# Limits on Simultaneous and Delayed Optical Emission from Well-localized Fast
Radio Bursts

Daichi Hiramatsu Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA  The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA Daichi Hiramatsu<EMAIL_ADDRESS>Edo Berger Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA  The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA  Brian D. Metzger Department of Physics and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street, New York, NY 10027, USA  Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA  Sebastian Gomez Space Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA  Allyson Bieryla Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA  Iair Arcavi School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel  CIFAR Azrieli Global Scholars program, CIFAR, 661 University Ave., Toronto, ON, M5G 1M1, Canada  D. Andrew Howell Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA  Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA  Ryan Mckinven Department of Physics, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada  McGill Space Institute, McGill University, 3550 rue University, Montréal, QC, H3A 2A7, Canada  Nozomu Tominaga National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Department of Astronomical Science, School of Physical Sciences, The Graduate University of Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Department of Physics, Faculty of Science and Engineering, Konan University, 8-9-1 Okamoto, Kobe, Hyogo 658-8501, Japan  Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan

(Received 2022 November 7; Revised 2022 December 17; Accepted 2022 December
22)

###### Abstract

We present the largest compilation to date of optical observations during and
following fast radio bursts (FRBs). The data set includes our dedicated
simultaneous and follow-up observations, as well as serendipitous archival
survey observations, for a sample of 15 well-localized FRBs: eight repeating
and seven one-off sources. Our simultaneous (and nearly simultaneous with a
$0.4$ s delay) optical observations of 13 (1) bursts from the repeating FRB
20220912A provide the deepest such limits to date for any extragalactic FRB,
reaching a luminosity limit of $\nu L_{\nu}\lesssim 10^{42}$ erg s-1
($\lesssim 2\times 10^{41}$ erg s-1) with $15$–$400$ s exposures; an optical-
flux-to-radio-fluence ratio of $f_{\rm opt}/F_{\rm radio}\lesssim 10^{-7}$
ms-1 ($\lesssim 10^{-8}$ ms-1); and flux ratio of $f_{\rm opt}/f_{\rm
radio}\lesssim 0.02$–$\lesssim 2\times 10^{-5}$ ($\lesssim 10^{-6}$) on
millisecond to second timescales. These simultaneous limits provide useful
constraints in the context of FRB emission models, such as the pulsar
magnetosphere and pulsar nebula models. Interpreting all available optical
limits in the context of the synchrotron maser model, we find that they
constrain the flare energies to $\lesssim 10^{43}$–$10^{49}$ erg (depending on
the distances of the various repeating FRBs, with $\lesssim 10^{39}$ erg for
the Galactic SGR 1935+2154). These limits are generally at least an order of
magnitude larger than those inferred from the FRBs themselves, although in the
case of FRB 20220912A our simultaneous and rapid follow-up observations
severely restrict the model parameter space. We conclude by exploring the
potential of future simultaneous and rapid-response observations with large
optical telescopes.

Radio transient sources (2008); Transient sources (1851); Magnetars (992);
Pulsars (1306)

††journal: ApJL††facilities: ADS, ATLAS, IRSA, FLWO (KeplerCam), LCO (MuSCAT3,
Sinistro), MMT (Binospec), NED, ZTF††software: Astropy (Astropy Collaboration
2018), atlas-fp
(https://gist.github.com/thespacedoctor/86777fa5a9567b7939e8d84fd8cf6a76),
barycorr (Eastman et al. 2010; https://github.com/tronsgaard/barycorr),
Cosimic-CONN (Xu et al., 2021a, 2022a, b), dustmaps (Green, 2018), lcogtsnpipe
(Valenti et al., 2016), Matplotlib (Hunter, 2007), NumPy (Oliphant, 2006),
PyZOGY (Guevel & Hosseinzadeh, 2017), SciPy (Virtanen et al., 2020), seaborn
(Waskom et al., 2020), SExtractor (Bertin & Arnouts, 1996)

2023 April 26

##  1 Introduction

Fast radio bursts (FRBs) are incredibly bright, millisecond-duration pulses at
GHz frequencies (e.g., Lorimer et al. 2007; see Petroff et al. 2019, 2022 for
reviews). Their dispersion measures (DMs), the integrated electron column
density along the line of sight, exceed the range of the Milky Way (MW) plus
its halo, implying an extragalactic origin. While most FRBs appear as one-off
events, a growing subset are known to repeat (e.g., Spitler et al. 2016; see
CHIME/FRB Collaboration et al. 2019 and Fonseca et al. 2020 for sample
studies), and it is possible that the entire population repeats on a wide
range of timescales (e.g., Nicholl et al. 2017; Ravi 2019; Cui et al. 2021).
To date, about 20 FRBs (both one-off and repeaters) have been precisely
localized through their radio emission, providing an initial, though
potentially biased, view of their host-galaxy environments (see, e.g.,
Chatterjee et al. 2017; Marcote et al. 2017; Tendulkar et al. 2017; see
further Heintz et al. 2020; Bhandari et al. 2022), and directly confirming
their extragalactic origin.

Despite the rapidly increasing sample size and the availability of some
localizations and host-galaxy properties, the physical origin(s) and
mechanism(s) of FRBs remain unknown, with dozens of proposed sources and
emission mechanisms published to date (see, e.g., Platts et al. 2019 for a
living list of theoretical FRB
models).111https://frbtheorycat.org/index.php/Main_Page Given the short
duration and noncatastrophic nature of the repeating FRBs, models involving
young neutron stars or black holes are particularly popular; the recent faint
FRB-like detection from the Galactic magnetar SGR 1935+2154 may support such a
picture (CHIME/FRB Collaboration et al., 2020a; Bochenek et al., 2020). Some
of the precisely localized extragalactic FRBs (e.g., FRB 20200120E in an M81
globular cluster; Bhardwaj et al. 2021; Kirsten et al. 2022a) show no
correlation with star formation, possibly indicating an origin from older
neutron stars, accretion-powered binaries (e.g., Sridhar et al. 2021), or from
young magnetars born from an older progenitor channel (e.g., accretion-induced
collapse of white dwarfs or the merger of two neutron stars; Margalit et al.
2019; Kremer et al. 2022).

A significant barrier to our understanding of FRBs is their sole detection in
the radio band (see, e.g., Chen et al. 2020; Nicastro et al. 2021 for reviews
on multiwavelength observations).222Temporally coincident X-ray bursts from
the Galactic SGR 1935+2154 (Mereghetti et al., 2020; Li et al., 2021a; Ridnaia
et al., 2021; Tavani et al., 2021) remain the only nonradio transient
detections. However, such bursts would not be detectable in extragalactic FRBs
due to their much larger distances (see e.g., Scholz et al. 2016, 2017, 2020;
Tavani et al. 2020 for X-ray nondetections of extragalactic FRBs). This
situation is reminiscent of the first two decades of gamma-ray burst (GRB)
research when only gamma-ray emission had been detected; it was only through
rapid multiwavelength detections of associated afterglows that the progenitors
and physics of GRBs were eventually uncovered.333A key distinction is that for
GRBs detections of afterglows at longer wavelengths were critical for precise
localizations and the identification of host galaxies (and hence a distance
scale and stellar population properties), whereas FRBs can be precisely
localized directly in the radio band.

In the same vein, rapid and deep optical follow-up of FRBs may shed light on
their physical mechanisms. FRB models predict a wide range of possible
luminosities and timescales for optical transient counterparts, ranging from
no emission at all, to luminous ($\gtrsim 10^{41}$ erg s-1) afterglows on a
millisecond timescale, to fainter ($\lesssim 10^{39}$ erg s-1) afterglows on
timescales of minutes to hours (e.g., Lyubarsky 2014; Beloborodov 2017;
Metzger et al. 2019; Yang et al. 2019; Margalit et al. 2020a, b; Beloborodov
2020). Previous optical follow-up attempts have suffered from the combination
of FRB poor localizations, large distances, delayed announcements of bursts,
and a limited sample of repeating FRBs, resulting in only weak constraints
(e.g., $\lesssim 10^{45}$ erg s-1 over a millisecond timescale to $\lesssim
10^{43}$ erg s-1 over a minute timescale; Hardy et al. 2017; MAGIC
Collaboration et al. 2018; Andreoni et al. 2020; Niino et al. 2022); dedicated
monitoring of a single burst from the well-localized repeating FRB 20180916B
($d_{L}\approx 150$ Mpc; Marcote et al. 2020) placed marginal constraints
($\lesssim 10^{40}$ erg s-1 over minute timescale) on models such as the
synchrotron maser with high burst energies and circumburst densities
(Kilpatrick et al., 2021).

