First Constraints on Compact Dark Matter from Fast Radio Burst Microstructure

Dark matter comprises 24$\%$ of the energy density of the Universe (Bennett et al., 2013), yet its indeterminate form represents one of the largest unsolved problems in astrophysics. Exotic particles from outside the standard model, such as Weakly Interacting Massive Particles (WIMPs) or axions have been invoked as possible explanations (see Bertone et al. (2005) for a review). However, some fraction of dark matter could reside in the Universe as compact objects, such as black holes or neutron stars.

Decades of extensive research has constrained the fraction of dark matter present in compact objects over a range of masses. Low mass objects ($10^{-7}M_{\odot}\lesssim M\lesssim 10$M_⊙) are excluded as the dominant form of dark matter in the Milky Way and environs based on the absence of stellar variability caused by gravitational microlensing (Tisserand et al., 2007; Wyrzykowski et al., 2011; Alcock et al., 1997). High mass objects ($\gtrsim 100$M_⊙) are excluded by the lack of expected kinematic perturbations to wide binary orbits and ultra faint dwarf galaxies (Quinn et al., 2009; Brandt, 2016).

The only population of compact objects that are not well constrained lie in the range of 10 to 100M_⊙. There is a known population of black holes in this mass range; gravitational wave observations by LIGO have detected several mergers of these black holes (LIGO Scientific Collaboration and Virgo Collaboration et al., 2019). Subsequent theories suggest that dark matter composed of $\sim 30$M_⊙ primordial black holes could explain the merger event rates observed by LIGO (Bird et al., 2016; Clesse & García-Bellido, 2017; Sasaki et al., 2016). Better constraints on the fraction of compact dark matter within the 10-100M_⊙ range could therefore be key in identifying some fraction of dark matter.

The strong gravitational lensing of extragalactic transients provides a way to either detect or to place more stringent constraints on dark matter. The strong lensing of type Ia supernovae has been used to limit the compact dark matter fraction to less than  35$\%$ for all objects more massive than 0.01M_⊙ (Zumalacarregui & Seljak, 2018). Recently, it has been realised that cosmological transients such as Gamma Ray Bursts (GRBs) and Fast Radio Bursts (FRBs) will allow constraints to be placed at a much higher significance (Ji et al., 2018; Laha, 2018).

In both cases strong lensing creates multiple images of the source. Unlike the gravitational lensing of quasars by foreground galaxies (Wong et al., 2019), the images formed by a compact object would be too close to be spatially resolved. However, the images of the source will arrive separated in time due to different gravitational and geometric time delays along each path. This temporal separation ($\Delta t$) is linearly dependent upon the lens mass, whereas the magnification ratio is mass independent. The lens mass and geometry can be constrained using these two observables. Due to the achromatic nature of gravitational lensing the same formalism initially suggested by Muñoz et al. (2016) for FRBs can be applied at all wavelengths. The formalism ignores the effect of physical optics, which becomes important when the Einstein radius of the lens is smaller than the Fresnel scale, $\sim\sqrt{D_{\rm eff}\lambda/2\pi}$, where $D_{\rm eff}$ is the effective distance to the lens. This occurs for lens masses less than $\sim 10^{-5}$M_⊙ at a frequency of 1 GHz, and is well below the masses considered here; hence a full wave optics treatment, such as that explored by Jow et al. (2020), is not yet warranted.

Several thousands of GRBs have been discovered at redshifts $z\sim 1$ by dedicated GRB observatories such as Swift, BATSE, and Fermi. The cosmological distances they traverse allow them to probe a large volume of the Universe for compact dark matter. GRBs have a broad temporal profile ranging from milliseconds to minutes (Ji et al., 2018), and as a result, distinguishing multiple images is more difficult as the time delay between signals lensed by a 10M_⊙ compact object will be less than the duration of the GRB. Ji et al. (2018) have proposed auto-correlating the light curve as a method of detecting lensing. They conclude, however, that current GRB observatories would need to reduce their noise power by at least an order of magnitude to be able to detect lensing in the 10-100M_⊙ mass range.

