Impact of a Locally Measured H₀ on the Interpretation of Cosmic-chronometer Data

Cosmological measurements usually rely on the use of integrated (luminosity or angular) distances, which unavoidably introduces a model dependence in the data themselves. For example, the use of Type Ia supernovae as "standard candles" to measure the distance scale relies on finding the "correct" light curve shape needed to determine their absolute luminosity. The parameters defining this estimator are optimized along with the parameters of the assumed cosmological model, necessitating a recalibration of the data for each and every model being tested (Perlmutter et al. 1998; Riess et al. 1998; Schmidt et al. 1998; Melia 2012; Wei et al. 2013). Galaxies evolving passively on a timescale much larger than their age difference can instead be used to measure the expansion rate H(z) using solely the local redshift–time derivative dz/dt (Jimenez & Loeb 2002). As we shall explain in greater detail below, these are typically massive (with a stellar content >10¹¹ M_⊙) early-type galaxies that formed over ∼90% of their stellar mass at high redshifts (z > 2–3, i.e., before the universe was ∼4 Gyr old) very rapidly (in only ∼0.1–0.3 Gyr) and have experienced only minor subsequent episodes of star formation. They are believed to be the oldest objects at all redshifts (Treu et al. 2005) and if we observe them at z ∼ 1 (i.e., when the universe was ∼7 Gyr old), we see a stellar population that mostly formed during the first ∼2%–10% of the galaxies' evolution. These so-called cosmic chronometers therefore avoid the need of pre-assuming a cosmological model in order to extract the data and can therefore form a powerful discriminant to test different expansion scenarios.

In previous applications, we (Melia & Maier 2013; Melia & McClintock 2015) used the then available catalog of sources to compare two specific models: the R_h = ct universe (Melia 2007, 2016a, 2017; Melia & Shevchuk 2012) and the standard (concordance) ΛCDM model. In these one-on-one comparative tests, the use of information criteria showed that the cosmic chronometers strongly preferred R_h = ct over ΛCDM. The significance of this result is that, whereas the standard model requires a transition from deceleration at large redshifts to an accelerated expansion today, the R_h = ct universe predicts a constant expansion rate at all redshifts.

Recently, however, five new valuable data points were added to the compilation of H(z) measurements at that critical redshift (z ∼ 0.5) where the transition is thought to have occurred (Moresco et al. 2016), thereby improving the statistical significance of the fits in the redshift range 0 ≲ z ≲ 1. Based on a study of this expanded sample, direct evidence for the existence of an epoch of cosmic re-acceleration was claimed to have been seen. To reach this result, however, the locally measured high value of H₀ had to be included in the data set. Yet this determination of the Hubble constant is completely distinct from all other measurements based on the cosmic chronometers themselves. In this paper, we argue that this step has unduly biased this recent analysis and that—contrary to its conclusion—the expanded sample in fact strengthens the case for a constant expansion rate in the redshift range 0 ≲ z ≲ 2.

When one ignores the effects of the local gravitational potential at the position of the observer, the value of H₀ measured directly from the redshift–distance relation of local sources is discrepant at a level of ∼2.4σ (or roughly 9%) relative to the Hubble constant inferred from fitting anisotropies in the cosmic microwave background (CMB) using ΛCDM (Marra et al. 2013; Planck Collaboration 2014). Generally speaking, our observations of the universe are made from a vantage point whose spacetime differs from the mean by a degree that is very difficult to probe with any precision (Valkenburg et al. 2014). Estimates of the cosmic variance created by local inhomogeneities are often based on the Hubble Bubble picture, in which a sphere of matter is carved out of the Friedmann–Robertson–Walker (FRW) background and is compressed or diluted in order to obtain a simple distribution of the ensuing inhomogeneity with a slightly different FRW spacetime.

Detailed analysis of the dependence of the measurement of H₀ on the local gravitational potential has shown that the Hubble Bubble effect can at least partially mitigate the ∼9% tension between the CMB and local Hubble constants (Marra et al. 2013). Compelling observational support for this interpretation of the disparity is provided by the fact that local measurements of H₀ using supernovae within 74 h⁻¹ Mpc (corresponding roughly to z = 0.023) is 6.5% ± 1.8% larger than the value of H₀ measured using supernovae outside of this region. Consequently, one can largely alleviate the Hubble Bubble effect by adopting a minimum redshift of 0.023 in the analysis of the expansion rate.