Here, we report the largest set of optical constraints to date, for a sample
of eight well-localized repeating FRBs (including the Galactic SGR 1935+2154)
and seven well-localized non-repeating FRBs, using an extensive set of
archival data, as well as our own dedicated follow-up observations of the
nearby repeating FRB 20200120E, and simultaneous observations of the newly
discovered and highly active FRB 20220912A (McKinven & CHIME/FRB
Collaboration, 2022), the most sensitive simultaneous observations to date for
any extragalactic FRB. The paper is structured as follows. We summarize the
FRB sample, optical observations, and data reduction in Section 2. In Section
3, we analyze and discuss the optical luminosity, flux, and fluence limits in
the context of the FRBs’ radio properties, and compare these to theoretical
models. We summarize our findings and conclude with a future outlook in
Section 4.

##  2 Sample and Observations

###  2.1 Fast Radio Burst Sample

Table 1: The Sample of Well-localized Repeating FRBs

FRB | R.A. | Decl. | Redshift |  $d_{L}$aaCalculated from the host redshift assuming a standard Lambda cold dark matter ($\Lambda$CDM) cosmology with $H_{0}=71.0$ km s-1 Mpc-1, $\Omega_{\Lambda}=0.7$, and $\Omega_{m}=0.3$, unless otherwise noted. |  $A_{V,{\rm MW}}$bbFrom Schlafly & Finkbeiner (2011), retrieved via the NASA/IPAC Infrared Science Archive (IRSA), unless otherwise noted. | EventsccOnly the bursts with reported time of arrival are included. | Frequency | DM | WidthddIf only two of the width ($t_{\rm FRB}$), flux ($f_{\rm radio}$), and fluence ($F_{\rm radio}$) are reported for a particular burst, the third parameter is estimated assuming $F_{\rm radio}\sim f_{\rm radio}t_{\rm FRB}$. | FluxddIf only two of the width ($t_{\rm FRB}$), flux ($f_{\rm radio}$), and fluence ($F_{\rm radio}$) are reported for a particular burst, the third parameter is estimated assuming $F_{\rm radio}\sim f_{\rm radio}t_{\rm FRB}$. | FluenceddIf only two of the width ($t_{\rm FRB}$), flux ($f_{\rm radio}$), and fluence ($F_{\rm radio}$) are reported for a particular burst, the third parameter is estimated assuming $F_{\rm radio}\sim f_{\rm radio}t_{\rm FRB}$.
---|---|---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) |  | (Mpc) | (mag) | (#) | (GHz) | (pc cm-3) | (ms) | (Jy) | (Jy ms)
20200120E | 149.4779140(3)eeBest localized with the European VLBI Network (EVN; Kirsten et al. 2022a). | +68.8169036(4)eeBest localized with the European VLBI Network (EVN; Kirsten et al. 2022a). |  $-0.00013$ffFrom Speights & Westpfahl (2012), retrieved via the NASA/IPAC Extragalactic Database. |  $3.6$ffFrom Speights & Westpfahl (2012), retrieved via the NASA/IPAC Extragalactic Database. | $0.200$ | 74ggData sources: CHIME (CHIME/FRB Public Database, Bhardwaj et al. 2021), the Deep Sapce Network (DSN; Majid et al. 2021), EVN (Kirsten et al., 2022a), Effelsberg (Nimmo et al., 2023). |  $0.40$–$2.3$ |  $87.7$–$88$ |  $0.014$–$0.70$ |  $0.10$–$60$ |  $0.04$–$2$
20180916B | 29.5031257(6)hhBest localized with EVN, and the host spectroscopic redshift measured with Gemini-North (Marcote et al., 2020). | +65.7167542(6)hhBest localized with EVN, and the host spectroscopic redshift measured with Gemini-North (Marcote et al., 2020). |  $0.0337$hhBest localized with EVN, and the host spectroscopic redshift measured with Gemini-North (Marcote et al., 2020). |  $149$hhBest localized with EVN, and the host spectroscopic redshift measured with Gemini-North (Marcote et al., 2020). | $2.712$ | 244iiData sources: CHIME (CHIME/FRB Public Database; CHIME/FRB Collaboration et al. 2019; CHIME/FRB Collaboration et al. 2020b; Pleunis et al. 2021), Apertif (Pastor-Marazuela et al., 2021), Effelsberg (Bethapudi et al., 2022), EVN (Marcote et al., 2020), Green Bank Telescope (GBT; Chawla et al. 2020; Sand et al. 2022), Low Frequency Array (LOFAR; Pastor-Marazuela et al. 2021; Pleunis et al. 2021), Medicina Northern Cross (MNC; Trudu et al. 2022), Sardinia (Pilia et al., 2020), upgraded Giant Metrewave Radio Telescope (uGMRT; Marthi et al. 2020; Pleunis et al. 2021; Sand et al. 2022), Very Large Array (VLA; Aggarwal & Realfast Collaboration 2020; Aggarwal et al. 2020). |  $0.12$–$4.9$ |  $343$–$356$ |  $0.3$–$158$ |  $0.12$–$20$ |  $0.08$–$300$
20220912A | 347.2704(6)jjBest localized with the Deep Synoptic Array (DSA), and the host spectroscopic redshift measured with Keck (Ravi et al., 2022a). | +48.7071(3)jjBest localized with the Deep Synoptic Array (DSA), and the host spectroscopic redshift measured with Keck (Ravi et al., 2022a). |  $0.077$jjBest localized with the Deep Synoptic Array (DSA), and the host spectroscopic redshift measured with Keck (Ravi et al., 2022a). | $344$ | $0.637$ | 72kkData sources: CHIME (CHIME/FRB VOEvent Service; McKinven & CHIME/FRB Collaboration 2022), Arecibo (Perera et al., 2022), Big Scanning Antenna (BSA; Fedorova & Rodin 2022), DSA (Ravi et al. 2022a; private communication), DSN (Rajwade et al., 2022), Lovell (Rajwade et al., 2022), MNC (Pelliciari et al., 2022), Stockert (Kirsten et al., 2022b; Herrmann, 2022), Tianlai (Yu et al., 2022). |  $0.11$–$2.3$ |  $218$–$228$ |  $0.8$–$300$ |  $3$–$290$ |  $1.5$–$900$
20201124A | 77.0146142(8)llBest localized with EVN (Nimmo et al., 2022), and the host spectroscopic redshift measured with MMT (Fong et al., 2021). | +26.0606959(7)llBest localized with EVN (Nimmo et al., 2022), and the host spectroscopic redshift measured with MMT (Fong et al., 2021). |  $0.0979$llBest localized with EVN (Nimmo et al., 2022), and the host spectroscopic redshift measured with MMT (Fong et al., 2021). | $444$ | $1.964$ | 2914mmData sources: CHIME (CHIME/FRB Public Database, CHIME/FRB Collabortion 2021; Lanman et al. 2022), Allen Telescope Array (ATA; Farah et al. 2021), Apertif (Atri et al., 2022), Australian Square Kilometre Array Pathfinder (ASKAP; Kumar et al. 2022), Effelsberg (Hilmarsson et al., 2021a), EVN (Nimmo et al., 2022), Five-hundred-meter Aperture Spherical radio Telescope (FAST; Feng et al. 2022b; Wang et al. 2022; Xu et al. 2022b; Zhang et al. 2022b), GBT (Feng et al., 2022b), Onsala (Kirsten et al., 2021b), Parkes (Kumar et al., 2022), Stockert (Herrmann, 2021), Usuda Deep Space Center (UDSC; Takefuji et al. 2022; Ikebe et al. 2023), uGMRT (Marthi et al., 2022), VLA (Ravi et al., 2022b), Westerbork (Ould-Boukattine et al., 2022a, b). |  $0.40$–$2.3$ |  $409$–$424$ |  $0.9$–$300$ |  $0.004$–$220$ |  $0.005$–$770$
20121102A | 82.994575(3)nnBest localized with EVN (Marcote et al., 2017), and the host spectroscopic redshift measured with Gemini-North (Tendulkar et al., 2017). | +33.147940(1)nnBest localized with EVN (Marcote et al., 2017), and the host spectroscopic redshift measured with Gemini-North (Tendulkar et al., 2017). |  $0.1927$nnBest localized with EVN (Marcote et al., 2017), and the host spectroscopic redshift measured with Gemini-North (Tendulkar et al., 2017). | $927$ | $2.098$ | 3658ooData sources: Arecibo (Spitler et al., 2014; Scholz et al., 2016; Spitler et al., 2016; Scholz et al., 2017; MAGIC Collaboration et al., 2018; Michilli et al., 2018; Gourdji et al., 2019; Hessels et al., 2019; Aggarwal et al., 2021; Hilmarsson et al., 2021b; Hewitt et al., 2022; Jahns et al., 2023), Apertif (Oostrum et al., 2017, 2020), CHIME (Josephy et al., 2019), DSN (Majid et al., 2020; Pearlman et al., 2020), Effelsberg (Hardy et al., 2017; Spitler et al., 2018; Houben et al., 2019; Cruces et al., 2021; Hilmarsson et al., 2021b), EVN (Marcote et al., 2017), FAST (Li et al., 2021b), GBT (Scholz et al., 2016, 2017; Gajjar et al., 2018; Michilli et al., 2018; Zhang et al., 2018; Hessels et al., 2019), Lovell (Rajwade et al., 2020), MeerKAT (Caleb et al., 2020), VLA (Chatterjee et al., 2017; Law et al., 2017; Hilmarsson et al., 2021b). |  $0.60$–$7.5$ |  $527$–$698$ |  $0.01$–$78.52$ |  $0.002$–$70$ |  $0.002$–$35$
20190520B | 240.51780(3)ppBest localized with VLA, and the host spectroscopic redshift measured with the Hale Telescope (Niu et al., 2022). |  $-$11.28814(2)ppBest localized with VLA, and the host spectroscopic redshift measured with the Hale Telescope (Niu et al., 2022). |  $0.241$ppBest localized with VLA, and the host spectroscopic redshift measured with the Hale Telescope (Niu et al., 2022). | $1192$ | $0.769$ | 230qqData sources: FAST (Niu et al., 2022), GBT (Anna-Thomas et al., 2022; Feng et al., 2022b), Parkes (Dai et al., 2022), VLA (Niu et al., 2022). |  $1.4$–$6.2$ |  $1164$–$1291$ |  $0.7$–$33.1$ |  $0.002$–$2$ |  $0.029$–$6$
20180301A | 93.2268(2)rrBest localized with VLA, and the host spectroscopic redshift measured with Keck (Bhandari et al., 2022). | +4.6711(2)rrBest localized with VLA, and the host spectroscopic redshift measured with Keck (Bhandari et al., 2022). |  $0.3304$rrBest localized with VLA, and the host spectroscopic redshift measured with Keck (Bhandari et al., 2022). | $1712$ | $1.231$ | 22ssData sources: Parkes (Price et al., 2019), FAST (Luo et al., 2020; Laha et al., 2022), VLA (Bhandari et al., 2022). |  $1.2$–$1.3$ |  $510$–$536$ |  $1.7$–$12.3$ |  $0.005$–$1.2$ |  $0.021$–$4.90$
SGR |  |  |  | (kpc) |  |  |  |  |  |  |
1935+2154 | 293.7317(2)ttLocalized with Chandra (Israel et al., 2016). | +21.8966(2)ttLocalized with Chandra (Israel et al., 2016). | GalacticttLocalized with Chandra (Israel et al., 2016). |  $9.0$uuMean distance of a wide range estimated with various techniques ($\sim 4.4$–$14.2$ kpc; e.g., Park et al. 2013; Pavlović et al. 2013, 2014; Surnis et al. 2016; Kothes et al. 2018; Mereghetti et al. 2020; Zhou et al. 2020). |  $7.2$vvExtinction at the mean distance from the Green et al. (2019) 3D dust map ($\sim 5.2-7.2$ mag over the whole distance range; caution that the dust map suffers a lack of bright main-sequence stars at the larger ditances), retrieved via dustmaps (Green, 2018). | 14wwData Sources: CHIME (CHIME/FRB Collaboration et al., 2020a; Pleunis & CHIME/FRB Collaboration, 2020; Dong & CHIME/FRB Collaboration, 2022; Pearlman & CHIME/FRB Collaboration, 2022), Survey for Transient Astronomical Radio Emission 2 (STARE2; Bochenek et al. 2020), FAST (Zhang et al., 2020), GBT (Maan et al., 2022), Large Phased Array (LPA; Rodin & Fedorova 2022), Yunnan (Huang et al., 2022), Wasterbork (Kirsten et al., 2021a). |  $0.11$–$5.6$ |  $313$–$333$ |  $0.22$–$2000$ |  $0.030$–$2.5\times 10^{6}$ |  $0.06$–$1.5\times 10^{6}$

