Cosmic Genesis - From Big Bang to Present

The Singularity

t = 0

All matter, energy, space and time compressed to infinite density

$$\rho \to \infty \text{ g/cm}^3, \quad T \to \infty \text{ K}$$

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$$

Einstein Field Equations break down

$l_P = \sqrt{\frac{\hbar G}{c^3}} = 1.6 \times 10^{-35}$ m

$t_P = \sqrt{\frac{\hbar G}{c^5}} = 5.4 \times 10^{-44}$ s

$E_P = \sqrt{\frac{\hbar c^5}{G}} = 1.2 \times 10^{19}$ GeV

Planck Epoch

t = 10⁻⁴³ s

T = 10³² K

Quantum gravity dominates. All four fundamental forces unified.

$$G_{unified} = G_{strong} = G_{weak} = G_{EM} = G_{gravity}$$

$$\Lambda_{GUT} \approx 10^{16} \text{ GeV}$$

Theory of Everything required

Grand Unification

t = 10⁻³⁶ s

T = 10²⁸ K

Gravity separates. Strong, weak, and EM forces remain unified in GUT.

$$SU(5) \text{ or } SO(10) \to SU(3) \times SU(2) \times U(1)$$

$$M_X \sim 10^{16} \text{ GeV}/c^2$$

X-boson mass scale, proton decay possible

p

p̄

CP

⤫

Baryogenesis

t = 10⁻³⁶ → 10⁻¹⁰ s

T = 10²⁸ → 10¹⁶ K

CP violation and out-of-equilibrium conditions create slight excess of baryons over antibaryons.

$$\eta = \frac{n_B - n_{\bar{B}}}{n_\gamma} \approx 6.1 \times 10^{-10}$$

$$\text{Sakharov conditions:}$$ $$1. \text{ Baryon number violation } (\Delta B \neq 0)$$ $$2. \text{ CP symmetry violation}$$ $$3. \text{ Departure from thermal equilibrium}$$

Baryon-to-photon ratio sets matter content

Cosmic Inflation

t = 10⁻³² s → 10⁻²⁸ s

T = 10²⁷ K → 10²² K

Exponential expansion driven by inflaton field. Universe grows by factor of 10²⁶.

$$a(t) = a_0 e^{H(t-t_0)}$$

$$H^2 = \frac{8\pi G}{3}\rho_\phi$$ $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi) \text{ GeV}^4$$

Hubble parameter H ≈ 10¹⁴ GeV during inflation

W±

Z⁰

γ

Electroweak Breaking

t = 10⁻¹² s

T = 10¹⁵ K (100 GeV)

Higgs mechanism gives mass to W and Z bosons. EM and weak forces separate.

$$SU(2)_L \times U(1)_Y \to U(1)_{EM}$$

$$m_W = \frac{gv}{2} = 80.4 \text{ GeV}/c^2$$ $$m_Z = \frac{\sqrt{g^2+g'^2}v}{2} = 91.2 \text{ GeV}/c^2$$

W and Z boson masses from Higgs VEV v = 246 GeV

u

d

s

→

u u d

u d d

Quark-Gluon Plasma → Hadron Formation

t = 10⁻⁶ s

T = 10¹² K (150 MeV)

Universe cools below QCD scale. Quark-gluon plasma undergoes confinement transition to form hadrons.

$$\Lambda_{QCD} \approx 200 \text{ MeV}$$

$$m_p = 938.3 \text{ MeV}/c^2$$ $$m_n = 939.6 \text{ MeV}/c^2$$

QCD scale and nucleon masses

ν_e

ν_μ

ν_τ

Neutrino Decoupling

t ≈ 1 s

T = 10¹⁰ K (1 MeV)

Neutrinos cease interacting with matter, forming cosmic neutrino background (CνB).

$$\Gamma_\nu = n \langle \sigma v \rangle < H$$

$$T_{\nu,0} = \left(\frac{4}{11}\right)^{1/3} T_{\gamma,0} \approx 1.95 \text{ K}$$

