Velocity-Weighted Stacking

• Velocity-weighted stacking is a technique that aligns spectral data using expected velocity fields to coherently add faint signals and enhance their detectability.
• It employs reference velocity fields from models or high S/N tracers to regrid data, reducing correlated noise and significantly increasing signal-to-noise ratios.
• Commonly applied in protoplanetary disks, galaxies, and kSZ cosmology, this method uncovers subtle spectral features to inform studies of kinematics and structure.

Velocity-weighted stacking is a class of techniques that enhance the detectability and measurement precision of weak spectral or imaging features in astrophysical systems by aligning data according to an ordered velocity field prior to stacking. These methods exploit known kinematic patterns—such as Keplerian rotation in disks, rotational or radial fields in galaxies, or line-of-sight velocities in large-scale structure—to coherently sum faint signals across spatial or object ensembles. Velocity-weighted stacking is now widely employed in analyses of molecular line spectra in protoplanetary and external galactic disks, as well as in kinematic Sunyaev–Zel'dovich (kSZ) effect studies in cosmology. The approach enhances the signal-to-noise ratio (S/N), reduces correlated noise, and can reveal spectral features otherwise undetectable in individual measurements. This article provides a comprehensive overview of the theoretical foundations, algorithmic procedures, practical implementation, S/N analytics, and typical pitfalls of velocity-weighted stacking across these domains.

1. Fundamental Principles of Velocity-Weighted Stacking

Velocity-weighted stacking leverages the existence of a predictable velocity structure within the target system. Each line of sight or spatial pixel provides a spectrum or observable feature centered at a different (known) velocity due to orbital motion, rotation, or peculiar velocities. By shifting each measurement onto a common velocity grid—aligned so that the expected centroid or physical feature of interest coincides—the true signal adds coherently upon stacking, while the noise, being generally uncorrelated between velocity channels and often partially decorrelated spatially, averages down as in conventional stacking.

In protoplanetary disk applications, the Keplerian nature of rotation prescribes the transformation for alignment so that emission from each disk location is shifted to a zeroed velocity offset (Yen et al., 2016). In extragalactic molecular spectroscopy, the alignment velocity field is commonly derived from higher S/N tracers (often low-$J$ CO or HI 21cm lines) that are assumed to share the kinematic field of fainter molecules (Neumann et al., 2023). In kSZ stacking, the methodology generalizes to align and weight CMB temperature anisotropies at galaxy locations using estimates of the galaxies' line-of-sight velocities, which encode the kSZ signal (Harscouet et al., 16 Dec 2025).

2. Mathematical Formalism and Alignment Transformations

The core mathematical operation is a velocity shift and aligned summation. Denoting each input spectrum or intensity as $S_i(v)$ at location $i$ or object $i$, and a reference velocity $v_{\rm ref}(i)$ supplied either by a physical model (e.g., Keplerian rotation) or an empirical prior (e.g., moment-1 map), the shifted spectrum is

$$S_i'(v) = S_i(v + \Delta v_i), \quad \Delta v_i = -v_{\rm ref}(i)$$

Stochastic noise in $S_i$ remains uncorrelated across most channels provided the shift is not an integer multiple of the original channel spacing.

In protoplanetary disks, the reference velocity is calculated using the disk parameters (inclination $i$, position angle $\mathrm{PA}$, central mass $M_*$, and systemic velocity $V_{\rm sys}$) and the spatial location of the pixel. The precise transform is

$$V_k(r, \phi) = \sqrt{\frac{GM_*}{r}} \sin i \cos \phi,\quad V_{\rm cen}(r, \phi) = V_{\rm sys} \pm V_k(r, \phi)$$

with spatial coordinates deprojected according to the disk geometry (Yen et al., 2016).

For extended galaxies, similar logic applies, but the velocity field may be mapped directly from high S/N lines. In kSZ stacking, the analogous operation is to cross-correlate the CMB map with galaxy locations, weighting by the reconstructed line-of-sight velocity estimates derived from redshift-space distortions or other large-scale structure data (Harscouet et al., 16 Dec 2025).

3. Algorithmic Workflow and Weighting Strategies

The typical stacking workflow encompasses the following steps:

1. Reference Field Determination: Determine or model the velocity field $v_{\rm ref}(x, y)$ at each position. This may be calculated (Keplerian disks), empirically determined (spectral moment or peak-finding), or externally supplied (galaxy velocity estimators).
2. Spectral Extraction and Velocity Alignment: Extract $S_i(v)$ at each location, compute the required shift $\Delta v_i$, and regrid/interpolate onto a common velocity axis such that the signal is aligned (Neumann et al., 2023).
3. Stacking Procedure: Form the weighted sum

$$S_{\rm stack}(v) = \frac{\sum_{i} w_i S_i'(v)}{\sum_i w_i}$$

Common choices for $w_i$ include inverse-variance weighting ($w_i = 1/\sigma_i^2$), equal weights for flux conservation, or priors-based brightness weights ($w_i = S_{\rm prior, i}$). The stacking can be performed on the entire ensemble or within physically-motivated bins (e.g., annuli in disks or radial bins in galaxies).

1. Signal Extraction: Integrate the stacked spectrum or image over the velocity/channel range of interest (typically defined from prior knowledge or from the stacked profile itself) to measure total intensity.
2. Uncertainty Estimation: The rms of the emission-free stacked spectrum, combined with the number of contributing pixels, provides an analytically-tractable estimate of the noise in the stacked quantity.

An explicit recipe is provided in (Neumann et al., 2023) and (Yen et al., 2016) for both imaging cubes and catalog-based stacking.

