Known Limitations & Scientific References¶
This page documents model assumptions, known limitations, verified components, and scientific references for the EMRI dark siren H₀ inference pipeline ([Chen2018], [Gray2020]).
For the H₀ posterior bias investigation timeline, see docs/H0_BIAS_RESOLUTION.md.
Model Assumptions¶
Assumption |
Where used |
Notes |
|---|---|---|
Flat ΛCDM (\(w_0=-1\), \(w_a=0\)) |
All distance integrals |
|
Gaussian measurement noise on \(d_L\) |
GW likelihood |
Uses the full Fisher-matrix covariance (per-source) in |
SNR threshold as detection proxy |
|
Threshold = 20; detailed waveform-parameter dependence not captured |
Uniform prior on \(H_0\) |
|
No prior declared; implicitly flat over \([H_\mathrm{min}, H_\mathrm{max}]\) |
LISA mission duration 5 years |
Waveform generation |
Feeds directly into TDI response and sky-averaged sensitivity |
Known Limitations¶
Items are ordered by severity. Each references the specific source location and carries a status tag: bug (incorrect formula or logic), design choice (deliberate simplification), or pending fix (acknowledged issue not yet addressed).
Limitation 1 — Comoving volume formula [FIXED]¶
File: master_thesis_code/datamodels/galaxy.py, master_thesis_code/physical_relations.py
The function computes the comoving volume element \(dV_c/dz\), not total volume \(V_c\).
The exponent 2 and \(4\pi\) prefactor were correct for the element, but the formula was
missing the \(1/E(z)\) factor. Fix applied: \(cv\_grid = 4\pi \cdot (c/H_0)^3 \cdot I(z)^2 / E(z)\)
([Hogg1999], Eq. 27). All methods renamed from comoving_volume to
comoving_volume_element for clarity. Standalone comoving_volume_element() in
physical_relations.py verified against astropy to 0.07% accuracy. Regression test added.
Limitation 2 — Fisher matrix derivative accuracy [FIXED]¶
File: master_thesis_code/parameter_estimation/parameter_estimation.py
The Fisher matrix previously used an \(O(\varepsilon)\) forward difference via
finite_difference_derivative(). This was replaced with the \(O(\varepsilon^4)\)
five-point stencil (five_point_stencil_derivative()) as the default in GPD Phase 10.
Controlled by the use_five_point_stencil constructor parameter (default True).
The forward difference remains available as a fallback but is no longer used in production.
References: [Vallisneri2008]; [CF1994].
Limitation 3 — Galactic confusion noise [FIXED]¶
File: master_thesis_code/LISA_configuration.py
Galactic confusion noise is now included in the LISA PSD via _confusion_noise() in
LisaTdiConfiguration, implementing [Babak2023] Eq. (17) with
observation-time-dependent knee frequency. Controlled by include_confusion_noise
parameter (default True). The constants from constants.py:77–83 are now used.
Limitation 4 — wCDM parameters silently ignored [bug · MEDIUM]¶
File: master_thesis_code/physical_relations.py:72
dist() accepts w_0 and w_a but passes them only to lambda_cdm_analytic_distance(),
which ignores them — the hypergeometric formula is exact only for flat ΛCDM (\(w_0=-1\),
\(w_a=0\)). No warning is raised. Any caller supplying non-fiducial dark-energy parameters
silently receives ΛCDM distances. The correct general formula would require numerical
integration via hubble_function(), which already implements the full CPL parameterisation.
Reference: [Hogg1999], Eq. (14–16).
Limitation 7 — Outdated fiducial cosmological parameters [design choice · LOW]¶
File: master_thesis_code/constants.py:29–30
Parameter |
Code |
|
|---|---|---|
\(\Omega_m\) |
0.25 |
0.3153 ± 0.0073 |
\(\Omega_\Lambda\) |
0.75 |
0.6847 ± 0.0073 |
\(h\) (simulation) |
0.73 |
0.6736 ± 0.0054 |
The WMAP-era values used here differ from the current Planck 2018 best fit by ~2σ in \(\Omega_m\) and ~1σ in \(h\). For a simulation intended to represent realistic LISA science the fiducial point should be updated.
Reference: [Planck2018], Table 2.
Limitation 8 — Galaxy redshift uncertainty has non-standard scaling [design choice · LOW]¶
File: master_thesis_code/datamodels/galaxy.py:64
redshift_uncertainty = min(0.013 * (1 + redshift) ** 3, 0.015)
The \((1+z)^3\) scaling grows rapidly and hits the cap of 0.015 at \(z \approx 0.14\), so all galaxies above that redshift are assigned identical uncertainty. Standard photometric redshift errors scale as \(\sigma_z \approx 0.05(1+z)\); spectroscopic errors as \(\sigma_z \approx 0.001(1+z)\). No reference for the cubic form is provided.
What Is Mathematically Correct¶
The following components have been verified against their cited references:
Luminosity distance hypergeometric integral (
physical_relations.py): the form \(\,{}_2F_1(1/3,1/2;4/3;-\Omega_m(1+z)^3/\Omega_\Lambda)\) is the correct analytic solution for flat ΛCDM. ✓LISA instrumental PSD (A/E channels) (
LISA_configuration.py): matches [Babak2023], Eqs. (8)–(11), excluding galactic confusion noise. ✓Noise-weighted inner product (
parameter_estimation.py): the factor-of-4 prefactor and one-sided PSD convention are correct; FFT normalisation is handled correctly viatrapzover the frequency axis. ✓Five-point stencil formula (
five_point_stencil_derivativeinparameter_estimation.py): the coefficients \((-1, 8, -8, 1)/12\varepsilon\) are the correct \(O(\varepsilon^4)\) centred finite difference. Now used by default in the Fisher matrix computation. ✓Redshifted mass conversion: \(M_z = M(1+z)\) and its inverse are correctly implemented in
physical_relations.py. ✓
Bibliography¶
Cutler, C. & Flanagan, É. E. (1994). Gravitational waves from merging compact binaries: How accurately can one extract the binary’s parameters from the inspiral waveform? Phys. Rev. D 49, 2658.
Vallisneri, M. (2008). Use and abuse of the Fisher information matrix in the assessment of gravitational-wave parameter-estimation prospects. Phys. Rev. D 77, 042001. arXiv:gr-qc/0703086.
Chen, H.-Y., Fishbach, M. & Holz, D. E. (2018). A two percent Hubble constant measurement from standard sirens within five years. Nature 562, 545–547. arXiv:1709.08079.
Gray, R. et al. (2020). Cosmological inference using gravitational wave standard sirens: A mock data challenge. Phys. Rev. D 101, 122001. arXiv:1908.06050.