Tuesday April 22
The study of Cosmology is the study of the Universe as a whole. This means there is a fundamental problem with the subject: We have only one Universe to observe. Thus, cosmology differs from any other scientific pursuit, in that we do not have, and cannot have, an ensemble of objects to study.
Because of this, we have to be very careful about distinguishing between the things we observe and the things we assume.
The Cosmological Principle states that the Universe is homogeneous and isotropic on large scales. Thus, it is a restatement of the first two points above.
The Copernican Principle states that we do not live in a preferred location of the Universe. And, more strongly, there *IS* no preferred location in the Universe. Thus, it combines the assumptions of homogeneity and universality.
Newton realized that the only way for a Universe ruled by his theory of gravity to be stable would be for it to be infinite. Otherwise it would tend to collapse on itself.
Edmund Halley pointed out that this would lead to a paradox, now known as Olbers's paradox (but first pointed out centuries earlier): An infinite Universe, randomly populated by stars, should have a sky that is everywhere as bright as the surface of a star. This follows because, in an infinite Universe, all sightlines will eventually reach a star.
The modern resolution of this paradox was apparently first suggested by Edgar Allen Poe: If the speed of light is finite, and the age of the Universe is finite, then the sky can be dark even in a spatially infinite Universe.
Slipher (1914) first pointed out that most of the "spiral nebulae" had substantial redshifts. Hubble (1922) determined the extragalactic nature of these nebulae. By the end of the decade he had estimated enough distances to realize that distance and velocity appeared to be linearly related:
The implication of this was that the Universe is expanding. This turns out to be a natural consequence of Einstein's General Relativistic Field equations. But the determination of the proper value Ho has been a matter of some long controversy.
In the last decade, the range of estimates has narrowed from a factor of two uncertainty to somewhere in the range 65-75 km/sec/Mpc. The struggle to determine this parameter was so long not only because distances are hard to measure, but because galaxies have substantial peculiar velocities as well as cosmological velocities.
Now, the Hubble constant is a measure of the current expansion rate of the Universe, and has dimension of inverse time. Thus we define the Hubble Time as the inverse of the Hubble constant. The adopted value of Ho above yields a Hubble time of 14 +/- 2 Gyr.
This should not be interpreted as "The Age of the Universe". It is a useful measure of the expansion timescale though. And in a Universe that is dominated by gravity, the Hubble time is a reasonable approximation to the Age of the Universe. We shall return to this point.
First, realize that if there is any matter in the Universe at all, the Hubble "constant" is actually time-variable. Gravity will slow the expansion, thus a basic cosmology will have a larger Hubble parameter at earlier times.
One can make a general energetics analysis with the above consideration to derive a result of the form
Here, k is a curvature term. If it is zero, the curvature is flat. If it is positive, the curvature is positive, and the Universe is closed. If it is negative, the curvature is negative, and the Universe is open. The omega-bar character represents a Co-Moving Distance:
Here r(t) represents a co-ordinate distance, and R(t) represents the scale factor. Thus the Co-Moving Distance does not change with the expansion of the Universe. By convention R(t0)=1. That is, we reference the scale at any past or future time to the scale now.
I derived or outlined a great deal of stuff in class. As an example, one can use standard Newtonian physics and the definiton of the scale factor to express the Hubble parameter as follows:
Here I summarize several of the other key results:
From this definition, the Universe if flat if Omega=1, open if Omega < 1, and closed if Omega > 1. Observations that attempt to determine the current baryon density of the Universe lead us to conclude that the baryon content of the Universe is insufficient to close the Universe by about two orders of magnitude:
One can use the relations above to examine how R(t), the scale factor, evolves in time. For a flat Universe, we have the following:
With k=0, and the density equal to the critical density, this gives us
One can integrate the above to derive the following result:
Similar analysis can be done for redshift, and the age of the Universe. For the age of the Universe, in the flat case:
And for the redshift we have
The open and closed cases are more complex, but behave as they must: In a closed Universe, the behavior is cyclic:
And in an open Universe R diverges with t:
For a flat Universe, the current age is 2/3 of the Hubble time. For a closed Universe, the age is less than this. For an open Universe it is greater than 2/3 of the Hubble time, but still less than the Hubble time. Only for the case of an empty Universe does the Hubble time equal the actual age of the Universe.
