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Dark Matter & Energy

Field: Cosmology

Hypothetical forms of matter and energy that are thought to account for the majority of the mass-energy content of the universe.

Sequence of Expressions

The fundamental relationship between total energy EE, rest mass mm, and the speed of light cc is given by:\nE=mc2E = m c^2 \nWhere EE is the relativistic energy (measured in Joules), mm is the invariant mass (measured in kilograms), and cc is the speed of light in vacuum.
Define the Einstein tensor GμνG_{\mu\nu} and the Stress-Energy tensor TμνT_{\mu\nu} for a spacetime manifold (M,gμν)(M, g_{\mu\nu}). The field equations are given by:\nGμν+Λgμν=8πTμνG_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu}
Assuming the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, the evolution of the scale factor a(t)a(t) is governed by the Friedmann equations:\n\n1. **First Friedmann Equation (Energy):**\n(a˙a)2=8πG3c2ρ−kc2a2+Λc23\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2} \rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3} \n\n2. **Second Friedmann Equation (Acceleration):**\na¨a=−4πG3c2(ρ+3Pc2)+Λ3\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^2} \left(\rho + \frac{3P}{c^2}\right) + \frac{\Lambda}{3}
Define the critical density ρc\rho_c and the dark energy density ρΛ\rho_{\Lambda}. The dark energy density parameter ΩΛ\Omega_{\Lambda} is defined as the ratio:\nΩΛ=ρΛρc\Omega_{\Lambda} = \frac{\rho_{\Lambda}}{\rho_c} \nFurthermore, the evolution of ΩΛ\Omega_{\Lambda} with the scale factor a(t)a(t) is constrained by the equation of state parameter ww: ρΛ∝a−3(1+w)\rho_{\Lambda} \propto a^{-3(1+w)}. For a cosmological constant, w=−1w=-1, implying ΩΛ=constant\Omega_{\Lambda} = constant.
Define the Hubble parameter H(t)H(t) as the proportionality constant relating the recession velocity vv of a galaxy to its distance dd via Hubble's Law:\nv(t)=H(t)d(t)v(t) = H(t) d(t) \nWhere v(t)=dotaad(t)v(t) = \frac{\text{dot{a}}}{\text{a}} d(t) is the recession velocity, and a(t)a(t) is the scale factor of the universe.
Consider the Friedmann equation governing the evolution of the scale factor a(t)a(t) in a homogeneous and isotropic universe:\n(a˙a)2=8πG3ρ−kc2a2\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi\text{G}}{3}\rho - \frac{\text{k}\text{c}^2}{a^2} \nWhere the total energy density ρ\rho is parameterized by the components of the Λ\LambdaCDM model:\nρ=ρma−3+ρra−4+ρΛ\rho = \rho_m a^{-3} + \rho_r a^{-4} + \rho_{\Lambda} \nAnd ρΛ\rho_{\Lambda} is the constant vacuum energy density associated with the cosmological constant Λ\Lambda.
General Relativity is formulated by varying the Einstein-Hilbert action SS with respect to the metric gμνg_{\mu\nu}. The action is:\nS=∫d4x−g[c16πGR+Lmatter]S = \int d^4x \sqrt{-g} \left[ \frac{c}{16\pi G} R + \mathcal{L}_{matter} \right]\nWhere RR is the Ricci scalar, and Lmatter\mathcal{L}_{matter} is the Lagrangian density of matter and fields. Varying SS yields the field equations:\nc16πG(Rμν−12Rgμν)+∂Lmatter∂gμν=0\frac{c}{16\pi G} (R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}) + \frac{\partial \mathcal{L}_{matter}}{\partial g_{\mu\nu}} = 0
The Cosmological Principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large scales. Mathematically, this implies that the spacetime metric gμνg_{\mu\nu} must take the form of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric:\nds2=−c2dt2+a(t)2[dr21−kr2+r2(dθ2+sin⁡2θdϕ2)]ds^2 = -c^2 dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - k r^2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right]\nWhere a(t)a(t) is the scale factor, and kk is the spatial curvature constant (k∈{−1,0,1}k\in\{-1, 0, 1\}). The assumption dictates that the physical laws and statistical properties are independent of the spatial coordinates (r,θ,ϕ)(r, \theta, \phi).
Define the equation of state parameter ww for dark energy as the ratio of its pressure pp to its energy density ρ\rho: \nw=pρw = \frac{p}{\rho} \nThis parameter governs the evolution of the energy density ρ\rho via the continuity equation:\nρ˙+3a˙a(+ρ+p)=0\dot{\rho} + 3\frac{\dot{a}}{a}(+\rho + p) = 0 \nFor a constant ww, the density evolves as ρ(a)=ρ0a−3(1+w)\rho(a) = \rho_0 a^{-3(1+w)}.