Main

A continuous monitoring campaign of FRB 121102 with the Five-hundred-meter Aperture Spherical radio Telescope (FAST9) has been carried out since August 2019. Between 29 August and 29 October 2019, we detected 1,652 independent burst events (Supplementary Table 1) in a total of 59.5 h, covering 1.05 GHz to 1.45 GHz with 98.304-μs sampling and 0.122-MHz frequency resolution. The total number of previously published bursts from this source was 347 (refs. 7,8 and http://www.frbcat.org). The flux limit of this burst sample is at least three times lower than those of previous observations. The cadence and depth of the observations allow for a statistical study of the repeating bursts, revealing several previously unseen characteristics. The burst statistics are shown as a function of time (Fig. 1), with the accumulated counts in 1-h bins in Fig. 1a and the day-to-day average burst rates in Fig. 1b. The burst rate peaked at 122 h−1 on 7 September and at 117 h−1 on 1 October, both measured over the respective averaged 1-h sessions. In both instances, the burst rate dropped precipitously afterwards. Burst energies are plotted against epoch (Fig. 1c), and together with the energy histogram (Fig. 1d), demonstrate bimodality that is itself a function of epoch.

Fig. 1: Detected bursts and temporal energy distribution during the observation campaign.
figure 1

a, The duration of each observation session (blue bar) and the cumulative number distribution of the bursts (red solid line). b, The rate (blue bar) and count (red dashed line) of the bursts detected. The grey shaded bars show days without observations. c, Time-dependent burst energy distribution. The blue dots are all of the 1,652 bursts, the red dots represent the average value for each observation session. The blue contour is the 2D kernel density estimation (KDE) of the bursts. d, The isotropic energy histogram of the bursts detected before MJD 58740 (14 September 2019); the red dashed line represents the KDE of this distribution.

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We measured the peak flux density, pulse width and fluence for each burst. Given the redshift z = 0.193 (ref. 5), we adopted the corresponding luminosity distance DL = 949 Mpc based on the latest cosmological parameters measured by the Planck team10 and calculated the isotropic equivalent energy of each burst at 1.25 GHz (Methods). The derived energies span more than three orders of magnitude, from below 1037 erg to near 1040 erg. Figure 2 presents the histogram of the bursts (bottom panel) and the cumulative counts as a function of energy (top panel). With a prominent peak and two broad bumps, the distribution cannot be fit by a single power law or a single log-normal function (Table 1). A satisfactory fit can be achieved with a log-normal distribution plus a generalized Cauchy function:

$$N(E)=\frac{{N}_{0}}{\sqrt{2\pi }{\sigma }_{E}E}\exp \left[\frac{-{(\log E-\log {E}_{0})}^{2}}{2{\sigma }_{E}^{2}}\right]+\frac{{\epsilon }_{E}}{1+{(E/{E}_{0})}^{{\alpha }_{E}}},$$
(1)

where ϵE = 0 for E < 1038 erg and ϵE = 1 for E > 1038 erg. The characteristic energy E0 is 4.8 × 1037 erg and is robust against uncertainties in detection threshold and choices of pipelines (Methods). The best-fit distribution index is αE = 1.85 ± 0.30. At high energies, a simple power law consistent with the results derived from previous bursts8,11 can describe the distribution reasonably well. It is clear that no single functional form can fit the data in the full energy range. At the low energy end, the log-normal distribution is consistent with a stochastic process. At the high-energy end, the generalized Cauchy function describes a steepening power law with an asymptotic slope of αE. Mathematically, a Cauchy distribution, also known as a Cauchy–Lorentz distribution, also describes the ratio of two independent, normally distributed random variables. The bimodality of the energy distribution is also time-dependent—the high-energy mode has more events before modified Julian date (MJD) 58740—which also helps to rule out that the bimodality is an artefact due to significant drifts in system calibration over time.

