Analytical approximations for primordial power spectra in a spatially closed emergent universe

The emergent universe scenario was proposed to solve the big bang singularity by suggesting that the universe originates from an Einstein static state and then evolves into a subsequently inflationary era. Thus, to find the relic of the existence of the Einstein static state becomes a crucial work. In this paper, we derive analytical approximation of the primordial power spectra and analyze the CMB TT-spectra for the spatially closed emergent universe. After analyzing the CMB TT-spectrum of the emergent universe scenario, we find that both the CMB TT-spectra produced by the Einstein static state followed by the ultraslow-roll inflationary epoch (method I) and by a special evolution of the scale factor in the emergent scenario as $a=a_{0}+A e^{H_{0}t}$(method II) are suppressed at $l<30$, and their spectra are nearly identical. Additionally, by comparing the spectra of the emergent universe scenario with the ones of the ultraslow-roll inflationary model in the closed universe, we find that the CMB TT-spectrum of the emergent universe is similar to the one of the inflationary model with the special case $\eta_t = \eta_{max}$.