Constraints on the minimally extended varying speed of light model using Pantheon+ dataset

The principles of physics should remain unchanged regardless of the chosen units or measurement instruments. This is exemplified by dimensionless quantities such as the fine structure constant, $\alpha$, which retains its value across different unit systems, as demonstrated in the Standard Model of particle physics. Conversely, dimensional constants like $\hbar$, $c$, $G$, $e$, and $k$ are human conventions whose numerical values vary depending on the units employed. Consequently, only dimensionless constants can be considered truly fundamental. Therefore, investigating potential variations over time in dimensionless fundamental constants is a valid scientific pursuit, while variations in dimensional constants like $c$ or $G$ depend on the chosen units and may result in discrepancies among observers. This perspective is supported by various studies Duff:2001ba ; Uzan:2002vq ; Ellis:2003pw ; Duff:2014mva , albeit within the framework of a static Universe or one existing at the present epoch Lee:2020zts ; Lee:2023bjz ; Lee:2024par .

In the Robertson-Walker (RW) metric, the expanding Universe is depicted as progressing from one hypersurface to another, with the scale factor increasing naturally, resulting in the cosmological redshift of various physical quantities, such as mass density, wavelength, and temperature. One can derive this metric from the cosmological principle and Weyl’s postulate. The redshift, defined as a function of the time-evolving cosmic scale factor $a(t)$, yields positive values for $z$ in our expanding Universe. Estimating the redshift of a galaxy involves analyzing the emission lines emitted by glowing gas within the galaxy. For example, the H$\alpha$ line, a red Balmer line of neutral hydrogen, has a rest wavelength of $6562$Å. If the observed wavelength of this line presently measures $8100$Å, it indicates that the galaxy is positioned at $z=0.234$ (i.e., $a=0.81$). Consequently, in an expanding Universe, the value of dimensional quantities, such as wavelength, varies depending on the observation time (i.e., cosmic time). Additionally, it has been observed that the temperature of the CMB decreases with the age of the Universe, scaling inversely with the scale factor as $T=T_{0}a^{-1}$ Islam01 ; Narlikar02 ; Hobson06 ; Roos15 . However, the RW model lacks a mechanism to determine cosmological time dilation (TD). The standard model of cosmology (SMC) makes an additional assumption, asserting that the speed of light is constant ($c$). This assumption arises from the dependence of SMC on general relativity (GR), which assumes that $c$ is invariant. As a result, the cosmological TD between two hypersurfaces at $t=t_{1}$ and $t=t_{2}$ vary in proportion to the inverse of the scale factors $a(t)$ at those specific times. However, if one allows a time-varying speed of light as proposed in this paper, this relationship may not hold anymore. Thus, establishing TD depends on experimental observations. Given the theoretical absence of cosmological TD, the relationship can be considered as a function of the scale factor, allowing the speed of light to be expressed as $c(t_{1})=(a(t_{1})/a(t_{2}))^{b/4}c(t_{2})$ in the meVSL model, where $b$ is a constant Lee:2020zts ; Lee:2023bjz ; Lee:2024par . Also, to preserve the Einstein Field Equations (EFE), the Einstein constant, denoted as $\kappa=8\pi G/c^{4}$, must remain constant even if both $c$ and $G$ undergo cosmological evolution. In the meVSL model, where $c=c_{0}a^{b/4}$, this implies $G=G_{0}a^{b}$ where $c_{0}$ and $G_{0}$ represent the current values of the speed of light and the gravitational constant, respectively Lee:2020zts ; Lee:2023bjz ; Lee:2024par .

In addition to models like meVSL, there is a model such as Co-varying Physical Couplings (CPC) Cuzinatto:2022mfe ; Cuzinatto:2022vvy ; Cuzinatto:2022dta , where physical constants vary with cosmic time. The CPC retains the EFEs with $G$, $c$, and $\Lambda$ treated as functions of spacetime. The interaction between the Bianchi identity and the requirement of stress-energy tensor conservation complicates the potential variations of these constants, which are constrained to co-vary according to the General Constraint (GC). Unlike meVSL, this model includes the dynamics of physical constants through the adoption of GC.

