Future Constraints on Dark Matter with Gravitationally Lensed Fast Radio Bursts Detected by BURSTT

Understanding the nature of dark matter is one of the most interesting questions in modern physics. Broadly, dark matter searches have focused on the subatomic matter (Weakly Interacting Massive Particles or WIMPs) and objects in the astrophysical mass domain (Massive Compact Halo Objects, or MACHOs). In the search for WIMPs, considerable efforts have been made. WIMPs are hypothetical particles that can be considered a candidate for dark matter. Recent sub-atomic experiments such as the LUX-ZEPLIN (LZ; Aalbers et al., 2022) experiment, DarkSide-50 experiment (The DarkSide-50 Collaboration et al., 2022) and XENONnT experiment (XENON Collaboration et al., 2023) carried by particle physicists have placed constraints on the parameter space of WIMPs (Kimball & Budker, 2023). Nevertheless, no conclusive evidence for WIMPs as a significant contributor to DM has been found to date. A particular method of gravitational microlensing has been used in recent decades to search for dark matter around our Milky Way galaxy in the form of MACHOS (Alcock et al., 2000). For example, microlensing surveys have derived constraints on mass ranges in the range of 10^-7 to 10 $M_{\odot}$ (e.g., Alcock et al., 2000; Tisserand et al., 2007). Moreover, constraints from the cosmic microwave background exclude masses $>$100 $M_{\odot}$ (Ali-Haïmoud & Kamionkowski, 2017). In MACHO and Expérience pour la Recherche d’Objets Sombres (EROS) (Alcock et al., 2000; Tisserand et al., 2007), steady background sources are monitored on timescales of days to weeks while their apparent brightness is monitored over time. Primordial Black Hole (PBH) distributions within the Local Group can thus be constrained by these searches. On the other hand, gravitational wave observations from mergers of compact binaries (Abbott et al., 2016) have recently revived interest in the possibility that dark matter is mainly compact matter (Laha, 2020), such as PBHs for which the most promising mass range for dark matter is 10-100 $M_{\odot}$.

One of the methods for detecting dark compact objects is through gravitational lensing of radio transients, such as fast radio bursts (FRBs) (Muñoz et al., 2016; Eichler, 2017; Katz et al., 2020; Wucknitz et al., 2021). FRBs are mysterious millisecond radio signals mostly emitted by extragalactic sources (Cordes & Chatterjee, 2019). Although we are not yet able to explain how these millisecond bursts originate or emit, their cosmological distance and abundance make them a good candidate for time-domain gravitational lensing searches. (Muñoz et al., 2016; Eichler, 2017; Katz et al., 2020; Wucknitz et al., 2021). As the FRB propagates around a foreground mass, coherent multiple images of the FRB are generated with small differences in the timing, which can be resolved in the time domain as interference fringes (Kader et al., 2022). Meanwhile, FRBs have been found in other galaxies (e.g., Chatterjee et al., 2017; Ravi et al., 2019; Bannister et al., 2019). Using FRBs as probes, one could constrain cosmological abundances of PBHs rather than local abundances in the future. The current challenge in searching for gravitationally lensed FRBs is that the detection rate of FRBs is still too low. Since the first discovery of FRBs more than a decade ago (Lorimer et al., 2007), astronomers have observed more and more FRBs every year. Among all of the search projects, the most successful project in finding FRBs has been CHIME (CHIME/FRB Collaboration et al., 2018). Due to its large field of view (FoV), CHIME has outperformed other radio arrays detecting about 600 FRBs since 2018 (Masui & Chime/Frb Collaboration, 2021), which is about 5/6 of the current observed number of FRBs. However, with this number of FRBs, no gravitationally lensed FRB has been confirmed. Leung et al. (2022); Kader et al. (2022) have observed no lenses from 172 bursts of 114 independent events of the CHIME catalog. They have placed an upper limit on the constraint of dark matter made of compact objects, such as PBHs. Laser Interferometer Gravitational-Wave Observatory (LIGO)/VIRGO interferometer has suggested a hint to the dark matter fraction (Abbott et al., 2016). Based on LIGO’s result, Bird et al. (2016) suggest the black hole merger rate is consistent with all dark matter being $\sim$ 30 $M_{\odot}$ black holes. If this is the case, about 1$-$2% of all FRBs will pass close enough to such a PBH to be microlensed (Muñoz et al., 2016) and 10⁴ FRBs would yield about 100$-$200 microlensed FRBs. If, on the other hand, no lenses are found in $10^{4}$ FRBs, the dark matter fraction in PBHs can be constrained to be below 1% (Muñoz et al., 2016).