Fig. 2: Burst rate distribution of the isotropic equivalent energy at 1.25 GHz for FRB 121102 bursts.
figure 2

The bimodal ‘lognormal (LN) (dashed blue) + Cauchy (solid blue)’ distribution is shown in red and a single power-law fit for bursts above a certain threshold E ≥ Eh = 3× 1038 erg is shown in black. The 90% detection completeness threshold is shown by the red dashed line, corresponding to E90 = 2.5 × 1037 erg for an assumed pulse width of 3 ms and scaling as the square root of the pulse width (Extended Data Fig. 6). The missed weak bursts below E90, as indicated by the upward arrow, will make the log-normal distribution wider, but will not affect the location of the peak E0 (Methods).

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Table 1 The fitted parameters of the isotropic equivalent energy distribution
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Each pulse time of arrival was transformed to the solar system barycentre using the DE405 ephemeris. No periodicity between 1 ms to 1,000 s could be found in the power spectrum calculated using either the phase-folding or the Lomb–Scargle periodogram (Methods).

The waiting time between two adjacent (detected) bursts is δt = ti + 1 − ti, where ti  + 1 and ti are the arrival times for the (i + 1)th and (i)th bursts, respectively. All waiting times were calculated for pulses within the same session to avoid long gaps of about 24 h. The distribution of the waiting times (Fig. 3) has a dominant feature that can be well fit by a log-normal function centred at 70 ± 12 s. Selecting only high-energy pulses E > 3 × 1038 erg, the peak moves to 220 s. The waiting time distribution and absence of periodicity are generally consistent with previous findings for FRB 121102 (refs. 7,8,12,13) and can be reproduced within the uncertainties by simulating bursts arriving randomly in time. For example, the location of the peak of the log-normal distribution can be obtained with a Monte Carlo simulation mimicking the sampling cadence and number of detections of the real observations (Methods). The peaks around 70 s and 220 s in the waiting time distribution are close to the average values for the respective samples (full and high energy). This is consistent with the waiting time distribution being a combination of a stochastic process and the lack of sampling for time scales longer than about 1,000 s. The secondary peak centred at approximately 3.4 ms, however, is most probably due to substructure of individual bursts, although some may be closely spaced, independent bursts.

Fig. 3: Waiting time distribution of the bursts.
figure 3

The grey bar and solid red curve show the distribution of waiting time and its log-normal (LN) fit. The high-energy component (E > 3 × 1038 erg) is shown by a solid purple line. The three fitted peak waiting times (blue dashed vertical lines) from left to right are 3.4 ± 1.0 ms, 70 ± 12 s, and 220 ± 100 s. The peaks around 70 s and 220 s in the waiting time distribution are close to the average values for the respective samples (full and high energy). This is consistent with a stochastic process (see the main text and Methods for further discussion).

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The waiting time distribution and absence of a periodicity are in sharp contrast to expectations from standard radio pulsars, which involve stable rotation and emission in narrow beams from a narrow range of altitudes. If FRB 121102 involves a rotating object, the periodicity can be erased if beam directions and altitudes are sufficiently stochastic, introducing scatter in arrival times and reducing any features in the power spectrum or waiting time distribution that would signify periodicity. Nevertheless, the 70-s waiting-time peak still places an upper bound on the underlying period.

The optimal dispersion measure (DM) of the bursts is constrained to 565.8 ± 0.9 pc cm−3 between MJD 58724 and MJD 58776 (Methods). This suggests that the DM of FRB 121102 has increased by about 5–8 pc cm−3 (or about 1.0–1.4%) compared to earlier detections14,15,16, confirming a trend seen before17 but this time with a larger significance level (Fig. 4). Combining all of the data, the averge slope is

$$\frac{d{\rm{DM}}}{dt}=+0.85\pm 0.10\,{\rm{pc}}\,{{\rm{cm}}}^{-3}\,{{\rm{yr}}}^{-1}.$$
(2)

The long-term trend relies heavily on earlier measurements, which is further explored in the Methods. This is inconsistent with the decreasing trend predicted for a freely expanding shell (for example, a supernova remnant) around the FRB source18, but is consistent with such a shell during the deceleration (Sedov–Taylor) phase19.

