Updated Cosmological Constraints in Extended Parameter Space with Planck PR4, DESI Baryon Acoustic Oscillations, and Supernovae: Dynamical Dark Energy, Neutrino Masses, Lensing Anomaly, and the Hubble Tension

While the six-parameter Λ Cold Dark Matter (ΛCDM) model has been widely considered the standard model of cosmology, the recent cosmological constraints from the Dark Energy Spectroscopic Instrument (DESI) collaboration (A. G. Adame et al. 2024) have brought into question whether dark energy (DE) can be described by a cosmological constant, i.e., Λ. One popular choice for an evolving or dynamical DE is the w₀ w_a CDM model, where the equation of state (EoS) of DE is described by the Chevallier–Polarski–Linder (CPL) parameterization (M. Chevallier & D. Polarski 2001; E. V. Linder 2003), given by w(z) ≡ w₀ + w_a z/(1 + z), where z is the redshift. One of the main results in the DESI collaboration paper shows that in the w₀ w_a CDM model, the cosmological constant is rejected by 2.5σ and 3.9σ when tested against the Pantheon+ (D. Brout et al. 2022) and the Dark Energy Survey Year 5 (DESY5; T. M. C. Abbott et al. 2024) supernovae (SN) data sets, respectively (when combined with DESI baryon acoustic oscillations (BAO), and cosmic wave background (CMB) data). The tremendous implication of said results have led to a large number of subsequent studies concerning DE (see, e.g., K. V. Berghaus et al. 2024; S. Bhattacharya et al. 2024; Y. Carloni et al. 2024; M. Cortês & A. R. Liddle 2024; K. S. Croker et al. 2024; B. R. Dinda & R. Maartens 2024; G. Efstathiou 2024; I. D. Gialamas et al. 2024; W. Giarè et al. 2024; J.-Q. Jiang et al. 2024b; K. Lodha et al. 2024; P. Mukherjee & A. A. Sen 2024; L. Orchard & V. H. Cárdenas 2024; Y.-H. Pang et al. 2024; C.-G. Park et al. 2024; O. F. Ramadan et al. 2024; J. A. Rebouças et al. 2024; N. Roy 2024; D. Shlivko & P. J. Steinhardt 2024; Y. Tada & T. Terada 2024; H. Wang & Y.-S. Piao 2024; W. J. Wolf et al. 2024; W. Yin 2024).

However, the DESI BAO collaboration results are not uncontested. In M. Cortês & A. R. Liddle (2024), the authors point out that while the fit to w(z) shows a dynamical nature, at the redshift ranges that are most strongly constrained by the data, the DE EoS remains close to the ΛCDM value of w(z) = −1, to a precision around ±0.02. In G. Efstathiou (2024), the author points out possible systematics present in the DESY5 SN data that might be leading to the preference for the non-ΛCDM behavior. In particular, the author finds that the SNe common to both the Pantheon+ and DESY5 samples have an offset of 0.04 mag. between low and high redshifts, and correcting this offset leads to the DESY5 data set being in agreement with ΛCDM, similar to Pantheon+. It is important to note that the analyses in the said paper have been done with SN data alone (i.e., not in combination with BAO and CMB).

