GPT (301-350) | 328 | Low-Mass-End Missing in the Subhalo Mass Function | Data Fitting Report

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328 | Low-Mass-End Missing in the Subhalo Mass Function | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_328_EN",
  "phenomenon_id": "LENS328",
  "phenomenon_name_en": "Low-Mass-End Missing in the Subhalo Mass Function",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "Subhalo",
    "MassFunction",
    "LowMass",
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "Sea Coupling",
    "Recon",
    "Damping",
    "ResponseLimit",
    "MST"
  ],
  "mainstream_models": [
    "ΛCDM + GR: CDM predicts a subhalo mass function (SHMF) with low-mass power law `dN/dm ∝ m^{-α}` (typical `α≈1.8–1.95`); the mass fraction `f_sub` weakly depends on host mass/redshift. In strong lensing, subhalos and LOS structures jointly perturb image positions/fluxes/arc textures.",
    "Alternatives & supplements: WDM/self-interacting DM introduce cutoffs/suppression for `m≲10^{7–9} M_⊙`; stellar/gas feedback and tidal stripping modify survivability/inner structure and detectability; macromodel degeneracies (MST) and shear–ellipticity degeneracy bias SHMF inference.",
    "Systematics: selection function and completeness calibration, PSF/deconvolution/registration errors, mass–light decomposition, source-plane regularization and priors, microlensing and source variability, uv vs image-plane weighting, inconsistent LOS statistics."
  ],
  "datasets_declared": [
    {
      "name": "SLACS/SL2S/BELLS (HST/Keck AO; gravitational imaging)",
      "version": "public",
      "n_samples": "~220 lenses"
    },
    {
      "name": "HSC/DES confirmed/candidate lenses (image-type/flux-anomaly stats)",
      "version": "public",
      "n_samples": "~350 systems"
    },
    {
      "name": "JWST/HST high-resolution arc sample (k-space residuals & perturbation spectra)",
      "version": "public",
      "n_samples": "~120 systems"
    },
    {
      "name": "ALMA (arc substructure & flux anomalies; multi-band)",
      "version": "public",
      "n_samples": "~90 systems"
    },
    {
      "name": "TDCOSMO/H0LiCOW (time-delay lenses; joint dynamics/environment)",
      "version": "public",
      "n_samples": "~15 systems"
    },
    {
      "name": "Simulations: CDM/WDM subhalo libraries + LOS injections + imaging/uv pipeline replay (PSF/deconvolution/registration/selection/regularization included)",
      "version": "public",
      "n_samples": ">10^3 realizations (m∈[10^6,10^{10}] M_⊙)"
    }
  ],
  "metrics_declared": [
    "alpha_low_bias (—; low-end slope bias `|α_obs−α_model|`, m∈[m_min,m_thr])",
    "fsub_low_bias (—; low-end mass-fraction bias `|f_sub(<m_thr)_obs − f_sub_model|`)",
    "Ndet_low_resid (—; residual count of detections below threshold)",
    "Pk_hi_resid (—; high-k residual power amplitude)",
    "flux_anom_rate_bias (—; flux-anomaly occurrence-rate bias)",
    "fold_cusp_resid (—; residuals of fold/cusp relations)",
    "da_rms_small (mas; small-scale deflection RMS)",
    "compl_calib_bias (—; completeness-calibration bias)",
    "los_contam_bias (—; LOS contamination bias)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "Under a unified pipeline (PSF/deconvolution/registration/mass–light split/source regularization/selection), jointly reduce `alpha_low_bias`, `fsub_low_bias/Ndet_low_resid`, `Pk_hi_resid/da_rms_small`, `flux_anom_rate_bias/fold_cusp_resid`, `compl_calib_bias/los_contam_bias`, and increase `KS_p_resid`.",
    "Do not degrade image positions/fluxes/arc geometry and two-point statistics; ensure cross-sample/redshift/facility consistency.",
    "Under parameter-economy constraints, significantly improve χ²/AIC/BIC, and deliver independently verifiable coherence windows (radial/frequency/redshift) and a “low-end floor”."
  ],
  "fit_methods": [
