Constraining Fundamental Constants with Fast Radio Bursts: Unveiling the Role of Energy Scale

Understanding physical mechanisms relies on the accurate determination of fundamental constants, although inherent limitations in experimental techniques introduce uncertainties into these measurements. This paper explores the uncertainties associated with measuring the fine-structure constant ($\alpha$) and the proton-to-electron mass ratio ($\mu$) using observed fast radio bursts (FRBs). We select 50 localized FRBs to quantify the effects of varying this fundamental coupling on the relation between dispersion measure and redshift. By leveraging independent measurements of dispersion measures and redshifts of these FRBs, we constrain the uncertainties in $\alpha$ and $\mu$ approximately to $\Delta\alpha/\alpha=1.99\times 10^{-5}$ and $\Delta\mu/\mu=-1.00\times 10^{-5}$ within the standard $\Lambda$CDM cosmological framework. Remarkably, these constraints improve nearly an order-of-magnitude when considering a dynamical dark energy model. This investigation not only yields one of the most stringent constraints on $\alpha$ and $\mu$ to date but also emphasizes the criticality of accounting for the energy scale of the system when formulating constraints on fundamental parameters.


INTRODUCTION
Fast radio bursts (FRBs) are short-duration (millisecond timescale) bright transient phenomena.These events manifest within the radio frequency spectrum, spanning from approximately 100 MHz to 8 GHz.Since the initial detection of the first FRB in 2007 by Lorimer et al. (2007), nearly 700 additional FRBs have been cataloged, with the majority of detections attributed to the Canadian Hydrogen Intensity Mapping Experiment (CHIME) telescope 1 .While a significant portion of these FRBs appear as non-repeating oneoff bursts, a subpopulation exhibit repeating behavior, albeit without apparent periodicity.Notably, only one FRB (FRB 20200428) has been conclusively traced to an origin within our own Milky Way galaxy, specifically associated with the Galactic soft gamma repeater SGR 1935 + 2154 (Bochenek et al. 2020a,b;CHIME/FRB Collaboration et al. 2020), while some others have been localized within their respective host galaxies.While various theoretical models invoking compact objects such as white dwarfs (WDs), neutron stars, or black holes, have been proposed to elucidate the characteristics of FRBs (Platts et al. 2019;Zhang 2020), the lack of coincident electromagnetic counterparts hinders the definitive identification of a single progenitor mechanism.Addressing this challenge, recent work by Kalita & Weltman (2023) suggests that the detection of continuous gravitational waves from FRB sites could help to constrain potential progenitor models, offering a promising avenue for future research.
FRBs, distinguished by their notable characteristics including high flux, very short pulse widths, large dispersion measures (DMs), and ★ E-mail: surajit.kalita@uct.ac.za 1 https://www.chime-frb.ca/catalogtheir capability to probe the intergalactic medium on cosmological scales, have emerged as valuable tools for investigating various astrophysical and cosmological studies.Leveraging FRB 150418, Bonetti et al. (2016) constrained the photon mass to   < 1.8×10 −14 eV c −2 , a limit which strengthens with the inclusion of numerous other FRBs (Wang et al. 2021;Lin et al. 2023).Moreover, different research groups have recently employed localized FRBs to estimate the Hubble constant (Hagstotz et al. 2022;Wu et al. 2022;James et al. 2022).Additionally, utilizing data from 12 localized FRBs, Reischke & Hagstotz (2023) established the most stringent constraint on the parameterized post-Newtonian parameter associated with the weak equivalence principle.Furthermore, employing the phenomenon of gravitational lensing in FRBs, Muñoz et al. (2016) derived constraints on the fraction of dark matter attributed to primordial black holes, which were subsequently refined by considering various factors such as FRB microstructures (Sammons et al. 2020), extended mass functions (Laha 2020), plasma lensing (Leung et al. 2022), and enhanced data accuracy (Liao et al. 2020;Kalita et al. 2023).
One of the key objectives in physics is to establish the universality of physical laws by testing fundamental couplings.Among them, the fine-structure constant, denoted by , and the proton-to-electron mass ratio, denoted by  are of particular interest.These parameters hold significance as they delineate the strength of the electromagnetic interaction between elementary charged particles.Moreover, being dimensionless quantities,  and  are independent of any specific system of units, rendering them fundamentally invariant across reference frames.
In this article, we investigate the constraints on the parameters  and  using observations of localized FRBs.We outline the structure of the article as follows.In Section 2, we explore the relationship between the mean DM as a function of source redshift and its dependency on the fluctuation of .In Section 3, we detail our data sample consisting of 50 localized FRBs.This data is subsequently employed to calculate the constraint on  under standard ΛCDM cosmology.We further extend this analysis by encompassing a cosmological model that incorporates dynamical dark energy.In Section 4, we proceed to calculate the constraint on  and subsequently followed by a comprehensive discussion of our results.Finally, we present our concluding remarks in Section 5.

