Likelihood-based inference of growth rate with SNIa velocity and galaxy density fields

 

 

 

 

ZTF meeting october 2025

 

 

ACCEL² simulation

Corentin Ravoux - CNRS/LPCA Clermont-Ferrand

 

with Bastien Carreres, Julian Bautista, Damiano Rosselli, Alex Kim and many others

Likelihood-based field-level estimator

ACCEL² simulation

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Likelihood-based field-level estimator

\begin{aligned} \mathcal{L}(p) \propto & \exp\left[-\frac{1}{2} v^T(p) C_{vv}(p)^{-1} v(p)\right] \\ \end{aligned}

Velocity from SNe Ia

Field covariance matrix computed from theory and coordinates

Parameters

Velocity example:

  • Purpose: Maximizing or sampling a likelihood computed from all coordinates of the data
  • p = fitted parameters: simultaneously cosmology (fσ8) and field parameters (e.g., Hubble diagram parameters α, β, M0 for SN)

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Computing the field covariance

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Combining densities (RSD) and peculiar velocities

\begin{aligned} \mathcal{L}(p) \propto & \exp\left[-\frac{1}{2} \right. \end{aligned}

Velocities

Density field from galaxy surveys

Full field covariance matrix

\delta
\delta
v
v
\Bigg]
-1
T
\left(C_{gg}\right)
\left(C_{gv}\right)
\left(C_{vv}\right)
\left(C_{gv}\right)

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

General covariance calculation

  • Objective: Improve/Extend the likelihood-based modeling
  • General form of field power spectrum
  • Algorithmically-oriented calculations (Hankel transforms)
  • Wide-angle effect accounted in the integration
C_{\mathrm{ab}}(\mathbf{r_a}, \mathbf{r_b}) = \frac{1}{(2 \pi)^3} \int_{\mathbf{k}} P_{\mathrm{ab}}(k, \mu_{\mathrm{a}}, \mu_{\mathrm{b}}) e^{i \mathbf{k}.\mathbf{r}} d^3 \mathbf{k}
P_{\mathrm{ab}}(k, \mu_{\mathrm{a}}, \mu_{\mathrm{b}}) = \sum_{n} w_{\mathrm{ab},n} F_{\mathrm{ab},n} (k, \mu_{\mathrm{a}}, \mu_{\mathrm{b}}) \mathcal{P}_{\mathrm{ab},n}(k)

gg, gv, vv

Parameters to fit

Terms to integrate analytically

Power spectra terms to integrate numerically

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

flip Field Level Inference Package

  •  python package for likelihood-based field-level inference:

 

 

https://github.com/corentinravoux/flip

  • Symbolic code generation for covariance model
  • Fast generation of field covariance
  • MCMC and Minuit fitter
  • General expression for vectors (SNIa, density, log distance...)
  • Survey geometry-dependent Fisher forecast
  • Covariance model evaluation

Flip article: arxiv:2501.16852

Contributors: Bastien Carreres, Damiano Rosselli and Alex G. Kim

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Field covariance models

Ravoux et al. 2025

  • Model comparison on a regular coordinate grid (Adams & Blake 2017, Adams & Blake 2020, Lai et al. 2022)

 

  • New wide-angle model developed (RC25): improved stability in integration and for wider fitting parameter ranges

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Validating flip

  • Field covariance comparison with pairV Anthony Carr & David Parkinson
  • Model validated on N-body simulations  Tyann Dummerchat

Disagreement due to different power spectrum model

Ravoux et al. 2025

Ravoux et al. 2025

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Application: combining DESI galaxy density and ZTF SNIa velocities on simulation

ACCEL² simulation

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

DESI

  • 4m multi-object spectrograph at Kitt Peak Observatory

DESI collaboration 2022

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

DESI

  • 5 μm positioning in few sec. for 5000 targets
  • All fibers connected to 10 spectrographs

Silber et al. 2022

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

DESI

  • BGS: Bright Galaxy Survey
  • LRG: Luminous Red Galaxy
  • ELG: Emission Line Galaxy
  • QSO: Quasar
    • z > 2.1: with a Lyman-α forest
    • z < 2.1: as tracers

63 million redshifts in 8 years

over 17,000 deg²

Primary science: BAO and RSD measurements

21

12

24

5

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Galaxy field

  • Runs of SNsim to mimic ZTF SNIa Y6 data set (following Carreres et al. 2023)

Supernovae velocities

https://github.com/bastiencarreres/snsim

  • AbacusSummit mocks matching DESI DR3 (5 years) BGS clustering and footprint

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Model for density & velocity fit

