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Luminosity function and selection function
The null correlation approach allows us to determine the shapes of the luminosity function f(M) and the selection function (m). For the true values of the cosmological parameters, the sample is described with the following probability density: dPw = fw(M)dM w(m)dm (3.23) where the functions fw(M) and w(m) are dened as follows: fw(M) / 10 5 Mf(M) (3.24) w(m) / 10.
They are evaluated from the cumulative distribution functions: Fw(M) = Z M 1 fw(M)dM (3.26) w(m) = Z m 1 w(m)dm where is the Heaviside function. To obtain the functions f(M) and (m), we compute fw(M) and w(m) by determining the derivatives of Fw(M) and w(m) and use Eqs.3.24 and 3.25. In order to avoid the arbitrary choice of the t, the derivatives of smooth approximations of Fst w (M) and st w(m) are determined by the linear interpolations. Then, the luminosity function f(M) and the technological function (m) are determined as follows: 8< : where rM and lM are the right and left intervals of the variable M in which contain pN 2 values of Mk > M and Mk < M respectively. The intervals rm and lm also contain pN 2 values of mk > m and mk < m respectively. As a result of the use of linear interpolation technique, the ( pN 2 ) points on the left side and the (pN 2 ) points on the right side are isolated and therefore, must be discarded in the estimation of the functions f(M) and (m). Figs.3.11 and which f(M) is a Gaussian function with an average M0 = 20, a standard deviation M = 0:15, a Gumbel function with mode M0 = 20 and a scale parameter M = 1:09 (see appendix A.2). The (m) function is a Heaviside function. The upper panels of these gures show the statistics of the luminosity function. The (left panel) illustrates the empirical cdf of the evaluated luminosity function represented by the green curve, it is compared to the theoretical cdf of the luminosity function represented by the red curve. The (right panel) shows the luminosity function with respect to the absolute magnitude. The red curve is the luminosity function used for the simulation of the sample and the green plus symbols represent the luminosity function evaluated by the interpolation technique. The lower panels show the statistics of the selection function. The (left panel) represents the empirical cdf of the evaluated selection function represented by the green curve and the theoretical cdf of the Heaviside function represented by the red curve. The (right panel) shows the technological function with respect to the apparent magnitude m. The function ‘(m) (m) (the green plus symbols) obtained is the same as the one used in the simulation, it is evaluated as a Heaviside function (in red). These gures show that the initial luminosity and technological functions used for the simulation are retrieved perfectly by this technique with hMi = 20:005 and M = 0:158.
Kolmogorov-Smirnov test
The Kolmogorov-Smirnov (KS) test is a nonparametric test that allows one to check whether deviations between the theoretical cdf and the empirical (step) cdf can be considered as random uctuations. The KS is based on the probability law of the maximum distance dened as follows: Dmax = Max(jcdfTheoretical cdfEmpiricalj) The result of Dmax is compared to the critical value, Dmax;, (for a signi- cance level ) obtained from the KS table with respect to the size of the sample. It allows one to decide if the null hypothesis is accepted (Dmax Dmax;) or rejected (Dmax Dmax;) for a given (by default, = 5%). Fig.3.13 shows the cdf of three distributions with increasing sample size (N) (from left to right) that can be compared to the theoretical cdf of a uniform distribution (solid line). It shows how the empirical and the theoretical cdf get closer when N grows. The KS test enables one to check whether specic functions f(M) and (m) t the databy comparing Fw(M) with Fst w (M) and w(m) with st w(m). For this purpose we started with the case of the luminosity function. If f(M) has a Gaussian form (App.A.1), Eq.3.24 shows that fw(M) is still Gaussian with a mean hMi dened in Eq.3.11 and a standard deviation M dened as the following:
The V/Vmax test
The V=Vmax test is well known statistical approaches that have been used in cosmology. The random variable V=Vmax has a uniform distribution between 0 to 1. Herein, we apply this test as described by [G. Bigot & R. Triay 1991]. We dene the uniform random variable: h(z;m) = V0(z) Vm(M) where V0(z) is the comoving volume at redshift z and Vm(M) is the comoving volume that the source of absolute magnitude M remains visible up to a redshift zmax at a limiting apparent magnitude mlim of the survey, it corresponds to V (zmax) dened in Eq.2.8 [Schmidt M., 1968]. The probability density of a complete sample reads: dPobs = 1 A (mlim m)(z)(zform z)f(M)dM dV (3.38).
where A denotes the normalization factor, zform the redshift of the formation epoch and the Heaviside function. A range of redshift is dened by the product of the Heaviside functions in the probability density in which a source of absolute magnitude M is observed. Following Eq.2.17 we dened this range by M(z): M(z) = (mlim M (z))(z)(zform z) (3.39) A little algebra shows that the probability density of M and h reads: dP / Vm(M) f(M)dM dh (3.40) After obtaining a set of candidate models (the NCC), we apply the KS test to test to these candidate models. That requires the calculation of: the absolute magnitude Mk, the maximum of redshift zmax (which corresponds to the limited apparent magnitude mlim), the maximum of volume V (zmax) and the volume V0(zk) according to Eqs.2.17, 2.7, 2.8 and 1.37, respectively.
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 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
