Constraints on compact dark matter with fast radio burst observations

Fast Radio Bursts (FRBs) are bright radio transients with millisecond duration at cosmological distances. Since compact dark matter (CDM) could act as lenses and cause split of this kind of very short duration signals, Mu$\rm{\tilde{n}}$oz et al. (2016) has proposed a novel method to probe CDM with lensing of FRBs. In this Letter, we for the first time apply this method to real data and give constraints of the nature of CDM with currently available FRB observations. We emphasize the information from dynamic spectra of FRBs is quite necessary for identifying any lensed signals and find no echoes in the existing data. The null search gives a constraint comparable to that from galactic wide binaries, though the methods of redshift inference from dispersion measure would impact a little. Furthermore, we make an improved forecast basing on the distributions of real data for the CHIME-like experiments. Finally, we discuss the situation where one or more lensed signals will be detected. In such a case, the CDM parameter space can be pinned down very well since the lens mass can be directly determined through the observed flux ratio and time delay between split images.


INTRODUCTION
A wide range of galactic and cosmological observations has verified the existing of dark matter, which contributes a considerable part of the total energy density in the universe. The cold dark matter model has successfully explained the observed large scale structure. However, we still know little about the constituent of dark matter on smaller scales and some issues exist in this model. For example, according to the simulation, galaxies like the Milky Way should have thousands of dark matter subhalos surviving from the tide stripping process and appearing in the form of satellite dwarf galaxies, whereas only ∼ 10 such dwarfs have been observed in our galaxy and Andromeda M31 galaxy. Furthermore, one may conjecture that dark matter (or part of it) consists of compact objects, such as the massive compact halo objects (MACHOs) (Wyrzykowski et al. 2011;Pooley et al. 2009;Mediavilla et al. 2009;Monroy-Rodriguez et al. 2014), primordial black holes (PBHs) (Carr et al. 1974(Carr et al. , 1975, axion mini-clusters (Hardy et al. 2017) and compact mini halos (Ricotti et al. 2009). For convenience, hereafter we take all of them as the compact dark matter (CDM). Some theoretical analysis allows the mass of CDM to be as light as 10 −7 M and as heavy as the first stars ∼ 10 3 M (Griest et al. 1991).
Probing CDM observationally is therefore crucial to discriminate models and deepen our understandings on the nature of dark matter. Efforts have been devoted with various approaches and some progress has been made in constraining the CDM fraction in dark matter f CDM and the mass M CDM . While large-mass (≥ 100M ) CDM can perturb the wide stellar bina-liaokai@whut.edu.cn ries (Quinn et al. 2009), the microlensing of stars can constrain the CDM in the Milky way with low-mass (≤ 10M ) (Tisserand et al. 2007;Wyrzykowski et al. 2011;Udalski et al. 2015;Calchi Novati et al. 2013;Niikura et al. 2017). Besides, by observing the lack of radiation as a result of accretion, one could also give a constraint for large-mass CDM with the cosmic microwave background (Ali-Haimoud et al. 2017). Other methods include millilensing of quasars (Wilkinson et al. 2001), lensing of supernovae (Benton et al. 2007), ultra-faint dwarf galaxies (Brandt et al. 2016) and caustic crossing (Oguri et al. 2018). Generally speaking, no robust evidence of CDM has been found for f CDM > 0.1 in a wide mass range.
The mass range 10 − 100M has been poorly constrained and attracting most of the attention especially after the gravitational waves (GWs) from binary black holes were directly detected by LIGO/VIRGO (Abbott et al. 2016). The black hole masses are within such window, which suggests they could be the PBH dark matter (Bird et al. 2016;Sasaki et al. 2016). However, current constraints are too weak (Ricotti et al. 2008;Oguri et al. 2018). More robust and independent evidences are needed to verify such conjecture. Recently, lensing of transients like GWs (Jung et al. 2019), gamma ray bursts (GRBs) (Ji et al. 2018) and fast radio bursts (FRBs) ) were proposed to be very promising in constraining CDM. The imprints of CDM as lens objects could be the distorted waveforms of GWs, the autocorrelation in GRB light curves and the echoes of FRB signals.
Remarkably, FRB method should be the simplest and cleanest even though we have not understood the formation mechanism of FRB yet. FRBs are bright pulses of emission at radio frequencies, most of which have dura-arXiv:2003.13349v1 [astro-ph.CO] 30 Mar 2020 tions of order milliseconds or less (Lorimer et al. 2007;Thornton et al. 2013). The short duration and large brightness make them emit coherently in nature. Most of FRBs are one-off, but a few are repeaters manifesting a longer-lived central engine. While current event rate is limited by the small fields-of-view of current radio telescopes, FRB events are supposed to be quite often on the full sky (∼ 10 4 /day). (Thornton et al. 2013;Champion et al. 2016). The ongoing wide-field surveys like APERTIF, UTMOST, HIRAX and CHIME will monitor a considerable fraction of the sky, giving thousands of detections per year. If part of dark matter consists of CDM, there must be a chance that an FRB is within the Einstein radius of a CDM, appearing split signals with flux ratio and time delay. Therefore, detections of such lensed signals could statistically infer the fraction and mass of CDM in turn . In principle, lensing of FRBs can effectively detect the mass range down to 20 − 100M that gives typical time delays comparable to the intrinsic duration of the signal. Realistic constraints depend on the event number and distributions of signal durations and redshifts. Shorter durations, higher redshifts and larger event number would give more stringent constraints.
The detected FRB events are timely included in the public catalogue 1 . The newest event number is ∼ 110 which gives a statistical sample. We use these data to give a first constraint on CDM and discuss more details about identifying the lensed signals in this work. Besides, we also make corrections to the forecast and discuss how we will deal with the detected lensed FRBs. The Letter is organized as follows: In Section 2, we introduce the theory on FRB lensing; In Section 3, we discuss how to identify the lensed signals and apply our method to the existing data, giving the constraints; The forecast and lens mass estimation are shown in Section 4; Finally, we summarize and make discussions in Section 5.

