Early Universe Physics Insensitive and Uncalibrated Cosmic Standards: Constraints on Ω_m and Implications for the Hubble Tension

The standard assumptions in our default analysis are as follows. First, to calculate the sound speed, we need the reduced baryon fraction Ω_b h², for which we use the big bang nucleosynthesis (BBN) constraint Ω_b h² = 0.0222 ± 0.0005 (Cooke et al. 2018). We also need to know z_* and z_d, which we separately obtain from Planck (Planck Collaboration et al. 2020) (z_* = 1089.99 ± 0.28 and z_* − z_d = 30.27 ± 0.57) and from ACT+WMAP (Aiola et al. 2020) (z_* = 1089.91 ± 0.42 and z_* − z_d = 29.95 ± 0.75). Next, as we are assuming the standard ΛCDM after recombination, the function E(z) and the sound speed c_s take their standard forms; see Equations (A3) and (A12). In particular, for the function E(z) we consider a universe with a cosmological constant filled with pressureless matter (CDM and baryons), photons, and neutrinos (one massless and two massive with a normal mass hierarchy).

The joint analysis of the θ_cmb likelihood and the BAO likelihood will break the degeneracy in the Ω_m–r_d H₀ plane and significantly improve the constraint on Ω_m compared to the uncalibrated BAO alone. We illustrate how this works as follows. Recall that the angular size of the sound horizon at the drag epoch is given by

$$\begin{eqnarray}&&{\theta }_{{\rm{d}}}(z)=\displaystyle \frac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{M}}}(z)},\end{eqnarray} \tag{ 2 }$$

where f_M defined in Equation (A1) is the Hubble-distance normalized comoving distance. BAO uses the angular size and the redshift span of the (drag-epoch) sound horizon, θ_d and ${\rm{\Delta }}{z}_{{r}_{{\rm{d}}}}$, at different effective redshifts to constrain Ω_m, which determines the relative change of the Hubble parameter with redshift; see Appendix A.2 for a discussion. With our standard assumptions, the difference ΔrH₀ is ∼6 × 10⁻⁴ according to Equation (1), given that BAO alone can already constrain Ω_m with a certain precision. With ΔrH₀ roughly known, the measurement of θ_cmb can be viewed as another measurement of θ_d at z_* (with an offset of ${\rm{\Delta }}\theta \equiv {\theta }_{{\rm{d}}}({z}_{* })-{\theta }_{\mathrm{cmb}}=\tfrac{{\rm{\Delta }}{{rH}}_{0}}{{f}_{{\rm{M}}}({z}_{* })}$). In Figure 1, we plot the measured or inferred values of θ_d at different redshifts along with the theoretical prediction according to the best-fit model. ⁸ We can see that the prediction matches the measured/inferred θ_d at different redshifts quite well. The precise measurement of θ_cmb (and then the inferred θ_d) adds a powerfully constraining anchor at a large redshift, so that the constraint on the relative change of the Hubble parameter can be significantly tightened, and hence so can the Ω_m in the standard ΛCDM model. For the same (comoving) r_d, the angular size θ_d asymptotically approaches the minimum value at large redshift, as indicated in the upper panel of Figure 1. This gives the advantage that z_* has to be significantly incorrect in order to affect the analysis.

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BAO measurements at different redshifts give different degeneracy directions in the Ω_m–r_d H₀ plane (Cuceu et al. 2019). This also applies to θ_cmb. In Figure 2, we plot different constraints in the Ω_m–r_d H₀ plane at different redshifts, including the one inferred from θ_cmb. At lower redshifts, BAO measurements are more sensitive to r_d H₀, because f_M(z) (and also E(z)) is not sensitive to Ω_m at low redshifts. This is shown in Figure 2 as the constraint from the six degree field galaxy survey and main galaxy sample (6df+MGS) (z_eff < 0.2) in the Ω_m–r_d H₀ plane is almost horizontal. At larger redshifts, BAO measurements are sensitive to both r_d H₀ and Ω_m; this is shown in Figure 2 as constraints from BAO at higher redshifts (z_eff > 0.3). The CMB constraint is diagonal in the Ω_m–r_d H₀ plane. Therefore, the tight constraint from θ_cmb breaks the degeneracy in the Ω_m–r_d H₀ plane of the BAO measurements at lower redshifts, and significantly improves the constraint on Ω_m.

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To constrain Ω_m, besides θ_cmb and uncalibrated BAO, we will also consider the Pantheon catalog of SNe Ia (without calibration). We denote the joint likelihood of all these UCS data as ${{ \mathcal L }}_{\mathrm{UCS}}$. In our standard analysis, the free parameters are Ω_m, r_* H₀, ${ \mathcal M }$, and h (a very weak dependence; see Appendix A.1), with some assumed prior on z_*, Δz_s, and Ω_b h².

