Cosmology By the end of this topic you should be able to: -describe Olbers’ paradox in Newtonian cosmology and how it is resolved; --describe the main features of the Big Bang and the expansion of the universe; --understand the significance of the cosmic background radiation; --state the meaning of the terms open universe and closed universe; --outline the theoretical possibliities for the evolution of the universe; --state the meaning and significance of the term critical density; --appreciate the importance of various forms of dark matter.

Cosmological principle The Universe appears to be full of structures. But if we look at it on avery large scale, we no longer see any. If we imagine cutting up the Universe into cubes of side 300 Mpc across, the interior of any one of these cubes would look the same as the interior of any other cube, anywhere else in the universe. This is called the homogeneity principle. On a large scale, the Universe looks uniform, i.e. the matter density of our local region, or lets say the amount of galaxies, stars, gas and dust per a certain volume is pretty much the same anywhere in the universe.

Cosmological principle Similarly, if we look in different directions, we see essentially the same thing. No direction is special in comparison with another. This is called the isotropy principle. These two principles, homogeneity and isotropy, make up what is called the cosmological principle. The cosmological principle implies that the Universe has no edge (fot if it did, the part of the Universe near the edge would look different from a part far from the edge, violating the homogeneity principle). Similarly, it implies that the Universe has no centre (for if it did, observing from the centre would show a different picture from observing from any other point, violating the principle of isotropy).

Newton’s universe Using an extreme version of the cosmological principle, Newton suggested that the Universe is infinite in extent, han no beginning and is static, meaning it has been uniform and isotropic at all times. However serious theoretical problems were posed for this model.

Olbers’ paradox Why is the night sky dark? Imagine a Universe that is infinite and contains an infinite number of stars more or less uniformly distributed in space. The very distant stars contribute very little light to an observer on Earth but there are vary many of them. Mathematically, let n stand for the number density of stars, that is the number of stars per unit volume of space. At a distance d from a star of luminosity L, the apparent brightness is The number of stars in a thin shell of thickness t a distance d from the observer is number density x volume. Hence the received energy per area per unit of time from all the stars in the thin shell is

Olbers’ paradox This is a constant (i.e., it does not depend on the distance to the shell d). Since there is an infinite number of such shells surrounding the observer, and since each contributes a constant amount of energy, the total energy received must be infinite, making the night sky infinitely bright, which it is not. Olbers’ paradox cannot be eliminated in Newton’s universe. The obvious way to try to solve the puzzle is to invoke absorption of the radiation from the intervening stars and the interstellar medium. This does not work, however, because in an eternal universe the interstellar medium would, in time, be heated up by the radiation it absorbed and would then itself radiate as much energy as it received, leading to the same difficulty.

Olbers’ paradox In a finite, expanding Universe, however, the radiation received by the observer is small and finite for two main reasons: 1.There is a finite number of stars and each has a finite lifetime. This means that stars have not been radiating forever, nor will the go on radiating forever. Their total radiation is thus small and finite. 2.Because of the finite age of the Universe, stars that are far away (beyond the ‘event horizon’) have not yet had time for their light to reach us. An additional reason that helps resolve Olbers’ paradox is the following: 3. The radiation received is redshifted and so contains less energy.

Hubble’s Law The dark lines in the absorption spectra of distant galaxies correspond to wavelengths that have been absorbed by the chemical elements in the outer layers of the galaxies. The positions of the dark lines are well known from experiments on Earth but the observed wavelengths from the galaxies, when compared to those measured on Earth, were found to be a bit longer; they were redshifted.

Hubble’s Law Hubble interpreted the redshift of the spectral lines as evidence of a velocity of a galaxy away from us. The faster the galaxy, the larger the redshift. Hubble’s observations thus suggest an expanding Universe with galaxies moving away from us and from each other. It also suggests that in the past the Universe was much smaller. The Universe appears to have started from a kind of explosion that set matter moving outward. It is important to realice that the universe is not expanding into empty space. The galaxies that are moving apart from us are not moving into another, previously unoccupied, part of the Universe. Space is being created in between the galaxies.

The cosmic background radiation In 1964, Penzias and Wilson, two radioastronomers working at Bell Laboratories, made a fundamental, if accidental, discovery. They used an antenna they had just designed to study radio signals from our galaxy. But the antenna was picking up a signal that persisted no matter what part of the sky the antenna was pointing at. This signal turn to have a black-body spectrum corresponding to a temperature of 2.7 K. The isotropy of this radiation indicated that it was not coming from any particular spot in the sky; rather it was radiation that was filling all space. This kind of radiation had been predicted on the basis of the Big Bang theory 30 years earlier by George Gamow. Penzias and Wilson realized that the radiation detected was the remnant of the hot explosion at the beginning of time. It was the afterglow of the enormous temperature that existed in the very early universe. As the universe has expanded, the temperature has kept falling to reach its present value of 2.7 K.