Taiwan plans to build a new radio telescope, the Bustling Universe Radio Survey Telescope in Taiwan (BURSTT; Lin et al., 2022) which is dedicated to detecting FRBs with an accurate localization capability ($\sim$ 1”). BURSTT will be the next frontier telescope in FRB science after CHIME. Because of the telescope’s unique fisheye design, it will have a massive 1 steradian(sr) FoV, meaning its survey area will be 25 times greater than CHIME, covering more FRB-like events. BURSTT will have a 256-antenna array in the first phase and it will be extended to a 2048-antenna array in the second phase. This improves the sensitivity by about 8 times. Moreover, BURSTT will conduct long-term monitoring observations to prevent missing any repeating FRBs. Regarding the time resolution, BURSTT has a ring buffer to record the voltage data, which the timing resolution is adjustable to be about a nanosecond (H.-H. Lin 2022, private communication). If we detect any gravitationally lensed nanosecond FRB, we can use it to constrain dark matter down to $10^{-4}M_{\odot}$ of the lensing mass (see Eq. 2 for more detail). The proposed BURSTT opens up the exciting possibility for the searches of gravitationally lensed FRBs which can help us constrain the fraction of dark matter better. In this work, we focus on the future prospect of BURSTT-2048. We aim to give an estimation of how many lensed FRBs will be detected by BURSTT-2048.

This paper is composed as follows; we describe the gravitational lensing model in Section 2. The predicted number of lensed FRBs to be detected by BURSTT and the calculation are presented in Section 3. Discussion in 4 and conclusion are in Section 5. Throughout this paper, we adopt the AB magnitude system and assume a cosmology with H₀ = 70 km s^-1Mpc^-1, $\Omega_{\Lambda}$ = 0.7, and $\Omega_{\text{M}}$ = 0.3 (Spergel et al., 2003).

In this section, for strong lensing by compact objects, we calculate the optical depth by calculating microlensing effects on a given FRB. These results allow us to calculate the number of lensed FRBs that can be expected when all dark matter is PBHs with a combination of different redshift distributions. We consider compact objects that may be modeled using a point-mass lens, which is the simplest kind of lens model from Muñoz et al. (2016), in which a PBH of mass $M_{L}$. We do not consider a mass spectrum but only fixed mass cases for $M_{L}$, which can be modeled as point lenses around the Einstein radius:

The (angular-diameter) distances from the source, to the lens, and between the source and the lens are represented by $D_{S},D_{L}$, and $D_{LS}$, respectively (Takahashi & Nakamura, 2003). With a point lens, two images are formed at positions $\theta_{\pm}=\left(\beta\pm\sqrt{\beta^{2}+4\theta_{E}^{2}}\right)/2$, where $\beta$ corresponds to impact angle (Muñoz et al., 2016). The time delay between these two images is

where $y\equiv\beta/\theta_{E}$ represents the normalized impact parameter, and the lens redshift is $z_{L}$ (Muñoz et al., 2016). In this equation, the time delay, $\Delta t$ is mainly affected by the lensing mass, $M_{L}$. In addition, we follow Muñoz et al. (2016) to define the $R_{f}$ as the flux ratio of the magnifications of both images $\mu_{+}$ and $\mu_{-}$; i.e.,

Following Muñoz et al. (2016), an FRB must meet three conditions to be considered strongly lensed. First, the brighter image between the two pulses has a signal-to-noise ratio of 10 or higher (Petroff et al., 2014). Second, the observed time delay is longer than some reference time $\overline{\Delta t}$ (e.g., observational resolution), and therefore a lower bound will be placed on the impact parameter $y>y_{\min}\left(M_{L},z_{L}\right)$, as calculated by Eq. 2. Finally, we require that the flux ratio $R_{f}$ is lower than a critical value for $\bar{R}_{f}$ (which we take to be redshift independent) so that both events are observed. By doing this, the impact parameter is forced to be smaller than (Muñoz et al., 2016):

Muñoz et al. (2016) calculated the probability for an FRB to be lensed as follows. The lensing optical depth of a source at redshift $z_{S}$ is given by

Here, $\chi(z)$ represents the comoving distance at redshift $z$, $n_{L}$ is the comoving number density of lenses comoving, and $\sigma$ is the lensing cross-section of a point lens with mass $M_{L}$, expressed as an annulus between the maximum and minimum impact parameters by the amount of mass. The equation is as follows

Using the Hubble parameter both at the redshift of the lens, $H\left(z_{L}\right)$, and present $H_{0}$, we can recast Eq. 5 as

where $\Omega_{c}=0.24$ is the cold-dark-matter density at present and $\Omega_{m}=\Omega_{c}+\Omega_{b}$. The only remaining dependence on the lens mass $M_{L}$ is through $y_{\min}$.

