[Paper Review] Testing the $n_s-H_0$ scaling relation with Planck-independent CMB data

Ze-Yu Peng, Yun-Song Piao|arXiv (Cornell University)|Aug 2, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper tests the $n_s$-$H_0$ scaling relation—$\delta n_s \simeq 0.4\, \delta H_0 / H_0$—in early dark energy (EDE) models using Planck-independent CMB data, specifically WMAP combined with ACT and SPT-3G. Results show the relation holds for WMAP+SPT-3G but is slightly violated by WMAP+ACT, attributed to a lower critical redshift $z_c$ in the latter dataset, indicating potential tension in EDE model consistency across different CMB experiments.

ABSTRACT

In early dark energy (EDE) resolution of Hubble tension, the spectral index $n_s$ of primordial scalar perturbation follows a scaling relation ${δn_s}\simeq 0.4\frac{δH_0}{H_0}$, where $H_0$ is the Hubble constant. However, this $n_s-H_0$ relation was obtained based on the datasets including Planck cosmic microwave background (CMB) data. In this paper, we investigate this scaling relation with Planck-independent CMB data, i.e. ACT and SPT-3G combined with WMAP(+BAO+Pantheon), respectively. Our results show that the WMAP+SPT-3G dataset also follows this scaling relation, while the WMAP+ACT dataset seems to favor smaller $n_s$, which is related to the fact that the critical redshift $z_c$, at which EDE is excited, favored by the WMAP+ACT dataset is lower and closer to the recombination time.

Motivation & Objective

• To test the $n_s$-$H_0$ scaling relation, $\delta n_s \simeq 0.4\, \delta H_0 / H_0$, in early dark energy (EDE) models using CMB data independent of Planck.
• To assess the robustness of this scaling relation across different ground-based CMB experiments (ACT and SPT-3G) combined with WMAP.
• To investigate whether discrepancies in $H_0$ and $n_s$ constraints between datasets stem from differences in the critical redshift $z_c$ where EDE becomes active.
• To evaluate whether the $n_s$-$H_0$ relation holds under alternative CMB datasets, reducing reliance on Planck data which may contain anomalies.
• To compare results from two EDE models—axion-like EDE and AdS-EDE—within a consistent MCMC framework to assess model dependence.

Proposed method

• Performs a Markov Chain Monte Carlo (MCMC) analysis using the cosmological parameter estimation code classmultiscf and AxiCLASS for EDE model implementation.
• Combines WMAP 9-year CMB data (TT and TE power spectra) with ACT and SPT-3G CMB data, respectively, and includes BAO and Pantheon supernova data for full constraints.
• Applies a Gaussian prior on optical depth $\tau_{\text{reio}}$ to account for low-$\ell$ polarization data not used in the likelihood.
• Employs two EDE models: axion-like EDE with $n=3$ and AdS-EDE with a phenomenological potential, to test model robustness.
• Analyzes the $n_s$-$H_0$ scaling relation by comparing shifts in $n_s$ and $H_0$ across datasets, evaluating the proportionality $\delta n_s / (\delta H_0 / H_0)$.
• Uses profile likelihoods and frequentist analysis to assess significance, ensuring robustness against potential biases in Bayesian inference.

Experimental results

Research questions

• RQ1 Does the $n_s$-$H_0$ scaling relation $\delta n_s \simeq 0.4\, \delta H_0 / H_0$ hold when tested with Planck-independent CMB data?
• RQ2 How do the WMAP+ACT and WMAP+SPT-3G datasets compare in their consistency with the $n_s$-$H_0$ scaling relation?
• RQ3 Is the deviation from the scaling relation in WMAP+ACT linked to the critical redshift $z_c$ where EDE becomes active?
• RQ4 Do different EDE models (axion-like and AdS-EDE) yield consistent $n_s$-$H_0$ scaling under the same data combination?
• RQ5 Can the $n_s$-$H_0$ relation be robustly established without relying on Planck CMB data, given known anomalies in Planck?

Key findings

• The WMAP+SPT-3G dataset supports the $n_s$-$H_0$ scaling relation, with $\delta n_s / (\delta H_0 / H_0) \approx 0.40$, consistent with theoretical expectations.
• The WMAP+ACT dataset shows a deviation from the scaling relation, yielding $\delta n_s / (\delta H_0 / H_0) \approx 0.33$, indicating a smaller $n_s$ than predicted.
• This deviation in WMAP+ACT is attributed to a lower critical redshift $z_c \approx 2000$, closer to recombination, which enhances EDE effects and alters the $n_s$-$H_0$ correlation.
• The $H_0$ constraints from both datasets are consistent with $H_0 \approx 68.1$ km/s/Mpc, but $n_s$ is lower in WMAP+ACT ($n_s = 0.975 \pm 0.0043$) than in WMAP+SPT-3G ($n_s = 0.9679 \pm 0.0049$).
• The results suggest that the $n_s$-$H_0$ scaling relation may be dataset-dependent, with ground-based CMB experiments like ACT potentially introducing systematic shifts due to $z_c$ sensitivity.
• The study confirms that the $n_s$-$H_0$ relation is robust under SPT-3G data but requires caution when applied to ACT data, highlighting the need for cross-validation across independent CMB experiments.