  • For velocity x density, use of a new wide-angle model implemented in flip
P_{\mathrm{vv}} = (aHf)^2 \frac{\mu_1 \mu_2}{k^2} P_{\mathrm{\theta\theta}}(k) D_u^2(k, \sigma_u)
P_{\mathrm{gg}}=\left[b^2 P_{\mathrm{mm}}(k)+b f (\mu_1^2 + \mu_2^2) P_{\mathrm{m\theta}}(k)+f^2 \mu_1^2 \mu_2^2 P_{\theta \theta}(k)\right] \exp \left[\frac{-k^2 (\mu_1^2 + \mu_2^2) \sigma_{\mathrm{g}}^2}{2}\right]
P_{\mathrm{gv}} = iaH \frac{\mu_2}{k^2} \left(bf P_{\mathrm{m\theta}}(k) + f^2 \mu_1^2 P_{\mathrm{\theta\theta}}(k) \right) \exp \left[\frac{-k^2 \mu_1^2 \sigma_{\mathrm{g}}^2}{2}\right] D_u(k, \sigma_u)
C_{\mathrm{ab}}(\mathbf{r_1}, \mathbf{r_2}) = \frac{1}{(2 \pi)^3} \int_{\mathbf{k}} P_{\mathrm{ab}}(k, \mu_1, \mu_2) e^{i \mathbf{k}.\mathbf{r}} d^3 \mathbf{k}

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

ZTF velocity fit

  • Velocity fit including Hubble diagram parameters (Carreres et al. 2023)

 

 

 

  • Variation of analysis parameters (max redshift, SNIa density...)
\hat{v}_i\left(\mathbf{p}_{\mathrm{HD}}\right)=-\frac{\ln (10) c}{5}\left(\frac{\left(1+z_i\right) c}{H\left(z_i\right) r\left(z_i\right)}-1\right)^{-1} \Delta \mu_i\left(\mathbf{p}_{\mathrm{HD}}\right)
\begin{split} \Delta \mu_i (\mathbf{p}_{\mathrm{HD}}) &=m_{B,i} - M_0 + \alpha x_{1,i} - \beta c_i \\ &- \left( 5 \log\left[(1+ z_{\mathrm{obs},i})D(z_{\mathrm{obs},i})\right] + 25 \right) \end{split}

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

DESI density fit

  • Density grid with pypower
  • Test varying all density gridding parameters: grid size, mesh cell size, interpolation scheme (ngc, cic, tsc, pcs), footprint (Y5 or full)
  • Mean minuit fit on 27 mocks with average error bar

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

DESIxZTF joint fit

  • SNIa velocity fit with Hubble diagram parameters and density field

Adding density improves fs8 constraints up to 60%

Different density configurations

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Summary of forecasts on simulations

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

DESIxZTF article

  • Aim: Test full method on end-of-survey simulations (different effort than DR2.5 realistic simulations and Raphael's work)
  • Paper that uses DESI BGS & ZTF simulations
  • DESI project already created
  • Aim to publish at the end of this year
  • 2 choices:
    • Include in ZTF x DESI agreement (not easy and not necessary for simulation)
    • External collaborator agreement for main contributors of the analysis and ZTF log

Corentin Ravoux                                                                                                                                                                                 ZTF meeting - October 6

Conclusion

  • flip = Framework for likelihood-based field-level inference
  • Growth rate analysis on simulation with ZTF SNIa and DESI density field

ACCEL² simulation

Survey dependent Fisher forecast

  • Field covariance matrix accounts for survey geometry
  • Comparing ZTF simulation (Carreres et al. 2023) with Fisher forecasts
  • Error ~ 30% closer to likelihood-based than standard Fisher

Damiano Rosselli

Howlett et al. 2017

A word on wide-angle in clustering

\mathbf{r_{1}}
\mathbf{r_{2}}
\mathbf{v_{1}}
\mathbf{v_{2}}
\mathbf{r} = \mathbf{r_{1}} - \mathbf{r_{2}}

Plane parallel models

Wide angle models

\mu = \mu_1 = \mu_2
\mathbf{r_{1}}
\mathbf{r_{2}}
\mathbf{v_{1}}
\mathbf{v_{2}}
\mathbf{r} = \mathbf{r_{1}} - \mathbf{r_{2}}
\theta
\phi
\mu_1 \neq \mu_2
\mu_{i} = \mathbf{k} \cdot \mathbf{r_{i}}
\theta

Plane parallel model

  • Express power spectrum model in a general way:
P_{\mathrm{ab}}(k, \mu) = \sum_{n} w_{\mathrm{ab},n} F_{\mathrm{ab},n} (k, \mu) \mathcal{P}_{\mathrm{ab},n}(k)
  • vv "simple" example:
P_{\mathrm{vv}} = (f\sigma_8)^2 \times \left(a^2 H^2 \frac{\mu^2}{k^2}\right) \times \left(P_{\mathrm{\theta\theta}}(k) D_u^2(k, \sigma_u)\right)