Special relativity (SR)’s universal Lorentz covariance, grounded in Minkowski spacetime, adequately upholds its principles Morin07 . In contrast, within GR, an inertial frame (IF) refers to one that is freely falling. While Lorentz invariant (LI) spacetime intervals can be established between events, the definition of a global time in GR is impeded by the absence of a universal IF. However, a global time can be delineated for the Universe, satisfying the Cosmological Principle (CP), enabling a foliation of spacetime into non-intersecting spacelike 3D surfaces. This description pertains to the Universe modeled by the RW metric Islam01 ; Narlikar02 ; Hobson06 ; Roos15 . The LI varying speed of light (VSL) model is plausible if the speed of light, denoted as $c$, remains locally constant at each given epoch but varies in cosmic time Lee:2020zts ; Lee:2023bjz ; Lee:2024par . In other words, in an expanding Universe, if the speed of light is expressed as a function of the scale factor, $c[a]$, then although its value changes akin to wavelengths at different epochs, such as $a_{1}$ and $a_{2}$, it maintains a constant local value at each epoch, ensuring LI and thereby preserving the validity of quantum mechanics and electromagnetism in accordance with SR during every epoch. Thus, the speed of light could change over cosmic time or remain constant within an expanding Universe, contingent upon its relationship with cosmological TD. Without explicit laws governing TD, the speed of light in the RW metric could potentially vary with cosmological time, akin to other physical properties such as mass density, temperature, and fundamental constants like the Planck constant Lee:2020zts ; Lee:2022heb ; Lee:2023bjz ; Lee:2024par . However, to construct a coherent model around this concept, the varying speed of light (VSL) needs to be incorporated into the Einstein field equations (EFEs) and resolved for solutions. Previous studies, notably within the framework of the minimally extended VSL (meVSL) model, have addressed such scenarios Lee:2020zts ; Lee:2023bjz ; Lee:2024par . One can freely select a local value for the speed of light as it merely entails a scaling of length units. As long as this local value remains constant on a given time hypersurface, it satisfies the SR to be consistent with local physics laws. Newton’s gravitational constant, $G$, could potentially vary. To avoid trivial unit rescaling, one must examine the concurrent variation of $c$, $G$, and possibly other physical constants Lee:2020zts ; Lee:2023bjz ; Lee:2024par .

Numerous endeavors have aimed to measure cosmological TD. One approach involves directly observing TD by analyzing the decay time of distant supernova (SN) light curves and spectra Leibundgut:1996qm ; SupernovaSearchTeam:1997gem ; Foley:2005qu ; Blondin:2007ua ; Blondin:2008mz . Another method entails measuring TD by examining the stretching of peak-to-peak timescales of gamma-ray bursters (GRBs) Norris:1993hda ; Wijers:1994qf ; Band:1994ee ; Meszaros:1995gj ; Lee:1996zu ; Chang:2001fy ; Crawford:2009be ; Zhang:2013yna ; Singh:2021jgr . Efforts have also been made to detect the TD effect in the light curves of quasars (QSOs) located at cosmological distances Hawkins:2001be ; Dai:2012wp . So far, no definitive detection of cosmic TD has been achieved, with conflicting results from different measurements.

The assessment of whether the speed of light alters due to the Universe’s expansion hinges solely on observation. Therefore, it is vital to observe how any changes in the speed of light affect cosmic scales. To achieve this, diverse observational methods have been utilized Lee:2020zts ; Lee:2024par .

First, there is the cosmic distance duality relation CDDR method among these methods. Etherington’s theorem, derived from the geodesic deviation equation, establishes reciprocity between the area distances of galaxies and observers, linked by the redshift factor $(1+z)$ under geometric invariance Etherington:1933pm . This theorem, applicable in spacetimes where photons follow null geodesics, forms the basis for the CDDR. By relating area distances to angular and luminosity distances, the CDDR offers a means to test the validity of the SMC  Ellis:1998ct ; Ellis:2007grg . Various tests of the CDDR using astrophysical and cosmological observations have been conducted to constrain VSL models More:2008uq ; Nair:2012dc ; Wu:2015prd ; Ma:2016bjt ; Martinelli:2020hud ; Holanda:2012ia ; Qi:2014zja ; Salzano:2014lra ; Lee:2021xwh ; Rodrigues:2021wyk ; Cuzinatto:2022mfe . Our analysis of the meVSL model suggests a potential deviation from the standard CDDR based on current data. However, with different priors for certain cosmological parameters, the current dataset aligns with the SMC, indicating no deviation from the expected CDDR Lee:2020zts ; Lee:2021xwh . Therefore, acquiring more precise data is essential to thoroughly investigate any deviations from the established CDDR and reaffirm the viability of the meVSL model.