Apart from DE, neutrinos are important constituents of the Universe. While neutrinos are massless in the standard model of particle physics, terrestrial neutrino oscillation experiments have strongly confirmed the presence of three different neutrino masses, of which at least two must be nonzero for neutrino-flavor oscillations to occur. Given the mass-squared splittings ${\rm{\Delta }}{m}_{21}^{2}\simeq 7.42\times {10}^{-5}$ eV² and $| {\rm{\Delta }}{m}_{31}^{2}| \simeq 2.51\times {10}^{-3}$ eV², there are two possible neutrino-mass orderings: normal ordering (∑m_ν > 0.057 eV) and inverted ordering (∑m_ν > 0.096 eV; I. Esteban et al. 2020). Apart from the evidence for dynamical DE, one of the main results of the DESI BAO collaboration paper (A. G. Adame et al. 2024) was the strongest bound on the sum of neutrino-mass parameter until then, ∑m_ν < 0.072 eV (95% confidence limit (CL)), in the minimal ΛCDM+ ∑m_ν model with a ∑m_ν > 0 prior against CMB+BAO data. This bound is much stronger than the bound of ∑m_ν < 0.18 eV circa 2016 (E. Giusarma et al. 2016), ∑m_ν < 0.12 eV circa 2018 (S. Vagnozzi et al. 2017, 2018; E. Giusarma et al. 2018; S. Roy Choudhury & S. Choubey 2018; S. Roy Choudhury & A. Naskar 2019; S. Roy Choudhury & S. Hannestad 2020; I. Tanseri et al. 2022), or the bound of ∑m_ν < 0.09 eV circa 2021 (E. Di Valentino et al. 2021). While the DESI bound is not strong enough to rule out inverted ordering at more than 2σ, later studies have put forward even stronger constraints with additional background probes (J.-Q. Jiang et al. 2024a; D. Wang et al. 2024), all using Planck Public Release (PR) 3–based likelihoods for CMB anisotropies. However, some other studies have shown that the DESI collaboration bound weakens somewhat when the new Planck PR4 likelihoods are used instead of the Planck PR3 (2018) ones (I. J. Allali & A. Notari 2024; D. Naredo-Tuero et al. 2024; H. Shao et al. 2024), primarily due to the absence of the lensing anomaly in Planck PR4 (M. Tristram et al. 2024). One also needs to take into account the fact that in the DESI collaboration results (A. G. Adame et al. 2024), the ΛCDM model is disfavored compared to the w₀ w_a CDM model, which begs the question whether one should take the bounds in the ΛCDM+∑m_ν model seriously. The dynamical DE parameters are correlated with ∑m_ν (especially w_a ) through the "geometric degeneracy" (S. Roy Choudhury & S. Hannestad 2020; see Equations (3.2) and (3.3) and the related text in this cited paper). The strong degeneracy between the DE EoS and neutrino masses was first noticed in S. Hannestad (2005). While it is known that the quintessence-like or non-phantom DE (w(z) ≥ −1 at all redshifts) leads to stronger-than-ΛCDM constraints on ∑m_ν (S. Vagnozzi et al. 2018; S. Roy Choudhury & A. Naskar 2019; S. Roy Choudhury & S. Hannestad 2020; J.-Q. Jiang et al. 2024a), the w₀−w_a parameter space preferred in the DESI collaboration paper in a w₀ w_a CDM model falls outside the non-phantom/quintessence region (A. G. Adame et al. 2024; M. Cortês & A. R. Liddle 2024), and the bounds obtained on the ∑m_ν are relaxed by more than a factor of 2 compared to the ones in the ΛCDM+∑m_ν model.

After neutrinos become nonrelativistic at late times (due to their small masses), they start contributing to the matter density of the Universe. They suppress the small-scale structure by avoiding clustering at small scales due to high thermal velocities. This, in turn, affects the lensing of the CMB photons, the effect of which shows up in the small scales (l ≥ 200) of the CMB power spectrum as an enhancement of the two-point correlation functions (J. Lesgourgues & S. Pastor 2012; S. Roy Choudhury 2020). In a particular model, the theoretical prediction for the gravitational potential (responsible for generating the weak lensing of the CMB) corresponds to A_lens = 1, where A_lens is the scaling of the lensing amplitude. When A_lens is varied as a cosmological parameter, the weak lensing is decoupled from the primary anisotropies that produce it and then scaled by the value of A_lens (E. Calabrese et al. 2008). Thus, A_lens serves as a consistency-check parameter. If the data favor A_lens > 1 in a particular model, it indicates a preference for more smoothing of the acoustic peaks in the power spectra (typically due to lensing) than what is theoretically expected. The "lensing anomaly" in the Planck PR3 (2018) high-l data (N. Aghanim et al. 2020a; Planck Team 2020) is well known (N. Aghanim et al. 2020b), where one finds A_lens = 1.18 ± 0.065 (68%) in the ΛCDM+A_lens model. This value of A_lens is more than 2σ away from the expected A_lens = 1. This lensing anomaly, however, has been shown to have reduced to less than 1σ in the same ΛCDM+A_lens model with the Planck PR4 (2020) likelihoods (M. Tristram et al. 2024). Since massive neutrinos suppress the matter power spectrum in small scales, increasing ∑m_ν decreases weak lensing of CMB photons, and thus, ∑m_ν and A_lens are strongly correlated. This provides a strong motivation to use Planck PR4 (2020) likelihoods instead of Planck PR3 (2018).

A_lens is also degenerate with the curvature density parameter Ω_k (E. Di Valentino et al. 2019), and the lensing anomaly is intimately related to the curvature tension (E. Di Valentino et al. 2019; W. Handley 2021), where Planck PR3 (2018) likelihoods alone preferred a closed Universe, and Ω_k = 0 was rejected at more than 2σ. However, the curvature tension goes away once other data sets like BAO and SNe are included in the analysis (N. Aghanim et al. 2020b). This is in sharp contrast to the lensing anomaly, which persists at more than 2σ even when other data sets are included with Planck PR3. Thus, in this work, we only consider the lensing anomaly while keeping Ω_k fixed at zero.