Fig. 4: DM evolution of FRB 121102.
figure 4

a, Temporal DM variation for FRB 121102 over a 9-year period. The solid grey line denotes the linear fit with slope +0.85 ± 0.10 pc cm−3 yr−1. b, The distribution of DM estimates for the brightest burst on each day of FAST observations. The red dashed line indicates the average DM of 565.6 pc cm−3 during the FAST observations, and the grey region shows the 95% confidence level.

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We detected no polarization in the bursts at 1.4 GHz, in contrast with higher frequency observations6 but consistent with previous results at similar frequencies17 (Methods).

The large sample of bursts sheds new light on theoretical models of FRBs. The isotropic equivalent energy distribution (or energy function) necessitates a bimodal fit, suggesting possibly more than one emission mechanism or emission site or beam shape. The log-normal distribution characterizes the weaker bursts, the generation of which may become less efficient below the characteristic energy scale of E0 ~ 4.8 × 1037 ergs. Some magnetar models do predict a luminosity lower bound for producing FRBs20,21, and the reported E0 value may be interpreted by adjusting parameters within these models.

As shown in Fig. 3, the distribution of the time intervals between bursts (henceforth referred to as the waiting times) is log-normal in form. This behaviour is similar to that observed in other astrophysical bursting events such as soft gamma-ray repeaters22,23,24,25,26. The extremely high burst rate revealed by our observational campaign poses challenges to some models invoking emission20,21 or persistent magnetosphere interactions27 are attractive options but require masking of the rotational periodicity by stochastic beaming, by large variations in emission altitude, or by propagation delays.

The frequent triggers of bursts also constrain coherent radiation models. In particular, the popular synchrotron maser models demand well-ordered magnetic field lines in the upstream regions of the shock28,29. Clustered FRBs require successive shocks propagating into the previously shocked medium, which is hot and probably with distorted magnetic field lines. The short waiting times therefore challenge these models regarding whether coherent emission can be emitted with such short waiting times.

The synchrotron maser mechanism is also very inefficient28,29. The total isotropic energy emitted in the 1,652 bursts that we report is 3.4 × 1041 erg. Adopting a typical radiative efficiency, the total isotropic energy output during our 47-day observational campaign is already about 37.6% of the available magnetar energy. The estimation would not change significantly even considering beaming effect (Methods). Conversely, coherent emission mechanisms that invoke a neutron star magnetosphere20,30 can radiate in the radio band much more efficiently, and are therefore preferred by the data.

Methods

Search procedures and burst energetics

We carried out blind searches using two separate software packages, Heimdall31 and Presto32, with the same search parameter space (DM = 400 to 650 pc cm−3, DM step of 0.2 pc cm−3, and threshold Speak/Noise ≥ 7). Burst candidates resulting from both searches were kept for further inspections. The dynamic spectra of all candidates in both the DM = 0 time series and the de-dispersed time series at 565 pc cm−3 were generated and then manually checked to ensure that the surviving candidates have a plausible dispersion sweep and to filter out radio frequency interference (RFI) events.

To obtain high-quality flux density and polarization calibration solutions, a 1 K equivalent noise calibration signal was injected before each session, which was used to scale data to Tsys units. The off-pulse brightness (mK s) of the first pulse in each session is shown in Extended Data Fig. 1 (bottom). The standard deviation of off-pulse brightness is constant within 6% for all observations. The variation in the off-pulse level comes mainly from the zenith angle dependence of the telescope gain. Kelvin units were then converted to mJy using the zenith angle-dependent gain curve, provided by the observatory through quasar measurements. The zenith-angle-dependent gain applied for each pulse is shown in Extended Data Fig. 1 (top). The red dots denote the average gain in each day. For most days, the pulses that have brightness closest to the average value were taken at zenith angles <15 degrees, which corresponds to a stable gain of 16 K Jy−1.

