Cosmography of the Minimally Extended Varying Speed of Light Model

1. Introduction

One can characterize the expansion of the Universe by the dimensionless scale factor $a\left(t\right)$. The evolution of the scale factor is determined by the Friedmann equations of general relativity, specifically for a spatially homogeneous and isotropic Universe. Recent empirical research indicates an accelerated expansion of the Universe. Essentially, this implies that the second derivative of the scale factor, denoted as $\ddot{a}\left(t\right)$, is positive, or equivalently, that the first derivative $\dot{a}\left(t\right)$ progressively increases over time. Moreover, theoretically, the scale factor can be Taylor expanded from the current epoch to a nearby time, and the dimensionless coefficients of these expansions are referred to as cosmographic parameters. Although the scale factor is not directly observable, it is possible to constrain cosmographic parameters based on cosmological observations. This approach, utilizing the kinematics of the scale factor, is called cosmography, allowing for the study of the cosmological evolution of the scale factor and, conversely, determining other physical quantities that govern the dynamics of the scale factor [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].

Relying solely on the cosmological principle, the method of cosmography serves as a kinematic description of the Universe’s evolution, specifically focusing on the dynamics of cosmological expansion. Cosmography provides a versatile framework for handling cosmological parameters, relying solely on the kinematics of the Universe. This model-independent approach eliminates the need for dependence on any specific theoretical model, enabling a more generalized analysis. Notably, it abstains from defining any model a priori, thereby avoiding the necessity to postulate gravity for determining the Universe’s dynamics. The effectiveness of cosmography is demonstrated by its ability to choose models consistent with cosmological observations. Cosmography primarily concentrates on the later phases of the Universe’s development. The Taylor expansions associated with cosmography typically pertain to the observable domain where $z\ll 1$, facilitating the establishment of constraints on the current Universe. In our discussion, we explore the modification of late-time cosmography to apply the varying speed of light models.

As evident from the above equations, the effects of the meVSL model can manifest in each cosmographic parameter. It can potentially influence the values of the density contrasts in the $\Lambda$CDM model. In the subsequent follow-up paper, we will discuss the impact of this varying speed-of-light theoretical model on the values of density contrasts, comparing it with the standard model using several cosmological observational datasets. In the following section, we compare the representations of cosmography in the meVSL model for each observable with those in the standard model, highlighting differences. In Section 3, we investigate the expression of the luminosity distance within the framework of the meVSL model. Section 4 describes how cosmographic parameters are represented by density contrasts when applying the LambdaCDM model. The final section concludes with a discussion.

2. Cosmolography of varying speed of light models

The Friedmann-Lemaître-Robertson-Walker (FLRW) metric of varying speed of light (VSL) models is given by

$$\begin{array}{c}\hfill d{s}^2=-c{\left(t\right)}^{2}d{t}^2+a{\left(t\right)}^2\left[\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\theta }^2+r^2{sin}^2\theta d{\varphi }^2\right],\end{array}$$

(1)

where k is a constant, related to curvature of space and $a\left(t\right)$ is the scale factor. It describes the evolution of the Universe. The coordinates $(t,r,\theta ,\varphi )$ of the RW metric are called comoving coordinates. The spatial coordinates of objects remain the same, but their physical (proper) distance grows with time as space expands. The cosmic time gives the time a comoving observer measures at $(r,\theta ,\varphi )=$ constant. Let the photon be emitted at time $t_{*}$ with $r=0$ and absorbed by an observer located at $r=r_0$ at the present epoch $t_0$. One can express this time as $t_{*}=t_0-\delta t$, where the time difference $\delta t$ is significantly smaller than the current cosmic time $t_0$. This relationship implies that the emitting galaxy is close to our location, and the time it takes for the photon to reach us is negligible compared to the current cosmic time. Thus, one can Taylor expand the scale factor about $t_0$

$$\begin{array}{cc}\hfill a\left(t_{*}\right)& =a\left(t_0\right)+\sum _{n=1}^{\infty }\frac{1}{n!}\frac{d^{n}a\left(t\right)}{d{t}^n}{|}_{t=t_0}\left(t_{*}-t_0\right)=a_0-{\dot{a}}_0\delta t+\frac{1}{2!}{\ddot{a}}_0\delta t^2-\frac{1}{3!}{\stackrel{⃛}{a}}_0\delta t^3+\frac{1}{4!}{a}_0^{\left(4\right)}\delta t^4+\cdots \hfill \\ \hfill & \equiv 1-H_0\delta t-\frac{1}{2}{q}_0{H}_0^2\delta t^2-\frac{1}{3!}{j}_0{H}_0^3\delta t^3+\frac{1}{4!}{s}_0{H}_0^4\delta t^4+\cdots ,\hfill \end{array}$$

