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Délivré par l'Université Toulouse III - Paul Sabatier Discipline ou spécialité : Astrophysique
JURY
Pr. Peter Von Ballmoos (Président du jury) Dr. Anne Ealet (rapporteuse)
Pr. Jim Bartlett (rapporteur) Pr. Ana Mourao (examinateur) Pr. Alain Blanchard (directeur de thèse) Pr. Yves Zolnierowski (co-directeur de thèse)
Ecole doctorale : S2UDE Unité de recherche : LATT
Directeur(s) de Thèse : Pr. Alain Blanchard (directeur), Pr. Yves Zolnierowski (co-directeur) Rapporteurs : Dr. Anne Ealet (OAM-Marseille), Pr. Jim Bartlett (APC - Paris)
Présentée et soutenue par Luis Ferramacho Le 3 Novembre 2008
This work focused on the constraints that can be made on cosmological models based on the standard cold dark matter and dark energy paradigm. With the increasing quality of different observational data, it is important to access how the astrophysical modeling, data treatment and assumptions taken can influence the final results on the cosmological parameters. Initially, I have studied the cosmological test that uses the principle of self-similarity in X-ray galaxy clusters, which translates into a supposed constant profile for the gas mass fraction at different redshifts that can be used to derive the “correct” cosmological parameters through the angular diameter distance. The obtained results are highly dependent on the scaled radius considered to perform the test, indicating that such cosmological probe is not consistent and should not be used in precision cosmology. The second part of this thesis was dedicated to implement joint constraints using three of the main observational probes, which are the CMB anisotropies, Type Ia Supernovae, and large scale structure. The constraints were performed using the Bayesian method of Markov chain Monte Carlo (MCMC). The obtained results show a great level of precision in all cosmological parameters, typically 3 − 5%. By relaxing some of the assumptions, like considering a flat Universe or a cosmological constant, the constraints remain highly stable on most parameters.
Again, possible systematics were studied when using these data. Using different de-scriptions (matter power spectrum or correlation function) and modelization on the LSS data can cause some the derived cosmological parameters to be shifted by a factor up to 1σ. Also, using compressed information from this data causes the error bars to be overestimated. At the same time, the possible evolution of the intrinsic luminosity of SNIa was constrained to be, at most, of the order of 0.1 mag at z = 0.5.
Finally, we have constrained a model of dynamical dark energy equation of state based on the quintessence paradigm, which involves a rapid transition between values of w = pDE/ρDE close to 0 at higher redshifts to a value close to −1 in the local universe.
We show that the data impose a transition at redshifts zt> 1.5 in all the studied cases,
Ce travail de thèse s’est concentré sur les contraintes qui peuvent être appliquées sur des modèles cosmologiques basés sur le paradigme standard de matière noire et énergie noire. Avec la qualité croissante des différentes données observationnelles, il est important de déterminer comment la modélisation astrophysique, le traitement de données et les hy-pothèses prises peuvent influencer les résultats finaux sur les paramètres cosmologiques. Dans un premier temps, j’ai etudié le test cosmologique qui utilise le principe de autosimilarité dans les amas de galaxies, principe qui se traduit dans un profil de la fraction de masse de gaz supposé constant à differents redshifts qui peut être utilisé pour dériver les “vrais” paramètres cosmologiques à travers la distance angulaire. Les résultats obtenus montrent une forte dépendance sur le rayon considéré pour effectuer ce test, indiquant que tel test cosmologique n’est pas consistant et ne devrait pas être utilisé en cosmologie de precision.
La seconde partie de cette thèse a été dédiée à l’implémentation des contraintes con-jointes en utilisant trois des principaux tests observationnels, qui sont les anisotropies du CMB, les supernovaes de type Ia (SNIa) et les structures à grande échelle (LSS). Les contraintes ont été effectuées en utilisant la méthode Bayesienne des chaînes de Markov Monte-Carlo. Les résultats obtenus montrent un grand niveau de précision sur tous les paramètres, typiquement 3 à 5 %. En relaxant quelques unes des hypothèses utilisées, tel que un univers plat ou une constante comologique pour décrire l’énergie noire, les constraintes restent très stables sur la pluspart des paramètres.
Par ailleurs, on a étudié des possibles effets systématiques quand on prend ces données. Le fait d’utiliser différentes descriptions (spectre de puissance ou fonction de corrélation) et modélisations sur les données des LSS peut causer un décalage sur certains paramètres cosmologiques par un facteur maximum de 1σ. En outre, utiliser de l’information com-pressée sur ces données a pour conséquence une surestimation des barres d’erreur. Dans le même temps, l’évolution possible de la luminosité intrinsèque des SNIa a été contrainte à une valeur maximale de l’ordre de 0.1 magnitude à z = 0.5.
Finallement, j’ai contraint un modèle dynamique pour l’équation d’état de l’énergie noire basée sur le paradigme de quintessence, et qui comprend une transition rapide entre les valeurs de w = pDE/ρDE proches de 0 à hauts redshifts à une valeur proche de
−1 dans l’univers local. Je montre que les données imposent une transition à redshifts zt> 1.5 pour touts les cas étudiés, et zt > 4 quand le w local est égal à −0.95.
The work presented in this manuscript was made much easier with the help of many different people whom I would like to thank. First, a special word for my thesis director, Alain Blanchard, for all the useful discussions, encouragement and many other aspects that helped me during the time period of my PhD. I feel that over the last 3 years, the possibility to work and discuss with someone with such an wide and sharp scientific vision as definitely helped me to better understand the problems and methods that are present in modern cosmology and large scale astrophysics, and developing a critical spirit of major importance when doing scientific research. I would also like to thank my co-director Yves Zolnierowski for the support and disponibility when discussing data analysis and for his review of this manuscript. I acknowledge Marian Douspis for his support and help during my first times in the LATT back in 2005. I am grateful to Fundação para a Ciência e Tecnologia (FCT-Portugal) for the financial support of this thesis work, through fellowship contract SFRH/BD/16416/2004.
A good environment in the office is always important, and I would like to thank to Edgardo Vidal and Enrique Montero for the useful discussions on astrophysics and other subjects.
This French adventure started in Paris, were I could count with the support and friendship of Rui Pereira. While in Toulouse I have met some people who helped me to integrate and adapt to this new country and who have become good friends. I would like to thank Marie, Florence, Ruben, Marion, Cindy, Guillaume, Julien, Ronan, Julie, Sidonie and Fabrice for their support.
A very special thought to my companion Mathilde Treguer, whom I had the chance to meet during my stay at Toulouse. I am deeply grateful for all her support, love and sharing of good moments but also for being there when things were not so shiny. This thesis would have been much harder to complete without her dedication and support.
To all my friends in Portugal, despite the fact of having less time and space to be with them, I thank for their support whenever I got back to Portugal. A special word to Ana Neves, Tiago, Artur, Guimas, Susana, Ana Luisa, Gato, Carmo, Nuno, Guillaume, Xana, David, Gabdiku, and of course, the ADIL from Faro: Vidal, Pinto, Fred, Manuca, Zé, Joao.
