The Most Probable Behaviour of the Dark Energy Equation of State Indicates a Thawing Quintessence Field: Tomographic Alcock-Paczyński Test with Redshift-Space Correlation Function II

TL;DR

This study extends the tomographic Alcock-Paczyński test to SDSS data to constrain the CPL dark energy equation of state without relying on CMB priors. By leveraging the full redshift-space two-point correlation function, its normalization, and Legendre multipoles, the authors extract the expansion history while mitigating amplitude evolution, calibrating nonlinear effects with Multiverse simulations and validating with Horizon Run 4 mocks. The results favor a slowly evolving dark energy (no phantom divide crossing up to $z\approx0.7$), consistent with a thawing quintessence scenario, especially when combined with low-redshift probes like Pantheon$+$ SN and DESI BAO; inclusion of CMB data moves the constraints toward a crossing at low redshift. The work demonstrates the robustness and potential of the AP test as a complementary probe to BAO and SNe, with future DESI DR2 data expected to tighten the constraints further.

Abstract

We apply an extended Alcock-Paczyński (AP) test to the Sloan Digital Sky Survey data to constrain the dark energy models with the Chevallier-Polarski-Linder (CPL) parametrization of the dark energy equation of state. The extended AP test method uses the full shape of redshift-space two-point correlation funcion(CF) as the standard shape in order to measure the expansion history of the universe. We calibrate the standard shape by using the cosmology-dependent nonlinear evolution of the CF shape in the Multiverse simulations. Further validation of the method and calibration of possible systematics are performed based on mock samples from the Horizon Run 4 simulation. Using the AP test alone, we constrain the flat CDM plus CPL-type dark energy model (flat $w^{\rm CPL}$CDM) to have $Ω_m=0.290_{-0.031}^{+0.029}$, $w_0=-0.800_{-0.100}^{+0.208}$, and $w_a=-0.238_{-0.972}^{+0.650}$. When combined with other results from the low-redshift universe, such as the Pantheon$+$ supernova compilation and DESI BAO data, the constraint on dark energy becomes $w_0=-0.857_{-0.042}^{+0.051}$, and $w_a=-0.153_{-0.356}^{+0.347}$. The best-fit $w^{CPL}(z)$ suggests no phantom-divide crossing at $z<0.7$, and the dark energy behaviour is consistent with a thawing quintessence field. It is only when the CMB data are combined with late-time cosmological probes that a phantom-divide crossing at low redshift is favored.

Figures (6)

• Figure 1: Comparison of the true and approximate CF. The solid line represents the true value obtained from precise numerical calculations in the cosmology with $\Omega_m=0.26$ and $w=-1.5$, and the dash-dotted line represents the approximation derived from calculation in $\Omega_m=0.31$ and $w=-1$ cosmology and then conversion to $\Omega_m=0.26$ and $w=-1.5$ cosmology using Equation \ref{['EQ:APD']}. The lower panel shows the difference between them; a constant value of 1 has been added to the CF in this comparison to avoid numerical instabilities near zero.
• Figure 2: Constraints on the flat $w^{\rm CPL}$CDM model from SDSS AP-only (magenta), DESI-Y3 BAO-only (DR2, grey), and SDSS AP+DESI BAO (orange) analyses. The sub-panels show the marginalized probability distribution function of each cosmological parameter. The likelihood contours from the BAO analysis and AP test exhibit some overlapping distributions in the parameter space. However, the AP test shows a good consistency with $w_a=0$ while BAO favors $w_a<0$.
• Figure 3: Similar to Figure \ref{['fig:result_APBAO']}, but Pantheon$+$ SNe I$a$ data (grey) is included for joint-constraints. The orange contours are from combining the constraints of three low-redshift probes of SDSS AP, DESI BAO, and Pantheon$+$ SN I$a$. The probability distribution of $w_0$ and $w_a$ from the joint analysis indicates that the dark energy equation of state is most likely to evolve slowly with no phantom-divide crossing up to $z=0.7$.
• Figure 4: The most probable evolution of the dark energy equation of state parameter $w(z)$ within the CPL parametrization, obtained from combining SDSS AP with DESI BAO and Pantheon$+$ SN data (orange line). The light orange contour represents the $68\%$ confidence region. The red dot denotes the Dong et al. (2023)'s measurement of $w_{\rm eff}$ within the flat $w$CDM paradigm derived from the same data. The horizontal error bar represents the redshift range of the observational samples. For comparison, the green line shows the corresponding constraints derived from CMB + DESI BAO + Pantheon$+$ SN2025PhRvD.112h3515A.
• Figure 5: Results for the posterior distributions of $w_0$ and $w_a$, derived from the combined constraints of SDSS AP, DESI BAO and Pantheon+ SN data on the flat $w^{\rm CPL}$CDM model, excluding (orange) and including CMB (red). The CMB has relatively much weaker constraining power and the favored region in the $w_0$-$w_a$ plane is greatly shifted from those of low-redshift probes. The inclusion of CMB data significantly pulls down the cosmological constraint away from $w_a=0$, driving it to a phantom-divide crossing scenario.