Fisher Forecasts

HIcosmo provides professional Fisher matrix analysis tools, supporting 21cm intensity mapping survey forecasts, multi-probe joint analysis, and parameter constraint predictions.

Fisher Matrix Fundamentals

Fisher Information Matrix

The Fisher information matrix quantifies the curvature of the likelihood function in parameter space, defined as:

\[F_{ij} = -\left\langle \frac{\partial^2 \ln \mathcal{L}}{\partial \theta_i \partial \theta_j} \right\rangle\]

For a Gaussian likelihood, the Fisher matrix can be computed from the covariance matrix \(C\) and the theoretical prediction \(\mu\):

\[F_{ij} = \frac{1}{2} \text{Tr}\left[ C^{-1} \frac{\partial C}{\partial \theta_i} C^{-1} \frac{\partial C}{\partial \theta_j} \right] + \frac{\partial \mu^T}{\partial \theta_i} C^{-1} \frac{\partial \mu}{\partial \theta_j}\]

Parameter Constraints

The \(1\sigma\) constraints on parameters are given by the covariance matrix:

\[\text{Cov}(\theta) = F^{-1}\]

\[\sigma(\theta_i) = \sqrt{(F^{-1})_{ii}}\]

Marginalized constraints: When nuisance parameters exist, the constraints on target parameters require integration over nuisance parameters:

\[\text{Cov}_{\text{target}} = (F^{-1})_{\text{target block}}\]

Conditional constraints: Constraints are tighter when other parameters are fixed:

\[\sigma_{\text{cond}}(\theta_i) = \frac{1}{\sqrt{F_{ii}}}\]

Quick Start

3 lines to complete a Fisher forecast:

from hicosmo.models import CPL
from hicosmo.fisher import IntensityMappingFisher

result = IntensityMappingFisher.forecast('ska1_mid_band2', CPL(), ['w0', 'wa'])
print(result)

IntensityMappingFisher Class

IntensityMappingFisher is the core class for 21cm intensity mapping Fisher forecasts, implementing the Bull et al. (2016) method.

Class Method Interface (Recommended)

from hicosmo.models import ILCDM
from hicosmo.fisher import IntensityMappingFisher

# 3 lines to complete a forecast
ilcdm = ILCDM(H0=67.36, Omega_m=0.3153, beta=0.001)
result = IntensityMappingFisher.forecast('tianlai', ilcdm, ['beta', 'H0'])
print(result)

Example Output:

======================================================================
Fisher Forecast Results: tianlai
======================================================================
Cosmology: ILCDM
Redshift bins: 3
Parameters: beta, H0

1σ Constraints:
----------------------------------------------------------------------
  σ(beta      ) =    1.340000e-02
  σ(H0        ) =    8.250000e-01
======================================================================

Instance Interface

Use the instance interface when multiple forecasts are needed:

from hicosmo.fisher import IntensityMappingFisher, load_survey
from hicosmo.models import CPL

# Load survey configuration
survey = load_survey('ska1_mid_band2')
cosmology = CPL(H0=67.36, Omega_m=0.3153, w0=-1.0, wa=0.0)

# Create Fisher calculator
fisher = IntensityMappingFisher(survey, cosmology)

# Multiple forecasts
result1 = fisher.parameter_forecast(['w0', 'wa'])
result2 = fisher.parameter_forecast(['H0', 'Omega_m'])

With Marginalization

Marginalize over nuisance parameters:

result = IntensityMappingFisher.forecast(
    'ska1_mid_band2',
    cpl_model,
    params=['w0', 'wa'],
    marginalize_over=['H0', 'Omega_m', 'gamma']
)

print(f"σ(w0) = {result.errors['w0']:.4f}")
print(f"σ(wa) = {result.errors['wa']:.4f}")

FisherForecastResult Object

The result object returned by forecast() contains:

Attribute	Type	Description
`params`	list	List of parameter names
`errors`	dict	\(1\sigma\) constraints `{param: σ}`
`fisher_matrix`	ndarray	Fisher matrix
`covariance`	ndarray	Covariance matrix
`correlation`	ndarray	Correlation matrix
`survey_name`	str	Survey name
`n_bins`	int	Number of redshift bins

# Access results
print(f"σ(w0) = {result.errors['w0']:.4f}")
print(f"w0-wa correlation coefficient: {result.correlation[0,1]:.3f}")

# Convert to dictionary (compatible with legacy API)
data = result.to_dict()

Survey Configurations

HIcosmo includes built-in configurations for multiple 21cm intensity mapping surveys.