The idea that we may be living in a local underdense Hubble Bubble has actually been considered since the 1990s (Turner et al. 1992; Suto et al. 1995; Shi et al. 1996; Shi & Turner 1998; Zehavi et al. 1998; Giovanelli et al. 1999). In every case, the implied variation on the local gravitational potential was shown to generate variance of the cosmological parameters, including the local Hubble constant. The general consensus from all of this work is that the locally measured value of H₀, though obtained in a model-independent fashion, is nonetheless high (compared to what one might expect in the context of other cosmological measurements) with systematic uncertainties that are difficult to ascertain with any precision.

For this principal reason, we believe that a test of cosmological models using measurements of H(z) is more robust when only the truly model-independent data obtained with cosmic chronometers are used, without the contamination introduced through the inclusion of a poorly understood measurement of H₀. Our approach is bolstered by the fact that the results of the model comparisons could not be more different when the locally measured H₀ is included versus when it is not, even with exactly the same sample of 30 cosmic-chronometer measurements. But if the expansion rate measured with this technique firmly points to an epoch of cosmic re-acceleration (Moresco et al. 2016), then this should be seen regardless of whether or not H₀ is added to the data set.

To demonstrate this point compellingly, we will here use the most recent sample of 30 cosmic-chronometer measurements to compare seven different models, including ΛCDM. In so doing, we shall re-affirm and strengthen the conclusions drawn from our previous two studies (Melia & Maier 2013; Melia & McClintock 2015)—that measurements of H(z) using cosmic chronometers strongly favor the constant expansion rate predicted by the R_h = ct universe over other models, including ΛCDM. The models we will compare are as follows.

• 1.  
  Standard flat ΛCDM model, in which the matter and dark energy densities are fixed by the condition ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=1-{{\rm{\Omega }}}_{{\rm{m}}}$. Throughout this paper, Ω_i is the energy density ρ_i of species i, scaled to today's critical density, ${\rho }_{c}\equiv 3{c}^{2}{H}_{0}^{2}/8\pi G$. In this model, the two free parameters are H₀ and Ω_m, and

  $$\begin{eqnarray}{H}^{{\rm{\Lambda }}\mathrm{CDM}}(z) & = & {H}_{0}[{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}\\ & & {+{{\rm{\Omega }}}_{{\rm{r}}}{(1+z)}^{4}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}]}^{1/2}.\end{eqnarray} \tag{ 1 }$$
• 2.  
  Concordance oΛCDM model with free parameters H₀, Ω_m, and Ω_Λ. In this model,

  $$\begin{eqnarray}{H}^{o{\rm{\Lambda }}\mathrm{CDM}}(z) & = & {H}_{0}[{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{r}}}{(1+z)}^{4}\\ & & +{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{k}{(1+z)}^{2}]}^{1/2}.\end{eqnarray} \tag{ 2 }$$
• 3.  
  Flat wCDM model, in which the matter and dark energy densities are fixed by the condition ${{\rm{\Omega }}}_{\mathrm{de}}=1-{{\rm{\Omega }}}_{{\rm{m}}}$, but with an unconstrained dark energy equation of state, ${w}_{\mathrm{de}}\equiv {p}_{\mathrm{de}}/{\rho }_{\mathrm{de}}$, where p_de is the pressure. The three free parameters here are H₀, Ω_m, and w_de, with an expansion rate given by

  $$\begin{eqnarray}{H}^{w\mathrm{CDM}}(z) & = & {H}_{0}[{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{r}}}{(1+z)}^{4}\\ & & +\,{{{\rm{\Omega }}}_{\mathrm{de}}{(1+z)}^{3(1+{w}_{\mathrm{de}})}]}^{1/2}.\end{eqnarray} \tag{ 3 }$$
• 4.  
  Einstein-de Sitter space, in which the cosmic fluid contains only matter. H₀ is the sole free parameter, and

