---
title: 'VDG∞: Galaxy Redshift-Space Clustering Model'
url: https://www.emergentmind.com/topics/velocity-difference-generator-vdg
type: topic
---

# VDG∞: Galaxy Redshift-Space Clustering Model

The velocity difference generator (VDG), specifically the VDG$_\infty$ model, is a theoretical framework for modeling the galaxy two-point correlation function (2PCF) in redshift space. It employs a cumulant expansion to capture the non-perturbative effects of galaxy peculiar velocities, particularly the non-Gaussian "Finger-of-God" (FoG) damping, enabling accurate cosmological parameter estimation at small spatial separations relevant for Stage-IV surveys such as those conducted by Euclid [2601.04780].

## 1. Formalism and Mathematical Structure

The VDG$_\infty$ model operates on the redshift-space moment
\[
\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle
\]
where $u_z(x)$ denotes the line-of-sight peculiar velocity at position $x$, $f$ is the linear growth rate, $\delta(x)$ is the density fluctuation, $k$ is the Fourier wavenumber, and $\mu$ is the cosine of the angle to the line-of-sight.

A damping prefactor, $D(k,\mu)$, is factored out to capture non-perturbative FoG effects, and the remainder is expanded to one loop. The final model for the redshift-space galaxy power spectrum has the form
\[
P^s_{gg}(k,\mu) = D(k,\mu) \Big[ P_{gg}(k) + 2\mu^2 f P_{g\theta}(k) + \mu^4 f^2 P_{\theta\theta}(k) + C_A(k,\mu) + C_B(k,\mu) \Big]
\]
where:
- $P_{gg}, P_{g\theta}, P_{\theta\theta}$ are the real-space galaxy and velocity power spectra,
- $C_A$ and $C_B$ are loop corrections treated with a bias expansion.

The distinctive feature of VDG$_\infty$ is the non-Gaussian damping kernel,
\[
D(k,\mu;\sigma_v,a_{\rm vir}) = \frac{1}{\sqrt{1-\lambda^2 a_{\rm vir}^2}} \exp\left(-\frac{\lambda^2 \sigma_v^2}{1-\lambda^2 a_{\rm vir}^2}\right)\,,
\]
where $\lambda = i f k \mu$, $\sigma_v^2$ is computed from the linear power spectrum,
\[
\sigma_v^2 = \frac{1}{3} \int \frac{d^3 q}{(2\pi)^3} \frac{P_L(q)}{q^2},
\]
and $a_{\rm vir}$ is a free kurtosis parameter encoding deviations from Gaussianity in the pairwise velocity distribution.

To obtain configuration-space multipoles, the model employs the transformations:
\[
\xi_\ell(s) = \frac{2\ell+1}{2} \int_{-1}^1 d\nu\,\mathcal L_\ell(\nu)\,\xi(s,\nu),
\quad
P_\ell(k) = \frac{2\ell+1}{2} \int_{-1}^1 d\mu\,\mathcal L_\ell(\mu)\,P^s_{gg}(k,\mu)
\]
followed by an FFTlog Hankel transform.

## 2. Parameterization and Model Components

The VDG$_\infty$ parameterization incorporates cosmological, bias, damping/FoG, and counterterm parameters:

| Category                 | Parameters                                                        | Features / Priors               |
|--------------------------|--------------------------------------------------------------------|---------------------------------|
| Cosmology / Linear Power | Fixed templates: $f, \sigma_8$; Full-shape: $h, 10^9 A_s, \omega_c$ (optionally $w_0$) | Vary per snapshot               |
| Galaxy Bias (EFT)        | Eulerian $b_1, b_2$; Non-local $\gamma_1 \equiv b_{G2}$, $\gamma_2 \equiv b_{\Gamma3}$ | Redshift-independent            |
| FoG / Non-Gaussianity    | $a_{\rm vir}$ (kurtosis), prior $\mathcal U[0, 10]$                | Marginalized                    |
| EFT Counterterms         | $c_0$, $c_2$, $c_4$ (multiply $\mathcal L_{0,2,4}(\mu)\,k^2 P_L(k)$), priors $\mathcal U[-500,500]$ | Redshift-independent            |

Bias and counterterm priors remain constant across redshifts, while cosmological parameters are allowed to vary with each simulation snapshot.

## 3. Computational Implementation

The computational pipeline for VDG$_\infty$ utilizes the following elements:
- Power spectra and correction terms ($C_A, C_B$) are generated by the COMET emulator, delivering $P_{gg}, P_{g\theta}, P_{\theta\theta}$ plus counterterms in $\sim 10$ ms.
- $D(k,\mu)$ is built directly in COMET, with $P^s_{gg}(k,\mu)$ multipoles $P_\ell(k)$ computed by Gaussian quadrature over $\mu$.
- Hankel transforms to configuration space are performed using FFTlog/Talman (as implemented in "hankl"), including a mild Gaussian damping factor $\exp[-(k/k_\text{cut})^2]$ ($k_\text{cut} = 0.25\, h$ Mpc$^{-1}$) to suppress ringing, relevant only for $s < 20\,h^{-1}$ Mpc.
- Parameter space is explored using PyMultiNest (3000 live points, efficiency 0.8, tolerance 0.5). The likelihood employs an analytic Gaussian covariance matrix iterated five times.

