Main

The Commensal Real-time ASKAP Fast Transients (CRAFT) survey on the Australian Square Kilometre Array Pathfinder (ASKAP) has commissioned a mode capable of localizing fast radio bursts (FRBs) with subarcsecond accuracy, thus enabling identification of their host galaxies and measurement of their redshifts z. ASKAP consists of 36 antennas equipped with phased array feeds, able to view 30 degrees2 on the sky. Bursts are detected by incoherently summing the total power signal of individual beams from each of the antennas. Bursts detected in the incoherent pipeline are subsequently localized interferometrically by triggering a download of voltage data from a 3.1-s-duration ring buffer that is correlated and imaged at high time resolution to provide the localizations15,16. The 6-km baselines of the array yield statistical position errors of approximately 10″(S/N)−1, where the final coherent signal-to-noise ratio of the burst, S/N, exceeds 50 for any burst whose signal-to-noise ratio in the incoherent pipeline is greater than 9. The resulting statistical (thermal) uncertainties are smaller than 0.2″. Systematic errors in these positions are typically smaller than 0.5″. At z = 0.5, 1″ corresponds to 5 kpc which is approximately the precision needed to associate an FRB with its host galaxy while reducing the chance coincidence probability to <1% (ref. 16).

We report the detection of four localized ASKAP bursts. Table 1 lists the burst properties, sky positions and host galaxy offsets, while Fig. 1 shows the host galaxy identifications (see also ref. 21 and Methods). Their dispersion measures (DMs) are well in excess of the 30−100 pc cm−3 contributions expected from the disk and halo of the Milky Way at high Galactic latitudes13,22, with the large excesses attributable to the intergalactic medium (IGM) and gas within each burst host galaxy. Two other ASKAP-detected bursts and their host galaxies were reported previously15,16 in addition to three other host-galaxy identifications14,17,18.

Table 1 Properties of FRBs interferometrically localized with ASKAP
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Fig. 1: Locations of FRBs relative to their host galaxies.
figure 1

a–f, Optical images of the host galaxies of six FRBs localized by ASKAP, including the four new bursts reported here. a, FRB 180924; b, FRB 181112; c, FRB 190102; d, FRB 190608; e, FRB 190611; and f, FRB 190711. HG, host galaxy. White ellipses denote the 90% confidence region of each burst position, including statistical uncertainty and phase referencing errors, while the red crosses mark the measured centroids of each host galaxy. The identification of the host galaxy of FRB 190611 is tentative. a–d, Deep VLT g-band images; e, f, deep GMOS i-band images.

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The precise localization of a set of FRBs to their host galaxies provides the first ensemble of DMFRB and zFRB measurements. The DMFRB measurement represents the electron density weighted by (1 + z)−1 integrating over all physical distance increments ds to a given FRB: \({{\rm{DM}}}_{{\rm{FRB}}}=\int {n}_{{\rm{e}}}\,{\rm{d}}s/(1+z)\) . Physically, we expect DMFRB to separate into four primary components:

$${{\rm{DM}}}_{{\rm{FRB}}}(z)={{\rm{DM}}}_{{\rm{MW}},{\rm{ISM}}}+{{\rm{DM}}}_{{\rm{MW}},{\rm{halo}}}+{{\rm{DM}}}_{{\rm{cosmic}}}(z)+{{\rm{DM}}}_{{\rm{host}}}(z)$$
(1)

with DMMW,ISM the contribution from our Galactic ISM, DMMW,halo the contribution from our Galactic halo13, DMhost the contribution from the host galaxy including its halo and any gas local to the event, and DMcosmic the contribution from all other extragalactic gas. Only DMcosmic, determined by its path length through the IGM and the increase in baryon density with look-back time, is expected to have a strong redshift dependence, although DMhost is weighted by (1 + zFRB)−1 and may correlate with age, for example, if host galaxies have systematically lower mass at earlier times.

Adopting our cosmological paradigm of a flat universe with matter and dark energy, the average value of DMcosmic to redshift zFRB is:

$$\langle {{\rm{D}}{\rm{M}}}_{{\rm{c}}{\rm{o}}{\rm{s}}{\rm{m}}{\rm{i}}{\rm{c}}}\rangle =\mathop{\mathop{\int }\limits^{{z}_{{\rm{F}}{\rm{R}}{\rm{B}}}}}\limits_{0}\frac{c{\bar{n}}_{{\rm{e}}}(z){\rm{d}}z}{{H}_{0}{(1+z)}^{2}\sqrt{{\varOmega }_{{\rm{m}}}{(1+z)}^{3}+{\varOmega }_{\Lambda }}}$$
(2)

with mean density \({\bar{n}}_{{\rm{e}}}={f}_{{\rm{d}}}{\rho }_{{\rm{b}}}(z){m}_{{\rm{p}}}^{-1}(1-{Y}_{{\rm{He}}}/2)\), where mp is the proton mass, YHe = 0.25 is the mass fraction of helium, assumed doubly ionized in this gas, fd(z) is the fraction of cosmic baryons in diffuse ionized gas (this accounts for dense baryonic phases, for example, stars and neutral gas; see Methods), ρb(z) = Ωbρc,0(1 + z)3, and Ωm and ΩΛ are the matter and dark energy densities today in units of \({\rho }_{{\rm{c}},0}=3{H}_{0}^{2}/8{\rm{\pi }}G\) where we parameterize Hubble’s constant H0 in terms of the dimensionless h70 = H0/(70 km s−1 Mpc−1).

The DMMW,ISM term (equation (1)) arises primarily from the so-called warm ionized medium of the Galaxy and is estimated from a model of this ISM component22,23. At the high Galactic latitudes (|b| > 33°) of the ASKAP sample, the value is DMMW,ISM ≈ 30 pc cm−3. The DMMW,halo term is not well constrained13, but is expected to be in the range of approximately 50−100 pc cm−3. Hereafter we assume DMMW,halo = 50 pc cm−3 and emphasize that the sum of its scatter and uncertainty are less than those of DMcosmic and DMhost, which we discuss below.

