Sampling the light curve of type-Ia SN and selection eects 

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Baryonic Acoustic Oscillations

The properties of the CMB guarantee the existence of a scale from which the universe becomes isotropic and homogeneous, as well as the measurement of the velocity of our galaxy with respect to the referential that it denes. Even though the temperature of the CMB is extremely uniform throughout the sky, the COBE satellite (Cosmic Background Explorer) detected tiny uctuations or temperature anisotropies in the CMB in 1992. The WMAP (Wilkinson Microwave Anisotropy Probe) satellite, launched in 2001, conrmed the COBE observation and mapped the temperature uctuations with a much higher resolution.
More recently, the Planck satellite mapped the CMB anisotropy with even higher accuracy (see Fig.1.1). In general agreement with the observation, the anisotropy in the CMB grew from the gravitational tension of small uctuations existent in the early Universe. These perturbations gave rise to acoustic oscillations in the photon-baryon uid before the recombination. During that period, the gravitational attraction between baryons tended to collapse the system and compress the photon-baryon uid whereas the photon pressure provided an opposite restoring force. This created sound waves that propagate in the uid of the primordial Universe. At the time of recombination, the photons were diused freely and these oscillations outlined in the CMB. A modelling of the dynamics of these structures in the primordial plasma makes it possible to interpret the angular uctuations of its temperature, which are observed by means of a spectral analysis, which provides estimates of cosmological parameters.
Moreover, these features have an imprint on baryonic structures at every stage of the evolution of the Universe called Baryonic Acoustic Oscillations (BAO). It is believed that the high density areas associated with the acoustic waves in the CMB condense to create the current structures. The size of structures, can be used to trace back the cosmological parameters [W. Hu 1997]. Fig.1.2 shows the power spectrum of the CMB obtained by Planck satellite.

The Dark Energy

Since the end of 90s, acceleration of cosmic expansion has been detected by using Hubble diagram of type-Ia supernovae thanks to Supernova Cosmology Project (SCP) [S. Perlmutter et al. 1999] and the High-Z supernova [A. G. Riess et al. 1998]. According to FLG model, such an acceleration results from either a positive zero cosmological constant or the an unknown gravitational source, an ad hoc alternative motivated by the cosmlogical constant problem in High Energy Physics. These two interpretations being integrated in a single one named « Dark Energy ». Fig.1.5 presents the condence contours of m and (see Sect.1.5) obtained by SCP using 42 SNe Ia. This section outlines the two frequent approaches used to explain this acceleration.

The Cosmological Constant

While in GR, the cosmological constant stands for a universal constant. From the point of view of particle physics, it would be the energy density of quantum uctuations of vacuum. Hence, two major obstacles appear: the cosmological constant problem and the problem of coincidence. The Cosmological constant problem According to the particle physics, the value of the energy density of the vacuum is estimated: vacuum (1018GeV )4 2:10110erg=cm3 (1.27).
On the other hand, observations of type-Ia SNe and CMB uctuations, the observed energy density is of order: obs (10􀀀12GeV )4 2:10􀀀10erg=cm3 (1.28) Therefore, the expected value is 120 order of magnitude larger than the observed energy density [S. Weinberg 1989], [S. Carroll 2001]: vacuum 10120obs (1.29) This disagreement is known as the Cosmological constant problem.

Hubble diagram and measure of distances

What Hubble did that led to the discovery of his law was the measure of both the distance and the velocity of galaxies in the nearby Universe (see Fig.1.7). He showed that galaxies recede with a velocity that increases proportionally with their distance, what gives the Hubble’s law: v = Hd (1.30).
where the recession velocity of galaxies v and their distance d. The constant of proportionality H is called the Hubble constant. Since its discovery, the Hubble diagram became a major tool for retrieving of cosmological information. The distribution of type-Ia supernovae in this diagram is used with (large samples of type-Ia supernovae) detected from surveys such as the SNLS, SDSS, HST and low-z surveys (Fig.1.8). The spectra of a source is seen redshifted.