In this Letter, we also focus on the Hubble tension, which is currently one of the most discussed topics in cosmology. The Cepheid-calibrated, Type-Ia-supernova (SN Ia)-based distance-ladder measurement of H₀ by the SH0ES collaboration provides H₀ = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹ (A. G. Riess et al. 2022; local Universe measurement). This result is discrepant at the level of 4.6σ with the Planck PR4 (2020) measurement of H₀ = 67.64 ± 0.52 km s⁻¹ Mpc⁻¹ (M. Tristram et al. 2024) in the ΛCDM model. This discrepancy is known as the "Hubble tension." The dynamical DE parameters are correlated with H₀ via the geometric degeneracy (G. Efstathiou & J. R. Bond 1999; S. Roy Choudhury & S. Hannestad 2020; S. Vagnozzi 2020) and might largely loosen the bounds on H₀ from CMB and BAO data. However, tighter constraints are expected when the uncalibrated SNe Ia data from Pantheon+ or DESY5 are included as these data sets constrain the late-time background evolution well, thereby partially breaking the degeneracy. At the same time, Hubble tension might also be alleviated to a certain extent by introducing extra radiation species in the early Universe by adding to the N_eff (the effective number of non-photon radiation species in the early Universe), as this increases the Hubble expansion rate in the pre-recombination era. However, N_eff also leads to a phase shift of the CMB power spectrum toward larger scales and a damping of the power spectrum that increases as we go to smaller scales (D. Baumann 2018). To completely resolve the Hubble tension, a contribution of ΔN_eff ≃ 1 is needed, which is currently strongly disfavored by Planck data (N. Aghanim et al. 2020b; S. Vagnozzi 2023), and this is true even in the presence of neutrino self-interactions, which can partially counter the effects of N_eff (S. Roy Choudhury et al. 2021, 2022; N. Bostan & S. Roy Choudhury 2024).

The S₈ tension, which is a discrepancy in the measurement of the amplitude of the matter clustering between low-redshift weak-lensing surveys and the high-redshift CMB experiments, is another puzzle in cosmology that has received significant attention in the cosmology community. In the ΛCDM model, with Planck PR3 likelihoods, the S₈ tension stands at the level of ∼2.4σ with Dark Energy Survey (DES) Year 3 analysis (N. Aghanim et al. 2020b; T. M. C. Abbott et al. 2022). However, with the Planck PR4 likelihoods, this tension drops to 1.5σ (M. Tristram et al. 2024). However, again, there has been no assessment of this tension in a dynamical DE cosmology with Planck PR4 likelihoods. Other than weak lensing, the disagreement with redshift space distortion measurements and Planck PR3 remains at a moderate 2.2σ level (R. C. Nunes & S. Vagnozzi 2021).

Given the above issues, it is important to have a closer look at the DESI collaboration results in A. G. Adame et al. (2024). In this study, one of our main aims is to check whether the preference for dynamical DE survives in a largely extended parameter space where we allow for variation in usual extensions to ΛCDM: sum of neutrino masses (∑m_ν ) and effective number of non-photon radiation species (N_eff), the scaling of the lensing amplitude (A_lens), and the running of the scalar spectral index (α_s ). This takes the total number of varying parameters to 12, which is double the number of parameters in the vanilla ΛCDM model. For previous studies in such largely extended parameter spaces, see E. Di Valentino et al. (2015, 2016, 2017, 2020), V. Poulin et al. (2018), and S. Roy Choudhury & A. Naskar (2019). Another major aim of this paper is to provide a robust bound on the ∑m_ν parameter, which can be used by the cosmology and particle physics community, by using an extended parameter space incorporating dynamical DE and also the scaling of the lensing amplitude parameter, A_lens, which is degenerate with ∑m_ν (S. Roy Choudhury & S. Hannestad 2020). Thus, including A_lens makes the bound obtained on ∑m_ν in this study more robust. At the same time, DE itself modifies the gravitational potential in the very late times, thus affecting the lensing of CMB photons. Until now, there has been no study in literature which has varied the A_lens parameter in a dynamical DE cosmology with Planck PR4 (2020) likelihoods. This provides us another motivation to include A_lens in our analysis to give a definitive answer regarding the lensing anomaly in a dynamical DE scenario. Also, we do not expect to completely resolve the Hubble tension with this simple extension to ΛCDM cosmology studied in this Letter. One of our main goals in this Letter is, rather, to assess the level of this Hubble discrepancy in our 12-parameter extended cosmological model with the new data sets. We also aim to assess the level of discrepancy with the weak-lensing measurement of the S₈ parameter in a dynamical DE cosmology. With the release of Planck PR4 likelihoods (2020): HiLLiPoP and LoLLiPoP (M. Tristram et al. 2024); Planck PR4 lensing combined with Atacama Cosmology Telescope (ACT) Data Release (DR) 6 lensing likelihoods (M. S. Madhavacheril et al. 2024); DESI DR1 BAO likelihoods (A. G. Adame et al. 2024); and the latest uncalibrated SN Ia likelihoods: Pantheon+ (D. Brout et al. 2022) and DESY5 (T. M. C. Abbott et al. 2024), we think it is timely to update the constraints in such an extended model. The results will be undoubtedly useful to the cosmology and particle physics community.