Figure 2 shows the theoretical curve for 〈DMcosmic〉 versus zFRB for the Planck15 cosmology20 and a model estimate of the scatter (90% interval) due to statistical variations in foreground cosmic structure (see Methods). Overplotted on the model are the estimated DMcosmic and measured zFRB values for all arcsecond-localized FRBs. We have estimated DMcosmic by subtracting the following from the measured DMFRB value: DMMW,ISM from the Galactic ISM model; our assumed DMMW,halo contribution; and an ansatz of DMhost = 50/(1 + z) pc cm−3 estimated from theoretical work and informed from the analysis below. We ignore FRB 121102 and FRB 190523 in most of the analysis that follows because of selection bias in their discovery, FRB 180916 owing to its low Galactic latitude (see Methods), and FRB 190611 because of its tentative association with a host galaxy (see Methods). The five ASKAP FRBs that remain comprise what we term the gold-standard sample. The agreement between model and data is striking. Effectively, the FRB measurements confirm the presence of baryons with the density estimated from the cosmic microwave background (CMB) and Big Bang nucleosynthesis (BBN), and these five measurements are consistent with all the missing baryons being present in the ionized IGM.

Fig. 2: The DM–redshift relation for localized FRBs.
figure 2

Data points are estimations of the cosmic dispersion measure (DMcosmic) versus FRB redshift (zFRB) for all current arcsecond- and subarcsecond-localized FRBs. The DMcosmic values are derived by correcting the observed dispersion measure DMFRB for the estimated contributions from our Galaxy and the FRB host galaxy (the latter assumed here to be 50(1 + z)−1 pc cm−3; see text for details). Coloured points represent the gold-standard sample on which our primary analysis is based. The solid line denotes the expected relation between DMcosmic and redshift for a universe based on the Plank15 cosmology (that is, Ωb = 0.0486 and H0 = 67.74 km s−1 Mpc−1). The shaded region encompasses 90% of the DMcosmic values from a model for ejective feedback in Galactic haloes that is motivated by some simulations (with F = 0.2 in equation (4) in Methods), illustrating that the observed scatter is largely consistent with the scatter from the IGM.

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This result motivated us to quantitatively test for the consistency of Ωbh70 with CMB and BBN measurements, simultaneously determining the uncertain host-galaxy contributions to DMFRB as well as the sightline-to-sightline variance in dispersion owing to the IGM. We do this by analysing the joint likelihood of our sample of five (seven, including FRBs 190523 and 190611) DMFRB, zFRB measurements against a four parameter model: one parameter for the large-scale structure scatter in DMcosmic, two parameters for DMhost (a mean and a scatter), and Ωbh70. Our model for the contributions to DMFRB starts with equations (1) and (2), and we develop parametric models for DMhost and the intrinsic scatter in DMcosmic. We again fix DMMW,halo = 50 pc cm−3 and adopt the Galactic ISM model22 for DMMW,ISM. Uncertainty in these values can be absorbed into our model for DMhost.

For DMcosmic, our model accounts for scatter in the electron column from foreground structures, which is largely caused by random variation in the number of haloes a given sightline intersects. Cosmological simulations show that this variation is sensitive to the extent to which galactic feedback redistributes baryons around galactic haloes11,16 and that the fractional standard deviation of the cosmic DM equals approximately11 Fz−1/2 for z < 1, where the parameter F quantifies the strength of the baryon feedback (0.1 being strong feedback and 0.4 being weak). Stronger feedback corresponds to situations in which feedback processes expel baryons to larger radii from their host galaxies or where more massive haloes are evacuated by such feedback. The formalism incorporates the effect of large-scale structure associated with voids and the intersection of sightlines with clusters. We find that a one-parameter model based on a physically motivated shape for the probability distribution of DMcosmic given F provides a successful description of a wide range of cosmological simulations (see Methods). Our form for the distribution is strongly asymmetric towards lower redshifts, admitting large DMcosmic values that we find are important in the estimation of Ωbh70.

We chose our model for DMhost to follow a log-normal distribution, characterized by a median exp(μ) and logarithmic width parameter σhost such that the standard deviation of the distribution is \(\exp (\mu ){{\rm{e}}}^{{\sigma }_{{\rm{host}}}^{2}/2}{({{\rm{e}}}^{{\sigma }_{{\rm{host}}}^{2}}-1)}^{1/2}\). We do not attempt to incorporate redshift-dependent evolution in the host-galaxy dispersion contribution, but we do scale the distribution of DMhost by the factor (1 + zhost)−1 applicable to a parcel of plasma at redshift zhost so that DMhost is interpreted as the dispersion in the rest frame of the host galaxy. Our choice of a log-normal distribution is conservative in that it allows for a tail extending to large positive values, which may not be present in our sample given our selection criteria (see Methods) and the burst locations relative to the host stellar surface density. We explore DMhost distributions with median values in the range exp(μ) = 20−200 pc cm−3 and σhost in the range 0.2−2.0.

Our final analysis compares the relative likelihood of models in a four-parameter space (Ωbh70, F, σhost, μ); see Methods for a Bayesian approach that yields similar constraints. Marginalizing the other parameters over ranges restricted by other physical constraints, we derive the constraints on Ωbh70 shown in Fig. 3 using our five-FRB gold-standard sample. The results are fully consistent with the joint CMB + BBN estimations and with only five (seven) burst redshifts the experiment yields a precision of σ(Ωbh70)/Ωbh70 = 0.31 (0.28) at the 68% confidence level, with F marginalized over the range [0.09, 0.32] (see Methods). This quantitative result for Ωbh70 substantiates our inference from the DM–z relation in Fig. 2 that the FRB ensemble has resolved the missing baryons problem. The ratio of our estimated Ωb to that from CMB and BBN measurements is \({1.1}_{-0.6}^{+0.5}\,{h}_{70}\). Formally, we exclude Ωbh70 < 0.02 (0.01) at the 98.6% (99.8%) confidence level. This constraint should improve considerably in the near term as ASKAP and other facilities acquire a larger sample of bursts with redshifts.