We calculated the isotropic equivalent burst energy, E, following equation 9 in ref. 33:

$$E=({10}^{39}\,{\rm{erg}})\frac{4\pi }{1+z}{\left(\frac{{D}_{L}}{{10}^{28}{\rm{cm}}}\right)}^{2}\left(\frac{{F}_{\nu }}{{\rm{Jy}}\cdot {\rm{ms}}}\right)\left(\frac{{\nu }_{c}}{{\rm{GHz}}}\right),$$
(3)

where Fν = Sν × Weq is the specific fluence in units of erg cm−2 Hz−1 or Jy⋅ms, Sν is the peak flux density that has been calibrated with the noise level of the baseline, with the amount of pulsed flux above the baseline then measured, giving the flux measurement for each pulse at a central frequency of νc = 1.25 GHz, Weq is the equivalent burst duration, and the luminosity distance DL = 949 Mpc corresponds to a redshift of z = 0.193 for FRB 121102 (ref. 5).

Detection threshold and completeness

The combined effect of the sometimes bandwidth-limited structure of FRB bursts and RFI events, particularly the satellite bands around 1.2 GHz, affects the actual sensitivity of detection. The representative 7-σ detection threshold in the FAST campaign is 0.015 Jy ms assuming a 1-ms-wide burst in terms of integrated flux (fluence), which is a few times more sensitive than that previously available from Arecibo.

To quantify the detection completeness, we differentiate two kinds of signal-to-noise ratios (SNR), namely, the peak flux SNRp and the fluence flux SNRf (corresponding to integrated flux and in turn energy). Even for the same pulses, SNRp and SNRf behave differently (Extended Data Figs. 3, right, 4). The complexities and the deviation from the radiometer equation mainly arise from two ‘non-Gaussian’ aspects in the detection processes. First, the detection software (for example, Heimdall or Presto) uses the pulse peak SNRp as the initial selection criterion. Then downsampling approximate matched filtering (often just a box-car sum) in time is used to maximize the SNRf by comparing a series of trial-smoothing results with different widths in time. The candidates are then subject to visual inspection. Second, the instrumental background is often RFI-limited (Extended Data Fig. 2), particularly when the pulse is weak and/or bandwidth-limited (where the the dispersed pulse covers only a fraction of the passband); for example, a ‘weak’ (peak flux < 7σ) but more ‘complete’ pulse, with a sweep covering the full band and a larger pulse width, is more easily identified through visual inspection.

To quantify these effects, we carried out an experiment by adding simulated pulses into real data (Extended Data Fig. 3, left). All pulses were generated assuming DM = 565 pc cm−3 and a Gaussian pulse profile (in frequency) and then sampled in time and frequency in exactly the same fashion as for the FAST data. The frequency bandwidth of pulses is sampled from a normal distribution, similar to that of the observed pulses. The pulse width (in time) was sampled from a log-normal distribution, similar to that of the observed FRB bursts. The SNRp of the injected pulses was defined as the peak flux divided by the measured RMS of the real data. More than 1,000 mock FRB bursts were injected into 20 min of FAST data. We then processed the simulated datasets through the same pipelines and procedures. As shown in Extended Data Fig. 3 (left), the simulated pulses with sufficient SNRf can be distinguished from RFI and thus detected, whereas weaker ones cannot. Histograms of the peak flux SNRp and the integrated flux SNRf are shown in Extended Data Fig. 3 (right). The behaviours were similar to the previously published simulations34.