As mentioned earlier, even uncalibrated, standard rulers and candles can put a constraint on Ω_m (Scolnic et al. 2018; Aylor et al. 2019). With θ_cmb added according to the method described above, the constraint is now much stronger. In Figure 3, we show the constraints on Ω_m set by different combinations of those UCS. We first reproduced the results from Pantheon SNe Ia and uncalibrated BAO, respectively, shown by the green error bars (1σ) in the left panel of Figure 3. In the middle panel, we show the constraints from the combination of θ_cmb and three selected late-time BAO. We adopt two different θ_cmb results, one from Planck temperature and polarization (TTTEEE+lowE) with ⁹

$$\begin{eqnarray}&&100{\theta }_{\mathrm{cmb}}^{\mathrm{Planck}}=1.04109\pm 0.00030,\end{eqnarray} \tag{ 3 }$$

shown in blue (Planck Collaboration et al. 2020); and the other from ACT+WMAP with

$$\begin{eqnarray}&&100{\theta }_{\mathrm{cmb}}^{\mathrm{ACT}+\mathrm{WMAP}}=1.04170\pm 0.00068,\end{eqnarray} \tag{ 4 }$$

shown in red (Aiola et al. 2020). Among these three constraints, the one from the combination of θ_cmb and the BAO from the Sloan Digital Sky Survey Data Release 12 galaxy clustering (SDSS DR12 GC; Alam et al. 2017) is the tightest, which is due to both the degeneracy breaking and the strong constraining power of this late-time BAO. Nonetheless, the constraints on Ω_m via those different combinations are consistent with each other and with the SN Ia-only and uncalibrated-BAO-only constraints.

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We then produce the joint constraint with all UCS. The final result is shown in the right panel of Figure 3 with

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{m}}}=0.302\pm 0.008,\end{eqnarray} \tag{ 5 }$$

using the Planck constraint on θ_cmb. This is a strong constraint on Ω_m. In fact, with the same data the full analysis in ΛCDM (assuming the standard modeling of the sound horizon) gives a constraint on Ω_m that is only a little tighter, Ω_m = 0.310 ± 0.006. Although the average is somewhat lower, our constraint on Ω_m is consistent with the full analysis that assumes the standard ΛCDM model for the entire cosmic history.

Compared to the full standard analysis of CMB anisotropy, our method using UCS is only based on the geometrical information from CMB anisotropy and relies on many fewer assumptions. First, as explained, our method is insensitive to early universe physics. Second, our method is less vulnerable to systematic bias in the CMB observations. Indeed, in the full analysis of the CMB, the matter fraction is constrained by the scale dependence of the CMB power spectra (Planck Collaboration et al. 2020). This renders the constraint of Ω_m correlated with, e.g., the spectral index n_s. Incidentally, there have been discussions on the possibility of some small inconsistency between the large and small scale spectra in the Planck CMB power spectra, which might be due to unaccounted for systematic errors or more radically some beyond the Standard Model physics (Addison et al. 2016); but also see Planck Collaboration et al. (2016) for a discussion. Our method is clearly free of those potential biases.

As we mentioned earlier, the combination of θ_cmb and the SDSS DR12 GC BAO provides the strongest constraint on Ω_m. If there is some systematic error in the SDSS DR12 GC BAO, the result would be biased. Therefore, we also produce a joint constraint on Ω_m using all UCS except for the SDSS DR12 GC, which is shown in the second to the right set of error bars in Figure 3. As we can see, even without the SDSS DR12 GC BAO, the result is fully consistent with the one using all UCS.

In the last subsection, we treat r_* as a free parameter, which have already relaxed the most of the strong assumptions of early universe physics. In our standard analysis, we have only made an assumption that the difference between r_* H₀ and r_d H₀ can be calculated according to the standard ΛCDM model. We have assumed some priors on z_* and Δz_s from CMB observations and Ω_b h² from BBN, so that we can calculate ΔrH₀. One may worry that these may render our results dependent upon early universe physics. We now show that results obtained based on this standard analysis are robust against changes caused by possible nonstandard early universe physics. We quantify how significantly these assumptions need to be modified to substantially affect our results. We also discuss several possibilities that may be contemplated in future model building to circumvent the arguments and constraints derived in this paper.

Recall that the underlying reason we can jointly analyze the θ_cmb and uncalibrated BAO likelihoods is that the two horizon scales (r_* and r_d) are very close to each other but the measurements are so separated in redshift that together they can provide a stringent constraint on the evolution of the post-recombination background. We discuss this in detail in the following.

First, the fact that the difference between r_* H₀ and r_d H₀ is very small is important and quite robust. Because it is small, the difference between the two angles θ_d (z_*) and θ_cmb is comparable to or even smaller than the error on θ_d (z_*) extrapolated from the BAO measurements alone. Therefore, the CMB measurement on θ_cmb can be viewed as another statistically significant measurement of the BAO sound horizon at a very high redshift with an uncertainty mainly caused by the uncertainty of Δθ.

Second, because the value of Ω_m continuously and monotonically impacts the evolution of the background from z ∼ 1000 to z ∼ 1, with the same relative errors, the constraining power on Ω_m from the data-derived θ_d values grows if the redshifts of these data are further separated. Consequently, since z_* is so high, the effect on the constraint on Ω_m due to a change in ΔrH₀ from nonstandard physics is further reduced. The above two points, ΔrH₀ being small and z_* being high, argue that our analysis is robust against the change in ΔrH₀ due to possibly unknown nonstandard physics.

