Redshift Asymmetry of Cosmic Voids

• The paper demonstrates that redshift asymmetry distribution quantifies the line-of-sight distortions in void–galaxy correlations induced by peculiar velocities using multipole expansion.
• Redshift asymmetry is defined as the systematic anisotropy in cosmic void profiles that encodes key cosmological information such as the growth rate, galaxy bias, and dark energy effects.
• The paper employs advanced simulation and observational pipelines, including void finders, stacking techniques, and PDF modeling, to minimize systematics and robustly constrain inflationary and gravitational models.

The redshift asymmetry distribution of cosmic voids quantifies the line-of-sight anisotropy in galaxy or matter flows around these large underdense regions caused by peculiar velocities in an expanding universe. Redshift-space distortions (RSD) induce systematic effects—typically squashing or stretching—in the observed void–galaxy (or void–matter) cross-correlation, which encode key dynamical and cosmological information including the linear growth rate, bias parameters, and imprints of inflationary physics. This topic is central in large-scale structure cosmology and is extensively modeled and measured in galaxy, quasar, and Lyman-α surveys. Current theoretical and observational efforts focus on robust estimators for the RSD-induced asymmetry, the universality of their statistical distributions, and their applications to probing gravity, dark energy, and inflation.

1. Physical Origin and Mathematical Formalism of Redshift Asymmetry in Voids

In redshift space, the observed position $s$ of a galaxy near a void differs from its real-space comoving position $x$ due to line-of-sight peculiar velocity $v_\parallel$: $s_\parallel = x_\parallel + v_\parallel/(aH)$. This mapping distorts spherical voids into ellipsoidal or more complex surfaces, with the cross-correlation $\xi^s(s_\parallel,s_\perp)$ between void centers and galaxies becoming anisotropic along the line of sight.

Linear theory predicts that, on large scales, the void–galaxy cross-correlation in redshift space is well described by the Kaiser formula: $\xi^s(s,\mu) = (1 + \beta\mu^2)^2\xi^r(s)$ where $\beta=f/b$ is the redshift-space distortion parameter ($f$ is the linear growth rate, $b$ is galaxy bias), $\mu$ is the cosine of the angle to the line of sight, and $\xi^r(s)$ is the real-space density profile (Ravoux, 2022). This leads to a characteristic quadrupole-to-monopole ratio: $$\frac{\xi_2^s(s)}{\xi_0^s(s)} = \frac{\frac{4}{3}\beta + \frac{4}{7}\beta^2}{1 + \frac{2}{3}\beta + \frac{1}{5}\beta^2}$$ which is independent of separation $s$ for linear scales (Hawken et al., 2016, Cai et al., 2016).

Redshift asymmetry can also be characterized by the line-of-sight elongation parameter, often denoted

$$A(r) = \frac{\xi_2^s(r)}{\xi_0^s(r) - \frac{3}{r^3}\int_0^r dr' r'^2 \xi_0^s(r')}$$

with canonical values $A = \frac{2f}{3+f}$ (Cai et al., 2016), or the axis ratio $q_z/q_\perp$ of isocontours (Chuang et al., 2016).

2. Measurement Techniques: Multipole Expansion, Stacking, and Asymmetry PDFs

Observationally, redshift-space asymmetry is extracted by various methods:

• Multipole Expansion: The 2D cross-correlation $\xi^s(s_\parallel,s_\perp)$ is expanded onto Legendre polynomials:

$\xi^s(s,\mu) = \sum_\ell \xi_\ell^s(s) P_\ell(\mu)$

The monopole $\xi_0^s$ and quadrupole $\xi_2^s$ dominate the signal, with the ratio sensitive to $f/b$ (Nadathur et al., 2017, Ravoux et al., 2022).