Fig. 3: The density of cosmic baryons derived from the FRB sample.
figure 3

The constraints on the IGM parameters Ωbh70 and F, and the host galaxy parameters exp(μ) and σhost, for a log-normal DM distribution are derived using the five gold-standard bursts (as described in the text and Methods). a, Corner plots displaying the probability of a given value of F, exp(μ) or σhost relative to its most likely value, and marginalized over the other parameters: heavy dashed lines represent the most likely values in each case. The green, dotted lines in the corner plots of F, eμ and σhost denote the relative likelihood of these parameters when Ωbh70 is constrained to the value set by the CMB + BBN measurements. The contours displayed are in increments of 10% of the peak value. b, Magnified view of the corner plot for Ωbh70 where the orange shaded region denotes the range to which Ωbh70 is confined by CMB + BBN measurements. The dotted and dot-dashed lines represent the 68% and 95% confidence intervals of each parameter, respectively. The distribution of Ωb is alternatively marginalized over the range of F indicated by cosmological simulations, [0.09, 0.32] (blue curve; see Methods), and over the entire range of F [0,0.5] investigated here (red curve).

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Additionally, analysis of our gold-standard sample mildly favours a median host galaxy contribution of about 100 pc cm−3 with a factor of two dispersion around this value (σhost ≈ 1). This quantifies our result that the host contributions are sufficiently small to not compromise the use of FRBs for cosmology and IGM science. Even with our current small sample, we are beginning to constrain viable models for the redistribution of the cosmic baryons by galactic feedback. If we adopt a prior on Ωbh70 from the CMB, BBN and supernovae surveys24, we find \(F={0.04}_{-0.04}^{+0.26}\) (68% confidence), and if we further include FRB 190523 and FRB 190611 we find \(F={0.23}_{-0.12}^{+0.27}\) (see Methods). A factor of two smaller error would start to differentiate between viable feedback scenarios (as discussed further in Methods), suggesting that FRBs have not only revealed that all the baryons are present but—with modestly larger samples—could constrain where they lie.

Methods

Sample selection

Analogously to cosmological studies of the distance ladder with supernovae, we wish to establish a strict set of criteria for the FRB sample to minimize biases while maximizing statistical power. On the latter point, we wish to construct the largest sample while avoiding events whose DM is dominated by non-cosmological effects (that is, host or Galactic gas). Regarding bias, the greatest concern is the association of a host galaxy with a given FRB on the basis of the DM–z relation, that is, adopting this relation as a prior to establish the host identity. This is a valuable practice when one aims to resolve the underlying host-galaxy population17,27 but would bias any cosmological study. Last, one must also be cognizant of biases related to triggering on FRB events. This practice is a complex function of the FRB fluence, its DM, and the pulse properties28.

With these issues in mind, we propose the following set of criteria (1–4) to generate a ‘gold-standard sample’ of FRBs for cosmological study:

  1. 1.

    To make a confident host-galaxy association, we require the probability of mis-identifications to be <1% without invoking the DM–z relation since to do so would introduce a bias. For this we require the 95% localization area to encompass one and only one galaxy unless multiple galaxies have a common redshift. By ‘encompass’ we include light from any part of the galaxy. In practice, this will require a localization to <1″ for z > 1, but becoming less stringent for less distant hosts. We propose an initial set of specific criteria as follows. (i) Define 95% areas for the localization and for each galaxy in the region. Call these L and G1, G2, and so on. (ii) Demand one and only one galaxy overlap L. The only exception is if z1 = z2. (iii) For the overlapping L and G, require that >50% of the smaller area lies within the larger. (iv) Do this for galaxies as faint as R = 25 (anything fainter is generally too difficult for a spectroscopic redshift anyhow).

  2. 2.

    The finite temporal and spectral resolution of the FRB survey causes a decline in sensitivity with increasing DM to the point that telescope resolution causes an effective threshold at DMcutoff at which point a burst would no longer have been detectable. A conservative approach would omit any burst with DMcutoff sufficiently low that it excludes a large (\(\gtrsim 30 \% \)) fraction of the total probability of p(DMtotal|z), on the grounds that it presents a biased probe of p(DMtotal|z). Although application of this criterion presupposes a DM–z relation and its probability density function (PDF), it does so only weakly. This point is addressed in detail below in the subsection on biases in the probability distribution.

    An event detected near to the sensitivity threshold is biased in the sense that the instrumental decline in sensitivity with increasing DM dictates that any burst detected near this threshold would not have been detectable at higher DMs. Thus, for a given redshift we are biased to finding events with DMs that are under-representative of the entire DM distribution at that redshift. Thus, only more luminous bursts, whose detection S/N is sufficiently high that DMcutoff exceeds the plausible range of DMtotal at that redshift are devoid of this bias.

  3. 3.

    FRB events with extreme properties (for example, high RM, large temporal broadening) will be excluded to minimize the impact of host galaxy and Galactic gas.

  4. 4.

    A cutoff is imposed on FRBs whose expected contribution from the disk component of the Milky Way ISM is large, to avoid large uncertainties in the subtraction of the Galactic ISM DM contribution. Models of the Galactic plasma distribution22 typically produce errors in known pulsar distances of the order of several tens of per cent (and much higher in some cases)29,30. To avoid DM errors in excess of about 100 pc cm−3 we restrict our sample to those bursts whose predicted DMISM values are less than 100 pc cm−3. A conservative application of this criterion restricts FRB detections to sight lines at high Galactic latitude, \(|b|\gtrsim 20^\circ \).

We acknowledge that all criteria 1–4 are subject to refinement as we learn more about FRB progenitors and their host galaxies.

Regarding criterion 3, a dominant contributor to the DM variance is the circumburst environment and the ISM of the host galaxy. Although it is not possible to make a precise estimate of this component, the burst RM, the amount of Faraday rotation exhibited by linearly polarized emission caused by its propagation through a magnetized plasma, presents a means of identifying those bursts whose radiation has probably propagated through a substantial (>100 pc cm−3) amount of matter in the host galaxy. For each burst the Milky Way contribution to RM for |b| > 10° is small (<250 rad m−2) and measurable31 and the IGM contribution is estimated32 to be about 1 rad m−2. Galactic haloes, similarly, have been inferred to make contributions of several tens of rad m−2 to the RM33 from radio-loud quasar observations, but our first analysis with an FRB16 yields RM < 10 rad m−2. A suitable cutoff due to host-galaxy ISM contamination is suggested by assuming the host-galaxy magnetic field strength is comparable to that of our Galaxy. Measurements of Faraday rotation and dispersion from pulsars in the Milky Way (see figure 3 in ref. 34) exhibit a mean trend DM = 1.55|RM|0.95 ≡ fDM(RM), where RM and DM are measured in their usual units of rad m−2 and pc cm−3 respectively. We find that the root-mean-square (r.m.s.) deviation of the actual DM values from their values predicted on the basis of this trend using |RM| are 69% of the DM (that is, the r.m.s. errors are 69% of the mean DM value: ⟨[DM − fDM(RM)]2/DM2⟩1/2 = 0.69). We further find that there is an 85% probability that the actual DM deviates from its predicted value by less than 0.9 times the actual DM value, and a 96% probability that the predicted value differs by less than 2.0 times the DM value. We therefore suggest that a cutoff criterion |RM − RMMW|observed < 100(1 + z)2 rad m−2 bounds the dispersion measure to \({\rm{D}}{\rm{M}}\lesssim 250(1+z)\,{\rm{p}}{\rm{c}}\,{{\rm{c}}{\rm{m}}}^{-3}\) with 85% confidence.