Note. — Measurements of each individual burst are published in its entirety in
machine-readable form.

We select a sample of FRBs that are well localized ($\lesssim 2\arcsec$) and
with known host-galaxy identifications,444See https://frbhosts.org for the up-
to-date list of FRB host-galaxy identifications located at decl.
$\gtrsim-30^{\circ}$ in order to have access to the radio burst measurements
from the Canadian Hydrogen Intensity Mapping Experiment/FRB (CHIME/FRB
Collaboration et al., 2018) Public Database555https://www.chime-frb.ca/ and
optical forced photometry from the Zwicky Transient Facility (ZTF; Bellm et
al. 2019; Graham et al. 2019) and Asteroid Terrestrial-impact Last Alert
System (ATLAS; Tonry et al. 2018; Smith et al. 2020)666Typical ZTF and ATLAS
PSF FWHMs are $2\arcsec$ and $4\arcsec$, respectively, comparable to and
larger than the adopted FRB localization error limit. — major FRB and optical
time-domain surveys. We also include any other radio and optical data sets
from the literature. We select both repeating and non-repeating FRBs; however,
we find that all of the optical limits for non-repeating FRBs are not
particularly constraining given the small number of bursts and large distances
(see Appendix A). Therefore, we focus on the repeating FRBs in the subsequent
analysis. Their properties and references are summarized in Table 1. We note
that the distance to the Galactic SGR 1935+2154 is not well constrained, which
also results in a large extinction uncertainty.

###  2.2 Optical Observations

For each FRB in our sample, we obtained optical photometry from the ZTF
forced-photometry service777https://ztfweb.ipac.caltech.edu/cgi-
bin/requestForcedPhotometry.cgi (Masci et al., 2019) in the $g$, $r$, and $i$
bands, and the ATLAS forced photometry server888https://fallingstar-
data.com/forcedphot/ (Shingles et al., 2021) in the $c$ and $o$ bands for the
entire time interval spanning all of the radio bursts for each FRB (i.e., from
the first to latest measured bursts). The detection significance ($\sigma$) of
optical measurements was determined from the ratio of measured flux ($f$) to
its error ($f_{\rm err}$). For any measurements above $3\sigma$, we visually
inspected the difference images and found that all were due to subtraction
artifacts (e.g., bad focus, world coordinate system offset, shutter problems).
Thus, we calculate the $3\sigma$ upper limits as $-2.5\,{\rm log}_{10}(3\times
f_{\rm err})+{\rm ZP}$, where “ZP” is the zero point in the AB magnitude
system.

In addition, we carried out dedicated monitoring observations of the nearby
FRB 20200120E in M81 (Bhardwaj et al., 2021; Kirsten et al., 2022a) with 300 s
exposures simultaneously in the $g$, $r$, $i$, and $z$ bands from 2021 March
19 to 2022 May 18 (UT dates are used throughout) roughly every 10 days with
MuSCAT3 (Narita et al., 2020) on the 2 m Faulkes Telescope North (Hawaii, USA)
in the Las Cumbres Observatory (LCO; Brown et al. 2013) network. These images
have typical $griz$-band $3\sigma$ limiting magnitudes of $22.8$, $22.9$,
$22.3$, and $22.0$, respectively. The March 19 observations, during which no
FRBs were announced, were used as template images, and image subtraction was
performed using lcogtsnpipe999https://github.com/LCOGT/lcogtsnpipe (Valenti et
al., 2016), a PyRAF-based photometric reduction pipeline, and
PyZOGY101010https://github.com/dguevel/PyZOGY (Guevel & Hosseinzadeh, 2017),
an implementation in Python of the subtraction algorithm described in Zackay
et al. (2016). The $griz$-band data were calibrated to AB magnitudes using the
13th Data Release of the Sloan Digital Sky Survey (SDSS; Albareti et al.
2017).

Figure 1: Top: schematic timeline of the KeplerCam and LCO simultaneous
exposures, as well as the temporally closest MMT/Binospec exposure with
respect to the FRB 20220912A bursts detected by CHIME and DSA (where $N$ is
the number of bursts detected within 1 s), color-coded by the observed
$r$-band limiting magnitude. The UT date labeled next to each point is its
exposure start time. In total, we covered 13 bursts in 10 simultaneous
exposures: one DSA detection each in two KeplerCam exposures, one CHIME
detection each in five KeplerCam exposures, and two CHIME detections each in
two KeplerCam and one LCO exposures. Bottom: simultaneous KeplerCam 60 s image
during the two CHIME detections (left), the corresponding PS1 template image
(middle), and the resulting difference image (right), with the DSA
localization region and host galaxy marked by the magenta ellipse ($\Delta{\rm
R.A}\sim 2\arcsec$ & $\Delta{\rm decl.}\sim 1\arcsec$; Ravi et al. 2022a) and
navy circle (Kron radius $=3\farcs 44$ from PS1 DR2; Flewelling et al. 2020),
respectively. No optical counterpart is identified down to a $r$-band limiting
magnitude of $21.1$.

