Model-independent confirmation of a constant speed of light over cosmological distances

Recent attempts at measuring the variation of $c$ using an assortment of standard candles and the redshift-dependent Hubble expansion rate inferred from the currently available catalog of cosmic chronometers have tended to show that the speed of light appears to be constant, at least up to $z\sim 2$. A notable exception is the use of high-redshift UV $+$ X-ray quasars, whose Hubble diagram seems to indicate a $\sim 2.7\sigma$ deviation of c from its value $c_0$ ($\equiv 2.99792458 \times 10^{10}$ cm s$^{-1}$) on Earth. We show in this paper, however, that this anomaly is due to an error in the derived relation between the luminosity distance, $D_L$, and $H(z)$ when $c$ is allowed to vary with redshift, and an imprecise calibration of the quasar catalog. When these deficiences are addressed correctly, one finds that $c/c_0=0.95 \pm 0.14$ in the redshift range $0\lesssim z\lesssim 2$, fully consistent with zero variation within the measurement errors.


INTRODUCTION
The first known measurement of the speed of light was carried out in 1676 by the Danish astronomer Olaus Roemer (Romer & Cohen 1940), who timed the eclipses of the Jovian moon Io and estimated that the delay of about 22 minutes seen on diametrically opposite sides of Earth's orbit around the Sun was due to light's travel time across that distance.Planetary distances were not well known back then, but he nevertheless obtained a value of ∼ 2 × 10 10 cm s −1 , not too different from the much more precise measurement 1 we have today, c 0 = 2.99792458 × 10 10 cm s −1 .(Throughout this paper, we shall use the symbol c 0 to denote the speed of light on Earth, to distinguish it from the possible redshift dependent value, c(z), inferred from high-redshift data.) Many have wondered whether c is constant over much larger distances, however, perhaps even in time, most critically over cosmic scales.And several attempts have already been made to determine whether c is in fact a true constant of nature.But the problem with measuring c across the Universe is that one cannot avoid using the Friedmann-Lemaître-Robertson-Walker (FLRW) metric to describe the background cosmology.Of course, all solutions to Einstein's equations in general relativity assume ab initio that c is constant.After all, the interval ds written in terms of the metric coefficients would not even exist if c were variable in space and/or time (Melia 2020).
So unfortunately one is faced with an inherent inconsistency ⋆ John Woodruff Simpson Fellow.E-mail: fmelia@email.arizona.edu 1 Actually, this value is exact when quoted in these units because, by a 1983 international agreement, a 'metre' is defined in terms of how far light travels in 1/299792458 seconds.See https://www.nist.gov/siredefinition/definitions-si-base-unitswhen attempting to measure the hypothesized variability of c using a metric whose validity requires that c be constant.In this paper, we shall not avoid this issue either, so our goal will not be to actually measure a possible time-dependence (or redshift-dependence) of c but, rather, to demonstrate whether the cosmological data selfconsistently imply that c is constant, as required by FLRW.
The consequences of a variable c were considered by Einstein himself (Einstein 1907), and many workers have followed suit, including most prominently Albrecht & Magueijo (1999), Barrow & Magueijo (1999), Barrow (1999) and Bassett et al. (2000), who proposed that c has varied across the evolutionary history of the Universe.Similar ideas were espoused and strongly supported by Moffat (2002), and further developed by others since then.
But most cosmological tests of the constancy of c have tended to be carried out in the context of ΛCDM.Some attempts at avoiding the presumption of a background cosmology have also been made, e.g., by Liu et al. (2023), who 'measured' the speed of light using cosmic chronometers to infer the Hubble constant, H(z), along with Type Ia supernovae and the UV-X-ray correlation of high-redshift quasars.Their results based on the Type Ia SNe showed strong consistency with a constant value for c.For the quasars, however, they demonstrated that the speed of light up to redshift z ∼ 2 satisfies the constraint c/c 0 = 1.19 ± 0.07, in terms of its local value on Earth, c 0 .Though the difference between this c and c 0 is less than 3σ, an inconsistency of ∼ 2.7σ nevertheless fails to affirm the supernova result and creates some tension.In addition, as noted earlier, one cannot really be satisfied with c varying like this when the analysis is carried out using the FLRW metric, which was not constructed for a variable c in the first place.As it stands, the outcome based on the use of quasars as standard candles either implies that their presumed UV-X-ray correlation is not sufficiently precise for this type of analysis (see § § 2 and 3 for more details), or perhaps that the assumed cosmic spacetime itself is inaccurate.
There appear to be mitigating factors with this conclusion, however.First, the relation derived by Liu et al. (2023) for the luminosity distance, D L , in terms of H(z), misses the contribution from dc(z)/dz itself, which introduces an inconsistency when the inferred value of c(z) is variable.Secondly, previous work with the UV-X-ray catalog of high-z quasars (Risaliti & Lusso 2015, 2019;Lusso et al. 2010;Lusso & Risaliti 2016) has suggested that the calibration of these sources for use as standard candles is more accurately carried out with the simultaneous optimization of all the parameters, including those characterizing the cosmology, be it ΛCDM or the more generic 'chosmographic' polynomial fit, rather than with the separate introduction of luminosity distances inferred from Type Ia supernovae (Melia 2019).
This type of analysis overlaps with our previous application of high-redshift quasars to model selection, a head-to-head comparison between ΛCDM and an alternative FLRW cosmology known as the R h = ct universe (Melia 2007;Melia & Shevchuk 2012;Melia 2019).This test uses a recently refined method of sampling the redshift-distance relationship (Risaliti & Lusso 2015) based on an inferred correlation between the X-ray and UV monochromatic quasar luminosities, following an earlier proposal by Avni & Tananbaum (1986).From this study emerged the conclusion that the use of 'external' calibrators rendered the high-z quasar Hubble diagram marginally inconsistent with the predictions of ΛCDM, while an internal calibration using the simultaneous optimization of the UV-X-ray correlation relation and the model's parameters produced greater self-consistency.Given that the Type Ia SNe produced results consistent with a constant c, while the quasar Hubble diagram did not, the role played by this problem with the calibration of the UV-X-ray correlation needs to be better understood.
Our goal in this paper is thus to revisit the use of high-z quasars and cosmic chronometers to test the constancy of c, but this time using a more valid expression for dc(z)/dz, along with a better calibration of the UV-X-ray correlation function.And in keeping with the spirit of avoiding cosmology-dependent factors as much as possible, we shall also restrict our attention to the generic 'cosmographic' polynomial fit to the data (Risaliti & Lusso 2019), rather than relying on the parameterization in ΛCDM.
We begin with a brief description of the background for this test in § 2, and then describe the quasar and cosmic-chronometer data and our method of analysis in § 3. We end with a discussion of our results and a brief conclusion in § 4.