A similar trend observed between the DM and the temporal smearing of Galactic pulsars caused by scattering35,36 can also be used to place upper bounds on the host contribution. Recent updates to this relation36 indicate that, on average, a pulse smearing time, τ, less than 33 ms (2 ms) limits DM to <200 pc cm−3 (300 pc cm−3) at 0.327 GHz (1 GHz). However the DM–τ relation exhibits ~0.8 dex variation about the trend (as discussed in the context of FRBs in the supplementary material in ref. 15), thus requiring τ < 5 ms to ensure a reasonable (~70%) confidence that the DM contribution is less than 200 pc cm−3. We caution that the use of τ as an indicator of the host-galaxy DM contribution is subject to considerable uncertainty, since neither the distances to the scattering material from the bursts, nor even the nature of the turbulence responsible for the temporal smearing observed in FRBs, is well established. The estimates presented here would be invalid, for instance, if the scattering were associated with the direct burst environment rather than the ISM of the host galaxy.

Adopting all the above criteria to the current set of FRBs with redshift estimates based on their association with galaxies (Table 1), we eliminate the following sources from cosmological analysis.

FRB 121102

We exclude the repeating FRB 12110214 from our analysis for two reasons: (a) the rotation measure of this burst is anomalously high37, being three orders of magnitude higher than other FRBs in this sample and indicating that this burst DM is likely contaminated by an abnormally high circumburst or host galaxy contribution, and (b) its location within 2° of the Galactic plane imparts a larger and probably less well constrained DM contribution from the Milky Way relative to the high Galactic latitude bursts detected by ASKAP.

FRB 190523

We have conducted the analysis both with and without FRB 190523. The host galaxy identification17 from the larger, 3″ × 8″ localization region, is more uncertain than the ASKAP FRB detections and was partially based on an assumed DM–redshift relation which presents a potential source of bias in our analysis.

FRB 171020

The identification of the host galaxy associated with FRB 171020 is predicated on a search volume confined to a specific distance based on an assumed DM–redshift relation27, and is therefore excluded from the sample. Moreover, it is difficult to ascribe a numerical value to the likelihood of a correct association in this instance.

FRB 190611

Our follow-up observations for FRB 190611 identify a galaxy at J212258.0-792350 with redshift z = 0.378 offset by ~2″ from the current estimate of the FRB localization. The large offset (~10 kpc at that redshift) and large systematic uncertainty in the FRB localization and the presence of a closer, faint source revealed by deep GMOS i-band imaging preclude a secure association at present. As with FRB 190523, we conduct our analysis both with and without this burst in our sample.

FRB imaging and astrometry

The procedure for characterizing the position and positional uncertainty of FRBs 190102, 190608, 190611 and 190711 followed that described in the supplementary material of refs. 15,16. For the purposes of extracting these observables, we use only the total intensity data.

For each FRB, raw voltage data for a suitable calibrator source was captured via the CRAFT pipeline in the hours following the burst detection. For FRB 190102 and FRB 190608, the source PKS 1934-638 was used, while for FRB 190611 and FRB 190711, it was PKS 0407-658. From these calibrator data and the FRB data, visibility data sets were produced using the DiFX correlator38. An initial coarse search for the FRB position used the DM, pulse duration, and approximate position from the incoherently summed FRB detection data, and after detection in the interferometric data a re-correlation was performed with revised position, DM, and pulse time/duration. Radio frequency interference (RFI) was mitigated for the FRB data set by subtracting visibilities from an adjacent time range surrounding the burst itself. Additionally, for each FRB a visibility data set and image was generated using the entire 3.1 s of raw voltage data, to identify background radio continuum sources whose positions could be compared to catalogue values and verify the astrometric accuracy.

Per-station frequency-dependent complex gain calibration was derived from the calibrator data set using the ParselTongue39-based pipeline described in ref. 15 and transferred to the FRB data sets, before imaging in the Common Astronomy Software Applications (CASA) package. Best-fit positions and uncertainties were the extracted for each source using the task JMFIT in the Astronomical Image Processing System (AIPS)40.

Statistical uncertainties on the FRB positions were less than 0.5″ in all cases. However, as discussed in refs. 15,16, the phase referenced FRB images will be subject to a systematic positional shift resulting from the spatial and temporal extrapolation of calibration solutions. The magnitude of this systematic shift can be estimated by comparing the positions of continuum sources in the field surrounding the FRBs to their catalogue values. The accuracy to which this can be performed depends on the number of continuum sources visible in the ASKAP continuum image and their brightness, as well as the degree to which their intrinsic source structure can be modelled (or neglected). For any given continuum source, the presence of unmodelled structure will act to shift the position of the source centroid and results in a measured offset between the ASKAP and reference positions, which perturbs the actual systematic positional shift. However, the direction of such a shift depends on the source structure, and hence should not be correlated between different continuum sources. For FRB 190102 and FRB 190611, observations made with the Australia Telescope Compact Array at a comparable frequency and angular resolution to the ASKAP image minimize the impact of source structure, but for FRB 190608, we made use of the Faint Images of the Radio Sky at Twenty centimetres (FIRST) survey41, which has angular resolution roughly twice that of the ASKAP images, and for FRB 190711 we used archival 5 GHz ATCA data.