(2)

where $a\left(t_0\right)\equiv a_0=1$ denotes the value of the scale factor at the present epoch. The so-called cosmographic parameters $H_0$, $q_0$, $j_0$, and $s_0$ are Hubble, deceleration, jerk, and snap parameters, respectively. They are dimensionless. In cosmology, the Hubble, deceleration, jerk, and snap parameters characterize the first four-time derivatives of the scale factor. These parameters elucidate the rate and acceleration/deceleration of cosmic expansion, serving as valuable tools for comprehending the behavior and expansion history of the Universe. Widely employed in observational data analysis and cosmological research, they aid in probing cosmological models and measuring significant quantities such as the density of matter in the Universe and the equation of state for dark energy. While the scale factor a is an unobservable quantity, the cosmological redshift z is an observable physical quantity. Moreover, one can also define the cosmological redshift in the vicinity of small values for $\delta z=z_{*}-z_0=z_{*}$ using these parameters as

$$\begin{array}{c}\hfill z_{*}=\frac{a_0}{a(t_0-\delta t)}-1\simeq H_0\delta t+\frac{1}{2!}\left(2+q_0\right){\left(H_0\delta t\right)}^2+\frac{1}{3!}\left(6+6q_0+j_0\right){\left(H_0\delta t\right)}^3+\frac{1}{4!}\\ \hfill \left(24+36q_0+6q_0^2+8j_0-s_0\right){\left(H_0\delta t\right)}^4+\cdots ,\end{array}$$

(3)

where $z_0$ is the present cosmological redshift and equals 0.

Now, let’s apply cosmographic parameters to observable quantities, starting with the Hubble parameter. The Hubble parameter in the Taylor series is

$$\begin{array}{cc}\hfill H\left(\delta z\right)& \simeq H\left(z_0\right)+\frac{dH}{dz}{|}_{z_0}\delta z+\frac{1}{2!}\frac{d^{2}H}{d{z}^2}{|}_{z_0}\delta z^2+\frac{1}{3!}\frac{d^{3}H}{d{z}^3}{|}_{z_0}\delta z^3+\cdots \hfill \\ \hfill & \simeq H_0\left[1+\left(1+q_0\right)\delta z+\frac{1}{2!}\left(j_0-q_0^2\right)\delta z^2+\frac{1}{3!}\left(3q_0^2+3q_0^3-4q_0{j}_0-3j_0-s_0\right)\delta z^3+\cdots \right].\hfill \end{array}$$

(4)

This result holds consistently for both the standard cosmology model (SCM) and VSL models.

Considering a null geodesic in an FLRW metric along the line of sight (LOS), the total LOS comoving distance can be obtained using Equation (1)

$$\begin{array}{c}\hfill D_c\left(r_0\right)\equiv {\int }_0^{r_0}\frac{dr}{\sqrt{1-k{r}^2}}\equiv S_k^{-1}\left[r_0\right]=\left\{\begin{array}{cc}\frac{1}{\sqrt{k}}{sin}^{-1}\left[\sqrt{k}{r}_0\right]& \hfill ,k>0\\ r_0& \hfill ,k=0\\ \frac{1}{\sqrt{\left|k\right|}}{sinh}^{-1}\left[\sqrt{\left|k\right|}{r}_0\right]& \hfill ,k<0\end{array}\right\}={\int }_{t_{*}}^{t_0}\frac{c\left(t\right)}{a\left(t\right)}dt={\int }_{z_0}^{z_{*}}\frac{c\left(z\right)}{H\left(z\right)}dz.\end{array}$$

(5)

Notably, in contrast to the SCM, the comoving distance $D_c$ is influenced by the z-dependent speed of light. Consequently, the transverse comoving distance $r_0$ can be derived from Equation (5)

$$\begin{array}{c}\hfill r_0\left(z_{*}\right)\equiv S_k\left[D_c\left(r_0\right)\right]=\left\{\begin{array}{cc}\frac{1}{\sqrt{k}}sin\left[\sqrt{k}{\int }_{z_0}^{z_{*}}\frac{cdz}{H\left(z\right)}\right]& \hfill ,k>0\\ {\int }_{z_0}^{z_{*}}\frac{cdz}{H\left(z\right)}=D_c& \hfill ,k=0\\ \frac{1}{\sqrt{\left|k\right|}}sinh\left[\sqrt{\left|k\right|}{\int }_{z_0}^{z_{*}}\frac{cdz}{H\left(z\right)}\right]& \hfill ,k<0\end{array}\right..\end{array}$$

