13 Fisher Error Estimation : A Simple Worked Example

As a simple example of computing a Fisher matrix and estimating parameter errors in a cosmological context, we consider the following toy problem. Imagine that we measure the comoving distance $\chi$ (in units of $c/H_0$) at a single redshift with a known measurement error $\sigma_\chi$. We’ll assume a simple cosmological model with only $\Omega_m$, and we’d like to estimate the Fisher matrix and the expected error on $\Omega_m$ from this single measurement.

In this case, the derivatives are analytically tractable, so you could compute the Fisher matrix with standard numerical integration. However, as a demonstration, we will continue to explore differentiable programming and compute the Fisher matrix using automatic differentiation. See previous notebooks for examples of using JAX for automatic differentiation in cosmology.

Let’s just write down the Fisher matrix for this simple case.

The general form of the Fisher matrix for a fixed covariance matrix $C$ and a model $\mu$ is \[ F_{ij} = \frac{\partial \mu^T}{\partial \theta_i} C^{-1} \frac{\partial \mu}{\partial \theta_j} \] where $\theta_i$ are the model parameters (in our case, just $\Omega_m$), $\mu$ is the model prediction for the observable (here, the comoving distance $\chi$), and $C$ is the covariance matrix of the measurement (here, just $\sigma_\chi^2$).

In our case, this simplifies to a single parameter and a single measurement, so the Fisher matrix is just a scalar: \[ F = \frac{1}{\sigma_\chi^2} \left( \frac{\partial \chi}{\partial \Omega_m} \right)^2 \] and the expected error on $\Omega_m$ is simply \[ \sigma_{\Omega_m} = \frac{1}{\sqrt{F}}. \]

Code

import jax
import jax.numpy as jnp
from jax import grad, vmap
import diffrax
import matplotlib.pyplot as plt

13.1 Cosmological Distance Functions

We use JAX and Diffrax to compute the comoving distance, following the approach in the Friedmann-Numerical-JAX notebook. The key advantage is that JAX gives us automatic differentiation for free.

Code

# Fiducial cosmology (flat ΛCDM)
Om_fid = 0.3
Ol_fid = 0.7

def EHubble(Om, z):
    """Dimensionless Hubble parameter E(z) = H(z)/H0 for flat ΛCDM."""
    Ol = 1.0 - Om  # Flat universe
    return jnp.sqrt(Om * (1 + z)**3 + Ol)

def comoving_radial_distance(Om, z_final, rtol=1e-8, atol=1e-10):
    """Comoving radial distance χ(z) in c/H0 units using Diffrax ODE solver."""

    def dchi_dz(z, chi, args):
        """ODE: dχ/dz = 1/E(z)"""
        return 1.0 / EHubble(Om, z)

    term = diffrax.ODETerm(dchi_dz)
    solver = diffrax.Dopri5()
    stepsize_controller = diffrax.PIDController(rtol=rtol, atol=atol)

    sol = diffrax.diffeqsolve(
        term, solver,
        t0=0.0, t1=z_final, dt0=0.01, y0=0.0,
        stepsize_controller=stepsize_controller,
    )
    return sol.ys[0]

13.2 Computing the Fisher Error

Now we use JAX’s automatic differentiation to compute $\partial \chi / \partial \Omega_m$. The Fisher error on $\Omega_m$ is then: \[ \sigma_{\Omega_m} = \frac{\sigma_\chi}{|\partial \chi / \partial \Omega_m|} \]

We’ll assume a fixed fractional error on the distance measurement, say 1%.

Code

# Compute derivative of χ with respect to Ωm
dchi_dOm = grad(comoving_radial_distance, argnums=0)

# Measurement error: assume 1% fractional error on distance
fractional_error = 0.01

def fisher_error_Om(Om, z):
    """Compute the Fisher error on Ωm from a distance measurement at redshift z."""
    chi = comoving_radial_distance(Om, z)
    deriv = dchi_dOm(Om, z)
    sigma_chi = fractional_error * chi
    # σ_Ωm = σ_χ / |∂χ/∂Ωm|
    return sigma_chi / jnp.abs(deriv)

# Test at a few redshifts
z_test = jnp.array([0.5, 1.0, 2.0, 3.0])
print("Fisher error on Ωm from 1% distance measurement:")
print("=" * 50)
print(f"{'z':>8} {'χ [c/H0]':>12} {'∂χ/∂Ωm':>12} {'σ(Ωm)':>12}")
print("-" * 50)
for z in z_test:
    chi = comoving_radial_distance(Om_fid, z)
    deriv = dchi_dOm(Om_fid, z)
    sigma_Om = fisher_error_Om(Om_fid, z)
    print(f"{z:8.1f} {chi:12.4f} {deriv:12.4f} {sigma_Om:12.4f}")

Fisher error on Ωm from 1% distance measurement:
==================================================
       z     χ [c/H0]       ∂χ/∂Ωm        σ(Ωm)
--------------------------------------------------
     0.5       0.4410      -0.1529       0.0288
     1.0       0.7714      -0.4577       0.0169
     2.0       1.2095      -1.0380       0.0117
     3.0       1.4840      -1.4596       0.0102

13.3 Fisher Error vs Redshift

Let’s plot how the constraining power on $\Omega_m$ changes with the redshift of the measurement. Higher redshift measurements probe larger distances where the sensitivity to $\Omega_m$ is greater.