  $$\begin{eqnarray}&&{H}^{\mathrm{EdS}}(z)={H}_{0}{(1+z)}^{3/2}.\end{eqnarray} \tag{ 4 }$$
• 5.  
  Friedmann model (which we call Friedmann I) with negative curvature. Here, Ω_m is fixed to be 0.3 and Ω_Λ = 0, implying a curvature term with ${{\rm{\Omega }}}_{{\rm{k}}}=1\ -{{\rm{\Omega }}}_{{\rm{m}}}=0.7$ (Baryshev & Teerikorpi 2012). In this case, H₀ is the sole free parameter, and

  $$\begin{eqnarray}{H}^{\mathrm{Friedmann}{\rm{I}}}(z) & = & {H}_{0}[{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}\\ & & {+{{\rm{\Omega }}}_{{\rm{k}}}{(1+z)}^{2}]}^{1/2}.\end{eqnarray} \tag{ 5 }$$
• 6.  
  Friedmann model (which we call Friedmann II) with negative curvature, but with Ω_m free and Ω_Λ = 0, implying a curvature term with ${{\rm{\Omega }}}_{{\rm{k}}}=1-{{\rm{\Omega }}}_{{\rm{m}}}$. In this case, the two free parameters are H₀ and Ω_m, and

  $$\begin{eqnarray}&&{H}^{\mathrm{Friedmann}\mathrm{II}}(z)={H}^{\mathrm{Friedmann}{\rm{I}}}(z).\end{eqnarray} \tag{ 6 }$$
• 7.  
  R_h = ct universe (an FRW cosmology with zero active mass). In this model, the total equation of state is $\rho \ +3p=0$, with ρ and p as the total energy density and pressure of the cosmic fluid, respectively(Melia 2007, 2016a, 2017; Melia & Shevchuk 2012). In this cosmology, H₀ is the sole free parameter, and

  $$\begin{eqnarray}&&{H}^{{R}_{{\rm{h}}}={ct}}(z)={H}_{0}(1+z).\end{eqnarray} \tag{ 7 }$$

We test these models collectively against measurements of the Hubble constant using the passively evolving galaxies introduced above, based on the observed 4000 Å break in their spectra. For old stellar populations, this break is a discontinuity in the spectral continuum due to metal absorption lines whose amplitude correlates linearly with the age and metal abundance of the stars (Moresco et al. 2016). It is weakly dependent on the star formation history and is basically unaffected by dust reddening. When the metallicity of these stars is known, it is possible to measure the age difference Δt of two nearby galaxies as proportional to the difference in their 4000 Å amplitudes. The slope of this proportionality depends on the metallicity. Then, together with the measured redshift difference Δz of these galaxies, one may find the Hubble constant at the average of their redshifts using the simple relation

$$\begin{eqnarray}&&H(z)=-\displaystyle \frac{1}{(1+z)}\,\displaystyle \frac{{dz}}{{dt}}\approx -\displaystyle \frac{1}{(1+z)}\,\displaystyle \frac{{\rm{\Delta }}z}{{\rm{\Delta }}t}.\end{eqnarray} \tag{ 8 }$$

Of course, there are several factors that may inhibit the accuracy of this procedure, possibly mitigating the value of using differential measurements of age in these systems. Fortunately, extensive tests (Moresco et al. 2016) have shown that the 4000 Å feature depends principally on the age and metallicity of the host galaxies, but only weakly on the assumed star formation history, initial mass function, a possible progenitor bias, and the so-called α-enhancement. The latter refers to the observation that these early passive galaxies have higher ratios of α elements to iron than Milky-Way type galaxies. The progenitor bias may arise due to a possible evolution of the mean redshift of formation as a function of redshift of the galaxy samples.

Of these effects, it turns out that only an uncertainty in the metallicity contributes noticeably to a systematic error σ_sys comparable to the statistical errors in the sample. Simulations have shown that the progenitor bias contributes at most only a few percentage points to σ_sys, while the impact of the initial mass function is insignificant. The difference between the 4000 Å amplitudes estimated in a single stellar population using a Chabrier or Salpeter initial mass function is less than 0.3% for all reasonable metallicities and less than 0.2% for the solar metallicity (Moresco et al. 2016). The difference in the 4000 Å amplitudes due to the α-enhancement is likewise extremely small, on average only ∼0.5%.

