Why should we use the Number of e-folds as the Time Variable in the Langevin Equations?

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A Brief Cosmological History

The cosmological redshift (1.8) gives a rule for the behaviour of a black-body spectrum of radiation with temperature Tγ. Indeed, since all photons redshift as exactly the same rate λ ∝ a−1, a system which starts out as a black-body stays as a black-body, with a temperature that decreases with expansion, Tγ ∝ a−1 . (1.44).
Therefore, when one goes backwards in time during the radiation era, temperature increases as 1/a. In particular, this means that the initial singularity is also a point of infinite temperature. This leads us to the standard hot Big Bang picture of the Universe: a cosmological singularity at finite time in the past, followed by a hot, radiation dominated expansion, during which the Universe  gradually cools down as T ∝ a−1 and the radiation dilutes, followed by a period of matter dominated expansion during which galaxies, stars and planets form. Finally, the vacuum energy inevitably dominates and the Universe enters a state of exponential expansion.
This simple picture allows us to infer the presence of a few notable events that we now briefly recap, and that are summarized in Fig. 1.7 with orders of magnitude about time, energy and temperature at which these events occur. As one goes backwards in time, one can check that energy or temperature increases.
Inflation takes place at t . 10−35 s and is the object of section 1.4 and chapter 2 (in this section, times are given as elapsed since the initial singularity). This is why we start out our description afterwards, when the Universe is made of a hot plasma containing the fundamental particles of the standard model, at t ∼ 10−35 s. At t ∼ 10−11 s occurs the electroweak phase transition which breaks the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present day electromagnetic field [43, 44, 45, 46, 47, 48, 49]. This transition may be important to understanding the asymmetry between the amount of matter and antimatter in the present Universe through a process of baryogenesis [50, 51, 52, 53, 54]. It occurs at the electroweak scale which is often taken to be at the Higgs vev, around 246 GeV.

The Big Bang Model Problems and Inflation

Now that we have introduced the main aspects of the standard cosmological model, it is worth mentionning that it raises unanswered questions, known as the hot big bang model problems. The horizon problem and the flatness problem deal with the fact that, interpreted within the standard model, observations lead to the conclusion that the Universe must have been extremely flat at early times, and homogeneous even over causally disconnected physical scales. They are not definite impossibility problems, but still serious fine-tuning issues. The monopole problem deals with the density of topological defects arising from symmetry breaking in the early Universe.
Described in the standard model, these defects should be well visible at present time, which is in contradiction with experimental investigation for them. In this section, we describe in details these three problems, and we show that even if very different in nature, they can all be solved by the introduction of a phase of accelerated expansion in the early Universe.

The Horizon Problem

One of the properties of the above described cosmology is that it is endowed with a causal horizon, i.e. a frontier that separates observable events from non observable ones. Said differently, the set of events causally connected to a reference point in space-time only form a bounded set. Since no physical process can act on scales larger than the horizon, we typically expect the Universe to be strongly inhomogeneous on those scales.
However, as we shall now see, within the cosmological scenario of the standard hot big bang model, the last scattering surface (with respect to us) spans over several causally disconnected pieces of space-time, that is, the size of the cosmological horizon at time of recombination is smaller than the diameter of the last scattering surface. The last scattering surface should therefore exhibit strong deviations from homogeneity, which is not the case. In this section, we thus calculate the size of the cosmological horizon [86, 87] in a FLRW space-time, and express the angular distance of the horizon at time of recombination, as seen by a current observer on Earth. For the reasons mentioned at the end of section 1.3.1, namely the fact that curvature can never have dominated the Universe content, we consider the case of a flat universe (K = 0). We take the origin of the r coordinate to be the location of Earth, i.e. “our” comoving coordinate is r = 0.

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Inflation as a Solution to the Horizon Problem

Let us assume that the epoch dominated by radiation can be interrupted during the period tin < t < tend, or equivalently zend < z < zin. During this interval, we assume that the Universe is dominated by an unknown ideal fluid X with an equation of state parameter wX. We now wonder whether there are values of zin, zend and wX such that the horizon problem can be solved in this new scenario. Let us thus redo the horizon diameter distance calculation in this new setup.