For the recently discovered highly active FRB 20220912A (McKinven & CHIME/FRB
Collaboration, 2022), we carried out monitoring observations during the CHIME
observing windows111111https://www.chime-frb.ca/astronomytools with three
facilities: a series of 40 s exposures and a series of 60 s exposures in the
$r$ band with KeplerCam (Szentgyorgyi et al., 2005) on the 1.2 m Telescope at
the Fred Lawrence Whipple Observatory (FLWO; Arizona, USA), reaching typical
$3\sigma$ limiting magnitudes of $20.5$ and $21.1$, respectively; a series of
400 s exposures in the $r$ band with the Sinistro camera on the 1 m telescope
at the McDonald Observatory (Texas, USA) in the LCO network, reaching a
typical $3\sigma$ limiting magnitude of $22.6$; and a series of 15 s exposures
in the $r$ band with Binospec (Fabricant et al., 2019) on the 6.5 m MMT
Observatory (Arizona, USA), reaching a typical $3\sigma$ limiting magnitude of
$23.2$. In total, we obtained 10 simultaneous optical observations during 13
FRB detections: nine with KeplerCam121212One of which was reported in
Hiramatsu et al. (2022). for the two Deep Synoptic Array (DSA) detections on
2022 October 18 and 25 (Ravi et al., 2022a) and the nine CHIME detections on
October 20, 25, and 30 and November 7 and 9 (CHIME VOEvent
Service131313https://www.chime-frb.ca/voevents); and one with LCO for the two
CHIME detections on October 22. These KeplerCam and LCO simultaneous
exposures, as well as a nearly simultaneous Binospec exposure 0.4 s after a
CHIME detection, are shown in Figure 1. Due to the uncertainty ($\lesssim$ a
few seconds) in the shutter opening time stamps (from the open command being
issued to the shutter being fully opened), the nearly simultaneous Binospec
exposure may indeed be simultaneous. We note this caveat in the following
analysis and discussion wherever appropriate.

Using a custom photometry pipeline, the KeplerCam, LCO, and Binospec data were
reduced and calibrated to AB magnitudes from the Pan-STARRS1141414The field is
not covered by SDSS. (PS1; Chambers et al. 2016) Data Release 2 (DR2;
Flewelling et al. 2020). Cosmic rays were identified and masked using Cosimic-
CONN151515https://github.com/cy-xu/cosmic-conn (Xu et al., 2021a, b, 2022a),
and image subtraction was performed against PS1 template images using PyZOGY.
For each simultaneous exposure, we stacked it with the subsequent exposures in
the same series to obtain deeper limits (typical limiting magnitudes of $21.5$
and $23$ for KeplerCam and LCO observations, respectively). An example image
subtraction is shown in Figure 1 for one of the simultaneous KeplerCam images.
We do not detect any transient source in the subtracted KeplerCam, LCO, and
Binospec images within any of the FRB localization regions. Thus, we report
their $3\sigma$ upper limits.

We correct all optical limits for the MW extinction assuming the Fitzpatrick
(1999) reddening law with $R_{V}=3.1$. After correcting the DM- and frequency-
dependent time delay (see Equation (1) of Cordes & Chatterjee 2019) and
referencing all the radio and optical measurements to the solar system
barycenter (Eastman et al., 2010),161616https://github.com/tronsgaard/barycorr
we find the temporally closest radio burst for each optical limit with respect
to the optical exposure midpoint to determine the time difference.

##  3 Analysis and Discussion

###  3.1 Optical Luminosity Limits

Figure 2: Optical luminosity limits (in the $g$, $c$, $r$, $o$, $i$, and $z$
bands from the top to bottom rows) with respect to the temporally closest FRB
detection for our sample of eight repeating FRBs (including the Galactic SGR
1935+2154). Representative limits are shown for the targeted high-speed
observations (TNT $i$ and $i+z$, MAGIC $U$, Tomo-e open) in the panels with
the closest filter effective wavelength, and similarly for LCO $R$ and BOOTES
$Z$ and clear bands. A typical luminosity error in each band for SGR 1935+2154
(due to the large extinction and distance uncertainties) is shown as the
magenta vertical bar. Our KeplerCam and LCO limits for FRB 20220912A (enclosed
by the black rectangle) are the deepest among the simultaneous observations of
extragalactic FRBs. Optical data sources: KeplerCam (this work for FRB
20220912A), LCO (this work and Lin et al. 2020 for FRBs 20200120E and
20220912A in $griz$, and SGR 1935+2154 in $R$, respectively), Binospec (this
work for FRB 20220912A), ZTF (Andreoni et al. 2021, 2020 for FRBs 20200120E
and 20180916B, respectively), ZTF forced-photometry service (Masci et al. 2019
and this work for all the FRBs), ATLAS forced photometry server (Shingles et
al. 2021 and this work for all the FRBs), APO (Kilpatrick et al. 2021 for FRB
20180916B), TNT (Hardy et al. 2017 for FRB 20121102A), MAGIC (MAGIC
Collaboration et al. 2018 for FRB 20121102A), Tomo-e (Niino et al. 2022 for
FRB 20190520B), and BOOTES (Lin et al. 2020 for SGR 1935+2154). (The data used
to create this figure are available.)

None of our simultaneous and follow-up observations or the archival
observations have led to a detected optical counterpart. We plot all of the
resulting KeplerCam, LCO, and Binospec luminosity limits, along with the
archival and literature sample in Figure 2. Most of the untargeted ZTF and
ATLAS observations occur $\gtrsim 10^{4}$ s before and/or after radio bursts,
with a wide luminosity limit range of $\sim 10^{35}$–$10^{45}$ erg s-1; the
deepest limits are for the Galactic magnetar SGR 1935+2154, given the much
smaller distance compared to the other extragalactic FRBs. An untargeted ZTF
$r$-band observation for FRB 20200120E (Andreoni et al., 2021) and the
targeted (but nonsimultaneous) $gri$-band observations with ARCTIC on the
Apache Point Observatory (APO) 3.5 m telescope (Huehnerhoff et al., 2016) for
FRB 20180916B (Kilpatrick et al., 2021) probe a luminosity range of $\sim
2\times 10^{38}-4\times 10^{40}$ erg s-1 within $\sim 10^{3}$ s of radio
bursts.

On a shorter timescale, several targeted high-speed simultaneous observations
— by ULTRASPEC on the 2.4 m Thai National Telescope (TNT; Tulloch & Dhillon
2011; Dhillon et al. 2014) for FRB 20121102A (Hardy et al., 2017); the central
pixel of the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescopes
(Lucarelli et al., 2008; Hassan et al., 2021) for FRB 20121102 (MAGIC
Collaboration et al., 2018); and Tomo-e Gozen on the Kiso 105 cm Schmidt
telescope (Sako et al., 2018) for FRB 20190520B (Niino et al., 2022) — span a
temporal range $\sim 1$–$10$ ms before/after radio bursts with a luminosity
upper limit range of $\sim 10^{45}$–$10^{47}$ erg s-1 in a single exposure
(longer and deeper stacked exposures are also shown in Figure 2). In the ZTF
archival search, we also find a simultaneous $30$ s $r$-band observation of
FRB 20121102A with a luminosity limit of $\sim 6\times 10^{43}$ erg s-1.

Among the simultaneous optical observations, our KeplerCam and LCO $r$-band
limits for FRB 20220912A are the deepest to date in terms of luminosity for
the extragalactic FRBs, $\sim(0.3$–$2.9)\times 10^{42}$ erg s-1 with
$30$–$400$ s exposures (i.e., excluding the Burst Observer and Optical
Transient Exploring System, BOOTES, $Z$-band limit of $\sim 1.8\times 10^{35}$
erg s-1 with a 60 s exposure for the Galactic SGR 1935+2154; Lin et al. 2020).
Our nearly simultaneous Binospec limit for FRB 20220912A is at $\sim 2\times
10^{41}$ erg s-1 with a $15$ s exposure started at $0.4$ s after a burst.

###  3.2 Optical-to-Radio Flux and Fluence Ratio Limits

Figure 3: Ratio of optical flux limit (in janskys; in the $g$, $c$, $r$, $o$,
$i$, and $z$ bands from the top to bottom rows) to radio fluence (in janskys
times millisecond) with respect to the temporally closest FRB detection for
our sample of eight repeating FRBs (including the Galactic SGR 1935+2154);
labels are as in Figure 2. In the case when the radio fluence of a particular
burst is not available, it is assumed to be the mean of the fluence
distribution of that particular repeating FRB (and shown as a transparent
point with an upper error bar corresponding to one standard deviation). The
dashed and dotted gray lines in each panel show $F_{\rm opt}/F_{\rm radio}=1$,
$0.1$, $0.01$, and $0.001$ assuming $F_{\rm opt}=f_{\rm opt}\times\Delta t$,
where $\Delta t$ is the time delay between the optical and radio observations.
Note that most of the simultaneous observations (KeplerCam, LCO, TNT, MAGIC,
Tome-e, BOOTES), targeted Binospec and APO, and some untargeted ZTF
observations are in the regime of $F_{\rm opt}/F_{\rm radio}\lesssim 1$. Our
KeplerCam and LCO limits for FRB 20220912A are comparable even to the BOOTES
simultaneous observation of SGR 1935+2154 ($f_{\rm opt}/F_{\rm radio}\lesssim
10^{-7}$ ms-1, or $F_{\rm opt}/F_{\rm radio}\lesssim 10^{-3}$). Our nearly
simultaneous Binospec limit is at $f_{\rm opt}/F_{\rm radio}\lesssim 10^{-8}$
ms-1, or $F_{\rm opt}/F_{\rm radio}\lesssim 10^{-4}$ at $\Delta t\approx 0.4$
s.

