{ "schema_version": 1, "artifact_id": "qft.artifact.curved-spacetime.quantum-fields-cosmology.transplanckian-boundary-benchmark", "owner_page_id": "qft.topic.cosmological-qft.transplanckian-initial-state-signatures", "title": "Fixed-slice boundary-EFT signal and excitation cost", "generated_by": "figures-src/curved-spacetime/transplanckian-boundary-benchmark.mjs", "source_revision": 1, "generated_on": "2026-08-23", "reader_question": "How does the same boundary-EFT parameter control an oscillatory two-point signal, particle occupation, and the accumulated excitation energy?", "takeaway": "At fixed initial time the leading signal envelope grows as x=p/Lambda, while occupation grows as x^2 and the cumulative relativistic excitation density across the controlled plotted band grows as x^6-x_min^6; moving toward the cutoff strengthens both the putative signal and its consistency cost.", "scope": "Leading local, deep-subhorizon massless-scalar benchmark on one fixed initial surface. The cumulative energy includes only the controlled plotted band x_min<=q<=x. It is not an observational forecast and does not assign a detection significance.", "frozen_inputs": { "c1": 0.8, "xMin": 0.1, "xMax": 0.25, "xStep": 0.0005, "phaseSlope": 207.34511513692635, "phaseSlopeDefinition": "chi=-2*a0*Lambda*eta0", "phaseConvention": "2*k*eta0=-chi*x", "sample_count": 301, "maximum_c1_x": 0.2, "maximum_leading_beta_magnitude": 0.1, "exact_de_sitter_lambda_over_h": 103.67255756846318, "minimum_exact_de_sitter_p_over_h": 10.367255756846319, "maximum_exact_de_sitter_h_over_p": 0.09645754126781535 }, "conventions": { "momentum_parameter": "x=p/Lambda with p=k/a0 on the initial slice eta0", "boundary_shift": "delta_kappa_phys=c1*p^2/Lambda and delta_kappa_eta=a0*delta_kappa_phys", "wronskian": "v*v_conjugate_prime-v_prime*v_conjugate=i", "positive_frequency_mode": "v_BD~exp(-i*k*eta)/sqrt(2*k) in the local subhorizon limit", "physical_robin_condition": "(n dot nabla+kappa_BD+delta_kappa_phys)phi=0", "fixed_slice_phase": "chi=-2*a0*Lambda*eta0>0, so 2*k*eta0=-chi*x", "late_de_sitter_amplitude": "The growing late massless-de-Sitter mode is proportional to alpha-beta.", "subtraction": "The excitation density is local-vacuum-subtracted and time averaged so oscillatory interference terms are excluded; its reported cumulative value integrates only x_min<=q<=x within the controlled plotted band.", "units": "hbar=c=1" }, "analytic_results": { "exact_finite_robin_ratio": "beta/alpha=-(delta_kappa_eta/k)*exp(-2*i*k*eta0)/(2*i+delta_kappa_eta/k)", "exact_finite_robin_ratio_in_x": "beta/alpha=-c1*x*exp(-2*i*k*eta0)/(2*i+c1*x)", "normalized_leading_beta": "beta=+(i*c1/2)*x*exp(-2*i*k*eta0)+O(x^2,x*H/p)", "late_power_shift": "deltaP/P=-c1*x*sin(2*k*eta0)+O(x^2,x*H/p)", "occupation": "n=|beta|^2=c1^2*x^2/4 at the retained order", "excitation_density_definition": "delta_rho_band(x)=Lambda^4*integral_xmin^x dq*q^3*n(q)/(2*pi^2)", "excitation_density": "delta_rho_band(x)/Lambda^4=c1^2*(x^6-x_min^6)/(48*pi^2)", "signal_beta_relation": "deltaP/P=-2*Re(beta) at leading order for late amplitude alpha-beta" }, "approximation_and_error_model": { "retained_terms": "The plotted beta and power modulation retain the first nonzero term in the local boundary deformation; n and the band-limited delta_rho use its squared magnitude only where the same deep-subhorizon approximation is asserted.", "omitted_terms": "Amplitude corrections are O(x^2,x*H/p): curvature is a relative O(H/p) correction to the O(x) boundary-induced amplitude, so it vanishes with the boundary deformation. For the frozen exact-de-Sitter phase choice, Lambda/H=33*pi and H/p<=0.0965 throughout the plotted interval.", "exact_check_role": "The exact flat-space finite-Robin solution checks the sign, phase, normalization, and small-x expansion. It