• Stacked Void Imaging: Isodensity contours of stacked voids expose the line-of-sight flattening or stretching as an axis ratio $A(z)$, expressible as $A(z)=1+v_p(HR)^{-1}$ for a shell of proper velocity $v_p$ (Maeda et al., 2011).
• Probability Density Functions (PDFs): The distribution of asymmetry measures (e.g., ellipticities, maximal line-of-sight redshift differences $\Delta z_\mathrm{max}$) over a void sample is modeled analytically or fit with parametric forms. For rotational asymmetry, the Gamma distribution

$$p(\Delta z_{\mathrm{max}};k,\theta) = \frac{(\Delta z_{\mathrm{max}})^{k-1}}{2\Gamma(2k)\theta^k} \exp\left[-(\Delta z_{\mathrm{max}}/\theta)^{1/2}\right]$$

offers a nearly universal fit across cosmologies, with $k\approx1.19$ and $\theta$ linearly dependent on $\Omega_m$ and $\sigma_8$ (Kang et al., 10 Jan 2026).

Systematics arising from random velocities, survey geometry, shot noise, and selection functions must be modeled and mitigated, often using large mock catalogs and stacking multiple voids to suppress Poisson noise (Shoji et al., 2012).

3. Universality and Cosmological Parameter Dependence

Simulations and analytic models demonstrate that many functional forms of void redshift asymmetry—e.g., the Gamma PDF of $\Delta z_{\mathrm{max}}$—are highly universal across flat $\Lambda$CDM and $w$CDM cosmologies. The functional form remains fixed with $k=1.19$, while the scale parameter $\theta$ is tightly controlled by linear combinations of $\Omega_m$ and $\sigma_8$. Pivotal coefficients ($\eta_1, \eta_2, \eta_3$) for the scale parameter are calibrated to high precision in simulation suites (e.g., AbacusSummit) (Kang et al., 10 Jan 2026).

The universality breaks only for models with running primordial spectral index $\alpha_s \ne 0$, in which $\theta$ diverges linearly from its prediction—positive (negative) $\alpha_s$ yield lower (higher) asymmetry than $\Lambda$CDM (Kang et al., 10 Jan 2026). Consequently, the void redshift asymmetry distribution provides a direct probe of inflationary mechanisms, independently from late-time cosmological parameters.

For classical growth-rate inference, the quadrupole-to-monopole ratio and axis ratio are sensitive to $f(z)$, while Alcock–Paczynski tests can disentangle dynamical and geometric distortions by comparing stacked void shapes (Shoji et al., 2012, Maeda et al., 2011). Measurements across redshift bins further enable tracking the evolution of $f\sigma_8$ and constraints on dark energy (Hawken et al., 2016, Ravoux, 2022).

4. Simulation and Observational Pipelines

State-of-the-art analyses employ:

• Void Finders: Algorithms such as empty-sphere searches (Hawken et al., 2016), ZOBOV, and the Hoyle–Vogeley method (Kang et al., 10 Jan 2026) to identify void centers in spectroscopic surveys.
• Redshift-space Realization: Assign mock galaxy positions according to velocities to simulate observed catalogs (Kang et al., 10 Jan 2026).
• Cross-correlation Estimation: Davis–Peebles estimator and similar pair counting over void–galaxy or void–pixel separations; ratio to random pairs yields unbiased two-point functions (Micheletti et al., 2014).
• Bisector Scans: For spinning voids, all bisector orientations in the projected plane are scanned to maximize the line-of-sight velocity asymmetry, allowing measurement of coherent rotational signatures (Kang et al., 10 Jan 2026).
• Stack Control: Voids are binned by effective radius to standardize asymmetry statistics across cosmological models (Kang et al., 10 Jan 2026).

Quantitative fits to the RSD parameter $\beta$ are achievable to a precision of $2-10\%$ in large survey volumes (Hawken et al., 2016, Ravoux, 2022, Nadathur et al., 2017), and the measured asymmetry profiles agree with linear and Gaussian streaming models, incorporating velocity dispersion when needed (Nadathur et al., 2017, Cai et al., 2016, Achitouv, 2017).