EFFECT OF VARIATION IN FINE-STRUCTURE CONSTANT ON REDSHIFT-DISPERSION MEASURE RELATION
One of the pivotal characteristics exhibited by FRBs is the dispersion sweep discernible in the frequency-time domain.This phenomenon arises due to the presence of intervening ionized plasma along the propagation path of the radio waves from the source to Earth.The extent of this dispersion is quantified by DM, which reflects the total column density of free electrons encountered by the signal.DM integrates contributions from various regions, including the Milky Way (MW) Galaxy, its circumgalactic halo, the intergalactic medium (IGM), and the host galaxy of the FRB.Mathematically it can be represented as where  S is the source redshift, while DM MW , DM Halo , DM IGM , and DM Host are DM contributions from our Galaxy, halo, IGM, and host galaxy, respectively.Owing to our reasonable understanding of the Galactic distribution of free electrons, DM MW is fairly modeled.Additionally, Prochaska & Zheng (2019) estimated that the Galactic halo contributes DM Halo ≈ 50 − 80 pc cm −3 independent of any contribution from the Galactic interstellar medium.The remaining components DM IGM and DM Host are largely unknown due to the challenges associated with their measurements.Nonetheless, the IllustrisTNG simulation shows that for non-repeating FRBs, the median turns out to be DM Host = 33(1 +  S ) 0.84 pc cm −3 , whereas for repeating FRBs, it alters to 35(1 +  S ) 1.08 pc cm −3 or 96(1 +  S ) 0.83 pc cm −3 depending whether the host is a dwarf galaxy or a spiral galaxy, respectively (Zhang et al. 2020).
The average DM IGM is given by Macquart et al. (2020) as where  is the speed of light,  is the Newton gravitational constant,  0 is the Hubble constant, Ω b is the baryonic matter density,  p is the proton mass,  IGM is the baryon mass fraction in the IGM, and () is the ionization fraction along the line of sight, given by with  e,H and  e,He respectively being the ionization fractions of the intergalactic hydrogen and helium, and  H = 3/4,  p = 1/4 their respective mass fractions.In our calculations, we consider  IGM = 0.85 following Cordes et al. (2022).The Hubble function  () encodes the information of underlying cosmology and for the Λ cold dark matter (ΛCDM) formalism, neglecting the contributions from radiation and curvature, it is given by  () =  0 √︃ Ω m (1 + ) 3 + Ω Λ with Ω m and Ω Λ respectively being the present matter and vacuum density fraction such that Ω m + Ω Λ = 1.
As outlined in the Introduction, the objective of this study is to impose constraints on .Let us first define dimensionless couplings  p =  2 p /ℏ and  e =  2 e /ℏ.Considering that the Planck mass is fixed while allowing for variations in the quantum chromodynamics (QCD) scale and particle masses, Coc et al. (2007)   In our analysis, we compare our data points with this mean DM value to put constraint on Δ/.

CONSTRAINING FINE-STRUCTURE CONSTANT USING FRB DATA
In this study, we analyze 50 FRBs that have been accurately localized within their host galaxies as of January 2024.In other words, the source redshift  S of these FRBs are measured.The details of these FRBs are given in Table 1.Utilizing the NE2001 model (Cordes & Lazio 2002), we compute the Galactic distribution of free electrons and subsequently determine DM MW for each of them.DM Halo is calculated using the model by Prochaska & Zheng (2019) as discussed in the previous section.To account for the contribution of DM Host , we divide these 50 FRBs in two categories.For those with measured (reported) DM Host , we directly incorporate these values, whereas for FRBs with unknown DM Host , we employ the Zhang et al. ( 2020) model (as described previously) to estimate these missing values.Substituting all these values in Eq. ( 1), we obtain DM IGM for each of these FRBs.2020) simulations appear to be lower in comparison to the currently available data, which indicate significantly larger host DMs (Theis et al. 2024).Hence DM IGM is attributed to a probability distribution, given by Macquart et al. (2020) as where with   Host being the median and   2 Host − 1  2 Host + 2 Host its variance.In our analysis, we choose values of  Host and  Host from Macquart et al. (2020).These distribution functions effectively capture the inherent uncertainties associated with DM IGM and DM Host values.It is also worth noting that due to precise localization of these FRBs, there is minimal uncertainty associated with their redshift values.
Our objective is to constrain Δ/ using this data sample, necessitating an assessment of the theoretical fidelity of the ⟨DM IGM ⟩ −  S curve over these data points.Consequently, we define the following joint likelihood function (Macquart et al. 2020)

