Delay Time Distribution Research Articles

Deep Synoptic Array Science: A Massive Elliptical Host Among Two Galaxy-cluster Fast Radio Bursts

The stellar population environments that are associated with fast radio burst (FRB) sources provide important insights for developing their progenitor theories. We expand the diversity of known FRB host environments by reporting two FRBs in massive galaxy clusters that were discovered by the Deep Synoptic Array (DSA-110) during its commissioning observations. FRB 20220914A has been localized to a star-forming, late-type galaxy at a redshift of 0.1139 with multiple starbursts at lookback times less than ∼3.5 Gyr in the A2310 galaxy cluster. Although the host galaxy of FRB 20220914A is similar to typical FRB hosts, the FRB 20220509G host stands out as a quiescent, early-type galaxy at a redshift of 0.0894 in the A2311 galaxy cluster. The discovery of FRBs in both late- and early-type galaxies adds to the body of evidence that the FRB sources have multiple formation channels. Therefore, even though FRB hosts are typically star-forming, there must exist formation channels that are consistent with old stellar population in galaxies. The varied star formation histories of the two FRB hosts that we report here indicate a wide delay-time distribution of FRB progenitors. Future work in constraining the FRB delay-time distribution, using the methods that we develop herein, will prove crucial in determining the evolutionary histories of FRB sources.

Cosmic Supernova Neutrino and Gamma-Ray Backgrounds in the MeV Regime

Cosmic multimessenger backgrounds include relic diffuse components created in the early universe and contributions from individual sources. Here, we study both type Ia supernovae (SNe Ia) and core-collapse supernovae (CCSNe) contributions to the diffuse neutrino and gamma-ray backgrounds in the MeV regime referred to as DSNB and DSGB respectively. We show that the diffuse SN Ia background is 106 times lower (for ) than the CCSN background making it negligible. Our predicted DSNB flux at earth in the 19.3–32 MeV regime is 0.36 ν cm−2 s−1. We also find that the DSNB flux in the energy range from 11.3 to 32 MeV varies by +29% with a change in the SFRD model from Madau & Fragos which yielded a minimum predicted flux, to the extragalactic background light reconstruction model (maximum predicted flux). The diffuse SN Ia gamma-ray background and its dependence on the progenitor supernova delay time distribution are also evaluated. Furthermore, we address the origin of the CGB (Cosmic Gamma-ray Background) in the 0.1–7 MeV regime by adding contributions from sources such as SNe Ia, CCSNe, radio-quiet Active Galactic Nuclei, Flat spectrum radio quasars (FSRQs) and Neutron star—neutron star mergers. We find that our modeled background (including uncertainties) matches the observed CGB above 1.0 MeV, but is a factor ≈2 lower than the observed flux in the 0.1–1.0 MeV range, highlighting the need for future MeV missions to establish the CGB spectrum more reliably, and to possibly identify additional sources or even source classes in the underexplored MeV band.

Europium enrichment and hierarchical formation of the Galactic halo

Context.The origin of the large star-to-star variation of the [Eu/Fe] ratios observed in the extremely metal-poor (at [Fe/H] ≤ −3) stars of the Galactic halo is still a matter of debate.Aims.In this paper, we explore this problem by putting our stochastic chemical evolution model in the hierarchical clustering framework, with the aim of explaining the observed spread in the halo.Methods.We compute the chemical enrichment of Eu occurring in the building blocks that have possibly formed the Galactic halo. In this framework, the enrichment from neutron star mergers can be influenced by the dynamics of the binary systems in the gravitational potential of the original host galaxy. In the least massive systems, the neutron stars can merge outside the host galaxy and so only a small fraction of newly produced Eu can be retained by the parent galaxy itself.Results.In the framework of this new scenario, the accreted merging neutron stars are able to explain the presence of stars with sub-solar [Eu/Fe] ratios at [Fe/H] ≤ −3, but only if we assume a delay time distribution for merging of the neutron stars ∝t−1.5. We confirm the correlation between the dispersion of [Eu/Fe] at a given metallicity and the fraction of massive stars which give origin to neutron star mergers. The mixed scenario, where both neutron star mergers and magneto-rotational supernovae do produce Eu, can explain the observed spread in the Eu abundance also for a delay time distribution for mergers going either as ∝t−1or ∝t−1.5.

