New Equations for Dark Matter and Energy

Dark Matter and Dark Energy: Constants, Parameters, and Theoretical Concepts

Cosmological Constants and Parameters

• Cosmological Constant (\(\Lambda\))
• Dark Matter Density Parameter (\(\Omega_{\text{DM}}\))
• Dark Energy Density Parameter (\(\Omega_\Lambda\))
• Critical Density (\(\rho_c\)): \(\rho_c = \frac{3H_0^2}{8\pi G}\)
• Hubble Constant (\(H_0\)): Approximately \(67.4 \pm 1.4 \, \text{km/s/Mpc}\)
• Gravitational Constant (\(G\)): \(6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}\)
• Speed of Light (\(c\)): \(2.99792458 \times 10^8 \, \text{m/s}\)
• Planck Constant (\(h\)): \(6.62607015 \times 10^{-34} \, \text{Js}\)
• Boltzmann Constant (\(k_B\)): \(1.380649 \times 10^{-23} \, \text{J/K}\)
• Dark Matter Particle Mass (Hypothetical, \(m_{\text{DM}}\))
• Baryonic Matter Density Parameter (\(\Omega_b\))
• Equation of State Parameter for Dark Energy (\(w\))
• Age of the Universe (\(t_0\))
• Curvature Parameter (\(\Omega_k\))
• Reionization Optical Depth (\(\tau\))
• Neutrino Mass Sum (\(\Sigma m_\nu\))
• Dark Matter Annihilation Cross-Section (\(\langle \sigma v \rangle\))
• Cosmic Microwave Background Temperature (\(T_{\text{CMB}}\)): Approximately 2.725 K
• Matter Power Spectrum Normalization (\(A_s\))
• Dark Matter Velocity Dispersion (\(\sigma_{\text{DM}}\))
• Deceleration Parameter (q)
• Redshift (z)
• Sound Horizon at Decoupling (\(r_s\))
• Neutrino Density Parameter (\(\Omega_\nu\))

Advanced Cosmological Parameters and Concepts

• Comoving Distance (\(D_c\)): \[ D_c = c \int_0^z \frac{dz'}{H(z')} \]
• Luminosity Distance (\(D_L\)): \[ D_L = (1+z) D_c \]
• Angular Diameter Distance (\(D_A\)): \[ D_A = \frac{D_c}{1+z} \]
• Hubble Parameter (\(H(z)\)): \[ H(z) = H_0 \sqrt{\Omega_m (1+z)^3 + \Omega_r (1+z)^4 + \Omega_k (1+z)^2 + \Omega_\Lambda} \]
• Scale Factor (\(a\)): \[ a = \frac{1}{1+z} \]
• Dark Matter Halo Mass Function: \[ n(M) dM = \frac{\rho_0}{M} f(\nu) \frac{d\nu}{dM} dM \]
• Effective Number of Neutrino Species (\(N_{\text{eff}}\))
• Dark Matter Halo Concentration Parameter (\(c\)): \[ c = \frac{R_{\text{vir}}}{R_s} \]
• Dark Matter Self-Interaction Cross-Section (\(\sigma/m\))
• Virial Theorem: \[ 2 \langle T \rangle = -\langle V \rangle \]
• Hierarchical Structure Formation: \[ \delta_M = \int_0^{\infty} \frac{dk}{k} \Delta^2(k, z) W^2(k, R) \]
• Friedmann Equations: \[ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho - \frac{k}{a^2} + \frac{\Lambda}{3} \] \[ \frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left(\rho + 3p\right) + \frac{\Lambda}{3} \]

