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Big Bang Theory

Field: Cosmology

The prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution.

Sequence of Expressions

Definition

Redshift

Define the emitted wavelength λem\lambda_{em} at time temt_{em} and the observed wavelength λobs\lambda_{obs} at time tobst_{obs}. Redshift (zz) is defined by the ratio of these wavelengths: \n\n1+z=λobsλem1 + z = \frac{\lambda_{obs}}{\lambda_{em}} \n\nIn the context of cosmic expansion, the ratio of wavelengths is directly proportional to the ratio of the scale factors at the respective times:\n\n1+z=a(tobs)a(tem)1 + z = \frac{a(t_{obs})}{a(t_{em})}
The Cosmic Microwave Background Radiation (CMBR) is modeled as a perfect blackbody spectrum. The energy density ρrad\rho_{rad} of this radiation is given by:\nρrad=aradT4\rho_{rad} = a_{rad} T^4where TT is the temperature and arad=4σca_{rad} = \frac{4\sigma}{c} is the radiation constant. The temperature T(z)T(z) at redshift zz is related to the present temperature T0≈2.7 KT_0 \approx 2.7 \text{ K} by the adiabatic expansion law:\nT(z)=T0(1+z)T(z) = T_0 (1 + z)This relationship confirms the thermal nature and redshift dependence of the afterglow.
Inflation is modeled by the action S=∫d4x−g(12MPl2R−12gμν∂μϕ∂νϕ−V(ϕ))S = \int d^4 x \sqrt{-g} \left( \frac{1}{2} M_{Pl}^2 R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi) \right). Assuming a slow-roll regime, the equation of motion for the inflaton field ϕ\phi is approximated by the Klein-Gordon equation in a curved background:\nϕ¨+3Hϕ˙+V′(ϕ)=0\ddot{\phi} + 3H \dot{\phi} + V'(\phi) = 0The slow-roll conditions require 12MPl2(V′(ϕ)V(ϕ))2≪1\frac{1}{2} M_{Pl}^2 \left(\frac{V'(\phi)}{V(\phi)}\right)^2 \ll 1 and MPl2∣V′′(ϕ)V(ϕ)∣≪1M_{Pl}^2 \left|\frac{V''(\phi)}{V(\phi)}\right| \ll 1, ensuring ϕ¨≪3Hϕ˙\ddot{\phi} \ll 3H \dot{\phi}.
The density parameter Ω\Omega quantifies the ratio of the actual energy density ρ\rho to the critical density ρc\rho_c. It is defined as: \nΩ=ρρc\Omega = \frac{\rho}{\rho_c}The critical density ρc\rho_c is derived from the Friedmann equation at t=t0t=t_0 and is given by:\nρc=3H028πG\rho_c = \frac{3H_0^2}{8\pi G}The total density parameter Ωtotal\Omega_{total} determines the geometry of the universe: Ωtotal=1\Omega_{total} = 1 implies a flat (Euclidean) geometry, Ωtotal>1\Omega_{total} > 1 implies a closed (spherical) geometry, and Ωtotal<1\Omega_{total} < 1 implies an open (hyperbolic) geometry.
Let a(t)a(t) be the scale factor of the universe, ρ\rho the total energy density, PP the pressure, kk the curvature parameter (k∈{−1,0,1}k\in\{-1, 0, 1\}), and Λ\Lambda the cosmological constant. The Friedmann equations are:\n\n(a˙a)2=8πG3ρ−kc2a2+Λc23(First Friedmann Equation)\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \quad \text{(First Friedmann Equation)}\n\na¨a=−4πG3(ρ+3P)+Λc23(Second Friedmann Equation)\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3P) + \frac{\Lambda c^2}{3} \quad \text{(Second Friedmann Equation)}\n\nThese equations govern the evolution of the scale factor a(t)a(t) based on the energy-momentum tensor TμνT_{\mu\nu} via Einstein's field equations.
Consider the Friedmann-Lemaître-Robertson-Walker (FLRW) metric for a homogeneous and isotropic universe: ds2=−c2dt2+a(t)2(dr21−kr2+r2dΩ2)ds^2 = -c^2 dt^2 + a(t)^2 \big( \frac{dr^2}{1 - k r^2} + r^2 d\Omega^2 \big). The evolution of the scale factor a(t)a(t) is governed by the first Friedmann equation:\n(a˙a)2=H(t)2=8πG3c2ρ(t)−kc2a(t)2\left(\frac{\dot{a}}{a}\right)^2 = H(t)^2 = \frac{8\pi G}{3c^2} \rho(t) - \frac{kc^2}{a(t)^2}where ρ(t)\rho(t) is the total energy density and kk is the curvature constant. The Hubble parameter H0H_0 is defined as the value of H(t)H(t) at the present time t0t_0, leading to the observed relation v=H0dv = H_0 d in the linear regime.
Define the recession velocity vv of a galaxy at distance dd from the observer, and the Hubble constant H0=a˙(t0)/a(t0)H_0 = \dot{a}(t_0)/a(t_0). Hubble's Law states the linear relationship:\n\nv=H0dv = H_0 d\n\nThis relationship implies that the Hubble parameter H(t)=a˙(t)/a(t)H(t) = \dot{a}(t)/a(t) is constant at the present time t0t_0 for the observed relationship.
For a first-order phase transition involving a scalar field ϕ\phi, the dynamics are governed by the potential energy density V(ϕ)V(\phi). The transition occurs when the free energy functional F(ϕ)F(\phi) develops multiple minima. The transition rate Γ\Gamma is determined by the barrier height ΔV\Delta V and the vacuum expectation value ϕ0\phi_0. The transition proceeds via bubble nucleation, characterized by the action S3S_3: \ndΓdt=Ae−S3/ℏ\frac{d\Gamma}{dt} = A e^{-S_3/\hbar}where S3=1H4ΔV(ϕfalse)−1H2σS_3 = \frac{1}{H^4} \Delta V(\phi_{false}) - \frac{1}{H^2} \sigma, and σ\sigma is the bubble wall tension.
The Cosmological Principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large scales. Mathematically, this implies that the metric tensor gμνg_{\mu\nu} can be written in the Friedmann-Lemaître-Robertson-Walker (FLRW) form, which depends only on time tt and the scale factor a(t)a(t), but not on the spatial coordinates (x,y,z)(x, y, z): \n\nds2=−c2dt2+a2(t)[dr21−kr2+r2(dθ2+sin⁡2θdϕ2)]ds^2 = -c^2 dt^2 + a^2(t) \left[ \frac{dr^2}{1 - k r^2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right]\n\nWhere kk is the spatial curvature parameter.
Big Bang Nucleosynthesis (BBN) predicts the primordial mass fraction of light elements. Let η\eta be the baryon-to-photon ratio (η=nb/nγ\eta = n_b / n_{\gamma}). The predicted mass fraction of Helium-4 (YpY_p) and Deuterium (D/HD/H) are functions of η\eta and the number of relativistic species NeffN_{eff}. For a given η\eta, the predicted Helium-4 mass fraction is approximately:\n\nYp≈0.24+0.012ln⁡(η10−10)Y_p \approx 0.24 + 0.012 \ln\left(\frac{\eta}{10^{-10}}\right) \n\nThis relationship provides a quantitative test of the early universe conditions.