## 4. Data Sets, Fitting Regime, and Results

The primary dataset consists of Flagship 1 H$\alpha$ galaxies in a $3780\,h^{-1}$ Mpc box at redshifts $z=\{0.9, 1.19, 1.53, 1.79\}$ with number densities ranging from $2.0$ to $0.3\times10^{-3} h^3$ Mpc$^{-3}$. Multipoles $\xi_{0,2,4}(s)$ are measured in $5\,h^{-1}$ Mpc bins over $0$–$200\,h^{-1}$ Mpc, with covariances derived analytically.

Fitting is performed under two schemes:
- **Template-fitting:** Linear power spectrum is fixed; fit parameters include $\{f,\,\sigma_8,\,q_\perp,\,q_\parallel\}$ and nuisance, for $s_\text{min}$ in $\{20,25,30,35,40\}\,h^{-1}\,$Mpc.
- **Full-shape analysis:** Recomputes $P_L(k)$ at each sample in $\{h,A_s,\omega_c\}$, includes same nuisance, bias, and counterterms.

Key results at $z=0.9$:
- VDG$_\infty$ provides unbiased parameter recovery down to $s_\text{min}=20\,h^{-1}\,$Mpc in both fits, with mean reduced $\chi^2 \simeq 1$ and figure-of-bias (FoB) $<1.87$.
- Recovered cosmological parameters ($\{h, n_s, \omega_c\}$) agree with fiducial values to $\sim 1\%$, with precision (FoM) $1.5\times 10^3$ and $\chi^2/\text{dof}=1.02\pm0.05$.
- Other models (EFT, CLPT, CLEFT) exhibit substantial bias or loss of accuracy at small $s_\text{min}$.

## 5. Comparative Performance with Other Theoretical Frameworks

The VDG$_\infty$ approach is contrasted with effective field theory (EFT), convolutional Lagrangian perturbation theory (CLPT), and its effective-field extension (CLEFT):

| Model   | Unbiased Scale (template/full-shape) $[h^{-1}\,\text{Mpc}]$ at $z=0.9$ | $\chi^2/\text{dof}$ | FoB  | FoM          |
|---------|------------------------------------------------------------------------|---------------------|------|--------------|
| VDG$_\infty$ | 20 / 20                                                            | $1.02\pm 0.05$      | 1.3  | $1.5\times 10^3$ |
| EFT     | 30 / 25–30                                                             | $\sim 1.5$          | 3.2  | –            |
| CLPT    | $>25$ / $>30$                                                          | 1.1                | 2.9  | –            |
| CLEFT   | 25 / 20                                                                | 1.04               | 1.7  | –            |

In both template and full-shape fits, only VDG$_\infty$ successfully yields unbiased cosmological inferences to $s_\text{min}=20\,h^{-1}\,$Mpc for the lowest redshift snapshot. This performance is matched only by CLEFT in the full-shape analysis.

## 6. Strengths, Limitations, and Ongoing Challenges

The VDG$_\infty$ framework possesses several notable advantages:
- The non-Gaussian damping $D(k, \mu)$ models FoG effects more accurately than Gaussian or Lorentzian EFT kernels.
- EFT counterterms are integrated to control ultraviolet sensitivity.
- Enables unbiased recovery of both growth and geometric parameters at the smallest scales tested ($s_{\min}$ down to $20\,h^{-1}$ Mpc).
- Achieves comparable or superior figures of merit to other models, despite more nuisance parameters.

Principal limitations and caveats:
- The method requires the COMET emulator for computational feasibility; direct perturbation theory is significantly slower.
- The kurtosis parameter $a_{\rm vir}$ is phenomenological, lacking a first-principles determination and requiring marginalization.
- There is an ongoing need to test VDG$_\infty$ against more realistic synthetic catalogs, including light-cone effects, redshift errors, and lensing magnification.
- In the presence of degeneracies (e.g., $n_s$ with bias in full-shape fits), supplementing VDG$_\infty$ with complementary statistics such as the bispectrum may be beneficial.

## 7. Role in Configuration-Space Analysis for Cosmological Surveys

For configuration-space analyses of the 2PCF multipoles $\xi_{0,2,4}(s)$ targeting precision cosmology with data from Euclid and similar spectroscopic galaxy surveys, VDG$_\infty$ is identified as the baseline model. It uniquely enables exploitation of small-scale data ($s\geq20\,h^{-1}$ Mpc) without introducing bias in inferred cosmological parameters. CLEFT is recommended as a cross-check within this context, with future work foreseen in extending these models to incorporate survey-specific systematics and more complex observational effects [2601.04780].

Source: https://www.emergentmind.com/topics/velocity-difference-generator-vdg