Physical Foundations of Cosmology

300 Gravitational instability in General Relativity In a nonexpanding universeH= 0 ,and the first equation exactly coincides wit ...

7.3 Hydrodynamical perturbations 301 Hubble radius. The behavior of the energy density perturbations, however, depends crucially ...

302 Gravitational instability in General Relativity √Perturbations on scales smaller than the Jeans lengthλJ∼cst,for which wkη ...

7.3 Hydrodynamical perturbations 303 For a plane wave perturbation,u∝exp(ikx),the solutions of (7.65) can easily be found in two ...

304 Gravitational instability in General Relativity η^2 δε ε 1 9 20 ∼ ⏐Φ⏐ ηeq ∼k −^1 η 9 10 1 2 Fig. 7.2. atη∼k−^1 .It follows f ...

7.3 Hydrodynamical perturbations 305 and it immediately follows from ( 7.73) that (^) f = ( 1 +wf 1 +wi )( 5 + 3 wi 5 + 3 wf ) ( ...

306 Gravitational instability in General Relativity where[X]±≡X+−X−denotes the jump of a variableXon#.The integrand in (7.78) is ...

7.3 Hydrodynamical perturbations 307 perturbations with the help of (7.51). HereδSis constant because the entropy per baryon is ...

308 Gravitational instability in General Relativity mode should describe inhomogeneities in the baryon distribution on a nearly ...

7.3 Hydrodynamical perturbations 309 by the time of equality. Forη>ηeq,the entropy perturbations evolve like the non- decayin ...

310 Gravitational instability in General Relativity universe with perfect fluidδT i j(T)=^0 ,(7.43) simplifies to ( h′′ij+ 2 Hh′ ...

7.4 Baryon–radiation plasma and cold dark matter 311 the anisotropy of the cosmic microwave background and to determine the tran ...

312 Gravitational instability in General Relativity is the projection operator. We will analyze small perturbations using the co ...

7.4 Baryon–radiation plasma and cold dark matter 313 satisfies the equation (δd− 3 )′+aui,i= 0. (7.107) Solving forui,iin terms ...

314 Gravitational instability in General Relativity whereis the Laplacian operator andcsis the speed of sound in the baryon– ra ...

7.4 Baryon–radiation plasma and cold dark matter 315 the amplitude of the gravitational potential and the radiation fluctuations ...

316 Gravitational instability in General Relativity perturbations.For example, the dark matter might initially be distributed in ...

7.4 Baryon–radiation plasma and cold dark matter 317 To find the general solution of the homogeneous equation (7.122) we employ ...

318 Gravitational instability in General Relativity is of order the photon diffusion scale at a given cosmological timet. To est ...

7.4 Baryon–radiation plasma and cold dark matter 319 Transfer functionsFor long-wavelength perturbations withkηr 1 ,the ampli- ...