Available Surveys

from hicosmo.fisher import list_available_surveys

surveys = list_available_surveys()
print(surveys)
# ['ska1_mid_band2', 'ska1_wide_band1', 'tianlai', 'bingo', 'meerkat', 'chime', ...]

Survey	Redshift Range	Features
ska1_mid_band2	0.0 - 0.5	SKA1-MID medium-deep, Band 2
ska1_wide_band1	0.4 - 1.0	SKA1-MID wide field, Band 1
tianlai	0.8 - 1.2	Tianlai experiment, cylinder array
bingo	0.13 - 0.45	Brazilian 21cm project
meerkat	0.0 - 0.6	South African MeerKAT array
chime	0.8 - 2.5	Canadian CHIME cylinder array

Loading a Survey

from hicosmo.fisher import load_survey

survey = load_survey('ska1_mid_band2')

# View survey information
print(f"Name: {survey.name}")
print(f"Description: {survey.description}")
print(f"Redshift bins: {len(survey.redshift_bins)}")
print(f"Number of dishes: {survey.instrument.ndish}")
print(f"Survey area: {survey.instrument.survey_area_deg2} deg2")

Survey Configuration Format

Surveys are configured in YAML format (located in hicosmo/fisher/surveys/):

name: ska1_mid_band2
description: "SKA1-MID Medium-Deep Band 2 survey"

# Hardware configuration
instrument:
  telescope_type: single_dish
  ndish: 133
  dish_diameter_m: 15.0
  nbeam: 1

# Observing strategy
observing:
  survey_area_deg2: 5000.0
  total_time_hours: 10000.0
  channel_width_hz: 50000.0

# Noise parameters
noise:
  system_temperature_K: 13.5
  sky_temperature:
    model: power_law
    T_ref_K: 25.0
    nu_ref_MHz: 408.0
    beta: 2.75

# HI tracers
hi_tracers:
  bias:
    model: polynomial
    coefficients: [0.67, 0.18, 0.05]
  density:
    model: polynomial
    coefficients: [0.00048, 0.00039, -0.000065]

# Redshift binning
redshift_bins:
  default_delta_z: 0.1
  centers: [0.05, 0.15, 0.25, 0.35, 0.45]

21cm Physics

Brightness Temperature

21cm brightness temperature formula (Santos et al. 2015):

\[T_b(z) = 27 \, \text{mK} \times h \times \frac{\Omega_{\text{HI}}}{10^{-3}} \times \frac{(1+z)^2}{E(z)}\]

Where \(\Omega_{\text{HI}}\) is the HI density parameter and \(E(z) = H(z)/H_0\).

Power Spectrum

The 21cm power spectrum is:

\[P_{21}(k, \mu, z) = T_b^2 \, b^2 \, (1 + \beta \mu^2)^2 \, P_m(k, z)\]

Where: - \(b(z)\) is the HI bias factor - \(\beta = f/b\) is the redshift-space distortion parameter - \(f = d\ln D/d\ln a\) is the growth rate - \(\mu = \cos\theta\) is the cosine of the angle to the line of sight - \(P_m(k,z)\) is the matter power spectrum

Effective Volume

The Fisher information-weighted effective volume:

\[V_{\text{eff}}(k, \mu, z) = V_{\text{survey}} \left[ \frac{P_{\text{signal}}}{P_{\text{signal}} + P_{\text{noise}}} \right]^2\]

Noise Power Spectrum

Instrumental noise power spectrum:

\[P_{\text{noise}} = \frac{\sigma_{\text{pix}}^2 V_{\text{pix}}}{W^2(k_\perp)}\]

Where \(W(k_\perp)\) is the beam window function and \(\sigma_{\text{pix}}\) is the pixel noise.

FisherMatrix Class

For general Fisher matrix computations (not 21cm-specific), use the FisherMatrix class.

