SYREN-NEW spectral-tilt response and photometric Fisher discrepancies

Main finding

The photometric comparison identifies the nonlinear response to the primordial spectral index ns as an important source of SYREN-NEW versus EE2 disagreement. The symbolic formula contains explicit ns dependence that strongly cancels the linear tilt response at some low-redshift nonlinear scales. Fixing ns in the saved LCDM Fisher matrices greatly reduces several marginalized-error differences.

This is evidence of a response mismatch and altered degeneracies, not proof of a transcription error in SYREN-NEW or proof that this derivative is the only source of the Fisher differences. EE2 is the comparison reference, not an assumed exact solution. A controlled derivative-swap experiment remains the cleanest way to isolate causality.

Analysis date: 2026-09-29. The full photometric notebook runs successfully after the earlier HMcode non-finite-grid handling fix; that operational issue is unrelated to the finite-response discrepancy studied here.

Inputs and numerical definitions

The originating notebook is SYREN-investigation_Photo-EE2-vs-matched.ipynb. Despite its historical filename, the inspected run uses the following photo profiles:

Case	Packaged profile
CLASS + EE2	class/ee2_boost_photo.yaml
CLASS + HMcode2020	class/hmcode2020_photo.yaml
SYREN-NEW	symbolic/syren_new_photo.yaml

They reside in cosmicjellyfish/configs/default_boltzmann_yaml_files/. The comparison is for total-matter power, not a separate cold+baryon prediction.

Parameter	Fiducial
10^9As	2.1
Omegam	0.32
Omegab	0.05
h	0.67
ns	0.96
mnu	0.06 eV
w0, wa	−1, 0

All other parameters, including primordial amplitude As, are fixed during the ns variation. The CLASS profiles use the matched three-degenerate-neutrino translation (N_ur=0.00641, mass-density factor 93.14). The wrapper passes ns correctly, and its sigma8-to-As conversion is not invoked when 10^9As is supplied.

The response plotted below is a central difference of logarithmic power:

\[R_{n_s}(k,z) = \frac{\partial\ln P(k,z)}{\partial n_s} \simeq \frac{\ln P(k,z;n_s+\Delta n_s)-\ln P(k,z;n_s-\Delta n_s)}{2\Delta n_s}.\]

The default half-step is 0.0096, the notebook’s 1% relative ns step. This is not differentiation with respect to log ns. The script also saves the central difference of power itself, dP_dns, which is distinct at finite step from P_fiducial * response.

The plotted grid is k=0.01–3 h/Mpc, at z=0,1,2. The public Pmm API receives k*h in 1/Mpc; returned Mpc³ values are multiplied by h³. Both source spline support and positive, finite output are checked. No extrapolated points are used.

Reproduce the figures

The maintained script is scripts/validation/plot_ns_power_response.py. An executable companion with explanations, displayed figures and response tables is SYREN-investigation_ns-response.ipynb. Its default path reads the saved artifacts; the optional recomputation cell invokes the same script rather than reimplementing the backends. It uses build_analysis_context and cosmo_functions.Pmm for all three cases, including native CLASS HMcode and CLASS linear power multiplied by EE2’s boost. It reuses the context’s fiducial cosmology and computes only the two shifted cosmologies per backend: nine total evaluations, without Fisher or angular-spectrum calculations.

From the repository root, with CLASS, EuclidEmulator2 and symbolic_pofk installed:

OMP_NUM_THREADS=1 uv run python scripts/validation/plot_ns_power_response.py

To redraw the saved arrays without running a cosmology backend:

uv run python scripts/validation/plot_ns_power_response.py --plot-only

To investigate a smaller step while preserving this reference figure set:

OMP_NUM_THREADS=1 uv run python scripts/validation/plot_ns_power_response.py \
  --ns-step 0.0001 --output-dir results/ns_response_small_step

--cosmo-model w0waCDM selects that configuration with fixed fiducial w0=-1, wa=0; it does not marginalize any parameters. Use a separate output directory for variants. Run --help for the redshift and k-grid options.