Light curves of BSM-induced neutrino echoes in the optically thin limit

High-energy neutrinos from astrophysical transients serve as a probe of neutrino physics beyond the Standard Model. In particular, nonstandard interaction of neutrinos with the cosmic neutrino background or dark matter (DM) may have imprints on not only their spectra but also the arrival and time-delay distributions. Assuming that the interaction occurs at most once during the neutrino propagation, we provide general analytic formulas for light curves of the neutrino echoes induced by BSM. The formulas can be used for constraining neutrino-neutrino scattering, neutrino-DM scattering, and other scattering processes experienced by relativistic particles.

New constraints on the Bray conservation-of-momentum natal kick model from multiple distinct observations

ABSTRACT Natal supernova kicks, the linear momentum compact remnants receive during their formation, are an essential part of binary population synthesis (BPS) models. Although these kicks are well supported by evidence, their underlying distributions and incorporation into BPS models are uncertain. In this work, we investigate the nature of natal kicks using a previously proposed analytical prescription where the strength of the kick is given by $v_\text{k}=\alpha \frac{m_\text{ejecta}}{m_\text{remnant}}+\beta \, \mathrm{km\, s}^{-1}$ , for free parameters α and β. We vary the free parameters over large ranges of possible values, comparing these synthetic populations simultaneously against four constraints: the merger rate of compact binary neutron star (BNS) systems, the period–eccentricity distribution of Galactic BNSs, the velocity distribution of single-star pulsars, and the likelihood for low ejecta mass supernovae to produce low-velocity kicks. We find that different samples of the parameter space satisfy each test, and only 1 per cent of the models satisfy all four constraints simultaneously. Although we cannot identify a single best kick model, we report $\alpha =115^{+40}_{-55}\, \mathrm{km\, s}^{-1}, \beta =15^{+10}_{-15}\, \mathrm{km\, s}^{-1}$ as the centre of the region of the parameter space that fulfils all of our constraints, and expect $\beta \ge 0\, \mathrm{km\, s}^{-1}$ as a further constraint. We also suggest further observations that will enable future refinement of the kick model. A sensitive test for the kick model will be the redshift evolution of the BNS merger rate since this is effectively a direct measure of the delay-time distribution for mergers. For our best-fitting values, we find that the peak of the BNS merger rate is the present day.

On the hosts of neutron star mergers in the nearby Universe

ABSTRACT Recently, the characterization of binary systems of neutron stars has become central in various fields such as gravitational waves, gamma-ray bursts (GRBs), and the chemical evolution of galaxies. In this work, we explore possible observational proxies that can be used to infer some characteristics of the delay time distribution (DTD) of neutron star mergers (NSMs). We construct a sample of model galaxies that fulfils the observed galaxy stellar mass function, star formation rate versus mass relation, and the cosmic star formation rate density. The star formation history of galaxies is described with a log-normal function characterized by two parameters: the position of the maximum and the width of the distribution. We assume a theoretical DTD that mainly depends on the lower limit and the slope of the distribution of the separations of the binary neutron star systems at birth. We find that the current rate of NSMs ($\mathcal {R}=320^{+490}_{-240}$ Gpc−3 yr−1) requires that ∼0.3 per cent of neutron star progenitors lives in binary systems with the right characteristics to lead to an NSM within a Hubble time. We explore the expected relations between the rate of NSMs and the properties of the host galaxy. We find that the most effective proxy for the shape of the DTD of NSMs is the current star formation activity of the typical host. At present, the fraction of short-GRBs observed in star-forming galaxies favours DTDs with at least ${\sim}40~{{\ \rm per\ cent}}$ of mergers within 100 Myr. This conclusion will be put on a stronger basis with larger samples of short-GRBs with host association (e.g. 600 events at z ≤ 1).

Robot Fault Detection Based on Big Data

In order to improve the reliability of robot electrical fault detection and diagnosis, the author proposes a robot electrical fault detection and diagnosis method based on deep learning. Taking the return power and active power as constraints, the electrical fault data collection of the robot is carried out. Taking the resonant inductance and resonant capacitance of the robot electrical equipment as identification parameters, we conduct electrical fault differential feature mining. The fault features are extracted according to the time-delay distribution sequence of the electrical fault data of the robot, and the electrical fault detection and diagnosis results are output by using the deep learning function. Simulation results show that the author's method has a high accuracy probability for robot electrical fault diagnosis. The author's method is on average 14.7% higher than the neural network-based method and 24.5% higher than the expert system-based method. The accuracy rate of the author's method for robot electrical fault diagnosis is high. The author’s method is 16.6% higher than the neural network-based method on average and 34.2% higher than the expert system-based method. It is proved that the robot electrical fault detection and diagnosis based on deep learning has high accuracy and short time.