I cannot finish without saying how grateful I am to my close family: mother, sister and brother-in-law for their unconditional support and for providing me with such a
1. Introduction 1 2. Introduction (Français) 3 3. Some basics on Modern Cosmology 5
3.1. Friedmann-Robertson-Walker Universe . . . 5
3.1.1. Time and Distances in an expanding Universe . . . 10
3.2. The mystery of Dark Energy . . . 14
3.2.1. Vacuum energy . . . 14
3.2.2. Anthropic solutions . . . 16
3.2.3. Quintessence . . . 16
3.2.4. Modification of gravity . . . 18
4. Cosmological probes 20 4.1. Cosmic Microwave Background Radiation . . . 20
4.1.1. Thermalization and recombination . . . 21
4.1.2. CMB anisotropies . . . 22
4.1.3. Initial conditions . . . 24
4.1.4. The physics of the CMB anisotropies . . . 25
4.1.5. Secondary anisotropies . . . 27
4.1.6. Polarization . . . 29
4.1.7. CMB and cosmological parameters . . . 30
4.1.8. Observational status . . . 32
4.2. Large scale structure . . . 33
4.2.1. Describing LSS . . . 34
4.2.2. Dynamics of linear perturbations . . . 36
4.2.3. Transfer function . . . 37
4.2.4. Baryonic acoustic oscillations . . . 38
4.2.5. Redshift surveys . . . 40
4.3.1. Mass estimations for galaxy clusters . . . 43
4.3.2. Cosmological tests with Clusters . . . 49
4.3.3. Gas fraction in clusters as geometrical test . . . 51
4.3.4. X-ray observations . . . 52
4.4. Type Ia Supernovae . . . 52
4.4.1. The lightcurve - maximum luminosity relation . . . 53
4.4.2. Systematic effects and cosmological application . . . 54
4.4.3. Observational status . . . 55
4.5. Weak lensing . . . 56
4.5.1. Systematics . . . 60
4.5.2. Results . . . 61
5. Statistical tools for parameter constraints 62 5.1. Probabilities and Likelihood . . . 62
5.2. The frequentist approach . . . 64
5.3. The Bayesian approach . . . 66
5.4. Markov Chains Monte Carlo . . . 69
5.4.1. Metropolis-Hastings algorithm . . . 69
5.4.2. Convergence . . . 70
5.4.3. Proposal density . . . 70
5.4.4. Results from the chain . . . 71
5.4.5. COSMOMC . . . 71
6. The gas mass fraction in X-ray clusters 72 6.1. The principles of a geometrical test . . . 72
6.2. Observed trends in the gas fraction profiles . . . 74
6.2.1. The temperature dependence of the gas mass fraction from a local sample of 39 local clusters. . . 75
6.3. Application of the fgas test using 16 high redshift clusters . . . 79
6.3.1. The sample . . . 81
6.3.2. High redshift gas fractions . . . 82
6.3.3. Testing for cosmological parameters . . . 83
6.3.4. Discussion . . . 86
7. What can we know from available data? 89 7.1. Matter correlation function and BAO . . . 90
7.1.1. Non-linear corrections on the matter distribution . . . 90
7.1.2. Redshift distortions and scale dependent bias . . . 92
7.1.3. Corrected correlation function and cosmological constraints . . . . 94
7.2.1. Correlation function, power spectrum and compressed data - a
comparative approach . . . 98
7.2.2. Combined constraints on C.D.M cosmology . . . 103
7.3. Constraining the evolution of SN Ia intrinsic luminosity . . . 109
7.3.1. SNIa Evolution - A phenomenological modelization . . . 109
7.3.2. Constraints on the evolution model . . . 110
8. Beyond ΛCDM 114 8.1. Parametrization of quintessence models . . . 114
8.2. Figure of merit for present day data . . . 118
8.3. Constraining rapid transition models . . . 121
8.3.1. The effect of dynamical dark energy on the large scale structure data . . . 122
8.3.2. Joint constraints . . . 124
9. Conclusion 132
10.Conclusion (Français) 135 A. Contours for all the performed constraints 138
Bibliography 163
3.1. Look-back time vs. redshift for different values of the cosmological para-meters, assuming H0 = 72 km s−1 Mpc−1. . . 12
4.1. CMB spectrum measured at different wavelengths. The dotted line shows a Plank distribution with T = 2.724 K. . . 22 4.2. The theoretical CMB temperature anisotropy power spectrum, calculated
for a standard ΛCDM model with CMBFAST. The regions of different physical effects are identified. Figure taken from [36]. . . 24 4.3. Temperature and polarization power spectra produced by adiabatic scalar
perturbations (left ) and tensor perturbations. The different modes of polarization are identified in the figure. Taken from [37]. . . 31 4.4. All sky map of the temperature anisotropies as seen by COBE (left ) and
WMAP (right ). . . 33 4.5. Transfer functions for various adiabatic and isocurvature models,
normal-ized to 1 at small k and with Ωm = 1,h = 0.5. A number of possible
matter contents is illustrated: pure baryons, pure CDM, pure HDM and a mixture of CDM and HDM. Taken from [3]. . . 39 4.6. Galaxy distribution in the nearby universe as observed by the SDSS (left )
and 2dF (right ). . . 41 4.7. Left: Correlation function of the SDSS LRG sample, clearly presenting a
baryonic acoustic peak. The different lines show the expected correlation for different values of Ωmh2. Taken form [56] Right: Matter power
spec-trum for the SDSS LRG and main samples. Continuous lines show the expected linear power spectrum and dashed lines include corrections for non-linear effects. Taken from [57] . . . 43 4.8. Left : Lightcurves of several local Type Ia supernovae as they were
ob-served in the B-band. Right : The same lightcurves after correction with a stretch factor. . . 53
4.9. SNIa results from Kowalski et al. 2008. Upper Left : Binned Hubble diagram of the supernovae distance moduli vs. redshift. The line rep-resents the best fit curve for a ΛCDM cosmology, which corresponds to Ωm = 0.28 and ΩΛ = 0.72. Lower left : Binned residuals with respect to
the best fit model. Right : Confidence contours for Ωm and ΩΛ at 1, 2 and
3 σ using the full data set and after two selection cuts to reject for outliers. 55 6.1. Figure taken from RSB00, showing the variation of the gas fraction with scaled
radius (δ is the overdensity). The left panel shows this profile in the case of the hydrostatic assumption and right panel for mass estimate derived from the NFW dark matter profile, with the EMN normalization. . . 74 6.2. Table taken from [136] with the physical proprieties of the 39 clusters.
The question mark denotes those clusters with some substructure in the ROSAT image. . . 76 6.3. Table taken from [136] with the results of fits and calculations made. . . 77
6.4. Temperature dependence of the gas fraction for the local sample, plotted for two
cosmological models and different increasing radii. Left panels are for a EdS (Ωm =
1,ΩΛ = 0) and right panels for a concordance model (Ωm = 0.3,ΩΛ = 0.7). Upper
panels represent the gas fraction for R2000, middle panels for R1000, and lower panels
for the virial radius. Lines correspond to the power law fits. . . 80
6.5. Taken from Lumb et al. 2004. Summary of cluster parameters for Ωm =
1,ΩΛ = 0, h0=0.5, q0=0.5. Spectral fitting errors, L, β and rc are 1σ
errors on one parameter. . . 82 6.6. Likelihood functions of the gas mass fraction test at different reference
radii, for a flat universe. Left panels: Assuming a constant normalizations AT M of 6.24 keV, ([64], top) and 11.4 keV (down). Top right : Assuming
a normalization AT M that depends on Ωm as AT M = 4.9Ω−0.75m KeV.
Lower right : Likelihood functions using normalization AT M=11.4 keV,
but performing the test at the same physical radii as if AT M = 6.24.