Assuming the phase referencing errors result in a simple translation of the FRB field image, we estimate the magnitude of this offset and its uncertainty with a weighted mean of the measured offsets for each of the continuum sources in the FRB field, after discarding any sources that were resolved in either the ASKAP image or the reference image. The magnitude of the offset ranged between 0 and 1.7 arcsec, with uncertainties ranging from 0.3 to 0.6 arcsec.

Host identification and spectroscopy

The optical spectroscopy and redshift determinations for FRB 180924 and FRB 181112 have been outlined previously15,16. Spectroscopy of the host galaxies of FRB 190102 and FRB 190611 was conducted using the FOcal Reducer and low dispersion Spectrograph 242 (FORS2) on the European Southern Observatory’s Very Large Telescope (VLT) on Cerro Paranal. FORS2 was configured with the GRIS_300I grism, an OG590 blocking filter, and a 1.3″ wide slit, yielding a resolution RFWHM ≈ 550. For FRB 190102 2 × 600 s exposures were obtained on 2019 March 25 ut, while for FRB 190611 2 × 1,350 s exposures were taken on 2019 July 12 ut. These and associated calibration images were processed with the PypeIt software package43 to derive flux and wavelength calibrated spectra.

For the host galaxy of FRB 190608, the optical spectrum from the seventh data release (DR7) of the Sloan Digital Sky Survey44 (SDSS) was retrieved from the IGMSPEC database45.

Imaging of the host galaxies of FRB 180924, FRB 181112 and FRB 190102 was undertaken using FORS2 on the VLT, while the FRB 190608 and FRB 190711 hosts were imaged with VLT/X-shooter46. Imaging of the host galaxies of FRB 190611 and FRB 190711 was undertaken using GMOS on Gemini-South47, from sets of 44 and 12 images of 100 s each in the i-band, respectively.

The FORS2 images were first reduced with ESO Reflex48, further processed in Python, and then co-added using a median combine in Montage49. The WCS solutions were updated with Astrometry.net50, with further adjustments performed by comparison with Gaia51 or Dark Energy Survey52 positions. The X-shooter images were reduced using a custom Python pipeline making use of the package CCDPROC53, including measures to cope with prominent fringe patterns in I-band; the images were then co-added and the astrometry adjusted with the same method as above. The GMOS images were reduced and co-added with PYRAF using standard procedures; the astrometry was adjusted with the same method stated above. Projected distances were estimated using the Javascript Cosmology Calculator54.

Two of the FRBs in the gold-standard sample, FRB 190608 and FRB 190711, have offsets larger than 1 arcsec from the galaxy light centroid. FRB 190608, however, is a z = 0.11 galaxy (that is, nearby) and the chance projection is even less than 0.3%. Regarding FRB 190711 we estimate a 6.1 × 10−3 probability that an unrelated galaxy is within a region out to a distance between the galaxy centroid and the outermost edge of the FRB error circle (for the measured R(AB) = 23.7 ± 0.2 mag as calibrated against the SkyMapper survey), and we estimate a probability p = 1.9 × 10−3 for an unrelated galaxy to be within the FRB error circle but below the detection limit of r = 25.5 mag. The remainder of the host-galaxy associations for each FRB have a probability P < 10−3 of a chance occurrence16.

The radio burst dynamic spectra and host-galaxy optical spectra are shown in Extended Data Fig. 1.

Estimating ⟨DMcosmic⟩

Central to the analysis is an estimate of the average DMcosmic value as a function of redshift and for a given cosmology, as defined in equation (2). Previous formulations55,56 have adopted similar definitions but with less precise considerations for fd(z), the fraction of cosmic baryons in diffuse ionized gas. Our formulation considers the redshift evolution of three dense baryonic components that will not contribute to ne: (1) stars; (2) stellar remnants (for example, white dwarfs, neutron stars); and (3) the neutral ISM of galaxies. For (1), we interpolate the empirically estimated stellar mass density estimates57. For (2), we adopt the estimation of ref. 58 which is 30% of the stellar mass. For (3), we assume the mass ratio of the ISM to stars is constant from z = 0−1 and adopt the present-day estimate58 of MISM/M* = 0.38. The model also allows for the partial ionization of helium but this is not relevant for the FRBs considered here. All of these calculations are encoded in Python in the public FRB repository (https://github.com/FRBs/FRB). Censuses of the gas and star evolution of baryons in z < 1 systems constrain the error in the fraction of neutral and non-diffuse baryons (that is, 1 − fd(z)) to about 30% at present. Thus, with this component constituting ~15% of the total baryon budget at z ≈ 0, the correction to Ωbh70 is uncertain at a level below 6%, well below the level of precision that investigation of the current FRB sample permits. We refer the reader to ref. 59 for a discussion of the constraints on fd(z) possible in future with a larger sample of FRBs.

Cosmological parameter estimation

The ASKAP FRB measurements and localizations afford a new opportunity to constrain our cosmological paradigm through estimations of DMcosmic and zFRB. The cosmic DM is governed primarily by the baryonic density Ωb and the expansion rate of the Universe, H0, and the fraction of baryons in the diffuse phase, fd(z). In the following, we will assume a flat cosmology with ΩΛ = 0.691 (Planck15). The expansion rate is dominated by this dark energy term for z < 0.7, so cosmological analysis of the ASKAP FRBs is not sensitive to the precise value of Ωm and, therefore, to a close approximation, ⟨DMcosmic⟩ ∝ ΩbH0. We therefore proceed to place a constraint on this product.

To construct a likelihood function \( {\mathcal L} \) from our FRB measurements, we build a model for DMcosmic and its uncertainty. The model is based primarily on the cosmological parameters, but it must also allow for a nuisance parameter which accounts for the DM of our Galactic halo and that of the host galaxy: DMMW,halo + DMhost. For the former term, theoretical models informed by observation suggest DMMW,halo ≈ 50 pc cm−3 with a small dispersion13,56, but we acknowledge that the mean value is poorly constrained. We expect the variance in these terms to be driven by DMhost, which follows from the large range in DM values observed for the ISM of our Galaxy22, even if one ignores whether FRBs occur in ‘special’ locations within a galaxy. Furthermore, the very high RM and the (probably related) large DM excess of FRB 121102 above DMcosmic implies at least one FRB with a large DMhost value37.

