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This theoretical work investigates different models to predict the redshift of fast radio bursts (FRBs) from their observed dispersion measure (DM) and other reported properties. We performed an extended revision of the FRBs with confirmed galaxy hosts in the literature and built the most updated catalog to date. With this sample of FRBs, we propose four models that relate the DM and
Fast radio bursts (FRBs) are bright radio transients with a typical fluence rate of ≳ 1 Jy per millisecond. Since the first discovery of the first FRB in 2007 by
The dispersion measure (DM) is one of the main observables studied for these radio signals. It is defined as the integrated column density of the free electrons along the sight line (
To date, about 50 of these FRBs have been localized, and their parent galaxies have been observed at different wavelengths, providing astronomers with more tools to characterize these transients and reveal their nature and origin. Nonetheless, identifying the FRB host galaxy is still quite challenging and demands efforts on multiple fronts due to their randomness and ultrashort duration; thus, the study of FRB redshift models from the observed dispersion measure is a pressing matter.
This work builds on these efforts and presents several models that relate DM (and other FRB physical properties) with their confirmed redshift. We aim to provide a robust method to predict transients’ redshift and help astronomers program their observational campaigns.
The remainder of this paper is arranged as follows: § 2 presents the FRB catalog used throughout the paper.
We set up an FRB catalog from the Transient Name Server (TNS) (
With the FRB host information, we create a subsample of FRBs with a confirmed galaxy host (thus, its redshift is known). There are 48 FRBs in this category, but only 24 FRBs are unique since 8 of these instances are repeaters. We treat the repeating FRBs as the same object and take the mean DM for each transient. The resulting FRB subsample is presented in
It is worth noting that FRBs with the † symbol in
List of FRBs with known galaxy hosts to date. Columns 1 to 6 correspond to the name of the FRB, right ascension, declination, DM, the error of the DM, and the reference associated with the reported FRB. The redshift and redshift errors of the optical counterpart of each FRB are presented in columns 7 and 8. Columns 9 to 11 show whether the FRB is a repeater, the peak flux, and the time width of each transient. The information in the last two columns was taken from
FRB  RA_{FRB}  DEC_{FRB}  DM  ΔDM  Reference 

Δ 
Repeater  Peak flux  Width 

(deg)  (deg)  (pc/cm^{3})  (pc/cm^{3})  (Jy)  (ms)  
20121102A  82.9946  33.1479  557.0  2.0 