In contrast, FRBs have temporal profiles ranging from tens of microseconds (Cho et al., 2020) to several milliseconds, which is often shorter than the anticipated delay ($\sim 1$ms) for lensing by compact objects in the mass range under consideration here. This enables multiple images to be clearly distinguished, hence rendering FRBs considerably cleaner probes of compact structure along their sightlines. FRBs are highly luminous, extragalactic radio pulses, and those such as FRB 181112 (Cho et al., 2020) with substructure on timescales of a few tens of microseconds provide, to date, the finest timescale probe of sightlines at cosmological distances. Moreover, a unique capability of radio interferometric observations of such bursts is their ability to directly capture the wavefield of the each FRB at extremely high time resolution (3 ns; see Cho et al., 2020). This affords a powerful new diagnostic of the presence of gravitational lensing. The wavefield, which is directly observable at radio wavelengths, of any pair of paths in the lensed signal should be correlated, whereas burst substructure intrinsic to the FRB would not.

Of the FRBs localised to host galaxies so far, all have been at redshifts $z<1$, placing the current sample generally closer in the Universe than GRBs (Ji et al., 2018; Coward et al., 2013; Bannister et al., 2019; Prochaska et al., 2019b). However, this limitation can be overcome by inferring source redshifts from the dispersion measures of non-localised FRBs (Macquart et al., 2020); the existence of FRBs with dispersion measures exceeding $2000\,$pc cm^-3 (e.g. Bhandari et al., 2018) ostensibly places some fraction of the population at $z>2$.

To detect strong lensing, the temporal separation must be sufficiently large to allow each image to be distinguished. This is constrained by the shortest distinct temporal structure in the signal. In this paper we examine the implications of the high-time resolution structure observed in the FRBs 180924 and 181112. In FRB 180924 the shortest timescale corresponds to its rise time of only $30\mu$s (Farah, 2020). FRB 181112 has recognisable temporal structure on the scale of $15\mu$s, the shortest structure observed in an extragalactic radio signal (Cho et al., 2020). The resolution of temporal structures of $\sim 10\mu$s enables searches for lensing at temporal separations an order of magnitude below those considered in previous treatments (0.1 ms; Muñoz et al., 2016; Laha, 2018). The S/N of recorded FRBs allows us to consider magnification ratios an order of magnitude above previous treatments ($<$5; Muñoz et al., 2016; Laha, 2018). Additionally, if the FRB passes close to an intervening galaxy, as it did for FRB 181112 (Prochaska et al., 2019b), it opens up the possibility of examining lensing attributable to a specific galaxy along the burst sightline, other than the host galaxy or the Milky Way. Muñoz et al. (2016) and Laha (2018) report that a sample of 10⁴ FRBs would be required to exclude the compact dark matter fraction to less than 1$\%$. Assuming a $\Lambda$CDM cosmology, we apply the same formalism to estimate the constraining potential of detected high time resolution FRBs comparable to FRBs 181112 and 180924.

In the weak field limit, where the gravitational potential $|\Phi|\ll c^{2}$, gravitational lensing can be modelled as an achromatic deflection of incident light by a thin screen. Under this treatment, a point mass lens will produce two images on the lens plane. The temporal separation, magnification ratio and position of these images are determined by the angular impact parameter of the source ($\beta$) normalised by the characteristic Einstein radius of the lens ($y=\beta/\theta_{E}$).

Here we briefly review previous theory as expounded by Muñoz et al. (2016) and Laha (2018). Following this formalism, the difference in arrival time between the images ($\Delta t$) and the ratio of each magnification ($R_{f}$) correspond to unique normalised angular impact parameters of the source $y_{\Delta t}$ and $y_{R_{f}}$, respectively. The relation between $y_{R_{f}}$ and $R_{f}$ can be expressed analytically as (Muñoz et al., 2016),

which is notably independent of the lens mass. Conversely, $y_{\Delta t}$ cannot be derived analytically and is found numerically from Muñoz et al. (2016):

where $M_{L}$ and $z_{L}$ are the mass and redshift of the lens, respectively.