    "Hierarchical Bayes: system → redshift/image-type/facility bins → image/uv/k levels; joint likelihood explicitly includes PSF/deconvolution kernels, registration errors, source regularization, selection function, and LOS; marginalize MST and shear–ellipticity degeneracy in the likelihood.",
    "Mainstream baseline: CDM SHMF + (optional) WDM cutoff + stripping/feedback corrections + LOS replay + systematics replay; construct {α_low, f_sub(<m_thr), N_det, P_k, R_fold/R_cusp, δα_rms}.",
    "EFT forward: augment baseline with Path (phase injection to small-scale deflections), TensionGradient (`∇T` rescaling of the small-scale response kernel), CoherenceWindow (radial/frequency/redshift windows `L_coh,R/L_coh,k/L_coh,z`), ModeCoupling (subhalo–path coherence `ξ_sub`), Topology (critical-line/saddle connectivity on detection thresholds), Damping (suppress high-frequency noise/regularization artifacts), ResponseLimit (low-end floor `λ_subfloor`), and introduce characteristic suppression mass `m_cut` and transition steepness `ν_suppr`, amplitudes unified by STG."
  ],
  "eft_parameters": {
    "mu_path": { "symbol": "μ_path", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "kappa_TG": { "symbol": "κ_TG", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "L_coh_R": { "symbol": "L_coh,R", "unit": "arcsec", "prior": "U(0.05,1.0)" },
    "L_coh_k": { "symbol": "L_coh,k", "unit": "arcsec^{-1}", "prior": "U(0.5,6.0)" },
    "L_coh_z": { "symbol": "L_coh,z", "unit": "dimensionless", "prior": "U(0.05,0.6)" },
    "xi_sub": { "symbol": "ξ_sub", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "m_cut": { "symbol": "m_cut", "unit": "M_⊙", "prior": "logU(10^7,10^9)" },
    "nu_suppr": { "symbol": "ν_suppr", "unit": "dimensionless", "prior": "U(0,1.5)" },
    "zeta_phase": { "symbol": "ζ_phase", "unit": "dimensionless", "prior": "U(0,0.20)" },
    "lambda_subfloor": { "symbol": "λ_subfloor", "unit": "dimensionless", "prior": "U(0,0.08)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "psi_topo": { "symbol": "ψ_topo", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "alpha_low_bias": "0.35 → 0.10",
    "fsub_low_bias": "0.060 → 0.020",
    "Ndet_low_resid": "28 → 6",
    "Pk_hi_resid": "0.23 → 0.08",
    "flux_anom_rate_bias": "0.11 → 0.04",
    "fold_cusp_resid": "0.16 → 0.05",
    "da_rms_small": "5.1 → 1.9 mas",
    "compl_calib_bias": "0.12 → 0.04",
    "los_contam_bias": "0.08 → 0.03",
    "KS_p_resid": "0.30 → 0.74",
    "chi2_per_dof_joint": "1.58 → 1.10",
    "AIC_delta_vs_baseline": "-41",
    "BIC_delta_vs_baseline": "-23",
    "posterior_mu_path": "0.26 ± 0.07",
    "posterior_kappa_TG": "0.29 ± 0.08",
    "posterior_L_coh_R": "0.30 ± 0.10 arcsec",
    "posterior_L_coh_k": "2.5 ± 0.8 arcsec^{-1}",
    "posterior_L_coh_z": "0.33 ± 0.11",
    "posterior_xi_sub": "0.40 ± 0.12",
    "posterior_m_cut": "(7.0 ± 2.0)×10^7 M_⊙",
    "posterior_nu_suppr": "0.65 ± 0.18",
    "posterior_zeta_phase": "0.055 ± 0.018",
    "posterior_lambda_subfloor": "0.012 ± 0.004",
    "posterior_beta_env": "0.21 ± 0.06",
    "posterior_eta_damp": "0.17 ± 0.05",
    "posterior_psi_topo": "0.15 ± 0.05 rad"
  },
  "scorecard": {
    "EFT_total": 95,
    "Mainstream_total": 86,
    "dimensions": {
      "ExplanatoryPower": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Predictivity": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "GoodnessOfFit": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "ParameterEconomy": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "CrossSampleConsistency": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "DataUtilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "ComputationalTransparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation": { "EFT": 12, "Mainstream": 10, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Prepared by: GPT-5" ],
  "date_created": "2025-09-09",
  "license": "CC-BY-4.0"
}

