"""v86 -- N_star from scalaron reheating: the external band sharpened to a point [P]. Until now N_star (e-folds at horizon exit) was an EXTERNAL reheating input, demoted explicitly in origin_theory.tex and registered only as a BAND N_star in [50, 60] in the frozen registry (predictions_frozen.json). But TFPT fixes the scalaron mass from the axioms, M_scal = c3^{7/2} Mbar = 3.06e13 GeV [I] (v28/v36/v80), and once the scalaron mass is fixed, STANDARD physics (no new TFPT input) fixes the reheating history of R^2 inflation: 1. scalaron decay rate into minimally coupled scalars Gamma = N_eff * M^3/(48 pi Mbar^2), N_eff = 4 (SM Higgs doublet) (Starobinsky 1980; Vilenkin 1985; Gorbunov & Panin 2010); 2. reheating at H = Gamma: rho_reh = 3 Gamma^2 Mbar^2, T_reh = (30 rho_reh / (pi^2 g_star))^{1/4}, g_star = 106.75; 3. the standard pivot-matching equation (w = 0 during scalaron oscillations, entropy conservation a^3 s = const afterwards): ln k_star = -N_star + (1/3) ln(rho_reh/rho_end) + ln(T0/T_reh) + (1/3) ln(g_{s,0}/g_star) + ln H_star . Solving self-consistently on the exact Starobinsky potential V = (3/4) M^2 Mbar^2 (1 - e^{-sqrt(2/3) phi/Mbar})^2: T_reh = 9.6e9 GeV, N_star(k=0.05/Mpc) = 51.44, n_s = 1 - 2/N_star = 0.9611, r = 12/N_star^2 = 0.00454 . Cross-check: at the older pivot k = 0.002/Mpc the chain gives N_star = 54.7, reproducing the known literature value (~54) for R^2-inflation reheating. Robustness (the point of the exercise): N_star depends only LOGARITHMICALLY on the decay-channel ambiguity -- N_eff from 1 to 100 moves N_star by < 0.8 e-folds, and even instantaneous reheating only allows N_star <= 55.7. So the former free band collapses to a point with an O(0.5) theoretical width. HONEST TYPING AND TENSION (do not oversell): * the whole chain is [P] -- standard imported physics (decay channel, g_star, w=0 oscillation phase), conditional on M_Starobinsky = M_scal; the only compiler input is M_scal; * the frozen registry is NOT touched: N_star = 51.4 lies inside the frozen band [50, 60], so the freeze discipline holds (v84 still passes); * consequence n_s = 0.9611 sits ~1 sigma BELOW Planck 2018 (0.9649 +- 0.0042) and INCREASES the known DESI-combined tension (~0.9743). This is recorded as an honest, falsifiable sharpening: if precision data settle at n_s >= 0.967 the scalaron-reheating chain (not just N_star) is wrong; * A_s COHERENCE (the sharper consequence): with M_scal fixed, A_s is also predicted, A_s(51.44) = 1.76e-9 = -11.4 sigma vs Planck 2.105(30)e-9. The A_s-preferred N_star = 56.2 exceeds even the instantaneous-reheating bound 55.6 by 0.6 e-folds (instantaneous gives 2.06e-9 = -1.5 sigma, compatible). So within standard physics the measured A_s REQUIRES near-instantaneous reheating, and the slow Higgs-channel point 51.4 is A_s-disfavoured. The [P] point is therefore CONDITIONAL on the decay channel; the frozen band [50, 60] remains the prediction surface of record. Nothing here is promoted. """ import mpmath as mp from tfpt_constants import check, summary, reset, c3, Mbar, PI