Our Letter is structured as follows: in Section 2, we describe our analysis methodology. In Section 3, we discuss our results from the statistical analysis. We conclude in Section 4.

We note the cosmological model, parameter sampling and plotting codes, and priors on parameters in Section 2.1. In Section 2.2, we discuss the cosmological data sets used in this Letter.

2.1. Cosmological Model and Parameter Sampling

Here is the parameter vector for this extended model with 12 parameters :

$\begin{eqnarray}\begin{array}{rcl}\theta & \equiv & \left[{\omega }_{c},\,{\omega }_{b},\,{{\rm{\Theta }}}_{s}^{* },\,\tau ,\,{n}_{s},\,\mathrm{ln}({10}^{10}{A}_{s}),\right.\\ & & \left.{w}_{0,\mathrm{DE}},{w}_{a,\mathrm{DE}},{N}_{{\rm{eff}}},\sum {m}_{\nu },{\alpha }_{s},{A}_{{\rm{lens}}}\right].\end{array}\end{eqnarray} \tag{ 1 }$

The first six parameters correspond to the ΛCDM model: the present-day cold dark matter energy density, ω_c ≡ Ω_c h²; the present-day baryon energy density, ω_b ≡ Ω_b h²; the reionization optical depth, τ; the scalar spectral index, n_s , and the amplitude of the primordial scalar power spectrum, A_s (both evaluated at the pivot scale k_* = 0.05 Mpc⁻¹); and ${{\rm{\Theta }}}_{s}^{* }$, which represents the ratio between the sound horizon and the angular diameter distance at the time of photon decoupling.

The rest of the six parameters are the ones with which we extend the ΛCDM cosmology. For the CPL parameterization for DE EoS, in this Letter, we use the notation (w_0,DE, w_a,DE) interchangeably with (w₀, w_a ). The other parameters, as noted in the Section 1, are the effective number of non-photon radiation species (N_eff), the sum of neutrino masses (∑m_ν ), the running of the scalar spectral index (α_s ), and the scaling of the lensing amplitude (A_lens).

We note here that we are using the degenerate hierarchy of neutrino masses (m_i = ∑m_ν /3 for i = 1, 2, 3, i.e., all three neutrino masses are equal), and we are using a prior ∑m_ν ≥ 0. This choice is suitable considering that cosmological data are only sensitive to the neutrino energy density, and hence, the total mass sum (J. Lesgourgues & S. Pastor 2012), and even the near future cosmological data, would not be able to detect the neutrino-mass splittings since they are very small (M. Archidiacono et al. 2020). Also, forecasts show that in the event of an actual detection of ∑m_ν , the assumption of degenerate hierarchy (instead of the true hierarchy) leads only to a negligible bias (M. Archidiacono et al. 2020). There is also no conclusive evidence for a particular neutrino-mass hierarchy even when cosmological results are combined with other terrestrial data sets from particle physics, like neutrino oscillations or beta decay (S. Gariazzo et al. 2022).

