Pairwise Momentum Estimator in kSZ Cosmology

• The pairwise momentum estimator is a statistical tool that quantifies the mean momentum difference between galaxy pairs using the kSZ effect.
• It leverages aperture-photometry-filtered CMB temperature differences and a geometric weighting scheme to isolate the velocity signal from noise.
• This method provides robust constraints on cosmological growth parameters by integrating high-resolution CMB data with extensive spectroscopic galaxy surveys via pipelines like Iskay2.

The pairwise momentum estimator is a statistical tool designed to extract the mean pairwise momentum of large-scale structure tracers, typically galaxies or clusters of galaxies, through measurements of the kinematic Sunyaev–Zel’dovich (kSZ) effect. The estimator operates by cross-correlating cosmic microwave background (CMB) temperature maps with spectroscopic galaxy catalogs, leveraging the statistical velocity field of halos to constrain cosmological parameters governing structure growth. Highly efficient pipelines such as “Iskay2” deliver robust kSZ momentum measurements by operationalizing this estimator for large datasets composed of high-resolution CMB surveys and spectroscopic galaxy catalogs (Gallardo et al., 23 Oct 2025).

1. Definition and Mathematical Formalism

The kSZ effect manifests as a Doppler shift in the CMB due to the bulk motion of ionized gas associated with massive halos. In the single-scatter, nonrelativistic regime, the kSZ temperature fluctuation toward a halo or galaxy $i$ is given by:

$$\Delta T_i^{\rm kSZ} = -T_0\,\frac{\sigma_T}{c}\,\tau_i\,v_i^{\rm los}$$

where $T_0$ is the mean CMB temperature, $\sigma_T$ is the Thomson cross-section, $c$ is the speed of light, $\tau_i$ is the line-of-sight optical depth, and $$v_i^{\rm los} = \mathbf{v}_i \cdot \hat{\mathbf{n}}_i$$ is the line-of-sight peculiar velocity component.

While individual $v_i^{\rm los}$ cannot be measured due to overwhelming CMB and noise backgrounds, the mean pairwise momentum at a given separation $r$ is accessible through differencing. The estimator is [Eq. 2.1, (Gallardo et al., 23 Oct 2025)]:

$$\hat p(r) = -\frac{\sum_{(i,j)\in r} c_{ij}\,[T_{AP}^i - T_{AP}^j]}{\sum_{(i,j)\in r} c_{ij}^2}, \qquad c_{ij} = \frac{1}{2}\,\widehat{\mathbf{r}_{ij}} \cdot (\hat{\mathbf{n}}_i - \hat{\mathbf{n}}_j)$$

The aperture-photometry-filtered temperature $T_{AP}^i$ is measured at the catalogued position of galaxy $i$. Sums run over all pairs with comoving separations in bin $r$.

This estimator is proportional to the mean pairwise momentum projected along the separation vector.

2. Derivation and Physical Interpretation

The estimator construction proceeds as follows:

• The kSZ signal for galaxy $i$ is approximated by the aperture-photometry-filtered quantity: $T_{AP}^i \approx \Delta T_i^{\rm kSZ}$.
• For a pair $(i, j)$, the difference $$dT_{ij} = T_{AP}^i - T_{AP}^j = -T_0 \frac{\sigma_T}{c} (\tau_i v_i^{\rm los} - \tau_j v_j^{\rm los})$$ is formed.
• This is projected along the axis connecting the pair with the geometrical weight $c_{ij}$, as above.
• By binning over pair separations and, in practice, assuming a mean optical depth $\bar\tau$ or applying per-halo weights, one constructs a maximum-likelihood estimator of mean pairwise momentum, sensitive to the velocity difference along the separation vector.

The pairwise estimator captures the physical signal $$\langle (v_i^{\rm los} - v_j^{\rm los}) c_{ij} \rangle$$, which is tied to the peculiar velocity field and therefore to the cosmic matter density and growth rate.

3. Pipeline Methodology and Data Processing

High-efficiency implementations such as “Iskay2” utilize a modular pipeline:

• CMB Map Preparation: Foreground-removal is achieved with multi-frequency component separation (e.g., ILC), harmonic-space filtering, and deconvolution of the beam and instrument transfer function. Real-space aperture-photometry (AP) applies a compensated filter, suppressing primary CMB and foreground residuals:

$$T_{AP}^i = \frac{1}{N_{\rm in}} \sum_{\theta < \theta_{\rm in}} T(\hat{\mathbf{n}}) - \frac{1}{N_{\rm out}} \sum_{\theta_{\rm in} < \theta < \theta_{\rm out}} T(\hat{\mathbf{n}})$$

with aperture radii chosen to optimize for target halos.

• Galaxy Catalog Processing: Spectroscopic catalogs (e.g., SDSS LRGs, DESI LRGs) are masked, redshift-cut, and mapped to comoving coordinates using a fiducial $\Lambda$CDM cosmology. Optional weights such as mass proxies or inverse selection functions can be used.
• Pairwise Sum Computation: Fast pair-counting (e.g., Corrfunc) identifies all pairs in $r \pm \Delta r/2$, $c_{ij}$ is computed from geometry, and temperature differences are accumulated.
• Beam and Transfer Corrections: All measurements are divided by the harmonic-space beam and map-making transfer functions. The AP filter transfer function, determined in simulation, is applied as an overall calibration.

