Quantum fluctuations and the spectrum of primordial perturbations 

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From the primordial spectrum to observations

In this chapter we will make the link from the inflationary perturbations to two observables today: the power spectrum of the CMB and the power spectrum of the large scale structure distribution. Also most of this chapter is inspired on Refs. [66, 71, 114, 133, 154, 221].

The initial conditions from inflation

In Chapter 1 we saw that the map from an averaged background metric to the perturbed physical metric suffers from a gauge dependence. Calculations are most safely performed in gauge-invariant variables or in a specified gauge. I repeat here the perturbed metric, (1.18), ds2 = a2(τ ) −(1 + 2A)dτ 2 − Bidτ dxi + [(1 + 2ψ)δij + 2Eij ]dxidxj . (2.1)
Let us focus on scalar perturbations. The two gauge-invariant quantities describing scalar perturbations are the Bardeen potentials,[11], in Fourier-space.

The Bardeen equation

We gave the definition of the Bardeen potentials, but to derive their evolution, we need to know the energy contents of the universe. To come to the one equa-tion that describes the evolution of the gravitational potential, we first need to define all matter perturbations. Remember that in a homogeneous isotropic back- ground we have Tµ = diag(−ρ,¯ P , P , P ). The most general perturbations can be parameterised as,
T00 = −ρ¯(1 + δ), (2.8)
0 ¯ − Bj ), (2.9)
Tj = (ρ¯ + P )(vj
j ¯ j , (2.10)
T0 = −(ρ¯ + P )v
i ¯ i i (2.11)
Tj = P (1 + πL)δj + Πj .
Here δ is the over density, πL is the isotropic pressure perturbation, vj represents the energy flux which is the time like eigenvector of Tµν with eigenvalue ρ = ρ¯(1 + δ), Bj is the previously defined metric perturbation, and Πij is the traceless part of the stress tensor.

Matter power spectrum

In the previous chapter we briefly introduced the power spectrum. The reason that we are interested in power spectra, is that they provide all the statistically interesting information about distributions, of e.g. matter of photons. In a sense, the power spectrum contains the information about the common origin of different points on the sky. When considering the origin of the universe, one will always look for similarities in different directions, or, correlations. The presence or absence of the correlations tells about certain properties of the initial state that was or was not shared by different regions. Let us continue to see how the power spectrum of the matter perturbations comes about from the perturbed metric.
On scales that are much smaller than the Hubble radius, the perturbations of the metric become very small in comparison with the perturbations in the stress- energy tensor. This has the consequence that the difference between {D(.), V } and {δ, v} becomes very small. In other words, the gauge dependence of energy and velocity measures becomes negligible. This can also be seen by realising that sub-Hubble means k/a > H in Eqs. (2.13–2.19). The implication is that on sub-Hubble scales we can simply treat a(τ )2Ψ as the newtonian potential a(τ )2ψ. Then Poisson’s equation relates the potential to an over-density, ∇2ψ = 4πGN δρ, (2.26) or in momentum space
Let us focus on the time behaviour and write δρ = δρ/ρ and ρ ∝ ρ0a(τ )−3, such that we find δρ ∝ a(τ )ψ. Hence, Pρ ∝ a(τ )2PΨ ∝ a(τ )2 PR. During radiation era, the power spectrum on sub-Hubble scales decreases as Pρ ∝ a(τ )−2, and during matter era grows as Pρ ∝ a(τ ).
To make a translation to the matter power spectrum is slightly more compli-cated. During the radiation era, Pρ describes the perturbations in the dominating energy density component, the photons (and neutrinos). The evolution of the non-relativistic matter is then determined by its behaviour in following the gravi-tational potential set by the photons, who themselves are not self-gravitating, as their pressure counteracts the gravity. It turns out that the perturbations in the matter grow as δm ∝ ln a during radiation domination. During matter domination they are the dominating energy, hence δm ∝ δρ ∝ a(τ )−2.
Modes that re-enter the Hubble radius during radiation era start growing ∝ ln a. Modes that enter during matter era, will only experience growth, the smaller modes experiencing the growth earlier as they enter earlier. Along this reasoning, the matter power spectrum must have a change in slope, corresponding to the modes that re-entered the Hubble radius during the matter-to-radiation equality. This is evident in Figure 2.2.
Note that in this reasoning we ignore the fact the the radiation-to-matter transition is gradual, which does have consequences that cannot be ignored (e.g., for a period the matter perturbations will grow logarithmically). Still, this gives a rough but good understanding of the matter power spectrum.