Both detection and fluence completeness are shown in Extended Data Fig. 4, which shows the recovered fraction of injected ‘FRB’ pulses as a function of burst fluence and duration. For reference, the 90 per cent detection completeness threshold for a characteristic width of 3 ms occurs at a fluence of 0.02 Jy ms, roughly corresponding to E90 = 2.5 × 1037 erg. At a given fluence, the longer durations tend to be more incomplete. Extended Data Figure 4 shows that the search has a high detection completeness (>95%) for burst durations of <20 ms when the burst fluence is >0.06 Jy ms, which corresponds to an energy of >7.5 × 1037 erg. Below this energy, the survey will begin to miss wider bursts. The current detections are consistent with event rate starting to drop below E0.

We produce a ‘reconstructed’ energy distribution, by adding the missing fraction back into the sample (Extended Data Fig. 5a) based on the simulated recovery rate. The reconstructed energy distribution has a wider log-normal distribution for the low-energy pulses, but the same peak E0. We reaffirm the existence and robustness of the peak location in this simulation.

Energy distribution

The histogram of burst energies exhibits two clearly separable bumps, which can be well fit by two log-normal functions (Extended Data Fig. 5b, top).

The burst rate versus burst energy distribution shows a broad bump centred around 4.8 × 1037 erg and a power-law-like distribution with a slope close to −2 for the high-energy tail. We first fit the distribution with a single power law, Cauchy, and log-normal function, respectively. The χ2 and R2 tests in Table 1 show that these single-component models cannot adequately describe the data.

We then test the hypothesis that the energy distribution can be described by a power-law function \(N(E)={N}_{1}{E}^{-{\alpha }_{E}}\) in certain energy range Ei ≤ E ≤ Ef. That is,

$${N}_{1}=\frac{{N}_{ev}(1-{\alpha }_{E})}{{E}_{f}^{1-{\alpha }_{E}}-{E}_{i}^{1-{\alpha }_{E}}},$$
(4)

where N1 is a normalization constant, and Nev is the total number of bursts included. For 3 × 1038 ≤ E ≤ 8 × 1039 erg, the energy function is consistent with a power law with αE = 1.85 ± 0.3. Previous studies also found power-law energy distributions for the Australian square kilometre array pathfinder (ASKAP) sample and all of the bursts from the FRB catalogue8,11,23,35,36,37,38. A bimodal distribution is clearly needed to properly cover the full energy range. A log-normal function plus a Cauchy function for the high-energy range can achieve a satisfactory fit, with a coefficient of determination R2 = 0.928.

Giant pulses from the Crab pulsar show a power-law amplitude distribution combined with a long tail of ‘supergiant pulses’39 due to underlying magnetospheric physics that might help to elucidate the energy distribution of FRB 121102. However, the Crab’s giant pulses are manifestly tied to the spin of the neutron star, whereas the bursts discussed here are not periodic. Therefore, a mechanism is needed to erase the periodicity in detected bursts; this could be caused by large variations in altitude or beaming that are not strictly tied to the rotating neutron star. It could also be caused by propagation effects in the circumsource medium.

The low-energy component is similar to distributions seen in many radio pulsars. At the high-energy end, the trend towards a power law with slope αE is reminiscent of giant pulses from the Crab pulsar. The bimodality of the energy distribution is also clearly time dependent, with the high-energy mode having more events before MJD 58740. The fewer high-energy events after that epoch may signify that the bimodality is due to time-dependent lensing but more plausibly might be an analogue of ‘mode changes’ commonly seen in long period radio pulsars where pulse components change their relative amplitudes and occurrence rates. Further observations can distinguish between these possibilities.

In Fig. 1, temporal variations are seen for the collective behaviour of each session. There are days with significantly brighter averages, although weak bursts are always present.

Analysis with a modified Cauchy function

A common Cauchy distribution is

$$p(x)=\frac{1}{\pi ({x}^{2}+1)}$$
(5)

Cauchy distribution can also be obtained from the distribution of the ratio of two independent normally distributed random variables with zero mean.