THE QUASAR UV-X-RAY CORRELATION
Adopting the cosmological principle, which assumes isotropy and homogeneity throughout the cosmic spacetime, we may write the metric using the Friedmann-Lemaître-Robertson-Walker (FLRW) ansatz, where a(t) is the expansion factor, t is the cosmic time, and (r, θ, φ) are the spatial coordinates in the comoving frame.In addition, dΩ 2 ≡ dθ 2 + sin 2 θ dφ 2 , and k is the spatial curvature constant, which we assume to be zero in accordance with most of the data available to us today (Planck Collaboration et al. 2016).
In this context, the luminosity distance may be written as where H(z) ≡ ȧ/a is the redshift-dependent Hubble parameter, and we have explicitly written the speed of light as c(z) to remind ourselves that derivatives of c cannot be ignored should it turn out to be variable.Throughout this paper, we shall use the form where H 0 is the Hubble constant.Then, from which we derive . (5) Very importantly, note that H 0 completely cancels out in this expression.This is an especially desirable feature of this analysis because of the growing disparity between the measurements of H 0 at low and high redshifts, creating a ∼ 4σ uncertainty in its value (Riess et al. 2021).
And to further simplify the analysis, we divide Equations ( 2) and ( 4) and the numerators and denominators in Equation ( 5) by the constant c 0 , yielding the differential equation for the 'normalized' speed of light κ ≡ c/c 0 : where and D′ L is correspondingly derived from DL (z).This allows us to solve for c(z)/c 0 without having to specify the actual value of c 0 locally.
Our test of the variation of c with redshift will therefore rely on two independent cosmological measurements: (i) the luminosity distance D L inferred from the high-z quasar Hubble diagram (Risaliti & Lusso 2019), from which we shall also calculate the derivative D ′ L , and (ii) the redshift-dependent Hubble parameter H(z) estimated using cosmic chronometers, as originally proposed by Jimenez & Loeb (2002) and Jimenez et al. (2003).
The quasar's UV (i.e., disk) emission is correlated with its Xray (i.e., coronal) emission according to the following parametrization: where L X and L UV are the rest-frame monochromatic luminosities at 2 keV and 2,500 Å, respectively, and γ and β are two parameters that we shall optimize simultaneously with those characterizing our generic cosmographic cosmology described below.We shall use base 10 logarithms throughout this paper.Given that the data contain the fluxes, rather than model-dependent luminosities, we use a slightly modified form of Equation ( 8), where the constant β subsumes the slope γ and intercept β, so that The luminosity distance in the cosmographic empirical fit we employ here is given by the expression based on a third-order polynomial with two constants, a 2 and a 3 , that we shall optimize along with the other free parameters (Risaliti & Lusso 2019), by minimizing the likelihood function ln .
(12) In this expression, the variance σi 2 ≡ δ 2 + σ 2 i is given in terms of a global intrinsic dispersion, δ, and the individual measurement errors σ i in (F X ) i (Risaliti & Lusso 2015), while the errors in (F UV ) i are insignificant compared to σ i and δ, so we ignore them in this application.In Equation ( 12), Φ is defined as (13) incorporating the measured fluxes (F X ) i and (F UV ) i at redshift z i .
Once the values of a 2 and a 3 have been optimized from the quasar data, we can also calculate the derivative of D L from Equation (11), yielding and we divide both D L and D ′ L by c 0 to obtain DL and D′ L to be used in Equation ( 6).