Solid lines correspond to the test applied at the virial radius. The other lines correspond to likelihood functions obtained for different radii: R1
(dot), R2 (dashed), R500 (dashed-dot), and R2500 (dashed-dot-dot). . . 85
6.7. The confidence contours for Ωm and ΩΛ at the 1, 2, and 3 σ level on 1
parameter using the cosmological test based on the gas fraction of distant and nearby clusters and assuming AT M = 6.24 keV. Left : Test performed
7.1. Correlation functions expressed in units of ξ(r)r2 accounting for successive corrections for non-linear effects and distortions. Black, dashed-dot: Cor-relation funcion derived directly from the linear power spectrum. Blue, dashed: Similar to the black curve, but considering the peak suppres-sion observed in numerical simulations. Violet, dot: Similar to blue, but adding the non-linear corrections from HALOFIT. Red, straight: Fully corrected correlation function. . . 96 7.2. Joint confidence contours in the Ωm - ΩΛ parameter space using the full
data of the correlation function, and considering h = 0.7, Ωb = 0.046 and
ns = 1. Regions represent the 1, 2 and 3 σ confidence intervals on each
parameter. . . 97 7.3. Posterior distributions for the vanilla parameters constrained in the MCMC
analysis with different approaches to the LRG data. Doted curves rep-resent the mean likelihood of the samples, while the solid lines show the fully marginalized posterior. The color code represents the 3 different ap-proaches: WMAP5+SNIa+P (k) (Black ), WMAP5+SNIa+ξ(r) (Blue), WMAP5+SNIa+A parameter (Red ). . . 101 7.4. Correlation functions computed using the mean posterior values for the
parameters constrained with the combination WMAP5+SNIa+ξ(r) (con-tinuous line) and the combination WMAP5+SNIa+P (k) (dashed line). The SDSS data from E05 is also plotted for a better visualization. . . . 102 7.5. Same as fig 7.3, but here the red curves correspond to constraints using
the full data of the correlation function, using the Q-model (eq. 7.18) to correct for non-linear effects and redshift space distortions. . . 103 7.6. Posterior distributions for the parameters constrained in our MCMC
analy-sis. Doted curves represent the mean likelihood of the samples, while solid lines show the fully marginalized posterior. The 4 parameter sets constrained are represented by different colors: Vanilla (black), Vanilla + Ωk(green), Vanilla + w (blue), Vanilla + Ωk+ w (red). . . 105
7.7. 2D constraints on relevant cosmological parameters when Vanilla+Ωk+ w
is considered. The curves show the marginalized 1σ and 2σ confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 107 7.8. Constraints in the ΩΛ − K plane from the Hubble diagram for a flat
cosmological model. Contours corresponds to 1 and 2 σ region on two parameters. . . 110 7.9. Posterior distribution for the SN evolution parameter K, for different
pa-rameter sets of CDM models. The color code is the same as the one used in Fig. 7.5. Evolution is systematically preferred, although the non evolving solution remains acceptable. . . 112
7.10. Joint 2D marginalized constraint on the dark energy equation of state parameter w (supposed constant) and the curvature density Ωk, assuming
evolution on the SNIa luminosity (blue) and considering no evolution (red ) . The contours show the confidence regions of 1 (darker) and 2 σ. . . 112 8.1. Some examples for the quintessence EoS parameter as a function of
red-shift. The continuous line shows the CPL parametrization with w0 = −1
and w1 = 1. The other curves represent the general transition
parame-trization with at = 1, w+ = 0, w− = −1 and different values for the
transition rate: Γ = 0.2 (dashed-dot ), Γ = 0.75 (long-dashed ), Γ = 0.85 (dashed ), Γ = 1 (dotted ) and Γ = 5 (dashed-dot-dot-dot ). . . 116 8.2. 2D contours at 68 % and 95% confidence level, for the parameters w0 =
1/2(w+ + w−) and w1 = 1/2(w+ − w−), setting zt = 1. The blue
con-tours correspond to Γ = 1 and the red concon-tours to Γ = 0.85. The green countour shows the same confidence regions using the standard CPL pa-rameterization. . . 119 8.3. 2D contours at 68 % and 95% confidence level on the parameters w0 and
w1 of CPL parametetrization, assuming an evolution in SNIa absolute
magitude (red ) given by eq 7.21 and without evolution (dotted blue). . . 121 8.4. Expected shape for the correlation function, in units of ξr2, assuming
rapid (step-like) transitions in the dark energy EoS at different redshifts, with w− = −1. Upper panel: w+ = −0.2. Middle panel: w+ = −0.1.
Lower panel: w+ = 0. Plotted curves correspond to different values for
the transition redshift: zt = 0.5 (dashed-dot ), zt = 1 (dashed ), zt = 2
(dot )and zt= +∞ (straight ). . . 123
8.5. Confidence regions at 68 % and 95 % level for the transition epoch at
and the density of the quintessence field ΩQ, considering w+ = −0.2
and w−= −1 and using different combinations of data:CMB (light blue),
CMB+SNIa (red ), CMB+P(k)(green) and CMB+SNIa+P(k)(blue). . . 126 8.6. Confidence regions at 68 % and 95 % level for the transition epoch atand
the amplitude of matter perturbations σ8, considering w+ = −0.2 and
w− = −1 and using different combinations of data: CMB+SNIa (red ),
CMB+P(k)(green) and CMB+SNIa+P(k)(blue). . . 127 8.7. Confidence regions at 68 % and 95 % level for the transition epoch at
and the density of the quintessence field ΩQ, considering w+ = −0.1 and
w− = −1 and using different combinations of data: CMB+SNIa (red ),
8.8. Confidence regions at 68 % and 95 % level for the transition epoch at
and the density of the quintessence field ΩQ, considering w+ = −0 and
w− = −1 and using different combinations of data: CMB+SNIa (red ),
CMB+P(k)(green) and CMB+SNIa+P(k)(blue). . . 129 8.9. Confidence regions at 68 % and 95 % level for the transition epoch atand
the density of the quintessence field ΩQ, considering w−as free parameter
and using different values for w+: 0 (green), -0.1(red ) and -0.2(blue). . . 131
8.10. Confidence regions at 68 % and 95 % level for the transition epoch atand
w−n using different values for w+: 0 (green), -0.1(red ) and -0.2(blue). . . 131
A.1. 2D constraints on relevant cosmological parameters when the Vanilla model and the full data on the correlation function of SDSS LRG. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 139 A.2. 2D constraints on relevant cosmological parameters when the Vanilla
model and the A parameter from E05 are considered. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 140 A.3. 2D constraints on relevant cosmological parameters when Vanilla model
and the full data on the matter power spectrum of SDSS LRG. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 141 A.4. 2D constraints on relevant cosmological parameters when Vanilla+Ωk is
considered. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 142 A.5. 2D constraints on relevant cosmological parameters when Vanilla+w is
considered. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 143 A.6. 2D constraints on relevant cosmological parameters when Vanilla+Ωk+ w
is considered. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 144 A.7. 2D constraints on relevant cosmological parameters when Vanilla model
is considered together with the evolution model for SNIa luminosity. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 145
A.8. 2D constraints on relevant cosmological parameters when Vanilla+Ωk is
considered together with the evolution model for SNIa luminosity. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 146 A.9. 2D constraints on relevant cosmological parameters when Vanilla+w is
considered together with the evolution model for SNIa luminosity. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 147 A.10.2D constraints on relevant cosmological parameters when Vanilla+Ωk+ w
is considered together with the evolution model for SNIa luminosity. The curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 148 A.11.2D constraints on relevant cosmological parameters when the Vanilla
model is considered together with the dark energy parameterization with at= 1 and Γ = 1. The curves show the marginalized 68% and 95%
con-fidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 149 A.12.2D constraints on relevant cosmological parameters when the Vanilla
model is considered together with the dark energy parameterization with at = 1 and Γ = 0.85. The curves show the marginalized 68% and 95%
confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 150 A.13.2D constraints on relevant cosmological parameters combining CMB data
+ SNIa. The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = −0.2 and Γ = 5. The
curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 151 A.14.2D constraints on relevant cosmological parameters combining CMB +
P(k). The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = −0.2 and Γ = 5. The curves show
the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 152 A.15.2D constraints on relevant cosmological parameters combining CMB +
P(k) + SNIa. The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = −0.2 and Γ = 5. The
curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 153
A.16.2D constraints on relevant cosmological parameters combining CMB data + SNIa. The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = −0.1 and Γ = 5. The
curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distributions for each parameter. . . 154 A.17.2D constraints on relevant cosmological parameters combining CMB +
P(k). The model considered is the Vanilla together with the dark energy parameterization set by w−= −1, w+ = −0.1 and Γ = 5. The curves show
the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 155 A.18.2D constraints on relevant cosmological parameters combining CMB +
