The Cosmic Microwave Background (CMB) is a relic of the universe when it was only 380,000 years old. Standard lore has it that the photons in the CMB have traveled freely to us since that time, so constitute an unvarnished picture – literally a picture – of the early universe.We have already seen some of the power of this picture, in that the constraints on cosmological parameters from the CMB, as shown for example in Fig. 8.16, are extraordinarily tight.

The purpose of this chapter is to remove one word from the preceding paragraph: “unvarnished.” In fact, the photons in the CMB do not travel completely freely to us; rather, their paths are distorted by the gravitational potentials along the line of sight. On the simplest level this means that CMB photons observed as emanating from a given direction actually started their 13-billion-year journey from a slightly different direction.We have seen this through the book and learned to ask: how can the deflections be observed?

Understanding the effect of CMB lensing, as these deflections are called, is not as simple as understanding how a point source is magnified or a galaxy is stretched. Rather, it is the statistics of the CMB that are distorted. So we have to rely on our growing understanding of the spectrum of fields like the CMB and observe the small distortions in the spectrum in order to infer information about the projected gravitational potential.

The Cosmic Microwave Background

Photons characterized by a blackbody distribution with temperature T = 2.725K permeate the universe. The peak of the intensity of a blackbody boson gas is at ν = 3kBT/h, where kB is the Boltzmann constant and h Planck's constant. In this case the peak is at 160 GHz, in the microwave region of the electromagnetic spectrum. So there is a microwave background. This radiation is a relic of the early universe and conveys information about the state of the universe at early times. A photon with a frequency today of 160 GHz would have had a larger frequency (shorter wavelength) earlier in the history of the universe, when the scale factor that dictates physical lengths was smaller.

Until now we have been thinking of the source and lens as being located at fixed positions, so that the effects – e.g., multiple images and magnifications – will be fixed in time. There are some situations, though, where this is not true, where the relative positions of the lenses and sources change with time. In those cases, the magnifications, for example, will also change with time. This is potentially observable: if a source is detected with a given flux at one time and a different flux at another, the differences might be caused by the lens moving and creating more magnification at different times.

The name for this phenomenon is microlensing because it was first coined in observations of distant QSOs whose light was lensed by an intervening galaxy. If the light rays passed close enough to a single star in the lensing galaxy, then the image was perturbed in a time-dependent fashion that will occupy us in this chapter.We can estimate the deflection angle for this type of event: 4MG/DLc 2 for a solar mass star a cosmological distance of 1 Gpc from us is 3×10−6 arcseconds. The deflection angles are very small; hence the moniker, microlensing. However, the name is often used to describe any time-dependent change in flux even if the deflections are not at the 10-6 arcsecond level.

Fig. 5.1 gives a generic example, with the background source traversing through the Einstein radius of the lens. Of course, the motion is relative, so this is equivalent to the more likely case where the lens is moving and the source remains fixed. In either case, the magnification of the source will reach a maximum value when the lens and source are closest to one another, an angular distance β min away. An observer will detect a rise and then fall of the flux from the object.

In this simple example of a point mass lens, the magnification as a function of the relative position of the two objects is shown in Fig. 5.2 for two different values of β min. This characteristic signature of brightening and then dimming offers an excellent prospect for detecting a object moving between us and a background source. Consider as an example a solar mass lens in our galaxy a distance of 5 kpc from us, so that its Einstein radius is of order 10−3 arcsecond.