Artifacts: numerical arrays and run metadata. Metadata records profile contents/hashes, source hashes, dependency versions, resolved fiducial backend parameters, units, grid and step. Array power has axes [backend, minus/fiducial/plus, linear/nonlinear, redshift, k]; response and dP_dns omit the variation axis, and boost_response also omits the spectrum axis. Backend order is recorded in labels.

Linear response

The fixed-As primordial tilt predicts R_ns = ln(k*h/0.05) for physical pivot 0.05/Mpc. The symbolic linear expression has this dependence exactly; its remaining correction factors are independent of ns for this evaluation path. The public interfaces interpolate power on finite grids, so their derivatives need not agree with the analytic curve to machine precision. Both CLASS cases share the same linear physics.

Nonlinear response

The lower panels show additive response differences from CLASS+EE2, not percent differences. Dividing by a reference derivative near its zero crossing would create misleading spikes. The broad low-redshift discrepancy away from zero crossings is the scientifically relevant feature.

The generated public-interface run gives the following nonlinear responses at Delta ns=0.0096:

z	k [h/Mpc]	CLASS + EE2	CLASS + HMcode2020	SYREN-NEW
0	0.2	0.584306	0.560257	0.367360
0	0.5	1.070631	1.022561	0.442848
0	1.0	1.304604	1.245154	0.781266
1	0.5	1.518832	1.541994	1.431714
1	1.0	2.016386	1.992768	2.017233
2	0.5	1.733373	1.278536	1.616727
2	1.0	2.432370	1.393950	2.307892

Additional observation, not yet diagnosed: this run also shows a substantial HMcode high-k response departure at z=2, despite much closer fiducial spectra. This should be checked with step-size and direct CLASS-accessor comparisons before attributing it to HMcode physics or interpolation. It is not explained by the SYREN-specific cancellation below. The plotted results are a single-run diagnostic, not a numerical-convergence certification of any backend.

Nonlinear-boost response

The boost response is R_ns(Pnl) - R_ns(Plin). Removing the common primordial tilt makes differences in the nonlinear corrections easier to identify.

Fiducial spectrum agreement

These spectra use the same fiducial and interpolation pipeline as the response figures. Good agreement in power values does not guarantee equally accurate parameter responses.

The sensitive terms in the symbolic formula

In upstream symbolic_pofk/syren_new.py, pnl_new_emulated constructs L = log10(Plin) and then log10(Pnl) = L + term2 + term3 - term4 - term5 - bias(k). Besides dependence through L, it has explicit ns dependence in the nonlinear corrections. The particularly sensitive term3 is

\[T_3 = \frac{(0.6264a-0.3035L+0.6069)\,k} {0.7882\Omega_m+0.4811k+1.4326n_s-1.8971+ (0.0271L+0.9635k^{0.0264})^{22.9213a-71.1658n_s}}.\]

At a=1, ns=0.96, its exponent is −45.397868. The term2 denominator also contains a power with exponent 27.6818*a - 24.8736*ns (3.803144 at this point). Large fitted coefficients alone are not evidence of a bug, but the differentiated terms exhibit a strong cancellation.

The direct-formula chain-rule decomposition at z=0, k=0.5 h/Mpc is:

Contribution to d ln Pnl / d ns	Value
Linear primordial tilt	+1.90210753
term2 via linear power L	−0.13125580
term2 explicit ns dependence	+0.01353874
term3 via linear power L	+0.57108368
term3 explicit ns dependence	−1.91242415
Total	+0.44304999

The explicit term3 response almost cancels the linear contribution. The final pnl_bias(k) correction is independent of ns at fixed h, so it cannot repair the logarithmic tilt response (although it rescales the absolute derivative).

For source provenance, consult the run metadata’s symbolic_pofk.syren_new entry. The integration layer is cosmicjellyfish/cosmology/symbolic_new.py. The maintained plotting script calls that implementation; it does not duplicate or modify the symbolic fit.

Direct-formula response and step-size check

The initial diagnosis evaluated the symbolic formula directly and compared it with exact linear tilt plus the EE2 boost response, before running the three-backend public-interface plotting script. At Delta ns=0.0001, z=0:

k [h/Mpc]	SYREN-NEW response	EE2 boost + exact linear response	SYREN / reference − 1
0.2	0.367503	0.584554	−37.1%
0.5	0.443050	1.074483	−58.8%
1.0	0.781669	1.302355	−40.0%

These numbers are not a fresh CLASS calculation and need not exactly match the public-interface figures, which use power interpolation and the notebook’s larger step. The discrepancy varies with redshift: at z=1, k=0.5, the direct response difference is approximately −6.1%; at z=1, k=1, approximately −0.6%.