P(k) + SNIa. The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = −0.1 and Γ = 5. The
curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 156 A.19.2D constraints on relevant cosmological parameters combining CMB data
+ SNIa. The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = 0 and Γ = 5. The curves
show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 157 A.20.2D constraints on relevant cosmological parameters combining CMB +
P(k). The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = 0 and Γ = 5. The curves show
the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 158 A.21.2D constraints on relevant cosmological parameters combining CMB +
P(k) + SNIa. The model considered is the Vanilla together with the dark energy parameterization with w− = −1, w+ = 0 and Γ = 5. The
curves show the marginalized 68% and 95% confidence regions. On top are shown the marginalized 1D distribution for each parameter. . . 159 A.22.2D constraints on relevant cosmological parameters combining CMB +
P(k) + SNIa. The model considered is the Vanilla together with the dark energy parameterization with w+ = −0.2 and Γ = 5. The curves show
the marginalized 68% and 95% confidence regions. On top it is shown the marginalized 1D distribution for each parameter. . . 160 A.23.2D constraints on relevant cosmological parameters combining CMB +
P(k)+ SNIa. The model considered is the Vanilla together with the dark energy parameterization with w+ = −0.1 and Γ = 5. The curves show
the marginalized 68% and 95% confidence regions. On top it is shown the marginalized 1D distribution for each parameter. . . 161
A.24.2D constraints on relevant cosmological parameters combining CMB + P(k)+SNIa. The model considered is the Vanilla model, using a dark energy parameterization with w+ = 0 and Γ = 5. The curves show the
marginalized 68% and 95% confidence regions. On top is is shown the marginalized 1D distribution for each parameter. . . 162
3.1. Different form of the potentials for Quintessence. Taken from [25] . . . . 18 4.1. Overview of recent weak lensing surveys, with the area of teh survey. The
obtained values for σ8 from the different references are shown for a fixed
value of Ωm. . . 61
7.1. Summary of the posterior distributions mean values for the different com-binations of data, with the corresponding 67% confidance intervals. . . . 100 7.2. Summary of the posterior distributions mean values for the different sets
of parameters constrained, with the corresponding 67% confidance intervals.106 7.3. Similar to Table 1, but including an evolution parameter for the
super-novae intrinsic luminosity. . . 111 8.1. Table resuming the mean posterior and 68% confidence intervals of the
constrained parameters, comparing the use of the CPL parametrization (1) with the parametrization 8.8 for two different values of Γ: (2)-Γ = 1, (3)-Γ = 0.85. We show for comparison the same results but when considering a constant EoS parameter w. . . 119 8.2. Table resuming the mean posterior and 68% confidence intervals of the
cosntrained parameters, whith different combinations of data and setting w+= −0.2 and w+ = −1. . . 125
8.3. Table resuming the mean posterior and 68% confidence intervals of the cosntrained parameters, whith different combinations of data and setting w+= −0.1 and w− = −1. . . 128
8.4. Table resuming the mean posterior and 68% confidence intervals of the constrained parameters, whith different combinations of data and setting w+= 0 and w− = −1. . . 129
8.5. Table resuming the mean posterior and 68% confidence intervals of the constrained parameters, for the full combination of data (CMB+P(k)+SNIa), considering w− as a free parameter and with three different values of w+ 130
Introduction
The scientific field of Cosmology has made some impressive progress over the last century in the quest for a consistent model that allows to understand the past history of the universe. Through the interchange of many theoretical ideas and observational facts, it has become possible to develop a coherent representation of the world at large scales, both in space and in time. In this representation, the universe is expanding its physical size since about 14000 million years ago. After the cosmic “birth”, known as Big-Bang, the content of the universe had time to cool down from its initial very hot and nearly homogeneous state, allowing for the creation of structures, such as atoms in a first way, followed much later by larger structures such as galaxies and galaxy clusters. At the end of the 1990’s, it was discovered that the expansion of the universe is accelerating, which seems to be caused by an unknown physical component called dark energy. Measuring the physical properties of this component of the Universe has then become one of the main objectives in Cosmology.
In this standard model of cosmology, the exact properties of the universe depend then on a few number of parameters, generally called the cosmological parameters, which describe different aspects such as the initial conditions after the Big-Bang or the en-ergy content of the various components of the Universe. In order to derive the correct value for these parameters, one has to confront our models with different astrophysical observations. In fact, with the technological development in telescopes, detectors, and informatics, we can observe today photons originated from very different sources at very different epochs in the universe history. Some of the most important astrophysical ob-servation that can be used in cosmology are the anisotropies of the Cosmic Microwave Background, Type Ia Supernovae, distribution of galaxies at large scales and galaxy clusters. For different reasons, which will be described in this thesis, each of these ob-servations can provide useful information on some or all of the relevant cosmological parameters. This information can be combined, allowing to increase the precision of our estimates on the cosmological parameters. However, and since we are interested in precision, it is important to study the way in which the modelization of the data and the assumptions made can influence the final results of the constraints. The same is to say that one needs to well study the systematic effects that can be present in a given
cosmological method or observation.
The motivation for this thesis has then a double side. On one hand, we will be interested in taking use of the most recent astronomical data to derive cosmological constraints on diverse parameters and models for the dark energy evolution. Surprisingly enough, most of the work on this subject in current literature seldom uses some type of compressed information, mainly on the baryonic acoustic oscillations seen in the galaxy correlation function from the SDSS LRG sample. We will then focus on using the full data set instead of the compressed information, which can only be done after modeling for non-linear effects and other corrections. On the other hand, we will study how these and other different modelizations, such as self-similarity in galaxy clusters or luminosity evolution in Type Ia supernovae, affect the final results on the cosmological parameters. The outline for this manuscript is as follows. In Chapter 3 we introduce the fundamen-tal equations that govern the dynamics and measurements in the expanding universe, and we discuss some of the proposed models for dark energy. In chapter 4, we present some of the main cosmological observations and methods that can be used to constrain cosmological models. In chapter 5, we present the statistical methods that are needed to estimate cosmological parameters from the large quantity of data available. Chapter 6 is dedicated to the study of the particular cosmological test that uses the gas mass fraction in galaxy clusters, focusing in the systematic uncertainty that it presents. In chapter 7, we derive combined cosmological constraints using three of the main cosmological obser-vations available, the WMAP 5 year data on the CMB anisotropies, the Hubble diagram of type Ia Supernovae and the distribution of Large Red Galaxies. We will show the effect of using either the correlation function or the matter power spectrum to describe the latter, the effect of considering different priors on some cosmological parameters, and constraint the possible evolution for the intrinsic luminosity of SNIa. Finally, in chapter 8, we will constraint different models of dynamical dark energy, involving slow and fast transitions for the DE equation of state parameter w.
Introduction (Français)
Le domaine scientifique de la Cosmologie a effectué un progrès impressionnant au cours du dernier siècle, dans la quête pour un modèle consistant qui permet de comprendre l’histoire passé de l’univers. Au travers de l’interaction de beaucoup d’idées théoriques et faits observationels, il a été possible de développer une représentation cohérente du monde aux très grandes échelles, spatiales et temporelles. Dans cette représentation, l’univers se trouve en expansion depuis environ 14000 millions d’années. Après la “nais-sance” cosmique, connue comme le Big-Bang, le contenu de l’univers a commencé à se refroidir de son état initial très chaud et presque homogène, permettant ainsi la forma-tion de structures, telles que les atomes dans un premier temps, suivies beaucoup plus tard par des grandes structures comme les galaxies et les amas de galaxies. A la fin des années 90, il a été decouvert que l’expansion de l’univers est en train d’accélerer, ce qui semble être causé par une composante physique inconnue, qui a reçue le nom de “énergie noire”. Mesurer les proprietés physiques de cette composante de l’univers est devenu alors un des principaux objectifs en cosmologie.