First, realize that if there is any matter in the Universe at all, the Hubble "constant" is actually time-variable. Gravity will slow the expansion, thus a basic cosmology will have a larger Hubble parameter at earlier times.
One can take this idea, and couch it in terms of energetics: We observe the Universe to be expanding. Gravity will work to slow down this expansion. If the total energy (kinetic plus gravitational) is negative, the expansion will eventually halt, and the Universe will recollapse. If the energy is positive, the Universe will expand forever. An examination of the limiting case, with total energy equal to 0, leads to a determination of the critical density:
If the actual density is larger than the critical density, the Universe will recollapse. If it is less than the critical density, the Universe will expand forever.
One can derive a relation between redshift and the density function as follows:
Notice that as z goes to infinity, Omega goes to 1. So the early Universe was very nearly flat, no matter what its curvature.
One final useful parameter is the deceleration parameter that is generally defined as follows:
In simple cosmologies, this reduces to the following
For a flat Universe q=0.5 (>0.5 for closed and <0.5 for open Universes).
Thursday April 24
The presence of matter causes spacetime to curve. This is akin to saying that the global curvature of the Universe is determined by the average mass density. Thus to study the properties of the Universe, we need to use the language of General Relativity. The metric to describe the geometry of the Universe is the Robertson-Walker Metric:
From this, one can derive the Friedmann equation relating the scale factor with the density:
Note that this is identical to the Newtonian result, with the proviso that the density now represents the sum of the matter and radiation densities. Note, also that the Friedmann equation does not allow for a static solution. The philosophical dominance of the notion of a static Universe that came down to us from the Greeks led Einstein to add a term to the Friedmann equation to allow for a static solution:
This Cosmological Constant acts as a negative pressure term to balance the attraction of gravity. Although Einstein later called the cosmological constant the worst mistake of his career, there is now substantial observational evidence for the existence of the cosmological constant. This has the result of changing the relationship between the density and deceleration parameters:
We will discuss this further next week. But for the rest of today, Lambda will be assumed to be zero.
That the early Universe was hot was realized in the 1930s, shortly after Hubble's demonstration of the expanding Universe. The first rigorous physical treatment of the early Universe was done by Gamow and collaborators in the mid 1940s. Gamow realized that the early Universe should have been hot and dense enough for nucleosynthesis to occur.
Gamow and collaborators further argued that the current temperature of the Universe should be a few Kelvin. This argument was based on the following pieces:
In a bit more detail, one take the energy-density form of the Planck law, and includes the scale-factor terms as follows:
Although the existence of the CMB was predicted in 1948, it was not searched for at the time. The technology to do so did not yet exist. By the mid-1960s, the prediction had been largely forgotten.
But the technology did exist then. And in 1965 the CMB was accidentally discovered by Penzias & Wilson in the course of work they were doing on satellite communication systems. They detected a weak, isotropic signal with a blackbody spectrum at a temperature of about 2.7 K. The modern value (from the COBE mission) is 2.726 +/- 0.005 K.
Let's consider some of the cosmological implications of the smooth CMB. One can write down the energy density of a black body as a function of temperature. Taking into account the relation between temperature and the scale factor, the equivalent mass density then looks like this:
But the matter density just goes as 1/R^3. Now, even if we ignore dark matter, the current mass density of matter in the Universe is orders of magnitude larger than the current equivalent mass density of the CMB photons. In other words, we live in a Matter Dominated Universe. But the steeper dependence of the radiation density on the scale factor means that at early enough times, the radiation density must have been greater than the matter density. In other words, at early times, there was a Radiation Dominated Universe.
For a flat Universe, with h=0.7, this occurs a few thousand years after the Big Bang.
Another crucial transition that happens somewhat later is the actual mechanism that produces the photons we detect in the CMB. At early times, the Universe is hot enough to be a plasma. As such, it is optically thick to photons. That is the radiation and the matter are coupled. When the temperature drops to about 3000 K, the electons and protons combine to form neutral hydrogen atoms, and the opacity of the Universe drops. The Universe becomes optically thin at this time, and the radiation and matter decouple from one another. The photons we detect in the CMB now were emitted at this epoch of decoupling (also often called the "epoch of recombination").
What we can observe is quite limited. We can observe positions, fluxes, and redshifts. So we have to put the formalism of cosmology into the framework of our observational restrictions.