Second, there is the Cosmic Chronometer (CC) method. The CC method involves observing two passively evolving galaxies, typically elliptical galaxies, assumed to have formed at the same cosmic epoch but observed at different redshifts  Jimenez:2001gg . This approach offers a model-independent means of measuring the Hubble parameter, $H(z)$, as a function of redshift, derived from spectroscopic surveys with high precision ($\sigma_{z}\leq 0.001$). The expansion rate of the meVSL model, or the Hubble parameter $H(z)$, is determined from the differential age evolution of the Universe $\Delta t$ within a given redshift interval ($dz$) Lee:2020zts ; Lee:2023bjz ; Lee:2024par

where $E^{(\textrm{SMC})}$ is the normalized Hubble parameter in the SMC model. Various methods exist for measuring $\Delta t$, including predicting its age based on the chemical composition of a stellar population or utilizing spectroscopic observables like the $4000$ Åbreak, known to be linearly related to the age of the stellar population Moresco:2010wh . Unlike many cosmological measurements that rely on integrated distances, the CC method determines the expansion rate $H(z)$ as a function of the redshift–time derivative $dz/dt$, making it a potent tool for testing different cosmological models Wei:2016ygr ; Ratsimbazafy:2017vga ; Wei:2019uss ; Moresco:2020fbm ; Vagnozzi:2020dfn ; Dhawan:2021mel ; Borghi:2021zsr ; Borghi:2021rft ; Banerjee:2022ynv ; Jalilvand:2022lfb ; Asimakis:2022jel ; Kumar:2022ypo ; Li:2022cbk .. This method proves particularly valuable for investigating VSL models Rodrigues:2021wyk . Both minimum $\chi^{2}$-analysis and maximum-likelihood analysis using the most recent CC data have been performed to constrain the parameter b of the meVSL model. However, the precision of the current CC data is insufficient to distinguish between the meVSL model and the SMC Lee:2023rqv .

Third, there are cosmological TDs observed from supernovae. The luminosity curve (LC) of a supernova (SN) offers valuable insights into its evolution, aiding in the classification and understanding of its properties. LC analysis helps determine crucial parameters such as peak luminosity, time to peak brightness, and rate of decline, particularly for SNe Ia, essential as standard candles in cosmology. Comparing LCs across distances enables investigation into cosmic expansion and TD, contributing to significant discoveries like accelerated expansion and dark energy. Wilson’s method involves comparing LCs of nearby and distant SNe, revealing TD effects due to light travel time through space Wilson:39 . This information is crucial for studying the Universe’s expansion rate and testing cosmological models, involving data collection, mathematical modeling, and comparison of observed TD with theoretical predictions. We derived a TD formula within the meVSL model, analyzing data from $13$ high-redshift SNe Ia to determine the exponent $b$ as $b=0.198\pm 0.415$ Blondin:2008mz ; Lee:2023ucu . While less precise than CC, our analysis indicates consistency with both SMC and the meVSL model. Thus, distinguishing between the two based on SNe TD data is challenging.

Fourth, there is the cosmography method. It employs a kinematic description of the Universe’s evolution based on the cosmological principle, emphasizing the dynamics of cosmic expansion. As a model-independent framework, it offers flexibility in managing cosmological parameters, allowing for generalized analysis unconstrained by preconceived models. By focusing on the later stages of cosmic evolution and utilizing Taylor expansions tailored to the observable domain where $z\ll 1$, cosmography imposes constraints on the present-day Universe. We adapt late-time cosmography to incorporate meVSL models SLee:24CG .

Lastly, in this paper, we aim to discuss the constraint on any evidence of the cosmic variation of the speed of light using Pantheon$+$ data Scolnic:2017caz . The Pantheon compilation comprises a total of $1048$ SNe Ia spanning a redshift range of $0.01$ to $2.3$. It includes $365$ spectroscopically confirmed SNe Ia from the Pan-STARRS1 (PS1) Medium Deep Survey, combined with a subset of $279$ PS1 SNe Ia (with redshifts ranging from $0.03$ to $0.68$) with reliable distance estimates obtained from various sources such as SDSS, SNLS, and HST samples. Cosmological models fitted to minimize the $\chi^{2}$ for flat $\Lambda$CDM and $\omega$CDM models, without accounting for systematic uncertainties on $\Omega_{m0}$, yield values of $0.284\pm 0.012$ and $0.350\pm 0.035$, respectively. The $1$-$\sigma$ constraint on $\omega$ for the $\omega$CDM model is $-1.251\pm 0.144$. The Pantheon dataset allows for a precise constraint of approximately 10% on $\Omega_{m0}$ and 12% on $\omega$ for the flat $\omega$CDM model, and about 4% on $\Omega_{m0}$ for the flat $\Lambda$CDM model.