Temperature anisotropies

With the information at hand, we can already give an estimate of the temperature anisotropies of the CMB. So far we focused on the gravitational evolution of the universe, although in principle the quantities describing the evolution of the radiation-matter mixture in the universe have been defined. Assuming that the evolution is known, and that photons decouple from other matter instantaneously,

Some ignored but known effects

So far we have only considered the Einstein equations and the rough approximation of describing everything as perfect fluids. We ignored the Boltzmann equations, thereby not paying attention to the differences between photons and baryons and dark matter particles. The difference in mean free path of photons and mean free path of massive particles, leads to diffusion damping. Around decoupling, which is not instantaneous in reality, the period during which the coupling between photons and baryons is not perfect induces damping of perturbations at a larger range of scales, this is the Silk damping.
The Universe reionises at low redshift (z ∼ 6 − 10), after which point the photons will interact again with matter, mostly leading to polarisation of the photon fluid. Besides, CMB photons cross galaxies and interact with high-energy electrons on these galaxies, the Sunyaev-Zel’dovich effect.
Besides polarisation, these effects go into too much detail to discuss in this brief overview.

The power spectrum

The CMB in approximation left from a two-dimensional sphere at a distance rdec from the observer. Its power spectrum is therefore best described in terms of spherical harmonics, the spherical counterpart of the Fourier transformation, taking into account the curvature on a sphere.

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CMB Polarisation

Although in this thesis we do not deal with the polarisation of the CMB, a future detection of certain modes of polarisation would strongly narrow down the allowed parameter space for inflation. We have seen that a detection of tensors is necessary in order to determine the scale of inflation, in its turn crucial for selecting models. If one could never measure this scale, one might argue that there is no more use in continuing the investigation of inflation, as it would become a ‘theory of everything’. Hence, even though there is no direct relation to the following chapters, an introduction to CMB polarisation is inevitable for the motivation for considering inflation on the whole.
As mentioned in Chapter 1, the scattering process that dominates the coupling between photons and massive matter around the time of decoupling, is Thomson scattering. The classical explanation of Thomson scattering is that an electron is accelerated in the oscillating electric field of an incident electromagnetic wave. The acceleration causes the electron in its turn to emit radiation, as the vibration of the electron changes its electric field in an oscillating fashion. Speaking strictly
classically, the electric amplitude of the wave observed at some vectorn from the electron, is determined by the component of the electric amplitude of the incident wave in the plane perpendicular to the vector describing the direction of the outgoing wave. If the electric field in the incoming wave is orthogonal to the plane of scattering, the outgoing wave will have the same electric field. This is illustrated in Figure 2.4.
Let us turn this classical description into a quantum mechanical picture, albeit still intuitive. Individual photons are described by an electric and a magnetic field, oscillating perpendicular to each other, both perpendicular to the direction of propagation of the photon. If the electron is non-relativistic (which is a condition for Thomson scattering, otherwise we would need the more general description of Compton scattering), the electron will not gain energy from the photon and will scatter the photon in a somewhat random direction. If we remember the classical picture of the electron vibrating along the oscillating electric field of the photon, from Figure 2.4 we can understand that most photons will be scattered in directions that preserve the direction of the electric field of the incoming photon. Or, photons scattered in the direction of the observer will originally mostly have had an electric component perpendicular to the plane of scattering.
Now we come to the point of polarisation of radiation, i.e., a correlation be-tween the individual polarisations in a group of photons. We understand that only photons with a polarisation perpendicular to the plane of scattering are scattered towards the observer. If, from the point of view of the electron, photons come from all directions with random polarisation, the observer will still see random polarisation in the scattered photons. If the electron sees a dipole in the incoming radiation, the observer will still see random polarisation, as photons from left and right of the electron (imaging the dipole to be oriented as such) average each other out when ending up in the beam towards the observer. If the electron ob-serves a quadrupole, e.g., more photons coming in from top and bottom and less from left and right, the observer will see a preferred polarisation in the photons, namely a preference for polarisation perpendicular to the plane of scattering of the photons that fell in from top and bottom on the electron. Hence, if there was a temperature quadrupole in the photon fluid at the time of last scattering, we will observe polarisation of the photon fluid today.