The PDF for DMhost has limited theoretical motivation. In the following, we assume a log-normal distribution which has two salient features: (1) it is positive definite; (2) it exhibits an asymmetric tail to large values. The latter property allows for high DMhost values that might arise from gas local to the FRB, for example, an H ii region or circumstellar medium. Formally, we adopt a log-normal distribution:

$${p}_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}}({{\rm{D}}{\rm{M}}}_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}}|\mu ,{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}})=\frac{1}{{(2{\rm{\pi }})}^{1/2}{\rm{D}}{\rm{M}}{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}}}\exp \,\left[-\frac{{(\log {\rm{D}}{\rm{M}}-\mu )}^{2}}{2{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}}^{2}}\right]$$
(3)

This distribution has a median value of eμ and variance \({{\rm{e}}}^{\mu +{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}}^{2}/2}{({{\rm{e}}}^{{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}}^{2}}-1)}^{1/2}\). We consider distributions with eμ in the range 20−200 pc cm−3 and σhost in the range 0.2−2.0. An illustrative set of these probability distribution functions for DMhost is shown in Extended Data Fig. 2. For consistency of interpretation of DMhost values from bursts at disparate redshifts, the probability distribution function is referenced to the rest frame of the host galaxy, so a correction DMhost → DMhost(1 + zFRB)−1 is applied and the distribution normalized accordingly, however, in practice this redshift correction factor varies only over the range 0.7 to 0.9 in the gold-standard sample. The inferred dispersion in DMhost is consistent with the expected range of host DMs given the galaxy type, morphology and orientation on the sky and the distance of the FRB from the galaxy centre. However, we are unable to state more than this at present, and remark that present estimates of the DMhost contributions towards specific localized FRBs15,18 with two quite different host galaxies are, respectively, in the range 30–81 pc cm−3 and <70 pc cm−3. This suggests that any correction on this basis could be small for our sample. A further interesting aspect of our measurements is that it is beginning to place limits on these corrections.

Altogether, \({{\rm{DM}}}_{{\rm{FRB}}}^{{\rm{model}}}={{\rm{DM}}}_{{\rm{cosmic}}}(z)+{{\rm{DM}}}_{{\rm{host}}}+{{\rm{DM}}}_{{\rm{MW}},{\rm{ISM}}}\) with the last quantity estimated from NE2001 based on the FRB coordinates; given the high Galactic latitudes of the present ASKAP sample we adopt a value DMMW,ISM = 30 pc cm−3 for these bursts. The mechanics of our treatment of the DMhost, DMMW,halo and DMMW,ISM terms is described in greater detail in equation (6).

The model probability distribution for DMcosmic is derived from theoretical treatments of the IGM and galaxy haloes11,13 with σDM dominated by the physical variance in DMcosmic. Extended Data Fig. 3 shows that comparison against the analytic form (as used in other IGM-related contexts60)

$${p}_{{\rm{c}}{\rm{o}}{\rm{s}}{\rm{m}}{\rm{i}}{\rm{c}}}(\varDelta )=A{\varDelta }^{-\beta }\exp \,\left[-\frac{{({\varDelta }^{-\alpha }-{C}_{0})}^{2}}{2{\alpha }^{2}{\sigma }_{{\rm{D}}{\rm{M}}}^{2}}\right]\,,\,\varDelta > 0$$
(4)

provides an excellent match to the DMcosmic distributions observed in our semi-analytic models and in a hydrodynamic simulation, where Δ ≡ DMcosmic/⟨DMcosmic⟩. The motivation for this form is that in the limit of small σDM, the distribution of DM should approach a Gaussian owing to the Gaussianity of structure on large scales (a non-negligible component of the variance of pcosmic(Δ) comes from tens of megaparsec structures) and in the low-σDM limit the halo gas is more diffuse and so the PDF approaches a Gaussian owing to the intersection of the line of sight with more structures. Conversely, when the variance is large, this PDF captures the large skew that results from a few large structures that contribute to the DM of many sightlines. The sharp low-DM cutoff in the distribution reflects the fact that a large component of the IGM is highly diffuse, and displays much less variance than the halo-related component, thus imposing a strict lower limit to the DM. The parameter β is related to the inner density profile of gas in haloes. If the 3D density profile scales as ρ ∝ r−α, β = (α + 1)/(α − 1) such that an isothermal profile with α = 2 has β = 3 and an inner slope of α = 1.5 has β = 5. Such slopes are consistent with those found in numerical simulations of intrahalo gas61,62. The indices α = 3 and β = 3 provide the best match to our models (although we find that pcosmic(Δ) is weakly sensitive to order unity changes in these parameters, with β = 3 having the most flexibility for our z = 0.11 measurement relative to β = 4). We use the parameter σDM in pcosmic(Δ) as an effective standard deviation even though formally the standard deviation with β = 3 diverges logarithmically. We find that σDM is closely tied to the true standard deviation when imposing motivated maximum cutoffs for Δ on the distribution. The mean of the distribution requires that ⟨Δ⟩ = 1, which fixes the remaining parameter C0 in pcosmic(Δ).

Extended Data Figure 3 shows models that use equation (4) for pcosmic relative to numerical calculations at redshifts that span the considered range. The solid curves are the previously described semi-analytic models11, which assume that haloes below the specified mass have been evacuated of gas, and the ‘swinds’ simulation of ref. 63. The dashed curves show the function evaluated for the best-fit σDM, and the dot-dashed curves adopt the parameterization σDM = Fz−0.5 and scale off the z = 0.5 best fit value for σDM yielding F of 0.09, 0.15 and 0.32 in our semi-analytic models in which haloes of 1014, 1013 and 1011 solar masses (M☉) are evacuated of their gas. The agreement of the dot-dashed curves with the solid numerical model curves demonstrates that σDM = Fz−0.5 approximates the evolution over the range of our measurements. This scaling is further motivated in the Euclidean limit, applicable for \(z\ll 1\), where \(\langle {{\rm{DM}}}_{{\rm{cosmic}}}\rangle ={n}_{{\rm{e}}}cz/H\) and \({\sigma }_{{\rm{DM}}}\approx {{\rm{DM}}}_{{\rm{halo}}}\sqrt{N}/\langle {{\rm{DM}}}_{{\rm{cosmic}}}\rangle \), where ne is the mean electron density and N is the number of haloes intersected, which is proportional to the path length probed or cz/H.