4. Theoretical Prediction of the Pairwise Momentum

In linear perturbation theory, the mean pairwise momentum is expressed as:

$$p_{\rm th}(r) = \langle [v_i^{\rm los} - v_j^{\rm los}]\,c_{ij} \rangle = -\frac{T_0\,\sigma_T\,\bar\tau}{c}\; \xi_{v\delta}(r)$$

where:

$$\xi_{v\delta}(r) = \langle v^{\rm los}(\mathbf{x} + \mathbf{r})\,\delta(\mathbf{x})\rangle = \int_0^\infty \frac{k^2\,dk}{2\pi^2}\,P_{v\delta}(k)\,\frac{j_1(kr)}{kr}$$

and the velocity–density cross-power spectrum in linear theory is:

$P_{v\delta}(k) = \frac{f\,H(z)}{k}\,P_{m}(k)$

with $f$ the logarithmic growth rate, $H(z)$ the Hubble parameter, and $P_m(k)$ the matter power spectrum. This links the observed signal to fundamental cosmological quantities, including $f \sigma_8$.

5. Covariance Estimation and Error Analysis

Covariance matrices and error bars on $\hat p(r)$ are evaluated using multiple approaches:

• Bootstrap or Jackknife: The survey footprint is divided into $N_{\rm sub}$ regions; samples are resampled (bootstrap) or omitted (jackknife), and $\hat p(r)$ is recomputed. The resulting covariance estimator is:

$$\mathbf{C}_{ab} = \frac{1}{N_{\rm res} - 1} \sum_{\alpha=1}^{N_{\rm res}} [\hat p_\alpha(r_a) - \overline{\hat p}(r_a)][\hat p_\alpha(r_b) - \overline{\hat p}(r_b)]$$

• Analytic (Gaussian) Approximation: Under the assumption that CMB plus detector noise dominates, the variance is:

$$\mathrm{Var}[\hat p(r)] \approx \frac{\sigma_T^2}{\sum_{(i,j)\in r} c_{ij}^2}$$

• Simulation-Based: Mock catalogs including CMB, instrument noise, and synthetic kSZ are processed, and the scatter of $\hat p(r)$ is measured.

6. Validation, Null Tests, and Performance Assessment

The Iskay2 pipeline validation includes:

• Comparison with Previous Pipelines: Results are cross-compared to earlier releases (Iskay1/C21) using ACT DR5 × SDSS LRGs, with pairwise momentum estimates agreeing within $<1\sigma$ across the separation range. Bootstrapped errors are consistent within statistical fluctuations.
• Null Tests: Several null tests are implemented:
  • Frequency shuffling (“swap” test) of CMB maps: $\hat p(r)$ consistent with zero.
  • Rotating galaxy positions by $90^\circ$ or randomizing redshifts: results consistent with null.
  • Processing simulations with known kSZ inputs and verifying the recovery of $\hat p(r)$.
• Reporting: Typically, the significance of nulls is quantified via reduced $\chi^2$, and detection significance is measured relative to noise expectations.

7. Survey Applications and Projected Sensitivities

Next-generation surveys are expected to substantially advance pairwise kSZ measurements:

• CMB Instruments: ACT DR6 ($\sim 10\,\mu$K-arcmin noise, $1.4'$ beam), SPT-3G, Simons Observatory ($6\,\mu$K-arcmin, $1'$ beam).
• Galaxy Catalogs: DESI LRG+ELG ($N_g \sim 10^6$, $\bar z \sim 0.8$), Euclid, SPHEREx.
• Signal-to-Noise Scaling:

$$\left( \frac{S}{N} \right)^2 \propto N_{\rm pairs}(r) \frac{\bar\tau^2}{\sigma_T^2} \sim N_g^2 \frac{\bar\tau^2}{\sigma_T^2} \frac{\Delta r}{V_{\rm box}}$$

For fixed sky area and aperture, $S/N$ grows as $N_g/\sigma_T$ (with galaxy counts) and inversely with map noise.

Survey Combination	Expected $S/N$ Range	Redshift/Scale
ACT DR6 + DESI	$\gtrsim 6\sigma$	$30 < r < 200$ Mpc/h
Simons Obs. + DESI	up to $\sim 15\sigma$	$30 < r < 200$ Mpc/h

Forecasts indicate that SO + DESI will enable few-percent constraints on $f\sigma_8$.

Recommended practices include matching AP-filter radius to the halo scale ($2$–$3'$ at $z\sim 0.6$), masking point sources and tSZ clusters, and using constrained ILC algorithms. With $N_g\sim10^6$, pairwise computations via Corrfunc run in $\mathcal{O}(10^1)$ minutes on multi-core nodes.

A plausible implication is that the pairwise momentum estimator, as implemented in robust pipelines such as Iskay2, is positioned to deliver precision growth-rate constraints leveraging the synergy of deep galaxy spectroscopy and high-resolution CMB data (Gallardo et al., 23 Oct 2025).