Treasure Maps for Detections of Extreme Energy Cosmic Rays

The origin of ultra-high-energy cosmic rays is a 60 yr old mystery. We show that with more events at the highest energies (above 150 EeV) it may be possible to limit the character of the sources and learn about the intervening magnetic fields. Individual sources become more prominent, relative to the background, as the horizon diminishes. An event-by-event, composition-dependent observatory would allow a “tomography” of the sources as different mass and energy groups probe different GZK horizons. A major goal here is to provide a methodology to distinguish between steady and transient or highly variable sources. Using recent Galactic magnetic field models, we calculate “treasure” sky maps to identify the most promising directions for detecting Extreme Energy Cosmic Rays doublets, events that are close in arrival time and direction. On this basis, we predict the incidence of doublets as a function of the nature of the source host galaxy. Based on the asymmetry in the distribution of time delays, we show that observation of doublets might distinguish source models. In particular, the Telescope Array hotspot could exhibit temporal variability as it is in a “magnetic window” of small time delays. These considerations could improve the use of data with existing facilities and the planning of future ones such as Global Cosmic Ray Observatory (GCOS).

Simulating neutrino echoes induced by secret neutrino interactions

New neutrino interactions beyond the Standard Model (BSM) have been of much interest in not only particle physics but also cosmology and astroparticle physics. We numerically investigate the time delay distribution of astrophysical neutrinos that interact with the cosmic neutrino background. Using the Monte Carlo method, we develop a framework that enables us to simulate the time-dependent energy spectra of high-energy neutrinos that experience even multiple scatterings en route and to handle the sharp increase in the cross section at the resonance energy. As an example, we focus on the case of secret neutrino interactions with a scalar mediator. While we find the excellent agreement between analytical and simulation results for small optical depths, our simulations enable us to study optically thick cases that are not described by the simplest analytic estimates. Our simulations are used to understand effects of cosmological redshifts, neutrino spectra and flavors. The developments will be useful for probing BSM neutrino interactions with not only current neutrino detectors such as IceCube and Super-Kamiokande but also future neutrino detectors such as IceCube-Gen2 and Hyper-Kamiokande.

Global stability of a multi-group delayed epidemic model with logistic growth

<abstract><p>In this work, we aim to investigate the mechanism of a multi-group epidemic model taking into account the influences of logistic growth and delay time distribution. Despite the importance of the logistic growth effect in such models, its consideration remains rare. We show that $ \mathcal{R}_0 $ has a crusher role in the global stability of a disease-free and endemic equilibria. That is, if $ \mathcal{R}_0 $ is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable, whereas, if $ \mathcal{R}_0 $ is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable. In addition, we construct suitable Lyapunov functions to investigate the global stability of disease-free and endemic equilibria. Finally, we introduce numerical simulations of the model.</p></abstract>

Methodology for calculating the delay time in the management system infocommunication networks based on the theory of mass service

The article is devoted to the consideration of delay time calculation methods in the M/M/m service system, the main components of the model, such as information flows and the service device, are investigated, and the delay time distribution formula based on Little’s law is improved. The importance of this indicator in determining the effectiveness of information communication systems and optimizing their functioning is noted. The article provides examples of delay time calculations and considers factors that affect this indicator, such as the workload of the telecommunications system and the characteristics of the service device. In addition, possible ways to improve the system are explored, including the use of buffers and more complex models. M/M/m is a system with m service lines, a Poisson input flow, an indicative service time distribution, and a First-Come-First-Serve (FCFS) discipline, which means that requests are served in the order in which they are received. A detailed analysis of the M/M/m model and the study of its main components allow a deeper understanding of the processes in mass service systems. The delay time in the M/M/m system is reflected in its direct impact on the quality of service and satisfaction of users of such networks

Replacement dynamics and the pathogenesis of the Alpha, Delta and Omicron variants of SARS-CoV-2.