Indeed, even if we artificially multiply Equation (1) by a factor of 110% mimicking some possible unknown modifications to our standard assumptions, the joint constraint on Ω_m only changes to 0.303 ± 0.008, compared to 0.302 ± 0.008 in our standard analysis, or

$$\begin{eqnarray}&&\displaystyle \frac{\delta ({{\rm{\Omega }}}_{{\rm{m}}}^{\mathrm{mean}})}{{{\rm{\Omega }}}_{{\rm{m}}}^{\mathrm{mean}}}=0.036\displaystyle \frac{\delta ({\rm{\Delta }}{{rH}}_{0})}{{\rm{\Delta }}{{rH}}_{0}}.\end{eqnarray} \tag{ 6 }$$

The small coefficient above verifies the conclusion that the constraint on Ω_m is insensitive to changes to ΔrH₀.

Third, even the small value of ΔrH₀ itself is quite robust and it is not easy for nonstandard models to change it substantially. While the form of Equation (1) is generally true, one may change ΔrH₀ by changing the sound speed, c_s, the function E(z), and the redshift span between z_d and z_*. For example, changing the Ω_b h² can change the sound speed. Regarding this, due to the consistency of the Ω_b h² constraint between Planck and BBN (Cooke et al. 2018; Fields et al. 2020; Planck Collaboration et al. 2020), and between different uses of information from CMB anisotropy (Motloch 2020), it is difficult for the reduced baryon fraction to be significantly different from Ω_b h² ∼ 0.0222. Since the Ω_b h² term only has a relatively small effect on c_s between z_d and z_*, it is even more difficult to change ΔrH₀ via the Ω_b h² term. Quantitatively, even if we artificially change Ω_b h² by 10% (∼ 4σ away from the mean value of our adopted prior), which we use to mimic effects from possible changes in the sound speed, ΔrH₀ only changes by ≲2%. For the E(z) function, if nonstandard physics occurred before recombination as in a number of currently proposed early universe nonstandard models, its functional form will not change. ¹⁰ Due to the narrow redshift span between z_d and z_*, it is difficult to substantially change ΔrH₀ from its standard value by changing E(z) with modified energy compositions. Also, although a change to the redshift span itself can directly affect ΔrH₀, to get a ∼ 10% change to ΔrH₀, we would need a ∼10% change of Δz_s, which is ∼ 5.5σ away from the mean value of Δz_s inferred from Planck. It is difficult to achieve this because the decoupling between photons and baryons are based on well known physics. The redshift span between recombination and the drag epoch cannot be substantially different from the standard value unless there are some other significant interactions between, e.g., baryon and dark matter, which would have led to observable features in CMB anisotropy (Dvorkin et al. 2014; Boddy & Gluscevic 2018; Xu et al. 2018; de Putter et al. 2019). In support, Jedamzik et al. (2021) finds that r_* ∼ 1.018r_d persists in a number of pre-recombination nonstandard models. In summary, it is not easy to make a noticeable change (like 10%) to ΔrH₀.

Besides the change of ΔrH₀, one may worry about the prior on z_* that we adopted. But as discussed in Appendix A.3, our result is not sensitive to the prior on z_*, because z_* is sufficiently large and f_M is approaching a constant (though dependent of Ω_m). To numerically verify this, we artificially shift the mean value of Planck result of z_* from 1090–1110, which can be achieved, e.g., in the presence of primordial magnetic fields (Jedamzik & Pogosian 2020). This only leads to an ∼ 0.2% change in the mean value of Ω_m (from 0.3021–0.3015).

Another possible concern is that we have adopted the constraint on θ_cmb based on the standard ΛCDM model. The uncertainty of θ_cmb from Planck is about 0.03%. Such a small uncertainty of θ_cmb may not represent the possible uncertainty due to some nonstandard early universe physics. For example, releasing the effective relativistic particle number, the Planck result becomes 100θ_cmb = 1.04136 ± 0.00060. And releasing the mass of neutrinos gives 100θ_cmb = 1.04105 ± 0.00032. So, the uncertainty of θ_cmb due to these potential nonstandard physics at early times is comparable to or larger than 0.03%. However, these uncertainties in θ_cmb will not substantially change our result in Ω_m for the following reason. Take the combination of θ_cmb and the SDSS DR12 GC BAO, for example. As shown in Figure 2, the uncertainty of Ω_m is mainly due to the width of the projection of the overlap between these two constraints in the Ω_m direction. As we can visualize in that figure, even if we increase the width of the black contour or shift it a bit horizontally, the projection in the Ω_m direction would not be substantially changed. Quantitatively, this insensitivity can be shown by alternatively using the result of θ_cmb from ACT+WMAP. We can see from Equations (3) and (4) that there is an ∼2σ (in terms of Planck's uncertainty) difference in the mean value of θ_cmb between Planck and ACT+WMAP, and also the uncertainty of θ_cmb for ACT+WMAP is more than twice that for Planck. But Figure 3 shows the resultant constraints on Ω_m are nearly identical.

To evade the constraint on the evolution of the post-recombination CMB by UCS, one would need (1) some mechanism that makes the expressions for the two sound horizon scales different prior to the recombination; or (2) some nonstandard physics occurring during the narrow gap between z_* and z_d that significantly changes the function E(z) or the sound speed c_s; or (3) a substantial change of the redshift span between recombination and the drag epoch. None of those possibilities are easy to achieve, but nonetheless could serve as guidelines if we would like to build models of counter examples.

Our UCS constraint on Ω_m is strong and insensitive to early universe physics. This can be used as a strong prior or be combined with other observations to break degeneracy in the background. We will do that in Section 4 to obtain several early universe physics insensitive constraints on H₀.

