Luminosity distance dispersion in Swiss-cheese cosmology as a function of the hole size distribution

TL;DR

The work tackles the stochastic dispersion of the luminosity distance $D_L$ caused by small-scale inhomogeneities along light paths, using an exact LTB Swiss-cheese cosmology with a power-law hole-size distribution. Light propagation is computed by integrating geodesic and Sachs equations through consecutive LTB holes, with mass compensation enforced at cheese-hole boundaries via Israel junction conditions. A Monte Carlo framework quantifies $D_L$ dispersion at fixed redshift, revealing a robust scaling $\sigma_{D_L} \propto z^{2.25} (R_{\min}/R_0)^{0.157(\gamma-\gamma_0)}$ and showing dominance by a few large voids; high-$z$ extension yields updated parameters and sizable dispersion relevant for 3G standard sirens. The results suggest subdominant contributions from sub-Mpc voids and highlight the potential of forward-modeling foreground matter to reduce residual dispersion in cosmological distance measurements.

Abstract

The luminosity distance-redshift ($D_{\rm L}$--$z$) relation derived from Type Ia supernovae (SNe Ia) yields evidence for a nonzero cosmological constant. SNe Ia analyses typically fit to the functional form $D_{\rm L}(z)$ derived theoretically from the homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Yet, the metric in the epoch relevant to SNe Ia measurements deviates slightly from FLRW due to gravitational clumping of mass into large-scale structures like filaments and voids, whose sizes span many orders of magnitude. The small deviation is modeled typically by scalar perturbations to the FLRW metric. Each line of sight to a SNe Ia passes through a random sequence of structures, so $D_{\rm L}$ differs stochastically from one line of sight to the next. Here, we calculate the $D_{\rm L}$ dispersion in an exact Lemaitre-Tolman-Bondi Swiss-cheese universe with a power-law hole size distribution, as a function of the lower cut-off $R_{\rm min}$ and logarithmic slope $γ$. We find that the standard deviation of $D_{\rm L}$ scales as $σ_{D_{\rm L}} \propto z^{2.25\pm0.01} (R_{\rm min}/24\pm1\,{\rm Mpc})^{(0.157\pm0.003)\left[γ- (1.16\pm0.02)\right]}$ for redshifts in the range $0.5 \lesssim z \lesssim 2.1$. The scaling shows that the $D_{\rm L}$ dispersion is dominated by a few large voids rather than the many small voids.

Figures (7)

• Figure 1: Comparison between the one-parameter Emmentaler density profile used in this paper (blue curve) and the five-parameter HSW density profile (orange curve). The Emmentaler profile is plotted with $c=0.25$, and the HSW density profile is plotted with $\delta=0.90$, $R_{\rm s}=0.36R$, $R_{\rm eff}=0.91R$$\alpha=2.0$, $\beta=7.1$, which is approximately mass-compensated. The vertical dotted line marks $r=R$, where light beams exit one LTB region and enter another (see Section \ref{['subsec:continuity_conditions']}).
• Figure 2: Schematic diagram showing continuity conditions between two adjacent Swiss-cheese holes. Darker shading corresponds to higher matter density. The geodesic (red arrow) exits hole $\mathcal{H}_{i}$ and enters hole $\mathcal{H}_{i+1}$ at point $\mathcal{O}$ (black dot), where the continuity conditions in Section \ref{['subsec:continuity_conditions']} are applied. The holes overlap slightly by design.
• Figure 3: Schematic diagram of distance measures in cosmology of a source (galaxy) measured by the observer (telescope) and how they relate to the state variables of the beam. Top panel: Angular diameter distance. Bottom panel: Luminosity distance.
• Figure 4: An illustrative example of the $D_{\rm L}$ dispersion near $z_{\rm target} = 2.1$. Top panel. Sample of 50 $(D_{\rm L}, z)$ pairs in the range $2.05 < z < 2.20$ in a Swiss-cheese universe with standard parameters (see Table \ref{['tab:SwissCheeseParams']}), $\gamma = 1.1$, $R_{\rm min} = 10$ Mpc, and $R_{\rm max} = 70$ Mpc (black points). The colour gradient displays the kernel density estimate of the normalized PDF $p(D_{\rm L}, z)$ (see colour bar at right). Although hard to discern by eye, the colour gradient is roughly an ellipse, not a line. The gray dashed line marks $z=z_{\rm target}$. Bottom panel. Slice of the PDF $p(D_{\rm L}, z_{\rm target})$ calculated by evaluating the kernel density estimate for $z=z_{\rm target}$. Note that $p(D_{\rm L}, z)$ is normalized on the $D_{\rm L}$-$z$ plane, but $p(D_{\rm L}, z_{\rm target})$ is an unnormalized slice of $p(D_{\rm L},z)$.
• Figure 5: Luminosity distance dispersion as a function of Swiss-cheese hole size distribution. Each panel displays the standard deviation $\sigma_{D_{\rm L}}$ as a function of the lower cut-off $R_{\rm min}$ (logarithmic vertical axis; $R_{\rm min}$ in units of Mpc) and exponent $\gamma$ (horizontal axis; dimensionless) of the power-law hole size distribution for one redshift in the range $0.5 \leq z_{\rm target} \leq 2.1$ per panel. In each panel, a pixel correspond to a Swiss-cheese universe with standard parameters (see Table \ref{['tab:SwissCheeseParams']}), $R_{\rm max} = 70\, {\rm Mpc}$, $R_{\rm min}$ in the range $-1 \leq \log_{10} (R_{\rm min} / 1\, {\rm Mpc}) \leq 1$, and $\gamma$ in the range $1.1 \leq \gamma \leq 3.1$. The colour of each pixel displays $\sigma_{D_{\rm L}}$ in units of Mpc in logarithmic scale (see colour bar at right). For each universe, $\sigma_{D_{\rm L}}$ is calculated using the method detailed in Section \ref{['sec:measuring_DL_dispersion']} from a sample of 50 $(z, D_{\rm L})$ pairs generated independently using the recipe in Section \ref{['sec:measuring_DL_dispersion']}. For all Swiss-cheese universes, we have $R_{\rm max} = 70\, {\rm Mpc}$, and all other parameters are listed in Table \ref{['tab:SwissCheeseParams']}.