To place the optical limits in the context of the individual FRB burst
properties, we plot the ratio of optical flux limit to FRB radio fluence,
$f_{\rm opt}/F_{\rm radio}$, in Figure 3. We make this parameter choice
because the optical burst duration ($t_{\rm opt}$) is not known. If the
fluence measurement of a particular radio burst is not yet published, it is
assumed to be the mean of the fluence distribution of the relevant repeating
FRB (e.g., the recently discovered FRB 20220912A). We find that the range of
flux-to-fluence ratio limits is $\sim 10^{-9}$–$0.1$ ms-1, on a timescale of
$\gtrsim 10^{3}$ s.

If we make a conservative assumption that the optical counterpart duration is
$t_{\rm opt}\approx\Delta t$, where $\Delta t$ is the time delay between an
FRB and an optical observation, we find that all of the targeted simultaneous
observations lie below the critical line of $f_{\rm opt}\Delta t/F_{\rm
radio}\approx F_{\rm opt}/F_{\rm radio}=1$. On the other hand, nearly all of
the untargeted and archival observations are well above the $F_{\rm
opt}/F_{\rm radio}=1$ line (Figure 3), meaning that they provide weak
constraints for any model in which the energy release per frequency in the
optical band is at most comparable to that in the radio bursts; exceptions are
some nonsimultaneous APO and untargeted ZTF observations for FRBs 20180916 and
20220912A, respectively. Our nearly simultaneous Binospec limit is at $f_{\rm
opt}/F_{\rm radio}\lesssim 10^{-8}$ ms-1 (or $F_{\rm opt}/F_{\rm
radio}\lesssim 10^{-4}$) at $\Delta t\approx 0.4$ s, which is the deepest for
any FRBs on this timescale.

Among the simultaneous optical observations, our KeplerCam and LCO $r$-band
limits for FRB 20220912A are the deepest to date in terms of $f_{\rm
opt}/F_{\rm radio}$ for any extragalactic FRB, and indeed comparable to limits
for the Galactic SGR 1935+2154, with $f_{\rm opt}/F_{\rm radio}\lesssim
10^{-7}$ ms-1. In terms of fluence ratios, our observations place a limit of
$F_{\rm opt}/F_{\rm radio}\lesssim 10^{-3}$. Our nearly simultaneous Binospec
limit would be the deepest if it were indeed simultaneous given the
aforementioned shutter timing uncertainty. We stress that these fluence ratio
limits are lower than the observed values for the Crab and Geminga pulsars
(e.g., Danilenko et al. 2011; Bühler & Blandford 2014), which may suggest a
different emission mechanism (although pulsars with lower fluence ratios have
also been observed; see, e.g., Niino et al. 2022 for a discussion for FRB
20190520B in this context).

###  3.3 Constraints on Fast Optical Burst Models

To constrain possible fast optical bursts on a comparable timescale to the
radio bursts, we define an effective optical flux limit on a timescale,
$t_{\rm opt}$ (see also Lyutikov & Lorimer 2016):

$f_{\rm eff,opt}=f_{\rm opt}\frac{T_{\rm exp}}{t_{\rm opt}},$ (1)

where $T_{\rm exp}$ is the exposure time. For the sample of FRBs with
simultaneous optical observations, we show $f_{\rm eff,opt}/f_{\rm radio}$ as
a function of $t_{\rm opt}$ in Figure 4. The previously published high-speed
observations of FRBs 20121102A and 20190520B only probe $f_{\rm
eff,opt}/f_{\rm radio}\gtrsim 0.1$ on millisecond timescales, and reach
$f_{\rm eff,opt}/f_{\rm radio}\sim 0.01$ only on $\gtrsim{\rm second}$
timescales due to the shallow integrated flux limit (i.e., apparent magnitude
limit).

Figure 4: Effective optical flux limit ($f_{\rm eff,opt}=f_{\rm opt}T_{\rm
exp}/t_{\rm opt}$) to radio flux ratio with respect to the optical burst
duration ($t_{\rm opt}$), color-coded by the exposure time ($T_{\rm exp}$),
for the sample of FRBs with simultaneous optical observations (including the
nearly simultaneous Binospec observation for FRB 20220912A). The quoted
magnitude limits are the extinction-corrected values for each FRB, and their
solid lines extend from the time of radio burst to the end of each exposure.
The dotted extension to the nearly simultaneous Binospec observation shows
$f_{\rm eff,opt}$ within the timing uncertainty. The shaded regions show the
flux error ranges of the Galactic SGR 1935+2154 due to the large extinction
uncertainty. Broad ranges of optical burst model predictions from Yang et al.
(2019) are also plotted as rectangular regions, where the duration of one-zone
models are shown with a width corresponding to the mean plus one standard
deviation of the $t_{\rm FRB}$ distribution of each FRB. Potential constraints
from future observations with Subaru HSSC are also plotted as dashed lines for
each FRB. Note that the optical limits of FRBs 20121102A and 20190520B barely
reach the pulsar nebula model parameter space, while our limits for FRB
20200912A and published limit for SGR 1935+2154 reach the pulsar magnetosphere
as well as the pulsar nebula model parameter spaces.

On the other hand, our KeplerCam and LCO observations for FRB 20220912A are
far more sensitive, reaching $f_{\rm eff,opt}/f_{\rm radio}\lesssim 0.02$ on a
milliseocnd timescale, and $\lesssim 3\times 10^{-5}$ on a second timescale.
These limits are only an order of magnitude higher than the limits for the
Galactic SGR 1935+2154. Our nearly simultaneous Binospec observation for FRB
20220912A reaches $f_{\rm eff,opt}/f_{\rm radio}\lesssim 10^{-6}$ on a second
timescale (and $\lesssim 10^{-3}$ on a millisecond timescale if it were indeed
simultaneous), which is on the same order as the SGR 1935+2154 limits.

We also plot in Figure 4 broad ranges of some optical burst models from Yang
et al. (2019).171717Optical emission models from the intrinsic FRB mechanisms
are not shown here since they are expected to have much lower $f_{\rm
opt}/f_{\rm radio}$ (e.g., $<10^{-9}$ and $<10^{-10}$ from the maser emission
and curvature radiation models, respectively; Kumar et al. 2017; Waxman 2017;
Lu & Kumar 2018; Lu et al. 2020). In these models the subsequent optical
bursts to FRBs originate from several inverse-Compton scattering processes
involving a highly magnetized central source, such as a young pulsar or
magnetar. One-zone models (e.g., pulsar magnetosphere and maser outflow) are
expected to have $t_{\rm opt}\approx t_{\rm FRB}$, while two-zone models could
result in a much longer timescale for the optical emission given the larger
scattering region than FRB emission region (e.g., $\sim$ $5000$ s for the
pulsar nebula model; Yang et al. 2019). The optical constraints for FRBs
20121102A and 20190520B from previous observations barely reach the upper end
of the pulsar nebula model with $t_{\rm opt}\sim 0.1$–$10$ s. Those of SGR
1935+2154 are well within the model predictions for pulsar magnetosphere and
nebula (see also Lin et al. 2020). For the first time for an extragalactic
FRB, our limits for FRB 20220912A also reach the model prediction ranges.
Within these models, these limits provide meaningful constraints on the
magnetic field of $B\lesssim 10^{14}$ G for magnetars or the spin period of
$P\lesssim 0.01$ s for young pulsars.

In Figure 4, we also show potential constraints from high-speed observations
with a large-aperture telescope, specifically the Subaru High-Speed Suprime-
Cam (HSSC), a planned upgrade for the Suprime-Cam instrument. These potential
future observations can provide limits of $f_{\rm eff,opt}/f_{\rm
radio}\lesssim 0.01$–$10^{-4}$ on millisecond to second timescales for FRBs
20121102A and 20190520B, reaching the pulsar magnetosphere and pulsar nebula
model parameter space. For FRB 20220912A and SGR 1935+2154, these observations
would probe $f_{\rm eff,opt}/f_{\rm radio}\lesssim 10^{-4}$–$10^{-5}$ on a
millisecond timescale to $10^{-7}$–$10^{-8}$ on a second timescale, reaching
even to the maser outflow model parameter space.

