Mathematical Equations

Mathematical FormulationThe GPF theory is defined by the action: S=∫d4x−g[12MP2F(ϕ)R−12gμν(∂μϕ)(∂νϕ)−V(ϕ)+LMatter],S = \int d^4x \sqrt{-g} \left[ \frac{1}{2} M_P^2 F(\phi) R - \frac{1}{2} g^{\mu\nu} (\partial_\mu \phi)(\partial_\nu \phi) - V(\phi) + \mathcal{L}_{\text{Matter}} \right],S = \int d^4x \sqrt{-g} \left[ \frac{1}{2} M_P^2 F(\phi) R - \frac{1}{2} g^{\mu\nu} (\partial_\mu \phi)(\partial_\nu \phi) - V(\phi) + \mathcal{L}_{\text{Matter}} \right], where MP=ℏc/8πGM_P = \sqrt{\hbar c / 8\pi G}M_P = \sqrt{\hbar c / 8\pi G} is the reduced Planck mass, ( R ) is the Ricci scalar, gμνg_{\mu\nu}g_{\mu\nu} is the metric tensor, and LMatter\mathcal{L}_{\text{Matter}}\mathcal{L}_{\text{Matter}} includes all matter fields (baryonic, electromagnetic, etc.) to ensure universal coupling. The functions are: F(ϕ)=1+ξϕ2MP2,V(ϕ)=12m2ϕ2+Λ+βϕT,F(\phi) = 1 + \xi \frac{\phi^2}{M_P^2}, \quad V(\phi) = \frac{1}{2} m^2 \phi^2 + \Lambda + \beta \phi T,F(\phi) = 1 + \xi \frac{\phi^2}{M_P^2}, \quad V(\phi) = \frac{1}{2} m^2 \phi^2 + \Lambda + \beta \phi T, with parameters ξ∼10−6\xi \sim 10^{-6}\xi \sim 10^{-6} , m∼10−22 eVm \sim 10^{-22} \, \text{eV}m \sim 10^{-22} \, \text{eV} (fuzzy dark matter), β∼10−5\beta \sim 10^{-5}\beta \sim 10^{-5} , Λ∼10−122MP4\Lambda \sim 10^{-122} M_P^4\Lambda \sim 10^{-122} M_P^4 , and T=gμνTμνMatter≈−ρmT = g^{\mu\nu} T_{\mu\nu}^{\text{Matter}} \approx -\rho_mT = g^{\mu\nu} T_{\mu\nu}^{\text{Matter}} \approx -\rho_m .The field equations are:Modified Einstein Equations:F(ϕ)Gμν+12MP2[∇μ∇νF(ϕ)−gμν□F(ϕ)]=8πG[TμνMatter+Tμνϕ],F(\phi) G_{\mu\nu} + \frac{1}{2} M_P^2 \left[ \nabla_\mu \nabla_\nu F(\phi) - g_{\mu\nu} \Box F(\phi) \right] = 8\pi G \left[ T_{\mu\nu}^{\text{Matter}} + T_{\mu\nu}^{\phi} \right],F(\phi) G_{\mu\nu} + \frac{1}{2} M_P^2 \left[ \nabla_\mu \nabla_\nu F(\phi) - g_{\mu\nu} \Box F(\phi) \right] = 8\pi G \left[ T_{\mu\nu}^{\text{Matter}} + T_{\mu\nu}^{\phi} \right], where TμνϕT_{\mu\nu}^{\phi}T_{\mu\nu}^{\phi} is the scalar field’s stress-energy tensor. Scalar Field Equation:□ϕ+m2ϕ−βρm=12MP2(2ξϕMP2)R.