The only factor other than metallicity that contributes noticeably to σ_sys is the star formation history. Again, simulations have shown that variations in the assumed star-forming rate can lead to ≲13% errors in the inferred value of Δz from measurements of the 4000 Å amplitudes (Moresco et al. 2016). Together, the combination of uncertainties in the star formation history and the metallicity contribute an overall error of about 20% to Δz and hence the inferred value of H(z).

The data are shown in Figure 1 (Simon et al. 2005; Stern et al. 2010; Moresco et al. 2012, 2016; Zhang et al. 2014; Moresco 2015). For each model, we find the set of parameters that optimize the fit and minimize the χ². The best-fit curves of four cosmologies are also shown in this figure, superimposed on the data. As we have discussed in earlier papers (Melia & Maier 2013; Melia & McClintock 2015), building on the sound arguments developed in Liddle (2004, 2007) and Liddle et al. (2006), among others, a fair statistical comparison between models with different formulations and numbers of free parameters must be based on the use of information criteria (Takeuchi 2000; Tan & Biswas 2012). In this context, the likelihood of model α being the best choice is given by the expression

$$\begin{eqnarray}&&{L}_{\alpha }=\displaystyle \frac{\exp (-{\mathrm{IC}}_{\alpha }/2)}{{\sum }_{i}\exp (-{\mathrm{IC}}_{i}/2)},\end{eqnarray} \tag{ 9 }$$

where IC_α is one of AIC_α, KIC_α, or BIC_α, and the sum in the denominator runs over all of the models being tested simultaneously. The Akaike Information Criterion is defined by $\mathrm{AIC}={\chi }^{2}+2f$, while the Kullback implementation has $\mathrm{KIC}={\chi }^{2}+3f$, and $\mathrm{BIC}={\chi }^{2}+f(\mathrm{ln}\,n)$ is the Bayes information criterion. In these expressions, f is the number of free parameters and n is the number of data points. All of these criteria punish models with a large number of free parameters because these are deemed to be fitting the noise. Also, as discussed in these earlier works, the BIC is the most appropriate criterion to use when n is large, as we have here (with 30 source measurements). But, for completeness, we tabulate the results of all three criteria (see Table 1). This tabulation includes the individual ICs and each model's likelihood, weighed against all the seven cosmologies being tested simultaneously, of being the best choice. Table 1 also includes the optimized parameters for models that have them, along with their 1σ uncertainties.

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Table 1.  Best-fitting Results in Different Cosmological Models

Model	H0	Ωm	ΩΛ	wde	${\chi }_{\mathrm{dof}}^{2}$	Bayes IC	Kullback IC	Akaike IC
	(km s−1 Mpc−1)					BIC Prob	KIC Prob	AIC Prob
Rh = ct	${62.3}_{-1.4}^{+1.5}$	⋯	⋯	⋯	0.57	20.02 0.50	19.62 0.44	18.62 0.30
ΛCDM	${68.2}_{-2.8}^{+2.4}$	${0.32}_{-0.05}^{+0.06}$	$1-{{\rm{\Omega }}}_{{\rm{m}}}$	⋯	0.52	21.30 0.26	20.50 0.28	18.50 0.32
Friedmann II	${61.8}_{-3.5}^{+3.6}$	${0.03}_{-0.17}^{+0.17}$	⋯	⋯	0.59	23.31 0.10	22.51 0.10	20.51 0.12
wCDM	${70.8}_{-6.9}^{+7.2}$	${0.32}_{-0.06}^{+0.05}$	⋯	$-{1.25}_{-0.65}^{+0.57}$	0.53	24.56 0.05	23.36 0.07	20.36 0.12
oΛCDM	${69.0}_{-6.7}^{+3.4}$	${0.36}_{-0.22}^{+0.19}$	${0.77}_{-0.48}^{+0.32}$	⋯	0.54	24.66 0.05	23.46 0.06	20.46 0.12
Friedmann I	${57.4}_{-1.3}^{+1.3}$	0.30 (fixed)	⋯	⋯	0.74	24.87 0.04	24.47 0.04	23.47 0.03
Einstein-de Sitter	${49.0}_{-1.1}^{+1.2}$	⋯	⋯	⋯	1.74	53.90 2E–8	53.50 2E–8	52.50 1E–8