New SARS-CoV-2 variants causing COVID-19 are a major risk to public health worldwide due to the potential for phenotypic change and increases in pathogenicity, transmissibility and/or vaccine escape. Recognising signatures of new variants in terms of replacing growth and severity are key to informing the public health response. To assess this, we aimed to investigate key time periods in the course of infection, hospitalisation and death, by variant. We linked datasets on contact tracing (Contact Tracing Advisory Service), testing (the Second-Generation Surveillance System) and hospitalisation (the Admitted Patient Care dataset) for the entire length of contact tracing in the England - from March 2020 to March 2022. We modelled, for England, time delay distributions using a Bayesian doubly interval censored modelling approach for the SARS-CoV-2 variants Alpha, Delta, Delta Plus (AY.4.2), Omicron BA.1 and Omicron BA.2. This was conducted for the incubation period, the time from infection to hospitalisation and hospitalisation to death. We further modelled the growth of novel variant replacement using a generalised additive model with a negative binomial error structure and the relationship between incubation period length and the risk of a fatality using a Bernoulli generalised linear model with a logit link. The mean incubation periods for each variant were: Alpha 4.19 (95% credible interval (CrI) 4.13-4.26) days; Delta 3.87 (95% CrI 3.82-3.93) days; Delta Plus 3.92 (95% CrI 3.87-3.98) days; Omicron BA.1 3.67 (95% CrI 3.61-3.72) days and Omicron BA.2 3.48 (95% CrI 3.43-3.53) days. The mean time from infection to hospitalisation was for Alpha 11.31 (95% CrI 11.20-11.41) days, Delta 10.36 (95% CrI 10.26-10.45) days and Omicron BA.1 11.54 (95% CrI 11.38-11.70) days. The mean time from hospitalisation to death was, for Alpha 14.31 (95% CrI 14.00-14.62) days; Delta 12.81 (95% CrI 12.62-13.00) days and Omicron BA.2 16.02 (95% CrI 15.46-16.60) days. The 95th percentile of the incubation periods were: Alpha 11.19 (95% CrI 10.92-11.48) days; Delta 9.97 (95% CrI 9.73-10.21) days; Delta Plus 9.99 (95% CrI 9.78-10.24) days; Omicron BA.1 9.45 (95% CrI 9.23-9.67) days and Omicron BA.2 8.83 (95% CrI 8.62-9.05) days. Shorter incubation periods were associated with greater fatality risk when adjusted for age, sex, variant, vaccination status, vaccination manufacturer and time since last dose with an odds ratio of 0.83 (95% confidence interval 0.82-0.83) (P value < 0.05). Variants of SARS-CoV-2 that have replaced previously dominant variants have had shorter incubation periods. Conversely co-existing variants have had very similar and non-distinct incubation period distributions. Shorter incubation periods reflect generation time advantage, with a reduction in the time to the peak infectious period, and may be a significant factor in novel variant replacing growth. Shorter times for admission to hospital and death were associated with variant severity - the most severe variant, Delta, led to significantly earlier hospitalisation, and death. These measures are likely important for future risk assessment of new variants, and their potential impact on population health.

Observational Inference on the Delay Time Distribution of Short Gamma-Ray Bursts

The delay time distribution of neutron star mergers provides critical insights into binary evolution processes and the merger rate evolution of compact object binaries. However, current observational constraints on this delay time distribution rely on the small sample of Galactic double neutron stars (with uncertain selection effects), a single multimessenger gravitational wave event, and indirect evidence of neutron star mergers based on r-process enrichment. We use a sample of 68 host galaxies of short gamma-ray bursts to place novel constraints on the delay time distribution and leverage this result to infer the merger rate evolution of compact object binaries containing neutron stars. We recover a power-law slope of (median and 90% credible interval) with α < −1.31 at 99% credibility, a minimum delay time of with at 99% credibility, and a maximum delay time constrained to at 99% credibility. We find these constraints to be broadly consistent with theoretical expectations, although our recovered power-law slope is substantially steeper than the conventional value of α = −1, and our minimum delay time is larger than the typically assumed value of 10 Myr. Pairing this cosmological probe of the fate of compact object binary systems with the Galactic population of double neutron stars will be crucial for understanding the unique selection effects governing both of these populations. In addition to probing a significantly larger redshift regime of neutron star mergers than possible with current gravitational wave detectors, complementing our results with future multimessenger gravitational wave events will also help determine if short gamma-ray bursts ubiquitously result from compact object binary mergers.

