Turning dispersion into signal: density-split analyses of pairwise velocities

Technical Summary: Turning Dispersion into Signal: Density-Split Analyses of Pairwise Velocities

Problem Statement
Pairwise velocities of large-scale structure encode critical information regarding the growth of structure and the law of gravity. These velocities are observable indirectly through redshift-space distortions (RSD) and the kinetic Sunyaev–Zeldovich (kSZ) effect. However, the pairwise velocity distribution (PVD) is typically broad and non-Gaussian. In conventional analyses, cosmologically useful information is primarily extracted from the mean streaming velocity, while the dispersion around this mean is treated as a nuisance parameter to be marginalized over. This approach effectively averages out the signal, reducing the constraining power of observations. The authors posit that this dispersion is not merely noise but contains structured information that, if properly understood, can be converted into a signal.

Methodology
The authors propose a "density-split" (DS) technique to analyze the PVD by decomposing the global distribution based on the local density environment of the halos/galaxies. Using the Quijote simulations, the methodology proceeds as follows:

1. Environmental Characterization: The halo field is smoothed using a top-hat window function with a radius of $R_{sm} = 25 \, \text{Mpc} \, h^{-1}$. The density contrast ($\delta_h$) is measured at the locations of the halos, creating a probability distribution function (PDF) that is non-Gaussian and skewed toward positive values.
2. Density Binning: The halo sample is split into distinct density bins (ranging from underdense to overdense regions).
3. Conditional PVD Measurement: The pairwise velocity distribution, $P(v_p, r | \delta_h)$, is measured for halo pairs within specific density bins and at various separation scales ($r$).
4. Cross-Bin Inclusion: To ensure information conservation, the analysis includes "cross-bin" pairs (pairs where the two halos belong to different density bins). For $N$ density bins, all $N(N+1)/2$ pair combinations are included in the data vector, ensuring the total pair count matches the conventional global two-point correlation function (2PCF) method.
5. Statistical Comparison: The information content of the DS method is compared against the conventional global pair-counting (2PCF) method using the cumulative and differential $\chi^2$ statistic, defined as $\chi^2 = \bar{v}_p C^{-1} (\bar{v}_p)^T$, where $C$ is the covariance matrix derived from 4000 simulation realizations.

Key Results

• Decomposition of the PVD: The global PVD is revealed to be a composite of a series of near-Gaussian distributions, each corresponding to a specific density environment. While the global distribution is broad and non-Gaussian, the conditional PVDs ($P(v_p | \delta_h)$) are well-fit by Gaussian functions with distinct means and dispersions.
• Sign Reversal and Cancellation: The mean streaming velocity $\bar{v}_p(r | \delta_h)$ changes sign depending on the environment. It is positive (outflow) in underdense regions ($\delta_h < 0$) and negative (infall) in overdense regions ($\delta_h > 0$). In the global PVD, these opposing velocities cancel each other out, resulting in a large dispersion. The DS method avoids this cancellation by isolating these components.
• Amplitude and Dispersion: The amplitude of the mean streaming velocity in overdense regions is approximately four times larger than the global mean (peaking at $\sim 800 \, \text{km s}^{-1}$ compared to $\sim 200 \, \text{km s}^{-1}$ for underdense outflows). Furthermore, the velocity dispersion is significantly smaller in low-density regions compared to the global average.
• Signal-to-Noise Improvement: The DS technique yields a substantial increase in the signal-to-noise ratio. Compared to the global 2PCF method, the total $\chi^2$ increases by factors of 1.4, 3.1, and 3.4 when using 2, 3, and 6 density bins, respectively. On large scales ($r \approx 150 \, \text{Mpc} \, h^{-1}$), the gain reaches a factor of 10.
• Linearity and Gaussianity: The local mean streaming velocities remain linearly coupled to the conditional two-point correlation functions over a wide range of scales. Notably, the improvement persists even on large scales where the global PVD is nearly Gaussian, suggesting that the benefit arises from avoiding the cancellation of opposing flows rather than solely from sampling non-Gaussian density PDFs.

Significance and Claims
The paper claims that the conventional treatment of velocity dispersion as a nuisance is suboptimal. By splitting the data according to local density environments, researchers can "turn dispersion into signal." The primary significance lies in the ability to recover multiple coherent components of outflow and infall velocities that are otherwise hidden in the global average.

The authors assert that this method offers substantial potential for improving cosmological constraints derived from RSD and kSZ observations. Specifically:

• For kSZ, the method allows for the separate observation of pairwise signals with opposite signs in underdense and overdense regions, information that is lost in global averaging.
• For RSD, the method provides a more explicit tracking of cosmological information by isolating the mean streaming velocities in different environments.
• Unlike previous density-split applications that sample density at random locations (which complicates modeling due to non-trivial mappings between real and redshift space), this approach evaluates density at the tracer locations. This ensures that the modeling of DS-RSD remains fundamentally similar to standard 2PCF modeling, with the primary additional factor being an environmental bias that is approximately linear on scales of tens of $\text{Mpc} \, h^{-1}$.

The authors conclude that while two-point statistics are sufficient for Gaussian density fields, they are insufficient for velocity fields, and the density-split technique offers a pathway to extract significantly more information from existing and future observational data.