Code

# Compute Fisher error over a range of redshifts
z_range = jnp.linspace(0.1, 5.0, 50)
fisher_error_vec = vmap(lambda z: fisher_error_Om(Om_fid, z))
sigma_Om_vals = fisher_error_vec(z_range)

# Also compute the derivative for reference
deriv_vec = vmap(lambda z: dchi_dOm(Om_fid, z))
deriv_vals = deriv_vec(z_range)

# Create figure
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Left panel: Fisher error
ax1.plot(z_range, sigma_Om_vals, 'b-', linewidth=2)
ax1.set_xlabel('Redshift $z$', fontsize=12)
ax1.set_ylabel(r'$\sigma_{\Omega_m}$', fontsize=12)
ax1.set_title(r'Fisher Error on $\Omega_m$ (1% distance error)', fontsize=12)
ax1.grid(True, alpha=0.3)
ax1.set_ylim(bottom=0)

# Right panel: derivative (sensitivity)
ax2.plot(z_range, jnp.abs(deriv_vals), 'r-', linewidth=2)
ax2.set_xlabel('Redshift $z$', fontsize=12)
ax2.set_ylabel(r'$|\partial \chi / \partial \Omega_m|$ [c/H$_0$]', fontsize=12)
ax2.set_title(r'Sensitivity of Distance to $\Omega_m$', fontsize=12)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

13.4 Interpretation

The plots show that:

Higher redshift = better constraints: The Fisher error decreases with redshift, meaning high-z distance measurements constrain $\Omega_m$ better than low-z ones (for fixed fractional error).
Why? The sensitivity $|\partial \chi / \partial \Omega_m|$ increases with redshift. At higher redshifts, the comoving distance has accumulated more “history” of the expansion, making it more sensitive to the matter density. At low redshift, $\chi \sim z + \dots$ and so, there is much less sensitivity to $\Omega_m$ at low redshift.
Practical implications: This is why cosmological surveys target high-redshift objects (Type Ia supernovae at $z \sim 1$, BAO at $z \sim 0.5-2$, CMB at $z \sim 1100$).

Of course, this is a simplified example. Real surveys must account for:

Multiple parameters (not just $\Omega_m$)
Correlations between measurements
Systematic errors