Effect of dynamical dark energy equation of state on constraining the fine-structure constant
The ΛCDM cosmological model, despite its simplicity and efficacy in encapsulating the Universe, confronts recent challenges.Foremost among these challenges is the Hubble tension, characterized by a disagreement between the ascertained values of  0 obtained from observations of early and late time cosmologies.Measurements of the cosmic microwave background (CMB) radiation by the Planck satellite suggest a value of approximately  0 = 67.36 ± 0.54 km s −1 Mpc −1 (Planck Collaboration et al. 2020), while using local distance indicators like type Ia supernovae and cepheid variable stars yields a higher value of  0 = 73.04±1.04km s −1 Mpc −1 (Riess et al. 2022).This discrepancy has motivated exploration of alternative physics and cosmological models (see comprehensive discussions by Di Valentino et al. (2021) and Kamionkowski & Riess (2023) on these models).
In this study, we consider a simple model with a dynamical dark where Assuming the dark energy equation of state varies with time and the function () is parameterized by Jassal et al. (2005) as where  0 and   are dimensionless parameters.This leads to the following modified form of  () Notably  0 = −1 and   = 0 recover the standard ΛCDM cosmology with () = −1.However, other parameter combinations can lead to () < −1 for certain redshifts.This modification to the cosmological model also impacts ⟨DM IGM ()⟩, which in turn affects the likelihood function of Eq. ( 9).Fig. 3 illustrates Δ/ at which the joint likelihood function is maximized for different combinations of  0 and   .We observe an improvement in the constraint on Δ/ for this dynamical dark energy equations of state.Notably, the most stringent constraint becomes Δ/ ≈ 5.4 × 10 −7 , nearly one order of magnitude lower than the case for  = −1.

DISCUSSION
This study investigates the fine-structure constant using DM values of localized FRBs.Analysis is conducted on a sample of 50 FRBs, whose DM IGM is determined by subtracting other contributions from the observed DM.These DM IGM values are then compared to ⟨DM IGM ⟩, which is theoretically linked to Δ/.A likelihood function is further formulated for maximization over Δ/, employing parameters derived from the ΛCDM cosmology.The resulting maximized value is found to be Δ/ ≈ 1.99 +12.93 −9.61 × 10 −5 at the 1 confidence level.Remarkably, this constraint gets a nearly order-ofmagnitude improvement while considering a dynamical dark energy model with () < −1 for certain redshift.It is worth mentioning that the limitations in our current understanding on the probability distributions of DM IGM and DM Host introduce inherent uncertainties in the derived constraints on the parameter .Consequently, these results should be interpreted with caution until more robust data allows for a more comprehensive characterization of the relevant distributions.
Moreover, using Eqs.( 4) and ( 5), one can obtain the following relation between the uncertainties of the fine-structure constant and proton-to-electron mass ratio Substituting Δ/ = 1.99 +12.93 −9.61 × 10 −5 , we obtain Δ/ = −1.00+3.81  −7.47 × 10 −5 , with potential improvement by another order of magnitude for the aforementioned dynamical dark energy model.These constraints on  and  outperform many previous reported constraints based on quasar or WD data (mentioned in the Introduction), despite the relatively large error bars attributable to the limited sample size (only 50 FRBs with known redshifts).Anticipated improvements in these error bars are foreseeable with increased data availability, particularly from additional localized FRBs.
Within the context of the ΛCDM cosmological model, we posit the existence of characteristic physical scales such as the Hubble scale, horizon scale, and baryon acoustic oscillation scale.Elucidating these scales is paramount for investigations into the large-scale structure and evolution of the universe.Deviations from the ΛCDM formalism (e.g. the aforementioned dynamical dark energy model) necessitate a modification of both the length and energy scales governing the system.This, in turn, entails a restructuring of the characteristic timescales across different cosmological epochs.This study underscores the criticality of considering the inherent scales of a system when establishing constraints on any given physical parameter.Such an approach is anticipated to play an instrumental role in rigorously testing various cosmological models and alternative theories of gravity.It is important to emphasize that we do not advocate for the primacy of any specific dynamical dark energy model.Rather we emphasize the utilization of observations across diverse energy regimes to constrain fundamental parameters.
FRBs offer the advantage of probing a comparatively lower redshift range than CMB or other traditional cosmological observations.Conversely, FRBs are among the rare astronomical phenomena (alongside quasars) detectable at relatively high redshifts, establishing their unique value as a cosmological probe.Furthermore, alteration of dark energy equation-of-state alter underlying physics (e.g., rate of acceleration of the Universe) at identical stages of the Universe's evolution.This inherently translates to alterations in the system's characteristic length scale, which is directly linked to its energy scale.Our findings demonstrate that the constraints imposed on  and  from FRB observations exhibit sensitivity to deviations from the ΛCDM cosmological model.Consequently, the system's energy scale plays a critical role in constraining these fundamental parameters.