DATA AND ANALYSIS
With their high luminosities, quasars represent promising cosmological probes out to larger redshifts than many other types of source.Their luminosity distance appears to follow a reliable, nonlinear correlation between their ultraviolet (UV) and X-ray monochromatic fluxes (Eq.8).But although this correlation has been known for over three decades (Avni & Tananbaum 1986), only recently has the impractically large dispersion in this relation been suppressed by refining the selection criteria and flux measurements (Risaliti & Lusso 2015, 2019;Lusso & Risaliti 2016).These improvements allow the quasars to be used as distance indicators out to redshifts ∼ 6.
After improving their selection techniques and flux measurements, Risaliti & Lusso (2019) constructed a final sample of 1598 quasars with accurate measurements of the intrinsic UV and X-ray fluxes in the redsfhit range 0 z 6.These are the sources we shall use for the analysis in this paper, though only a portion of this redshift range will actually be matched to the more restricted redshift coverage (i.e., z 2) of the cosmic chronometers.
As noted above, we simultaneously optimize all of the quasar  1 shows that the optimization of γ via the use of internal calibration produces results fully consistent with this range.But as noted earlier, the slight differences between the values of these parameters optimized this way compare to their estimation using external calibrators are sufficient to mitigate the tension-first pointed out by Risaliti & Lusso (2019)-between the quasar Hubble diagram and the predictions of ΛCDM.