# Fisher Error Estimation : A Simple Worked Example As a simple example of computing a Fisher matrix and estimating parameter errors in a cosmological context, we consider the following toy problem. Imagine that we measure the comoving distance $\chi$ (in units of $c/H_0$) at a single redshift with a known measurement error $\sigma_\chi$. We'll assume a simple cosmological model with only $\Omega_m$, and we'd like to estimate the Fisher matrix and the expected error on $\Omega_m$ from this single measurement. In this case, the derivatives are analytically tractable, so you could compute the Fisher matrix with standard numerical integration. However, as a demonstration, we will continue to explore differentiable programming and compute the Fisher matrix using automatic differentiation. See previous notebooks for examples of using JAX for automatic differentiation in cosmology. Let's just write down the Fisher matrix for this simple case. The general form of the Fisher matrix for a fixed covariance matrix $C$ and a model $\mu$ is $$ F_{ij} = \frac{\partial \mu^T}{\partial \theta_i} C^{-1} \frac{\partial \mu}{\partial \theta_j} $$ where $\theta_i$ are the model parameters (in our case, just $\Omega_m$), $\mu$ is the model prediction for the observable (here, the comoving distance $\chi$), and $C$ is the covariance matrix of the measurement (here, just $\sigma_\chi^2$). In our case, this simplifies to a single parameter and a single measurement, so the Fisher matrix is just a scalar: $$ F = \frac{1}{\sigma_\chi^2} \left( \frac{\partial \chi}{\partial \Omega_m} \right)^2 $$ and the expected error on $\Omega_m$ is simply $$ \sigma_{\Omega_m} = \frac{1}{\sqrt{F}}. $$ ```{python} import jax import jax.numpy as jnp from jax import grad, vmap import diffrax import matplotlib.pyplot as plt ``` ## Cosmological Distance Functions We use JAX and Diffrax to compute the comoving distance, following the approach in the Friedmann-Numerical-JAX notebook. The key advantage is that JAX gives us automatic differentiation for free. ```{python} # Fiducial cosmology (flat ΛCDM) Om_fid = 0.3 Ol_fid = 0.7 def EHubble(Om, z): """Dimensionless Hubble parameter E(z) = H(z)/H0 for flat ΛCDM.""" Ol = 1.0 - Om # Flat universe return jnp.sqrt(Om * (1 + z)**3 + Ol) def comoving_radial_distance(Om, z_final, rtol=1e-8, atol=1e-10): """Comoving radial distance χ(z) in c/H0 units using Diffrax ODE solver.""" def dchi_dz(z, chi, args): """ODE: dχ/dz = 1/E(z)""" return 1.0 / EHubble(Om, z) term = diffrax.ODETerm(dchi_dz) solver = diffrax.Dopri5() stepsize_controller = diffrax.PIDController(rtol=rtol, atol=atol) sol = diffrax.diffeqsolve( term, solver, t0=0.0, t1=z_final, dt0=0.01, y0=0.0, stepsize_controller=stepsize_controller, ) return sol.ys[0] ``` ## Computing the Fisher Error Now we use JAX's automatic differentiation to compute $\partial \chi / \partial \Omega_m$. The Fisher error on $\Omega_m$ is then: $$ \sigma_{\Omega_m} = \frac{\sigma_\chi}{|\partial \chi / \partial \Omega_m|} $$ We'll assume a fixed fractional error on the distance measurement, say 1%. ```{python} # Compute derivative of χ with respect to Ωm dchi_dOm = grad(comoving_radial_distance, argnums=0) # Measurement error: assume 1% fractional error on distance fractional_error = 0.01 def fisher_error_Om(Om, z): """Compute the Fisher error on Ωm from a distance measurement at redshift z.""" chi = comoving_radial_distance(Om, z) deriv = dchi_dOm(Om, z) sigma_chi = fractional_error * chi # σ_Ωm = σ_χ / |∂χ/∂Ωm| return sigma_chi / jnp.abs(deriv) # Test at a few redshifts z_test = jnp.array([0.5, 1.0, 2.0, 3.0]) print("Fisher error on Ωm from 1% distance measurement:") print("=" * 50) print(f"{'z':>8} {'χ [c/H0]':>12} {'∂χ/∂Ωm':>12} {'σ(Ωm)':>12}") print("-" * 50) for z in z_test: chi = comoving_radial_distance(Om_fid, z) deriv = dchi_dOm(Om_fid, z) sigma_Om = fisher_error_Om(Om_fid, z) print(f"{z:8.1f} {chi:12.4f} {deriv:12.4f} {sigma_Om:12.4f}") ``` ## Fisher Error vs Redshift Let's plot how the constraining power on $\Omega_m$ changes with the redshift of the measurement. Higher redshift measurements probe larger distances where the sensitivity to $\Omega_m$ is greater. ```{python} # Compute Fisher error over a range of redshifts z_range = jnp.linspace(0.1, 5.0, 50) fisher_error_vec = vmap(lambda z: fisher_error_Om(Om_fid, z)) sigma_Om_vals = fisher_error_vec(z_range) # Also compute the derivative for reference deriv_vec = vmap(lambda z: dchi_dOm(Om_fid, z)) deriv_vals = deriv_vec(z_range) # Create figure fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5)) # Left panel: Fisher error ax1.plot(z_range, sigma_Om_vals, 'b-', linewidth=2) ax1.set_xlabel('Redshift $z$', fontsize=12) ax1.set_ylabel(r'$\sigma_{\Omega_m}$', fontsize=12) ax1.set_title(r'Fisher Error on $\Omega_m$ (1% distance error)', fontsize=12) ax1.grid(True, alpha=0.3) ax1.set_ylim(bottom=0) # Right panel: derivative (sensitivity) ax2.plot(z_range, jnp.abs(deriv_vals), 'r-', linewidth=2) ax2.set_xlabel('Redshift $z$', fontsize=12) ax2.set_ylabel(r'$|\partial \chi / \partial \Omega_m|$ [c/H$_0$]', fontsize=12) ax2.set_title(r'Sensitivity of Distance to $\Omega_m$', fontsize=12) ax2.grid(True, alpha=0.3) plt.tight_layout() plt.show() ``` ## Interpretation The plots show that: 1. **Higher redshift = better constraints**: The Fisher error decreases with redshift, meaning high-z distance measurements constrain $\Omega_m$ better than low-z ones (for fixed fractional error). 2. **Why?** The sensitivity $|\partial \chi / \partial \Omega_m|$ increases with redshift. At higher redshifts, the comoving distance has accumulated more "history" of the expansion, making it more sensitive to the matter density. At low redshift, $\chi \sim z + \dots$ and so, there is much less sensitivity to $\Omega_m$ at low redshift. 3. **Practical implications**: This is why cosmological surveys target high-redshift objects (Type Ia supernovae at $z \sim 1$, BAO at $z \sim 0.5-2$, CMB at $z \sim 1100$). Of course, this is a simplified example. Real surveys must account for: - Multiple parameters (not just $\Omega_m$) - Correlations between measurements - Systematic errors