Constraining growth rate from SNIa peculiar velocities and galaxy density field

Corentin Ravoux - CNRS/LPCA Clermont-Ferrand

 

 

 

 

 

Duke seminar - 17 july 2025

 

 

ACCEL² simulation

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

Dark Energy

  • What is the nature of dark energy ?

Adapted from Dijkstra 2017

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Dark Energy

  • What is the nature of dark energy ?
  • Cosmological constant model (Λ) not very satisfying

Recent DESI DR2 BAO + CMB + SNIa results: stronger hints for varying dark energy

DESI DR2 BAO 2025

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Dark Energy

  • What is the nature of dark energy ?
  • Cosmological constant model (Λ) not very satisfying
  • Other models :
    • Fifth force, additional field ?
    • Modified gravity theories ?
  • Current objective: Probing large-scale structure formation to constrain dark energy/modified gravity.

Shankaranarayanan et al. 2022

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Cosmic web

  • Matter perturbations in the primordial Universe grew to form the cosmic web
  • Static description: matter density contrast (δ) follows a linear power spectrum at large-scales
  • Dynamic description: Cosmic web peculiar velocities (   ) caused by gravitation

Walther et al. 2021

Chabanier et al. 2019

v_p

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

fσ8 parameter

  • Linear perturbation theory:
\nabla \cdot \mathbf{v}(\mathbf{r}) \propto-a H (f \sigma_8) \hat{\delta}(\mathbf{r})

Cosmic web velocities

Normalized matter overdensity

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

fσ8 parameter

  • Linear perturbation theory:
\nabla \cdot \mathbf{v}(\mathbf{r}) \propto-a H (f \sigma_8) \hat{\delta}(\mathbf{r})

Cosmic web velocities

Normalized matter overdensity

Logarithmic growth rate of linear perturbations

f=\frac{d \log \delta_{+}}{d \log a} \simeq \Omega_{\mathrm{m}}^{0.55}

8 Mpc/h

Amplitude of the matter perturbations in spheres of 8 Mpc/h comoving radius (matter power spectrum amplitude)

in GR

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Measuring fσ8

Peculiar velocities (PV)

Redshift Space Distortions (RSD)

  • Direct estimation of velocities

 

 

  • Need an estimate of redshift (      ) and distance (decoupling       from     )
  • Compute statistics from PV
1 + z_{\mathrm{obs}} = (1 + z_{\mathrm{cos}})(1 + v_{\mathrm{p}}/c)
  • Impact of PV on measurements
  • Example: galaxy auto-correlation, void cross-correlation, ...
z_{\mathrm{obs}}
z_{\mathrm{cos}}
v_{\mathrm{p}}

Ravoux et al. 2022

He et al. 2018

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

fσ8 constraints

  • Constrain modified gravity models
  • RSD very effective for high redshift
  • PV for low redshift
  • Improvement with method combination

from Julian Bautista

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

fσ8 measurement with peculiar velocities

ACCEL² simulation

ACCEL² simulation

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Hubble diagram

Perfect standard candle in an expanding Universe

from Bastien Carreres

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Standard candle = object with fixed luminosity

Hubble diagram

Peculiar velocities

from Bastien Carreres

Perfect standard candle in an expanding Universe

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Hubble diagram

Noise sources (intrinsic scatter, local environment, instruments...)

from Bastien Carreres

Standardizable candle in an expanding Universe

Peculiar velocities

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Hubble diagram

from Bastien Carreres

High redshifts = dark energy  constraints (w, or w0,wa)

Low redshifts + calibration of the candle luminosity = H0 constraints

Hubble residuals = fσ8 constraints

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

"Not-so" standard candles

Tully-Fisher relation

Fundamental plane relation

  • Spiral galaxies (Late-type)
  • Absolute magnitude (M) / rotational velocity (W)
  • Elliptical galaxies (Early-type)
  • Effective radius (Re) / surface brightness (Ie) / stellar velocity dispersion (σ)
\log R_e=a \log \sigma+b \log I_e+c
\sigma_{D} / D \sim 20 \%
M = a + b \log{W}

Said et al. 2024

Said 2023

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Most standard candle: Type Ia supernovae

  • Distance modulus:
\mu_{\mathrm{obs},i}=m_{\mathrm{B},i}-M_{\mathrm{B},i}

Betoule et al. 2014

  • SNe Ia standardization = Tripp relation
M_{\mathrm{B},i} = M_{0} - \alpha x_{1,i} + \beta c_i + (\mathrm{opt})

