Course Physics 122

Analysis and interpretation of cosmological observations

Luca Amendola

Definition of apparent and absolute magnitude. Luminosity distance. Examples of real surveys: LCRS, CfA.

The luminosity function, the selection function, magnitude- and volume-limited samples. Number counts, counts in magnitude. Estimation of the luminosity function.

Probability distribution functions, moments, generating functions,cumulants. Moments of a sum of random variables. Poisson sampling and Poisson noise.

Conditional probabilities. Conditional density. Correlation function (CF). Estimators of the CF. The CF of a planar and a cluster distribution.

The power spectrum (PS). The Wiener-Khintchin theorem. Estimators of the power spectrum: the window function, the shot-noise correction. Finite-size effects.

Higher order CF. The angular correlation function. The hierarchical hypothesis.

Biasing. The CF of threshold regions. Observational results on the CF, the PS, and the higher-order moments. Normalization of the power spectrum.

From the PS to the moments. Variance of the moments.

Homogeneity and fractality. Definition of fractal. The CF and PS of a fractal. Comparison to real data.

Bayes' theorem. The likelihood function method. An application: cell variance , CF.

Review of the standard Friedmann universe with a scalar field component. Energy-momentum tensor, four-velocity, equivalence. Matter- and radiation-dominated eras. Attractors in an exponential potential.

Gauge choice. Perturbed quantities. Scalar, vector and tensor perturbations. The synchronous and Newtonian gauge. The Fourier transform of the perturbations.

The Newtonian limit. The peculiar velocity field. The Jeans scale.

Introduction to Mathematica. A Mathematica algebraic code for tensor manipulation.

The full set of equations in synchronous gauge for matter, radiation and a scalar field. Adiabatic initial conditions. Behavior at short and long scales in the various cosmological epochs.

The transfer function, the CDM spectrum, normalization.

The velocity field in linear perturbation theory. The redshift correction at small and large scales. The non-linear correction.

Anisotropies in the CMB. The Sachs-Wolfe(SW) effect, the integrated Sachs-Wolfe effect, the acoustic oscillations, the adiabatic perturbations, the last scattering surface thickness, the Sunyiaev-Zeldovich effect..

Expansion in spherical harmonics. The multipole spectrum. The analytic SW multipole spectrum for a general power- law. The CMB normalization of the power spectrum.

The program CMBFAST.