Module 8 · Weeks 14–15 · Graduate

Observational Constraints on Baryogenesis Models

Baryogenesis scenarios make sharp, falsifiable predictions for cosmology and particle physics experiments. We review the current observational landscape — Planck CMB, BBN, electric dipole moments, proton decay, and neutrinoless double-beta decay — and the sensitivity projections for the next generation of experiments.

1. Planck CMB & BBN: \(\eta\) to 1% Precision

Two completely independent probes converge:

Planck CMB (T \u223C 0.26 eV, t = 380 kyr)

\(\Omega_b h^2 = 0.02237 \pm 0.00015\)

\(\eta = (6.12 \pm 0.04)\times 10^{-10}\)

From acoustic oscillations; angular scale of peaks measures \u03A9_b, ratio measures \u03B7.

BBN (T \u223C 0.1 MeV, t = 3 min)

\(D/H = (2.527 \pm 0.030)\times 10^{-5}\) (Cooke 2018)

\(Y_p = 0.2449 \pm 0.0040\) (Aver 2015)

D/H is most sensitive to \u03B7; agrees with Planck.

A residual tension: \(^7\mathrm{Li}/\mathrm{H}\) observed \((1.6\pm 0.3)\times 10^{-10}\) vs BBN prediction \(5.2\times 10^{-10}\). This is the “lithium problem”; its resolution may involve stellar depletion or new physics but doesn't affect the baryogenesis context.

2. Electric Dipole Moments: Probes of BSM CP Violation

Any permanent EDM of an elementary particle violates P and T (hence CP by the CPT theorem). The SM contribution to the electron EDM is \(|d_e^{SM}| \lesssim 10^{-38}\) e\u00B7cm (Khriplovich). Current experiments:

Neutron EDM

\(|d_n| < 1.8\times 10^{-26}\)

e\u00B7cm (nEDM, PSI 2020)

Constrains \u03B8_QCD < 10\u207B\u00B9\u2070

Electron EDM (ACME III)

\(|d_e| < 4.1\times 10^{-30}\)

e\u00B7cm (ThO molecule, 2018)

Rules out much EWBG parameter space

HfF+ (JILA 2023)

\(|d_e| < 4.1\times 10^{-30}\)

e\u00B7cm (trapped ion)

Competitive; future ACME IV projects 10\u207B\u00B3\u00B9

EWBG requires new CP phases to close the 10-orders-of-magnitude shortfall in SM CP violation. These phases generate electron and neutron EDMs at the 10 \(^{-28}\)–10 \(^{-30}\) level, making EDM experiments the most sensitive laboratory tests of electroweak baryogenesis.

3. Proton Decay: Tests of GUT Baryogenesis

GUT baryogenesis requires baryon-number-violating dimension-6 operators from X/Y boson exchange. These same operators mediate proton decay, with rates

\[\Gamma_p \sim \frac{\alpha_X^2 m_p^5}{M_X^4},\qquad \tau_p \sim 10^{35}~\mathrm{yr}\cdot\left(\frac{M_X}{10^{16}~\mathrm{GeV}}\right)^4.\]

Super-Kamiokande (current)

\(\tau(p\to e^+\pi^0) > 1.6\times 10^{34}\) yr
\(\tau(p\to \bar\nu K^+) > 5.9\times 10^{33}\) yr
Excludes minimal SU(5)

Hyper-Kamiokande (2027+)

Sensitivity \(\tau(p\to e^+\pi^0) \sim 10^{35}\) yr
Tests SUSY SU(5)
Tests SO(10) with realistic thresholds

4. Neutrinoless Double-Beta Decay: Majorana Nature

Leptogenesis requires Majorana neutrinos. The unambiguous test is neutrinoless double-beta decay ( \(0\nu\beta\beta\)): \((A, Z) \to (A, Z+2) + 2 e^-\). The half-life is

\[\frac{1}{T^{0\nu}_{1/2}} = G^{0\nu}|M^{0\nu}|^2\, \frac{|m_{\beta\beta}|^2}{m_e^2}\]

where the effective Majorana mass is

\[m_{\beta\beta} = \left|\sum_i U_{ei}^2\, m_i\right|.\]

Current bounds: KamLAND-Zen 2023 gives \(T_{1/2}^{0\nu}(^{136}\mathrm{Xe}) > 2.3\times 10^{26}\) yr, implying \(m_{\beta\beta} < 36\)– \(156\) meV. Future LEGEND-1000 and nEXO project down to \(m_{\beta\beta} \sim 10\) meV, covering the inverted hierarchy. Confirmation would establish Majorana nature and validate leptogenesis.