In our standard analysis the constraint in the H₀–Ω_m plane is actually not exactly along the Ω_m direction. That is because the normalized comoving distance f_M has a weak dependence on the Hubble constant via the radiation and the massive neutrino terms; see Appendix A.1 for a discussion. A principal component analysis shows that

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{m}}}}{0.3}{\left(\displaystyle \frac{h}{0.7}\right)}^{-0.08}=1.0060\pm 0.0258.\end{eqnarray} \tag{ 7 }$$

Therefore, the constraint exhibits a small positive correlation in the H₀–Ω_m plane. In situations where joint analyses using MCMC with other data sets are unavailable (such as that with γ-ray optical depth, which shall be discussed), we will use Equation (7) as a prior to set weights in parameter chains obtained from other analyses in order to approximate the joint result. In practice, we find that using either Equations (5) or (7) as a prior only makes a negligible difference.

Here, we discuss in detail the framework and the UCS data we used in this work. We normalize the (transverse) comoving distance d_M by the Hubble distance (1/H₀),

$$\begin{eqnarray}&&{d}_{{\rm{M}}}(z)=\displaystyle \frac{1}{{H}_{0}}{f}_{{\rm{M}}}(z;{{\rm{\Omega }}}_{{\rm{m}}},\cdots ),\end{eqnarray} \tag{ A1 }$$

with f_M(z; Ω_m, ⋯) dependent on a cosmological model. We assume the standard spatially flat ΛCDM model after recombination and

$$\begin{eqnarray}&&{f}_{{\rm{M}}}(z;{{\rm{\Omega }}}_{{\rm{m}}},\cdots )={\int }_{0}^{z}\displaystyle \frac{{dz}^{\prime} }{E(z^{\prime} )},\end{eqnarray} \tag{ A2 }$$

$$\begin{eqnarray}&&E(z)=\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{{\rm{m}}}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{r}}}{\left(1+z\right)}^{4}+\sum _{i}{{\rm{\Omega }}}_{{\nu }_{i}}{e}_{i}(z)},\end{eqnarray} \tag{ A3 }$$

with the constraint $1={{\rm{\Omega }}}_{{\rm{m}}}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{{\rm{r}}}+{\sum }_{i}{{\rm{\Omega }}}_{{\nu }_{i}}$. The subscript "m" stands for matter, "Λ" for the cosmological constant, "r" for radiation (photon and massless neutrino), and "ν_i " for massive neutrinos where i runs through the neutrino species. The function e_i (z) represents the redshift dependence of the evolution of the density fraction of the massive neutrinos and can be well approximated by (W. Lin et al. 2021, in preparation)

$$\begin{eqnarray}&&{e}_{i}(z)=\displaystyle \frac{1}{{a}^{4}}{\left(\displaystyle \frac{{a}^{n}+{a}_{{\rm{T}},i}^{n}}{1+{a}_{{\rm{T}},i}^{n}}\right)}^{\tfrac{1}{n}},\end{eqnarray} \tag{ A4 }$$

where $a=\tfrac{1}{1+z}$ is the scale factor, n = 1.8367, ${a}_{{\rm{T}},i}=3.1515\tfrac{{T}_{\nu }^{0}}{{m}_{{\nu }_{i}}}$, ${T}_{\nu }^{0}=1.676\times {10}^{-4}$ eV, and ${m}_{{\nu }_{i}}$ is the ith neutrino mass. The radiation and neutrino terms only have some small contribution to f_M at redshifts close to recombination, but we include them for completeness. In our standard analysis, we consider one massless and two massive neutrinos in the normal hierarchy with constraints on the difference of the mass-squared ${\rm{\Delta }}{m}_{12}^{2}=7.9\,\times {10}^{-5}\,{\mathrm{eV}}^{2}$ and ${\rm{\Delta }}{m}_{13}^{2}=2.2\times {10}^{-3}\,{\mathrm{eV}}^{2}$ (Maltoni et al. 2004). With the measured CMB temperature today and the assumption of thermal neutrino relics, we have ${{\rm{\Omega }}}_{{\rm{r}}}={{\rm{\Omega }}}_{\gamma }+{{\rm{\Omega }}}_{\nu }^{\mathrm{massless}}\,=3.0337{h}^{-2}\times {10}^{-5}$ and ${{\rm{\Omega }}}_{{\nu }_{i}}=\tfrac{{m}_{{\nu }_{i}}}{93.14\,{h}^{2}\,\mathrm{eV}}$, where $h\,\equiv \tfrac{{H}_{0}}{100\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}}$ (Lesgourgues & Pastor 2012). While we have ignored the uncertainties of the mass-squared difference and the mass of the lightest neutrino, we found that our results are insensitive to them for a reasonable range of the lowest neutrino mass for both hierarchy schemes. This is because the radiation and neutrino terms only have small effects on f_M at high redshifts close to recombination. For a reasonable range of the mass of the lightest neutrino, all neutrinos were relativistic before z ∼ 100 when they have (small) effects on f_M. In fact the effects of the radiation and the neutrinos terms are so small that our constraints on Ω_m are essentially unchanged even if we completely ignore these two terms.