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It is clear from this comparison that when the high locally measured value of H₀ is excluded from the data compilation, the cosmic chronometers favor the R_h = ct cosmology, which predicts a constant expansion rate over the redshift range 0 ≲ z ≲ 2. Flat ΛCDM is second on the list, but only because we assumed prior values for w_de = −1 and k = 0. In principle, all of the free parameters in each model should be optimized using solely the cosmic chronometers for a statistically fair comparison. And we can see that when fewer prior values are assumed, e.g., as for wCDM and oΛCDM, their probabilities drop considerably.

From the compilation in Table 1, we can also see how sensitive each of the model fits is to the choice of H₀. Notice, for example, that in the case of R_h = ct, which has only one free parameter (the Hubble constant itself), the optimized value of H₀ is very tightly constrained. A $1-\sigma$ variation may be generated with a mere change of only ±1.5 km s⁻¹ Mpc⁻¹. On the other hand, in a model such as wCDM, which has three free parameters—and therefore more flexibility—a $1-\sigma$ variation requires a change in H₀ of about ±7 km s⁻¹ Mpc⁻¹ from its optimum value.

In spite of this clear separation in the model outcomes, however, a possible concern with this analysis is the fact that all of the ${\chi }_{\mathrm{dof}}^{2}$ values listed in Table 1, with the exception of Einstein-de Sitter, are significantly smaller than 1, suggesting that the errors in Figure 1 may be overestimated. This may mitigate the tension between certain models and the data, perhaps even producing a biased likelihood of some cosmologies relative to the others. To test whether our conclusions are affected in this way, we carried out a parallel comparative analysis of these measurements by artificially reducing the errors by a factor (equal to 0.75, as it turns out) that makes these ${\chi }_{\mathrm{dof}}^{2}$ values approximately equal to 1 for all but the Einstein-de Sitter universe. The results are shown in Table 2. The likelihoods have indeed changed somewhat, and wCDM and ΛCDM are disfavored less, though the information criteria still prefer R_h = ct. On the other hand, the BIC probabilities for R_h = ct and standard flat ΛCDM are now indistinguishable. But, on the basis of these results with artificially reduced errors, one still cannot conclude that the cosmic chronometers favor an accelerating universe. This outcome is qualitatively similar to that based on the analysis of the published measurements and their errors.

Table 2.  Best-fitting Results with Errors artificially Reduced by 25% to Make ${\chi }_{\mathrm{dof}}^{2}\sim 1$ in Most Cases

Model	H0	Ωm	ΩΛ	wde	${\chi }_{\mathrm{dof}}^{2}$	Bayes IC	Kullback IC	Akaike IC
	(km s−1 Mpc−1)					BIC Prob	KIC Prob	AIC Prob
ΛCDM	${68.2}_{-2.1}^{+1.9}$	${0.32}_{-0.03}^{+0.04}$	$1-{{\rm{\Omega }}}_{{\rm{m}}}$	⋯	0.92	32.58 0.42	31.78 0.43	29.78 0.42
Rh = ct	${62.4}_{-1.1}^{+1.1}$	⋯	⋯	⋯	1.02	32.95 0.35	32.55 0.29	31.55 0.17
wCDM	${70.8}_{-4.8}^{+6.4}$	${0.32}_{-0.05}^{+0.04}$	⋯	$-{1.25}_{-0.59}^{+0.37}$	0.95	35.73 0.09	34.53 0.11	31.53 0.17
oΛCDM	${69.0}_{-4.7}^{+2.9}$	${0.36}_{-0.17}^{+0.14}$	${0.77}_{-0.38}^{+0.22}$	⋯	0.95	35.91 0.08	34.71 0.10	31.71 0.16
Friedmann II	${61.8}_{-2.6}^{+2.7}$	${0.03}_{-0.11}^{+0.13}$	⋯	⋯	1.05	36.16 0.07	35.36 0.07	33.36 0.07
Friedmann I	${57.4}_{-1.0}^{+1.0}$	0.30 (fixed)	⋯	⋯	1.32	41.57 0.005	41.17 0.004	40.17 0.002
Einstein-de Sitter	${49.1}_{-0.8}^{+0.9}$	⋯	⋯	⋯	3.10	93.18 3E–14	92.78 2E–14	91.78 1E–14

