Non-linear covariance matrix errors and cosmological parameter uncertainties for Euclid-like surveys 

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Cosmological horizons

Given that light has a finite speed, if the universe is finite in time there can be a maximum distance that a photon can travel. In particular, if the universe is finite in the past the maximum distance that a photon can have traveled at a given time t is:
which is called the particle horizon, or simply the horizon. This can be thought of as the radius of the sphere enclosing all events that have been in causal contact with us. We can also define an event horizon, that represents the region of space that will come in causal contact with us in the future:
For models in which a grows faster than t this quantity is finite.

Thermal history of the universe

From Eq.(2.42) we see that r evolves more rapidly with a than m. Since today we observe that r;0 m;0 (see §2.5), there is a moment in the past when the two components had the same energy density. This moment is called equivalence and is estimated to be of the order of zeq ’ 5 103. Before equivalence, the energy density of radiation was dominating the energy bud-get of the universe, thus driving the evolution of a(t). This period is thus called the radiation era, while the one after equivalence is called the matter dominated era.
Today, the characteristic time between two collisions of photons with matter particles is much longer than the characteristic time of the expansion, so matter and radiation components are not in thermal equilibrium. This condition can be written as:
where n is the mean number density of matter particles and T is the Thom-son scattering cross-section. In all viable cosmologies coll evolves more rapidly than exp, so we can find a redshift in the past where matter and ra-diation were in thermal equilibrium. This redshift is estimated to zdec ’ 103, where dec stands for decoupling, and is in general zdec < zeq, so that decou-pling happens during matter era.
From Eq.(2.31) we can derive the scaling of the temperature of matter and radiation with respect to the scale factor. For the matter component this gives:
For the radiation component, if we approximate its spectrum to a black body, the energy density is r c2 = SB T 4, where SB is the Stephan-Boltzmann constant. Thus:
After decoupling the temperature of the two components evolves indepen-dently following these scalings. Before decoupling the two fluids were tightly coupled by electromagnetic interactions and had a common temperature T , whose evolution can again be recovered from the adiabatic condition:
which is estimated to rad;0 = 1; 35 108 ( m;0 h2) 1, where h is the value of the Hubble parameter today normalised to 100 km s 1 Mpc 1. Knowing that at decoupling rad 1 Eq.(2.52) can be integrated to obtain:
The temperature of the universe also determines the matter constituents. At the beginning of the matter era, temperatures are so high that all the atoms are fully ionised. As the temperature drops, the interactions between photons and electrons become less frequent and the electrons start to com-bine with protons to make Hydrogen atoms. The time between the first combination and the moment in which Hydrogen is completely neutral is called recombination. The drop in density of free electrons makes the Thom-son scattering inefficient and photons are almost completely free to travel through the mainly empty space. This sea of photons released at decoupling is what we observe today as the CMB. As soon as the first stars start form-ing their surroundings are re-ionised by the emitted radiation and part of the CMB photons are re-absorbed. In fact, CMB observations can provide constraints on the optical depth of the medium at the time of re-ionisation.
At the beginning of the radiation era, temperature is above the thresh-old that triggers nuclear reactions. The efficient reactions in this era are the p + and the + reactions, where p is a proton and is a 4Helium nucleus. Since stable elements with atomic number of 5 or 8 do not exist, only elements lighter than 7Li are formed. In fact, density in this era is not high enough for unstable 8Be atoms to efficiently combine with particles to form 12 C atoms. This process is called Big Bang Nucleosynthesis and is supported by observations of the abundance of Hydrogen and Helium. In particular, the abundance of Helium today is too high to be explained by nucleosynthesis in the interior of stars. There is though a problem with the prediction of the abundance of 7 Li, which yields values three times higher than the measured ones (but see [130]).
Going further back in time, the temperature of the universe can reach a value Tx where:
where mx is the mass of a given particle type x, for example electrons. Above this temperature a significant amount of couples e+ e are created, and sim-ilarly at higher temperatures couples of heavier leptons, then hadrons and quarks are created. The following eras are defined:
• Lepton era: kB Te = 0; 5 MeV < kB T < kB T = 130 MeV
• Hadron era: kB T < kB T < 200 300 MeV
• Quark era: kB T > 200 300 MeV