In our standard treatment, f_M mainly depends on Ω_m. From our adopted normalization (i.e., Equation (A1)) we shall see more clearly the degeneracy of H₀ with the scale of the sound horizon (r_* or r_d) and with the absolution magnitude of SNe Ia, M₀. However, it is worth pointing out that f_M still weakly depends on h via the radiation and neutrino terms, especially at higher redshifts. ¹⁴ As a consequence, our constraints will not be totally uninformative in H₀; see Section 3.3.

Note that in this work, we have assumed the standard ΛCDM for cosmic time after recombination, but our analyses can be generalized and used to test post-recombination nonstandard models where Equation (A2) is replaced by the corresponding normalized comoving distance. Such a test of post-recombination nonstandard models will have the advantage of being robust against any potentially unknown early universe physics while retaining a strong constraining power.

The late-time BAO measurements are compressed information obtained from two-point correlation functions of tracers of the underlying matter fluctuation. With r_d the sound horizon scale at the end of the drag epoch (at redshift z_d), measurements can be given by a pair of quantities that are the angular size ${\theta }_{{\rm{d}}}=\tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{M}}}(z)}$ and the redshift span ${\rm{\Delta }}{z}_{{r}_{{\rm{d}}}}={r}_{{\rm{d}}}{H}_{0}E(z)$ of the baryon acoustic sound horizon when placed perpendicular to and along the light of sight. Alternatively, the set $\left[\tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{V}}}(z)},\,{F}_{\mathrm{AP}}(z)=\tfrac{{\rm{\Delta }}{z}_{{r}_{{\rm{d}}}}}{{\theta }_{{\rm{d}}}}=E(z){f}_{{\rm{M}}}(z)\right]$ will be given or can be constructed from $({\theta }_{{\rm{d}}},\,{\rm{\Delta }}{z}_{{r}_{{\rm{d}}}})$. Here,

$$\begin{eqnarray}&&{f}_{{\rm{V}}}(z)\equiv {\left[\displaystyle \frac{z}{E(z)}{\left[{f}_{{\rm{M}}}(z)\right]}^{2}\right]}^{1/3},\end{eqnarray} \tag{ A5 }$$

which can be interpreted as the cubic root of the differential comoving volume (w.r.t. the solid angle and logarithmic redshift) normalized by ${(1/{H}_{0})}^{3}$. The quantity $\tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{V}}}(z)}$ can be interpreted in the following way. Imagine a simple cubic lattice formed in the universe with the comoving length of each dimension equal to the size of the drag sound horizon. The comoving volume element of each lattice cell is ${r}_{{\rm{d}}}^{\,3}$. Then, ${\left(\tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{V}}}(z)}\right)}^{3}$ is (the inverse of) the number of those volume elements seen per solid angle per logarithmic redshift. The quantity F_AP(z) is the ratio between ${\rm{\Delta }}{z}_{{r}_{{\rm{d}}}}$ and θ_d, which is useful as it contains no dependence of ΔrH₀.

It is worth pointing out that r_d always goes with H₀ in those BAO measurements, while f_M and f_V are nearly independent of H₀. So, when r_d is treated as a free parameter, it is degenerate with H₀; also see Aylor et al. (2019). We therefore treat r_d H₀ as one free parameter when analyzing the BAO data.

In Table 2, we summarize some BAO measurements obtained or inferred from (in ascending order with effective redshift) the 6dF galaxy survey (Beutler et al. 2011), SDSS DR7 MGS (Ross et al. 2015), SDSS DR12 GC (Alam et al. 2017), SDSS DR16 luminous red galaxy, and emission line galaxy samples (Luminous Red Galaxy (LRG) and Emission Line Galaxy)) (Wang et al. 2020), and eBOSS DR16 quasars (QSO) and Lyα forest (Lyα) auto and cross correlations (Blomqvist et al. 2019). There are correlations between θ_d and ${\rm{\Delta }}{z}_{{r}_{{\rm{d}}}}$, which are shown in the last column of Table 2. In particular, there are correlations among the three SDSS DR12 GC BAO measurements with

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{\rho }}}_{\mathrm{SDSSDR}12\mathrm{GC}}({{\boldsymbol{\theta }}}_{d},{\boldsymbol{\Delta }}{z}_{{r}_{{\rm{d}}}})\\ =\,\left(\begin{array}{cccccc}1 & -0.084 & 0.47 & -0.042 & 0.17 & -0.014\\ -0.084 & 1 & -0.012 & 0.41 & 0.012 & 0.16\\ 0.46 & -0.012 & 1 & -0.027 & 0.50 & -0.022\\ -0.042 & 0.41 & -0.027 & 1-0.0085 & 0.47 & \\ 0.17 & 0.012 & 0.50 & -0.0085 & 1 & 0.050\\ -0.014 & 0.16 & -0.022 & 0.47 & 0.050 & 1\end{array}\right)\end{array}.\end{eqnarray} \tag{ A6 }$$

Table 2. BAO Measurements Obtained or Inferred from Wang et al. (2020), Blomqvist et al. (2019), Alam et al. (2017), Ross et al. (2015), and Beutler et al. (2011)