While our analytic parameterization describes the distribution of DMcosmic both in semi-analytic models and numerical simulations, we use the more flexible semi-analytic models to set the marginalization range in F that is used for some constraints on Ωbh70. Here we argue that the considered semi-analytic models shown in Extended Data Fig. 3 span the likely range of possible feedback scenarios. These models approximate haloes as retaining their gas in a manner that traces the dark matter above some mass threshold. This approximates the picture in many simulations61,62 and analytic models64,65 in which the fraction of halo gas retained is a strongly increasing function of halo mass before saturating at unity. Furthermore, gas that is outside haloes is less effective at contributing variance: take the example where gas is distributed out to a distance R around a halo. The probability a sightline intersects this gas scales as R2, leading to less shot noise for larger R, while the contribution of each individual system scales as R−2, leading to a smaller contribution for larger R. This picture motivates the semi-analytic model’s approximation that ejected gas diffusely traces large-scale structure11.

Simulations and models generally find that haloes below threshold masses in the range of about (1011−1013)M☉ are evacuated of gas61,62,64,65, although some implementations of stellar quasar feedback can result in different predictions66. Halo gas in M > 1014M☉ haloes is constrained by X-ray observations to mostly reside within such haloes67. Our strongest feedback model, in which F = 0.09, pushes up against this observational limit. Our model with the weakest feedback assumes that dwarf-galaxy-sized haloes with 1011M☉ retain their gas and yields F = 0.32 (and we find that F is just marginally larger if the 1010M☉ haloes of the smallest dwarf galaxies retain their gas, haloes that would be just massive enough to overcome the pressure of the IGM and retain their gas68). Thus our models span the range of likely feedback scenarios.

Given this semi-analytic formalism, we proceed to estimate the model likelihood by computing the joint likelihoods of all FRBs:

$$ {\mathcal L} =\mathop{\prod }\limits_{i=1}^{{N}_{{\rm{FRBs}}}}{P}_{i}({\rm{DM}}{{\prime} }_{{\rm{FRB}}}|{z}_{i})$$
(5)

where \({P}_{i}({\rm{DM}}{{\prime} }_{{\rm{FRB}}}|{z}_{i})\) is the probability of the total observed DMFRB corrected for the Galaxy:

$${\rm{DM}}{{\prime} }_{{\rm{FRB}}}\equiv {{\rm{DM}}}_{{\rm{FRB}}}-{{\rm{DM}}}_{{\rm{MW}},{\rm{ISM}}}-{{\rm{DM}}}_{{\rm{MW}},{\rm{halo}}}={{\rm{DM}}}_{{\rm{host}}}+{{\rm{DM}}}_{{\rm{cosmic}}}$$
(6)

For a burst at a given zi and the model parameters we have:

$${P}_{i}({\rm{DM}}{{\prime} }_{{\rm{FRB}},{i}}|{z}_{i})\,=\underset{0}{\overset{{\rm{DM}}{{\prime} }_{{\rm{FRB}}}}{\int }}{p}_{{\rm{host}}}({{\rm{DM}}}_{{\rm{host}}}|\mu ,{\sigma }_{{\rm{host}}}){p}_{{\rm{cosmic}}}({\rm{DM}}{{\prime} }_{{\rm{FRB}},{i}}-{{\rm{DM}}}_{{\rm{host}}},{z}_{i}){{\rm{dDM}}}_{{\rm{host}}}$$
(7)

and phost(DMhost|μ, σhost) the PDF for DMhost. With the likelihood function defined we construct a grid of ΩbH0, F, μ and σhost values and marginalize over the last three to obtain the constraint on Ωbh70. These results are presented in Fig. 3 in the main text for the gold-standard sample. Extended Data Fig. 4 presents the results of the same analysis when FRBs 190523 and 190611 are included in the data set.

To place confidence intervals on ΩbH0 and σDM, we use the likelihood ratio test statistic \({\mathscr{D}}\):

$${\mathscr{D}}({\varOmega }_{{\rm{b}}}{h}_{70},F,\mu ,{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}})=2\log {{\mathcal{L}}}_{max}-2\log {\mathcal{L}}({\varOmega }_{{\rm{b}}}{h}_{70},F,\mu ,{\sigma }_{{\rm{h}}{\rm{o}}{\rm{s}}{\rm{t}}})$$
(8)

where \({ {\mathcal L} }_{{\rm{\max }}}\) is the maximum value of \( {\mathcal L} \), that is, for parameters maximizing the likelihood. According to Wilks’ theorem, for a sufficiently large number of FRBs, \({\mathscr{D}}\) will be distributed according to a \({\chi }_{n}^{2}\) distribution with n = 4 degrees of freedom69. If the cumulative distribution function of the \({\chi }_{4}^{2}(x)\) distribution is CDF(x), solving CDF(x) = p constrains the Ωbh70, F, μ, σhost parameter space to the region \({\mathscr{D}}\le x\) at confidence level (C.L.) p.

Uncertainties in these confidence estimates are probably dominated by systematic effects in the sample selection, and small number statistics. To test both, we extend the gold-standard sample of five bursts to include FRBs 190523 and 190611. The resulting analysis is shown in Extended Data Fig. 4. Compared with Fig. 3, the inclusion of two further bursts shifts the maximum-likelihood estimate for Ωbh70 at 68% C.L. from \({0.051}_{-0.015}^{+0.014}\) to \({0.042}_{-0.012}^{+0.011}\), that is, consistent with the original uncertainties. This does not mean that there is no systematic bias, nor that Wilks’ theorem holds precisely for our sample, but rather that any such effects are minor compared to the inherent uncertainties from our small sample size of localized bursts.

Accounting for biases in P(DM,z)

The cosmological evolution of the FRB population, and its intrinsic luminosity function, can strongly influence the observed/expected distribution of FRBs in redshift–DM space70, P(DM,z). We therefore perform our likelihood maximization over P(DM|z) only. This discards the information contained in the redshifts of our detected FRBs, but makes the procedure more robust against factors influencing the redshift distribution.

The remaining bias comes from changing sensitivity as a function of DM. This can be either direct, through DM-smearing within frequency channels, or indirect, through increased scatter broadening associated with the same gaseous structures causing the observed DM.