The Proper Distance between any two objects follows from the Robertson-Walker metric. For a flat Universe the proper distance is equal to the co-ordinate distance. For a closed Universe it is greater than the co-ordinate distance, and for an open Universe it is less.
Considering the closed Universe case for a moment, notice that there is a maximum possible co-moving distance. This is analogous to the distance half-way around a great circle on a sphere. If one goes beyond that point, one is getting closer to the starting point again. The analogy breaks down, as the Universe is not static. A photon starting out at the big bang will travel one full circuit of a closed Universe in the amount of time it takes the Universe to reach its maximum scale, and then recollapse to zero scale.
This brings up a subtle, but important point: Given a finite age for the Universe, we can only see a finite volume of the Universe, even if the Universe is infinite. Furthermore, the co-moving volume that we can see must increase with time. This follows from examining the R-W metric in the case of a photon travelling to the origin (us) for some amount of time. The exact result one reaches depends on the choice of cosmological parameters. But the general result is that for a Universe with any amount of matter (Omega greater than 0), the scale factor expands more slowly than the speed of light. For a flat Universe, R goes as t^2/3. For Omega = 0, R goes as t^1. Thus the causally connected co-moving volume increases with time.
Examination of the Robertson-Walker metric shows that the Newtonian result for the redshift holds under general relativity:
One can also derive relationships between comoving and proper distances and redshift. For the flat Universe one has:
Note that as z->0, this recovers Hubble's law. And as z-> infinity, it recovers the horizon scale. In fact, one can derive similar relations for any cosmology, and recover both the Hubble law and horizon scale in the limits of small and large z. One can also derive a totally general relation between co-moving distance, redshift, and the deceleration parameter. The following holds even in cosmologies with a cosmological constant:
Tuesday April 29
A source of some known intrinsic luminosity L emits photons. We detect a flux F. Thus the Luminosity Distance is defined as
Now, if we say the source is at the origin, and we are at some comoving distance from the source, we can express the flux we observe as follows:
Where the two factors of (1+z) come from 1) redshift and 2) time dilation. This means we can express the luminosity distance as
This is what we actually measure. We cannot measure co-ordinate or co-moving distances directly.
This can be re-cast into an expression involving Ho and qo (from the relations for the comoving distance) as follows:
This form holds even in the case of a non-zero Cosmological Constant. Note that the first term is just the Hubble Law. The higher order terms indicate the curvature away from a linear Hubble Law for large redshift. This means that we can use the observed relationship between dl and z to test the value of qo.
The best modern method to make this test is observations of SNIa's at redshifts of z~0.5 to 1. Such observations are founded on the assumption that SNIa's are true standard candles. That is that they do not suffer from luminosity evolution. This appears to be reasonable, but is far from clearly proven. Many people are studying the problem because of the result of supernova test for cosmology. The implication of the data is that the expansion rate of the Universe is accelerating. That is that the Cosmological Constant is non-zero, and acts as a repulsive term in the large-scale dynamics of the Universe.
Another classic observational test of cosomolgy is the Angular Size Test. If one has a population of sources of a fixed (and known!) proper linear diameter, one can evaluate how the angular diameter will change as a function of redshift. Beginning with the Robertson-Walker metric, one arrives at the following general statement:
Recalling the discussion of proper length from above, one can express the angular size as a function of redshift in terms of the cosmological parameters as follows:
Note that this function has a minimum at some value of z that is determined by the value of the deceleration parameter. Thus one can, in principle, use this test to measure the deceleration parameter. The problem has always been in defining a sample of objects that have a standard linear size that does not change with redshift. The best results to date have used the sizes of powerful radio sources, but the systematics have always been too large to place meaningful constraints on cosmology.
Decoupling occurs at t~100000 yrs, T~3000 K. Before that, not much happened until we get back to the epoch of Primordial Nucleosynthesis. This occurs at t~4 minutes, T~10E9 K.
Before this, the Universe was composed of protons, neutrons, electrons, and photons, plus backgrounds of neutrinos and gravitons. This holds, as we look further back, until t~10E-4 sec, and T~10E12 K. Prior this this, the density of matter was high enough that the Universe was opaque to neutrinos, and the temperature was high enough that both electons and positrons existed in nearly equal numbers. The density and temperature were also high enough to allow reaction that kept the number densities of protons and neutrons roughly equal.