In Section II, we give a brief overview of the meVSL model, delving into its theoretical foundations and implications for cosmological phenomena. Specifically, we explore the luminosity distance predictions derived from this model, shedding light on its unique characteristics and potential effects for observational data. In Section III, we elucidate the statistical methods employed in this paper. Additionally, a portion of the Mathematica file used for this purpose is included in the appendix. In Section IV, we provide a brief explanation of the temporal variations of the speed of light and Newton’s constant used in this paper, along with a concise interpretation for the $\Lambda$CDM model for comparison. The investigation into models capable of implementing variations in the speed of light across various $\omega$CDM models is conducted in Section V. Research on potential models within the CPL framework of the meVSL model is performed in Section VI. In Section VII, we offer various possible constraints on the temporal variations of the speed of light and Newton’s constant within the meVSL model, comparing them with existing observational constraints. Finally, in Section VIII, we distill our findings and insights into a comprehensive summary, drawing actionable conclusions and outlining avenues for future research and exploration in the realm of cosmology and fundamental physics.

The conceptualization of the four-dimensional spacetime of a spatially homogeneous and isotropic, expanding universe entails envisioning it as a seamless continuum composed of homogeneous and isotropic spatial hypersurfaces evolving dynamically over cosmic time Islam01 ; Narlikar02 ; Hobson06 ; Roos15 . At the heart of this framework lies the RW metric, elegantly expressed by

where $c(t)$ denotes the speed of light and $\gamma_{ij}(\vec{x})$ signifies the time-independent spatial metric defining the hypersurface, while $a(t)$ governs the scale factor dictating the relationship between physical distance and comoving distance. We adopt that the speed of light is a function of cosmic time, deviating from the conventional RW metric. We already show that there is no contradiction in this assumption as long as we adopt the CP and Weyl’s postulate Lee:2023bjz ; Lee:2024par . The derivation of redshift involves utilizing the geodesic equation for a light wave, where $ds^{2}=0$ as represented by Equation (2). The consistency of $dl_{3\textrm{D}}$ over time is maintained by exclusively employing comoving coordinates. Building upon this foundation, we arrive at the expression for radial light signals:

where $dt_{i}=1/\nu(t_{i})$ denotes the time interval between successive crests of light at $t_{i}$ (i.e., the inverse of the frequency $\nu_{i}$ at $t_{i}$) and $b$ characterizes the deviation of $c$ from the constant value. Thus, the RW metric naturally allows the VSL models if we remove the traditional assumption on the cosmological TD Lee:2020zts ; Lee:2023bjz ; Lee:2024par .

Additionally, the introduction of $\chi=D_{\text{C}}$ as the comoving distance as

where $d_{\text{L}}$ is the so-called the Hubble free luminosity distance and $f_{k}(\chi)=\sinh(\sqrt{-k}\chi)/\sqrt{-k}=D_{\text{M}}$ as the transverse comoving distance

with $c_{0}=2.9979\times 10^{5}$ km/s and $H_{0}=100\textrm{h}$ km/Mpc/s representing the present values of the speed of light and the Hubble parameter, respectively SLee:24CG ; Tiesinga:2022 . In a recent local measurement by the SH0ES collaboration, utilizing Cepheid-calibrated SNeIa, the Hubble constant was reported to be approximately $73$ km/s/Mpc Breuval:2024lsv , contrasting with the prediction of around $67$ km/s/Mpc by the standard $\Lambda$CDM model based on observations of the CMB Tristram:2023haj .

