In this chapter we discuss the most important second-order effect on CMB anisotropies and polarization. Patches of higher or lower CMB temperature are modified and polarization patterns are distorted when they propagate through an inhomogeneous gravitational field. The content of this chapter is strongly inspired by the excellent review by Lewis & Challinor (2006) on the subject.

An introduction to lensing

On their path from the last scattering surface into our antennas, the CMB photons are deflected by the perturbed gravitational field. If the CMB were perfectly isotropic, the net effect of this deflection would vanish, since, by the conservation of photon number, as many photons would be deflected out of a small solid angle as into it. On the other hand, if there is no perturbation in the gravitational field, the latter is perfectly isotropic and the effect also vanishes. Hence, gravitational lensing of the CMB is a second-order effect and we have not discussed it within linear perturbation theory.

To estimate the effect let us consider the CMB temperature in a point n in the sky, T(n). If the direction n is deflected by a small angle α, we receive the temperature T(n) from the direction n′ = n + α. Note that, since α is a vector normal to n also n′ is a unit vector to first order in α.

So far we have only studied the evolution of perturbations assuming that the initial conditions are fixed and given once and for all. Now we want to study how classical perturbations are generated out of quantum fluctuations during a simple inflationary phase. The fact that inflation generates a nearly scale-invariant spectrum of scalar perturbations in good agreement with the observations of the cosmic microwave background is to be considered as its greatest success. The solution of the flatness and entropy problems with an inflationary phase are actually ‘post-dictions’ while the scale-invariant spectrum of scalar perturbations was first predicted in Mukhanov & Chibishov (1982) and Mukhanov et al. (1992), long before its discovery by the COBE satellite by Smoot et al. (1992). It represents therefore a real prediction of inflation. There are also other models for structure formation which predict a scaleinvariant spectrum of fluctuations but which disagree with the detailed spectrum of CMB fluctuations such as topological defects (Durrer et al., 2002).

In this chapter we first study perturbations in a FL universe filled with a scalar field. Next we discuss the generation of fluctuations during inflation. We especially determine the spectral index of scalar and tensor perturbations and the ratio of their amplitudes in the slow roll approximation. This will lead us to the well known consistency relation for slow roll inflation. We study in detail the simple case of one scalar field, the ‘inflaton’.

Cosmology, the quest concerning the Universe as a whole, has been a primary interest of human study since the beginnings of mankind. For a long time our ideas about the Universe were dominated by religious beliefs – tales of creation. Only since the advent of general relativity in 1915 have we had a scientific theory at hand that might be capable of describing the Universe. Soon after Einstein's first attempt of a static universe, Hubble and collaborators (Hubble, 1929) discovered that the observable Universe is expanding. This together with the discovery of the cosmic microwave background (CMB) by Penzias and Wilson (Nobel prize 1978) has established the theory of an expanding and cooling universe which started in a ‘big bang’.

For a long time observations that have led to the determination of cosmological parameters, such as the rate of expansion, the so-called Hubble parameter, the mean matter density of the Universe, or its curvature, have been very sparse and we could only determine the order of magnitude of these parameters.

During the last decade this situation has changed significantly and cosmology has entered an era of precision measurements. This major breakthrough is to a large extent due to precise measurement and analysis of the CMB. In this book I develop the theory which is used to analyse and understand measurements of the CMB, especially of its anisotropies and polarization, but also its frequency spectrum.

We conducted an optical survey (Keck Telescope, 3,700–7,000 Å) of 24 high-metallicity (Z) starburst galaxies to investigate whether high-Zenvironments favor the formation of Wolf–Rayet (WR) stars. We searched for the presence of the He II 4686 Å line produced by the massive WR stars. We detected this feature in six galaxies (25% of the sample). We also used a stellar-population-synthesis code to determine their ages. We find that (i) all galaxies hosting considerable numbers of WR stars are very young systems, with ages log(t) > 8, with t in years; (ii) not all young star-forming galaxies host WR stars, or at least that population cannot be detected in their integrated spectra; and (iii) most galaxies hosting WR populations are found in interacting systems. We for the first time detect WR populations in galaxies ESO 485-G003, NGC 6090, and NGC 2798.

Reports of high metallicities in galactic systems have always been controversial. I disuss whether observational claims both for nebulae and for stars are well-founded, and try to form a rational view of just how metal-rich some regions of galaxies do become. Metallicity is linked to the evolution of star formation in a galaxy through the yield, the mass of metals produced each time star formation locks up unit mass of interstellar material. The mechanisms by which real or apparent high yields might be achieved are examined – global and local gas flows, poor mixing, star formation and metallicity effects in stellar evolution. As perhaps expected, it turns out to be not so easy to ‘get rich’, quickly or otherwise – suggesting that sorting out the lingering uncertainties in the abundance analysis of H ii regions and stars remains a priority.

We review some of the models of chemical evolution of ellipticals and bulges of spirals. In particular, we focus on the star-formation histories of ellipticals and their influence on chemical properties such as [α/Fe] versus [Fe/H], galactic mass and visual magnitudes. By comparing models with observational properties, we can constrain the timescales for the formation of these galaxies. The observational properties of stellar populations suggest that the more-massive ellipticals formed on a shorter timescale than less-massive ones, in the sense that both the star-formation rate and the mass-assembly rate, strictly linked properties, were greater for the most-massive objects. Observational properties of true bulges seem to suggest that they are very similar to ellipticals and that they formed on a very short timescale: for the bulge of the Milky Way we suggest a timescale of 0.1 Gyr. This leads us to conclude that the bulge evolved in a quite independent way from the Galactic disk.

We present results for the chemical evolution of the Galactic bulge in the context of an inside-out formation model of the Galaxy. A supernovadriven wind was also included in analogy with elliptical galaxies. New observations of chemical-abundance ratios and the metallicity distribution have been employed in order to check the model results. We confirm previous findings that the bulge formed on a very short timescale with quite a high star-formation efficiency and an initial mass function more skewed towards high masses than the one suitable for the Solar neighbourhood. A certain amount of primary nitrogen from massive stars might be required in order to reproduce the nitrogen data at low and intermediate metallicities.