Physical Foundations of Cosmology

340 Inflation II: origin of the primordial inhomogeneities in an inflationary phase δφsFsφ ̇ 0 + 1 2 A φ ̇ 0 H ( kcs Ha ) 2 . ( ...

8.3 Quantum cosmological perturbations 341 whereOˆ ≡Oˆ()is a time-independent operator to be determined. Using the first equati ...

342 Inflation II: origin of the primordial inhomogeneities Problem 8.11Considering only the physical (nonfictitious) mode of a l ...

8.3 Quantum cosmological perturbations 343 Substituting vk=rkexp(iαk) into (8.89) we infer that the real functionsrkandαkobey th ...

344 Inflation II: origin of the primordial inhomogeneities the following expansion for the operator ˆ: ˆ(η,x)=^4 π(ε+p) 1 / 2 √ ...

8.3 Quantum cosmological perturbations 345 We can use (8.69) to simplify this expression during inflation: uk(η)− Ak 4 π(ε+p)^1 ...

346 Inflation II: origin of the primordial inhomogeneities δφ m a 0 =ai a 2 >a 1 af>a 3 >a 2 ∝ln(λph H) 3 π a 1 ∼ aiaf ...

8.3 Quantum cosmological perturbations 347 by the power law,δ 2 (k)∝knS−^1 , and thus characterize it by the spectral indexnS. ...

348 Inflation II: origin of the primordial inhomogeneities no preferred position in space? Quantum mechanical unitary evolution ...

8.4 Gravitational waves from inflation 349 Rewritten in terms of the new variable vk= √ eijeij 32 π ahk, (8.114) the action beco ...

350 Inflation II: origin of the primordial inhomogeneities π λph ph HΛ−^1 − 1 HΛ HΛ−^1 ηi η 1 η=η 1 η=ηf η=ηi HΛ−^1 ηi ηf 8 ∝λ δ ...

8.4 Gravitational waves from inflation 351 and hence the spectrum of the gravitational waves is also slightly tilted to the red. ...

352 Inflation II: origin of the primordial inhomogeneities λph δh H 0 −^1 zeq H 0 −^1 HΛzeq −1/2 2 ph ph ∝λ ∝λ ∼ −^1 ∼HΛ Fig. 8. ...

8.5 Self-reproduction of the universe 353 quantum fluctuations H−^1 ∆φq ∼H Fig. 8.7. the size of this domain grows toH−^1 exp(H ...

354 Inflation II: origin of the primordial inhomogeneities 1 t m−1/2 φ Fig. 8.8. self-reproduction scale,φrep∼m−^1 /^2 , quantum ...

8.6 Inflation as a theory with predictive power 355 The condition of flatness is not as “natural” as it might appear at first gl ...

9 Cosmic microwave background anisotropies After recombination, the primordial radiation freely streams through the universe wit ...

9.1 Basics 357 parameters that control the change of perturbation amplitudes after they enter the Hubble scale. The purpose of t ...

358 Cosmic microwave background anisotropies It isinvariantunder general coordinate transformations. To prove this, let us go to ...

9.1 Basics 359 Since phase volume is invariant under coordinate transformations,fis a spacetime scalar. In the absence of partic ...