G_STAR = mp.mpf('106.75') # SM relativistic dof at T_reh ~ 1e10 GeV G_S0 = mp.mpf('3.91') # entropy dof today T0_GEV = mp.mpf('2.3486e-13') # 2.7255 K GEV_PER_INV_MPC = mp.mpf('1.9733e-14') / mp.mpf('3.0857e24') N_EFF_HIGGS = 4 # real scalar dof of the SM Higgs doublet def starobinsky(): """Exact Starobinsky potential data in units Mbar = 1.""" m2 = (c3**(mp.mpf(7) / 2))**2 # (M/Mbar)^2 = c3^7 b = mp.sqrt(mp.mpf(2) / 3) V = lambda f: mp.mpf(3) / 4 * m2 * (1 - mp.e**(-b * f))**2 Vp = lambda f: mp.mpf(3) / 4 * m2 * 2 * (1 - mp.e**(-b * f)) * b * mp.e**(-b * f) eps = lambda f: mp.mpf(1) / 2 * (Vp(f) / V(f))**2 f_end = mp.findroot(lambda f: eps(f) - 1, mp.mpf('0.6')) return V, Vp, f_end def nstar_chain(n_eff, kstar_inv_mpc, rho_reh_override=None): """Solve the pivot-matching equation self-consistently; returns (N_star, T_reh, Gamma).""" V, Vp, f_end = starobinsky() M = c3**(mp.mpf(7) / 2) * Mbar gamma = n_eff * M**3 / (48 * PI * Mbar**2) rho_end = mp.mpf(4) / 3 * V(f_end) * Mbar**4 rho_reh = rho_reh_override if rho_reh_override is not None \ else 3 * gamma**2 * Mbar**2 t_reh = (30 * rho_reh / (PI**2 * G_STAR))**mp.mpf('0.25') kstar = mp.mpf(kstar_inv_mpc) * GEV_PER_INV_MPC def n_of_phi(f): return mp.quad(lambda x: V(x) / Vp(x), [f_end, f]) def phi_of_n(n): return mp.findroot(lambda f: n_of_phi(f) - n, mp.mpf('5.0')) def mismatch(ns): v_st = V(phi_of_n(ns)) * Mbar**4 h_st = mp.sqrt(v_st / 3) / Mbar return (-ns + mp.log(rho_reh / rho_end) / 3 + mp.log(T0_GEV / t_reh) + mp.log(G_S0 / G_STAR) / 3 + mp.log(h_st)) - mp.log(kstar) return mp.findroot(mismatch, mp.mpf('54')), t_reh, gamma def run(): reset() mp.mp.dps = 30 print("v86 N_star from scalaron reheating (band -> point, [P])") M = c3**(mp.mpf(7) / 2) * Mbar check("compiler input: M_scal = c3^{7/2} Mbar = 3.0600e13 GeV [I]", M, mp.mpf('3.0599695e13'), tol=mp.mpf('1e-7')) V, Vp, f_end = starobinsky() check("end of slow roll: V_end^{1/4} = 5.881e15 GeV", (V(f_end))**mp.mpf('0.25') * Mbar, mp.mpf('5.88089e15'), tol=mp.mpf('1e-5')) # baseline: SM Higgs doublet channel ns, t_reh, gamma = nstar_chain(N_EFF_HIGGS, '0.05') check("scalaron width Gamma = 4 M^3/(48 pi Mbar^2) = 128.1 GeV [P]", gamma, mp.mpf('128.14'), tol=mp.mpf('1e-3')) check("reheating temperature T_reh = 9.55e9 GeV [P]", t_reh, mp.mpf('9.5505e9'), tol=mp.mpf('1e-3')) check("N_star(k=0.05/Mpc) = 51.44 (the former free band, now a point) [P]", ns, mp.mpf('51.4406'), tol=mp.mpf('1e-4')) check("consequence n_s = 1 - 2/N_star = 0.9611 [P]", 1 - 2 / ns, mp.mpf('0.961120'), tol=mp.mpf('1e-5')) check("consequence r = 12/N_star^2 = 0.00454 [P] (CMB-S4/LiteBIRD target)", 12 / ns**2, mp.mpf('0.0045352'), tol=mp.mpf('1e-4')) # literature cross-check at the older pivot ns002, _, _ = nstar_chain(N_EFF_HIGGS, '0.002') check("cross-check k=0.002/Mpc: N_star = 54.7 (reproduces the known " "R^2-reheating literature value ~54)", ns002, mp.mpf('54.6602'), tol=mp.mpf('1e-4')) # robustness: only log sensitivity to the decay-channel ambiguity ns_lo, _, _ = nstar_chain(1, '0.05') ns_hi, _, _ = nstar_chain(100, '0.05') check("channel robustness: N_eff 1 -> 100 moves N_star by < 0.8 e-folds " "(51.21 .. 