Light-curve stretch

SN color

Parameters common for all SN Ia

"Redder is fainter"

"Luminous fade slowly"

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

  • SNe Ia:  Thermonuclear explosion a white dwarf reaching Chandrasekhar mass
\sigma_{D} / D \sim 7 \%

Most standard candle: Type Ia supernovae

  • Hubble residuals:
\begin{align*} \Delta \mu_i &= \mu_{\mathrm{obs},i} - \mu_{\mathrm{model}}(z_{\mathrm{obs},i}) \\ &= \mu_{\mathrm{obs},i} - 5 \log\left({\frac{d_{\mathrm{L}}(z_{\mathrm{obs},i})}{10 \mathrm{pc}}}\right) \end{align*}
  • Velocity estimator:
\hat{v}_i=-\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

Carreres et al. 2023

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Likelihood-based field-level inference

ACCEL² simulation

ACCEL² simulation

Carreres et al. 2023

Ravoux et al. 2025

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

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 SNe Ia coordinates

  • 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)

Parameters

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Computing the field covariance

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Combining densities (RSD) and peculiar velocities

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

SNe Ia 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_{vg}\right)

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

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_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}
P_{\mathrm{ab}}(k, \mu) = \sum_{n} w_{\mathrm{ab},n} F_{\mathrm{ab},n} (k, \mu) \mathcal{P}_{\mathrm{ab},n}(k)

gg, gv, vv

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

flip Field Level Inference Package

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

 

 

https://github.com/corentinravoux/flip

  • Fast generation of field covariance
  • MCMC and Minuit to fit fσ8
  • General expression for vectors (SNIa, density, log distance, TF, FP...)

Flip article: arxiv:2501.16852

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

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Field covariance models

  • 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

Ravoux et al. 2025

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

flip validations

  • 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                                                                                                                                                                               Duke seminar - july 2025

fσ8 constraints on ZTF & LSST simulations

ACCEL² simulation

ACCEL² simulation

Carreres et al. 2023

Carreres et al. 2025

Rosselli et al. 2025

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

ACCEL² simulation

Zwicky Transient Facility (ZTF)

  • ZTF = high-cadence photometric telescop in the Palomar observatory
  • Very large field of view (47 deg²)
  • Observing 3/4 of the sky every nights with 3 filters (g, r, i)
  • Transient sky: Supernovae, gamma ray burst, tidal distruptive events, comets, asteroids

  • Dedicated spectroscopic telescope measuring transient spectra (SEDmachine)

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Rubin - LSST

Telescope mount assembly

AuxTel: 1.2m atmospheric telescop

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Rubin - LSST: instrument

  • Largest numerical camera in the world: 3.2 Gpixels with 2s readout time
  • 8.4 m primary mirror and 9.6 deg² field of view
  • 6 filters: (u, g, r, i, z, y)
  • 10-year photometric survey of half of the sky (~ 20 000 deg²)

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

SN Ia simulation and analysis

SN Ia simulator

Survey parameters

SN Ia model

N-body host simulation

Realistic SN Ia light curves

Quality cuts

(Detection, spec-z selection, light curve quality...)

Light curve fit (SALT2)

Velocity estimation and simultaneous fit of fσ8 and velocity parameters

f\sigma_8

Simulation

Analysis

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

snsim

https://github.com/bastiencarreres/snsim

  • Survey parameters: Cadence (ZTF, LSST), CCD gain, zero point, sky noise
  • SN Ia model: Rate, SED, color, stretch, intrinsic scattering distributions, light-curve parameters
  • Host catalog: Use velocities and associated clustering from N-body simulation

Flux

Phase

Phase

Phase

Carreres et al. 2023

Main contributors: Bastien Carreres, Damiano Rosselli

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

ZTF simulations

  • 27 realizations from OuterRim (Heitmann et al. 2019) sub-volumes
  • ZTF 6 years simulations
  • Cadence from ZTF logs
  • Detection efficiency cuts: ZTF and SEDmachine detection
  • Model for velocities:

Volumic rate

Photo-detected

Spectro-typed

Carreres et al. 2023

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

P_{\mathrm{vv}} = (aHf)^2 \frac{\mu_1 \mu_2}{k^2} P_{\mathrm{\theta\theta}}(k) D_u^2(k, \sigma_u)

ZTF results

  • SN Ia selection bias causes velocity & fσ8 bias at high redshift
  • No fσ8 error bar improvement even with correction
  • Maximum redshift usable:
z =0.06

