Mysteries of the Cosmic Void Unveiled

Mysteries of the Cosmic Void Unveiled

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  • Prompt: Depict in 3D an arbitrary vector field $V_a(\bfx)$ can then be expanded as \begin{equation} V_a(\bfx) = \int \frac{d^3k}{(2\pi)^3} \left[ \tilde V^L(\bfk) \Psi^{L,\bfk}_a(\bfx) + \tilde V^1(\bfk) \Psi^{1,\bfk}_a(\bfx) + \tilde V^2(\bfk) \Psi^{2,\bfk}_a(\bfx) \right], \label{eqn:vectorexpansion} \end{equation} in terms of Fourier expansion coefficients, \begin{eqnarray} \tilde V^L(\bfk) = \int \, d^3x\, V^a(\bfx) \left[\Psi^{L,\bfk}_a(\bfx)\right]^* = -\int\, d^3x\,\left[\Psi^{\bfk}(\bfx) \right]^* \frac{1}{k} \nabla^a V_a(\bfx), \nn \\ \tilde V^1(\bfk) = \int \, d^3x\, V^a(\bfx) \left[\Psi^{1,\bfk}_a(\bfx)\right]^* = \int\, d^3x\, \left[\Psi^{\bfk}(\bfx) \right]^* \frac{1}{| \bfk \times \hatz |} \epsilon_{abc} \hat z^a \nabla^b V^c(\bfx), \nn \\ \tilde V^2(\bfk) = \int \, d^3x\, V^a(\bfx) \left[\Psi^{2,\bfk}_a(\bfx)\right]^* = \int\, d^3x\, \left[\Psi^{\bfk}(\bfx)\right]^* \frac{-i}{k| \bfk \times \hatz |} \hat{z}^a (\nabla_a \nabla_b - g_{ab} \nabla^2) V^b(\bfx). \end{eqnarray}
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