Introduction and Overview
The extraordinary measurements of the CMB temperature anisotropy by both the COBE and WMAP satellites have elevated the state of the art to an unprecedented height. The era of precision cosmology is here. But if there is one thing we have learned from these extraordinarily precise observations of the light streaming to us from merely 400,000 years after the birth of the universe it is that the more we think we know, the more we realize what we do not know.
The Composition of the Universe
The composition of the universe (from Sean Carroll)
The above pie chart represents the status of the current "concordance" model for the breakdown of the composition of the universe, even though it is fair to say that the origin of its specific form is still unknown (in fact, that is what we are here to talk about). Represented in yellow is the familiar stuff: baryons (the same stuff you and I are made of) protons, neutrons; in red is some other kind of matter that acts like baryons in terms of gravitational attraction, but very differently when it comes to interacting with baryons: dark matter. In the blue is something even more bizarre and even less familiar that scientists (most notably Michael Turner) have come to refer to as the dark energy. So what we have here is the interesting phenomenon that we (or the people who care to even ask the question) do not know what 95% of the universe is made of, and as far as we can tell, it is completely invisible. For the record, though, what is meant by invisible is merely that this strange stuff (the dark matter and dark energy) does not interact with light in the same way that all of the normal day-to-day baryons do.
So what do we do? Well, we do what any curious being would do: we start guessing and then check to see if we guessed correctly.
The CMB
The WMAP CMB sky map (from the WMAP Official website)
The CMB is most likely a familiar topic, but it is worth a quick review of its origin and detection before we jump head-first into its ability to shed light on the gritty details of the universe as a whole.
When the universe was very, very young (and here we mean young with respect to cosmological time scales...i.e. tens and hundreds of thousands of years old compared to its current age of about 14 billion) it was also very, very hot. In fact, it was so hot that the electrons and protons (basically, the stuff you and I are made of with some neutrons thrown into the mix) could not even settle down enough to bind together to form hydrogen or helium or other neutral light elements that require only a few protons, electrons and neutrons, only bare nuclei of these atoms could float around in the cosmos. The reason for this is that there were also photons in this "primordial soup" and since both the density and temperature of this plasma were quite large, every time an electron and nucleus (usually just a proton) had the chance to get together and bind to form an atom of neutral hydrogen, a photon would come and smash into the atom and break it apart. Of course, this is a simplistic view of the state of things in the first few tens and hundreds of thousands of years after the big bang; there must also have been a great deal of dark matter (of which we are still clueless as to its composition) floating around exerting a gravitational pull as well as neutrinos (cousins of the electron) and some complicated kinematics that describe the overall motion of this plasma.
The catch to this story, though, is that we now know that the universe expands, and so this hot plasma does not last forever. As the universe expands, it also cools. This means, of course, that at some point all of those photons that are dissociating hydrogen atoms as soon as they form will fall below some minimum energy and cease to be able to break apart the hydrogen atoms. At this point, which is inaptly named recombination (this is, really, the first such combination to occur...) the photons are not longer energetic enough to smash into anything and so free stream from this surface of last scattering (SLS) to us today.
Once again, there is a catch. Even though the photon-baryon plasma that we described above was a very uniform substance due to the high frequency with which the particles were interacting (and thus in all in equilibrium), it is still a fluid and therefore still subject to all of the properties that any fluid possesses.
CMB Anisotropy
The CMB Power Spectrum (from the WMAP Science Team)
A fluid will move if given the right sort of push, just like almost anything else. Drop a pebble into a very calm pond and we all know what happens: small ripples develop and spread across the water. Keep your eye trained on one spot in the pond and you see an oscillation of the height of the surface of the water around some average height. Now, instead of a ripple on the surface, imagine a sound wave traveling through the water of the pond. Again focusing on one point in the water, what you see is not an oscillation of the height of the water around some average value, but instead you see the oscillation of the density of the water around some average density. This is a sound wave: successive compressions and rarefactions of the local density of the fluid.
Since the photon-baryon plasma operates as a nearly ideal fluid, it will respond to dynamical effects in much the same way as the water in a pond. Any initial fluctuations (or perturbations) in the gravitational field that affects the motions of the particles in the plasma has the ability to induce some variation in the local temperature of the photons and baryons. To see this, recall that we noted that the dark matter that is embedded in the fluid will be affecting the photon-baryon plasma through its gravitational effects. Specifically, the photon-baryon fluid will be slightly compressed when it is near a clump of this dark matter (due to the attractive force of gravity). However, since there is some pressure associated with this fluid (as with any) its pressure begins to rise in response to this compression and eventually becomes strong enough to rebound and expand.
A qualitative picture of the acoustic oscillations in the CMB (from Wayne Hu, astro-ph/0002520)
This situation is very much like the case of two heavy balls placed on a spring: squeeze them together and then let move freely. Due to the repulsive force of the spring, they will initially move away from each other. After a certain distance the spring begins to pull instead of push and it attempts to pull the balls back together just to start the oscillation over again. Qualitatively, the very same process of compression and rarefaction is taking place in the photon-baryon fluid just before recombination. We have set up a "sound" wave in the primordial soup! It is precisely this oscillation in the density of the photon-baryon plasma that leads to the power spectrum that is shown at the beginning of this section.
The importance of this phenomenon will come to bear very soon. In fact, we will see that a generalization of this phenomenon is at the very heart of the effect we wish to discuss, the integrated Sachs-Wolfe effect.
Overview
Now that we have introduced the CMB at a very basic level, we are ready to jump head first into the gritty details of the integrated Sachs-Wolfe effect (ISW), the topic of this tutorial. For more details regarding the origin of these ubiquitous photons from the early universe and the consequences thereof, at varying levels of rigor and difficulty, see Wayne Hu's Physics of Microwave Background Anisotropies, or even his class notes. Our discussion will mimic some approximation of the following progression:
I. Review of the ordinary Sachs-Wolfe (SW) effect, of which the ISW effect is a special case.
II. Introduction of the ISW effect and the mathematics involved in its quantitative predictions.
III. A discussion of the implications of the ISW effect and its direct relationship to the dark energy.
IV. Brief introduction to the proposed methods for detecting he ISW effect and complications which arise.
V. A review of the current status of ISW detection and implications for future experiments and cosmology.
•