Within this conceptual framework, the timelike worldlines of constant space delineate the threading, while the spacelike hypersurfaces of constant time define the slicing within the four-dimensional spacetime. Each spacelike threading corresponds to a homogeneous universe at a given epoch, with the slicing being orthogonal to these hypersurfaces, offering a natural arrangement conducive to the definition of constant physical quantities such as density, temperature, and the speed of light on each spacelike hypersurface Islam01 ; Narlikar02 ; Hobson06 ; Roos15 .Thus, our choice of coordinates emerges organically, rendering alternative considerations unnecessary. Furthermore, the derivation of the Ricci tensors and Ricci scalar curvature from the provided metric in Eq. (2) further enriches the understanding of the underlying spacetime dynamics and its mathematical representation Lee:2020zts .Specifically, the expressions for $R_{00}$ and $R_{ii}$ are given by

In cosmology, one treats matter as a perfect fluid, defined by its total mass density $\rho$ and isotropic pressure $P$. This density is its rest frame mass density for a perfect fluid. Within the framework of GR, the stress-energy tensor describes this perfect fluid, given by

where $U^{\mu}$ represents its four-velocity. When the fluid is in motion, a set of fundamental (i.e. comoving) observers is considered comoving with it, characterized by a four-velocity denoted as $U^{\mu}=(c,0,0,0)$  Islam01 ; Hobson06 . Once one establishes the metric and the stress-energy tensor, the subsequent step involves solving the Einstein Field Equations (EFEs) to elucidate the dynamics of the scale factor in the metric. These equations govern the dynamics of expansion, including the speed and acceleration of the Universe’s expansion as observed between two fundamental observers. Thus, the energy-momentum tensor of the $i$-component perfect fluid with the equation of state $\omega_{i}$ is given by Lee:2020zts

where $c_{0}$ is the present value of the speed of light, $\rho_{i0}$ is the present value of mass density of the i-component, and we use $a_{0}=1$. One can derive Friedmann equations including the general DE from Eqs. (6)-(9)

where $\rho_{\textrm{r}0}$, $\rho_{\textrm{m}0}$, and $\rho_{\textrm{DE}0}$ represent the mass-density of radiation (photon and neutrino), matter (baryon and DM), and dark energy, respectively, at the present epoch Lee:2020zts ; Lee:2023bjz . The Chevallier-Polarski-Linder (CPL) parametrization, which assumes $\omega_{\textrm{DE}}$ to be a linear function of the scale factor $a$, is presented in Eq. (14) Chevallier:2000qy ; Linder:2002et . We also define the $\omega$ model when $\omega_{a}=0$.

We examine constraints on the evidence for cosmic variation in the speed of light using the Pantheon$+$ dataset Scolnic:2017caz . The Pantheon compilation includes $1048$ SNe Ia covering a redshift range $0.01\leq z\leq 2.3$. This dataset incorporates $365$ spectroscopically confirmed SNe Ia from the Pan-STARRS1 (PS1) Medium Deep Survey, along with a subset of $279$ PS1 SNe Ia (with redshifts ranging from $0.03$ to $0.68$) with reliable distance estimates derived from various sources, including SDSS, SNLS, and HST samples. To determine cosmological parameters using $H(z)$, higher-redshift SNe Ia are employed, and the degenerate parameters $\mathcal{M}$ are typically marginalized as nuisance parameters SDSS:2014iwm ; Pan-STARRS1:2017jku . For example, Eq. (25) is used to construct and minimize $\bar{\chi}^{2}(\Omega_{\text{m}0})\equiv\int d\mathcal{M}\chi^{2}(\mathcal{M}\,,\Omega_{\text{m}0})$ where the degenerate combination provided in the same equation Arjona:2018jhh ; Kazantzidis:2020tko . However, marginalizing the parameter $\mathcal{M}$ can result in the loss of valuable physical information regarding potential spatial variations of $H_{0}$ and/or temporal variations of the absolute magnitude $\mathcal{M}$. For instance, an $\mathcal{M}$ value that evolves with redshift, resulting in low $\mathcal{M}$ values at low $z$, could indicate either higher local values of $H_{0}$ due to a local matter underdensity or lower values of the absolute magnitude $\mathcal{M}$ in recent cosmological times, possibly caused by a time variation of Newton’s constant. Specifically, since we are studying the potential variation of the speed of light with cosmic time, we do not marginalize other parameters in this manuscript when we use the minimal $\chi^{2}$ method Shanks:2018rka ; Shanks:2019inu ; Lukovic:2019ryg ; Bohringer:2019tyj . The chi-squared ($\chi^{2}$) represents a weighted summation of squared deviations, given by