Parameters to fit

Terms to integrate analytically

Power spectra terms to integrate numerically

Plane parallel model

  • Without loosing generality we can express the covariance:
C_{\mathrm{ab}}(\mathbf{r_1}, \mathbf{r_2}) = \sum_n w_{\mathrm{ab},n} \sum_{\ell} N_{\mathrm{ab},\ell}(\phi) \mathcal{H}_\ell\left[\mathcal{P}_{\mathrm{ab},n}(k)M_{\mathrm{ab},n}^{\ell}(k)\right](r)
M_{\mathrm{ab},n}^{\ell}(k) = \frac{1}{2}\int_{-1}^{1}d\mu^\prime F_{\mathrm{ab},n}(k, \mu^\prime)L_{\ell}(\mu^\prime)
N_{\mathrm{ab},\ell}(\phi) =(2\ell +1)L_\ell(\cos\left(\pi - \phi\right))

Plane parallel model

C_{\mathrm{ab}}(\mathbf{r_1}, \mathbf{r_2}) = \sum_n w_{\mathrm{ab},n} \sum_{\ell} N_{\mathrm{ab},\ell}(\phi) \mathcal{H}_\ell\left[\mathcal{P}_{\mathrm{ab},n}(k)M_{\mathrm{ab},n}^{\ell}(k)\right](r)

Linear sum of matrix with parameters to fit as coefficients

  • Without loosing generality we can express the covariance:

Plane parallel model

C_{\mathrm{ab}}(\mathbf{r_1}, \mathbf{r_2}) = \sum_n w_{\mathrm{ab},n} \sum_{\ell} N_{\mathrm{ab},\ell}(\phi) \mathcal{H}_\ell\left[\mathcal{P}_{\mathrm{ab},n}(k)M_{\mathrm{ab},n}^{\ell}(k)\right](r)

Hankel transform

\mathcal{H}_\ell\left[f(k)\right](r) = i^\ell \int_{0}^\infty\frac{k^2 dk}{2\pi^2}j_\ell(kr)f(k)
  • Most important part: Algorithmically optimized way to compute power spectrum integral, with FFTLog algorithm
  • Without loosing generality we can express the covariance:

Plane parallel model

  • Without loosing generality we can express the covariance:
C_{\mathrm{ab}}(\mathbf{r_1}, \mathbf{r_2}) = \sum_n w_{\mathrm{ab},n} \sum_{\ell} N_{\mathrm{ab},\ell}(\phi) \mathcal{H}_\ell\left[\mathcal{P}_{\mathrm{ab},n}(k)M_{\mathrm{ab},n}^{\ell}(k)\right](r)

Terms pre-computed analytically with symbolic code generaltion

  • More complicated formalism derived and implemented for wide-angle
M_{\mathrm{ab},n}^{\ell}(k) = \frac{1}{2}\int_{-1}^{1}d\mu^\prime F_{\mathrm{ab},n}(k, \mu^\prime)L_{\ell}(\mu^\prime)
N_{\mathrm{ab},\ell}(\phi) =(2\ell +1)L_\ell(\cos\left(\pi - \phi\right))

Wide angle model

\xi_{\mathrm{ab},\ell}^{i}(r) = \sum_j N_{\mathrm{ab},\ell}^{i,j} (\theta, \phi) \left[ i^{\ell} \int_k \frac{k^2 dk}{2 \pi^2} P_{\mathrm{ab},i} (k) M_{\mathrm{ab},\ell}^{i,j} (k) j_{\ell}(k r)\right]
C_{\mathrm{ab}}(\mathbf{r_1}, \mathbf{r_2}) = \sum_i A_{\mathrm{ab},i} \sum_l \xi_{\mathrm{ab},\ell}^{i}(r)
M_{\mathrm{ab},\ell}^{i,j=(\ell_1,\ell_2)} (k) = \int_{\mu_1 = -1}^1 \int_{\mu_2 = -1}^1 \frac{1}{4} L_{\ell_1}(\mu_1)L_{\ell_2}(\mu_2) B_{\mathrm{ab},i} (k, \mu_1, \mu_2) d \mu_1 d \mu_2
N_{\mathrm{ab},\ell}^{i,j=(\ell_1,\ell_2)} (\theta, \phi) = (4 \pi)^2 \sum_{m, m_1, m_2} G_{l, l_1, l_2}^{m, m_1, m_2} Y_{\ell m}(\hat{\mathbf{r}})^* Y_{\ell_1 m_1}(\hat{\mathbf{r_1}})^* Y_{\ell_2 m_2}(\hat{\mathbf{r_2}})^*
G_{m_1, m_2, m_3}^{l_1, l_2, l_3}=\sqrt{\frac{\left(2 l_1+1\right)\left(2 l_2+1\right)\left(2 l_3+1\right)}{4 \pi}} \left(\begin{array}{ccc} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{lll} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array}\right)
P_{\mathrm{ab}}(k, \mu_1, \mu_2) = \sum_{i} A_{\mathrm{ab},i} B_{\mathrm{ab},i} (k, \mu_1, \mu_2) P_{\mathrm{ab},i}(k)

linear sum of matrix