Physical Foundations of Cosmology

320 Gravitational instability in General Relativity noting that, at earlier times when the wavelength of the mode exceeds the cu ...

7.4 Baryon–radiation plasma and cold dark matter 321 whereη∗=ηeq/( √ 2 −1).Neglecting the decaying mode and using the relation b ...

8 Inflation II: origin of the primordial inhomogeneities One of the central issues of contemporary cosmology is the explanation ...

8.1 Characterizing perturbations 323 8.1 Characterizing perturbations At a given moment in time small inhomogeneities can be cha ...

324 Inflation II: origin of the primordial inhomogeneities In the continuous limit, asV→∞, the sum in (8.1) is replaced by the i ...

8.2 Perturbations on inflation (slow-roll approximation) 325 Thus, in the case of the Gaussian random process we need to know on ...

326 Inflation II: origin of the primordial inhomogeneities (compare with (5.24)). To linear order in metric perturbations andδφi ...

8.2 Perturbations on inflation (slow-roll approximation) 327 λph∼a/k, grows. For the modes we will be interested in, the physica ...

328 Inflation II: origin of the primordial inhomogeneities V∼L^3. Assuming that the field is nearly homogeneous within this volu ...

8.2 Perturbations on inflation (slow-roll approximation) 329 wavenumberk, the typical amplitude of fluctuations is of order δφ(k ...

330 Inflation II: origin of the primordial inhomogeneities Since 3H^2  8 πVduring inflation, we obtain the equation d(yV) dt = ...

8.2 Perturbations on inflation (slow-roll approximation) 331 scales is δ ( k,tf ) ∼Akk^3 /^2 ∼ ( H V V,φ ) k∼Ha ∼ ( V^3 /^2 V,φ ...

332 Inflation II: origin of the primordial inhomogeneities Problem 8.4Show that the scalar field perturbations in the synchronou ...

8.2 Perturbations on inflation (slow-roll approximation) 333 corresponding to gravitational waves can be estimated on dimensiona ...

334 Inflation II: origin of the primordial inhomogeneities In a universe which undergoes a stage of accelerated expansion, the H ...

8.3 Quantum cosmological perturbations 335 and in many cases (8.43) can be rearranged to givep=p(ε), the equation of state for a ...

336 Inflation II: origin of the primordial inhomogeneities and theδT 00 component is δT 00 =δε=ε,XδX+ε,φδφ=ε,X ( δX−X′ 0 δφ φ 0 ...

8.3 Quantum cosmological perturbations 337 Finally, in terms of the new variables u≡  4 π(ε+p)^1 /^2 ,v≡ √ ε,Xa ( δφ+ φ′ 0 H  ...

338 Inflation II: origin of the primordial inhomogeneities Taking into account that ε+p= 2 Xp,X= 1 a^2 φ 0 ′^2 p,X (8.64) and su ...

8.3 Quantum cosmological perturbations 339 stage is over we must use (8.67) and (8.68) directly. Inflation is usually followed b ...