Assume that X and Y are independent of each other and obey a normal distribution:

$${f}_{X|Y(x)}=\frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}$$
(6)

Then the probability density of Z = X/Y is:

$$\begin{array}{c}{f}_{X/Y}(z)={\int }_{-\infty }^{\infty }|y|f(y,\,yz)dy\,\,\,\,\,\,\,\,\,\\ \,\,\,\,\,\,\,\,={\int }_{-\infty }^{\infty }|y|f(y)f(yz)dy\\ \,\,\,\,\,\,\,=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|y|{e}^{-{y}^{2}/2}{e}^{-{y}^{2}{z}^{2}/2}dy\\ \,\,\,\,\,\,\,\,=\frac{1}{\pi }{\int }_{0}^{\infty }y\,{e}^{-({z}^{2}+1){y}^{2}/2}dy\\ \,\,\,\,\,\,\,=\frac{1}{2\pi }{\int }_{0}^{\infty }{e}^{-({z}^{2}+1)u/2}du\\ \,\,\,\,\,\,\,=\frac{1}{2\pi }\frac{2}{({z}^{2}+1)}\\ \,\,\,\,\,\,\,\,=\frac{1}{\pi ({z}^{2}+1)}\end{array}$$
(7)

Hence, if we change the index 2 of the normal distribution to alpha, then the Cauchy distribution will be the general Cauchy function:

$$p(x)=\frac{1}{\pi ({x}^{\alpha }+1)}$$
(8)

The generalized Cauchy function can describe the ratio between two normally distributed variables. The best-fit index of 1.85 is close to 2 with σ ~ 0.3, which could suggest correlated events for generating strong bursts.

Width distribution

The pulse width Weq is shown against flux density and the pulse width distribution of the bursts (Extended Data Fig. 6, left and right, respectively). The equivalent width Weq is defined as the width of a rectangular burst that has the same area as the bursts, with the height of peak flux density denoted as Speak. In our sample, several pulses might be described as multiple components in a single burst, if there is ‘bridge’ emission (higher than 5σ) between pulses for the bursts with a complex time-frequency structure. This results in some bursts having overestimated equivalent widths. The computed equivalent widths range from 0.43 ms to about 40 ms, consistent with a log-normal distribution centred around approximately 4 ms. This is consistent with the known statistical properties of repeating FRBs40,41.

Monte Carlo simulations of the waiting time distribution

Following the exact setup of the observations, including starting time, duration, sampling rate and pulse burst rate, we generate random times of arrival through Monte Carlo simulations. The simulation was performed three times, with numbers of generated pulses of 100 × 1,652 ~ 1.6e5, 1 × 1,652 = 1652 and 0.2 × 1,652 ~ 330. The distributions of different waiting time sets are shown in Extended Data Fig. 7. The log-normal distribution appears in the randomly generated waiting time distribution, and centred at 0.62 s, 61.89 s and 272.04 s.

The peak waiting time of the log-normal distribution increases as the number of bursts in the simulated sample decreases. Among the 1,652 pulses of FRB 121102, 296 have higher energy than 3 × 1038 erg, which accounts for about one-fifth of the total pulses. The peak waiting time of these 296 pulses is 220 s, which is close to the peak waiting time from the generated 0.2 × 1,652 ~ 330 pulses. The main feature of waiting is the log-normal distribution centred at 70 s, which is close to the simulated distribution with 1,652 generated pulses.

Our simulations suggest that the observed log-normal distributions of waiting time centred at 70 s and 220 s, though not an instrumental effect, are nevertheless consistent with emission from a source that emits FRBs randomly or other ‘masking’ factors, such as rotating attitudes.

Periodicity search

The Lomb–Scargle periodogram method42,43 is widely used to identify periodicities in data that are not uniformly sampled.

We apply this method to the times of arrival of FRB121102 to determine whether there is a possible period. If the bursts of FRB121102 do have a period, folding the burst arrival times according to this period would show clustering in burst phase.