We also note here that since we are varying the running of the scalar spectral index (${\alpha }_{s}\equiv {{dn}}_{s}/d\mathrm{ln}k$, where k is the wavenumber), we are, in turn, assuming a standard running power-law model for the primordial primordial power spectrum of scalar perturbations, given by

$\begin{eqnarray}&&{\bf{ln}}{{ \mathcal P }}_{s}(k)=\mathrm{ln}\,{A}_{s}+({n}_{s}-1)\,\mathrm{ln}\left(\displaystyle \frac{k}{{k}_{* }}\right)+\displaystyle \frac{{\alpha }_{s}}{2}{\left[\mathrm{ln}\left(\displaystyle \frac{k}{{k}_{* }}\right)\right]}^{2}.\end{eqnarray} \tag{ 2 }$

A small value of log₁₀∣α_s ∣ = −3.2 is naturally expected from slow-roll inflationary models (J. Garcia-Bellido & D. Roest 2014). However, it can be larger in certain other inflationary scenarios (see, e.g., D. J. H. Chung et al. 2003; R. Easther & H. Peiris 2006; K. Kohri & T. Matsuda 2015).

Parameter Sampling. We use the cosmological inference code Cobaya (J. Torrado & A. Lewis 2019, 2021) for all the Markov Chain Monte Carlo analyses in this Letter. For theoretical cosmology calculations, we use the Boltzmann solver CAMB (A. Lewis et al. 2000; C. Howlett et al. 2012). When utilizing the combined Planck PR4+ACT DR6 lensing likelihood, we apply the higher-precision settings recommended by ACT. We use the Gelman and Rubin statistics (S. P. Brooks & A. Gelman 1998) to estimate the convergence of chains. All our chains reached the convergence criterion of R − 1 < 0.01. We use GetDist (A. Lewis 2019) to derive the parameter constraints and plot the figures presented in this Letter. We employ broad flat priors on the cosmological parameters, as given in the Table 1.

Table 1. Flat Priors on the Main Cosmological Parameters Constrained in This Letter

Parameter	Prior
Ωb h2	[0.005, 0.1]
Ωc h2	[0.001, 0.99]
τ	[0.01, 0.8]
ns	[0.8, 1.2]
$\mathrm{ln}({10}^{10}{A}_{s})$	[1.61, 3.91]
100 ${{\rm{\Theta }}}_{s}^{* }$	[0.5, 10]
w0,DE	[−3, 1]
wa,DE	[−2, 2]
Neff	[2, 5]
∑mν (eV)	[0, 5]
αs	[−0.1, 0.1]
Alens	[0.1, 2]

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2.2. Data Sets

The key results from our cosmological parameter estimation analyses are presented in Table 2. Here, we describe our results regarding the cosmological parameters:

Table 2. Bounds on Cosmological Parameters in the 12-parameter Extended Model

Parameter	Planck PR4	Planck PR4	Planck PR4	Planck PR4	Planck
		+ Lensing	+ Lensing +DESI	+Lensing +DESI + PAN+	Lensing + DESI + DESY5
Ωb h2	0.02227 ± 0.00024	0.02235 ± 0.00022	0.02237 ± 0.00020	0.02242 ± 0.00020	0.02239 ± 0.00021
Ωc h2	0.1184 ± 0.0030	0.1193 ± 0.0028	0.1192 ± 0.0027	0.1191 ± 0.0029	0.1190 ± 0.0028
τ	0.0579 ± 0.0065	0.0582 ± 0.0064	0.0583 ± 0.0062	0.0585 ± 0.0066	0.0584 ± 0.0064
ns	0.967 ± 0.010	0.971 ± 0.010	0.971 ± 0.009	0.974 ± 0.009	0.972 ± 0.009
$\mathrm{ln}({10}^{10}{A}_{s})$	3.038 ± 0.017	3.043 ± 0.015	3.044 ± 0.016	3.043 ± 0.016	3.042 ± 0.016
100 ${{\rm{\Theta }}}_{s}^{* }$	1.04079 ± 0.00042	1.040758 ± 0.00038	1.04079 ± 0.00039	1.04081 ± 0.00040	1.04082 ± 0.00039
∑mν (eV)	<0.700 (2σ)	${0.211}_{-0.21}^{+0.060}$ (1σ),	${0.173}_{-0.12}^{+0.086}$ (1σ),	<0.292 (2σ)	${0.151}_{-0.14}^{+0.055}$ (1σ),
		<0.450 (2σ)	<0.339 (2σ)		<0.318 (2σ)
Neff	3.03 ± 0.21	3.09 ± 0.20	3.10 ± 0.18	3.13 ± 0.20	3.10 ± 0.19
w0,DE	$-{01.09}_{-0.80}^{+0.67}$	-0.96 ± 0.64	$-{0.45}_{-0.22}^{+0.32}$	−0.848 ± 0.066	$-{0.745}_{-0.078}^{+0.070}$
wa,DE	unconstrained	unconstrained	< − 0.37 (2σ)	$-{0.61}_{-0.29}^{+0.36}$	$-{0.95}_{-0.34}^{+0.41}$
αs	−0.0053 ± 0.0078	−0.0031 ± 0.0075	−0.0031 ± 0.0072	−0.0022 ± 0.0074	−0.0030 ± 0.0074
Alens	${1.094}_{-0.11}^{+0.066}$	${1.084}_{-0.090}^{+0.044}$	${1.061}_{-0.052}^{+0.046}$	${1.064}_{-0.053}^{+0.045}$	${1.063}_{-0.053}^{+0.046}$
H0 (km s−1 Mpc−1)	${71}_{-20}^{+10}$	${73}_{-20}^{+10}$	${64.6}_{-3.3}^{+2.3}$	68.1 ± 1.1	67.2 ± 1.1
S8	${0.774}_{-0.046}^{+0.051}$	0.787 ± 0.040	0.822 ± 0.022	${0.809}_{-0.017}^{+0.020}$	${0.811}_{-0.018}^{+0.022}$

