## A study of linearized density perturbations in the universe |

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4.1 in terms 4xGPos adiabatic perturbations assumption that grad baryon-dominated universe baryonic based on adiabatic canonical form characteristic curve coefficient matrix consider constant corresponding eigenvector corresponds to yi dark matter differential equation 4.3 dimensions div grad equation 3.4 equations of motion Figure B.l find a Riemann find characteristics form a fundamental fundamental set g and h galaxy formation theory given by equations gives gradients are linearly gravitational initial conditions integral equations invariants and solutions known function linearly independent Meszaros Meszaros's paper order equations ordinary differential equation original variables partial differential equation Phase plots real and distinct real valued solutions reduce system 4.5 reduce the system result rewrite Riemann invariants saddle point simplified single travelling wave solve the system spatial neighbourhood speed of sound summarizes the linearization system given system is unstable travelling wave solution ugrad)u unperturbed background velocity of sound wave equation Wi(t Wronskian xi-plane y-axis corresponds zero