Labels	References	zeff	$\tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{V}}}(z)}$	FAP	${\theta }_{{\rm{d}}}\equiv \tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{M}}}(z)}$	Δz ≡ rd H0 × E(z)	$\rho ({\theta }_{{\rm{d}}},{\rm{\Delta }}{z}_{{r}_{{\rm{d}}}})$
6df	Beutler et al. (2011)	0.106	0.336 ± 0.015	N/A	N/A	N/A	N/A
MGS	Ross et al. (2015)	0.15	0.224 ± 0.0084	N/A	N/A	N/A	N/A
SDSS DR12 GC	Alam et al. (2017)	0.38	0.1002 ± 0.0011	0.410 ± 0.016	0.0977 ± 0.0016	0.0400 ± 0.0012	Equation (A6)
		0.51	0.07893 ± 0.00080	0.599 ± 0.021	0.0748 ± 0.0011	0.0448 ± 0.0011	Equation (A6)
		0.68	0.06898 ± 0.00071	0.761 ± 0.027	0.0641 ± 0.0010	0.0488 ± 0.0012	Equation (A6)
LRG and LEG	Wang et al. (2020)	0.77	0.05707 ± 0.00090	0.960 ± 0.035	0.0530 ± 0.0011	0.0509 ± 0.0015	8.02 × 10−4
Quasar	Blomqvist et al. (2019)	1.52	0.0383 ± 0.0017	N/A	N/A	N/A	N/A
Lyα-quasar	Blomqvist et al. (2019)	2.35	0.03256 ± 0.00065	4.13 ± 0.20	0.02698 ± 0.0009	0.1115 ± 0.0028	0.645

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The uncalibrated BAO likelihood reads as

$$\begin{eqnarray}&&\mathrm{ln}{{ \mathcal L }}_{{\rm{u}}.{\rm{c}}.\ \mathrm{bao}}=\displaystyle \sum _{i}-\frac{1}{2}{\left({{\boldsymbol{\Theta }}}_{i}-{{\boldsymbol{Q}}}_{{\boldsymbol{i}}}\right)}^{T}{{\boldsymbol{C}}}_{{\boldsymbol{i}}}^{-1}({{\boldsymbol{\Theta }}}_{i}-{{\boldsymbol{Q}}}_{{\boldsymbol{i}}})+\mathrm{const}.,\end{eqnarray} \tag{ A7 }$$

where Θ_i (Ω_m, r_d H₀) and Q _i are predictions and measurements of $({\theta }_{{\rm{d}}},\,{\rm{\Delta }}{z}_{{r}_{{\rm{d}}}})$ or $\left(\tfrac{{r}_{{\rm{d}}}{H}_{0}}{{f}_{{\rm{V}}}(z)},\,{F}_{\mathrm{AP}}(z)\right)$, respectively, and C _i is the covariance matrix and the index i denotes different BAO measurements.

The photon-baryon plasma perturbation also manifests as acoustic oscillations in the CMB temperature and polarization angular power spectra. These oscillations correspond to a sharply defined angular scale (θ_cmb), which has been robustly and precisely measured and is almost independent of the underlying cosmological model (Planck Collaboration et al. 2020). It reads as ${\theta }_{\mathrm{cmb}}={r}_{* }/{d}_{{\rm{M}}}^{\mathrm{rec}}$, where r_* is the comoving sound horizon at recombination and ${d}_{{\rm{M}}}^{\mathrm{rec}}$ is the comoving distance at recombination. Plugging in Equation (A1) we have ${\theta }_{\mathrm{cmb}}=\tfrac{{r}_{* }{H}_{0}}{{f}_{M}({z}_{* })}$. So, θ_cmb mainly depends on r_* H₀ and Ω_m. It weakly depends on h via the radiation and neutrino terms in E(z) within f_M(z); see Appendix A.1. It also depends on z_*. We first assume a prior on z_* obtained from the Planck baseline constraint based on the standard ΛCDM model (Planck Collaboration et al. 2020) or from ACT+WMAP (Aiola et al. 2020). But since ${f}_{{\rm{M}}}(z\to \infty )\to \mathrm{constant}$, corresponding to the particle horizon normalized by the Hubble distance and z_* is sufficiently large, the dependence of θ_cmb on z_* is very weak. Therefore, our results would be barely changed for a reasonable range of z_* and we verify this in Section 3.2.

The θ_cmb likelihood reads as

$$\begin{eqnarray}&&\mathrm{ln}{{ \mathcal L }}_{{\theta }_{\mathrm{cmb}}}=-\frac{1}{2}{({{\boldsymbol{\theta }}}_{{\rm{p}}}-{{\boldsymbol{\theta }}}_{{\rm{o}}})}^{T}{{\boldsymbol{C}}}^{-1}({{\boldsymbol{\theta }}}_{{\rm{p}}}-{{\boldsymbol{\theta }}}_{{\rm{o}}})+\mathrm{const}.,\end{eqnarray} \tag{ A8 }$$

where θ = (θ_cmb, z_*) and the subscripts "o" and "p" denote "observation" and "prediction," respectively. Here, z_* only serves as a nuisance parameter, and, since the derived constraint on θ_cmb correlates with z_* for both Planck and ACT+WMAP, we have included their correlation in the θ_cmb likelihood Equation (A8). Thus, the parameters contained in the θ_cmb likelihood are r_* H₀, Ω_m, and h (a very weak dependence).