5. Constraints Landscape

The constraint landscape spans 25 orders of magnitude in energy, from sub-eV neutrino oscillations to proton decay at the GUT scale.

6. Future Experimental Outlook

DUNE (2028+)δ_CP in PMNS to 10°; supernova ν; proton decay

Hyper-Kamiokande (2027+)Proton decay τ ∼ 10³⁵ yr; δ_CP to 10°

LEGEND-1000 (2028+)0νββ 目標 m_ββ ≲ 20 meV

ACME IV / HfF+ upgrade|d_e| ≲ 10⁻³¹ e·cm, kills most EWBG models

n2EDM @ PSI|d_n| ≲ 10⁻²⁷ e·cm; tests θ_QCD ≲ 10⁻¹¹

LISA (2035)stochastic GW from 1st-order PT at EW scale

FCC-ee (2040+)Higgs self-coupling to 1%; heavy ν searches

FCC-hh (2050+)≥1 TeV scalar singlets; Higgs triple-coupling

Cosmic Explorer / ETPrimordial GW; phase-transition background

Simulation: Global Constraint Analysis

The following simulation combines the BBN likelihood for \(\eta\), the current and future EDM bounds in the plane of BSM CP phases, the proton-decay lifetime projection, and the 0 \(\nu\beta\beta\) Majorana mass bound as a function of the lightest neutrino mass.

Python

script.py162 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec

plt.rcParams['font.family'] = 'monospace'
BG='#0a0a1a'; INDIGO='#818cf8'; PURPLE='#c4b5fd'; WARN='#ef4444'; GOLD='#f59e0b'; GREEN='#34d399'

fig = plt.figure(figsize=(14, 12), facecolor=BG)
fig.suptitle('Observational constraints on baryogenesis',
             color='white', fontsize=16, fontweight='bold', y=0.98)
gs = gridspec.GridSpec(3, 2, figure=fig, hspace=0.45, wspace=0.35)

# ============================================================
# Plot 1: BBN likelihood for eta
# ============================================================
ax1 = fig.add_subplot(gs[0, 0], facecolor=BG)
eta_arr = np.linspace(5.0, 7.0, 400)
eta10 = eta_arr

# Approximate (for schematic): Gaussian likelihoods
L_BBN_DH    = np.exp(-0.5*((eta10 - 6.14)/0.15)**2)
L_BBN_Yp    = np.exp(-0.5*((eta10 - 5.80)/0.35)**2)
L_CMB       = np.exp(-0.5*((eta10 - 6.12)/0.04)**2)
combined    = L_BBN_DH * L_BBN_Yp * L_CMB
combined   /= np.trapezoid(combined, eta10)

ax1.plot(eta10, L_BBN_DH/L_BBN_DH.max(), color=INDIGO, lw=2, label='D/H')
ax1.plot(eta10, L_BBN_Yp/L_BBN_Yp.max(), color=GOLD,   lw=2, label='Y_p')
ax1.plot(eta10, L_CMB/L_CMB.max(),       color=GREEN,  lw=2, label='CMB')
ax1.plot(eta10, combined/combined.max(), color=WARN,   lw=2.5, label='Combined')
ax1.axvline(6.12, color='white', lw=0.8, ls=':')
ax1.set_xlabel('η × 10¹⁰', color=PURPLE)
ax1.set_ylabel('relative likelihood', color=PURPLE)
ax1.set_title('Baryon-to-photon ratio: BBN + CMB combined fit', color='white', fontsize=11)
ax1.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax1.tick_params(colors=PURPLE)
ax1.grid(True, alpha=0.15, color=PURPLE)
for sp in ax1.spines.values(): sp.set_color('#312e81')

