Module 0 · Week 1 · Graduate

Cosmological Context: The Baryon Asymmetry Problem

The observed universe contains galaxies, stars, planets, and people — all made of matter, essentially none of antimatter. This module quantifies that asymmetry, sets it against the thermal history of the early universe, and shows why the Standard Model alone cannot explain it.

1. Quantifying the Asymmetry

Define the baryon-to-photon ratio in the present universe as

\[\eta \equiv \frac{n_B - n_{\bar B}}{n_\gamma}.\]

Because the universe contains essentially no antibaryons today, to excellent approximation \(\eta \approx n_B / n_\gamma\). Two completely independent measurements pin down this number:

1. CMB (Planck 2018)

The acoustic peaks encode the baryon density parameter \(\Omega_b h^2 = 0.02237 \pm 0.00015\), giving

\(\eta_{\mathrm{CMB}} = (6.12 \pm 0.04) \times 10^{-10}\)

2. BBN (primordial light elements)

D/H, ⁴He, ³He abundances at T ∼ 0.1 MeV depend on \(\eta\):

\(\eta_{\mathrm{BBN}} = (5.8\text{-}6.5) \times 10^{-10}\)

The agreement between CMB and BBN — over 10 decades of energy and more than 300,000 years of cosmic evolution — is one of the great triumphs of the hot Big Bang model. It nails down \(\eta\) to a few percent.

A useful alternative: the entropy ratio

The photon number density dilutes with expansion, but so does the total entropy density \(s = (2\pi^2/45)\,g_{*s}\, T^3\). The ratio

\[Y_B \equiv \frac{n_B - n_{\bar B}}{s} \approx \frac{\eta}{7.04} \approx 8.7\times 10^{-11}\]

is conserved in a comoving volume (once baryon number is conserved). This is the natural quantity for theorists tracking baryogenesis through cosmic expansion.

2. Why the Standard Model Alone Fails

The Standard Model (SM) contains CP violation in the CKM matrix, baryon-number violation via sphalerons, and a crossover electroweak transition. In principle, it satisfies all three Sakharov conditions. But the magnitude of each is hopelessly small.

CP violation in the SM: Jarlskog invariant

The CKM contribution to any CP-violating observable is proportional to the Jarlskog invariant

\[J_{CP} = \mathrm{Im}\!\left[V_{us} V_{cb} V_{ub}^* V_{cs}^*\right] \approx 3.1 \times 10^{-5}.\]

To generate a CP asymmetry at the electroweak scale, this must be multiplied by a suppression factor from the mass differences of the quarks it couples:

\[\delta_{CP}^{SM} \sim J_{CP}\,\frac{(m_t^2 - m_c^2)(m_c^2 - m_u^2)(m_t^2 - m_u^2)(m_b^2 - m_s^2)(m_s^2 - m_d^2)(m_b^2 - m_d^2)}{T^{12}}\bigg|_{T\sim 100~\mathrm{GeV}}.\]

Numerically \(\delta_{CP}^{SM} \sim 10^{-20}\), about 10 orders of magnitude too small to explain \(\eta \sim 6\times10^{-10}\).

Crossover vs. first-order transition

With the measured Higgs mass \(m_h = 125\) GeV, lattice results show the EW transition is a smooth crossover, not a first-order phase transition. Therefore the “departure from equilibrium” required by Sakharov is absent. Sphalerons remain in thermal equilibrium throughout the crossover, washing out any asymmetry generated.

Conclusion: the SM alone produces \(\eta_{SM} \lesssim 10^{-20}\), while the observed value is \(6\times 10^{-10}\). New physics is required.

3. Thermal History of the Early Universe

For a radiation-dominated universe with effective number of relativistic degrees of freedom \(g_*\), the Friedmann equation \(H^2 = \frac{8\pi G}{3}\rho\) combined with \(\rho = \frac{\pi^2}{30}g_* T^4\) gives

\[H(T) = \sqrt{\frac{8\pi^3 G\, g_*}{90}}\, T^2 = 1.66\, \sqrt{g_*}\,\frac{T^2}{M_{Pl}}.\]

Integrating \(H = -\dot T/T\) (using \(g_{*s} T^3 a^3 = \mathrm{const}\) and radiation \(a \propto t^{1/2}\)) gives the time-temperature relation

\[t = \frac{1}{2H} \;\Longrightarrow\; T(t) \approx 1~\mathrm{MeV}\,\left(\frac{1~\mathrm{s}}{t}\right)^{1/2}.\]

Derivation of T(t)

Starting from \(a\propto t^{1/2}\) and \(T\propto a^{-1}\) (for constant \(g_{*s}\)), we get \(T\propto t^{-1/2}\). The Friedmann equation then fixes the normalization: \(H = 1/(2t)\) gives

\[t = \frac{1}{2H} = \frac{1}{2\cdot 1.66\sqrt{g_*}}\,\frac{M_{Pl}}{T^2}.\]

Plugging \(M_{Pl} = 1.22\times 10^{19}\) GeV and \(g_* \approx 10.75\) at T = 1 MeV gives the quoted normalization.