Periodograms of bursts from FRB121102 for periods ranging from 1 ms to 100 d are shown in Extended Data Fig. 8; the left panel covers periods from 0.01 d to 100 d and the right panel periods from 1 ms to 1,000 s. Of the five peaks in the left panel, four are at periods of 0.998 d, 0.499 d, 0.333 d and 0.153 d, corresponding to the daily sampling and its higher harmonics. The fifth peak at about 24 d yields a non-random but broad distribution of burst phases (Extended Data Fig. 8, bottom left) that most probably reflects non-uniform detections over the 47-d data span. The marginally significant peak at 10.575 ± 0.008 ms in the top-right panel appears to be related to a large multiple of the original 98.304-μs sample interval of the data. Folding with that period does not show any concentration in pulse phase (Extended Data Fig. 8, bottom right).

In addition to a search for a constant period over the 47-d dataset, we also searched for periods (P) between 1 ms and 1,000 s accompanied by a period derivative (\(\dot{P}\)) between 10−12 and 10−2 ss−1 and the same negative \(\dot{P}\) range to fold all of the pulses. This also did not reveal any underlying period. In addition, we divided the pulses according to energy with dividing lines at 5 × 1037 erg and 3 × 1038 erg, and found that the pulses in different energy intervals do not have significant periodicity. All observing sessions fall in the predicted active phase of FRB 121102 (refs. 44,45), thus the addition of this sample does not alter the 157-d period found there.

DM variation

Before we can study the detailed emission characteristics of the bursts, the optimum DM should be determined. De-dispersed pulse profiles were created for each DM trial between 500 and 650 pc cm−3 with a step size of 0.05 pc cm−3, using the single pulse search tools in Presto32. Gaussians (multiple when necessary) were fitted to the profiles. The derivative of each Gaussian was then squared. For multiple components, the squared profiles were summed. The optimum DM was then identified according to the maximization of the area under the squared derivative profiles, thus maximizing the structures in the frequency integrated burst profile. The typical DM optimization method and de-dispersed profiles are shown in Extended Data Fig. 9.

The resulting histogram distribution of DMs are shown in Extended Data Fig. 10 (left), the optimal value is 565.8 ± 0.3 ± 0.76 pc cm−3, between MJD 58724 and MJD 58776. The two uncertainties are statistical error and systematic error. The latter is estimated by measuring the ΔDM that results in a DM time delay across the whole band equal to half the equivalent width of the bursts (the typical value is 1.5 ms). This suggests that the DM of FRB 121102 has increased by 5–8 pc cm−3 (1.0–1.4%) with more than 20σ significance compared with earlier detections from MJD 57364; ref. 14 reported the optimal value to be 558.6 ± 0.3 ± 1.4 pc cm−3 with similar methodology. Furthermore, the measured DM values and their uncertainties of the bursts are shown in Fig. 4 as a function of individual observations. DM apparently increased over the past 6 years15,16 and is found to be consistent with a DM growth rate of +0.85 ± 0.10 pc cm−3 yr−1 in equation (2).

To inspect the reliability of this DM variation trend, we divided the DM measurement into three time bins according to the dates of the event and generated mock DM values in each bin, based on the mean and the standard deviation of the measured DMs in the respective time bins. The null hypothesis was then tested based on the generated DMs under the assumption that DM does not change over time. For each set of generated DM, a slope was fitted. Based on 20,000 trials, the σ of the resulting slope distribution is 0.38 pc cm−3 yr−1, shown in Extended Data Fig. 10 (right). Thus, the fitted DM growth rate of 0.85 could result from a null hypothesis sample at a 2.6% probability, slightly better than 2σ. This is apparently less significant than the simple fitting, but probably more realistic.

