Physical Foundations of Cosmology

280 Gravitational instability in Newtonian theory These new coordinates can be used until the trajectories of the matter element ...

6.4 Beyond linear approximation 281 Problem 6.7Prove that tr ( J ̇·J−^1 ) =(lnJ) , (6.87) whereJ(q,t)≡detJ. After substitution ...

282 Gravitational instability in Newtonian theory Problem 6.8Verify that the solution of (6.93) can be written in the following ...

6.4 Beyond linear approximation 283 factor of εm ε(R→∞) = 9 π^2 16  5. 55 , (6.101) the matter detaches from the Hubble flow an ...

284 Gravitational instability in Newtonian theory If we ignore vector perturbations thenfi=∂ψ/∂qi, whereψis the potential for th ...

6.4 Beyond linear approximation 285 The decaying mode soon becomes negligible and does not influence the evolution even in the n ...

286 Gravitational instability in Newtonian theory substituting (6.113) into (6.86), we find ε(q,t)= ε 0 [ 1 − ( (αβ+αγ+βγ)δi^2 − ...

6.4 Beyond linear approximation 287 density, which is a higher-dimensional analog of a mountain peak,α∼β∼γ. Hence the surroundin ...

288 Gravitational instability in Newtonian theory are formed. The walls connect the filaments. It is clear that in those places ...

7 Gravitational instability in General Relativity The Newtonian analysis of gravitational instability has limitations. It clearl ...

290 Gravitational instability in General Relativity x t t= const, ε =const t∼=const, ε∼=const Fig. 7.1. the hypersurfaces of con ...

7.1 Perturbations and gauge-invariant variables 291 gauge-invariant variables. The relation between the different coordinate sys ...

292 Gravitational instability in General Relativity components and there are four constraints). Thus we have ten functions altog ...

7.1 Perturbations and gauge-invariant variables 293 Problem 7.1Consider a 4-scalarq(xρ)=(^0 )q(xρ)+δq,where(^0 )qis its back- gr ...

294 Gravitational instability in General Relativity span the two-dimensional space of the physical perturbations, are ≡φ− 1 a [ ...

7.1 Perturbations and gauge-invariant variables 295 rotational motions. The correspondingcovariantcomponents of the rotational v ...

296 Gravitational instability in General Relativity class of synchronous coordinate systems. From (7.18), it follows that if the ...

7.2 Equations for cosmological perturbations 297 7.2 Equations for cosmological perturbations To derive the equations for the pe ...

298 Gravitational instability in General Relativity The components ofδT α βcan also be decomposed into scalar, vector and tensor ...

7.3 Hydrodynamical perturbations 299 Vector perturbations The equations for the vector perturbations take the forms Vi= 16 πGa^ ...