Big bang theory The discovery of the expanding universe by Hubble implies a definite beginning of the universe, some 14 billion years ago. The size of the universe at that time was infinitesimally small and the temperature and pressure enormous. These conditions create the picture of a gigantic explosion at t=0, which set matter moving outwards. Billions of years later we see the remnant of this explosion in the receding motion of the distant galaxies. It is important to understand that the Big Bang was not an explosion that took place at a specific time in the past somewhere in the universe. At the time of the Big Bang the space in which the matter of the universe resides was created as well. Thus, the Big Bang happened about 14 billion years ago everywhere in the universe (the universe then being a point).

Big bang theory The main experimental evidence in support of the Big Bang theory: The expansion of the universe – The universe is now observed to expand. Hence in the past the universe had a smaller size. This points to a picture of an ‘explosion’ that set the universe moving outward. The cosmic background radiation – Today we observe the background radiation at 2.7K. This is conisistent with a small, hot universe in the distant past, which began to cool down as it expanded. Helium abundance – It is a prediction of the Big Bang model that there should be an abundance of helium in the universe, of about 25% by mass. Measurements of helium abundance today give a number that is never less than 25%. It is very difficult to account for this lower bound on helium in such different measurements if we do not accept the cosmological explanation of helium formation.

The development of the Universe Mathematically, the expansion of the universe can be described in terms of a scale factor of the universe in the following way. If the distance between two galaxies was x 0 at some arbitrary time, then the separation of these two galaxies at some time t later is given by the expression x(t) = R(t) x 0. The function R(t) is called the scale factor of the universe and is of basic importance to cosmology. It is sometimes referred to loosely as the radius of the universe. Note that this is a scalar function, not a vector, indicating the standard assumption about the isotropy and homogeneity of the universe on a large scale.

The development of the Universe It is a basic problem in cosmology to discover what this scale factor R(t) is. Application of the laws of general relativity results in three possibilities for R(t). The first possibility is that R(t) starts from zero, increases to a maximum value and then decreases back to zero again. The universe collapses after an initial period of expansion. This is called a closed universe.

The develpment of the Universe In the second possibility, the scale factor R(t) increases without limit – the universe continues to expand forever. This is called an open universe. The third possibility is that the universe does expand forever, but the rate of expansion decreases. This is called a flat universe.

The development of the Universe The three solutions of Einstein’s equations for the evolution of the universe. The present time is indicated by “now”. Notice that, depending on which solution is taken, the age of the universe is different. In other words, different solutions imply a different age. Which of the three possibilities is actually realized depends on the value of the mass density of the universe, ρ, relative to a critical density whose value is about ρ c ≈ kg/m 3.

The developtment of the Universe If ρ < ρ c, the universe expands forever at a slowing rate. The universe is called open. If ρ = ρ c, the universe expands forever at a slowing rate that approaches zero. The universe is called flat. If ρ > ρ c, the universe collapses after a period of expansion. The universe is called closed. General relativity actually gives an additional interpretation to the three different scenarios for R(t). General relativity says that the geometry of the universe (i.e. the rules of geometry) depend on the amount of mass in the universe. The mass in the universe bends or curves the space and time in the universe. The amount of bending depends on how much mass there is. The case ρ ρ c corresponds to a closed universe, with a finite volume and a curvature similar to that of a sphere. Finally, ρ = ρ c corresponds to an open, infinite, but flat universe, analogous to the surface of an ordinary plane.

Dark matter To measure the mass density of the universe means measuring the mass of galaxies within a large volume of space and dividing that mass by the volume. There is an immediate problem in all of this in that we know there exists “dark matter”, matter that we cannot see. The determination of the density of the universe is difficult.

Dark matter Dark matter could be in the form of brown dwarfs and other similar cold objects, but the existence of more exotic possibilities is also hypothesized. MACHOS (massive compact halo objects), for example black and brown dwarfs. WIMPs (weakly interacting massive particles), for example neutrinos (almost excluded), neutralinos, gravitinos, axions, axinos, etc.

Dark energy The discussion before is based on the standard Big Bang model of the universe and now is outdated. Since 1998 it has been known that distant supernovas are moving away from us at much faster speeds than those expected based on the standard Big Bang model. Based on gravitation alone, we would expect a deceleration in the speed of recession of distant objects. The data says, instead, that the speed is increasing. What is causing this acceleration?

Dark energy It appears that the universe is filled with a kind of all-permeating vacuum energy called dark energy. The presence of this “energy” creates a kind of repulsive force that not only counteracts the effect of gravity on a large scale, but actually dominates it, causing acceleration in distant objects rather than the expected deceleration. The domination of the effects of dark energy over gravity appears to have started about 5 billion years ago. It thus appears that, even though the present density of the universe is now believed to equate the critical density, the universe follows the pattern shown in the figure. There is now convincing evidence that ρ = ρ c based on detailed radiation undertaken by the Wilkinson Microwave Anisotropy Probe (WMAP)

Mass-energy composition The mass-energy density of the universe is believed to be made out of approximately 73% dark energy and only 27% matter. And of this matter in the universe, 85% is estimated to be dark matter (i.e. 85% of the 27% matter), leaving a miniscle fraction (15% of the 27% matter) about 4% accounted for by ordinary matter.

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