Basic Usage

from hicosmo.fisher import FisherMatrix, FisherMatrixConfig

# Configuration
config = FisherMatrixConfig(
    differentiation_method='automatic',  # Use JAX automatic differentiation
    regularization=True
)

# Create calculator
calculator = FisherMatrix(config)

# Compute Fisher matrix
fisher_matrix, param_names = calculator.compute_fisher_matrix(
    likelihood_func=my_likelihood,
    parameters=params,
    fiducial_values={'H0': 67.36, 'Omega_m': 0.315}
)

Configuration Options

from hicosmo.fisher import FisherMatrixConfig

config = FisherMatrixConfig(
    # Differentiation method
    differentiation_method='automatic',  # 'automatic', 'numerical', 'hybrid'
    step_size=1e-6,
    fallback_to_numerical=True,

    # Regularization
    regularization=True,
    reg_lambda=1e-10,
    svd_threshold=1e-12,

    # Validation
    check_symmetry=True,
    check_positive_definite=True,

    # Performance
    vectorize_derivatives=True,
    cache_derivatives=True
)

Matrix Combination

Combine multi-probe Fisher matrices:

# Compute Fisher matrices for each probe
fisher_bao, params_bao = calculator.compute_fisher_matrix(bao_likelihood, ...)
fisher_sn, params_sn = calculator.compute_fisher_matrix(sn_likelihood, ...)

# Combine
combined_fisher, combined_params = calculator.combine_fisher_matrices(
    [(fisher_bao, params_bao), (fisher_sn, params_sn)],
    probe_names=['BAO', 'SN']
)

Parameter Marginalization

# Marginalize over nuisance parameters
marginalized_fisher, remaining_params = calculator.marginalize_parameters(
    fisher_matrix,
    param_names,
    marginalize_over=['M_B', 'delta_M']
)

Dark Energy Figure of Merit

The constraining power of dark energy experiments is quantified by the Figure of Merit (FoM):

\[\text{FoM} = \frac{1}{\sqrt{\det(\text{Cov}_{w_0, w_a})}}\]

This is equivalent to the inverse of the area enclosed by the \(1\sigma\) contour in the \(w_0\)-\(w_a\) plane.

from hicosmo.models import CPL
from hicosmo.fisher import IntensityMappingFisher
import numpy as np

# Forecast w0-wa constraints
cpl = CPL(H0=67.36, Omega_m=0.3153, w0=-1.0, wa=0.0)
result = IntensityMappingFisher.forecast('ska1_mid_band2', cpl, ['w0', 'wa'])

# Compute FoM
fom = 1.0 / np.sqrt(np.linalg.det(result.covariance))
print(f"Dark Energy FoM: {fom:.1f}")

Growth Parameter Forecasts

HIcosmo supports growth parameter forecasts for modified gravity theories.

Growth Parameterization

The growth rate is parameterized as:

\[f(z) = \Omega_m(z)^{\gamma_0 + \gamma_1 z}\]

The deviation parameters are:

\[\eta(z) = \eta_0 + \eta_1 z\]

In General Relativity: \(\gamma_0 = 0.55\), \(\gamma_1 = \eta_0 = \eta_1 = 0\).

Growth Scenarios

HIcosmo implements the four growth scenarios from Bull et al. (2016):

from hicosmo.fisher.intensity_mapping import GrowthScenario

# Available scenarios
scenarios = [
    'gr',           # General Relativity (no free growth parameters)
    'gamma0_free',  # gamma_0 free, others fixed
    'gamma1_free',  # gamma_1 free, others fixed
    'eta0_free',    # eta_0 free, others fixed
    'eta1_free',    # eta_1 free, others fixed
    'all_free',     # All growth parameters free
]

# Use a growth scenario
scenario = GrowthScenario('gamma0_free')
print(f"Free parameters: {scenario.free_params}")
print(f"Fixed parameters: {scenario.fixed_params}")

Visualization

Visualizing Fisher forecast results:

from hicosmo.visualization import Plotter, plot_fisher_contours
import numpy as np

# Method 1: Using Plotter.from_fisher
plotter = Plotter.from_fisher(
    mean=result.cosmology_summary['fiducial'],
    covariance=result.covariance,
    param_names=result.params
)
plotter.corner(filename='forecast.pdf')

# Method 2: Using standalone function
plot_fisher_contours(
    mean=[-1.0, 0.0],
    covariance=result.covariance,
    param_names=['w0', 'wa'],
    filename='de_forecast.pdf'
)