Timeline from the Planck era to recombination. Baryogenesis plausibly occurs between the GUT scale and the electroweak scale; the tiny asymmetry then survives and seeds all structure.

4. Key Epochs in the Baryogenesis Narrative

Each of the following eras is a potential stage for some aspect of baryogenesis or a key constraint on it.

Planck (T ~ 10¹⁹ GeV)Quantum gravity epoch. Currently outside controlled theory; a complete UV-complete theory is required to calculate initial conditions.

Inflation (T_reh ~ 10¹⁴-10¹⁶ GeV)Dilutes all pre-existing relics to zero. Any baryons produced before inflation are washed out; post-inflation reheating produces the thermal bath.

GUT era (T ~ 10¹⁶ GeV)Hypothetical grand unified symmetry broken. Heavy X-bosons decay and may generate baryon number. Limited by reheating temperature.

Leptogenesis (T ~ 10⁹-10¹³ GeV)Right-handed neutrinos decay out of equilibrium. Produces lepton asymmetry that sphalerons convert to baryon asymmetry.

Electroweak (T ~ 100 GeV)Sphalerons active above T_c; decouple around T ~ 130 GeV. Electroweak baryogenesis requires new physics for first-order transition.

QCD (T ~ 150 MeV)Quarks bind into hadrons. Baryon number is already fixed by this point; QCD just repackages it.

BBN (T ~ 0.1 MeV, t ~ 3 min)Light-element synthesis. Sensitive to baryon number; constrains η to ~5-10%.

CMB (T = 0.26 eV, t = 380 kyr)Acoustic oscillations encode Ω_b; independent determination of η.

Simulation: Thermal History, BBN Constraints, and the Asymmetry

The simulation below (a) plots T(t) across the radiation-dominated era with the baryogenesis windows highlighted, (b) shows the Hubble rate H(T) and the relevant interaction rates, and (c) displays the BBN light-element abundances as functions of \(\eta\), with the observed band overlaid.

Python

script.py198 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'

# --- Physical constants (natural units, GeV) ---
M_PL = 1.221e19        # Planck mass in GeV
G_STAR_SM = 106.75     # high-T SM degrees of freedom
eta_obs = 6.12e-10     # Planck 2018 central value
eta_err = 0.04e-10

fig = plt.figure(figsize=(14, 12), facecolor=BG)
fig.suptitle('Thermal History of the Early Universe & The Baryon Asymmetry',
             color='white', fontsize=16, fontweight='bold', y=0.98)

gs = gridspec.GridSpec(3, 2, figure=fig, hspace=0.45, wspace=0.32)

# ============================================================
# Plot 1: T(t) across radiation era
# ============================================================
ax1 = fig.add_subplot(gs[0, :], facecolor=BG)

# Build log-spaced T grid from 10^16 GeV down to 10^-4 GeV
T = np.logspace(-4, 16, 500)  # GeV
# g*(T) - simple step-function approximation
def g_star(T):
    g = np.where(T > 175,    106.75,        # above top
          np.where(T > 80,    96.25,
          np.where(T > 4,     86.25,
          np.where(T > 1,     75.75,
          np.where(T > 0.1,   61.75,
          np.where(T > 0.005, 17.25,
          np.where(T > 5e-4,  10.75, 3.36)))))))
    return g

gs_T = g_star(T)
H = 1.66 * np.sqrt(gs_T) * T**2 / M_PL   # GeV
t = 1.0 / (2 * H)                          # 1/GeV; convert to seconds: 1 GeV^-1 = 6.58e-25 s
t_sec = t * 6.58e-25