# ============================================================
# Plot 2: EDM bounds
# ============================================================
ax2 = fig.add_subplot(gs[0, 1], facecolor=BG)
years = np.array([1990, 2000, 2010, 2015, 2018, 2023, 2028, 2033])
de_bounds = np.array([1e-26, 1.6e-27, 1.1e-28, 8.7e-29, 1.1e-29, 4.1e-30, 1e-30, 1e-31])
dn_bounds = np.array([1.2e-25, 6.3e-26, 2.9e-26, 3e-26, 3e-26, 1.8e-26, 1e-27, 1e-28])
ax2.semilogy(years, de_bounds, 'o-', color=INDIGO, lw=2, ms=6, label='|d_e| bound')
ax2.semilogy(years, dn_bounds, 's-', color=GOLD,   lw=2, ms=6, label='|d_n| bound')
ax2.axhspan(1e-29, 1e-28, color=WARN, alpha=0.15, label='EWBG target')
ax2.set_xlabel('year', color=PURPLE)
ax2.set_ylabel('EDM bound [e·cm]', color=PURPLE)
ax2.set_title('EDM experimental history and projections', color='white', fontsize=11)
ax2.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax2.tick_params(colors=PURPLE)
ax2.grid(True, alpha=0.15, color=PURPLE)
for sp in ax2.spines.values(): sp.set_color('#312e81')

# ============================================================
# Plot 3: proton decay lifetime projection vs M_X
# ============================================================
ax3 = fig.add_subplot(gs[1, 0], facecolor=BG)
MX = np.logspace(14, 17, 200)
# rough formula tau ~ (1/(alpha_X^2)) M_X^4 / m_p^5,  alpha_X ~ 1/40
alpha_X = 1/40.0
m_p = 0.938   # GeV
tau_yr = (MX**4 / (alpha_X**2 * m_p**5)) * 6.58e-25 / (3.15e7)   # GeV^-1 to yr
ax3.loglog(MX, tau_yr, color=INDIGO, lw=2.5)
ax3.axhline(1.6e34, color=WARN, lw=2, ls='--', label='Super-K 2020')
ax3.axhline(1e35,   color=GOLD, lw=2, ls='--', label='Hyper-K projection')
ax3.fill_between(MX, 1e34, 1e36, where=(tau_yr > 1.6e34), color=GREEN, alpha=0.1)
ax3.set_xlabel('M_X [GeV]', color=PURPLE)
ax3.set_ylabel('τ(p → e⁺ π⁰) [yr]', color=PURPLE)
ax3.set_title('Proton lifetime vs GUT scale', color='white', fontsize=11)
ax3.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax3.tick_params(colors=PURPLE)
ax3.grid(True, alpha=0.15, color=PURPLE)
for sp in ax3.spines.values(): sp.set_color('#312e81')

# ============================================================
# Plot 4: 0nubb: m_bb vs lightest mass (NH and IH)
# ============================================================
ax4 = fig.add_subplot(gs[1, 1], facecolor=BG)
m_min = np.logspace(-4, 0, 200)   # eV

# Normal hierarchy
dm21_sq = 7.4e-5
dm31_sq = 2.5e-3
s12 = 0.307; s13 = 0.022; s23 = 0.545  # sin^2 theta
c12 = 1-s12; c13 = 1-s13

m1 = m_min
m2 = np.sqrt(m_min**2 + dm21_sq)
m3 = np.sqrt(m_min**2 + dm31_sq)
# Vary Majorana phases over 0..pi
mbb_NH_max = np.abs(c12*c13*m1 + s12*c13*m2 + s13*m3)
mbb_NH_min = np.abs(c12*c13*m1 - s12*c13*m2 - s13*m3)

# Inverted hierarchy
m3_IH = m_min
m1_IH = np.sqrt(m_min**2 + dm31_sq - dm21_sq)
m2_IH = np.sqrt(m_min**2 + dm31_sq)
mbb_IH_max = np.abs(c12*c13*m1_IH + s12*c13*m2_IH + s13*m3_IH)
mbb_IH_min = np.abs(c12*c13*m1_IH - s12*c13*m2_IH - s13*m3_IH)

ax4.fill_between(m_min, mbb_NH_min, mbb_NH_max, color=INDIGO, alpha=0.5, label='Normal hierarchy')
ax4.fill_between(m_min, mbb_IH_min, mbb_IH_max, color=GOLD,   alpha=0.5, label='Inverted hierarchy')