2019). A quick inspection of Table
Note that the Hubble constant H 0 is not independent of β.One can easily see this from Equation ( 9), where β and H 0 combine to produce the 'single variable' β − 2(γ − 1) log 10 H 0 .For the purpose of optimization, we have therefore subsumed H 0 into the parameter β.To make the results easy to interpret, however, we assume a fiducial value H 0 = 70 km s −1 Mpc −1 , and the optimization for β shown in Table 1 corresponds to this choice.For a different Hubble parameter, H ′ 0 , the optimized value of β in Table 1 would be changed by the amount ∆ β = 2(γ − 1) log 10 (H ′ 0 /H 0 ).Given the degeneracy between β and H 0 , the actual value of the Hubble constant does not affect the calibration of the UV-X-ray correlation function, i.e., it has no impact on any of the other optimized variables, γ, δ, a 2 and a 3 .And to reiterate, H 0 completely cancels out from the righthand side of Equation ( 6), so this analysis is independent of the contentious Hubble constant.
The expansion rate of the Universe, H(z), and the integrated function z 0 du/H(u), are obtained directly from the redshift-time derivative dz/dt, using at any redshift z 0. The quantity dz/dt may be measured from the differential age evolution of passively evolving galaxies, without the need to assume any particular cosmological model (Jimenez & Loeb 2002;Jimenez et al. 2003).These galaxies are commonly referred to as 'cosmic chronometers.'A recent sample of 32 cosmic-chronometer measurements (see Ruan et al. 2019 and references therein) is shown in Table 2.
We integrate Equation ( 6) over redshift, starting at z = 0.09 (the first entry in Table 2) where c(0) is assumed to have the value c 0 , i.e., where κ = 1.0, using the empirically derived quantities DL ,

D′
L , H(z) and z 0 du/H(u).Estimating the errors of the 'measured' value of κ at any given redshift is complicated, however, in part because the errors incurred with the integration of the function 1/H(z) over z are correlated with the errors in H(z) itself.To address this difficulty, we instead estimate the error in κ(z) at each given value of z, based on 10,000 Monte Carlo simulations utilizing the measured uncertainties in the coefficients γ, δ, β, a 2 and a 3 in Table 1, and the 1σ errors quoted in Table 2.All of these variables are assumed to be distributed normally with dispersions set equal to their reported errors.
In Figures 1 and 2 we show the resultant distributions in κ(z) at two specific redshifts, z = 0.48 and 1.53.Though not perfectly normal, these distributions are nevertheless matched quite well by Gaussian fits (solid curves), from which the 1σ errors may be extracted.We find σ c/c 0 ≈ 0.125 in both cases.Table 3 lists the value of κ(z) calculated in this fashion throughout the redshift range 0 z 2. As one may easily confirm, this ratio's deviation from 1 is always well within the inferred 1σ error (see also Fig. 3).In other words, the speed of light estimated with this approach appears to be fully consistent with c 0 all the way out to redshift ∼ 2, when the Universe was roughly one-third as old as it is today.