(6)

Given that the acceleration of the Universe is a relatively recent phenomenon, we can focus our analysis on the vicinity of small values for the redshift interval $\delta z=z_{*}-z_0=z_{*}$ in Equation (6). For a short redshift interval $\delta z\equiv z_{*}\simeq 0$, one can expand the Taylor series of $r_0$ around $z_0$

$$\begin{array}{c}\hfill r_0\left(\delta z\right)\simeq \left\{\begin{array}{cc}{\int }_{z_0}^{z_0+\delta z}\frac{c\left(z\right)dz}{H\left(z\right)}-\frac{k}{3!}{\left({\int }_{z_0}^{z_0+\delta z}\frac{c\left(z\right)dz}{H\left(z\right)}\right)}^3& \hfill ,k>0\\ {\int }_{z_0}^{z_0+\delta z}\frac{c\left(z\right)dz}{H\left(z\right)}=r_0& \hfill ,k=0\\ {\int }_{z_0}^{z_0+\delta z}\frac{c\left(z\right)dz}{H\left(z\right)}-\frac{k}{3!}{\left({\int }_{z_0}^{z_0+\delta z}\frac{c\left(z\right)dz}{H\left(z\right)}\right)}^3& \hfill ,k<0\end{array}\right..\end{array}$$

(7)

Measuring distances to cosmological objects stands as the primary method for probing the cosmic metric and deciphering the expansion history of the Universe. In the meVSL model, the comoving distance at redshift $z_{*}$ is given by

$$\begin{array}{cc}\hfill D_c\left(r_0\right)& =\hfill \\ \hfill {\int }_{z_0}^{z_0+\delta z}\frac{c\left(z\right)dz}{H\left(z\right)}\simeq {\int }_{z_0}^{z_0+\delta z}\frac{c_0}{H_0}dz\left[1+B_0\delta z+C_0\delta z^2+D_0\delta z^3+\cdots \right]=\\ \hfill \frac{c_0}{H_0}\delta z\left[1+\frac{B_0}{2}\delta z+\frac{C_0}{3}\delta z^2+\frac{D_0}{4}\delta z^3+\cdots \right],\end{array}$$

(8)

where

$$\begin{array}{cc}\hfill B_0& =-\frac{H_0^{\prime }}{H_0}+\frac{c_0^{\prime }}{c_0}=-\left(q_0+1+\frac{b}{4}\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill C_0& =\frac{1}{2!}\left[-\frac{H_0^{″}}{H_0}+2{\left(\frac{H_0^{\prime }}{H_0}\right)}^2+\frac{c_0^{″}}{c_0}-2\frac{c_0^{\prime }}{c_0}\frac{H_0^{\prime }}{H_0}\right]=\frac{1}{2}\left[-j_0+3q_0^2+4q_0+2+\frac{b}{4}\left(\frac{b}{4}+2q_0+3\right)\right],\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill D_0& =\frac{1}{3!}\left[-\frac{H_0^{‴}}{H_0}+6\frac{H_0^{″}}{H_0}\frac{H_0^{\prime }}{H_0}-6{\left(\frac{H_0^{\prime }}{H_0}\right)}^3+\frac{c_0^{‴}}{c_0}-3\frac{c_0^{″}}{c_0}\frac{H_0^{\prime }}{H_0}-3\frac{c_0^{\prime }}{c_0}\left(\frac{H_0^{\prime \prime }}{H_0}-2{\left(\frac{H_0^{\prime }}{H_0}\right)}^2\right)\right]\hfill \\ \hfill & =\frac{1}{3!}\left[s_0+9j_0+10q_0{j}_0-15q_0^3-27q_0^2-18q_0-6+\frac{b}{4}\left(\left(3j_0-9q_0^2-15q_0-11\right)-\frac{3}{4}(q_0+2)b-\frac{1}{16}{b}^2\right)\right].\hfill \end{array}$$

(11)