Spatial resolution analysis of passive bistatic radar using low-orbit navigation augmented signals

Aiming at the problem that the traditional monostatic ambiguity function is inconvenient to analyze the spatial resolution of the passive bistatic radar, this paper proposes a method to analyze it in a given airspace by using the spatial distribution of the ambiguity function. This method converts the time-delay Doppler distribution of the traditional ambiguity function into the spatial distribution near the reference target, which makes it more suitable for the intuitive analysis of the spatial resolution capability of the passive bistatic radar. This new method is used to analyze the target resolution capability in the designated airspace when Navigation augmented signals transmitted by the low-orbit communication satellite is used as the illuminator of opportunity. The distribution of the ambiguity function is transformed from a pushpin shape under the monostatic time delay-Doppler shift to a flat “sandwich biscuit” shape in space. The shape is closely related to the geometric relationship of the bistatic radar. Through the spatial distribution of the ambiguity function, the target resolution in a given airspace can be directly observed, which is convenient for analyzing the spatial resolution capability of the signal under different spatial geometric relationships. This also provides convenience for the optimal selection of the location of the illuminator of opportunity and receiving station.

Short GRB Host Galaxies. II. A Legacy Sample of Redshifts, Stellar Population Properties, and Implications for Their Neutron Star Merger Origins

We present the stellar population properties of 69 short gamma-ray burst (GRB) host galaxies, representing the largest uniformly modeled sample to date. Using the Prospector stellar population inference code, we jointly fit photometry and/or spectroscopy of each host galaxy. We find a population median redshift of (68% confidence), including nine photometric redshifts at z ≳ 1. We further find a median mass-weighted age of t m = Gyr, stellar mass of log(M */M ⊙) = , star formation rate of SFR = M ⊙ yr−1, stellar metallicity of log(Z */Z ⊙) = , and dust attenuation of mag (68% confidence). Overall, the majority of short GRB hosts are star-forming (≈84%), with small fractions that are either transitioning (≈6%) or quiescent (≈10%); however, we observe a much larger fraction (≈40%) of quiescent and transitioning hosts at z ≲ 0.25, commensurate with galaxy evolution. We find that short GRB hosts populate the star-forming main sequence of normal field galaxies, but do not include as many high-mass galaxies as the general galaxy population, implying that their binary neutron star (BNS) merger progenitors are dependent on a combination of host star formation and stellar mass. The distribution of ages and redshifts implies a broad delay-time distribution, with a fast-merging channel at z > 1 and a decreased neutron star binary formation efficiency from high to low redshifts. If short GRB hosts are representative of BNS merger hosts within the horizon of current gravitational wave detectors, these results can inform future searches for electromagnetic counterparts. All of the data and modeling products are available on the Broadband Repository for Investigating Gamma-ray burst Host Traits website.

Stochastic dynamics and ribosome-RNAP interactions in transcription-translation coupling

Under certain cellular conditions, transcription and mRNA translation in prokaryotes appear to be “coupled,” in which the formation of mRNA transcript and production of its associated protein are temporally correlated. Such transcription-translation coupling (TTC) has been evoked as a mechanism that speeds up the overall process, provides protection against premature termination, and/or regulates the timing of transcript and protein formation. What molecular mechanisms underlie ribosome-RNAP coupling and how they can perform these functions have not been explicitly modeled. We develop and analyze a continuous-time stochastic model that incorporates ribosome and RNAP elongation rates, initiation and termination rates, RNAP pausing, and direct ribosome and RNAP interactions (exclusion and binding). Our model predicts how distributions of delay times depend on these molecular features of transcription and translation. We also propose additional measures for TTC: a direct ribosome-RNAP binding probability and the fraction of time the translation-transcription process is “protected” from attack by transcription-terminating proteins. These metrics quantify different aspects of TTC and differentially depend on parameters of known molecular processes. We use our metrics to reveal how and when our model can exhibit either acceleration or deceleration of transcription, as well as protection from termination. Our detailed mechanistic model provides a basis for designing new experimental assays that can better elucidate the mechanisms of TTC.