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Einstein–de Sitter universe

The Einstein–de Sitter (EdS) model describes a flat universe without cosmo-logical constant. In this model the Friedmann equations for a single com-ponent can be integrated analytically. We will review here the main results because as we will see in §2.5 these results can be used to approximate the evolution of the universe at given epochs.
Let us rewrite Eq.(2.33) in different forms that will be useful in the fol-lowing. By dividing Eq.(2.33) by a0 and evaluating it today we get:

Table of contents :

1 Introduction 
1.1 A brief history of modern Cosmology
1.2 State-of-the-art Cosmology
1.2.1 Dark Matter
1.2.2 Dark Energy
1.3 Plan of the thesis
I Theoretical Cosmology 
2 Cosmological models 
2.1 Distances in cosmology
2.2 Hubble’s law
2.3 Hot Big Bang model
2.3.1 Big Bang singularity
2.3.2 Cosmological horizons
2.3.3 Thermal history of the universe
2.4 Einstein–de Sitter universe
2.5 CDM scenario
2.6 Scalar field models of Dark Energy
2.6.1 Quintessence
2.6.2 K-essence
3 Structure formation 
3.1 Statistical treatment of perturbations
3.2 Linear Perturbation Theory
3.2.1 Power spectrum evolution in linear regime
3.3 Non-linear regime
3.3.1 The Zel’dovich approximation
3.3.2 Spherical collapse
4 Numerical simulations 
4.1 N-body techniques
4.2 Eulerian hydrodynamics
4.3 Generation of initial conditions
4.4 Power spectrum estimation
II Cosmology with large volume galaxy surveys 
5 Cosmological dependence of the large scale structure of the universe 
5.1 Galaxy power spectrum
5.2 Baryon Acoustic Oscillations
5.2.1 Non-linear effects on BAO
5.3 Bayesian methods for cosmological parameter inference
5.3.1 Maximum Likelihood Estimator
5.3.2 Fisher matrix
5.3.3 Information content of galaxy surveys
5.4 Contributions of this thesis
6 Matter power spectrum covariance matrices from the DEUSPUR simulations 
6.1 N-body dataset
6.2 Power Spectrum & Covariance Matrix Estimators
6.3 Numerical Simulation Mass Resolution Errors
6.4 Fourier Mode Correlations
6.5 Probability Distribution of the Power Spectrum Estimator
7 Non-linear covariance matrix errors and cosmological parameter uncertainties for Euclid-like surveys 
7.1 Signal-to-Noise
7.2 Covariance and Precision Matrix Errors
7.2.1 Precision Matrix Bias
7.2.2 Variance of Sample Covariance
7.2.3 Variance of Sample Precision Matrix
7.3 Covariance Estimation Errors and Parameter Forecast
III Numerical methods for Dark Energy simulations
8 Numerical methods for clustering Dark Energy simulations 
8.1 Euler Equations for Dark Energy Fluids
8.2 The Riemann Problem
8.2.1 Generalised Riemann Invariants
8.2.2 Rarefaction Waves
8.2.3 Shock Waves
8.2.4 Exact Riemann Solver
8.2.5 Approximate Riemann Solvers
8.2.6 Specificities of the Riemann Problem for Dark Fluids
8.3 Wave structure in 3D
8.4 Upwind Numerical Schemes
8.4.1 Conservative Hyperbolic Methods
8.4.2 MUSCL-Hancock Method
8.4.3 Piecewise Linear Method (PLM)
8.4.4 Piecewise Parabolic Method (PPM)
8.5 Euler Equations in Supercomoving Variables
8.6 Euler Equations in Spherical Symmetry
9 Future prospects 

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