5. Key Results and Applications in Large Surveys

Recent analyses report:

• VIPERS ($0.55 < z < 0.9$): $\beta=0.423^{+0.104}_{-0.108}$, $f\sigma_8 = 0.296^{+0.075}_{-0.078}$ at $z\sim0.73$; anisotropy matches the Gaussian streaming model (Hawken et al., 2016).
• eBOSS-Ly$\alpha$ ($z>2$): $\beta=0.52\pm0.05$ from void–Ly$\alpha$ forest cross-correlations, demonstrating viability of high-redshift growth measurement (Ravoux, 2022, Ravoux et al., 2022).
• Large N-body Simulations: Asymmetry ratio $A\approx0.30$ for void–halo cross-correlation in linear regime; PDF is Gaussian in large samples (Cai et al., 2016). Voids with more prominent walls (“void-in-cloud”) show diverse distortion signatures.
• Inflationary Constraints: The distribution $p(\Delta z_\mathrm{max})$ matches the Gamma form unless $\alpha_s\ne0$; residuals $\Delta\theta$ scale nearly linearly with running parameter, enabling independent diagnostics of inflation (Kang et al., 10 Jan 2026).

Stacked void analysis, ellipticity PDFs, and multipole ratios now yield competitive precision on cosmological parameters, with the added benefit of environmental sensitivity distinct from classical galaxy RSD analyses (Shoji et al., 2012, Nadathur et al., 2017, Achitouv, 2017).

6. Limitations, Systematics, and Future Prospects

Key challenges include:

• Random Velocity Dispersion: On sub-void scales, random motions can reverse or mask the coherent streaming asymmetry, necessitating careful modeling (Nadathur et al., 2017, Cai et al., 2016).
• Poisson Noise and Sampling: Small number of void member galaxies washes out ellipticity and asymmetry PDFs, requiring stacking and large void catalogs (Shoji et al., 2012).
• Survey Geometry and Tomography: Finite sight-line density in Ly$\alpha$ mapping introduces geometric quadrupoles; correction via "shuffled" mocks is essential (Ravoux et al., 2022).
• Void Finder Sensitivity: The bias parameter $\beta_v$ for voids depends on the details of void construction, generally distinct from the galaxy value $\beta_g$ unless voids are linear tracers (Chuang et al., 2016).

Forthcoming surveys (DESI, WEAVE-QSO, LSST) will expand tomographic resolution and sample sizes, reducing errors in asymmetry measurement to sub-5% and enabling robust multi-redshift growth-rate and inflationary tests (Ravoux, 2022, Kang et al., 10 Jan 2026). Improved reconstructions and velocity mapping techniques promise enhanced void detection fidelity, especially in the low-density regime where deviations from vanilla gravity or inflation may emerge.

7. Theoretical Implications and Cosmological Significance

The redshift asymmetry distribution around voids is a uniquely sensitive probe of the velocity field in underdense regions, reflecting both local dynamical properties (outflow, compensation wall structure, streaming) and fundamental cosmological parameters, including the growth rate $f$, clustering amplitude $\sigma_8$, and the running of the primordial spectral index $\alpha_s$.

Its universality in standard cosmologies permits the measurement of departures attributable exclusively to non-standard inflation or modified gravity, decoupled from late-time dark energy parameters. The ellipticity PDF of stacked voids, axis ratios, and multipole decomposition complement existing galaxy clustering and BAO metrics, extending growth and expansion history constraints into previously inaccessible redshift and environmental ranges.

In total, the redshift asymmetry distribution of cosmic voids—parameterized via multipole ratios, axis ratios, maximal velocity differences, and robust PDFs—constitutes a powerful observational and theoretical tool for precision cosmology (Hawken et al., 2016, Ravoux, 2022, Ravoux et al., 2022, Kang et al., 10 Jan 2026, Cai et al., 2016, Nadathur et al., 2017, Micheletti et al., 2014, Chuang et al., 2016, Shoji et al., 2012, Maeda et al., 2011, Paul et al., 5 Aug 2025, Achitouv, 2017).