0.1927    Y  1,375  557 
20171020A  333.75  −19.6667  114.1  0.2 

0.008672    N  1,297  114.1 
20180301A  93.2292  4.6711  536.0  5.0 

0.3305    Y  1,352  522 
20180916B  29.5031  65.7168  348.8  1.62 

0.0337    Y  603.9  347.8 
20180924B  326.1052  −40.9000  362.16  0.06 

0.3214    N  1,320  361.42 
20181030A  163.2  73.74  103.5  1.62 

0.0039    Y  703.7  101.9 
20181112A  327.3485  −52.9709  589.0  0.03 

0.4755    N  1,272.5  589.27 
20190102C  322.4157  −79.4757  364.545  0.3 

0.2913    N  1,320  363.6 
20190520B  240.5167  −11.2883  1204.7  4.0 

0.241  0.001  Y  1,375  1,202 
20190523A  207.0650  72.4697  760.8  0.6 

0.66  2.0  N  1,411  760.8 
20190608B  334.0199  −7.8982  340.05  0.5 

0.1178    N  1,320  338.7 
20190611B  320.7455  −79.3976  332.63  0.2 

0.3778    N  1,320  321.4 
20190711A  329.4195  −80.3580  592.6  0.4 

0.5217    Y  1,272.5  593.1 
20190714A  183.9797  −13.0210  504.13  2.0 

0.2365    N  1,272.5  504 
20191001A  323.0  −54.6667  507.9  0.04 

0.234    N  920.5  506.92 
20191228A  344.4292  −29.5942  297.5  0.05 

0.2432    N  1,272.5  297.9 
20200120E  146.25  68.77  87.82  1.62 

0.00014    Y  600  88.96 
20200430A  229.7064  12.3769  380.1  0.4 

0.1608    N  864.5  380.1 
20200906A  53.4958  −14.0831  577.8  0.2 

0.3688    N  864.5  577.8 
20201124A  76.99  26.19  413.52  3.23 

0.0979    Y  600  410.83 
20210117A  339.9792  −16.1517  728.95  0.36 

0.214  0.001  N  1,271.5  730 
20220610A  351.0  −33.5167  1458.1  0.2 

1.016  0.002  N  1,271.5  1,458.1 
20220509G^{†}  282.6700  70.2438  269.53  10.0 

0.0894    N    270.26 
20220914A^{†}  282.0568  73.3369  631.29  10.0 

0.1139    N    630.703 
Based on the observed dispersion measure (and ΔDM) of the 22 known FRBs with galaxy hosts, we exclude the last two FRBs presented in
Model A:
The simplest model to find an estimated redshift from DM is through a linear relationship, a reasonable assumption since the dispersion measure is an indicator of the distance, and the distance grows linearly with
The bestfit parameters for the linear model are
DM–redshift relationship for localized FRBs predicted by the linear model (refer to Eq.
Interestingly,
The following two models assume that the IGM term is subdominant in Eq.
Model B:
Inspired by works from
The parameters that best adjust to the observed data for this function are
DM–redshift relationship for FRBs with known galaxy hosts in the log parabolic model (see Eq.
Model C:
Work from
It is worth mentioning that
DM–redshift relationship for FRBs with known galaxy hosts in the powerlaw model or model C, as described in Eq.
Model D:
In order to capture the fluctuations in the observed DM reported by
(i) Transforming the coordinates reported in
(ii) Under the assumption that RA, DEC, and
(iii) With the interpolation mentioned above, we calculated DM from z and the angular coordinates. Nevertheless, our primary objective is to determine
(iv) We define an error function Err
(v) We repeat steps (i) to (iv) to calculate the upper and lower values of the redshift errors Δ
The free parameters that warrant the minimum Err in the interpolator are
Predictions of the FRB redshift given in the models. The first column shows the name of the FRB, the second column shows the redshift of the host galaxy, and the third to sixth columns show the predicted redshifts with models A, B, C, and D with their corresponding errors.
FRB 






20121102A  0.1927 




20171020A  0.008672  – 

– 

20180301A  0.3305 




20180916B  0.0337 




20180924B  0.3214 




20181030A  0.0039  – 

– 

20181112A  0.4755 




20190102C  0.2913 




20190520B  0.241±0.001 




20190523A  0.66±2.0 




20190608B  0.1178 




20190611B  0.3778 




20190711A  0.5217 




20190714A  0.2365 




20191001A  0.234 




20191228A  0.2432 




20200120E  0.00014  – 

– 

20200430A  0.1608 




20200906A  0.3688 




20201124A  0.0979 




20210117A  0.214±0.001 




20220610A  1.016±0.002 




To measure the success rate of the models, we compute the median of the redshift difference for each model as follows:
The AIC is defined by the expression:
The BIC is defined as
In order to rank the proposed models, we demand three conditions that need to be fulfilled simultaneously: i) the lowest median of the redshift difference
Statistical metrics implemented to quantify the success rate of each model’s predictions.
Metric  Model A  Model B  Model C  Model D 


0.13  0.07  0.13  0.11 

33.52  2.03 × 10^{−5}  218.94  433.22 
AIC  −3.02  27.61  −6.78  −8.14 
BIC  −0.84  30.88  −4.60  −5.96 
Revisiting the three criteria demanded to rank the performance of the models, we found that model D (i.e., the interpolation) has the best success rate at fitting the observed data since it shows the largest likelihood while minimizing the AIC and BIC, in addition to offering the smallest median difference between the “true” and the predicted redshifts reported in
We find some insightful pointers on the performance of the models based on the outcome from metrics (
On the other hand, the linear and powerlaw models are susceptible to small values of DM, and they cannot predict physical redshifts in a few instances (represented by – in
Moreover, the model with the worst performance in predicting the redshift for these FRBs is the logparabolic function. There are very large residuals for some
Furthermore, we highlight that all models fail to predict data points with large DM and small
Additionally, we remark on one strength of model D compared with models A, B, and C: this is the only one that can predict significant differences in the dispersion measure from similar redshifts. Focus on the following pairs of FRBs: 20180301A–20180924B (
We also report the predicted redshifts of the FRBs 20220509G and 20220914A in
Redshifts predicted by the four models for FRBs hosted by galaxy clusters.







20220509G  0.0894 




20220914A  0.1139 




Finally, we highlight that improving the theoretical models by adding physical complexity to them is critical to recovering the variety of the transients. We will update our predictions and bestfit parameters with incoming data from newly confirmed FRB galaxy hosts.
DM vs. SFR from the FRB and their galaxy hosts. The black data points are the 22 FRBs with confirmed hosts, and the dark red line is the functional form, as described in Eq.
The bestfit parameters for the DM–SFR relationship are
On the other hand, we examine a possible association between the stellar mass of the putative galaxies and the observed DM in
DM vs. stellar mass (M_{
s
}) from the FRB and their galaxy hosts. The black data points are the 22 FRBs with a confirmed host, and the blue line is the functional fit, as proposed in Eq.
The best values of the function (
We have presented the most updated catalog of FRBs with confirmed galaxy hosts and their properties, along with four models that relate the DM and redshift
All the models are physically motivated: the linear model (A) is inspired by the Macquart relation. It assumes that DM should be linear with
We implement the likelihood, the median difference of the redshifts, and the Akaike and Bayesian information criteria to measure the success rate of each model at predicting
Finally, we investigate possible correlations between the observed DM and other known properties of the host galaxies: the SFR and their reported mass (M_{ s }). We found a very weak correlation between the variables in the cases analyzed in this work. However, observational evidence indicates that detecting FRBs would be preferred in active environments such as galaxies with high star formation. Therefore, there is room for improvement in this aspect, with an increasing number of FRBs detected with current and future telescopes devoted to understanding these transients.
The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.
EPM: writing–original draft and writing–review and editing. LG: writing–original draft and writing–review and editing.
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported by Fundación Universitaria Los Libertadores programme “Eleventh Annual Internal Call for Research and Artistic and Cultural Creation Projects 2023,” project “Análogos cosmológicos de la ecuación de difusión” (grant number: ING0923).
The authors would like to thank Universidad ECCI and Fundación Universitaria Los Libertadores for granting them the resources to develop this project. They are also grateful to the reviewers for their valuable comments, which helped improve the quality of our manuscript. This research made use of Matplotlib (
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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