Present-day neutrino temperature, n_ν ≈ 112 cm⁻³ per species

p + n → D + γ

D + D → ³He + n

³He + n → ⁴He + γ

H 75%

⁴He 25%

D 0.01%

⁷Li 10⁻⁹

Big Bang Nucleosynthesis

t = 1 s → 20 min

T = 10⁹ K → 10⁸ K

Light element synthesis. Protons and neutrons fuse to form deuterium, helium, lithium.

$$n/p = e^{-\Delta m c^2/kT} \approx 1/7$$

$$Y_p = \frac{2(n/p)}{1+(n/p)} \approx 0.25$$

Neutron-proton ratio determines helium abundance

Radiation: ρ_r ∝ a⁻⁴

Matter: ρ_m ∝ a⁻³

Matter-Radiation Equality

t = 50 kyr

T = 9,000 K

Matter density equals radiation density. Universe becomes matter-dominated.

$$\rho_m = \rho_r$$

$$z_{eq} = \frac{\Omega_m h^2}{2.5 \times 10^{-5}} - 1 + \Delta z_{eq} \approx 3,400 \pm 100$$

$$\Delta z_{eq} = \frac{0.3 \Omega_b h^2}{2.5 \times 10^{-5}}$$

Equality redshift with baryon correction term

H⁺

He²⁺

e⁻

γ

→

H

He

γ

Recombination Era

t = 380 kyr

T = 3,000 K

Electrons combine with nuclei to form neutral atoms. CMB photons decouple.

$$X_e = \frac{n_e}{n_H} \approx \sqrt{\frac{\pi}{12}}\frac{m_e c^2}{kT}e^{-E_I/kT}$$

$$z_{rec} \approx 1090$$ $$T_{CMB,0} = 2.725 \text{ K}$$

Saha equation for ionization fraction

Dark Ages

t = 380 kyr → 2 Myr

T = 3,000 K → 60 K

No stars yet. Dark matter halos grow via gravitational collapse.

$$\delta(z) = \delta_0 \frac{g(z)}{1+z}$$

$$M_{halo} \sim 10^6 M_\odot \text{ (first star formation)}$$

Linear growth of density perturbations

100-300 M☉

H → He → C → O → Si → Fe

First Stars (Pop III)

t = 200 Myr

T = 60 K

First stars ignite. Metal-free, very massive. End in supernovae, create heavy elements.

$$M_{Jeans} = \left(\frac{5kT}{G\mu m_H}\right)^{3/2}\left(\frac{3}{4\pi\rho}\right)^{1/2}$$

$$t_{MS} \approx 3 \text{ Myr} \left(\frac{M}{100M_\odot}\right)^{-2.5}$$

Jeans mass and main sequence lifetime

Epoch of Reionization

t = 400 Myr → 1 Byr

T = 30 K → 19 K

UV photons from first stars and galaxies ionize the IGM. Universe becomes transparent again to UV.

$$\dot{n}_{ion} = \epsilon f_{esc} \dot{n}_* - \alpha n_e^2 - \frac{n_{HI}}{t_{rec}}$$

$$z_{reion} \approx 7.7 \pm 0.7$$ $$\tau_e = 0.054 \pm 0.007$$

Ionization balance and optical depth to reionization

Galaxy Formation

t = 500 Myr → 1 Byr

T = 30 K

First galaxies form in dark matter halos. Gas cools and forms stars.

$$M_{halo} \sim 10^{11} - 10^{12} M_\odot$$

$$SFR \propto M_{gas}^{1.4} \text{ (Kennicutt-Schmidt law)}$$

Typical halo masses and star formation rate

$L_{Edd} = \frac{4\pi GMm_p c}{\sigma_T} = 1.3 \times 10^{38} \left(\frac{M}{M_\odot}\right)$ erg/s

Supermassive Black Holes

t = 1 Byr

T = 19 K

Supermassive black holes grow at galaxy centers. Quasars light up the universe.

$$M_{BH} \approx 10^6 - 10^{10} M_\odot$$

$$r_s = \frac{2GM}{c^2}$$ $$t_{growth} = \frac{M}{\dot{M}} = \frac{\epsilon c^2}{\lambda L_{Edd}} t_{Salpeter}$$

Schwarzschild radius and growth timescale

→

Structure Formation

t = 1 Byr → 13.8 Byr

T = 19 K → 2.725 K

Hierarchical structure formation. Small objects merge to form larger structures in ΛCDM model.