Table of contents :

I. Cosmology and Inflation 
1. The Cosmological Standard Model 
1.1. The Homogeneous and Isotropic Universe
1.1.1. The Friedmann-Lemaˆıtre-Robertson-Walker Metric
1.1.1.1. The Hubble Law
1.1.1.2. Redshift and Comoving Coordinates
1.1.1.3. Einstein Equations
1.1.2. An Expanding Universe
1.1.2.1. Friedmann and Raychaudhuri Equations
1.1.2.2. The Newtonian Expanding Sphere
1.1.2.3. Constant Equations of State
1.2. The Present Composition of the Universe
1.3. The History of the Universe: the Hot Big Bang Model
1.3.1. Dominant Constituant
1.3.2. Age of the Universe
1.3.3. A Brief Cosmological History
1.4. The Big Bang Model Problems and Inflation
1.4.1. The Horizon Problem
1.4.1.1. Cosmological Horizon
1.4.1.2. Angular Distance to the Horizon
1.4.1.3. Formulation of the Problem
1.4.1.4. Inflation as a Solution to the Horizon Problem
1.4.1.5. Heuristic Understanding: Conformal Diagrams
1.4.2. The Flatness Problem
1.4.2.1. Formulation of the Problem
1.4.2.2. Inflation as a Solution to the Flatness Problem
1.4.3. The Monopole Problem
1.4.3.1. Formulation of the Problem
1.4.3.2. Inflation as a Solution to the Monopole Problem
1.A. FLRW Christoffel symbols, Einstein tensor and Geodesics
1.B. Numerical Values of Cosmological Parameters
2. Cosmological Inflation 
2.1. Single-Field Inflation
2.2. The Slow-Roll Approximation
2.2.1. The Slow-Roll Parameters
2.2.2. The Slow-Roll Trajectory
2.2.3. Next-to-Leading Orders in Slow Roll
2.3. Inflationary Perturbations
2.3.1. Basic Formalism
2.3.1.1. SVT Decomposition
2.3.1.2. Gauge Invariant Variables
2.3.1.3. Equation of Motion for the Perturbations
2.3.2. Quantization in the Schr¨odinger Picture
2.3.3. The Power Spectrum
2.3.4. Power Spectrum at Leading Order in Slow Roll
2.4. Stochastic Inflation
2.4.1. Heuristic Derivation of the Langevin Equations
2.4.1.1. Coarse-Grained Field
2.4.1.2. Split Klein-Gordon Equation
2.4.1.3. Stochastic Processes
2.4.1.4. Noise Moments
2.4.1.5. Window Function and Coloured Noises
2.4.1.6. Case of a de Sitter Background
2.4.1.7. Case of a Light Field
2.4.1.8. Case of a Slow-Rolling Field
2.4.2. Why should we use the Number of e-folds as the Time Variable in the Langevin Equations?
2.4.2.1. Steady-State Distributions
2.4.2.2. Perturbations Equation derived from the Background Equation .
2.4.2.3. Stochastic Inflation and QFT on Curved Space-Times
2.4.3. Stochastic Inflation and the Scalar Power Spectrum
2.4.3.1. The δN Formalism
2.4.3.2. Stochastic Inflation and Number of e-folds
2.4.3.3. Ending Point Probability
2.4.3.4. Mean Number of e-folds
2.4.3.5. Number of e-folds Dispersion
2.4.3.6. Scalar Power Spectrum
2.4.3.7. Scalar Spectral Index
2.4.3.8. A Complete Example: Large Field Inflation
2.4.3.9. Discussion
II. Results and Publications
3. Inflationary Predictions and Comparison with “Big Data” Observations 
3.1. “Horizon-Flow off-track for Inflation” (article)
3.2. “Encyclopædia Inflationaris” (article)
3.3. “The Best Inflationary Models After Planck” (article)
3.4. “Compatibility of Planck and BICEP2 in the Light of Inflation” (article)
3.5. “K-inflationry Power Spectra at Second Order” (article)
4. Quantum Aspects of Inflation and the Stochastic Formalism 
4.1. “Stochastic Effects in Hybrid Inflation” (article)
4.2. “Recursive Stochastic Effects in Valley Hybrid Inflation” (article)
4.3. “Cosmological Inflation and the Quantum Measurement Problem” (article)
Conclusion
List of figures
List of tables
Bibliography 

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