DISCUSSION
The use of quasars for cosmological analysis out to a redshift exceeding ∼ 6 has already been firmly established, notably for the optimization of cosmological parameters in the standard model (Risaliti & Lusso 2019), and for model selection between competing cosmologies, such as ΛCDM and R h = ct (Melia 2019).The interesting suggestion to also use them for the purpose of testing the constancy of c over cosmological distances (Liu et al. 2023) has pointed to a possible anomaly, however, revealing a ∼ 2.7σ deviation of the speed of light from its value measured on Earth.
This tension mirrors a similar inconsistency identified by Risaliti & Lusso (2019) between the inferred quasar Hubble diagram and the predictions of ΛCDM.What is more puzzling, though, is that the variation of c(z)/c 0 identified by Liu et al. (2023) is based on the use of a cosmographic polynomial fit, not directly related to ΛCDM.In principle, this tension would thus appear to be more general, not tied to any particular choice of background cosmology.
But though our previous comparative test between ΛCDM and R h = ct concluded that the quasar data favour the latter over the former, we also demonstrated that the tension between these data and the standard model is largely mitigated when one uses internal calibration (via the simultaneous optimization of all the parameters) instead of external sources, such as Type Ia supernovae.It also appears that an incorrect relation between D L and H(z) was used in the previous measurement of c(z)/c 0 , given that terms involving dc/dz cannot be ignored when the outcome points to a variable c(z).
We have thus retested the presumed constancy of c over cosmological distances using the high-z quasar observations together with the expansion rate H(z) inferred from cosmic chronometers, using the alternative calibration of the quasar data and an updated expression for d(c/c 0 )/dz.We now find no evidence for a variation of c, based on these observations, all the way out to z ∼ 2.
It is important to note that this outcome confirms the result already discussed by Liu et al. (2023) based on the use of Type Ia supernova data.In both cases, the profile of c(z) with redshift suggests no variation over cosmic scales and times.This conclusion is relevant to the larger question of how confident we should be in the use of the FLRW metric to describe the cosmic spacetime as opposed to possible alternative interpretations of cosmic redshift based on the assumption that c has varied in our past (Albrecht & Magueijo 1999;Barrow & Magueijo 1999;Bassett et al. 2000;Moffat 2002).
Thus, in addition to these conclusions based on the use of Type Ia SNe and quasars, it would be very useful to redo the analysis utilizing Equations ( 1)-( 5) with other classes of sources, particularly yielding other measures of distance and/or age.For example, strong gravitational lenses provide us with the means of 'measuring' the ratio of angular-diameter distances to the lens and background source (see, e.g., Wei & Melia 2020).A clear benefit of this approach is the elimination of H 0 from the analysis, which avoids the current uncertainty with its value.When combined with the cosmic chronometer measurements of H(z), these sources should provide an important affirmation of the constancy of c over cosmic distances analogously to the work reported in Liu et al. (2023) and in this paper.A similar analysis may also be feasible with the use of HII galaxies to construct the Hubble diagram (Yennapureddy & Melia 2017), from which the redshift dependence of c may be inferred along the lines described in this paper.This work is underway and its results will be reported elsewhere.

CONCLUSION
In closing, we reiterate the very important feature of this work that it is completely independent of the value of H 0 .This is crucial because it now appears that we do not have a clear understanding of how this parameter is to be measured most accurately (Riess et al. 2021).In addition, our use of a cosmographic fit for D L and D ′ L has rendered these results completely free of any presumed cosmolog-ical model, other than their generic dependence on the viability of the FLRW metric.It thus appears that the assumption of a constant speed of light, required for the derivation of the FLRW spacetime in the first place, is fully consistent with all of the data we have at our disposal today.

Figure 3 .
Figure3.Solid black curve: the speed of light, c(z), in units of c (i.e., the normalized speed κ(z) defined in the text) as a function of redshift, calculated from the integration of Equation (5).Yellow shaded region: the ±1σ error estimated with 10,000 Monte Carlo simulations, as illustrated in Figs1 and 2. By comparison, the dashed line corresponds to c(z)/c 0 = 1.The measured value of c(z)/c 0 is fully consistent with zero variation within the measurement errors.

Table 1 .
Optimization of the UV-X-ray correlation in high-z quasars

Table 2 .
Hubble parameter H(z) from cosmic chronometers (Avni & Tananbaum 1986;Just et al. 2007uation (9), using solely the quasar data on their own.This avoids any possible contamination from outside calibrators, and is in fact analogous to what one does with Type Ia SNe, where the so-called 'nuisance' parameters shaping the SN lightcurve are optimized along with parameters of the cosmological model itself.The results of this fitting are shown in Table1, including the polynomial coefficients a 2 and a 3 in the cosmographic expression for D L .Previous work has shown that γ ∼ 0.5 − 0.7 when external sources are used to calibrate the quasar data(Avni & Tananbaum 1986;Just et al. 2007; Young et al. 2010; Lusso et al. 2010; Risaliti & Lusso 2015; Lusso & Risaliti 2016; Risaliti & LussoFigure 1. Distribution of κ(z) ≡ c(z)/c 0 values at z = 0.48 based on 10,000Monte Carlo simulations using the measured uncertainties in the coefficients γ, δ, β, a 2 , and a 3 and the 1σ errors in Table2.The dispersion corresponding to the best fit Gaussian (solid black curve) is σ = 0.133.

Table 3 .
Speed of light in units of c 0 as a function of redshift