$$\delta_c = 1.686 \text{ (spherical collapse threshold)}$$

$$\sigma_8 = 0.811 \pm 0.006 \text{ (Planck 2018)}$$

Critical density contrast and power spectrum normalization

☿

♀

🌍

♂

♃

♄

⛢

♆

Solar System Formation

t = 9.2 Byr (4.6 Bya)

T = 8 K

Solar nebula collapse forms Sun and planets. Rocky planets inner, gas giants outer.

$$M_{\odot} = 1.989 \times 10^{30} \text{ kg}$$

$$r_{snow} \approx 2.7 \text{ AU}, \quad T_{snow} \approx 150 \text{ K}$$

Solar mass and snow line location

10⁻³⁵ m

Quantum Gravity

Theoretical Framework

Planck Scale Physics

Unification of quantum mechanics and general relativity at the Planck scale.

$$l_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.616 \times 10^{-35} \text{ m}$$

$$t_P = \sqrt{\frac{\hbar G}{c^5}} \approx 5.391 \times 10^{-44} \text{ s}$$

$$E_P = \sqrt{\frac{\hbar c^5}{G}} \approx 1.956 \times 10^9 \text{ J}$$

Planck length, time, and energy scales where quantum gravity dominates

String Theory

Theoretical Framework

11-Dimensional Physics

Fundamental particles as vibrating strings in 11-dimensional spacetime with compactified extra dimensions.

$$S = \frac{1}{4\pi\alpha'} \int d^2\sigma \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X_\mu$$

$$\alpha' = \frac{l_s^2}{2\pi} \approx 10^{-66} \text{ m}^2$$

$$d = 11 \text{ (M-theory dimensions)}$$

String action, string length parameter, and M-theory dimensional structure

Large Scale Structure

t = 13.8 Byr (Present)

T = 2.725 K

Cosmic web fully formed. Galaxy clusters, filaments, and voids define structure.

$$\Omega_m = 0.315 \pm 0.007$$ $$\Omega_\Lambda = 0.685 \pm 0.007$$

$$H_0 = 67.4 \pm 0.5 \text{ km s}^{-1}\text{ Mpc}^{-1}$$

Current cosmological parameters (Planck 2018)

150 Mpc

Cosmological Observations

t = 13.8 Byr (Present)

T = 2.725 K

Modern precision cosmology. CMB, BAO, SNe Ia constrain cosmological parameters.

$$n_s = 0.9649 \pm 0.0042, \quad r < 0.056$$

$$N_{eff} = 2.99 \pm 0.17, \quad \sum m_\nu < 0.12 \text{ eV}$$

Spectral index, tensor-scalar ratio, neutrino parameters

LIGO/Virgo

LISA

PTA

Gravitational Wave Background

All cosmic epochs

GWs from inflation, phase transitions, and astrophysical sources. New window on the universe.

$$h_{ij} = A_{ij} e^{ik \cdot x}$$ $$\Omega_{GW}(f) = \frac{1}{\rho_c} \frac{d\rho_{GW}}{d \ln f}$$

$$f_{LIGO} \sim 10-10^3 \text{ Hz}, \quad f_{LISA} \sim 10^{-4}-10^{-1} \text{ Hz}$$

GW strain and energy density spectrum

Λ

Dark Energy Era

t > 13.8 Byr

T → 0 K

Universe accelerates forever. Galaxies drift apart. Heat death approaches.

$$\ddot{a} = -\frac{4\pi G}{3}(\rho + 3p)a + \frac{\Lambda c^2}{3}a$$

$$w = \frac{p}{\rho c^2} = -1 \text{ (cosmological constant)}$$

Acceleration equation and dark energy equation of state

Λ

α

α_s

Multiverse Theory

t = Theoretical

T = Various

Eternal inflation creates infinite bubble universes with different physical constants.

$$\mathcal{L} = -\frac{1}{2}(\partial_\mu \phi)^2 - V(\phi) + \frac{\xi}{2} R \phi^2$$

$$P(\Lambda) = \frac{d\text{Vol}}{d\Lambda} \cdot P_{\text{obs}}(\Lambda)$$

Inflaton Lagrangian and anthropic probability distribution