Given a normalized observed distribution function $N(z)$ for FRBs with redshift, $z$, (Masui & Chime/Frb Collaboration, 2021; Hashimoto et al., 2022), we can calculate their integrated optical depth $\bar{\tau}\left(M_{L}\right)$, due to compact objects of mass $M_{L}$, as

Here, $\tau$ is included because it becomes approximately the same as the lensing probability when $\tau$ $\ll$ 1. We show this quantity in Fig. 1 for different time delays with a redshift cutoff at $z_{\rm cut}=0.5$ and $f_{\mathrm{DM}}$ = 1 (assuming all the dark matter was composed of PBHs). Muñoz et al. (2016) adopted a $z$ = 0.5 cutoff redshift because their redshift distribution function fits well with the observed redshift distribution of FRB samples (FRBCAT, Petroff et al., 2016) if $z$ = 0.5 is chosen. We applied the redshift cutoff at $z_{\rm cut}=0.5$ as Muñoz et al. (2016) since the redshift distributions from both works peak at $z$ $\sim$ 0.5. In this figure, $N(z)=1$ is assumed. We take an empirically derived $N(z)$ from CHIME FRBs into account in Section 3.2. It is shown that the distribution shifts to a smaller lensing mass as the time delay reduces. It is important to note that, even if all the dark matter consisted of a single mass $M_{L}$, there will be no unique value for the lensing time delays induced on FRBs because of the different redshifts of lenses and impact parameters.

BURSTT is a proposed instrument tailored for detecting FRB-like events with accurate arcsecond localization and high cadence (Lin et al., 2022). BURSTT will observe 25 times more sky than CHIME, because of its unique fisheye design and FoV of 1 sr. BURSTT’s modular nature allows it to be expanded by adding additional antennas or outrigger stations. The main station could be upgraded from 256 antennas to 2048 antennas in the next phase. The system equivalent flux density (SEFD) will improve about 8 times from $\sim$ 5,000 Jy in the first phase to $\sim$ 600 Jy in the second phase. This would improve the sensitivity to detect more bursts, which improves the detection rate. At the same time, BURSTT is an interferometer with baselines of over 100 km and has an astrometric accuracy for FRB localizations of better than 1”, more precise than that of CHIME (2- and 25-arcsec precision between the CHIME and the CHIME Pathfinder; Leung et al., 2021), ASKAP (arcsecond accuracy, Bhandari et al., 2020), and Deep Synoptic Array (DSA, $\pm 2.5"$ accuracy; Ravi et al., 2019). This allows us to discover a large sample of bright FRB-like events with more accurate localization, including those located close to the Earth. In this section, we use parameters from BURSTT-2048 and CHIME to calculate a predicted number of gravitationally lensed FRB to be detected by the proposed BURSTT-2048.

In this work, it has been demonstrated that coherent FRB lensing can constrain the constituents of the cosmological dark matter, e.g., PBHs with the parameters of BURSTT-2048. We consider compact objects which can be modeled by a point-mass lens (Muñoz et al., 2016). Each FRB in the 536-FRB CHIME catalog is evaluated for optical depth $\tau$. To obtain the integrated optical depth in each redshift bin, we add the individual optical depth $\tau$ into each bin. Each of these numbers represents the number of lensed FRBs detected by BURSTT for each redshift bin. In our estimation, we assume if all the dark matter is composed of PBHs ($f_{\mathrm{DM}}=1$), BURSTT-2048 can detect up to $\sim$ 2 lensed FRBs out of a total number $\sim$ 1,700 FRBs per annum. With this amount of lensed FRBs and BURSTT’s ability to detect nanosecond bursts, we can constrain PBHs as dark matter in the $10^{-4}-100M_{\odot}$ range.