The transverse comoving distance, denoted as $r_0\left(\delta z\right)$, can be derived from Equations (7) and (8)

$$\begin{array}{c}\hfill r_0\left(\delta z\right)\simeq \frac{c_0\delta z}{H_0}\left[1+\frac{1}{2!}{B}_0\delta z+\frac{1}{3!}\left(2C_0+{\Omega }_k\right)\delta z^2+\frac{1}{4!}\left(6D_0+6B_0{\Omega }_k\right)\delta z^3+\cdots \right].\end{array}$$

(12)

The luminosity distance ($d_L$) for an object is expressed as $d_L={(L/4\pi F)}^{1/2}$, where L denotes the luminosity of the observed object (assumed to be known for high-redshift supernovae), and F represents the energy flux received from the object. This formula describes the luminosity distance in a dynamic, homogeneous, isotropic spacetime. Expressing the luminosity distance as a function of $\delta z$ involves Equations (3) and (12)

$$\begin{array}{c}\hfill d_L\left(\delta z\right)=(1+\delta z)r_0\left(\delta z\right)\simeq \frac{c_0\delta z}{H_0}[1+\frac{1}{2!}(2+B_0)\delta z+\frac{1}{3!}\\ \hfill \left(3B_0+2C_0+{\Omega }_k\right)\delta z^2+\frac{1}{4!}(8C_0+6D_0+(4+6B_0{\Omega }_k)\delta z^3+\cdots ]\equiv \frac{D_L\left(\delta z\right)}{H_0},\end{array}$$

(13)

introducing the concept of the Hubble-free luminosity distance $D_L$, which remains independent of $H_0$. These results remain consistent with those found in references [1,2,12], barring the impact of the time-varying speed of light. Consequently, by eliminating the derivation terms associated with c, the obtained results match those of the reference. Notably, our approach doesn’t necessitate relying on lookback time to arrive at these outcomes.

3. Observation

The distance modulus ($\mu$) is the logarithmic measure of the ratio between an astronomical object’s intrinsic and observed brightness, used for determining its distance. When dealing with standard candles like Cepheid variables or Type Ia supernovae (SNe Ia), the distance modulus is widely used in astronomy to quantify the distance to celestial objects. It represents the difference between the apparent magnitude (i.e., observed brightness), $m\left(z\right)$ (preferably corrected for interstellar absorption effects), and the absolute magnitude (i.e., intrinsic brightness), M, of a standard candle. The luminous distance measured in megaparsecs (Mpc), $d_L\left(z\right)$, is connected to the distance modulus through

$$\begin{array}{c}\hfill \mu \left(z\right)=m\left(z\right)-M=5{log}_{10}\left[\frac{d_L\left(z\right)}{\left(\mathrm{Mpc}\right)}\right]+25.\end{array}$$

(14)

By replacing the luminosity distance in the above equation with the Hubble-free luminosity distance, $D_L\left(z\right)$, defined in Equation (13), we obtain

$$\begin{array}{c}\hfill \mu \left(z\right)=5{log}_{10}\left[\frac{D_L\left(z\right)}{H_0\left(\mathrm{Mpc}\right)}\right]+25=5{log}_{10}\left[\frac{D_L\left(z\right)}{\left(\mathrm{km}/\mathrm{s}\right)}\right]-5{log}_{10}\left[\frac{H_0}{\left(\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\right)}\right]+25.\end{array}$$

(15)

Equations (14) and (15) enable us to express the Hubble parameter in units of (km/s/Mpc) as

$$\begin{array}{c}\hfill {log}_{10}\left[\frac{H_0}{\left(\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\right)}\right]=0.2M+{log}_{10}\left[\frac{D_L\left(z\right)}{\left(\mathrm{km}/\mathrm{s}\right)}\right]-0.2m\left(z\right)+5\equiv 0.2M+a_B\left(z\right)+5,\end{array}$$

(16)

where the Hubble intercept parameter, denoted as $a_B$ and employed in the SH0ES (Supernova H0 for the Equation of State) analysis, defines the x-intercept ($m=0$) of a Hubble diagram plotting $0.2m$ against a modified ${log}_{10}{D}_L$ term [31]. At low redshifts ($z\ll 1$), one can determine $a_B$ using Equations (13) and (16),

$$\begin{array}{c}\hfill a_B\approx {log}_{10}[\frac{cz}{\left(\mathrm{km}/\mathrm{s}\right)}(1+\frac{1}{2!}(2+B_0)z+\frac{1}{3!}\left(3B_0+2C_0+{\Omega }_k\right)z^2+\frac{1}{4!}\\ \hfill (8C_0+6D_0+\left(4+6B_0\right){\Omega }_k)z^3+\cdots \left)\right]-0.2m\left(z\right),\end{array}$$

(17)

where $a_B$ is measured from a group of SNe Ia with known redshifts and magnitudes $\left(z,m\left(0\right)\right)$.