Note. Marginalized limits are given at 68% C.L., whereas upper limits are given at 95% C.L. Note that H₀ and S₈ are derived parameters.

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3.1.  w_0,DE and w_a,DE

As can be seen from Figures 1 and 2, the dynamical DE EoS parameters remain poorly constrained without the use of SNe Ia data sets. In fact, w_a is unconstrained from both sides with CMB-only data. Very interestingly, we find that when we combine CMB and BAO data with Pantheon+, the cosmological constant (w_0,DE = −1, w_a,DE = 0) is allowed at 2σ. This is a significant result considering that the exclusion of the cosmological constant in the DESI BAO collaboration paper (A. G. Adame et al. 2024) at more than 2σ in a more restrictive eight-parameter w₀ w_a CDM model. Our result implies that invoking simple extensions to w₀ wa CDM cosmology can reconcile the cosmological constant again with the data. We hypothesize that there might be two distinct reasons for this: (i) the degeneracy with the additional parameters might expand the allowed w₀–w_a parameter space, and (ii) the Planck PR4 likelihoods might be partially responsible given that the DESI collaboration results are derived with Planck PR3. With the DESY5 SNe Ia data, we, however, find that the cosmological constant is rejected at more than 2σ, and the same is true for non-phantom/quintessence-like DE (w(z) ≥ −1 at all redshifts). However, we advise caution, since, as we discussed in Section 1, the DESY5 measurements might have unknown systematics as pointed out in G. Efstathiou (2024). Therefore, it can be concluded that the results from the DESI BAO collaboration regarding the evidence for dynamical nature of the DE EoS are not yet definitive.

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3.2. ∑m_ν

The 1D marginalized posterior distributions of ∑m_ν for various data set combinations are given in Figure 3, whereas in Figure 4, we present the 68% and 95% contour plots between ∑m_ν and the DE EoS parameters. While we do not have a 2σ detection of nonzero neutrino masses, we find that each of the 1D posterior distributions have a peak, and for three of the five data set combinations studied in this Letter, we get a 1σ detection. This is encouraging since it points to potential future detection of nonzero neutrino masses with cosmological data. At the same time, the bounds on ∑m_ν presented in this Letter are much more relaxed compared to the bounds in the ΛCDM model in A. G. Adame et al. (2024). While w_0,DE and ∑m_ν are only weakly correlated, ∑m_ν has a strong negative correlation with w_a,DE. This is one of the primary reasons for the more relaxed bounds. There are other parameters that are degenerate with ∑m_ν , as we show later. However, inclusion of those degenerate parameters also makes these bounds more robust, especially given the current uncertainty involving the nature of DE. With Planck PR4 data, the bound is quite relaxed at ∑m_ν < 0.700 eV (95%). This bound improves step by step as we include lensing, BAO, and SN data. With Planck PR4+lensing+DESI+PAN+, we get a bound of ∑m_ν < 0.292 eV (95%), whereas with DESY5 instead of PAN+, the bound is very similar: ∑m_ν < 0.318 eV (95%). Given the robustness of these bounds, we suggest that the cosmology and particle physics community use this bound of ∑m_ν ≲ 0.3 eV (95%) as a reference.