Dans ce modèle standard, les propriétés exactes de l’univers dependent d’un cer-tain nombre de grandeurs, nommés généralement comme paramètres cosmologiques, qui décrivent différents aspects tels que les conditions initiales après le Big-Bang ou le con-tenu énergetique des différents composants: baryons, matière noire, énegie noire. De façon a éstimer la meilleur valeur pour ces paramètres, on doit forcément confronter les modèles à différentes observations astrophysiques. En fait, avec le développment tech-nologique en télescopes, détecteurs et informatique, on peut aujourd’hui observer des photons émis par beaucoup de sources astrophysiques differentes et à très differentes époques de l’histoire de l’univers. Quelques des plus importantes observations qui peu-vent être utilisées pour la cosmologie sont les anisotropies du fond de radiation cosmique en micro-ondes (CMB), les supernovae de type Ia, la distribuition des galaxies à large échelle et les amas de galaxies. Pour différentes raisons, qui seront décrites dans cette thèse, chaqu’une de ces observations peut contribuer à fournir une information utile sur certains ou tous les paramètres cosmologiques relevants. Cette information peut être combinée statistiquement, permettant ansi d’augmenter la precision de nos estimations sur les paramètres cosmologiques. Cependant, et comme nous sommes interessés en
pré-cision, il est important d’étudier comment la modelisation des données et les hypothèses prises influencent les résultats finaux des contraintes. C’est à dire qu’on doit étudier les effets systématiques qui peuvent être prèsents dans une méthode ou observation cosmologique donnée.
La motivation pour cette thèse à ainsi un double sens. D’abord, on sera interessé par utiliser les plus récentes données pour dériver des contraintes cosmologiques sur divers paramètres et des modèles d’évolution de l’énergie noire. Un peu étonnamment, la plupart des travaux sur ce sujet dans la litérature utilisent quelque type d’information compressé, principalement sur les oscillations acoustiques baryoniques observées dans la fonction de corrélation des galaxies du échantillon SDSS LRG. On va alors se focaliser en utiliser l’ensemble complet des données, au lieu d’information compressée, ce qui de-mande une correcte modélisation des effects non-linéaires et des autres corrections. Dans le même temps, on se propose d’étudier comment ces et d’autres modélisations, comme par exemple, l’autosimilarité dans les amas de galaxies et l’évolution de la luminosité des SNIa, affectent le résultat final sur les paramètres cosmologiques.
Le plan de ce manuscrit est donc le suivant. Dans le chapitre 3, on introduit les équations fondamentales qui controlent la dynamique et les mesures dans un univers en expansion, en discutant aussi quelques modèles proposés pour l’énergie noire. Dans le chapitre 4, on présente quelques unes des principales observations et méthodes qui peuvent être utilisées pour contraindre les modéles cosmologiques. Dans le chapitre 5, on présente les méthodes statistiques dont on a besoin pour extraire l’information des grandes quantités de données disponibles. Le chapitre 6 est dedié à l’étude du test comologique qui utilise le profil attendu pour la fraction de gaz dans les amas de galaxies, et en particulier, étudier les incertitudes systématiques. Dans le chapitre 7, on dérive des contraintes cosmologiques en combinant trois des principales observations disponibles: les données de WMAP5 et ACBAR sur les anisotropies du CMB, le plus grand echantillon de SNIa publié et la distribution des grandes galaxies rouges observées dans le survey SDSS. On y étudie en particulier la difference entre l’utilisation de la fonction de corrélation et le spectre de puissance, l’effet de considérer différentes pri-ors dans certains paramètres cosmologiques et contraindre le possible évolution de la luminosité intrinsèque des SNIa. Finallement, dans le chapitre 8, differents modèles d’évolution dynamique pour l’énergie noire, qui invoquent des transitions rapides ou lentes du paramètre de l’équation d’état w, seront contraints.
Some basics on Modern Cosmology
Science is a great game. It is inspiring and refreshing. The playing field is the universe itself. - Isidor Isaac Rubi -. In this chapter, I present the theoretical and observational basis of modern cosmology, which have permitted to obtain a strong “concordance model”. After this brief review, I discuss the main challenges that come with the existence of the dark energy component and the importance of having good constrains on cosmological parameters.
3.1. Friedmann-Robertson-Walker Universe
The main paradigm in modern cosmology is that the Universe as we know it was born from a cosmic expansion that started nearly 14000 million years ago. In its initial state, right after the Big Bang, the Universe was extremely hot, where matter and radiation formed a primordial fluid in thermal equilibrium. With the expansion of the space-time, this fluid started to cool down, allowing for electrons to be captured by protons and light nuclei to form the first atoms, and the existing radiation to be liberated and free stream through the Universe. The observation of the CMB radiation by Penzias & Wilson in 1964 [8] provided the first observation of this “relic” radiation from the primordial Universe. The small (1 to 105) anisotropies observed in CMB radiation, corresponding to equally small perturbations in the primordial fluid, which are the seeds to the large scale structures we observe in the close Universe, such as galaxies and galaxy clusters, formed through gravitational collapse. The fact that the CMB radiation is, in a very good approximation, homogeneous and isotropic, together with the observations from high redshift galaxy surveys supports the assumption that at large enough scales, the Universe is homogeneous and isotropic. This assumption, known as Cosmological Principle means that the space-time structure of the Universe can be described by the Friedman-Lemaître-Robertson-Walker (FLRW) metric,
ds2 = −dt2+ R(t)2 dr2 1 − kr2 + r 2 dθ2+ r2sin2(θ)dϕ2 (3.1) where t is the cosmic time, r, θ, φ are spatial comoving coordinates and R(t) is the radius of curvature of the universe. This factor which is sort of a “ruler” that defines a standard scale in the Universe, and finally :
K = k
R2 (3.2)
defines the curvature of the Universe, so the metric can also be written: ds2 = −dt2+ a(t)2 dr2 1 − Kr2 + r 2dθ2+ r2sin2(θ)dϕ2 (3.3) in which a(t) = R(t)/R0 is the scale factor with a(t0) = 1.
If k = +1, the Universe is closed, meaning that an observer which travels along a fixed direction will eventually return to the point of departure. The 3D universe is similar to the surface of a sphere: finite, but unbounded. If k = −1, the Universe is open, meaning that its geometry is hyperbolic and is eventually infinite in extent. If k = 0, the 3D Universe can also be infinite in extent but with an euclidean geometry.
Once we have a metric to describe the large scale Universe, we can use it to relate the expansion of the Universe with the observed Hubble law, which states that galaxies separated by a distance d are receding from each other at velocity v = Hd, where H is the Hubble parameter. At small separations, the recessional velocity is given by the Doppler effect,
νemit
νobs
≡ 1 + z ' 1 + v
c, (3.4) which defines the redshift z in terms of the shift in frequency of the spectral lines. For larger distances, and since photons travel along null geodesics, we get directly from the metric that
r =
Z cdt
a(t). (3.5) The comoving distance r is an invariant quantity, while the above integration in time seems to depend on temit to tobs. Actually, photons emitted at later temit are received at
a later time tobs, but these changes in temit and tobs leaves the integral unchanged, since
translated to the frequencies, yielding: νemit νobs ≡ 1 + z = a(tobs) a(temit) . (3.6)
By defining a(tO) = 1, where t0 is present time, we get a simple a(t) = (1 + z)−1.
The redshift is thus a direct measure of the relative scale of the Universe between the time when light was emitted and today and it is one of the most important observable quantities in cosmology.