51.98)", ns_hi - ns_lo < mp.mpf('0.8') and ns_lo > 51 and ns_hi < 52) # absolute upper bound: instantaneous reheating rho_end = mp.mpf(4) / 3 * V(f_end) * Mbar**4 ns_inst, _, _ = nstar_chain(N_EFF_HIGGS, '0.05', rho_reh_override=rho_end) check("instantaneous-reheating upper bound: N_star <= 55.7", ns_inst, mp.mpf('55.6125'), tol=mp.mpf('1e-4')) # freeze discipline: the sharpened point lies INSIDE the frozen band; # predictions_frozen.json is untouched (v84 remains the lock) check("freeze discipline: 50 < N_star = 51.4 < 60 (inside the frozen " "registry band; registry NOT modified)", mp.mpf(50) < ns < mp.mpf(60)) # honest tension, recorded not hidden: n_s = 0.9611 is ~1 sigma below # Planck 2018 (0.9649 +- 0.0042) and increases the DESI-combined tension planck_ns, planck_sig = mp.mpf('0.9649'), mp.mpf('0.0042') pull = (planck_ns - (1 - 2 / ns)) / planck_sig check("honest tension: n_s pull vs Planck 2018 = +0.90 sigma (recorded; " "kill: robust n_s >= 0.967 falsifies the reheating chain)", pull, mp.mpf('0.90'), tol=mp.mpf('0.02')) # A_s coherence (the sharper consequence -- recorded, not hidden): # with M_scal fixed, A_s = N_star^2 c3^7/(24 pi^2) is ALSO predicted. planck_As, planck_As_sig = mp.mpf('2.105e-9'), mp.mpf('0.030e-9') As = lambda n: n**2 * c3**7 / (24 * PI**2) check("A_s coherence: A_s(N_star=51.44) = 1.764e-9 -> -11.4 sigma vs " "Planck (the Higgs-channel chain UNDERPREDICTS A_s; recorded)", (As(ns) - planck_As) / planck_As_sig, mp.mpf('-11.38'), tol=mp.mpf('1e-2')) nA = mp.sqrt(planck_As * 24 * PI**2 / c3**7) check("A_s-preferred N_star = 56.20 exceeds even the instantaneous-" "reheating bound 55.61 by 0.59 e-folds", nA, mp.mpf('56.198'), tol=mp.mpf('1e-4')) check("instantaneous limit: A_s(55.61) = 2.061e-9 = -1.5 sigma " "(compatible) => within standard physics the A_s match REQUIRES " "near-instantaneous reheating; the slow Higgs-channel point " "N_star=51.4 is A_s-disfavoured. VERDICT: the [P] point is " "conditional on the decay channel; the frozen band [50,60] stays " "the prediction surface of record", (As(ns_inst) - planck_As) / planck_As_sig > mp.mpf('-1.6')) # the dichotomy, quantified: at fixed N_star = 51.4 the measured A_s # would need M_scal raised by +9.2% -- but M_scal = c3^{7/2} Mbar is # [I]-locked (exponent 7 fourfold forced). So the future discriminator: # either a fast preheating mechanism [P] (then N_star -> 55-56 and the # c3^7 normalisation stands), or precision cosmology pins slow reheating # and the exponent-7 reading fails. Both outcomes are decisive. M_needed = mp.sqrt(planck_As / As(ns)) check("dichotomy quantified: matching A_s at N_star=51.4 needs " "M_scal x 1.092 (+9.2%) -- impossible at locked c3^{7/2} => " "either fast preheating [P] or the exponent-7 normalisation " "fails (decisive future discriminator)", M_needed, mp.mpf('1.0924'), tol=mp.mpf('1e-3')) return summary("v86 N_star reheating") if __name__ == "__main__": raise SystemExit(1 if run() else 0)