ax1.loglog(t_sec, T * 1e9, color=INDIGO, lw=2.5, label='T(t) in MeV')  # convert GeV to MeV for y-scale
ax1.axvspan(1e-36, 1e-24, color='#6366f1', alpha=0.12, label='Baryogenesis window')
ax1.axvspan(1e-10, 1e-10, color='red', alpha=0.0)
# Mark key events
events = [
    (1e-36, 1e22, 'Inflation end'),
    (1e-28, 1e21, 'GUT era'),
    (1e-24, 1e18, 'Leptogenesis'),
    (1e-10, 1e11, 'EW PT'),
    (1e-5, 1.5e8, 'QCD PT'),
    (1.0, 1e6, 'BBN'),
    (1e13, 2.6e-1, 'CMB'),
]
for t_val, T_val, lab in events:
    ax1.axvline(t_val, color=GOLD, lw=0.7, alpha=0.5, ls='--')
    ax1.text(t_val, T_val, ' ' + lab, color=GOLD, fontsize=8, rotation=0, va='bottom')

ax1.set_xlabel('time  t  [s]', color=PURPLE, fontsize=10)
ax1.set_ylabel('temperature  T  [MeV]', color=PURPLE, fontsize=10)
ax1.set_title('Temperature vs Time in the Radiation-Dominated Era (log-log)',
              color='white', fontsize=12)
ax1.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#1e3a5f', loc='upper right')
ax1.tick_params(colors=PURPLE, labelsize=8)
ax1.grid(True, alpha=0.15, color=PURPLE)
for sp in ax1.spines.values(): sp.set_color('#312e81')

# ============================================================
# Plot 2: Hubble rate vs temperature
# ============================================================
ax2 = fig.add_subplot(gs[1, 0], facecolor=BG)
ax2.loglog(T, H, color=INDIGO, lw=2, label='H(T)')
ax2.axvline(1e2,  color=GOLD,  lw=1, ls='--', alpha=0.7)
ax2.axvline(1e9,  color=GREEN, lw=1, ls='--', alpha=0.7)
ax2.text(1.5e2, 1e-15, 'EW',  color=GOLD, fontsize=9)
ax2.text(1.5e9, 1e-15, 'Leptogenesis', color=GREEN, fontsize=9)
ax2.set_xlabel('T  [GeV]', color=PURPLE)
ax2.set_ylabel('H  [GeV]', color=PURPLE)
ax2.set_title('Hubble rate  H = 1.66 \u221Ag\u22C6 T\u00B2 / M_Pl', color='white', fontsize=11)
ax2.tick_params(colors=PURPLE, labelsize=8)
ax2.grid(True, alpha=0.15, color=PURPLE)
for sp in ax2.spines.values(): sp.set_color('#312e81')

# Interaction rates for reference
# sphaleron: Gamma_sph ~ alpha_W^5 T^4 / T^3 = alpha_W^5 T (per particle); with log corrections
alpha_W = 1/30.0
Gamma_sph = 250 * alpha_W**5 * T  # rough numerical factor from lattice
mask = (T > 80) & (T < 1e5)
ax2.loglog(T[mask], Gamma_sph[mask], color=WARN, lw=1.5, ls=':', label='\u0393_sph (per particle)')
ax2.legend(fontsize=8, facecolor='#0f172a', labelcolor='white', edgecolor='#1e3a5f')

# ============================================================
# Plot 3: g_*(T)
# ============================================================
ax3 = fig.add_subplot(gs[1, 1], facecolor=BG)
ax3.semilogx(T, gs_T, color=PURPLE, lw=2)
ax3.fill_between(T, 0, gs_T, color=INDIGO, alpha=0.15)
ax3.set_xlabel('T  [GeV]', color=PURPLE)
ax3.set_ylabel('g\u2099 effective d.o.f.', color=PURPLE)
ax3.set_title('Effective Relativistic Degrees of Freedom g\u22C6(T)', color='white', fontsize=11)
ax3.tick_params(colors=PURPLE, labelsize=8)
ax3.grid(True, alpha=0.15, color=PURPLE)
for sp in ax3.spines.values(): sp.set_color('#312e81')