Let us now return to FLRW metric and the cosmological principle. At the large scales involved, it is presumed that the Einstein field equations can describe all the dynamic effects that control the expansion of space-time. In order to solve them one needs to have the form of the energy-momentum tensor of the Universe as a whole. Following the assumptions of homogeneity and isotropy, the content of the Universe can be seen as a perfect fluid, which leads to:
Tµν = (ρ + p)uµuν + P gµν, (3.7)
where ρ and p represent the fluid’s energy density and pressure, respectively and the fluid’s 4-velocity is the the time-like 4-vector uν = (1/a, 0, 0, 0). The above tensor can then be used together with the FLRW metric in the Einstein field equations, where we consider the most generic form which includes the cosmological constant Λ:
Rµν −
1
2Rgµν+ Λgµν = 8πG
c4 Tµν. (3.8)
The obtained results are the two Friedmann-Lemaître (FL) equations: H2 = 8πG 3 ρ + Λ 3 − kc2 R2 (3.9) 3a¨ a = −4πG ρ + 3p c2 + Λ . (3.10) In these equations G is the gravitational constant and we redefine the Hubble pa-rameter as H = ˙a/a. These two equations encompass all the dynamic features of the Universe, since they can be solved to get a(t). They show that the past and future evolution of the expansion are dependent only on the energy density ρ, pressure p and the curvature k. The first quantities, for all the components that enter in the overall fluid, are related through the equation of state (EoS) p = wρ, where w is called the equation of state parameter and it’s value depends on the type of particle: w = 0 for non relativistic particles (dust and cold dark matter), w = 1/3 for relativistic particles (photons) and w = −1 for a cosmological constant (vacuum energy). I will come back to
the cosmological constant later. Furthermore, and since the different components do not interact with each other, their energy momentum tensor must satisfy the conservation law, Tν
µ;ν = 0, which together with the FL equations translates into a third equation
that controls the evolution of the density of each component: ˙
ρi = −3H(ρi+ pi) (3.11)
This equation can be solved to obtain an explicit dependence of each density on the factor scale:
ρm ∝ a−3 (non-relativistic matter) (3.12)
ρrad ∝ a−4 (radiation) (3.13)
ρΛ= constant (cosmological constant) (3.14)
Physically, the energy density of non-relativistic matter is diluted by the growth of the physical volume in 3D-space, while for radiation an extra factor of a−1 comes in from the redshifting of the particles energy. Thus, and since a is growing due to the expansion, at early enough time the Universe is radiation dominated. One can define the equality time as the time at which the two contributions are equal (ρm = ρrad), after which the
Universe becomes matter dominated. Therefore: aeq a0 = ρrad ρm t0 ≈ 3 × 10−3 (3.15) which translates to redshift (3.6) as:
zeq≈ 3000 (3.16)
The subscript 0 indicates that the quantity is evaluated today. The numerical esti-mate comes from the measurement of the present day radiation density in the form of microwave background: ρrad = 7.94 × 10−34 TCM B 2.734 K 4 g/cm3 (3.17) The matter density of today’s Universe is determined by observations such as CMB, large scale structure and type Ia supernovae as we will see further in this thesis. For the case of a cosmological constant, since its density remains constant, its contribution in the early universe can be neglected when compared to the high matter and radiation densities. However, if Λ 6= 0, the late Universe evolution will be dominated by the cosmological constant term in the Friedmann equations. In that case,
a(t) ∝ e √
Λ
3t , (3.18)
and the expansion will become exponential in time.
From the Friedmann equation (3.9), one can define a critical density that corresponds to a flat Universe (k = 0):
ρc =
3H2 0
8πG , (3.19) where H0 is the present Hubble parameter which is usually written as H0 = 100 h km
s−1 Mpc−1 h being the reduced Hubble parameter which is of order unity. This implies that the critical energy today evaluates to
ρc ≈ 1.88 × 10−29h2 g cm−3 . (3.20)
In terms of the critical density, the first FL equation (3.9) can be rewritten using the normalized density parameters Ωi ≡ ρi/ρc, where the subscript i runs over all possible
energy sources. For matter, radiation, vacuum energy and curvature, these are: Ωm = 8πGρm 3H2 0 Ωrad = 8πGρrad 3H2 0 (3.21) ΩΛ = Λ 3H2 0 Ωk = − k a2 0H02 (3.22) With these definitions, the first FL equation can be re-written as:
H2(a) = H02 ΩΛ+ Ωk a20 a2 + Ωm a30 a3 + Ωrad a40 a4 , (3.23) and therefore, the first FL equation today becomes
ΩΛ+ Ωk+ Ωm+ Ωrad = 1 , (3.24)
also known as the “cosmic sum” rule. The radiation content of the Universe today comes mainly from the CMB, which is a perfect blackbody ration with a temperature of 2.724 K. Using 3.17 we have:
Ωrad = 2.4 × 10−5h−2 , (3.25)
which means that we can totally neglect the contribution of Ωrad to the total energy
density of the Universe today. Moreover, recent measurements from CMB radiation indicate that the Universe is spatially flat to a high degree of precision, which means that Ωk is also negligibly small. In this case, the main contributions to equation 3.24
written in the short form: Ωm+ ΩΛ = 1.
Another important parameter that can be derived from the second Friedmann equation is the deceleration parameter:
q0 ≡ − ¨ aa ˙a2|t0 = Ωm 2 + Ωrad− ΩΛ (3.26) By its own definition, this parameter measures the rate of change in the cosmic expansion velocity and the choice of sign was made so that a positive value of q0 corresponds to
¨
a < 0, meaning that the universe’s expansion is decelerating. A negative value of q0
would mean that the relative velocity of any two points in the universe is increasing with time. The above equation shows that q0 is negative if the universe contains a sufficiently
large dark energy component. This seems to be the case, since recent measures point to q0 ≈ −0.6 and an accelerated expansion if the universe.
3.1.1. Time and Distances in an expanding Universe
We have seen above how we can physically describe our expanding universe. In order to test and constrain these models and its parameters, astronomers need to measure the past evolution of the Universe, which means looking at distant structures and back in time. The redshift z measured directly by observation gives information on the receding velocity of the galaxies but one needs to know how this observable quantity relates to the distance and the time between the light emission and today.
Let’s take the definition of the Hubble parameter, which together with eq. 3.6 yields: H ≡ ˙a a = a0 a(t) d dt a(t) a0 = (1 + z)d dt 1 1 + z = −1 1 + z dz dt (3.27) Inserting eq.3.23 in this equation allows to obtain:
dt dz = − 1 H0 (1 + z)−1 [Ωm(1 + z)3+ Ωk(1 + z)2+ ΩΛ]1/2 , (3.28) where we can use the Friedman equation Ωk = 1 − Ωm − ΩΛ to reduce the number of
free parameters: dt dz = −H −1 0 (1 + z) −1 [(1 + z)2(1 + Ωmz) − z(2 + z)ΩΛ]−1/2, (3.29)
Finally, we can integrate the above equation to obtain a general expression for the look-back time, which is the time difference between the present time t0 and the time
t0− t1 = 1 H0 Z z1 0 (1 + z)−1 [(1 + z)2(1 + Ω mz) − z(2 + z)ΩΛ]1/2 dz (3.30) By setting z1 → ∞, one can get the age of the Universe as a function of the
cosmologi-cal parameters (neglecting the “short” period of some thousand years where the radiation density had an important contribution and the even shorter period of inflation). Fig. 3.1 shows the evolution of the look-back time with the redshift for different combinations of Ωm and ΩΛ.