When the temperature fell to T~10E10 K, the neutrinos decoupled, the positrons annihilated, along with *almost* all the electrons, and neutron production shut down. As neutrons are unstable, they began to decay away. By the time the temperature cooled enough for nucleosynthesis to occur, the neutron to proton ratio had dropped from ~1 to about 1/7.
Nucleosynthesis proceded until all the neutrons were gone (either used up, or decayed away), and the temperature got to cool for further reactions among the light nuclei. The final mass fraction of helium is about 0.23, just from the availability of neutrons. But the mass fractions of the other light nuclei (mainly deuterium, helium-3 and lithium-7) are very sensitive functions of the baryon density of the Universe at the time of nucleosynthesis (and thus of the current baryon density!). This is actually the strongest constraint on the value of Omega_b ~ 0.04.
We see structure at substantial density contrast in the current Universe. How did this structure come to exist?
The goal of COBE was not just to measure the CMB temperature (it did that in the first five minutes of data taking). It was really to search for evidence of structure in the CMB. The structure we see in the Universe today must have had its origins in the structure imprinted on the CMB. And yet for 30 years all attempts to detect structure in the CMB failed. COBE found evidence for structure at a level of about a part in 10E5.
The observed structure in the Universe has evolved from the very modest structure we observe imprinted on the CMB. In detail, structure has evolved from the density fluctuations present in the early Universe. Such fluctuations can be in one of two forms: Adiabatic Fluctuations, in which there is an actual density enhancement (or deficit) that exists in both the matter and radiation densities. And Isothermal Fluctuations, in which there is no density fluctuation. An example of such a fluctuation might be a region with anomolously high Helium abundance (but normal total matter density). This fluctuation would have a lower than average *particle* density. But as the mass density is normal, and the radiation is coupled to the matter, it is "frozen in" until decoupling. After decoupling, it becomes a region of low pressure, and material flows in to normalize the pressure with its surroundings. This will lead to a mass density enhancement.
Consider how an adiabatic fluctuation should behave in the early Universe. Assume that it has a size that is larger than the horizon scale, and also that it represents a density enhancement. Also assume a flat Universe. From this, one can derive a relationship for the density enhancement as follows:
Note that the k here represents the LOCAL curvature imposed by the density enhancement.
Given the relationships between density, the scale factor, and time in the radiation and matter dominated eras, this results in density enhancements that grow with time.
Eventually, the horizon scale grows to encompass the entire fluctuation. Thus the fluctuation is now causally connected, and can evolve coherantly. Thus one can treat its behavior with the physics of the Jeans Mass approximation. This differs from the sort of treatment one makes for star formation, as the Universe is still radiation dominated. The result of this is that the Jeans mass remains about an order of magnitude higher than the Horizon mass as long as the Universe is radiation dominated. Thus the fluctuations don't collapse.
Once the Universe becomes matter dominated, the sound speed remains high, as the matter is still coupled to the radiation. But as the matter now dominates the dynamics, it tries to collapse. It cannot do so because of radiation pressure. Thus the fluctuation begins to oscillate acoustically. The oscillations allow diffusion from the fluctuations. Thus small-scale fluctuations damp out before the Universe decouples. The prediction is that all adiabatic fluctuations with masses less than ~10e13 M Solar will damp out during the period of acoustic oscillation. Any remaining fluctuations on smaller mass scales must therefore be isothermal fluctuations.
As soon as decoupling happens, the sound-speed of the gas goes from roughly the speed of light to the sound speed for a classical ideal gas. This causes the Jeans mass to drop from ~10e18 M Solar down to about ~10e6 M Solar.
Tuesday May 1
Notice that there are two natural mass-scales the come out of this set of arguments. One is the mass-scale of the smallest surviving adiabatic fluctuations, around 10e13 M Solar. Or about the mass of a giant elliptical galaxy, or small galaxy group. The other is the mass-scale of the Jeans mass after decoupling, around 10e6 M Solar. Or about the mass of a globular cluster. These scales seem unlikely to be coincidental.
One of the dramatic advances in cosmology over the last two decades has been the discovery of anisotropy in the microwave background. This began with the COBE maps, but the COBE resolution is actually larger than the horizon scale at decoupling. So the COBE maps can't tell us much about the inhomogeneities that led to structure in the Universe today. The current state of the art is the WMAP data, with angular resolution of about 18 arcminutes. This is important because another way of testing our models of cosmology is to study the strength of the anisotropies in the microwave background radiation as a function of angular scale. This is typically done by plotting the amplitude of the measured temperature fluctuation versus the wavenumber of the spherical harmonic that the measurement is made at. The wavenumber scales inversely with the angular scale, so that larger wavenumbers correspond to smaller angular scales.
