Optical depth to reionization in a Universe with multiple inhomogeneous domains

1. Problem Statement

The standard cosmological model ($\Lambda$CDM) assumes a homogeneous and isotropic universe described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. However, recent observations reveal significant matter inhomogeneities on large scales (e.g., giant arcs, voids) and persistent tensions between early-universe and late-universe measurements, most notably the Hubble tension ($H_0$ discrepancy of $4.7\sigma$ to $6.5\sigma$).

Additionally, the optical depth to reionization ($\tau_{\text{reion}}$), a critical parameter describing the scattering of Cosmic Microwave Background (CMB) photons by free electrons, shows slight tensions between different observational datasets and model assumptions. The standard $\Lambda$CDM model, when constrained by low-redshift supernova data (PantheonPlus+SH0ES), predicts a $\tau_{\text{reion}}$ value that deviates from the Planck CMB constraints.

The authors investigate whether backreaction effects arising from matter inhomogeneities can resolve these discrepancies. Specifically, they ask: Can a cosmological model incorporating the backreaction of inhomogeneous matter distributions yield a value for $\tau_{\text{reion}}$ that aligns better with Planck observations while simultaneously alleviating the Hubble tension, without invoking exotic physics?

2. Methodology

A. Theoretical Framework: Buchert Averaging

The study employs Buchert's averaging formalism, which averages scalar quantities (like density and expansion rate) over space-like hypersurfaces to derive effective cosmological evolution equations.

• Multi-Domain Model: Instead of simplified two-domain models, the authors construct a multi-domain spacetime model consisting of a global domain $D$ divided into multiple disjoint subregions:
  • Overdense regions ($F_{oi}$): Modeled as matter-dominated FLRW regions with positive curvature.
  • Underdense regions ($F_{ui}$): Modeled as almost-empty, matter-dominated FLRW regions with negative curvature (voids).
• Kinematical Backreaction ($Q_D$): The formalism introduces a backreaction term $Q_D$ representing the variance in expansion rates and shear between subdomains. The global evolution is driven by the interplay of these subdomain Hubble parameters ($H_l$) and their volume fractions ($\lambda_l$).
• Parameterization: The model is characterized by six parameters:
  • $h$: Dimensionless Hubble parameter ($H_0 = 100h$).
  • $n$: Total number of subregions (fixed at $n=100$).
  • $\mu_u, \sigma_u$: Mean and standard deviation of the deceleration parameter for underdense regions.
  • $\mu_o, \sigma_o$: Mean and standard deviation of the deceleration parameter for overdense regions.
  • Volume fractions are assigned via Gaussian distributions for the deceleration parameters.

B. Optical Depth Calculation

The optical depth to reionization is calculated using the integral:
$$\tau_{\text{reion}} = n_{H,0} c \sigma_T \int_0^{z_{\text{max}}} \frac{(1+z')^2 \chi(z')}{H(z')} dz'$$

• $H(z)$: Derived from the backreaction model's effective scale factor $a_D(t)$ and the covariant scheme relating theoretical scale factors to observational redshift and angular diameter distance.
• $\chi(z)$: The ionization fraction is modeled using the standard tanh parameterization (Planck collaboration standard), assuming a reionization midpoint $z_{\text{reion}} = 7.68$.

C. Statistical Analysis

• Dataset: The authors use the PantheonPlus+SH0ES Type Ia supernova dataset (distance modulus vs. redshift) to constrain the model parameters.
• MCMC: A Markov Chain Monte Carlo (MCMC) analysis is performed to obtain marginalized posterior distributions for the model parameters ($\mu_u, \sigma_u, \mu_o, \sigma_o, h, M$).
• Comparison: The derived $\tau_{\text{reion}}$ and $H_0$ values from the backreaction model are compared against:
  1. Standard $\Lambda$CDM results constrained by the same supernova data.
  2. Standard $\Lambda$CDM results constrained by Planck CMB data.
  3. Direct Planck PR3 observational constraints on $\tau_{\text{reion}}$.

3. Key Contributions

1. Multi-Domain Backreaction Model: The paper moves beyond toy models by implementing a realistic multi-domain structure with Gaussian distributions of subregion properties, providing a more robust test of backreaction effects on cosmological observables.
2. Unified Pipeline for $\tau_{\text{reion}}$: The authors develop a consistent pipeline where the background expansion history (modified by backreaction) directly feeds into the calculation of the optical depth, ensuring coherence between the cosmological model and the reionization physics.
3. Parameter Sensitivity Analysis: The study identifies that the optical depth is most sensitive to the parameters governing underdense regions (specifically $\mu_u$), which occupy the majority of the cosmic volume (91% in their setup), while overdense parameters have negligible impact.
4. Resolution of Tensions: The work demonstrates that the backreaction model can simultaneously address the Hubble tension and the $\tau_{\text{reion}}$ discrepancy without requiring dark energy equation of state modifications ($\omega < -1$) or other exotic physics.

4. Key Results

• Optical Depth ($\tau_{\text{reion}}$):

  • Using PantheonPlus+SH0ES data, the backreaction model yields:
    $$\tau_{\text{reion}} = 0.0581^{+0.0105}_{-0.0096} \quad (68\% \text{ CL})$$
  • This value is in excellent agreement with the Planck PR3 constraint ($\tau_{\text{reion}} = 0.0544^{+0.0070}_{-0.0081}$) and other observational estimates.
  • In contrast, the standard $\Lambda$CDM model constrained by the same supernova data yields a lower value ($\approx 0.0476$), which is in tension with Planck.

• Hubble Constant ($H_0$):

  • The backreaction model constrained by PantheonPlus+SH0ES gives:
    $$H_0 = 71.46^{+0.89}_{-1.1} \text{ km s}^{-1} \text{Mpc}^{-1}$$
  • This value lies between the Planck CMB value ($\approx 67.3$) and the standard $\Lambda$CDM supernova value ($\approx 73.5$), significantly reducing the Hubble tension (bringing the supernova-derived $H_0$ within $2\sigma$ of the Planck value).

• Parameter Constraints:

  • The MCMC analysis tightly constrains the underdense mean parameter: $\mu_u = 0.01308^{+0.00012}_{-0.0031}$.
  • The dimensionless Hubble parameter is constrained to $h = 0.7146^{+0.0089}_{-0.011}$.

5. Significance

This study provides a compelling alternative to the standard $\Lambda$CDM paradigm by demonstrating that spatial inhomogeneities, when treated via Buchert averaging, can naturally modify the expansion history of the universe.

• Physical Consistency: It achieves better agreement with CMB-derived reionization optical depth and reduces the Hubble tension without introducing new fields, exotic dark energy, or modifying General Relativity.
• Observational Relevance: The results suggest that the "smooth universe" assumption of FLRW may be insufficient for precision cosmology, and that the backreaction of cosmic voids and structures plays a measurable role in interpreting supernova and CMB data.
• Future Directions: The authors note that while their results are robust using supernova data, incorporating additional datasets (BAO, DESI, CMB lensing) could further tighten constraints on the backreaction parameters, potentially solidifying this approach as a viable solution to current cosmological tensions.