Let’s now see how we can measure large distances in an Universe that is expanding. In cosmology, there are many ways to specify the distance between two points, because in the expanding Universe, the distances between comoving objects are constantly changing and observers in Earth look back in time as they look far away. Consider then a light source that emits a certain number of photons N γ at a given instant temit from a point
with a coordinate r in the cosmic frame. The proper distance between that source and the earth would be, if it existed at that time, is a(temit)r. Therefore, if the Universe
wouldn’t be expanding and if it were flat, a telescope on Earth with collect surface A would observe a photon flux of NγA/4π(a(temit)r)2. In fact, due to expansion, at the
observation instant tobs, the spherical surface centered at the emission point and passing
through Earth increased to 4π(a(tobs)r)2, and so:
Ndetected =
NγA
4π(a(tobs)r)2
. (3.31) Two other factors must be taken into account. The first one is that the photons are redshifted by a factor 1 + z = a(tobs)/a(temit), which means that the photons energy is
reduced by that factor. The second one is that, if the time interval between emission of flashes was δt, then this will increase by a the same factor of 1 + z for the time interval of the observations. Consequently, the total observed power per unit surface in a telescope, or received flux will be:
F = Lemit 4πa(t0)2r2(1 + z)2 ≡ Lemit 4πd2 L , (3.32) defining in this way the luminosity distance as:
dl =
r Lemit 4πFobs
= a0r(1 + z) . (3.33)
For a photon trajectory, the quantities a, r and t are related by the equation that defines a null radial geodesic in the FLRW metric (dθ = dφ = 0):
dr dt =
√
1 − kr2
Figure 3.1.: Look-back time vs. redshift for different values of the cosmological parame-ters, assuming H0 = 72 km s−1 Mpc−1.
By multiplying the above equation by a0 and using (3.6), we can write:
a0
dr √
1 − kr2 = (1 + z)dt . (3.35)
Using eq. 3.29 to change variables and integrating in both sides, we get: a0 Z r1 0 dr √ 1 − kr2 = Z z1 0 dz H0[(1 + z)2(1 + Ωmz) − z(2 + z)ΩΛ]1/2 , (3.36) where r1 is the radial coordinate of the light source and z1 its observed redshift. The
integration of the left-hand side of this equation yields: arcsin(r1 √ k) √ k if k > 0 r1 if k = 0 arcsinh(r1 √ −k) √ −k if k < 0 ,
and so, by taking the definition Ωk= k/(a20H02) and comparing with eq.3.33, one gets
cosmological parameters: dL(z; H0, Ωm, ΩΛ) = (1 + z) H0p|Ωk| S p|Ωk| Z z 0 F (z0; Ωm, ΩΛ)dz0 , (3.37) with F (z; Ωm, ΩΛ) = [(1 + z)2(1 + Ωmz) − z(2 + z)ΩΛ]−1/2 (3.38)
The function S(x) is defined as sin(x) for Ωk < 0, sinh(x) for Ωk > 0 and S(x) = x for
Ωk = 0. In this case the factorp|Ωk| is removed from (3.37). The above equations show
that we can effectively determine H0,Ωm and ΩΛ by measuring the luminosity distance
for objects at different redshifts. In order to determine the luminosity distance of an object, one needs to know in advance the emitted luminosity. These objects are known as standard candles and type Ia supernovae are the main example of such an object. We will come back to SN Ia further in this thesis.
The luminosity distance dL is not the only distance measure that can be computed
using the observable properties of astrophysical objects. Suppose that instead of a standard candle, you observe a standard ruler. A standard ruler is an object whose proper length l is known. In most cases, it is convenient to choose the standard ruler as an object that is tightly bound together, by gravity or some other influence, and hence is not expanding along with the universe as a whole. Suppose an object with constant proper length l that is aligned perpendicular to the line of sight. An angular interval δθ is measured between the ends of the “ruler”, together with a redshift z for the light emitted from the object. If δθ << 1, and if the length l is known, then we can compute a distance to the standard ruler using the small-angle formula
dA=
l
δθ (3.39)
This function of l and δθ is called the angular-diameter distance and is also related to the expansion history of the universe. Consider now a “standard ruler” object located at the comoving coordinate r. At a time temit, the object emits the light observed on
Earth at t0. The distance ds between the two ends of the object, measured at the time
when the light was emitted, can be found directly from the FLRW metric (3.3), since dr = dφ = 0:
ds = a(temit)rδθ (3.40)
However, for a “standard ruler” whose length l is known, one can set ds = l, which leads to:
l = a(temit)rδθ =
a0rδθ
Thus, the angular diameter distance dA to the object is dA ≡ l δθ = a0r 1 + z . (3.42) By comparing with equation 3.33, we get a simple relation between the angular-diameter distance and the luminosity distance:
dA =
dL
(1 + z)2 . (3.43)
Measuring these two quantities can provide a powerful test to the content of the Universe and dark energy properties. This is the principle behind the use of Supernovae type Ia and baryonic acoustic oscillations to test cosmology as it will be shown bellow in this chapter.
3.2. The mystery of Dark Energy
We have seen above that the FL equations can be written with constant term - the “cos-mological constant”. Historically, this term was introduced by Einstein in order to obtain a static solution to the equations that govern the universe dynamics. The discovery by Hubble that the Universe is expanding eliminated the empirical need for a static model, which led Einstein to renegade the cosmological constant, describing it as “the greatest blunder in my career”. However, the disappearance of the original motivation for intro-ducing the cosmological constant did not change its status as a legitimate addition to the gravitational field equations, or as a parameter to be constrained by observations. The only way to purge Λ from cosmological discussion would be to measure all of the other terms in eq.3.24 to sufficient precision to be able to conclude that the Λ/3 term is negligibly small in comparison, a feat that has not been reached to the present day. In fact, and as we will see below, there is better reason than ever to believe that Λ is actually nonzero, and Einstein may not have blundered after all. So, if Λ is present in the field equations that describe the universe, one has to understand what does this term account for, what is its physical origin. In order to give Λ a physical meaning, one needs to identify some component of the Universe whose energy density ΩΛ remains constant
as the Universe expands or contracts. One of the most logical candidates is the vacuum energy.
3.2.1. Vacuum energy
In classical physics, the idea of a vacuum having energy is nonsense. The vacuum, from the classical viewpoint, contains nothing, and so it has no energy. In quantum physics, however, the vacuum is not a sterile void. The Heisenberg uncertainty principle allows
particle-antiparticle pairs to spontaneously appear and then annihilate in an otherwise empty vacuum. The total energy ∆E and lifetime ∆t of these pairs must satisfy the relation:
∆ E∆t ≤ h; . (3.44) This phenomenon is known as zero point vacuum fluctuations and in a pioneering work, Y.B.Zeldovich [7] was the first to point out that in Minkowski space-time, Lorentz invariance constrains the energy-momentum tensor of zero point vacuum fluctuations to be proportional to the Minkowski metric (Tvac
µν ∝ ηµν). This relation can be generalized
to the case of curved space-time with metric gµν. The principle of general covariance
requires that Tµνvac∝ gµν, which has then the form of a cosmological constant. This
im-plies that in General Relativity, since the gravitational field couples through the Einstein equations with all kinds of energy, the vacuum energy contributes to the total curvature of space-time. Moreover, the pressure exerted by the vacuum fluctuations can be shown to be pvac = −ρvac. The vacuum energy tensor constitutes thus a perfect candidate
to explain the cosmological constant, since they both are mathematically equivalent in terms of their effect in the Universe expansion. Attempts to compute the value of the vacuum energy density led to very large or divergent results. For each mode of a quantum field there is a zero-point energy ~ω/2, so that the total energy density of the quantum vacuum is given by
ρvac= 1 2 X f ields gi Z ∞ 0 √ k2+ m2 d 3k (2π)3 ' X f ields gik4max 16π2 (3.45)
where gi accounts for the number of degrees of freedom of the field (the sign of giis
+ for bosons and - for fermions), and the sum runs all over the possible quantum fields (quarks, leptons, gauge fields, etc). Here kmax is an imposed momentum cutoff,
be-cause the integral diverges quadratically. Taking the cutoff to be at the Plank scale (≈ 1019GeV), where quantum field theory in a classical spacetime metric is expected to
break down, the zero-point energy density exceeds the critical density (ρc) by some 120
orders of magnitude! If we set the cutoff scale at the QCD phase transition, we find ρQCDvac ≈ 10−3GeV4
. Since current observations indicate that the dark energy density is of the same order of magnitude as the critical density, this very large discrepancy is known as the cosmological constant problem. To illustrate the magnitude of the problem, if the energy density contributed by just one quantum field is to be at most the critical density, then the cutoff kmax must be smaller then 0.01 eV, which is well bellow any
energy scale where one could appeal to ignorance of physics beyond.
