Implicit Likelihood Inference and $z$-Binned Reconstruction of Dark Energy $w(z)$
In this paper, to reconstruct the equation of state (EOS) of dark energy (DE) $w(z)$ with the redshift binning method, we first introduce a $w_i$CDM model with a piecewise-constant EOS in $7$ redshift bins. Then, we turn to the Learning the Universe Implicit Likelihood Inference (LtU-ILI) pipeline to perform a multi-round ILI of $w_i$ from the cosmological data combination, including $TT$, $TE$, $EE$ and lensing power spectra of Planck 2018, distance ratios of DESI DR2 and corrected apparent magnitudes of SNIa from Pantheon+ sample. More precisely, we build the Cosmic Microwave Background (CMB) power spectrum, Baryon Acoustic Oscillation (BAO) distance ratio and Type Ia Supernovae (SNIa) apparent magnitude simulators by $\mathtt{CLASS}$ and embed them into the LtU-ILI pipeline. And, using Sequential Neural Likelihood Estimation (SNLE), we sequentially train neural networks with $6$ rounds of total $6\times20000$ simulations to target a ``black box'' likelihood of our forward model $w_i$CDM. Finally, with the estimated posteriors of $w_i$, we find that except for the unconstrained $w_5$ and $w_6$ (the last two bins), our reconstruction of $w(z)$ marginally favors dynamical DE in the first bin and is consistent with the cosmological constant at $68\%$ C.L. in the other bins.