The two sound horizon scales, r_* at recombination and r_d at the end of the drag epoch, are closely related to each other. They are given by

$$\begin{eqnarray}&&{r}_{* }{H}_{0}={\int }_{{z}_{* }}^{\infty }\displaystyle \frac{{c}_{{\rm{s}}}(z)}{E(z)}{dz},\end{eqnarray} \tag{ A9 }$$

$$\begin{eqnarray}&&{r}_{{\rm{d}}}{H}_{0}={\int }_{{z}_{{\rm{d}}}}^{\infty }\displaystyle \frac{{c}_{{\rm{s}}}(z)}{E(z)}{dz},\end{eqnarray} \tag{ A10 }$$

where c_s is the sound speed. A number of early universe nonstandard models/considerations have been proposed to make both r_* and r_d smaller to mitigate the Hubble tension between Planck and the Cepheid-based local measurement, e.g., Poulin et al. (2019), Kreisch et al. (2020), and Jedamzik & Pogosian (2020). But those models (in fact, almost all early universe nonstandard models) shift the two horizons by approximately the same constant, so their difference remains intact and small. (See discussions in Section 3.2 for the details and possible exceptions.) This permits us to jointly analyze the uncalibrated BAO and the θ_cmb likelihoods by treating r_* H₀ as a free parameter and adopting some reasonable and model insensitive assumptions about the calculation of the difference between r_* H₀ and r_d H₀.

To do this, we take the difference between Equations (A10) and (A9),

$$\begin{eqnarray}&&{\rm{\Delta }}{{rH}}_{0}\equiv ({r}_{{\rm{d}}}-{r}_{* }){H}_{0}={\int }_{{z}_{d}}^{{z}_{* }}\displaystyle \frac{{c}_{{\rm{s}}}(z)}{E(z)}{dz}.\end{eqnarray} \tag{ A11 }$$

In our standard analysis, we first assume that ΔrH₀ can be modeled in the same way as in the standard ΛCDM model, so that we have the sound speed given by

$$\begin{eqnarray}&&{c}_{{\rm{s}}}(z;{{\rm{\Omega }}}_{{\rm{b}}}{h}^{2})=\displaystyle \frac{1}{\sqrt{3\left(1+\tfrac{1}{1+z}\tfrac{3\,{{\rm{\Omega }}}_{{\rm{b}}}{h}^{2}}{4\,{{\rm{\Omega }}}_{\gamma }{h}^{2}}\right)}},\end{eqnarray} \tag{ A12 }$$

and E(z) given by Equation (A3). With Ω_γ h² = 2.47 × 10⁻⁵, c_s(z) is determined if Ω_b h² is given. In the standard analysis, we assume a BBN prior on Ω_b h² = 0.0222 ± 0.0005. In the θ_cmb likelihood, we already assume a prior on z_* obtained from Planck or ACT+WMAP, so the only thing needed to model ΔrH₀ is the redshift difference between z_* and z_d. In the standard analysis, we also assume a prior on Δz_s ≡ z_* − z_d obtained from Planck (Δz_s = 30.26 ± 0.57) or ACT+WMAP (Δz_s = 29.95 ± 0.75). Then, the joint θ_cmb and uncalibrated BAO likelihood, i.e., $\mathrm{ln}{{ \mathcal L }}_{{\theta }_{\mathrm{cmb}}}+\mathrm{ln}{{ \mathcal L }}_{{\rm{u}}.{\rm{c}}.\ \mathrm{bao}}$, contain parameters r_* H₀, Ω_m, and h (a very weak dependence), with some assumed priors on z_*, Δz_s, and Ω_b h².

In a Friedmann–Lemaître–Robertson–Walker universe, the luminosity distance (observed in the heliocentric frame) is related to the comoving distance by d_L = (1 + z_hel)d_M(z_cmb) (see, e.g., Equation (2) in DES Collaboration et al. 2019) and we have

$$\begin{eqnarray}&&m=5{\mathrm{log}}_{10}\left[(1+{z}_{\mathrm{hel}}){f}_{{\rm{M}}}({z}_{\mathrm{cmb}})\right]+{ \mathcal M },\end{eqnarray} \tag{ A13 }$$

where z_hel and z_cmb are the redshifts of SNe Ia observed in the heliocentric frame and in the CMB frame, m is the apparent magnitude of the SN Ia, and ${ \mathcal M }$ is defined as

$$\begin{eqnarray}&&{ \mathcal M }\equiv {M}_{0}-5{\mathrm{log}}_{10}(10\,\mathrm{pc}\times {H}_{0}),\end{eqnarray} \tag{ A14 }$$

with M₀ the absolute magnitude of SNe Ia at some reference stretch and color. We treat ${ \mathcal M }$ as a free parameter in the SN Ia likelihood analysis. The SN Ia likelihood reads as

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}{{ \mathcal L }}_{\mathrm{SN}} & = & -\frac{1}{2}{\left({{\boldsymbol{m}}}_{{\rm{o}}}-{{\boldsymbol{m}}}_{{\rm{p}}}\right)}^{{\rm{T}}}\times {\left({{\boldsymbol{C}}}_{\mathrm{stat}}+{{\boldsymbol{C}}}_{\mathrm{sys}}\right)}^{-1}\\ & & \times \,({{\boldsymbol{m}}}_{{\rm{o}}}-{{\boldsymbol{m}}}_{{\rm{p}}})+\mathrm{const}.,\end{array}\end{eqnarray} \tag{ A15 }$$

where m _o and ${{\boldsymbol{m}}}_{{\rm{p}}}({{\rm{\Omega }}}_{{\rm{m}}},{ \mathcal M })$ are the observed and predicted apparent magnitudes, and C _stat and C _sys are the statistic and systematic covariance matrices. In our standard analyses, the SN Ia likelihood has two free parameters, Ω_m and ${ \mathcal M }$. When analyzing the SN Ia data alone, we fixed h = 0.7 for the radiation and neutrino terms. Since the relevant redshifts are low and these terms are very small, ignoring them leads to almost the same constraint on Ω_m and ${ \mathcal M }$. In this work, we use the Pantheon compilation of SNe Ia data (Scolnic et al. 2018).