Over k=[0.03,0.05,0.1,0.2,0.5,1,3] h/Mpc and z=[0,1,2], decreasing the direct SYREN half-step from 0.0096 to 0.0001 changed the log response by at most 0.00078737. Decreasing 0.001 to 0.0001 changed it by at most 8.459e-6. This is far smaller than the low-z response mismatch. It is a scoped step-size check of SYREN’s direct expression, not a convergence claim for every backend or the full projected forecast.

The general distinction is straightforward: if P_syren = P_ref * exp(epsilon), then R_syren - R_ref = d epsilon / d ns. A small value error at the fiducial can coexist with a substantial slope error.

Evidence from the saved photometric Fisher matrices

The existing artifacts under results/photo_class_ee2_vs_syren_new_minimal_matched/ were inspected, not recomputed. Their names follow CosmicFish_v1.3.1_{LCDM|w0waCDM}_photo_{class_ee2|class_hmcode2020|symbolic_syren_new}_GCphWL_FM.txt. Parameter ordering was read from each accompanying .paramnames file.

Fixing ns means deleting its row and column from the full Fisher matrix, then inverting the retained block and marginalizing all other retained parameters, including survey nuisances. It does not mean deleting a covariance row/column.

LCDM parameter	SYREN vs EE2 marginalized-error difference, ns free	With ns fixed
As	−34.73%	−4.84%
Omegam	−3.82%	−1.52%
Omegab	−21.97%	−8.22%
h	−33.76%	−7.37%

This supports an important role for ns degeneracies, without establishing sole causality. A weaker individual response can yield tighter marginalized constraints if its shape changes parameter correlations. For example, SYREN’s fully conditional LCDM ns error (1/sqrt(F_ns,ns)) is 8.53% larger than EE2’s, while its marginalized ns error is 33.78% smaller.

Why w0waCDM can show closer relative agreement

Adding dark-energy parameters changes the degeneracy structure. Absolute constraints weaken, while the relative discrepancy between backends can shrink:

Marginalized sigma(ns)	CLASS + EE2	CLASS + HMcode2020	SYREN-NEW
LCDM	0.015057	0.015340	0.009971
w0waCDM	0.015418	0.019250	0.013607

The fully conditional errors of the five shared cosmological parameters are unchanged between these two saved model cases for each backend. Closer w0wa percentage agreement is therefore not evidence that the underlying SYREN derivative became more accurate.

Notebook plotting interpretation

Two details in the inspected notebook matter when reading its saved plots:

1. The notebook’s compare_constraints call uses compare_to_index=1 (HMcode2020) even though its axis label and introductory prose say EE2; compare_to_index=0 would select EE2, and its printed numerical dictionary does compare against EE2. The spectrum helper has the same mismatch. All figures on this page explicitly use CLASS + EE2 as the reference.

2. The darker “unmarginalized” error shading is computed after omitted nuisance parameters have been marginalized by the plotting helper. It holds the other displayed cosmological parameters fixed; it is not the original full-matrix conditional error. The LCDM SYREN–EE2 ns difference is approximately 10.27% by this plotted definition, versus 8.53% for the full conditional error.

These are observations about the saved notebook version, not changes made to it.

Interpretation and next discriminating check

The combined evidence supports a nonlinear tilt-response mismatch in the symbolic fit and an associated change in Fisher degeneracies. Neither decreasing the finite-difference step nor adjusting an ns-independent multiplicative bias is expected to remove the measured direct-formula mismatch.

The next controlled experiment should replace only the ns angular-spectrum derivative, keeping the covariance, remaining derivatives, parameter ordering, nuisance treatment and priors common. Comparing the resulting marginalized errors would quantify the causal contribution of that one derivative. This experiment has not been performed here. Removing or retuning the symbolic term ad hoc would change the calibrated model and is not justified by this diagnostic alone.