To detect gravitational lensing, we require the normalised angular impact parameter to be within the observable range ($y_{\text{min}}$–$y_{\text{max}}$). This range is defined by two conditions: (1) The associated time delay calculated from eq. (2) must be less than the maximum observable time delay $\Delta t_{\text{max}}$ and greater than the minimum distinguishable separation $\Delta t_{\text{min}}$. The length of the observation sets $\Delta t_{\text{max}}$, and $\Delta t_{\text{min}}$ is set by the structure in the pulse profile (Muñoz et al., 2016). (2) The magnification ratio must be below the maximum ($\bar{R_{f}}$) set by the detection threshold (Muñoz et al., 2016).

For the thin screen approximation to be valid, the gravitational field at the impact parameter must also satisfy the weak field condition:

where $R_{S}$ and $D_{L}$ are respectively, the Schwarzschild radius and angular diameter distance of the lens. $y_{\text{min}}$ and $y_{\text{max}}$ define the annulus of the cross section to observable lensing. This cross section can then be used to calculate the observable lensing optical depth. Details on this calculation are provided in the following subsections for different environments. If the fraction of all dark matter that is compact ($f_{\text{DM}}$) is assumed to be constant, the probability of observing lensing ($P_{L}$) at least once in a set of $N$ FRBs can then be calculated as

where $\tau_{i}$ is the optical depth of the ith FRB in the set. To exclude compact dark matter fractions of $\geq f_{\text{DM}}$ with 95$\%$ confidence, we require a null observation of lensing in a set of FRBs with a cumulative observable lensing optical depth of 3.0.

The determination of the redshift of an FRB, either by localisation or by inference from its dispersion measure (Macquart et al., 2020), allows the formalism outlined in Section 2 to be applied. Here, we calculate the halo and IGM lensing optical depth for localised FRBs 181112 (Prochaska et al., 2019b) and 180924 (Bannister et al., 2019). The temporal microstructure of these bursts has been resolved, enabling us to probe to the minimum value of $y_{\text{min}}$ allowed by the burst structure. FRBs 181112 and 180924 probe a similar range of masses ($0.1$M${}_{\odot}\lesssim$M$\lesssim 10^{4}$M_⊙) due to their similar minimum and maximum temporal separations (Table 1). Over this range of masses, eq. (3) is satisfied, and the strong field region is orders of magnitude smaller than the spatial scale probed by a temporal separation of $10\,\mu$s. This is the scale of the smallest distinguishable temporal separation amongst known FRBs; therefore, the weak field approximation is valid for all cases considered here.

The spectra of FRB 181112 shown in Fig. 1 (see also Cho et al., 2020) exhibits multi-peaked structure that could potentially be explained by gravitational lensing. Indeed, if the two major peaks are assumed to be two images, the temporal profile is consistent with gravitational lensing by a $\sim 10$M_⊙ compact object in the halo of the foreground galaxy (hence referred to as FG 181112). Cho et al. (2020) test for the presence of microlensing by searching for correlations in the burst wavefield with time; in the case of FRB 181112 no fringes between sub-pulses were seen, suggesting the pulse multiplicity is more likely intrinsic to the burst, rather than multiple lensed copies of the same burst. However, the absence of a correlation is not definitive since other effects, notably due to differences in any turbulent cold plasma encountered along the slightly separated sightlines of the lensed images, could scatter the radiation in different manners, and thus destroy the phase coherence between the lensed signals. However, in the present instance Cho et al. (2020) also find that the polarization properties of the sub-bursts differ in detail, particularly in their circular polarization, an effect which is difficult to attribute to lensing ⁴⁴4We can exclude circular polarization differences due to the existence of any relativistic plasma from a neutron star along the sightline, except at the source (where its presence would be irrelevant for the present argument). The lens mass of $10$M_⊙ required to explain the sub-burst time delays is significantly above the largest observed neutron star mass of 2.14M_⊙ (Cromartie et al., 2020), ruling out neutron stars as potential lens candidates, and the effects of any relativistic plasma associated with them..

