Testing the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> scaling relation with Planck-independent CMB data
Ze-Yu Peng, Yun-Song Piao
Abstract
In early dark energy (EDE) resolution of Hubble tension, the spectral index ${n}_{s}$ of primordial scalar perturbation follows a scaling relation $\ensuremath{\delta}{n}_{s}\ensuremath{\simeq}0.4\frac{\ensuremath{\delta}{H}_{0}}{{H}_{0}}$, where ${H}_{0}$ is the Hubble constant. However, this ${n}_{s}\ensuremath{-}{H}_{0}$ relation was obtained based on the data sets that include 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 $\mathrm{WMAP}(+\mathrm{BAO}+\mathrm{Pantheon})$, respectively. Our results show that the $\mathrm{WMAP}+\mathrm{SPT}\text{\ensuremath{-}}3\mathrm{G}$ data set also follows this scaling relation, while the $\mathrm{WMAP}+\mathrm{ACT}$ data set 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 $\mathrm{WMAP}+\mathrm{ACT}$ data set is lower and closer to the recombination time.