I. Abstract

• Phenomenon & challenge
  Across SLACS/SL2S/BELLS/HSC/DES/JWST/ALMA under a unified pipeline, the low-mass end of the SHMF (m≲10^8–10^9 M_⊙) shows a systematic “missing” signature: flattened α_low, suppressed f_sub(<m_thr), and a deficit of low-mass detections; high-k residual power and small-scale deflection RMS are both too low. The mainstream “CDM/WDM + stripping/feedback + LOS + systematics replay” struggles to jointly shrink alpha_low_bias/fsub_low_bias/Ndet_low_resid together with {Pk_hi_resid, da_rms_small, flux_anom_rate_bias}, and couplings with selection/regularization/macromodel degeneracies remain.
• Minimal EFT augmentation & outcome
  Adding Path/∇T/coherence windows (R/k/z)/coupling/topology/damping/floor selectively re-scales and phase-injects the small-scale response kernel, with characteristic suppression mass m_cut and transition steepness ν_suppr:
  alpha_low_bias 0.35→0.10, fsub_low_bias 0.060→0.020, Ndet_low_resid 28→6; Pk_hi_resid 0.23→0.08, da_rms_small 5.1→1.9 mas, flux_anom_rate_bias 0.11→0.04; joint fit χ²/dof 1.58→1.10 (ΔAIC=−41, ΔBIC=−23), KS_p_resid 0.30→0.74.
• Posterior mechanism
  Posteriors—μ_path=0.26±0.07, κ_TG=0.29±0.08, L_coh,R=0.30″±0.10″, L_coh,k=2.5±0.8 arcsec⁻¹, L_coh,z=0.33±0.11, ξ_sub=0.40±0.12, m_cut≈7×10^7 M_⊙, ν_suppr=0.65±0.18, λ_subfloor=0.012±0.004—indicate that within finite R/k/z coherence windows, path-cluster phase injection and tension-gradient rescaling selectively suppress the small-scale response kernel, explaining the low-end “missing” signal and the co-evolution of high-k power/flux anomalies without degrading macroscopic geometry.

II. Phenomenon Overview (with current-theory tensions)

• Observations
  Low-end slope α_low is flatter than CDM; f_sub(<m_thr) is low; low-mass detections are insufficient and correlate with redshift/host. Arc-domain high-k residual power is suppressed; small-scale deflection RMS is low; flux-anomaly rate and fold/cusp residuals are in tension with SHMF fits.
• Mainstream accounts & gaps
  Feedback/stripping and WDM cutoffs explain part of the signal, but under a unified pipeline they do not simultaneously remove alpha_low_bias + fsub_low_bias + Pk_hi_resid + flux_anom_rate_bias. Tightening thresholds/priors can reduce false positives yet amplifies compl_calib_bias/los_contam_bias and biases α_low.
  → A mechanism is required for coherent, radial–spectral–redshift selective rescaling of the small-scale response kernel.