###  3.4 Constraints on the Synchrotron Maser Model

To place the luminosity limits on a longer timescale into context, we compare
them with theoretical light curves from the synchrotron maser model (Metzger
et al., 2019; Margalit et al., 2020a, b). In this model, relativistic plasma
ejected from a central engine (e.g., a magnetar) creates a shock in the
external magnetized medium (produced in previous flare events), resulting in
an FRB from synchrotron maser emission (Plotnikov & Sironi, 2019). The model
predicts that a broadband synchrotron afterglow will follow the FRB and peak
in gamma- to X-rays on a millisecond timescale and in optical on minute to
hour timescales. We provide details of the model (including the relevant
equations) in Appendix B and show model light curves in Figure 8. Assuming an
external medium with a constant (shell-like) density distribution, the
afterglow light curves are parameterized by the flare energy ($E_{\rm flare}$)
ejecting the relativistic plasma and the external density ($n_{\rm ext}$) at
the shock front; we note that the density structure could in reality be more
complex, either wind-like ($\propto r^{-2}$) or a combination of multiple
structures (e.g., see Cooper et al. 2022 for the case of SGR 1935+2154).

Figure 5: FRB-based estimates (points) and optical-based constraints (shaded
regions) on the external density ($n_{\rm ext}$) and flare energy ($E_{\rm
flare}$) in the synchrotron maser model (Figure 8). The contours mark the
number of most stringent optical limits (i.e., deeper limits with shorter
delays; from left to right in each panel: 1, 5, 25, 50, 100, and 200) used to
constrain the parameter space, while the gray dashed and dotted lines show the
possible constraints from future observations: $60$–$300$ s exposures
(limiting magnitude of $20$–$22$) with $1$–$5$ minutes delay using a 1 m class
telescope (e.g., KeplerCam and LCO), and simultaneous $0.1$–$1$ s exposure
(limiting magnitude of $20$–$22$) using an 8 m class telescope (e.g., Subaru
HSSC). Note that optical limits already reach the high-$E_{\rm flare}$ ends of
the distributions for FRB 20180916B and SGR 1935+2154, and down to the
mid-$E_{\rm flare}$ range ($\approx 65\%$) of the distribution of FRB
20220912A with our KeplerCam, LCO, and Binospec observations. The future
observations have potential to reach FRBs 20200120E and 20201124A, and to
constrain SGR 1935+2154 even further.

We explore $E_{\rm flare}$ and $n_{\rm ext}$ ranges of $10^{37}$–$10^{50}$ erg
and $10$–$10^{7}$ cm-3, respectively, for each FRB, except the Galactic SGR
1935+2154, where we use a lower $E_{\rm flare}$ bound of $10^{32}$ erg. We
calculate a model light curve (in terms of specific luminosity at each optical
filter effective wavelength) for every combination of $E_{\rm flare}$ and
$n_{\rm ext}$ and compare each optical limit with the integrated average over
the exposure time. If the model light curve is brighter than the optical
limit, we consider the specific combination of $E_{\rm flare}$ and $n_{\rm
ext}$ parameter space as constrained. The results are shown in Figure 5, along
with FRB-based estimates of $E_{\rm flare}$ and $n_{\rm ext}$ for each burst
(see Equations (B6)–(B8)). The model light curves, and thus the optical
constraints, are more sensitive to $E_{\rm flare}$ than to $n_{\rm ext}$ since
the characteristic synchrotron frequency, above which the light curves decline
exponentially, is set by $E_{\rm flare}$ (see Equations (B1)–(B5)).

As shown in Figure 5, the existing optical constraints already reach the
high-$E_{\rm flare}$ ends of the distributions for FRB 20180916B ($\gtrsim$
$10^{44}$ erg; see also Kilpatrick et al. 2021) and SGR 1935+2154 ($\gtrsim$
$10^{39}$ erg; see also Cooper et al. 2022). The combination of our
simultaneous KeplerCam and LCO observations, and nearly simultaneous deep
Binospec observation (with only a $0.4$ s delay), are the main drivers for
constraining the FRB 20220912A parameter space ($\gtrsim$ $10^{45}$ erg)
covering $\approx 65\%$ of its radio bursts observed to date (the ones with
published flux/fluence information).

In Figure 5, we also show the potential constraints from future observations.
Follow-up $60$-$300$ s observations with $1$–$5$ minutes delay using a 1 m
class telescope (e.g., KeplerCam and LCO; limiting magnitude of $20$–$22$)
have the potential to reach the high-$E_{\rm flare}$ end of the distribution
of FRB 20201124A ($\sim$ $10^{46}$ erg). Similarly, simultaneous $0.1$–$1$ sec
observations using an 8 m class telescope (e.g., Subaru HSSC) could
potentially constrain the high-$E_{\rm flare}$ end and mid-$E_{\rm flare}$
range of the distributions of FRB 20200120E ($\sim$ $10^{39}$ erg) and SGR
1935+2154 ($\sim$ $10^{36}$ erg), respectively, if we could target them in an
active phase.

##  4 Summary and Conclusions

We presented the most constraining limits on transient optical counterparts of
FRBs to date, from dedicated simultaneous observations of 13 bursts from the
repeating FRB 20220912A, and a nearly simultaneous observation with a 6.5 m
MMT/Binospec delayed by only 0.4 s (which may indeed be simultaneous given the
shutter timing uncertainty). We further presented our dedicated follow-up
observations, and compiled all available serendipitous archival and published
observations following bursts from a sample of eight repeating FRBs (including
the Galactic SGR 1935+2154) and seven non-repeating FRBs. This data set, and
in particular our simultaneous observations of FRB 20220912A, provides the
largest study to date of optical emission associated with FRBs. None of the
FRBs studied here presented detectable transient optical counterparts. Our key
findings are as follows:

  1. 1.

Nearly all archival optical observations have a delay of $\gtrsim 10^{4}$ s
relative to the time of bursts, while targeted observations have shorter and
even up to no delays providing some constraining limits.

  2. 2.

Previous simultaneous observations provided optical luminosity limits of
$\lesssim 10^{45}$ erg s-1 (with the exception of SGR 1935+2154 with $\lesssim
10^{35}$ erg s-1), while our simultaneous KeplerCam and LCO observations
provide limits of $\lesssim 10^{42}$ erg s-1, a three orders of magnitude
improvement. Our nearly simultaneous Binospec observation reaches a limit of
$\lesssim 2\times 10^{41}$ erg s-1.

  3. 3.

Normalizing the optical fluence limits (using the time delay as a proxy for
$t_{\rm opt}$) by the radio fluences, most archival observations have $F_{\rm
opt}/F_{\rm radio}\gtrsim 1$, and therefore place only weak constraints on the
relative energy per frequency in the optical and radio. On the other hand, our
simultaneous KeplerCam and LCO observations and nearly simultaneous deep
Binospec observations place more meaningful limits of $F_{\rm opt}/F_{\rm
radio}\lesssim 10^{-3}$ and $\lesssim 10^{-4}$, respectively (even comparable
to/deeper than the limits obtained for SGR 1935+2154).

  4. 4.

Comparing the simultaneous limits (normalized by radio flux), we find that
previous observations did not constrain models for simultaneous optical bursts
(with the exception of one observation of SGR 1935+2154), while our KeplerCam
and LCO and nearly simultaneous Binospec observations of FRB 20220912A place
the first meaningful constraints on model predictions, ruling out portions of
the parameter space for some models.

  5. 5.

Interpreting the optical luminosity limits in the context of the synchrotron
maser model, and comparing to the distributions of $E_{\rm flare}$ and $n_{\rm
ext}$ inferred from the radio bursts for each repeating FRB, we find that most
optical limits are not constraining (except the one limit for SGR 1935+2154),
but our simultaneous and targeted observations of FRB 20220912A rule out the
bulk of the parameter space for this model.

Given the estimated high burst rate of FRB 20220912A ($\sim$ a few hundreds
per hour; Kirsten et al. 2022b; Bhusare et al. 2022; Feng et al. 2022a;
McKinven & CHIME/FRB Collaboration 2022; Zhang et al. 2022a), and the fact
that most searches have not published arrival times for their detected bursts,
it is possible that we have simultaneous coverage for additional bursts
(including multiple bursts in a single exposure added coherently, as in the
KeplerCam and LCO exposures), which would place tighter constraints on the
fast optical burst and synchrotron maser models. This will be explored once
all the FRB measurements (e.g., arrival times, duration, flux, and fluence)
become public. Indeed, to enable this type of work on a rapid timescale, we
advocate for FRB search efforts to make these basic properties public in real
time.

