shall see ω ∼ ω b , thus ν ω for the parameters in our range of interest. Therefore, for the purpose of this work, we may neglect the imaginary part of Eq. (3).
where ω γ is the photon frequency, while ω A is the DP frequency. On the other hand, when m 2 A ω 2 p the frequencies are The positive-sign solution in Eq. (3) behaves as the photon for m 2 A < ω 2 p , while for m 2 A > ω 2 p is the negative-sign solution that represents the photon frequency.

III. FAST RADIO BURSTS AND DARK PHOTON DARK MATTER
Due to the interaction between photon and DP, the frequency of the former when it travels through the IGM is no longer ω p , but given by Eq. (3).
The observed time delay ∆t obs for FRB photons with different energies may have the following contributions [91,100] ∆t obs = ∆t int + ∆t LIV + ∆t spe + ∆t DM + ∆t grav , (9) where ∆t int is the intrinsic astrophysical time delay, ∆t LIV represents the time delay due to (possible) Lorentz invariance violation, ∆t spe is the time delay caused by photons with a non-zero rest mass, ∆t grav is the Shapiro time delay and ∆t DM is the time delay from the dispersion by the line-of-sight free electron content.
We are interested only in the time delay due to dispersion by free electrons, thus we can ignore all other sources of delay. This time delay ∆t DM , in turn, has contributions due to the host galaxy, the IGM and the Milky Way. However, the host galaxy is usually unknown and the contribution from the Milky Way is much smaller than the one from the IGM [101,102], so that we can consider the the limit where the dispersion measure time delay is solely due to the IGM ∆t DM ≈ ∆t IGM . This limit is translated to conservative bounds on the other contributions in Eq. (9), as constraints on the photon mass [103] or the WEP [91], which can be even more constrained if other contributions are taken into account.
The IGM magnetic effect on the dispersion velocity of photons can be ignored because the Larmor frequency is much smaller than the plasma frequency. The time delay due to the IGM plasma on two photons with frequencies ν l and ν h is [103] ∆t ωγ = ν 2 where ν γ,0 = ω γ,0 /(2π), and we will adopt the cosmological parameters from the Planck satellite, Ω m = 0.315, Ω Λ = 0.685, H 0 = 100h km s −1 Mpc −1 , and h = 0.674 [104]. The correspondent increase in the time delay caused by IGM when DP DM is present is where ∆t IGM is the time delay when ω γ = ω p .
In order obtain ∆t DP , we use five detected FRBs and two combinations of FRBs and gamma-ray bursts (GRB): 3 • FRB 110220 was discovered by the 64-m Parkes telescope [105], localized to coordinates (J2000) RA = 22h34m, Dec = −12 • 24 for photons ranging between 1.2 GHz and 1.5 GHz, and whose inferred redshift of 0.81 was estimated through its dispersion measure value.
• FRB 150418 was also detected by the Parkes telescope [115] in the frequency range 1.2 -1.5 GHz. Although its redshift was claimed to be measured [115], its localization was contested [116]. More recently its redshift was constrained [117] and the result is similar to the original claim.   (10) for ω γ = ω p , for seven detected FRBs.
• FRB 190523 was detected by the Deep Synoptic Array ten-antenna prototype (between frequencies 1.3 GHz and 1.5 GHz) [119] and was localized to J2000 coordinates RA = 13h48m15.6s and Dec = +72 • 28 11 . Its redshift of 0.66 is also one of the few that were determined.
The correspondent observed time delay, the time delay caused by IGM effect for the frequency ω γ = ω p , and the (inferred or measured) redshift are shown in Table I. Using the seven FRBs, we apply Eqs. (3) and (12) to obtain the contribution ∆t DP , whose results are presented in Fig. 2. The observed time delay follows this hierarchical relation ∆t obs ≥ ∆t ωγ ≥ ∆t IGM , and only the positive branch ω + can reach a value that would give an increase in ∆t IGM , possible only for m A < ω p . On the other hand, for m A > ω p the extra time delay is |∆t DP | ≈ ε 2 ∆t IGM , which in turn is much smaller than the one for the ultra-light DP region, considering the still allowed parameter space in Fig. 1. The results are very similar for all FRBs used and we plot only one of them. The extra time delay ∆t DP is very small to have a considerable influence on Eq. (9), therefore it does not impact the existing bounds from the other terms in ∆t obs .

IV. CONCLUSIONS
In this paper we have investigated light DP DM using seven detected FRBs. These observations lie in the redshift range 0.1 < z < 0.8 and have observed time delays between 0.1 s < ∆t obs ≤ 1 s. Due to the mixing between the photon and DP, the photon frequency is no longer equal to the plasma frequency of the IGM, but depends on the DP mass and the kinetic-mixing parameter. Therefore, taking the conservative scenario where the time delay between radio photons of different frequencies caused by their dispersion through the electron plasma is solely due to the IGM, we obtained the possible contribution to this time delay from DP DM. The  [59,60], which include several cosmological and astrophysical constraints [10,49,[59][60][61][62][63][120][121][122], where the vertical blue line is the plasma frequency today.
FIG. 2: Extra time delay ∆t DP (12) due to DP DM for the region m A ≤ ωp. The region m A > ωp has even smaller values for the allowed range of parameters. It is shown in red the existing limits, as in Fig. 1. results are practically insensitive to the FRBs, and the corresponding extra time delays ∆t DP are very small to have an impact in near future observations. The extra time delay ∆t obs − ∆t IGM of FRB photons have been used to constrain other contributions in Eq. (9). Therefore, DP DM does not give a considerable increase in ∆t IGM , which otherwise could change existing limits on photon mass or the WEP, for instance.

ACKNOWLEDGMENTS
We thank Samuel Witte and Gonzalo Alonso-lvarez for comments. This work was supported by CAPES (process 88881.162206/2017-01) and Alexander von Humboldt Foundation.