Table of contents :

I Some of the foundations of modern cosmology 
1 The basics of inflation
1.1 Anisotropy, expansion and its implications
1.2 Inflation to the rescue
1.3 The simplest model of inflation
1.4 Quantum fluctuations and the spectrum of primordial perturbations
1.5 Reheating
1.6 Difficulties for inflation
2 From the primordial spectrum to observations
2.1 The initial conditions from inflation
2.2 The Bardeen equation
2.3 Matter power spectrum
2.4 Temperature anisotropies
2.4.1 Some ignored but known effects
2.4.2 The power spectrum
2.5 CMB Polarisation
2.6 Non-Gaussianity
II Constraints from primordial perturbations 
3 The inflaton potential: probing V (φ)
3.1 Introduction
3.2 Fitting the primordial spectrum
3.3 Fitting the scalar potential
3.4 Conclusions
4 The inflaton potential: probing H(φ)
4.1 Introduction
5 The inflaton potential: Slow Roll vs. Numerics
5.1 Introduction
5.2 Background and perturbations in single field inflation
5.2.1 Exact spectra via mode equation
5.2.2 Approximation I
5.2.3 Approximation II
5.3 Constraints from current data
5.3.1 Expansion in H vs. expansion in H2
5.3.2 Approximations vs. exact spectra
5.4 Why the difference?
5.4.1 The prior issue
5.4.2 Comparison of accuracy
5.5 Discussion
6 The inflaton potential: Prior dependence of parameters
6.1 Introduction
6.2 Priors and posteriors
6.2.1 Importance sampling
6.2.2 Cosmological parameters
6.3 Flat prior on HInf
6.4 Conclusion
III Constraints from secondary anisotropies 
7 The integrated Sachs-Wolfe effect: constraining the neutrino mass
7.1 Introduction
7.2 The galaxy-ISW correlation in the presence of neutrino mass
7.2.1 Definitions
7.2.2 Effect of neutrino masses
7.2.3 Detectability
7.3 An MCMC analysis of mock data
7.4 Conclusions
8 The Rees-Sciama effect: apparent acceleration from structure formation in trouble
8.1 Introduction
8.2 The Model
8.2.1 The metric and geodesics
8.2.2 Dimensions and configurations
8.2.3 Different methods for the angular diameter distance
8.2.4 Using 2D geodesic equations in a 3D setup
8.3 The CMB
8.3.1 Temperature maps and their power spectra
8.3.2 Numerical limitations
8.3.3 Discussion
8.4 Angular diameter distance – redshift relation
8.4.1 The same maximum distance for all hole sizes up to 1.75 Gpc
8.4.2 Distribution in different directions
8.4.3 Dependence on the size of holes
8.5 Discussion and conclusion
General Conclusion 
A Vacuum states
A.1 Vacuum in Minkowsky space
A.2 Ambiguity in curved space
A.3 Conformal vacuum
A.4 Bunch-Davies vacuum
B The Limber approximation


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