CONCLUSION
Our work stands out for providing not only some of the tightest constraints on  and  using localized FRBs, but also sheds light on how these constraints depend on the specific theory of gravity employed and its characteristic energy scale.Despite the current count of detected FRBs remaining relatively modest, projections estimate an anticipated rate of occurrence ranging from approximately 100 to 1000 events per day (Champion et al. 2016).Ongoing efforts, bolstered by forthcoming observational facilities such as the Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX), the Deep Synoptic Array (DSA)-2000, and the Bustling Universe Radio Survey Telescope in Taiwan (BURSTT), promise a significant enhancement in the detection rates for FRBs in the near future.Consequently, this advancement will facilitate further refinement of the aforementioned constraints.

Figure 1 .
Figure 1.The host redshift values for the localized FRBs are plotted against their estimated DM contribution from IGM along with their error bars.The black solid line depicts the ⟨DM IGM ( S ) ⟩ as a function of  S assuming ΛCDM cosmology with  0 = 73 km s −1 Mpc −1 , Ω m = 0.30966, Ω Λ = 1 − Ω m , and Ω b = 0.04897.Green shaded regions represent 1 and 2 confidence regions of DM IGM .
Fig.1shows  S values of these FRBs plotted against the calculated DM IGM and their errorbars along with ⟨DM IGM ( S )⟩ assuming standard ΛCDM cosmology with  0 = 73 km s −1 Mpc −1 , Ω m = 0.30966, Ω Λ = 1 − Ω m , and Ω b = 0.04897.It also depicts 1 and 2 confidence regions of ⟨DM IGM ⟩ following the results obtained by Jaroszynski (2019) using the Illustris simulation which takes into account for the inhomogeneous distribution of ionized gas in the IGM.It is important to note that some DM Host values derived fromZhang et al. ( IGM = DM IGM /⟨DM IGM ⟩,  DM is its standard deviation, ,  1 ,  2 , and  0 are model parameters.The values of these parameters are chosen from Zhang et al. (2021) which is based on IllustrisTNG simulation.Similarly, the inherent difficulty in measuring DM Host , introduce significant uncertainty regarding its exact distribution to characterize its statistical properties.Thus following Macquart et al. (2020), we adopt the following log-normal probability distribution  Host (DM Host ) = 1 √ 2DM Host  Host exp − (ln DM Host −  Host ) Fig.2illustrates the probability density function of the joint likelihood of all 50 localized FRBs with respect to Δ/ alongside different confidence intervals.This likelihood function is maximized for Δ/ ≈ 1.99+12.93−9.61 × 10 −5 within 1 confidence level, thereby establishing the most robust constraint on  derived from the localized FRB dataset within the framework of the ΛCDM cosmology.

Figure 2 .
Figure 2. Probability distribution of joint likelihood function with respect to Δ/ along with its 1 and 2 confidence intervals.

Figure 3 .
Figure 3. Variation of Δ/ for different dark energy equation of states.The colorbar shows the values of Δ/ for certain combinations of  0 and   .

Table 1 .
List of all localized FRBs (until January 2024).DM MW contribution is calculated based on NE2001 model.FRBs in bold indicate that their DM Host values are reported.