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Events in the space-time diagram

With the motivation in mind to use events (such as light emissions from astronomical objects) to probe the geometry of space-time, one has to specify their intrinsic properties that characterize a single family. They are assumed to be not very dierent from one another in terms of their intrinsic luminosity(standard candles). Moreover, we assume that they do not show evolutionary eects and that they are uniformly distributed in space. These characteristics stand for random variables with probability densities who are dened by speci c working hypotheses. The data are face to selection eects in observation, and a selection function is used at this purpose. We check the eciency of our statistical method on simulation samples as a representation of the real data. The main selection eect, it has been described by Malmquist (see Sect.2.4), depends only on the apparent magnitude. Therefore, a selection function must be used in the probability density in order to take it into account among other selection eects in observation.

Modelling a sample of quasars

Given that the quasars are detected at high redshift, therefore the use of the Hubble diagram with this type of objects will be a good tool to obtain the cosmological information. The farthest quasar observed up to now is the quasar ULAS J112001.48+064124.3 detected at redshift z = 7:085 [E. Momjian et al 2013]. As usual, a non evolution of quasars is assumed on interpreting the nonlinear Hubble diagram. In the following, we assume H = 70 Km:Mpc􀀀1:s􀀀1 [D. N. Spergel et al 2013].A

Table of contents :

1 Basics of Observational Cosmology 
1.1 Introduction
1.2 Newtonian cosmology
1.3 Einstein’s cosmology
1.4 The Friedmann-Lemaître-Gamow model
1.4.1 The Friedmann’s equations
1.4.2 Primordial Universe
1.5 Contents of the Universe
1.6 Baryonic Acoustic Oscillations
1.7 Dark matter
1.8 The Dark Energy
1.8.1 Models with Scalar Fields
1.8.2 The Cosmological Constant
1.9 Hubble diagram and measure of distances
1.9.1 The Comoving Distance
1.9.2 The Age and the Conformal Time
1.9.3 The Luminosity Distance
1.10 The Magnitude Systems
1.10.1 The Apparent Magnitude
1.10.2 The Vega System
1.10.3 The AB System
1.11 Constrain the cosmological parameters
1.12 Conclusion
2 Quasars samples 
2.1 Introduction
2.2 Spectrum of quasar
2.3 Events in the space-time diagram
2.4 The selection eect
2.5 Modelling a sample of quasars
2.5.1 Statistical modelling
2.5.2 Simulation technique
2.5.3 The k-correction
2.6 Conclusion
3 The null correlation technique 
3.1 Introduction
3.2 Weighting factors
3.3 Luminosity function and selection function
3.4 Kolmogorov-Smirnov test
3.5 The V/Vmax test
3.5.1 Calculation of the V/Vmax terms
3.6 Conclusion
4 Application to quasars data 
4.1 Introduction
4.2 The Sloan Digital Sky Survey data
4.2.1 The k-correction
4.3 Results
4.3.1 The null correlation test
4.3.2 The V/Vmax test
4.3.3 Estimation of the luminosity function and the selection function
4.4 Inferences on cosmological expansion based on QSOs
4.5 Conclusion
5 Application to supernovae type Ia sample 
5.1 Introduction
5.2 Type Ia Supernovae as standard candles
5.2.1 The lightcurve of type-Ia SN
5.2.2 Standardisation of type-Ia SN
5.2.3 Modelling of the Supernova event
5.2.4 5.2.4 Sampling the light curve of type-Ia SN and selection eects
5.2.5 Calibration statistics
5.2.6 Simulation
5.3 The null correlation test on the supernova sample
5.3.1 Luminosity function and Selection function
5.3.2 Precision and error
5.4 Application to the JLA sample
5.4.1 Description of samples
5.4.2 Results
5.5 Conclusion
6 Résumé en français 
6.1 La cosmologie moderne
6.2 Simulation d’échantillon de quasar
6.3 La technique de corrélation nulle
6.4 Résultats avec les données quasars
6.5 Modélisation d’échantillon de supernova et résultats sur les donn ées de SDSS-II/SNLS3
Appendices 
A Probability density functions 
B Calculation of the weighting factor 
C Calculation of V(M) 
D Code Python 

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