ax4.axhspan(0.036, 0.156, color=WARN, alpha=0.25, label='KamLAND-Zen 2023')
ax4.axhspan(0.01, 0.036, color=GREEN, alpha=0.20, label='LEGEND-1000 target')

ax4.set_xscale('log'); ax4.set_yscale('log')
ax4.set_xlabel('lightest ν mass [eV]', color=PURPLE)
ax4.set_ylabel('|m_ββ| [eV]', color=PURPLE)
ax4.set_title('Effective Majorana mass m_ββ vs lightest mass', color='white', fontsize=11)
ax4.legend(fontsize=8, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax4.tick_params(colors=PURPLE)
ax4.grid(True, alpha=0.15, color=PURPLE)
for sp in ax4.spines.values(): sp.set_color('#312e81')

# ============================================================
# Plot 5: Summary bar chart of baryogenesis scenarios
# ============================================================
ax5 = fig.add_subplot(gs[2, :], facecolor=BG)
scenarios = ['SM (crossover)', 'GUT baryogenesis', 'EW BG (2HDM)', 'Leptogenesis', 'Resonant Lepto', 'Affleck-Dine']
viability = [0.01, 0.4, 0.6, 0.95, 0.7, 0.6]   # subjective score
testability = [0.9, 0.3, 0.95, 0.5, 0.8, 0.3]

x = np.arange(len(scenarios))
ax5.bar(x-0.2, viability,    0.4, color=INDIGO, alpha=0.85, label='theoretical viability')
ax5.bar(x+0.2, testability,  0.4, color=GOLD,   alpha=0.85, label='near-term testability')
ax5.set_xticks(x)
ax5.set_xticklabels(scenarios, color=PURPLE, fontsize=10)
ax5.set_ylim(0, 1.05)
ax5.set_ylabel('relative score (subjective)', color=PURPLE)
ax5.set_title('Baryogenesis scenarios: viability and testability snapshot', color='white', fontsize=12)
ax5.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax5.tick_params(colors=PURPLE)
ax5.grid(True, alpha=0.15, color=PURPLE)
for sp in ax5.spines.values(): sp.set_color('#312e81')

# ----- console -----
print('=== Combined eta best fit ===')
idx = np.argmax(combined)
mean = np.trapezoid(eta10*combined, eta10)
var = np.trapezoid((eta10-mean)**2 * combined, eta10)
print(f'  eta_10 best = {eta10[idx]:.3f}  mean = {mean:.3f}  sigma = {np.sqrt(var):.3f}')

print('\n=== Projected EDM reach ===')
print(f'  ACME III (2018):  |d_e| < 4.1e-30 e.cm')
print(f'  ACME IV target:   |d_e| < 1e-31 e.cm')
print(f'  EWBG target:      |d_e| ~ 1e-29 to 1e-28 e.cm  (reachable)')

print('\n=== 0nubb target ===')
print(f'  KamLAND-Zen 2023: m_bb < 36-156 meV')
print(f'  LEGEND-1000:      m_bb target < 20 meV (covers IH)')
print(f'  Inverted hierarchy band: m_bb = 15-50 meV')

plt.savefig('output.png', dpi=110, bbox_inches='tight', facecolor=BG)
print('\nSaved output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

7. Antimatter in Cosmic Rays: AMS-02, GAPS

Could there be pockets of antimatter left over in the universe, rather than a uniform matter-antimatter asymmetry? Observational constraints are tight:

No \(\gamma\)-ray emission from matter-antimatter annihilation along boundary regions: Steigman (1976); EGRET limits imply no antimatter domains within \(\sim 10^{20}\) m.
AMS-02 on ISS: \(\bar p/p \sim 10^{-4}\) consistent with secondary production from cosmic ray collisions; a few candidate antideuteron events reported (2016), but statistics marginal.
Future GAPS balloon: sensitivity to cosmic antideuterons at the \(10^{-6}\) level, a smoking-gun signature of primordial antimatter if found.
No antimatter stars seen: spectrum of an anti-H atom is indistinguishable from H, but any antimatter cloud would annihilate on the surrounding ISM.

Taken together, the universe is globally matter-dominated to a volume \(\gtrsim\)Hubble scale.