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3.3.  A_lens

This measurement represents a crucial test of the consistency of the standard cosmological model as deviations from A_lens = 1 could indicate new physics, such as changes in the growth of cosmic structures or modifications to gravity. The 1D marginalized posterior distributions of A_lens and the correlation plots with ∑m_ν are given in Figure 5. As expected, we find that A_lens has a strong positive correlation with the ∑m_ν parameter. More importantly, we find that all data set combinations yield constraints on A_lens that are consistent with the theoretical expectation of A_lens = 1 at less than 2σ. With Planck PR4+lensing+DESI+PAN+, we obtained a constraint of ${A}_{\mathrm{lens}}={1.064}_{-0.053}^{+0.045}$ (68%). We get the almost same bound with DESY5 instead of PAN+. This highlights the reliability of our conclusions across different SN samples. The absence of a significant deviation in the lensing amplitude implies that there is no strong evidence for a lensing anomaly within the context of a dynamical DE cosmology when using the latest Planck PR4 likelihoods. This is an important finding, especially when contrasted with earlier studies that employed Planck PR3 likelihoods, which had reported indications of a potential lensing anomaly.

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3.4.  N_eff

The 1D marginalized posterior distributions of N_eff and the correlation plots with ∑m_ν are given in Figure 6. We find that all the obtained bounds on N_eff are fully consistent at <1σ level with the theoretical value of N_eff assuming standard model of particle physics, i.e., ${N}_{\mathrm{eff}}^{\mathrm{SM}}=3.044$ (K. Akita & M. Yamaguchi 2020; J. Froustey et al. 2020; J. J. Bennett et al. 2021). We also find that while N_eff does not have any significant correlation with ∑m_ν , adding the SN Ia data sets leads to a slight tilt in the correlation direction in the 2D contour plots.

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3.5.  α_s

The 1D marginalized posterior distributions of α_s for different data combinations are given in Figure 7. We find that all the obtained bounds on α_s are consistent with α_s = 0 within 1σ. This lack of significant running implies that more complex inflationary models with multiple fields or nonstandard dynamics may not be necessary to explain the observed data.

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3.6.  H₀

The 1D marginalized posterior distributions of H₀ are given in Figure 8, and the 2D correlation plots with N_eff and ∑m_ν are given in Figure 9. We notice that CMB data alone cannot constrain H₀, and essentially a very large range of H₀ is allowed when only CMB data are used. The key reason for this is the strong degeneracy with the DE EoS parameters through the geometric degeneracy (G. Efstathiou & J. R. Bond 1999; S. Roy Choudhury & S. Hannestad 2020). However, we find that H₀ is also strongly correlated with ∑m_ν in the CMB-only data, through the same geometric degeneracy. But this degeneracy gets partially broken with DESI BAO data, and it is broken even further with the use of SN data, as can be clearly seen from the bottom panel of Figure 9. This degeneracy-breaking quality of BAO and SNe Ia have been studied in literature previously (see, e.g., S. Roy Choudhury & S. Choubey 2018). On the other hand, CMB-only data produce no visible degeneracy in the H₀–N_eff plot since this degeneracy is weakened by the presence of the other degeneracies stated above. However, once the DE EoS parameter space is well constrained with the use of BAO and SNe Ia, we see the clear, strong, positive correlation between H₀ and N_eff in the top panel of Figure 9. Regarding the Hubble tension, we note that there is a 3.2σ tension between Planck PR4+lensing+DESI+PAN+ and the SH0ES (A. G. Riess et al. 2022) measurements, which increases to 3.9σ when DESY5 SNe Ia are used instead of PAN+. Thus, it is clear that the simple extensions of ΛCDM studied in this Letter are not sufficient to resolve the Hubble tension completely, i.e., below the 2σ level. Therefore, the resolution of Hubble tension requires more complicated new physics (assuming data systematics are not responsible for the tension).

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3.7.  S₈

Here, we concentrate on the tension associated with the amplitude of matter clustering in the late Universe, which is quantified by the parameter ${S}_{8}\equiv {\sigma }_{8}{\left(\tfrac{{{\rm{\Omega }}}_{m}}{0.3}\right)}^{0.5}$. In this expression, σ₈ represents the root mean square of the amplitude of matter perturbations smoothed over a scale of 8h⁻¹ Mpc, where h is the Hubble constant in units of 100 km s⁻¹ Mpc⁻¹ and Ω_m denotes the current matter density parameter. With Planck PR4+lensing+DESI+PAN+ we obtain a bound of ${S}_{8}={0.809}_{-0.017}^{+0.020}$. This result is discrepant at the level of ≃1.4σ with the measurement derived from galaxy clustering and weak lensing from the DES Year 3 analysis, S₈ = 0.776 ± 0.017 (for ΛCDM with fixed ∑m_ν ; T. M. C. Abbott et al. 2022). While the Planck PR3 data set is in a ∼2.4σ tension with DES Year 3 results (N. Aghanim et al. 2020b; T. M. C. Abbott et al. 2022), the reduction in this S₈ tension with Planck PR4 likelihoods is consistent with a previous result in literature (M. Tristram et al. 2024), and the additional parameters considered in this Letter might be increasing the error bar on S₈, thereby reducing the tension further, albeit slightly. In Figure 10, we provide the 2D contour plots between Ω_m and $\sqrt{0.3}{S}_{8}\equiv {\sigma }_{8}{{\rm{\Omega }}}_{m}^{0.5}$.