Theoretical Models and Concepts

• Cosmic Inflation
• String Theory and Dark Matter
• Dark Energy from Extra Dimensions
• Fifth Force: \[ F = G \frac{m_1 m_2}{r^2} \left(1 + \alpha e^{-r/\lambda}\right) \]
• Clustering of Dark Matter: \[ P(k) = A_s k^n T^2(k) D^2(z) \]
• TeV-Scale Dark Matter
• Mirror Matter
• Primordial Non-Gaussianity: \[ \Phi(\mathbf{k}) = \Phi_{\text{G}}(\mathbf{k}) + f_{\text{NL}} \left[ \Phi_{\text{G}}^2(\mathbf{k}) - \langle \Phi_{\text{G}}^2 \rangle \right] \]
• Dark Energy Clustering: \[ \delta \rho_{\Lambda} = \rho_{\Lambda} \delta_{\Lambda} \]
• Quantum Cosmology: \[ \hat{H} \Psi[h_{ij}, \phi] = 0 \]
• Scalar-Tensor Theories: \[ G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} + \frac{\omega(\phi)}{\phi} \left( \nabla_\mu \nabla_\nu \phi - g_{\mu\nu} \Box \phi \right) \]
• Chaplygin Gas Model: \[ p = -\frac{A}{\rho^\alpha} \]
• F(R) Gravity: \[ F'(R) R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} F(R) + \left( g_{\mu\nu} \Box - \nabla_\mu \nabla_\nu \right) F'(R) = \frac{8\pi G}{c^4} T_{\mu\nu} \]
• Braneworld Gravity
• Chameleon Mechanism: \[ \Box \phi = \frac{\partial V_{\text{eff}}(\phi)}{\partial \phi} \]
• Fuzzy Dark Matter: \[ \rho_{\text{FDM}} \propto \sin^2(m \phi t + \delta) \]
• Mass-Varying Neutrinos (MaVaNs): \[ m_{\nu} = m_0 \left( \frac{\phi}{\phi_0} \right)^n \]
• Cosmological Constant Problem: \[ \rho_{\Lambda, \text{obs}} \approx 10^{-29} \, \text{g/cm}^3 \]
• Axion Dark Matter: \[ \phi'' + 2 \mathcal{H} \phi' + m_a^2 \phi = 0 \]
• Primordial Black Holes (PBHs)
• Modified Gravity (MOG)

Observational Techniques and Surveys

• CMB Anisotropies: \[ \Delta T(\theta, \phi) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\theta, \phi) \]
• CMB Polarization: \[ P = Q + iU \]
• Reionization Epoch
• Large-Scale Structure Surveys
• Sunyaev-Zel'dovich Effect (SZ Effect): \[ \frac{\Delta T}{T} = y \left( x \frac{e^x + 1}{e^x - 1} - 4 \right) \]
• Integrated Sachs-Wolfe Effect (ISW Effect): \[ \frac{\Delta T}{T} = \int \frac{\partial \Phi}{\partial t} d\eta \]
• 21 cm Line Observations: \[ T_b = 27 x_{\text{HI}} (1+\delta) \left( \frac{H_0}{H(z)} \right) \left( \frac{1+z}{10} \right)^{1/2} \left( 1 - \frac{T_{\text{CMB}}}{T_s} \right) \, \text{mK} \]
• Lyman-Alpha Forest: \[ F(\lambda) = e^{-\tau(\lambda)} \]
• Gravitational Wave Astronomy: \[ h(t) = \frac{4G}{c^2} \frac{\mu}{d} \left( \frac{G M}{r} \right) \cos(\omega t) \]
• Weak Gravitational Lensing Shear Power Spectrum: \[ C_\ell^{\kappa \kappa} = \int_0^{\chi_H} d\chi \, \frac{W^2(\chi)}{\chi^2} P_\delta \left(\frac{\ell}{\chi}, \chi \right) \]

High-Energy Physics and Dark Matter

• Axion Dark Matter: \[ m_a \approx 6 \, \mu\text{eV} \left( \frac{10^{12} \, \text{GeV}}{f_a} \right) \]
• Supersymmetric Dark Matter (Neutralino)
• Kaluza-Klein Dark Matter: \[ m_n = \frac{n}{R} \]
• Sterile Neutrinos: \[ \Omega_{\nu_s} h^2 = \frac{m_{\nu_s}}{94 \, \text{eV}} \]
• Thermal Relic Density: \[ \Omega_{\text{DM}} h^2 \approx \frac{1.07 \times 10^9 \, \text{GeV}^{1}}{g_*^{1/2} m_{\text{Pl}} \langle \sigma v \rangle} \]

Quantum Field Theory and Cosmology

• Vacuum Energy Density: \[ \rho_{\text{vac}} = \frac{\Lambda}{8\pi G} \]
• Renormalization Group Flow: \[ \frac{d \lambda}{d \ln \mu} = \beta(\lambda) \]
• Effective Field Theory: \[ \mathcal{L}_{\text{eff}} = \mathcal{L}_0 + \sum_i \frac{c_i}{\Lambda^i} \mathcal{O}_i \]
• Quantum Vacuum Fluctuations: \[ \langle 0 | T_{\mu\nu} | 0 \rangle = \rho_{\text{vac}} g_{\mu\nu} \]