Although initially investigated for other reasons, supersymmetry (SUSY) turns out to have a significant relevance in the cosmological constant problem, although it only seems to solve it halfway. In a SUSY world, every fermion in the Standard model of particle
physics has an equal mass SUSY bosonic partner, so that fermionic and bosonic contribu-tions for the zero-point energy density would exactly cancel. However, suppersymmetry is yet to be observed in nature: no SUSY particles have been detected in collider ex-periments, which means that supersymmetry must be broken at low energies. If SUSY is spontaneously broken at mass scale M , the imperfect cancellations would generate a finite vacuum energy density ρvac ≈ M4. Since the lower limit for the breaking of SUSY
is currently at M ∼ 1 TeV , one still has a discrepancy of 60 orders of magnitude to current observations. Hence, even in the SUSY frame, on needs an exceptionally bizarre fine tuning on cancellation mechanisms in order to explain the observed discrepancy.
3.2.2. Anthropic solutions
The cosmological consequences of the small (Λ/8πG ∼ 10−43GeV4) cosmological constant observed are rather instructive. Unless the value of Λ lies in a very small window, the universe would be very different to what we are used to. For instance, a negative value of Λ (Λ/8πG < −10−43GeV4) would cause the universe to re-collapse (the effect of Λ would be attractive instead of being repulsive) less than a 1000 million years after the big bang - a period which is too short for galaxies to form and life to emerge. On the other hand, a positive and large Λ/8πG > 10−43GeV4 would cause an accelerated expansion of the universe much before the present epoch, thereby inhibiting structure formation and the emergence of life.
This very narrow window in Λ which allows life to emerge has led some cosmologists to propose anthropic arguments for the existence of a small cosmological constant [9], [10], [11]. One such possibility can be expressed by: “if our big bang is just one of many big bangs, then it is natural that some of this big bangs should have a vacuum energy in the narrow range where galaxies can form, and of course it is just these big bangs that give origin to universes where astronomers and physicists can wonder about vacuum energy”. One can treat this principle in a probabilistic way, but this solution is just partial without an underlying theory that allows for multiple big bangs with different values of Λ. The recent developments in string theory seem to provide a natural framework where these issues can be addressed [12].
3.2.3. Quintessence
A non-anthropic solution to the cosmological constant problem can be solved if a new degree of freedom is introduced, a scalar field φ, which can make the equation of state time-dependent. For a scalar field φ, with a Lagrangian density L = 12∂νφ∂νφ − V (φ),
ρ = ˙ φ2 2 + V (φ) (3.46) p = ˙ φ2 2 − V (φ) (3.47) where φ is assumed to be spatially homogeneous, ˙φ2/2 is the kinetic energy term and
V (φ) is the potential energy. The evolution of the field is governed by the Klein-Gordon equation of motion, ¨ φ + 3H ˙φ + dV dφ = 0, (3.48) with H = 8πG 3 ρm+ ρrad+ ˙ φ2 2 + V φ ! (3.49) The equation of state for quintessence dark energy can be described by the parameter:
w = ˙ φ2/2 − V (φ) ˙ φ2/2 + V (φ) = −1 + ˙φ2/2V 1 + ˙φ2/2V (3.50)
If the scalar field evolves slowly, ˙φ2/2V 1, then w ≈ −1 and the scalar field behaves like a slowly varying vacuum energy, with ρvac(t) ≈ V [φ(t)]. In general, from the previous
equation, w can take on any value between -1 (rolling vary slowly) and +1 (evolving vary rapidly) and varies with time (redshift). Obviously, one is interested in models with w(t0) < −1/3, since those are the ones that can explain the observed accelerated
expansion of the Universe.
Potentials which are sufficiently steep to satisfy Γ ≡ V00V /(V0)2 ≥ 1 have the
in-teresting property that scalar fields evolving in such a potential approach a common evolutionary path from a wide range of initial conditions [23]. In these so-called ’tracker’ models, the kinetic and potential energies initially scale with a constant ratio, and there-fore the equation of state is constant or slowly varying, following (tracking) the value of the dominant background component (matter or radiation). At late times, the field leaves the tracker solution, the potential energy starts dominating over the kinetic one and the equation of state tends to negative values. During this final phase the field becomes the dominant component of the Universe, the evolution of the field is more or less independent of the initial conditions so the fine tuning problem is alleviated and the field drives the accelerated expansion.
There has been a profusion of proposed models for the quintessential potential, which are resumed in table 3.2.3.
Quintessence Potential Reference
V0exp (−λφ) Ratra & Peebles (1988) [13], Wetterich (1988) [14],
Ferreira & Joyce (1998) [15] m2φ2, λφ4 Frieman et al (1995) [16] V0/φα, α > 0 Ratra & Peebles (1988) [13]
V0exp (λφ2)/φα Brax & Martin (1999,2000) [18],[17]
V0(cosh λφ − 1)p Sahni & Wang (2000) [19]
V0sinh−α(λφ) Sahni & Starobinsky (2000) [20], Ureña-López & Matos (2000) [21]
V0(eακφ+ eβκφ) Barreiro, Copeland & Nunes (2000) [22]
V0(exp Mp/φ − 1) Zlatev, Wang & Steinhardt (1999) [23]
V0[(φ − B)α+ A]e−λφ Albrecht & Skordis (2000) [24]
Table 3.1.: Different form of the potentials for Quintessence. Taken from [25]
3.2.4. Modification of gravity
A very different approach to explain the observed accelerated expansion is to consider the existence of new gravitational physics rather than dark energy. The main hypothesis for these theories is that General Relativity is only accurate at small scales and has to be modified at cosmological distances. Assuming that the spacetime geometry can still be fully described by the metric, there are two ways to change the large-scale governing equations: either to modify the Friedmann equations or the equations that govern the growth of the density perturbations into large scale structure. Some variety of ideas has appeared within this context, from models derived from higher dimension theories and string theories [26], [27] to phenomenological modifications of the Einstein-Hilbert Lagrangian of General Relativity [28] .
The standard GR cosmological model agrees well with the abundances of light elements from big bang nucleosynthesis (BBN) and the evolution of the spectrum of primordial
density fluctuations, yielding the observed power spectrum of temperature anisotropies of the CMB. Also, the age of the Universe and the power spectrum of large scale structure agree reasonably well with the standard cosmological model. Therefore, all changes in the Friedmann equations must reduce to the GR form at early times of the Universe and show their effect only at late times. One of the best-studied examples of such models is the Dvali, Gabadadze & Porrati [26] brane-world model, where gravity leaks off the 4-dimensional Minkowski brane into a 5-D bulk Minkowski space-time. The first Friedmann equation for this model is given by:
H2 −H rc
= 8πG
3 ρ , (3.51) where rc is a length scale related to the crossover between the 5-D and 4-D regimes. This
equation reduced to the convectional Friedmann equation at early times, but at late times asymptotes to a brane self-accelerated phase. If one sets the crossover distance scale to be on the order of H0−1, DGP model could account for today’s cosmic acceleration. While attractive, there is no observational way to distinguish between this model and a dark energy component that evolves from w = −1/2 at z 1 to w = −1 at the distant future [29]. Moreover, it is not clear that a consistent model with such a dynamical behavior can exist [30].
![Figure 6.3.: Table taken from [136] with the results of fits and calculations made.](https://thumb-eu.123doks.com/thumbv2/123doknet/2143983.8976/94.892.142.745.446.993/figure-table-taken-results-fits-calculations.webp)