# ============================================================
# Plot 4: BBN light-element abundances vs eta
# ============================================================
ax4 = fig.add_subplot(gs[2, :], facecolor=BG)

eta_arr = np.logspace(-10.5, -9.0, 200)
eta10 = eta_arr * 1e10

# Approximate BBN scaling relations (Cyburt, Fields, Olive 2016)
# D/H vs eta
DH     = 2.45e-5 * (6.0 / eta10)**1.6
# He4 mass fraction Yp (weak dependence)
Yp     = 0.2485 + 0.013 * np.log(eta10/6.0)
# 3He/H - mild decrease with eta
He3H   = 1.0e-5 * (6.0/eta10)**0.58
# 7Li/H - steep increase
Li7H   = 5.2e-10 * (eta10/6.0)**2.1

ax4b = ax4.twinx()
ax4.loglog (eta10, DH,   color=INDIGO,  lw=2, label='D/H')
ax4.loglog (eta10, He3H, color=PURPLE,  lw=2, label='\u00B3He/H')
ax4.loglog (eta10, Li7H, color=WARN,    lw=2, label='\u2077Li/H')
ax4b.plot  (eta10, Yp,   color=GREEN,   lw=2, label='Y_p (He-4)')

# Observational bands
ax4.fill_between(eta10, 2.4e-5, 2.6e-5, color=INDIGO, alpha=0.25)
ax4.fill_between(eta10, 1.5e-10, 2.5e-10, color=WARN, alpha=0.25)
ax4b.fill_between(eta10, 0.245, 0.253, color=GREEN, alpha=0.20)

# CMB eta band
ax4.axvspan(6.08, 6.16, color=GOLD, alpha=0.20, label='Planck CMB \u03B7')

ax4.set_xlabel('\u03B7\u2081\u2080 = 10\u00B9\u2070 \u00B7 (n_B - n_{B-bar})/n_\u03B3', color=PURPLE, fontsize=10)
ax4.set_ylabel('abundance  X/H', color=PURPLE, fontsize=10)
ax4b.set_ylabel('Y_p (He-4 mass fraction)', color=GREEN, fontsize=10)
ax4.set_xscale('log')
ax4.set_xlim(3, 15)
ax4.set_ylim(1e-11, 1e-3)
ax4b.set_ylim(0.22, 0.28)
ax4.set_title('BBN Constraint on \u03B7: primordial abundances vs baryon-to-photon ratio',
              color='white', fontsize=12)

lines1, labels1 = ax4.get_legend_handles_labels()
lines2, labels2 = ax4b.get_legend_handles_labels()
ax4.legend(lines1 + lines2, labels1 + labels2, fontsize=8,
           facecolor='#0f172a', labelcolor='white', edgecolor='#1e3a5f', loc='lower left')
ax4.tick_params(colors=PURPLE, labelsize=8)
ax4b.tick_params(colors=GREEN, labelsize=8)
ax4.grid(True, alpha=0.15, color=PURPLE)
for sp in ax4.spines.values():  sp.set_color('#312e81')
for sp in ax4b.spines.values(): sp.set_color('#312e81')

# Numerical summary
print('\n=== Thermal history summary ===')
print(f'Planck mass:       M_Pl = {M_PL:.3e} GeV')
print(f'Observed asymmetry: eta = {eta_obs:.2e} +/- {eta_err:.0e}')
Y_B = eta_obs / 7.04
print(f'Entropy-normalized: Y_B = eta/7.04 = {Y_B:.3e}')
print()
print('=== Time-temperature relation (radiation dominated) ===')
for Tref_GeV in [1e16, 1e12, 100, 1e-3, 1e-4]:
    gs_val = g_star(np.array([Tref_GeV]))[0]
    H_val = 1.66 * np.sqrt(gs_val) * Tref_GeV**2 / M_PL
    t_val_sec = 6.58e-25 / (2*H_val)
    print(f'  T = {Tref_GeV:.2e} GeV  |  g* = {gs_val:.2f}  |  H = {H_val:.3e} GeV  |  t = {t_val_sec:.3e} s')

print()
print('=== SM CP violation estimate ===')
# Jarlskog invariant numerical
J_CP = 3.1e-5
# quark-mass suppression factors (in GeV^2)
mt, mc, mu = 173, 1.27, 0.0022
mb, ms, md = 4.18, 0.093, 0.0047
T_EW = 100.0
suppression = (mt**2 - mc**2)*(mc**2 - mu**2)*(mt**2 - mu**2) * \
              (mb**2 - ms**2)*(ms**2 - md**2)*(mb**2 - md**2) / T_EW**12
delta_CP_SM = J_CP * suppression
print(f'  J_CP            = {J_CP:.2e}')
print(f'  Quark-mass supp = {suppression:.2e}')
print(f'  delta_CP_SM     = {delta_CP_SM:.2e}')
print(f'  Required        ~ 1e-10')
print(f'  Shortfall       ~ {1e-10/delta_CP_SM:.2e}  <- orders of magnitude too small')