Polarization characteristics

The polarization was calibrated by correcting for differential gains and phases between the receivers through separate measurements of a noise diode injected at an angle of 45° from the linear receivers. The circular polarization of a few brightest bursts is consistent with noise, lower than a few per cent of the total intensity, which agrees with ref. 6. We searched for the rotation measure (RM) from −6.0 × 105 to 6.0 × 105 rad m−2, a range that is much larger than the RM ~ 105 rad m−2 reported in ref. 6, but no significant peak was found in the Faraday spectrum. The linear polarization becomes negligible at L band compared to almost 100% linear polarization at C band reported in ref. 6. For our data, we estimate the depolarization fraction fdepol using

$${f}_{{\rm{depol}}}=1-\frac{\sin (\varDelta \theta )}{\varDelta \theta },$$
(9)

where the intra-channel Faraday rotation Δθ is given by

$$\Delta \theta =\frac{2{{\rm{RM}}}_{{\rm{obs}}}{c}^{2}\Delta \nu }{{\nu }_{c}^{3}},$$
(10)

where c is the speed of light, Δν is the channel width, and νc is the central channel observing frequency. Taking RMobs = 105 rad m−2 reported in ref. 6, Δν = 0.122 MHz and νc = 1.25 GHz for our data, we get fdepol = 20%. Although the depolarization fraction is not negligible, the non-detection of the linear polarization cannot be caused by depolarization assuming RMobs < 105 rad m−2.

We are confident about the non-detection. We have applied the same analysis procedures to bright pulsars and retrieved expected results. The same procedure has been used in multiple publications46,47,48. Our non-detection does not conflict with the previous almost 100% linear polarization because all previous polarization detections were accomplished at frequency bands higher than L band (about 1.4 GHz). Polarization measurements at 4–8 GHz were published in ref. 6 and at 3–8 GHz in ref. 49. Unless the RM was somehow much larger during the FAST observations compared to previous determinations, the non-detection appears to require strong frequency evolution of the linear polarization.

Energy budget constraint on the synchrotron maser magnetar model

The total isotropic energy emitted in the 1,652 bursts reported in this paper is 3.4 × 1041 erg. We consider that each FRB has a beaming factor of fb = δΩ/4π < 1 (where δΩ is the solid angle of the emission of individual burst). If these individual bursts are isotropically distributed in sky, even though the energy budget for each FRB is smaller by a factor of fb, there would be also approximately \({f}_{b}^{-1}\) more undetected bursts (whose emission beams elsewhere) so that the total energy is not changed27. Adopting a typical radiative efficiency η ~ 10−4η−4 from numerical simulations50 (which could be even lower in view of the very rapid repetition rate), the total energy emitted solely during the approximately 60 h of observation spanning 47 d is already \( \sim 6.4\times {10}^{46}{\eta }_{-4}^{-1}\) erg (considering FAST only observed about 60 h during these 47 d and assuming that the observed rate applies to the epochs of no observations as well). The total magnetic energy of a magnetar with a surface magnetic field strength B = 1015B15 G is ~(1/6)B2R3 ≃ 1.7 × 1047 erg. The total energy emitted during our observational campaign is already about 37.6% of the available magnetar energy. One possible way to avoid this criticism is to argue that there is a ‘global beaming factor’ Fb = ΔΩ/4π that is smaller than unity but greater than fb to describe the beaming angle of all emitted FRBs. This factor could be of the order 0.1 for pulsar-like emission due to the geometry defined by the magnetosphere configuration of the central source. However, for relativistic shocks invoked in the synchrotron maser models, Fb would not be much less than unity due to the lack of a collimation mechanism for a Poynting-flux-dominated outflow.

In view of the fact that FRB 121102 has already been active for nearly a decade and that many faint bursts such as the ones reported in this paper have escaped detection from previous telescopes, we believe that the synchrotron maser model is significantly challenged from the energy budget point of view. A previous analysis using data observed by the Green Bank Telescope7 obtained a similar result51.Even if this global beaming factor Fb = 0.1 is assumed, the released energy during this active period is already 3.8% of the total magnetar energy budget. This disfavors the synchrotron maser model and any model that invokes a low radio radiative efficiency.