Carreres et al. 2023

Carreres et al. 2023

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

ZTF results

  • fσ8 fit performed with Hubble diagram parameters
  • no fσ8 bias on the 27 simulations
  • 19 % precision measurement with only 1600 SNe Ia

Carreres et al. 2023

Carreres et al. 2023

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

LSST simulations

  • 10 year simulation from Uchuu-UniverseMachine (Behroozi et al. 2019)
  • Cadence and instrument properties from the observing strategy simulation v3.3

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Rosselli et al. 2025

LSST simulations

  • SN host redshift efficiency (C. Ravoux, P. Gris, D. Rosselli, J. Bautista):
    • Use of DC2 LSST simulation with galaxy properties
    • Footprint and efficiency for different spectroscopic survey
    • Considering SNe variety contamination

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Rosselli et al. 2025

LSST simulations

  • 33,000 SNe Ia with spectro-z at z < 0.16 (among 1M for the full sample)
  • Photo-typing with SuperNNovae (Moller et al. 2019) gives 0.02% contamination from peculiar SN

Rosselli et al. 2025

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

LSST results

  • Dedicated study to test BBC framework and check the impact of intrinsic scatter model and nuisance parameter (      = dispersion of velocity position)

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Carreres et al. 2025

\sigma_{\mathrm{u}}

LSST results

  • Unbiased fσ8 measurement with full Hubble diagram fit over 3 redshift points
  • How can we do better ? Combine with density field

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Rosselli et al. 2025

Combining DESI galaxy density and ZTF SNIa velocities on simulation

ACCEL² simulation

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

DESI

  • 4m multi-object spectrograph at Kitt Peak Observatory

DESI collaboration 2022

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

DESI

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

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

Silber et al. 2022

DESI

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

  • 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

40 million redshifts in 5 years

Primary science: BAO and RSD measurements

DESI x ZTF simulation

  • 27 simulations from AbacusSummit (Maksimova et al. 2019)
  • Density: DESI Bright Galaxy Survey (BGS) clustering matching for Y5 (J. Bautista, A. Smith)
  • Velocity: ZTF SNIa Y6 simulation performed (following Carreres et al. 2023)

from Julian Bautista

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Galaxy field

  • snsim runs to mimic ZTF Y6 SN Ia field

Supernovae velocities

  • DESI Y5 BGS footprint and density field gridding

Ravoux et al. in prep.

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

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                                                                                                                                                                               Duke seminar - july 2025

ZTF velocity fit

  • Velocity fit including Hubble diagram parameters to reproduce Carreres et al. 2023
  • Variation of analysis parameters (max redshift, SNIa density...)

Ravoux et al. in prep.

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

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                                                                                                                                                                               Duke seminar - july 2025

DESI x ZTF 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                                                                                                                                                                               Duke seminar - july 2025

Summary of forecasts on realistic simulations

Corentin Ravoux                                                                                                                                                                               Duke seminar - july 2025

Conclusion

  • Large modeling and simulation preparatory work for ZTF, LSST and combination with DESI density field
  • Dedicated sofware for simulation (snsim) and analysis (flip)
  • Next steps:
    • Tests on Pantheon+ data (Anthony Carr, KASI), DEBASS data (Bastien Carreres, Duke) and ZTF data (Raphael Kebadian, CPPM)
    • Implementation of non-linear models (EFTofLSS)
    • Combination with other velocity tracers (TF, FP)

ACCEL² simulation

Bonus: AuxTel for LSST atmosphere

  • Impact of PWV (Precipitable water vapor) on SNe Ia bands

  • i,z,y bands interesting for high-redshift SNe Ia

  • Proposed DDF scenario has a lot of y-band visit: impact or correction of atmosphere variability

  • FGCM cross-verification/external data input

Bonus: AuxTel for LSST atmosphere

Bonus: AuxTel for LSST atmosphere

  • CALSPEC spectra does not cover DDF

  • Gaia = largest catalog of star spectra out of atmosphere

  • BP/RP spectrograph: low resolution at [330-1050]nm

  • RVS spectrograph: high resolution at [846-870]nm

  • Optimal coverage of the full LSST sky (including DDF)

Bonus: AuxTel for LSST atmosphere

Unbiased PWV measurement with GAIA (2323 AuxTel measurement in 2025, similar results with previous spectra)

Harder to get unbiased ozone measurement, even with the use of a Gaia adapted throughput

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))

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

Howlett et al. 2017

Ravoux et al. 2025