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

Appendix A: Natural Units and Useful Conversions

Cosmological and particle physics calculations use natural units where \(\hbar = c = k_B = 1\). Every quantity becomes a power of energy (GeV). The conversions most used in baryogenesis:

Energy ↔ time, length

1 GeV \(^{-1}\) = 6.58 \u00D7 10 \(^{-25}\) s
1 GeV \(^{-1}\) = 1.97 \u00D7 10 \(^{-14}\) cm
1 eV = 1.16 \u00D7 10 \(^{4}\) K

Key mass scales

\(M_{Pl} = (\hbar c/G)^{1/2}\) = 1.22 \u00D7 10 \(^{19}\) GeV
Reduced Planck mass \(\bar M_{Pl} = M_{Pl}/\sqrt{8\pi}\) = 2.4 \u00D7 10 \(^{18}\) GeV
Electroweak vev v = 246 GeV
QCD scale \(\Lambda_{QCD}\) \u2248 200 MeV

The entropy density and photon number density at temperature T are

\[s(T) = \frac{2\pi^2}{45} g_{*s}(T)\,T^3, \qquad n_\gamma(T) = \frac{2\zeta(3)}{\pi^2}\,T^3 \approx 0.24\,T^3.\]

Today, with T_\u03B3 = 2.725 K = 2.35 \u00D7 10 \(^{-4}\) eV, these give \(n_{\gamma,0} \approx 411~\mathrm{cm}^{-3}\) and \(s_0 \approx 2890~\mathrm{cm}^{-3}\).

Appendix A2: Entropy of the Universe Today

Putting together the pieces, the present-day entropy density in photons and neutrinos is

\[s_0 = \frac{2\pi^2}{45} g_{*s,0} T_\gamma^3, \quad g_{*s,0} = 2 + \frac{7}{8}\cdot 2\cdot 3\cdot \left(\frac{T_\nu}{T_\gamma}\right)^3 = 3.94.\]

With \(T_\gamma = 2.725\) K: \(s_0 \approx 2890\) cm \(^{-3}\) and \(n_{\gamma,0} \approx 411\) cm \(^{-3}\). The ratio \(s_0/n_{\gamma,0} \approx 7.04\) converts the baryon-to-photon ratio \(\eta\) to the entropy-normalized yield \(Y_B = \eta/7.04 \approx 8.7\times 10^{-11}\). This is the fundamental number any baryogenesis mechanism must reproduce.

Appendix B: From Ω_b to η

The CMB measures the physical baryon density \(\omega_b \equiv \Omega_b h^2\), which is related to the baryon-to-photon ratio via

\[\eta = \frac{n_B}{n_\gamma}\bigg|_{today} = \frac{\rho_b / m_p}{n_\gamma} = \frac{\omega_b\,\rho_{crit,0}/h^2}{m_p\, n_{\gamma,0}}.\]

Plugging in \(\rho_{crit,0}/h^2 = 1.88\times 10^{-29}~\mathrm{g/cm}^3\), \(m_p = 1.67\times10^{-24}\) g, and \(n_{\gamma,0} = 411~\mathrm{cm}^{-3}\):

\[\eta \approx 2.74\times 10^{-8}\,\omega_b \approx 6.12\times 10^{-10}\quad \text{for}\quad \omega_b = 0.02237.\]

CMB acoustic oscillations measure \(\omega_b\) via the ratio of odd to even peak heights (higher \(\omega_b\) enhances compression, suppresses rarefaction), giving the current precision of 0.7% on \(\omega_b\) and hence on \(\eta\).

References

Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters”, A&A 641, A6 (2020).
Cyburt, R. H., Fields, B. D., Olive, K. A. & Yeh, T.-H., “Big Bang Nucleosynthesis: 2015”, Rev. Mod. Phys. 88, 015004 (2016).
Kolb, E. W. & Turner, M. S., The Early Universe (Addison-Wesley, 1990), Chapters 1–3.
Mukhanov, V., Physical Foundations of Cosmology (Cambridge, 2005), Chapter 3 (thermal history).
Huet, P. & Sather, E., “Electroweak baryogenesis and standard model CP violation”, Phys. Rev. D 51, 379 (1995) — the \(10^{-20}\) SM estimate.
Gavela, M. B. et al., “Standard model CP violation and baryon asymmetry”, Mod. Phys. Lett. A 9, 795 (1994).

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