As recorded in Table 1, FRB 181112 had an extremely narrow pulse profile, with its shortest temporal structure being 15 $\mu s$. FRB 180924 had an extended scattering timescale of 580$\,\mu$s but a short rise-time of 30$\,\mu$s. Were any delayed lensed signal present, it would also have a sharp 30$\,\mu$s rise time which would have been detectable within the tail of the overall pulse envelope. The maximum magnification ratios ($\bar{R_{f}}$) and redshifts are similar for each burst. To calculate $\bar{R_{f}}$, the S/N (signal-to-noise ratio) of the primary peak is divided by the detection threshold (3$\sigma$). A key difference between the two is that FRB 181112 passed within 29kpc of FG 181112, allowing it to probe a longer path through a galactic halo. In the following optical depth calculations these parameters are used to determine $y_{\text{min}}$ and $y_{\text{max}}$ from the equations defined in Section 2. For all following calculations, we use values for $H_{0}$ and the cosmological density parameters from the Planck 2018 results (Planck Collaboration et al., 2018).

As a consequence of the greatly improved temporal resolution of FRBs 181112 and 180924 we have been able to probe to much smaller mass scales than considered in previous treatments. Longer observation of FRBs would allow greater maximum temporal separations to be observed, extending the mass independent regime of any constraints to higher masses. Improvements to sensitivity will boost S/N and increase $y_{\text{max}}$ in the magnification-limited regime. A larger $y_{\text{max}}$ yields a larger cross section and consequently a greater observable lensing optical depth, thus providing a more sensitive probe to small scale structure.

FRBs captured at high time resolution represent an opportunity to explore fine structure of galaxy halos and clusters on unprecedented scales. We have focused here on the potential for FRBs to detect compact objects and derived simple constraints on non-baryonic dark matter models. The favoured $\Lambda$CDM cosmology is well known for its success describing the large scale structure of our Universe, but it faces a number of challenges on length scales below 1 Mpc (Bullock & Boylan-Kolchin, 2017). To meet these challenges, the substructure of dark halos must be understood, and, as shown, high time resolution FRBs provide us with the means to do so by directly constraining the inner scale of hierarchically clustered dark matter.

To exclude lensing with 95$\%$ confidence, a cumulative optical depth of 3.0 is required. From Figures 2 and 3, we estimate the optical depth probed by an FRB similar to those considered here at a range of compact dark matter fractions. The cumulative optical depth probed by a set of FRBs is simply the summation of their individual optical depths as per eq. (4). Hence, we can predict the number of FRBs that would be required to make a desired constraint. The number required varies with $f_{\text{DM}}$, the desired confidence level and the assumed distribution (e.g. in halos or distributed throughout the cosmic web). The cumulative optical depth required is non–linear with the desired level of confidence, and, hence, a lesser constraint of 80-90$\%$ would require a sample of 54-77$\%$ the size, respectively. Conversely, the cumulative optical depth required, is linear with the compact dark matter fraction, i.e. to exclude the compact dark matter fraction with the same confidence to below 0.5$f_{\text{DM}}$ requires a sample twice the size.

In summary, recent FRBs detections, made at high time resolution, have revealed the potential of FRBs to probe dark matter within our Universe. The fact that FRBs have narrower temporal structure than previously assumed in gravitational lensing studies, allows searches for smaller lens masses than previously considered. The probability of observing halo lensing, in an FRB similar to 181112, is $\sim 0.017$ (assuming $f_{DM}\leq$0.35). To exclude $f_{DM}\geq 0.35$, in galaxy halos, would require a sample of $\sim 170$ FRBs like FRB 181112. The probability of observing lensing anywhere along the sightline, in an FRB similar to FRB 181112 or FRB 180924, is $\sim 0.023$ (assuming $f_{DM}\leq$0.35). This is a lower limit, in the sense that a large fraction of FRBs have dispersion measures that place them at higher redshifts than these two bursts and it ignores the possibility that the sample of already detected bursts favours lensed events through magnification bias. Thus, it is possible that a significant number of the sample of $>100$ FRBs known to date have been lensed, although the lower time resolution and lower S/N of a large fraction of these previous detections would substantially hinder the discoverability of any lensing signal. To exclude $f_{DM}\geq 0.35$, in the IGM, would require detection of $\sim 130$ FRBs similar to FRB 181112 or FRB 180924. Finally, we conclude that when distributed as a uniform field of compact objects, the volume filling factor of dark matter in FG 181112 is likely insufficient to contribute to the temporal scatter-broadening of FRBs on nanosecond to microsecond timescales. However, the gravitational scattering of FRBs does present a promising probe of hierarchically clustered dark matter.