f(R) gravity — the fRn1 suite

The f(R) (n = 1) twin campaign re-runs the 64 ΛCDM design cosmologies with a fifth parameter logf_R0 = log10(|f_R0|) ∈ [−7, −4], reusing the matched ΛCDM initial-condition seeds. Address it with gravity="fRn1" and a 5-parameter θ:

th5 = [0.31, 0.677, 0.967, 0.83, -5.0]   # [Omega_m, h, n_s, S_8, logf_R0]
xi  = reg.predict("xi_mm", "fRn1", 0.25, th5)

Every acceptance test of the ΛCDM sections runs unchanged at the fRn1 key.

How the f(R) emulators are built

Most f(R) emulators model the modified-gravity boost over the matched ΛCDM twin, not the absolute quantity, because the shared seeds let the difference cancel cosmic variance:

\[X_{f(R)}(\theta_5) = B(\theta_5)\, X_{\Lambda\mathrm{CDM}}(\theta_5[:4])\]

Property	f(R) representation
hmf	multiplicative boost (MGBoostEmulator)
pk_mm	multiplicative boost
xi_mm	additive Δξ boost
vel_c40, vel_c04, vel_c22	log-ratio boost
vel_c20, vel_c02, vel_c12, vel_c30	additive δ boost, arcsinh
vel_m10	direct 5-parameter GP
b_cum	direct 5-parameter peak-height GP

Boost emulators record the sha256 checksum of their ΛCDM base, fixing exactly which file is used; retraining the base forces the checksum to be re-recorded. The choice of boost vs direct GP per property is evidence-based — see Representations.

Fiducial references

Two 100-box fiducials at the GR fiducial background, with independent seeds (not limited by the 5-box sampling floor):

• F5n1 — logf_R0 = −5 (get_cosmology("fRn1", 0)).
• F6n1 — logf_R0 = −6 (get_cosmology("fRn1", -1)).

They are sharp held-out tests for the hmf/matter/velocity sectors. For the halo-clustering chain (b_cum + the D surface) they are calibration points since the 2026-07 accuracy campaign — their measured b(>M) and threshold cubes enter the training (the design alone cannot constrain the b mass-slope; see bias and halo clustering).

Accuracy (z = 0.25)

• Composed boosts 0.5–0.9 % LOO.
• hmf fiducial OOS 0.3–0.5 % with the screened tail resolved (see HMF).
• ξ_hh design-sweep band medians 1.1–2.2 % to 60 Mpc/h at median |χ| ≤ 1 (binned radial convention); fiducial calibration consistency χ_rms 4.3/3.8 vs the 100-box SEM.
• Velocity moments: aggregated fiducial test χ 0.94–1.73 over all eight moments (the maximum is the fourth-order c04 — the signal-weighted infill, imodel 68–72, repaired the formerly-worst c02, interior LOO 2.03→1.57, and trains the c02/c20/c12 boosts and all ΛCDM moments). c12, excluded 2026-07-04 when its narrow-grid boost with a noise-scaled transform failed the aggregation test, was re-tested on the wide bins with the family's plain additive-arcsinh boost and admitted 2026-07-15: interior LOO χ 1.51, fiducial aggregation χ 1.24, independent-seed χ 1.53 — in-family on every ruler (the carrier-only route it replaces missed the F5n1 c12 response at χ ≈ 14). On the wide bins the boosts' design-interior LOO sits at χ ≈ 1.3–1.9 — the noisy fourth moments ran up to χ ≈ 7 at the retired narrow binning — per-emulator χ in the manifest.
• Independent-seed test: 25 F5n1 fiducial boxes (6–30, measured for the purpose, never used in any calibration). Correction off: χ 2.5–4.3, 68–94% coherent — the shared-seed offset is real in f(R). Shipped correction on (calibrated purely on ΛCDM boxes): χ 1.1–2.3 on the 2.2× tighter 25-box error bars — the seed-bias correction transfers across gravity, including to the newly admitted c12 (1.53). The 2026-07-06 run adjudicated c02: its fiducial deviation is a genuine r = 3–10 design-sampling error (it persisted against independent seeds, where the design-only c02 at 2.09 was indistinguishable from m10 and c20); the supplement-trained c02 now reads 1.60 there. Evidence: registration_widea/vel_indepseed_oos.json (refreshed 2026-07-15).

Improving the f(R) emulators with new simulations

The high even velocity moments (c02, c40, c22, c04) plateau because 64 design points are sparse in 5D. A campaign-era learning-curve experiment (velocity_frn1_learning_curve.py, now in git history) showed the design is still in the undersampled regime: adding new f(R) + ΛCDM simulation pairs gives roughly a 10–15 % χ reduction for +8 pairs and 20–25 % for +16, enough to bring c40/c02/c22 near the floor but not, by itself, to break the c04 plateau (which needs tens of pairs).

A sequential maximin infill design provides an ordered list of new 5D cosmologies, each filling the largest remaining hole, biased toward the chameleon screening transition (logf_R0 ~ −5.5 to −6, where the response is steepest):

micromamba run -n cosemu python3 haloemu/properties/generate_frn1_infill.py \
    --n-new 64 --out doc/frn1_infill_design.csv

Each row is one simulation pair — the f(R) run at the five parameters plus a matched-seed ΛCDM twin at the four background parameters, with new IC seeds (ibox ≥ 6). Run the top K for whatever budget you have. Full recipe and numbers: doc/frn1_new_runs_velocity.tex and the methods paper.

Caveats specific to f(R)

• The frozen-seed large-scale offset affects ξ / ξ_hh for fRn1 as for ΛCDM — confirmed at both independent-seed fiducials. See Caveats.
• Near the GR limit logf_R0 → −7 the f(R) response → ΛCDM, so the boost carries little information there.