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We have presented updated constraints on cosmological parameters within a 12-parameter framework that extends the standard six-parameter ΛCDM model to include additional degrees of freedom: the parameters for dynamical DE (w_0,DE, w_a,DE), the sum of neutrino masses (∑m_ν ), the effective number of non-photon radiation species (N_eff), a scaling parameter for the lensing amplitude (A_lens), and the running of the scalar spectral index (α_s ). For CMB data, we utilized the latest Planck PR4 (2020) HiLLiPoP and LoLLiPoP likelihoods, in conjunction with Planck PR4+ACT DR6 lensing and Planck 2018 low-ℓ TT likelihoods. Additionally, we incorporated the latest DESI DR1 BAO likelihoods, as well as uncalibrated SN Ia data from the Pantheon+ and DESY5 samples.

Our primary findings are as follows:

• 1.  
  In contrast to the conclusions of the DESI collaboration, we found that the combination of CMB+BAO+Pantheon+ data does not exclude a cosmological constant at more than 2σ, whereas the combination of CMB+BAO+DESY5 data excludes it at a significance level greater than 2σ. Therefore, the evidence for a dynamical DE component is not yet compelling or robust. This discrepancy may be driven by potential systematics in the DESY5 SN sample, as has been suggested in a recent study (G. Efstathiou 2024).
• 2.  
  We observed that for certain data combinations, there is a 1σ level indication for the sum of neutrino masses, ∑m_ν , although this does not reach the 2σ level detection threshold. Furthermore, all data set combinations yielded a posterior peak, indicating a promising trend toward a possible future detection of nonzero neutrino masses. At the same time, we established a robust upper bound of ∑m_ν < 0.292 eV at 95% C.L. with CMB+BAO+Pantheon+, which is slightly more relaxed at ∑m_ν < 0.318 eV with DESY5 SN data. Given the current uncertainties regarding the true underlying cosmological model, we recommend that this bound of ∑m_ν ≲ 0.3 eV (95%) is used by the cosmology and particle physics communities as a reference.
• 3.  
  Combining CMB, DESI BAO, and Pantheon+ data, we found that the scaling of the lensing amplitude parameter has the following constraint: ${A}_{\mathrm{lens}}={1.064}_{-0.053}^{+0.045}$ (68%). With DESY5 SN data instead of Pantheon+, we obtained a similar constraint. These results are consistent with the expected theoretical prediction of A_lens = 1 at the 2σ level (though not at 1σ). This implies that there is no significant lensing anomaly within the context of a dynamical DE cosmology when using Planck PR4 likelihoods (contrary to earlier findings in literature with Planck PR3–based likelihoods). The discrepancy between the PR3- and PR4-based results suggests that improvements in data quality and analysis techniques play a crucial role in resolving such anomalies. Therefore, our findings contribute to the ongoing efforts to refine our understanding of cosmic structure formation and the potential role of new physics in cosmology.
• 4.  
  Despite considering a significantly extended cosmological model, the Hubble tension persists at a significance level of 3.2σ with CMB+BAO+Pantheon+ data combination, whereas the tension is at 3.9σ with CMB+BAO+DESY5. Therefore, the simple extensions to ΛCDM explored in this study are insufficient to resolve the Hubble tension to below the 2σ level.
• 5.  
  The S₈ tension with the DES Year 3 analysis is reduced to ≃1.1σ with CMB+BAO+SNe data. Reduction in tension to 1.5σ from the 2σ level (with Planck PR3) has been previously documented with Planck PR4 in ΛCDM (M. Tristram et al. 2024). Another reason for further reduction to 1.4σ can be attributed to the inclusion of additional parameters (which increase the error bars on S₈) in our Letter.

We acknowledge the use of the HPC facility at ASIAA (https://hpc.tiara.sinica.edu.tw/) where the numerical analyses were done. T.O. acknowledges support from the Taiwan National Science and Technology Council under grant Nos. NSTC 112-2112-M-001-034- and NSTC 113-2112-M-001-011-.