We further investigate the potential of future follow-up and coordinated
simultaneous observations using $1$–$8$ m class telescopes on a wide range of
timescales from milliseconds to minutes. We find that these observations have
the potential to better constrain optical emission from FRBs compared to the
bulk of serendipitous archival observations. In particular, based on the
experience and constraining limits from our KeplerCam, LCO, and Binospec
observations, as well as previous targeted observations (e.g, Hardy et al.
2017; MAGIC Collaboration et al. 2018; Kilpatrick et al. 2021; Niino et al.
2022), we advocate for an approach of simultaneous monitoring of FRBs during
CHIME (or other facility) observing windows, especially when repeating FRBs
are in a particularly active phase.

We are grateful to Charles Kilpatrick, Wenbin Lu, Lachlan Marnoch, Sandro
Mereghetti, Eli Waxman, and Luca Zampieri for useful discussions and comments,
to Joscha Jahns for providing Arecibo burst information of FRB 20121102A, to
Vikram Ravi for providing DSA burst information of FRB 20220912A, to Daniel
Fabricant, Jan Kansky, and Benjamin Weiner for scheduling the MMT Binospec
observations and providing the instrument details, and to Griffin Hosseinzadeh
for providing the basis of our custom photometry pipeline. The Berger Time-
Domain research group at Harvard is supported by the NSF and NASA. I.A. is a
CIFAR Azrieli Global Scholar in the Gravity and the Extreme Universe Program
and acknowledges support from that program, from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(grant agreement No. 852097), from the Israel Science Foundation (grant No.
2752/19), from the United States–Israel Binational Science Foundation (BSF),
and from the Israeli Council for Higher Education Alon Fellowship. This work
makes use of observations from KeplerCam on the 1.2 m telescope at the Fred
Lawrence Whipple Observatory. Observations reported here were obtained at the
MMT Observatory, a joint facility of the Smithsonian Institution and the
University of Arizona. This work makes use of observations from the Las
Cumbres Observatory global telescope network. This paper is based in part on
observations made with the MuSCAT3 instrument, developed by the Astrobiology
Center and under financial support by JSPS KAKENHI (grant No. JP18H05439) and
JST PRESTO (grant No. JPMJPR1775), at Faulkes Telescope North on Maui, HI,
operated by the Las Cumbres Observatory. The authors wish to recognize and
acknowledge the very significant cultural role and reverence that the summit
of Haleakal$\bar{\text{a}}$ has always had within the indigenous Hawaiian
community. We are most fortunate to have the opportunity to conduct
observations from the mountain. This research has made use of the NASA
Astrophysics Data System (ADS), the NASA/IPAC Extragalactic Database (NED),
and NASA/IPAC Infrared Science Archive (IRSA, which is funded by NASA and
operated by the California Institute of Technology), and IRAF (which is
distributed by the National Optical Astronomy Observatory, NOAO, operated by
the Association of Universities for Research in Astronomy, AURA, Inc., under
cooperative agreement with the NSF). We acknowledge use of the CHIME/FRB
Public Database and the VOEvent Service, provided at https://www.chime-frb.ca/
by the CHIME/FRB Collaboration. This work has made use of data from the Zwicky
Transient Facility (ZTF). ZTF is supported by NSF grant No. AST-1440341 and a
collaboration including Caltech, IPAC, the Weizmann Institute for Science, the
Oskar Klein Center at Stockholm University, the University of Maryland, the
University of Washington, Deutsches Elektronen-Synchrotron and Humboldt
University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan,
the University of Wisconsin at Milwaukee, and Lawrence Berkeley National
Laboratories. Operations are conducted by COO, IPAC, and UW. The ZTF forced-
photometry service was funded under the Heising-Simons Foundation grant No.
12540303 (PI: Graham). This work has made use of data from the Asteroid
Terrestrial-impact Last Alert System (ATLAS) project. ATLAS is primarily
funded to search for near-Earth asteroids through NASA grant Nos. NN12AR55G,
80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO search include images
and catalogs from the survey area. This work was partially funded by Kepler/K2
grant No. J1944/80NSSC19K0112 and HST grant No. GO-15889, and STFC grant Nos.
ST/T000198/1 and ST/S006109/1. The ATLAS science products have been made
possible through the contributions of the University of Hawaii Institute for
Astronomy, the Queen’s University Belfast, the Space Telescope Science
Institute, the South African Astronomical Observatory, and The Millennium
Institute of Astrophysics (MAS), Chile. The PS1 and the PS1 public science
archives have been made possible through contributions by the Institute for
Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-
Planck Society and its participating institutes, the Max Planck Institute for
Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial
Physics, Garching, the Johns Hopkins University, Durham University, the
University of Edinburgh, the Queen’s University Belfast, the Harvard-
Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global
Telescope Network Incorporated, the National Central University of Taiwan, the
Space Telescope Science Institute, NASA under grant No. NNX08AR22G issued
through the Planetary Science Division of the NASA Science Mission
Directorate, NSF grant No. AST-1238877, the University of Maryland, Eotvos
Lorand University, the Los Alamos National Laboratory, and the Gordon and
Betty Moore Foundation. . .

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##  Appendix A Optical Constraints for Well-localized Non-repeating FRBs

We summarize the well-localized non-repeating FRB sample and their references
in Table 2. The same luminosity and flux analyses as for the repeating FRBs
are performed and shown in Figures 6 and 7, except for the simultaneous flux
analysis since no such measurements are available given the unpredictable
nature of one-off FRBs. We find that none of the optical limits are
particularly constraining. Nevertheless, we also compare the luminosity limits
to the light curves from the synchrotron maser model (although its application
is somewhat questionable given the one-off nature of these events), concluding
that only the very high-$E_{\rm flare}$ end ($\gtrsim 10^{48}$ erg) could be
constrained. These constraints are at least three orders of magnitude larger
than the FRB-based estimates (where available).

Table 2: Well-localized Non-repeating FRB Sample

FRB | R.A. | Decl. | Redshift |  $dL$aaCalculated from the host redshift assuming a standard $\Lambda$CDM cosmology with $H_{0}=71.0$ km s-1 Mpc-1, $\Omega_{\Lambda}=0.7$, and $\Omega_{m}=0.3$. |  $A_{V,{\rm MW}}$bbFrom Schlafly & Finkbeiner (2011), retrieved via IRSA. | Frequency | DM | WidthccIf only two of the width ($t_{\rm FRB}$), flux ($f_{\rm radio}$), and fluence ($F_{\rm radio}$) are reported for a particular burst, the third parameter is estimated assuming $F_{\rm radio}\sim f_{\rm radio}t_{\rm FRB}$. | FluxccIf only two of the width ($t_{\rm FRB}$), flux ($f_{\rm radio}$), and fluence ($F_{\rm radio}$) are reported for a particular burst, the third parameter is estimated assuming $F_{\rm radio}\sim f_{\rm radio}t_{\rm FRB}$. | FluenceccIf only two of the width ($t_{\rm FRB}$), flux ($f_{\rm radio}$), and fluence ($F_{\rm radio}$) are reported for a particular burst, the third parameter is estimated assuming $F_{\rm radio}\sim f_{\rm radio}t_{\rm FRB}$.
---|---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (z) | (Mpc) | (mag) | (GHz) | (pc cm-3) | (ms) | (Jy) | (Jy ms)
20190608BddLocalized with ASKAP and the host spectroscopic redshift measured from an archival SDSS spectrum (Day et al., 2020; Macquart et al., 2020). | 334.01988(1) | -7.89825(8) | $0.1178$ | $541$ | $0.125$ | $1.295$ | $338.7(5)$ | $6.0(8)$ | $\sim 4.3$ | $26(4)$
20200430AeeLocalized with ASKAP and the host spectroscopic redshift measured with Nordic Optical Telescope (NOT; Heintz et al. 2020; Kumar et al. 2020). | 229.70642(8) | +12.3769(3) | $0.160$ | $755$ | $0.112$ | $0.8645$ | $380.1(4)$ | – | – | $35(4)$
20190714AffLocalized with ASKAP and the host spectroscopic redshift measured with Keck (Bhandari et al., 2019; Heintz et al., 2020). | 183.9797(1) | -13.02103(8) | $0.2365$ | $1167$ | $0.166$ | $1.2725$ | $504(2)$ | – | – | $8(2)$
20191228AggLocalized with ASKAP and the host spectroscopic redshift measured with Keck (Shannon et al., 2019; Bhandari et al., 2022). | 344.4304(3) | -29.5941(3) | $0.2432$ | $1204$ | $0.065$ | $1.2715$ | $297.5(5)$ | $2.3(6)$ | $\sim 17$ | $40^{+100}_{-40}$
20200906AhhLocalized with ASKAP and the host spectroscopic redshift measured with Keck (Bhandari et al., 2022). | 53.4962(1) | -14.0832(2) | $0.3688$ | $1946$ | $0.126$ | $0.8645$ | $577.8(2)$ | $6.0(6)$ | $\sim 9.8$ | $59^{+25}_{-10}$
20190614DiiLocalized with VLA and the host photometric redshift estimated from spectral energy distribution fits (Law et al., 2020). | 65.07554(8) | +73.70675(8) | $0.6$ | $3480$ | $0.371$ | $1.4$ | $959(5)$ | $\sim 5.0$ | $0.12(1)$ | $0.62(7)$
20190523AjjLocalized with DSA and the host spectroscopic redshift measured with Keck (Ravi et al., 2019). | 207.0650(1) | +72.4697(6) | $0.660$ | $3908$ | $0.050$ | $1.530$ | $760.8(6)$ | $0.42(5)$ | $\gtrsim 660$ | $\gtrsim 280$

Note. — Measurements of each individual burst are published in its entirety in
machine-readable form.