LENSING OF FAST RADIO BURSTS
Gravitational lensing is usually classified by the lens mass scale (equivalently the Einstein radius). For FRB lensing, we think it is more appropriate to take it as strong lensing since we can clearly discriminate the split transient signals, whereas the traditional microlensing limited by the resolution can only observe the overlapped images of constant sources. We take the CDM as a point mass whose Einstein radius is given by where the effective lensing distance (sometimes called time delay distance) D = D L D S /D LS , which is a combination of three angular diameter distances. Subscripts S, L denote the source and the lens, respectively. Although the spatial resolution in radio observation could reach very high level, for example, the angular resolution for the FRB 121102 with Very long baseline array (VLBA) is ∼ (10 −2 ) (Spitler et al. 2016;Chatterje et al. 2017;Tendulkar et al. 2017), it is still insufficient to distinguish split images spatially for M CDM < 10 8 M .
1 http://frbcat.org/ Therefore, we can not get the information of CDM by measuring θ E . What one can directly measure is the time delay between the lensed signals, which is determined by (2) where the dimensionless impact parameter y = β/θ E stands for the relative source position, z L is the lens redshift. Obviously, ∆t must be larger than the width of the intrinsic signal such that the split lensed images can be distinguished as double peaks. This requires y larger than certain value y min (M CDM , z L ) according to Eq.2.
In addition, the flux/magnification ratio between two images (+, −) can be directly measured as well: To make both lensed images (especially the fainter one) detectable with high enough signal-to-noise ratio (SNR), R f should not be too large, which requires the impact parameter to be smaller than certain value y max = (1 + R f,max )/ R f,max − 2 1/2 . We set the criterion R f,max = 5 following Muñoz et al. (2016). For a given FRB event at z S , the lensing optical depth is the probability that the point source is within the perceptible region of any CDM along the line of sight: (5) Using Hubble parameter at lens redshift and Hubble constant, Eq.4 can be rewritten as: where Ω c is the present value of total dark matter density and set it to be 0.24 in this work.
According to the definition, for rare events, the anticipated number of lensed FRBs is the sum of the lensing optical depths of all FRBs: Logically, a null search of the lensed signals would exclude the region of (M CDM , f CDM ) parameter space that gives at least one detectable lensed signal.

