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	physical_relations.py; wCDM infrastructure exists but is untested
Gaussian measurement noise on \(d_L\)	GW likelihood	Width hardcoded to 10% of \(d_L\) in Pipeline A (see Limitation 5); Pipeline B uses Fisher covariance
SNR threshold as detection proxy	parameter_estimation.py, bayesian_inference.py	Threshold = 20; detailed waveform-parameter dependence not captured
Uniform prior on \(H_0\)	bayesian_inference.py	No prior declared; implicitly flat over \([H_\mathrm{min}, H_\mathrm{max}]\)
Synthetic galaxy catalog	Pipeline A	Galaxies drawn uniformly in log-mass and from comoving volume; GLADE used in Pipeline B
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 5 — GW likelihood distance uncertainty hardcoded at 10% [design choice · MEDIUM]

File: master_thesis_code/bayesian_inference/bayesian_inference.py

Applies to Pipeline A only. Pipeline B (BayesianStatistics in bayesian_statistics.py) uses the full Fisher-matrix covariance.

The Pipeline A GW likelihood uses \(\sigma_{d_L} = 0.1\,d_L\) for every event (via FRACTIONAL_LUMINOSITY_ERROR). The simulation already computes the actual per-source Cramér–Rao bound on \(d_L\) (stored as delta_luminosity_distance_delta_luminosity_distance in the CSV output). Using a source-by-source uncertainty from the Fisher matrix would make nearby, well-localised events contribute more sharply to the \(H_0\) posterior, as they physically should.

Limitation 6 — Two Bayesian pipelines with inconsistent formulations [design choice · IMPORTANT]

Files: master_thesis_code/bayesian_inference/bayesian_inference.py (Pipeline A); master_thesis_code/bayesian_inference/bayesian_statistics.py (Pipeline B)

Pipeline A (synthetic catalog) marginalises over a continuous redshift grid with a scalar Gaussian likelihood on \(d_L\) and a simplified selection correction. Pipeline B (GLADE catalog) constructs a full multivariate Gaussian likelihood over \((\varphi, \theta, d_L/d_L^\mathrm{pred})\) using the actual Fisher-matrix covariance and a simulation-based detection-probability estimate. The two formulations are not mathematically equivalent and would yield different posteriors on identical data. Pipeline B is the science-grade implementation; Pipeline A is a development-only cross-check.

Limitation 7 — Outdated fiducial cosmological parameters [design choice · LOW]

File: master_thesis_code/constants.py:29–30

Parameter	Code	[Planck2018]
\(\Omega_m\)	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 via trapz over the frequency axis. ✓

• Five-point stencil formula (five_point_stencil_derivative in parameter_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. ✓

• Bayesian selection-effects correction (Pipeline A): the ratio numerator/denominator where the denominator integrates \(p_\mathrm{det}(z,H_0)\,p(z|\mathrm{cat})\) correctly implements the Loredo–Mandel selection-bias correction for the marginalised likelihood. ✓

• Redshifted mass conversion: \(M_z = M(1+z)\) and its inverse are correctly implemented in physical_relations.py. ✓

Bibliography

[Hogg1999] (1,2)

Hogg, D. W. (1999). Distance measures in cosmology. arXiv:astro-ph/9905116.

[Babak2023] (1,2)

Babak, S. et al. (2023). LISA sensitivity and SNR calculations. arXiv:2303.15929.

[CF1994]

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.

[Vallisneri2008]

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.

[Chen2018]

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.

[Gray2020]

Gray, R. et al. (2020). Cosmological inference using gravitational wave standard sirens: A mock data challenge. Phys. Rev. D 101, 122001. arXiv:1908.06050.

[Planck2018] (1,2)

Planck Collaboration (2020). Planck 2018 results VI: Cosmological parameters. Astron. Astrophys. 641, A6. arXiv:1807.06209.