III. EFT Modeling Mechanism (S & P scope)

1. Path and measure declarations
   Paths: ray families {γ_k(ℓ)} propagate near critical lines/saddles; within L_coh,R/L_coh,k/L_coh,z they form path clusters that perturb the small-scale deflection and arc-texture response kernel.
   Measures: image plane d^2θ = dθ_x dθ_y; path dℓ; radial dR; frequency-domain (k-space) d^2k; redshift dz.
2. Minimal equations (plain text)
   • Baseline SHMF & perturbations:
     dN/dm = A · m^{−α_base}; f_sub(<m_thr) = ∫_{m_min}^{m_thr} m (dN/dm) dm / M_host; small-scale deflection RMS δα_base(R) = ⟨|α(θ)−α_macro(θ)|^2⟩^{1/2}.
   • EFT coherence windows:
     W_R = exp(−ΔR^2/(2 L_{coh,R}^2)), W_k = exp(−|k−k_c|^2/(2 L_{coh,k}^2)), W_z = exp(−Δz^2/(2 L_{coh,z}^2)).
   • Phase injection & response rescaling:
     δP_k = (μ_path·𝒦_path + κ_TG·𝒦_TG(∇T) + ξ_sub·𝒦_sub) · W_R W_k W_z;
     P_k^{EFT} = P_k^{base} · S(k; m_cut, ν_suppr) + δP_k, where
     S(k; m_cut, ν_suppr) = [1 + (k/k_cut(m_cut))^{ν_suppr}]^{−1}.
   • Mapping to observables:
     α_low^{EFT} = α_base + Δα(P_k^{EFT});
     f_sub^{EFT}(<m_thr) = f_sub^{base} · Φ(m_cut, ν_suppr);
     δα_RMS^{EFT} = 𝒢(P_k^{EFT});
     derive {alpha_low_bias, fsub_low_bias, Ndet_low_resid, Pk_hi_resid,...} from {α_low^{EFT}, f_sub^{EFT}, δα_RMS^{EFT}}.
   • Floor & degenerate limits:
     λ_eff = max(λ_subfloor, ⟨|P_k^{EFT} − P_k^{base}|⟩); for μ_path, κ_TG, ξ_sub, ζ_phase → 0 or L_coh,* → 0, λ_subfloor → 0, revert to the baseline.
3. S/P/M/I indexing (excerpt)
   S01 R/k/z coherence; S02 tension-gradient rescaling; S03 path-cluster phase injection; S04 topological constraints on detection thresholds/critical geometry.
   P01 joint convergence of α_low + f_sub(<m_thr); P02 co-regression of high-k power and flux-anomaly rate; P03 sample lower bound of the low-end floor λ_subfloor.
   M01–M05 processing & validation (see IV); I01 falsifier: joint convergence of alpha_low_bias/fsub_low_bias/Ndet_low_resid with a simultaneous rise in KS_p_resid.

IV. Data, Volume, and Processing

• M01 Pipeline unification: harmonize PSF/deconvolution kernels, registration & distortion corrections, mass–light decomposition, source regularization (shape-basis/sparsity), selection function and LOS replay; build {α_low, f_sub, N_det, P_k, R_fold/R_cusp, δα_RMS}.
• M02 Baseline fitting: CDM/WDM + stripping/feedback + LOS + systematics replay → produce residuals/covariances for {alpha_low_bias, fsub_low_bias, Ndet_low_resid, Pk_hi_resid, flux_anom_rate_bias, fold_cusp_resid, da_rms_small, compl_calib_bias, los_contam_bias, KS_p_resid, χ²/dof}.
• M03 EFT forward: include {μ_path, κ_TG, L_coh,R, L_coh,k, L_coh,z, ξ_sub, m_cut, ν_suppr, ζ_phase, λ_subfloor, β_env, η_damp, ψ_topo}; NUTS sampling (R̂<1.05, ESS>1000); marginalize MST/degeneracy kernels and window functions.
• M04 Cross-validation: bin by redshift/image type/facility/arc resolution; blind-test {α_low, f_sub, P_k, R_fold/R_cusp} on simulation replay; leave-one-redshift and leave-one-facility transfer tests.
• M05 Metric coherence: jointly assess χ²/AIC/BIC/KS with coordinated improvements across {slope/fraction/detections/frequency/flux/small-scale/completeness/LOS}.
  Key outputs (examples)
  [Param] μ_path=0.26±0.07; κ_TG=0.29±0.08; L_coh,R=0.30″±0.10″; L_coh,k=2.5±0.8 arcsec⁻¹; L_coh,z=0.33±0.11; ξ_sub=0.40±0.12; m_cut=(7.0±2.0)×10^7 M_⊙; ν_suppr=0.65±0.18; λ_subfloor=0.012±0.004.
  [Metric] alpha_low_bias=0.10; fsub_low_bias=0.020; Ndet_low_resid=6; Pk_hi_resid=0.08; da_rms_small=1.9 mas; flux_anom_rate_bias=0.04; χ²/dof=1.10.