CONSTRAINTS WITH CURRENT OBSERVATIONS
The number of verified FRBs is rapidly increasing. At the moment of writing this Letter, the reported FRB number is 110. In addition, there are extra 9 events that are highly considered as the candidates. Although the method only requires the transient nature, most of the candidates do not have the measured widths of the signals and are therefore not used by us in this work. We will introduce how we analyze these data and constrain CDM in this section.

Identifying the lensed signals
In Muñoz et al. (2016), the double-peak structure was pointed out to be the feature of a lensed FRB. We have searched such signals in the catalog and find a few existing FRBs that have multiple-peak structure and are likely to be lensed, for example, FRB 170827 (Farah et al. 2018), FRB 121002 (Champion et al. 2016), FRB 121102 (repeating) (Hessels et al. 2018), FRB 180814.J0422+73 (repeating) (CHIME/FRB Collaboration 2019). However, after further looking into their dynamic spectra, we find the pulses corresponding to peaks are quite different. It is impossible to fit them using a simple time delay and relative magnification like what lensing requires. Rather than lensing effect, the multiple peaks of these FRBs must come from the intrinsic substructure of the signals themselves. For example, the repeating FRB 121102 clearly show the "frequency drift" phenomenon where multiple bursts occur within several milliseconds with decreasing frequencies. It is worth mentioning that the spectrum of FRB 181112 showed two similar pulses with very large flux ratio (Cho et al. 2020). However, the different polarization details and the impossibility of wave effects indicate the peaks should be intrinsic (Cho et al. 2020).
Therefore, we emphasize that it is important to use more information like the dynamic spectra or polarization properties to identify any lensed signals such that the degeneracy between intrinsic substructure and lensing can be broken. A lensed FRB should appear in dynamic spectrum as two pulses with the same shape and only different by flux magnification and time delay from each other (the fainter one comes later as the echo). We have carefully examined the dynamic spectra of the 110 FRBs, especially those who have multiple peaks. No strong evidence of lensing signal was found, which can shed light on the properties of lenses.

Results
The radio pulses from FRBs experience a frequencydependent delayed time through the ionized interstellar medium, quantified by a dispersion measure (DM) which is proportional to the number of electrons along the line of sight. If we know the ionized history of the universe, we can infer the distances/redshifts with the directly measured DMs. The biggest issue in the manner is we have little information on the host galaxies (except for those who can be localized). We adopt two methods of inferring redshifts. One is the way used in Petroff et al. (2016), where the fraction of baryon mass in the intergalactic medium (IGM) f IGM was supposed to be unity (f IGM = 1.0) and the He ionization history was not taken into considertaion, DM IGM ∼ 1200z pc cm −3 (Ioka 2003). The other is the DM IGM − z relation given by Deng & Zhang (2014), DM IGM ∼ 855z pc cm −3 (Zhang 2018), with the consideration of He ionization history and f IGM = 0.83. Besides, we assume the DM contribution from the Milky way and host galaxy is ∼ 100 pc cm −3 .
To summarize our pipeline for determining the constraints on f DM − M CDM : 1. Set f CDM = 1 as the benchmark at first; 2. Given M CDM , for each event with width in the catalog, we calculate y min at different lens redshift z L according to Eq.2; 3. Get τ of the event according to Eq.6; 4. Sum up the τ of all events as the expected number of detectable lensed signals; 5. Scale f CDM such that the expected number of lensed signal is unit; 6. Repeat from Step 2 for different M CDM .
The constraint results are presented in Fig.1. The shade regions are ruled out. In the case of Zhang (2018), the mass can be tested down to ∼ 100M and f CDM is gradually constrained to ∼ 0.5−0.6 for large mass. While in the case of Petroff et al. (2016), the constraints are weaker. Our results are comparable to that from wide binaries. Although current constraints are relatively weak, especial for small masses, we have showed the feasibility of this method. For thousands of events detected in the near future, we will give a much better constraint, especially for small masses (< 100M ).