4. Parameters

Cosmography proves to be a valuable tool in constraining cosmological parameters [32,33,34,35]. The Hubble parameter in the meVSL model is derived from the modified Friedmann equation, incorporating the $\Lambda$CDM model [36]

$$\begin{array}{cc}\hfill & H^2\equiv {\left(\frac{\dot{a}}{a}\right)}^2=\left[\frac{8\pi G_0}{3}{\rho }_{\mathrm{m}}+\frac{\Lambda c_0^2}{3}-k\frac{c_0^2}{a^2}\right]a^{b/2},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & 1=\left[\frac{8\pi G_0}{3H^2}{\rho }_{\mathrm{m}}+\frac{\Lambda c_0^2}{3H^2}-\frac{k{c}_0^2}{a^2{H}^2}\right]a^{b/2}\equiv \left[{\Omega }_{\mathrm{m}}+{\Omega }_{\Lambda }+{\Omega }_k\right]a^{b/2},\hfill \end{array}$$

(19)

where one defines the density contrasts of each component. Using these density contrasts, the coefficients in Equations (9) to () can be expressed as

$$\begin{array}{cc}\hfill B_0& =-\left(\frac{3}{2}{\Omega }_{\mathrm{m}0}+{\Omega }_{k0}\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill C_0& =\frac{27}{4}{\Omega }_{\mathrm{m}0}^2+3\left(3{\Omega }_{k0}-1\right){\Omega }_{\mathrm{m}0}+\left(3{\Omega }_{k0}-1\right){\Omega }_{k0},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill D_0& =-\frac{405}{8}{\Omega }_{\mathrm{m}0}^3-\frac{81}{2}\left(\frac{5}{2}{\Omega }_{k0}-1\right){\Omega }_{\mathrm{m}0}^2-\frac{3}{2}\left(45{\Omega }_{k0}^2-27{\Omega }_{k0}+2\right){\Omega }_{\mathrm{m}0}-3{\Omega }_{k0}^2\left(5{\Omega }_{k0}-3\right).\hfill \end{array}$$

(22)

One can also use density contrasts of the $\Lambda$CDM model to express cosmographic parameters as

$$\begin{array}{cc}\hfill q_0& =\frac{3}{2}{\Omega }_{\mathrm{m}0}+{\Omega }_{k0}-1-\frac{b}{4},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill j_0& =1-{\Omega }_{k0}+\frac{b}{8}\left[b+6-4\left(3{\Omega }_{\mathrm{m}0}+2{\Omega }_{k0}\right)\right],\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill s_0& =1+\left({\Omega }_{k0}+\frac{3}{2}{\Omega }_{\mathrm{m}0}-2\right){\Omega }_{k0}-\frac{9}{2}{\Omega }_{\mathrm{m}0}+\frac{b}{8}\left[12+8\left({\Omega }_{k0}+3{\Omega }_{\mathrm{m}0}-2\right){\Omega }_{k0}+3{\Omega }_{\mathrm{m}0}\left(6{\Omega }_{\mathrm{m}0}-1\right)\right]\hfill \\ & +\frac{b^2}{16}\left(11-18{\Omega }_{k0}-27{\Omega }_{\mathrm{m}0}\right)+\frac{3}{32}{b}^3.\hfill \end{array}$$

(25)

As evident from the above equations, the effects of the meVSL model can manifest in each cosmographic parameter. It can potentially influence the values of the density contrasts in the $\Lambda$CDM model. In the subsequent follow-up paper, we will discuss the impact of this varying speed-of-light theoretical model on the values of density contrasts, comparing it with the standard model using several cosmological observational datasets.

5. Discussion

It has been well-established that cosmography provides a model-independent means of constraining cosmological parameters. However, in this paper, we demonstrated that if we apply cosmography to a varying speed of light model, the predictions may differ from those offered by the standard cosmological model, depending on cosmological observables. Specifically, we have presented predictions for the minimally extended varying speed of light model in this paper. According to these predictions, cosmological parameters obtained from Hubble parameters or photometric distances can yield different values in the standard model and the meVSL model. We intend to delve deeper into this theoretical possibility through empirical investigations using specific cosmological observations.