6b. Summary Table of Current Bounds

Observable	Current Bound	Near-Future Target	Relevance
η	(6.12 ± 0.04) × 10⁻¹⁰	Planck 2018 final	Overall constraint
\|d_e\|	< 4.1 × 10⁻³⁰ e·cm	< 10⁻³¹	EWBG
\|d_n\|	< 1.8 × 10⁻²⁶ e·cm	< 10⁻²⁷	EWBG, strong CP
τ(p→e⁺π⁰)	> 1.6 × 10³⁴ yr	> 10³⁵	GUT baryogenesis
m_ββ	< 36-156 meV	< 20 meV	Leptogenesis
δ_CP (PMNS)	≈1.5 rad (NuFit)	±10°	Leptogenesis
\|θ_QCD\|	< 10⁻¹⁰	< 10⁻¹¹	Strong CP, axion
Ω_GW(1 mHz)	n/a	≲1 × 10⁻¹² h²	EWBG phase transition

7b. Exotic Baryogenesis Scenarios

A handful of more exotic ideas deserve mention:

Primordial black hole Hawking evaporation

PBHs smaller than \(\sim 10^{15}\) g Hawking-evaporate by today. If they carry initial CP-violating asymmetry, their radiation can imprint a net baryon number on the universe. Constraints from gamma-ray background limit this mechanism to restricted parameter space.

Spontaneous baryogenesis

A rolling scalar with derivative coupling to the baryon current (Cohen-Kaplan 1987) can generate an effective chemical potential for baryon number even in thermal equilibrium. Requires B-violating interactions still active.

Cold electroweak baryogenesis

In models where reheating after inflation is followed by a tachyonic instability, the Higgs oscillates coherently around the broken minimum, producing strong out-of-equilibrium conditions without a first-order transition.

8. From η to Galactic Matter Density

The baryon-to-photon ratio today translates to a baryon number density:

\[n_B^0 = \eta\, n_\gamma^0 \approx (6.1\times 10^{-10})\times 411~\mathrm{cm}^{-3} = 2.5\times 10^{-7}~\mathrm{cm}^{-3}.\]

The corresponding mass density is \(\rho_B^0 = m_p\, n_B^0 \approx 4.2\times 10^{-31}\) g/cm \(^3\), matching \(\Omega_b = 0.049\) with h = 0.67. In the Milky Way disk, the baryonic number density is enhanced by a factor \(\sim 10^{6}\) through gravitational collapse, yielding \(\sim 0.2\) baryons/cm \(^3\) near the Sun.

The observed 0.1-1 per cm \(^3\) average in the galactic disk, and the corresponding stars-per-galaxy counts, all trace back to the tiny \(\eta \sim 6\times 10^{-10}\) primordial asymmetry: every structure we see is a consequence of Sakharov's conditions being satisfied some time before T = 10 \(^{-4}\) eV.

References

Planck Collaboration, A&A 641, A6 (2020).
Cooke, R. J. et al., “One percent determination of the primordial deuterium abundance”, ApJ 855, 102 (2018).
Aver, E. et al., “The effects of He I 10830 \u00C5 on helium abundance determinations”, JCAP 07, 011 (2015).
ACME Collaboration, “Improved limit on the electric dipole moment of the electron”, Nature 562, 355 (2018).
Roussy, T. S. et al. (JILA HfF+), Science 381, 46 (2023).
nEDM Collaboration @ PSI, “Measurement of the permanent electric dipole moment of the neutron”, Phys. Rev. Lett. 124, 081803 (2020).
Super-Kamiokande Collaboration, Phys. Rev. D 102, 112011 (2020).
KamLAND-Zen Collaboration, Phys. Rev. Lett. 130, 051801 (2023).
Particle Data Group, “Review of Particle Physics”, Prog. Theor. Exp. Phys. 2022, 083C01.

Where to Go Next

Having covered the observational context, the three Sakharov conditions, B violation via sphalerons, CP violation in CKM/PMNS, departure from equilibrium via Boltzmann evolution, and four baryogenesis scenarios, you are now equipped to read the research literature. Suggested next steps:

Riotto's lectures (hep-ph/9807454): deeper theoretical foundation.
Davidson, Nardi & Nir review (Phys. Rept. 2008): definitive leptogenesis reference.
Buchmuller, Di Bari, Plumacher (2005): Boltzmann-equation details.
Cline lectures (hep-ph/0609145): modern review including GW probes.
The Cosmology, QFT, and Particle Physics courses on this site.

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