Multi-Survey Comparison

from hicosmo.models import CPL
from hicosmo.fisher import IntensityMappingFisher
from hicosmo.visualization import Plotter

cpl = CPL()
surveys = ['ska1_mid_band2', 'tianlai', 'bingo']
results = []

for survey in surveys:
    result = IntensityMappingFisher.forecast(survey, cpl, ['w0', 'wa'])
    results.append(result)

# Comparison plot (using Plotter.from_fisher multiple times)
# Or manually generate Gaussian samples and use Plotter

Complete Examples

SKA1 Dark Energy Forecast

from hicosmo.models import CPL
from hicosmo.fisher import IntensityMappingFisher, list_available_surveys
import numpy as np

# Define the cosmological model
fiducial = CPL(
    H0=67.36,
    Omega_m=0.3153,
    Omega_b=0.0493,
    w0=-1.0,
    wa=0.0,
    sigma8=0.811,
    n_s=0.9649
)

# Forecast w0-wa constraints
result = IntensityMappingFisher.forecast(
    'ska1_mid_band2',
    fiducial,
    params=['w0', 'wa'],
    marginalize_over=['H0', 'Omega_m']
)

# Print results
print(result)

# Compute FoM
fom = 1.0 / np.sqrt(np.linalg.det(result.covariance))
print(f"\nDark Energy FoM: {fom:.1f}")

# Correlation coefficient
print(f"w0-wa correlation: {result.correlation[0,1]:.3f}")

Interacting Dark Energy Forecast

from hicosmo.models import ILCDM
from hicosmo.fisher import IntensityMappingFisher

# Interacting dark energy model
ilcdm = ILCDM(
    H0=67.36,
    Omega_m=0.3153,
    beta=0.001  # Interaction strength
)

# Tianlai constraints on beta
result = IntensityMappingFisher.forecast(
    'tianlai',
    ilcdm,
    params=['beta', 'H0', 'Omega_m']
)

print(f"Tianlai constraints on the interaction parameter:")
print(f"sigma(beta) = {result.errors['beta']:.4e}")

Multi-Survey Joint Analysis

from hicosmo.models import CPL
from hicosmo.fisher import IntensityMappingFisher, load_survey, FisherMatrix
import numpy as np

cpl = CPL()
surveys = ['ska1_mid_band2', 'tianlai', 'bingo']

# Collect Fisher matrices from each survey
fisher_matrices = []
for survey_name in surveys:
    result = IntensityMappingFisher.forecast(
        survey_name, cpl, ['w0', 'wa', 'H0', 'Omega_m']
    )
    fisher_matrices.append((result.fisher_matrix, result.params))

# Combine Fisher matrices
calculator = FisherMatrix()
combined_fisher, combined_params = calculator.combine_fisher_matrices(
    fisher_matrices,
    probe_names=surveys
)

# Compute joint constraints
combined_cov = np.linalg.inv(combined_fisher)
combined_errors = np.sqrt(np.diag(combined_cov))

print("Joint constraints:")
for i, param in enumerate(combined_params):
    print(f"  sigma({param}) = {combined_errors[i]:.4e}")

Performance Benchmarks

Operation	Time	Notes
Single-survey Fisher forecast	~0.5s	5 redshift bins
Full parameter forecast (with marginalization)	~1.2s	6 parameters
Automatic differentiation Hessian	~0.1s	JAX JIT

FAQ

Q: How to customize a survey configuration?

Create a YAML file and use IntensityMappingSurvey.from_file():

from hicosmo.fisher import IntensityMappingSurvey

survey = IntensityMappingSurvey.from_file('my_survey.yaml')

Q: What if the Fisher matrix is not positive definite?

Enable regularization:

from hicosmo.fisher import FisherMatrixConfig

config = FisherMatrixConfig(
    regularization=True,
    reg_lambda=1e-8
)

Q: How to compare different cosmological models?

Use the same survey configuration with different models:

from hicosmo.models import LCDM, wCDM, CPL

survey = 'ska1_mid_band2'

result_lcdm = IntensityMappingFisher.forecast(survey, LCDM(), ['H0', 'Omega_m'])
result_wcdm = IntensityMappingFisher.forecast(survey, wCDM(), ['H0', 'Omega_m', 'w0'])
result_cpl = IntensityMappingFisher.forecast(survey, CPL(), ['H0', 'Omega_m', 'w0', 'wa'])

Next Steps

Visualization: Display Fisher forecast results
Cosmological Models: Learn about available models