Here, we present a fast algorithm ¹⁵ to compute the above SN Ia likelihood, suitable for data sufficiently compact in the redshift space like the Pantheon compilation. The calculation of m _p involves many evaluations of f_M(z); each needs an integration over redshift from 0 to z (see Equation (A2)). For the Pantheon compilation, there are 1048 SNe Ia so that there are 1048 integrals to evaluate. To reduce computational cost, usually the comoving distance is first evaluated for some redshift grids and then interpolation is used to get the comoving distance at each observed redshift. Here, we present an even faster method essentially without a loss of accuracy. This is to take the advantage of the fact that all SN Ia are quite evenly and compactly distributed within 0.01 ≲ z ≲ 2.3, using a Runge–Kutta-like algorithm as follows. We first sort the data in a redshift-ascending order and use i to denote the ith SN Ia. The redshift separation between every adjoining pair of data points is usually small. This allows us to calculate f_M(z_i+1) by adding a small increase to f_M(z_i ). For the first SN Ia with the smallest redshift, we calculate f_M(z₁) using the integral Equation (A2). Then beginning with the second SN Ia, we successively calculate f_M(z_i+1) as

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{\rm{M}}}({z}_{i+1}) & = & {f}_{{\rm{M}}}({z}_{i})+{\rm{\Delta }}{z}_{i,i+1}\displaystyle \sum _{j=1}^{n}{k}_{j},\\ {k}_{j} & = & \displaystyle \frac{{c}_{j}}{E({z}_{i}+{b}_{j}{\rm{\Delta }}{z}_{i,i+1})},\end{array}\end{eqnarray} \tag{ A16 }$$

where Δz_i,i+1 ≡ z_i+1 − z_i and E(z) is defined in Equation (A3). In other words, we are replacing each integral for i ≥ 2 by an addition. The number of k_j terms needed, n, and the coefficients c_j and b_j are determined by the required order of accuracy. The more terms included in Equation (A16), the more accurate but slower the algorithm becomes. In Table 3, we list the values of c and b that allow the error of f_M(z_i+1) − f_M(z_i ) to be of the order ${ \mathcal O }({\rm{\Delta }}{z}_{i,i+1}^{3})$ for including one term, ${ \mathcal O }({\rm{\Delta }}{z}_{i,i+1}^{5})$ for two terms, and ${ \mathcal O }({\rm{\Delta }}{z}_{i,i+1}^{7})$ for three terms. If Δz_i,i+1 > 0.1, we force the algorithm to use Equation (A2) to calculate f_M(z_i+1). We tested our algorithm on a python code, and found that it is about an order of magnitude faster than the algorithm that uses Equation (A2) with the function quad in the scipy package to calculate f_M(z). While having such a high speed, our algorithm does not lose any accuracy. For the case including two k terms in Equation (A16), the numerical difference ($| {\rm{\Delta }}\mathrm{ln}{ \mathcal L }|$) between our approximation and the traditional method is well below 10⁻⁵. Considering both accuracy and speed, we use the two-term algorithm in this work. For the Pantheon SN Ia compilation, we reproduced Ω_m = 0.298 ± 0.022 in the standard ΛCDM model (Scolnic et al. 2018).

Table 3. List of Coefficients for One, Two, and Three Terms to be Included in Equation (A16)

	One Term	Two Terms	Three Terms
Coefficients	c1 = 1, ${b}_{1}=\tfrac{1}{2}$	${c}_{1}=\tfrac{1}{2}$, ${b}_{1}=\tfrac{1}{3-\sqrt{3}}$, ${c}_{2}=\tfrac{1}{2}$, ${b}_{2}=\tfrac{1}{3+\sqrt{3}}$	${c}_{1}=\tfrac{4}{9}$, ${b}_{1}=\tfrac{1}{2}$, c2 = 5/18, ${b}_{2}=\tfrac{1}{5-\sqrt{15}}$, c3 = 5/18, ${b}_{3}=\tfrac{1}{5+\sqrt{15}}$
Orders [accumulated]	${ \mathcal O }({\rm{\Delta }}{z}_{i,i+1}^{3})$ $[{ \mathcal O }({\rm{\Delta }}{z}^{2})]$	${ \mathcal O }({\rm{\Delta }}{z}_{i,i+1}^{5})$ $[{ \mathcal O }({\rm{\Delta }}{z}^{4})]$	${ \mathcal O }({\rm{\Delta }}{z}_{i,i+1}^{7}$ $[{ \mathcal O }({\rm{\Delta }}{z}^{6})]$
Speeds up by	∼14 times	∼8 times	∼6 times

Note. The two-term approximation is what we use with considerations of both numerical accuracy and speed.

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