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Adding five new measurements of the Hubble parameter H(z) with the cosmic-chronometer approach, Moresco et al. (2016) claimed to obtain the first cosmology-independent constraint on the transition redshift, showing the existence of an epoch of cosmic re-acceleration. This result, however, relies heavily on the use of a Gaussian prior on the Hubble constant, H₀ = 73 ± 2.4 km s⁻¹ Mpc⁻¹. It is generally recognized that this H₀ is not an accurate representation of the smoothed background H₀ if the local expansion rate is influenced by a Hubble Bubble. The inclusion of this high locally measured value of H₀ unfairly biases the results by contaminating the cosmic-chronometer data with a measurement whose systematics are unknown.

In this regard, we point out that all of the optimized values of H₀ in Table 1 are smaller than the locally measured Hubble constant, even for ΛCDM. In fact, were one to use the cosmic chronometers to infer H₀ in the context of the standard model, one would actually find a remarkable consistency with the value (67.8 ± 0.9 km s⁻¹ Mpc⁻¹) measured by Planck (Planck Collaboration 2014). Since the same cosmology is being referenced for the interpretation of these two disparate sets of data, this consistency with the value of H₀ seen at intermediate and very high redshifts adds some support to our thesis in this paper that the locally measured Hubble constant is not a true representation of its large-scale smoothed value.

Still, the fact that the optimized value of H₀ in R_h = ct is notably smaller than that in the standard model merits some attention. The Hubble constant cannot be measured directly using SNe Ia (since H₀ is degenerate with the absolute SN magnitude, which is also free). The most accurate determination of H₀ we have today is from the anisotropies in the CMB (Planck Collaboration 2014). The chronometer-measured value of H₀ in R_h = ct is about 8% smaller than the Planck measurement. But the CMB value of H₀ is itself model dependent and the temperature fluctuation spectrum has not yet been analyzed using R_h = ct. All we have at the moment is the Hubble constant optimized for ΛCDM, so it is too early to tell whether the chronometer and CMB measurements of H₀ are consistent with each other in the ${R}_{{\rm{h}}}={ct}$ model. What we do know from simulations is that a local Hubble Bubble can probably modify the large-scale value of H₀ by possibly ±2–3 km s⁻¹ Mpc⁻¹ (Marra et al. 2013; Ben-Dayan et al. 2014); so, to be compatible with the low value of H₀ reported here for R_h = ct, the future measurement of the Hubble constant in this model based on CMB anisotropies ought to be ≲65 km s⁻¹ Mpc⁻¹.

It is also worth noting that some local measurements of H₀ do actually suggest a value smaller than that (i.e., 73 km s⁻¹ Mpc⁻¹) employed by Moresco et al. (2016). For example, Tammann & Reindl (2013) used red-giant branch stars in the halos of local galaxies to calibrate the SN Ia luminosity and inferred a Hubble constant ${H}_{0}={64.0}_{-2.0}^{+1.6}$ km s⁻¹ Mpc⁻¹, while the (now somewhat dated) SN HST Project yielded a value ${H}_{0}={62.3}_{-5.0}^{+1.3}$ (Sandage et al. 2006). Still, the majority of local measurements of H₀ suggest a higher value, so the key question for R_h = ct remains how the cosmic-chronometer measurements will compare with those based on model fits to the CMB temperature anisotropies.

With these caveats in mind, we have demonstrated in this paper that the inference of re-acceleration is reversed when the additional H₀ datum is excluded. We have shown that, used on their own, without the prior on H₀, the latest compilation of 30 model-independent cosmic-chronometer measurements prefer a constant expansion rate over an accelerated one out to z ∼ 2.