Toward a Precision Measurement of Binary Black Holes Formation Channels Using Gravitational Waves and Emission Lines

The formation of compact objects—neutron stars, black holes, and supermassive black holes—and their connection to the chemical composition of galaxies is one of the central questions in astrophysics. We propose a novel data-driven, multi-messenger technique to address this question by exploiting the inevitable correlation between gravitational waves and atomic/molecular emission-line signals. For a fiducial probability distribution function of time delays, this method can provide a measurement of the minimum delay time of 0.5 Gyr and a power-law index of κ = 1 with a standard deviation of 0.12 (and 0.45) and 0.06 (and 0.34), respectively, from five years of LIGO–Virgo–KAGRA observations in synergy with SPHEREx line intensity mapping (and DESI emission-line galaxies). Such measurements will provide data-driven, multi-messenger constraints on the delay time distribution which is currently not well known.

Ultrawide Black Hole—Neutron Star Binaries as a Possible Source for Gravitational Waves and Short Gamma-Ray Bursts

The third observing run of the LIGO/Virgo/KARGA collaboration reported a few neutron star–black hole (NSBH) merger events. While NSBH mergers have yet to receive extensive theoretical attention, they may have a promising electromagnetic signature in the form of short gamma-ray bursts. Here we show that NSBH dynamical mergers can naturally form from ultrawide binaries in the field. Flyby gravitational interactions with other neighbors in the galaxy in these ultrawide systems may result in high eccentricity that drives the binary into a merger. We show that this process can result in a merger rate at the order of ∼10 Gpc−3 yr−1 (∼1 Gpc−3 yr−1) for elliptical (spiral) galaxies. This channel predicts a higher merger rate with a higher velocity dispersion of the host galaxy, a delay time distribution that is steeper than uniform but shallower than 1/t, and a higher merger rate for a lower black hole to neutron star mass ratio.

A global test of jet structure and delay time distribution of short-duration gamma-ray bursts

ABSTRACTThe multimessenger joint observations of GW170817 and GRB170817A shed new light on the study of short-duration gamma-ray bursts (SGRBs). Not only did it substantiate the assumption that SGRBs originate from binary neutron star (BNS) mergers, but it also confirms that the jet generated by this type of merger must be structured, hence the observed energy of an SGRB depends on the viewing angle from the observer. However, the precise structure of the jet is still subject to debate. Moreover, whether a single unified jet model can be applied to all SGRBs is not known. Another uncertainty is the delay time-scale of BNS mergers with respect to star-formation history of the Universe. In this paper, we conduct a global test of both delay and jet models of BNS mergers across a wide parameter space with simulated SGRBs. We compare the simulated peak flux, redshift, and luminosity distributions with the observed ones and test the goodness-of-fit for a set of models and parameter combinations. Our simulations suggest that GW170817/GRB 170817A and all SGRBs can be understood within the framework of a universal structured jet viewed at different viewing angles. Furthermore, model invoking a jet plus cocoon structure with a lognormal delay time-scale is most favoured. Some other combinations (e.g. a Gaussian delay with a power-law jet model) are also acceptable. However, the Gaussian delay with Gaussian jet model and the entire set of power-law delay models are disfavoured.

The redshift dependence of black hole mass distribution: is it reliable for standard sirens cosmology?

ABSTRACT An upper limit on the mass of a black hole set by the pair-instability supernovae (PISN) process can be useful in inferring the redshift of the gravitational wave (GW) sources by lifting the degeneracy between mass and redshift. However, for this technique to work, it is essential that the PISN mass scale is redshift independent or at least has a predictable redshift dependence. We show that the observed PISN mass scale can get smeared and the position of the PISN mass scale is likely to exhibit a strong redshift dependence due to a combined effect from the non-zero value of the delay time between the formation of a star and the merging of two black holes and the metallicity dependence of PISN mass scale. Due to the unknown form of the delay-time distribution, the redshift dependence of the PISN mass cutoff of the binary black holes (BBHs) cannot be well characterized and will exhibit a large variation with the change in redshift. As a result, the use of a fixed PISN mass scale to infer the redshift of the BBHs from the observed masses will be systematically biased. Though this uncertainty is not severe for the third observation run conducted by the LIGO–Virgo–KAGRA collaboration, in the future this uncertainty will cause a systematic error in the redshift inferred from the PISN mass scale. The corresponding systematic error will be a bottleneck in achieving a few per cent precision measurements of the cosmological parameters using this method in the future.