4. Signal-to-Noise Enhancement and Noise Decorrelation

Velocity-weighted stacking produces S/N gains through two principal effects: (a) the narrowing of the effective line profile after velocity alignment, which concentrates the emission into fewer velocity channels, and (b) the partial decorrelation of noise, particularly in spatially correlated (e.g., interferometric) data. For a line whose initial width in an unaligned stack is $\Delta V_{\rm na}$, and which is reduced to $\Delta V_a$ post-alignment:

$$R_{\rm sn} \simeq \frac{\Delta V_{\rm na}}{\Delta V_a} \sqrt{\frac{A_b}{A_{\rm dep}}}$$

where $A_b$ is the synthesized beam area and $A_{\rm dep}$ is the decorrelated area determined by the velocity gradients across the source (Yen et al., 2016).

Empirical studies show that peak S/N in molecular line stacking can increase by factors of 4–5, equivalent to an order-of-magnitude reduction in required integration time at fixed detection threshold (Yen et al., 2016, Neumann et al., 2023). The improvement in integrated S/N (integrated over velocity) scales as $\sqrt{A_b / A_{\rm dep}}$, and for typical data cubes with well-resolved kinematics and sufficient prior coverage, noise decorrelation is substantial.

Robust uncertainty propagation is possible using standard error scaling in the number of co-added, aligned pixels or spectra. Simulations match the analytic predictions within ~10% accuracy for most practical configurations (Neumann et al., 2023).

5. Applications and Extensions in Astrophysics and Cosmology

Protoplanetary and Extragalactic Disks: Velocity-aligned stacking is now routine in studies of molecular line emission in disks where the kinematic field is well-understood. For example, applied to the ALMA dataset for HD 163296, velocity-weighted stacking revealed previously undetectable H$_2$CO transitions and allowed the measurement of spatial intensity distributions of faint tracers (Yen et al., 2016). In external galaxies, stacking enables recovery of lines below individual detection limits when using a kinematic prior (e.g., from CO), with high fidelity for combined interferometric + total power datasets (Neumann et al., 2023).

kSZ Stacking in Cosmology: In the analysis of the kinematic Sunyaev–Zel'dovich signal, real-space velocity-weighted stacking—implemented as an aperture photometry of the CMB anisotropy field weighted by galaxy velocities—is now analytically unified and subsumed by the pseudo-$C_\ell$ cross-spectral estimators (Harscouet et al., 16 Dec 2025). These approaches give access to both the real-space stack and the full harmonic-space information, enabling fast, accurate covariance estimation and efficient computational scaling for galaxy samples of $N_{\rm gal} \sim 10^5$ and above.

Recent developments clarify the theoretical modeling of the velocity-weighted kSZ angular power spectrum, requiring the inclusion of multiple terms: the dominant 1-halo and 2-halo convolution terms, disconnected cross-terms, connected trispectrum (non-Gaussian) components, and negligible longitudinal velocity contributions (Wayland et al., 23 Sep 2025). The signal is sensitive to galaxy and electron bias parameters, velocity reconstruction fidelity, and satellite fractions. All relevant terms must be modeled for unbiased constraints from next-generation CMB experiments.

6. Practical Implementation and Best Practices

Implementation of velocity-weighted stacking requires high-fidelity estimation of the velocity field, consistent data regridding, careful uncertainty accounting, and attention to flux conservation. Best practices, as recommended in (Neumann et al., 2023) and (Yen et al., 2016), include:

• Use of a high S/N prior tracing the target kinematics, ideally covering the full spatial region of interest.
• Equal-weight stacking for flux conservation; use inverse-variance or brightness weighting when the measurement goal is maximization of S/N for detection.
• Inclusion of single-dish (total power) data in interferometric studies to mitigate loss of extended emission.
• Defining stacking bins to ensure adequate prior detection fractions (e.g., $F_{\rm det} \gtrsim 0.3–0.5$) to avoid bias from undetected pixels.
• Conservative significance thresholds ($\gtrsim 3\sigma$) when reporting new detections from stacks.

The public PyStacker Python package implements these steps for radio-interferometric data, offering a reproducible framework for velocity-weighted stacking tasks (Neumann et al., 2023).

7. Limitations, Assumptions, and Generalizations

Velocity-weighted stacking assumes a coherent and well-characterized velocity field. Deviations from order—arising from turbulent motions, warps, misestimation of the kinematic parameters (such as disk inclination, central mass, or systemic velocity), or structural asymmetries—lead to incomplete alignment, residual line broadening, and a reduction in the achievable S/N gain (Yen et al., 2016). For protoplanetary and extragalactic disks, this provides a potential route to parameter optimization by maximizing the stack S/N as a function of these inputs.

The methodology can also be applied beyond rotating disks to any astrophysical scenario where the kinematic pattern is known or can be modeled: for example, infalling envelopes, galaxy rotation fields inferred from HI velocity maps, outflows with known gradients, or even large-scale structure velocity fields in the context of cosmological cross-correlations (Harscouet et al., 16 Dec 2025).

In all applications, care must be taken in the treatment of nondetections, the propagation of uncertainties (especially when stacking submaps with incomplete prior coverage), and the interpretation of brightness-weighted versus flux-conserving mean quantities.

In summary, velocity-weighted stacking constitutes a model-light but highly effective technique for extracting weak signals in the presence of complex, ordered kinematic backgrounds. Its mathematical rigor, capacity for S/N enhancement, and adaptability across observational domains make it central to modern data analysis in astrophysics and cosmology (Yen et al., 2016, Neumann et al., 2023, Harscouet et al., 16 Dec 2025, Wayland et al., 23 Sep 2025).