We wish to compute the dispersion measure limit, DMcutoff, at which a given FRB would have been undetectable. The S/N of a detected burst depends on its intrinsic (or scatter-broadened) width, w, the time resolution of the detection system, tres, and the amount of dispersion measure time smearing between adjacent 1 MHz spectral channels, tsmear(DM). The resulting width of the pulse is:

$$\Delta {t}_{{\rm{obs}}}({\rm{DM}})=\sqrt{{w}^{2}+{t}_{{\rm{res}}}^{2}+{t}_{{\rm{smear}}}^{2}({\rm{DM}})}$$
(9)

We compute DMcutoff such that the burst, detected by our system with a signal-to-noise ratio of s0 at a DMobs would have fallen below our detection threshold of sd = 9.0σ. For each burst we thus solve

$${s}_{{\rm{d}}}={s}_{0}\sqrt{\frac{\Delta {t}_{{\rm{obs}}}({{\rm{DM}}}_{{\rm{obs}}})}{\Delta {t}_{{\rm{obs}}}({{\rm{DM}}}_{{\rm{cutoff}}})}}$$
(10)

Extended Data Table 1 lists the DM, widths, time resolution, detection S/N values and derived DMcutoff values for each of the bursts in our sample.

MCMC analysis

To complement the likelihood analysis presented in the main text, we have performed Bayesian inference of a model constructed to describe the DM and redshift measurements of the FRBs. The model consists of four parameters describing two PDFs for distinct components of the dispersion measure: (i) DMcosmic, which describes the extragalactic dispersion measure including both the diffuse IGM and the gas associated with intervening galactic haloes; and (ii) DMhost, which describes ionized gas associated with the host galaxy (we assume a fixed DMMW,halo value of 50 pc cm−3 for the Galactic halo). We parameterize the former PDF with equation (4), that is, pcosmic(Δ) with Δ ≡ DMcosmic/⟨DMcosmic⟩ and ⟨DMcosmic⟩ the average value for the assumed cosmology (equation (2)). The foregoing subsection on cosmology and host-galaxy parameter estimation describes theoretical treatments that motivate one to adopt α = 3 and β = 3 in equation (4) and to adopt the functional form of σDM = F/z1/2 for its dispersion parameter. For ⟨DMcosmic⟩, we modulate its amplitude via the product Ωbh70. Therefore, pcosmic(Δ) is governed by two free parameters: F and Ωbh70.

We adopt the same phost(DMhost|μ, σhost) PDF described earlier, with free parameters exp(μ) and σhost. From these two PDFs we construct a likelihood function for the set of observed FRBs using equations (5) and (7). Note that measurement uncertainty in DMFRB does not enter into the evaluation of \( {\mathcal L} \) because the dispersion from DMcosmic and DMhost are much greater. Put another way, our model is constructed to describe the observed distribution of \({\rm{DM}}{{\prime} }_{{\rm{FRB}}}\) values with an anticipated dispersion substantially exceeding the uncertainty in individual DMFRB measurements (typically <1 pc cm−3).

Effectively, two of the parameters (Ωbh70, μ) set the amplitude of the DM–z relation and two describe its dispersion (F, σhost). We anticipate a degeneracy between each set although if DMhost is approximately independent of redshift then this apparent degeneracy may be resolved. Only the dispersion in DMcosmic, parameterized by F, allows for large negative excursions from the mean relation. Lastly, we introduce priors for the four parameters based on a combination of experimentation, physical expectation and scientific motivation. For Ωbh70, the scientific focus of this manuscript, we adopt a uniform prior ranging from 0.015−0.095 which easily spans the Planck15 estimate. For F, we adopt a uniform prior in the interval (0.01, 0.5), a larger range than anticipated by our models in the frequentist analysis presented above. Regarding exp(μ), we adopt a uniform prior in the interval [20, 200] pc cm−3. We consider lower values for the mean to be non-physical and we will find that larger values are disfavoured by the observations. Lastly, we assume a uniform prior for σhost in the interval [0.2, 2]. The larger σhost values give non-negligible probability for DMhost values in excess of 1,000 pc cm−3. Future observations, especially an ensemble of FRBs at low redshift, will better inform these priors on μ and σhost.

Adopting the above likelihood and priors, we performed a Bayesian inference of the four parameters using the gold-standard sample of FRB measurements and standard MCMC techniques. These were performed with the PYMC3 software package using slice sampling and four independent chains of 40,000 samples after a tuning period of 2,000 samples. Extended Data Fig. 5 presents a corner plot of the combined samples. A principal result is that the data yield a Ωbh70 distribution fully consistent with the independent estimates from the CMB, BBN and supernovae. Quantitatively, the Ωbh70 samples have a median value of 0.056 and a 68% confidence interval spanning [0.046, 0.066] (see Extended Data Table 2). Taken strictly, at 95% confidence these FRB measurements require a universe with at least 70% of the baryons inferred from BBN and CMB analysis. These results hold despite the weak priors placed on the PDF for DMhost, but we warn that they are dependent on the value assumed for DMMW,halo.

Extended Data Figure 5 also reveals the anticipated anti-correlations between μ and Ωbh70 and (to a lesser extent) F and σhost. We expect these to weaken as the FRB sample grows in size and redshift range. Lastly, we note that the F and μ parameters have maximal probability at one edge of their assumed prior intervals. Values of F that are on the higher side of the considered range (a range that spans the possible model space) are modestly favoured. For μ, we consider 20 pc cm−3 to be the lowest sensible mean contribution from the host galaxy (which could also mean a lower value for the Galactic halo than adopted here).

The frequentist analysis in the main text and this Bayesian MCMC analysis agree very well on the gold-standard sample. The most notable differences are that the MCMC analysis prefers a distribution for exp(μ) that is more peaked to smaller exp(μ) values and one for F that peaks towards larger values, although with no value for exp(μ) or F strongly preferred by either analysis. When the parameters are not well constrained one would not expect perfect agreement between the methods, as, for example, the Bayesian analysis is sensitive to our prior on exp(μ) when this parameter is not well constrained. It is expected that the differences between the two methods will become smaller with more data. Already for the seven-burst sample (Extended Data Fig. 4), the distribution for F in the frequentist analysis is more similar to the MCMC analysis of the gold-standard sample.