Figure 6: Same as Figure 2, but for the non-repeating FRBs (summarized in
Table 2). Note the much less constraining luminosity and temporal ranges,
compared to those of the repeating FRBs, given the small number of bursts and
their large distances. Optical data sources: LCO (Núñez et al. 2021 for FRBs
20190608B and 20190714A in the $r$ band), ZTF forced-photometry service (Masci
et al. 2019 and this work for all the FRBs), ATLAS forced photometry server
(Shingles et al. 2021 and this work for all the FRBs). (The data used to
create this figure are available.) Figure 7: Same as Figure 3, but for the
non-repeating FRBs. Note that all available observations are in $F_{\rm
opt}/F_{\rm radio}\gtrsim 1$ regime.

##  Appendix B Synchrotron Maser Model

With slight modifications, we follow Kilpatrick et al. (2021) for the
application of the synchrotron maser model of Metzger et al. (2019), Margalit
et al. (2020a), and Margalit et al. (2020b) to their optical limits on the
well-localized repeating FRB 20180916B. In this model, the prompt radio
emission (i.e., FRB) and afterglow gamma-ray to optical emission originate
from the shock interaction between magnetar/pulsar-ejected plasma and the
highly magnetized external medium. The characteristic synchrotron frequency
($\nu_{\rm syn}$) is set by the flare energy ($E_{\rm flare}$):

$h\nu_{\rm syn}(t_{\rm FRB})=57\,{\rm
MeV}\left(\frac{\sigma}{0.1}\right)^{1/2}\left(\frac{E_{\rm
flare}}{10^{43}\,{\rm erg}}\right)^{1/2}\left(\frac{t_{\rm FRB}}{{\rm
ms}}\right)^{-3/2},$ (B1)

where $t_{\rm FRB}$ is the FRB duration (or width) and $\sigma$ is the
fractional magnetization (see Equations (56) and (57) of Metzger et al. 2019).
In this work, we assume a fiducial $\sigma=0.3$ (Plotnikov & Sironi 2019). The
time evolution of the synchrotron frequency is as follows:

$\nu_{\rm syn}(t)=\begin{cases}\nu_{\rm syn}(t_{\rm
FRB})\,\left(\frac{t}{t_{\rm FRB}}\right)^{-1}&\text{for $t<t_{\rm FRB}$}\\\
\nu_{\rm syn}(t_{\rm FRB})\,\left(\frac{t}{t_{\rm
FRB}}\right)^{-3/2}&\text{for $t>t_{\rm FRB}$}.\end{cases}$ (B2)

Assuming a constant (shell-like) external density ($n_{\rm ext}$) at the shock
front, the cooling frequency ($\nu_{\rm cool}$) can be expressed as a function
of time (see Equations (32) and (60) of Metzger et al. 2019 and also Equation
(4) of Kilpatrick et al. 2021):

$h\nu_{\rm cool}(t)=2.3\,{\rm
keV}\left(\frac{\sigma}{0.1}\right)^{-3/2}\left(\frac{n_{\rm
ext}}{10^{3}\,{\rm cm}^{-3}}\right)^{-1}\left(\frac{t}{10^{-3}\,{\rm
s}}\right)^{-1/2}.$ (B3)

Then, the peak luminosity ($L_{\rm peak}$) and specific luminosity ($L_{\nu}$)
can be written as a function of frequency (see Equations (63) and (64) of
Metzger et al. 2019):

$L_{\rm peak}(t)=10^{45}\,{\rm erg\,s}^{-1}\left(\frac{E_{\rm
flare}}{10^{43}\,{\rm erg}}\right)\left(\frac{t}{10^{-3}\,{\rm
s}}\right)^{-1},$ (B4)

$\nu L_{\nu}(t)=\begin{cases}L_{\rm peak}\left(\frac{\nu}{\nu_{\rm
cool}}\right)^{4/3}\left(\frac{\nu_{\rm cool}}{\nu_{\rm
syn}}\right)^{1/2}\propto t^{1/6}&\text{for $\nu<\nu_{\rm cool}$}\\\ L_{\rm
peak}\left(\frac{\nu}{\nu_{\rm sync}}\right)^{1/2}\propto t^{-1/4}&\text{for
$\nu_{\rm cool}<\nu<\nu_{\rm syn}$}\\\ L_{\rm peak}\,e^{-(\nu/\nu_{\rm
syn}-1)}\propto t^{-1}\,e^{-t^{3/2}}&\text{for $\nu_{\rm syn}<\nu$}\\\
\end{cases}$ (B5)

where we adopt an exponential cutoff above $\nu_{\rm syn}$, as in Kilpatrick
et al. (2021). The time dependence is shown for $t>t_{\rm FRB}$.
Representative model light curves in the $r$ band ($\nu=5\times 10^{14}$ Hz)
for various choices of $E_{\rm flare}$ and $n_{\rm ext}$ are shown in Figure
8.

Figure 8: Frequency and luminosity evolution (in the $r$ band) of the
synchrotron maser model (Metzger et al., 2019; Margalit et al., 2020a, b) with
an FRB duration $t_{\rm FRB}=1$ ms, assuming a shell-like density distribution
and fractional magnetization $\sigma=0.3$ (see Equations (B1)–(B5)). The light
curves exhibit changing evolution in each frequency regime ($\nu_{r}<\nu_{\rm
cool}$, $\nu_{\rm cool}<\nu_{r}<\nu_{\rm syn}$, and $\nu_{\rm syn}<\nu_{r}$).
Left: models with a fixed density of $n_{\rm ext}=2000$ cm-3, color-coded by
$E_{\rm flare}$. Right: models with a fixed flare energy of $E_{\rm
flare}=10^{46}$ erg, color-coded by $n_{\rm ext}$.

From FRB measurements of the radio fluence ($F_{\rm radio}$) and frequency
($\nu_{\rm obs}$), the isotropic radio energy ($E_{\rm radio}$) and flare
energy can be estimated as

$E_{\rm radio}=9.5\times 10^{31}\,{\rm
erg}\left(\frac{4\pi}{1+z}\right)\left(\frac{dL}{{\rm
Mpc}}\right)^{2}\left(\frac{F_{\rm radio}\cdot\nu_{\rm obs}}{{\rm
Jy\,ms}\cdot{\rm GHz}}\right),$ (B6) $\frac{E_{\rm flare}}{E_{\rm
radio}}=3.3\times
10^{4}\left(\frac{f_{e}}{0.5}\right)^{-1/5}\left(\frac{f_{\xi}}{10^{-3}}\right)^{-4/5}\left(\frac{\nu_{\rm
obs}\cdot t_{\rm FRB}}{{\rm GHz\cdot ms}}\right)^{1/5},$ (B7)

where $f_{e}$ and $f_{\xi}$ are the number density ratio of electrons to ions
in the upstream medium and the synchrotron maser efficiency, respectively (see
Equation (14) of Margalit et al. 2020a). In this work, we assume fiducial
$f_{e}=0.5$ and $f_{\xi}=10^{-3}$ (Plotnikov & Sironi, 2019). The external
density can also be estimated from FRB measurements as

$\begin{split}n_{\rm ext}&=420\,{\rm
cm}^{-3}\left(\frac{m_{\star}}{m_{e}}\right)^{2/15}\left(\frac{f_{e}}{0.5}\right)^{-11/15}\left(\frac{f_{\xi}}{10^{-3}}\right)^{-4/15}\\\
&\times\left(\frac{\nu_{\rm obs}}{{\rm GHz}}\right)^{31/30}\left(\frac{t_{\rm
FRB}}{{\rm ms}}\right)^{2/5}\left(\frac{E_{\rm radio}}{10^{40}{\rm
erg}}\right)^{-1/3},\end{split}$ (B8)

where $m_{\star}$ and $m_{e}$ are the masses of upstream particles and
electrons (see Equation (8) of Margalit et al. 2020b). In this work, we assume
a pair plasma (i.e., $m_{\star}=m_{e}$). Then, these FRB-based estimates for
$E_{\rm flare}$ and $n_{\rm ext}$ can be compared with the optical
constraints, as in Figure 5.