is not a claim that the dimension-five boundary operator is trustworthy nonperturbatively.", "frozen_domain": "0.10<=x<=0.25, |c1*x|<=0.20, and H/p<=0.0965 for the frozen exact-de-Sitter phase choice; the upper end is below but visibly approaches the EFT cutoff x=1.", "uncertainty": "The curves contain no statistical error bars. Their limitation is analytic truncation, not sampling uncertainty.", "observational_scope": "No transfer function, survey likelihood, instrumental response, or cosmological-parameter fit is included; no observational inference is licensed." }, "numerical_method": { "sampling": "Uniform deterministic x grid including both endpoints; every plotted sample is serialized in CSV and JSON.", "independent_energy_check": "Three-point Gauss-Legendre quadrature of q^3*|beta(q)|^2/(2*pi^2) on x_min<=q<=x, exact for the resulting degree-five polynomial, compared with the closed band-cumulative formula.", "exact_robin_check": "Solve the finite boundary ratio algebraically, normalize alpha and beta to |alpha|^2-|beta|^2=1, and compare the exact late |alpha-beta|^2-1 with the stated leading term.", "negative_control": "Repeat representative samples with c1=0 and require every signal, mixing, occupation, and energy output to vanish." }, "scientific_references": [ { "authors": "H. Collins and R. Holman", "title": "Renormalization of initial conditions and the trans-Planckian problem of inflation", "journal": "Physical Review D", "volume": "71", "article": "085009", "year": 2005, "doi": "10.1103/PhysRevD.71.085009", "url": "https://doi.org/10.1103/PhysRevD.71.085009", "use": "Boundary-effective-action organization and initial-surface renormalization context; no figure or numerical data were copied." }, { "authors": "K. Schalm, G. Shiu, and J. P. van der Schaar", "title": "Decoupling in an expanding universe: Boundary renormalization-group flow affects initial conditions for inflation", "journal": "Journal of High Energy Physics", "volume": "2004", "issue": "04", "article": "076", "year": 2004, "doi": "10.1088/1126-6708/2004/04/076", "url": "https://doi.org/10.1088/1126-6708/2004/04/076", "use": "Boundary-EFT and fixed-initial-surface context; no figure or numerical data were copied." } ], "thresholds": { "complexPhaseUnitResidual": 5e-15, "exactBoundaryResidual": 5e-15, "exactNormalizationResidual": 5e-15, "exactOccupationResidual": 5e-15, "leadingBetaIdentityResidual": 5e-15, "leadingSignalIdentityResidual": 5e-15, "energyQuadratureRelativeError": 5e-13, "bandEnergyOriginAbsoluteResidual": 1e-18, "zeroControlAbsoluteResidual": 1e-15, "exactLeadingSignalDifferenceOverY2": 1.01, "maximumLeadingBetaMagnitude": 0.101, "maximumDeSitterHOverP": 0.1 }, "diagnostics": { "maximum_phase_unit_residual": 2.220446049250313e-16, "maximum_exact_boundary_residual": 3.925231146709438e-17, "maximum_exact_normalization_residual": 4.440892098500626e-16, "maximum_exact_occupation_residual": 6.071532165918825e-18, "maximum_leading_beta_identity_residual": 3.469446951953614e-18, "maximum_leading_signal_identity_residual": 0, "maximum_energy_quadrature_relative_error": 2.0216563955657265e-15, "maximum_exact_leading_signal_difference": 0.03765687197351755, "maximum_exact_leading_signal_difference_over_y2": 0.9999975326009368, "maximum_leading_beta_magnitude": 0.1, "maximum_envelope": 0.2, "maximum_occupation": 0.010000000000000002, "maximum_cumulative_delta_rho_over_lambda4": 3.284706122205039e-7, "band_energy_origin_absolute_residual": 0, "maximum_de_sitter_h_over_p": 0.09645754126781535, "energy_monotone_non_decreasing": true }, "checks": { "phase_unitarity": true, "exact_robin_boundary_condition": true, "exact_bogoliubov_normalization": true, "exact_occupation_relation": true, "leading_beta_relation": true, 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