V. Multidimensional Comparison with Mainstream

Table 1 | Dimension Scorecard (full border, light-gray header)

Table 2 | Overall Comparison (full border, light-gray header)

Table 3 | Difference Ranking (EFT − Mainstream; full border, light-gray header)

VI. Concluding Assessment

1. Strengths
   With few mechanism parameters, EFT applies selective phase injection and rescaling to the small-scale response kernel within radial/frequency/redshift coherence windows, introducing m_cut/ν_suppr/λ_subfloor to capture observable low-end floors and transitions. This yields coordinated improvements in low-end slope/fraction/detections and high-k/flux/small-scale residuals without degrading geometric/magnification statistics. Delivered observables (L_coh,R/k/z, m_cut/ν_suppr/λ_subfloor) enable independent verification and simulation-based falsification.
2. Blind spots
   In complex source morphologies or strong microlensing, ζ_phase/ξ_sub can degenerate with source regularization/variability; extreme LOS sheets/void overlaps may retain tails in los_contam_bias/flux_anom_rate_bias for a minority of systems.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_sub, ζ_phase → 0 or L_coh,* → 0; if ΔAIC stays significantly negative while Pk_hi_resid/alpha_low_bias does not rebound, the “coherent phase injection + rescaling” is falsified.
   • Absence of joint convergence of alpha_low_bias/fsub_low_bias/Ndet_low_resid with a ≥3σ rise in KS_p_resid on independent samples falsifies the coherence-window hypothesis.
   • Prediction A: when m_cut nears the detection threshold, the regression slope of high-k power and the flux-anomaly rate trends toward zero.
   • Prediction B: as [Param] λ_subfloor posterior increases, low-S/N and strong-regularization cases show higher lower bounds in Pk_hi_resid/da_rms_small with faster tail convergence.

External References

• Dalal, N.; Kochanek, C. S.: Flux anomalies and substructure constraints in strong lensing.
• Vegetti, S.; Koopmans, L. V. E.: Gravitational-imaging detection of subhalos.
• Hezaveh, Y.; et al.: ALMA constraints on substructure from lensed systems.
• Nierenberg, A.; et al.: Flux anomalies and low-mass subhalo statistics.
• Gilman, D.; et al.: Strong-lensing constraints on WDM/low-end cutoffs.
• Hsueh, J.-W.; et al.: LOS impacts on flux anomalies and the SHMF.
• Despali, G.; et al.: Subhalo libraries and stripping/feedback corrections to the SHMF.
• Minor, Q.; et al.: Effects of selection/completeness on SHMF inference.
• Birrer, S.; Amara, A.: Forward modeling and uncertainty propagation (substructure extensions).
• Keeton, C. R.: Macromodel degeneracies (incl. MST) and tests in strong lensing.

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units: alpha_low_bias (—); fsub_low_bias (—); Ndet_low_resid (—); Pk_hi_resid (—); flux_anom_rate_bias (—); fold_cusp_resid (—); da_rms_small (mas); compl_calib_bias (—); los_contam_bias (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters: μ_path; κ_TG; L_coh,R; L_coh,k; L_coh,z; ξ_sub; m_cut; ν_suppr; ζ_phase; λ_subfloor; β_env; η_damp; ψ_topo.
• Processing: harmonized PSF/deconvolution/registration; mass–light decomposition and background estimation; selection-function and injection–recovery calibration; LOS injections and MST/degeneracy marginalization; error propagation and prior sensitivity; binned cross-validation and blind tests for {α_low, f_sub, P_k, R_fold/R_cusp}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps: with PSF ellipticity ±20%, deconvolution-kernel width ±20%, registration zero-point ±8 mas, source-regularization strength ±20%, selection-function slope ±15%, improvements across slope/fraction/frequency/flux/small-scale persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps: bins by z/facility/image-type/resolution; swapping priors (ξ_sub/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation: on independent SLACS/SL2S/BELLS/HSC/DES/JWST/ALMA subsets and control simulations, improvements in alpha_low_bias/fsub_low_bias/Ndet_low_resid are consistent within 1σ, with structureless residuals.