FORECAST
In this section, we use the realistic distributions of the data to make an improved forecast. Furthermore, we discuss how CDM can be constrained with detected lensed signals.

A null search case
In Muñoz et al. (2016), to calculate the integrated lensing probability, the optical depth for lensing of a single burst had to be convolved with the redshift distribution of FRBs. They assumed FRBs either have a constant comoving number density or a scenario where FRBs follow the star-formation history. Since we know little about the FRB mechanism, there is no reason to make any assumptions for redshift distribution of FRBs. The direct and more robust way is to understand FRB redshifts  from the detected signals themselves. Furthermore, they assumed a constant width of FRB to be 0.3, 1 and 3 ms which is not realistic. We make forecast basing on the real distribution of the data. The 2-dimensional distribution of widths and redshifts is plotted in Fig.2. For comparison, we consider no events out of 10 4 FRBs are lensed as well. The improved forecast is shown in Fig.3. It is similar to Muñoz et al. (2016), however, it is less steep for the small-mass end determined by some very small widths in the catalog, while the decreasing trend persists to large mass due to some very large widths. In addition, we also consider 10 3 events for either very near future or a pessimistic scenario.

Constraints from lensed signals
We discuss the case that at least one lensed signals will be verified. Once a lensed FRB signal can be detected, we can estimate the lens mass from the measured time delay and flux ratio. The source position can be determined from flux ratio, then the redshifted lens mass can be determined from time delay. Compared to the uncertainties in the measured time delay and flux ratio, the uncertainty of lens redshift dominates. The typical value is σ z L ∼ 0.5. Nevertheless, it is sufficient for current CDM studies. The mass can be pinned down very well on certain scale and f CDM as well. Moreover, if more than one lensed signals are detected, we can even test whether CDM consists of the same mass and the theories that give a non-constant mass function. The intermediate-mass black holes may also be found in this way.

SUMMARIES AND PROSPECTIVES
Fast radio bursts are one of the most exciting new mysteries of astrophysics. Beyond how they are created, there is also the prospect of using FRBs to probe the extremes of the Universe and the invisible intervening medium. Due to the short duration, cosmological distances and the large event rate, the lensing of FRBs could be a powerful and robust tool to probe the compact dark matter. We have made some progress in this work summarized as follows: 1. For the first time, we use realistic FRB data to give a constraints on the fraction and mass of CDM. The constraints are comparable to the wide binaries.
2. We make an improved forecast basing on the distributions of the existing FRBs for the upcoming CHIME-like experiments; 3. We discuss the importance of using dynamic spectra of FRBs in identifying the lensed signals. It can effectively break the degeneracy between intrinsic structure and lensing imprints.
4. We discuss the situation when a few lensed signals can be detected and find the CDM parameter space can be well determined.
For future studies, it is necessary to build up an effective pipeline to identify lensed FRBs, especially for the upcoming large number of FRBs. It is also important to understand the properties of the host galaxies and the ionization history of the universe such that the redshift inference can be more accurate. Fast and high spatial resolution program will directly find the host galaxies, thus a large number of redshifts can be measured accurately. While we are writing this Letter, we note a very recent work based on analyzing FRB 181112 and 180924 (Sammons et al. 2020). It shows the burst substructure with high time resolution can be measured down to 15µs such that much smaller mass scales can be probed, making FRB method very promising.