This result may seem surprising at first, but it merely confirms the outcome of many previous comparative tests between ΛCDM and the constant expansion rate cosmology R_h = ct. This type of analysis has been carried out using diverse sets of data, at low, intermediate, and high redshifts. The R_h = ct model appears to be favored over ΛCDM by all the observations used so far. A brief survey of the results includes the following sample. (1) When the supernovae in the Supernova Legacy Survey Sample are correctly recalibrated for each model being tested, these favor ${R}_{{\rm{h}}}={ct}$ over ΛCDM with a BIC likelihood of ∼90% versus ∼10% (Wei et al. 2015a). (2) According to the quasar Hubble diagram and the Alcock-Pacźynski test (López-Corredoira et al. 2016; Melia & López-Corredoira 2017), R_h = ct is more likely to be correct than ΛCDM. (3) Based on the presumed constancy of the gas-mass fraction in clusters, the BIC favors R_h = ct over ΛCDM with a likelihood of ∼95% versus only ∼5% (Melia 2016b). (4) While ΛCDM cannot account for the appearance of high-z quasars without some anomalous seed formation or greatly super-Eddington accretion, none of which have ever been seen, their formation and growth are fully consistent with the timeline predicted in R_h = ct (Melia 2013). (5) A similar time-compression problem arises with the emergence of high-z galaxies in ΛCDM, though not in R_h = ct (Melia 2014a). (6) Based on the age–redshift relationship of old passive galaxies, the BIC favors R_h = ct over ΛCDM with a likelihood of ∼81% versus ∼19% (Wei et al. 2015b). (7) Whereas the inferred probability of ΛCDM accounting for the CMB angular correlation function is <0.3%, R_h = ct fits it much better, including the absence of correlation at angles greater than 60° (Melia 2014b). This tension between ΛCDM and observations of the CMB may be quite serious because the lack of any large-angle correlation is inconsistent with inflationary scenarios. Yet the standard model would not survive without inflation to fix the horizon problem. In contrast, the R_h = ct universe does not have or need inflation.

This is only a small sample of the many tests completed thus far, but already it spans a broad range of data, some at low redshifts, others at high redshifts. In every case, the R_h = ct cosmology has been favored over ΛCDM. Even so, the challenge of establishing the viability of R_h = ct is ongoing. In this paper, we have added to the observational evidence in its favor, but several important issues need to resolved. It remains to be seen whether Big Bang nucleosynthesis (BBN) in this model can correctly reproduce the light elements. Initial attempts at simulating BBN with a constant expansion rate have been very promising, showing that the well-known ⁷Li anomaly plaguing the standard model may be solved by the slower burning taking place with such an expansion scenario (Benoit-Lévy & Chardin 2012). Of course, this would not be known for sure until the BBN calculations will have been carried out correctly for the conditions in R_h = ct. Also, although the anisotropies in the CMB have been analyzed for R_h = ct in terms of their angular correlation function, they have yet to be used to calculate a power spectrum in this cosmology. As is well known, the ability of ΛCDM to accurately account for the CMB power spectrum provides strong support in its favor. This analysis, however, is beyond the scope of the present work and its results will be reported elsewhere.

We are grateful to the anonymous referee for his insightful comments that have led to an improvement in the manuscript. FM is grateful to PMO in Nanjing, China, for its hospitality while this work was being carried out. We acknowledge partial support from the National Basic Research Program ("973" Program) of China (grant Nos. 2014CB845800), the National Natural Science Foundation of China (grant Nos. 11322328, 11433009, 11673068, and 11603076), the Youth Innovation Promotion Association (2011231 and 2017366), the Key Research Program of Frontier Sciences (QYZDB-SSW-SYS005), the Strategic Priority Research Program "Multi-waveband gravitational wave Universe" (grant No. XDB23000000) of the Chinese Academy of Sciences, and the Natural Science Foundation of Jiangsu Province (grant No. BK20161096). This work was also partially supported by grant 2012T1J0011 from The Chinese Academy of Sciences Visiting Professorships for Senior International Scientists, and grant GDJ20120491013 from the Chinese State Administration of Foreign Experts Affairs.