The basic pattern of fluctuation amplitudes expected is that there should be a main peak at about the horizon scale at the time of decoupling (about 1 degree), with higher-order peaks of lower amplitude at scales that correspond to the rarifaction (even peaks) and compression (odd peaks) of acoustic oscillations that begin when the Universe becomes matter dominated, but the matter is still coupled to the radiation.
If the Universe is open, then the distance to the surface of last scattering is greater than it would be in the flat case, so the peaks should all be shifted to higher wavenumber (smaller angular scale). And, because the density of everything is lower, the amplitudes of the peaks should be higher than in a flat Universe.
If the baryon density of the Universe is higher than expected, (but the total density still gives a flat Universe), then the peaks associated with the compression cycles of the oscillations (the odd peaks) will be amplified, and the rarifaction peaks (even peaks) will be suppressed.
Finally, if there is a cosmological constant, the higher order peaks will be suppressed, relative to the first peak, and the large-scale power-spectrum will have a somewhat different shape than the flat, lambda=0 model predicts.
The best analysis of the CMB combines the all-sky COBE and WMAP data with small-scale maps at higher angular resolution. The current data are best fit with a globally flat model, with a cosmological constant that is close to that found by the SN Ia teams. This is an especially important comparison because the two sorts of data probe the cosmogical parameters in different ways. As I noted above, the SNe data really probe the difference of Omega_lambda - Omega_matter. While the CMB anisotropies probe their sum:
This means that the two constraints can be combined into a global constraint on both parameters. This is the best current data on the independent values of Omega_M~0.3, and Omega_L~0.7.
Inflationary Cosmology was developed to address two major incompletenesses in the standard Big Band cosmology. These are
As z goes to infinity, Omega goes to 1. Now, if Omega_0 = 0.5, this implies that at t~10e-4 (at the end of pair-production),
Why is Omega_0 so close to 1 if it isn't equal to 1? This is an example of a fine-tuning problem.
Inflation posits that the Universe underwent an exponential increase in the scale factor at very early times, due to a phase transition in the vacuum energy driven by strong-electroweak symmetry breaking. The scale increase is predicted to have been a factor of 10e50 or so. This will drive Omega -> 1 naturally (imagine how flat a balloon would look if you increased its diameter by 50 orders of magnitude). It will also result in our entire observable Universe having once been part of a small, causally connected region in the pre-inflationary Universe, thus solving the horizon problem.
Why should such a thing happen?
We live in a Universe that is governed by four fundamental forces: Gravity, Electromagnitism, and the Weak and Strong Nuclear forces. It is expected that these forces all represent aspects of a single fundamental force that governs interactions at the very high energy scales that obtain in the early Universe.
As the Universe cools, the four forces we now measure break off at various energy scales, in a process called spontaneous symmetry breaking. We see examples of such behavior quite commonly: When water freezes it goes from a higher-energy symmetric state to a lower-energy asymmetric state. One cannot tell, a priori, what orientation the crystal lattice of ice will have as one cools water down to its freezing point.
In fact, we have tested this idea experimentally at some level. The last spontaneous symmetry breaking is that between Electromagnetism and the Weak Nuclear force. That physics was described in the early 1970s, and was demonstrated to be correct by the observation of the Weak Vector Bosons in collider experiments in the 1990s. The electroweak symmetry breaking in the early Universe is what leads to the neutrino background, and what causes the equilibrium between neutron production and decay to break at t~10e-11 seconds.
Gravity, the weakest force, is the first to part company with the other three (I'll return to that story in a bit). After the gravity symmetry breaking, the Universe contains photons, quarks, leptons, strong vector bosons, and gravitons. Expansion procedes until the energy scale of Grand Unified Symmetry Breaking, when the strong force seperates from the Electroweak force. Although we have not managed to do experiments at this energy level, theory predicts that the symmetry breaking occurs at particle energies of around 10e13 GeV, corresponding to a temperature of T~10e27 K, or a cosmological time of t~10e-34 sec.
According to inflation theory, when the Universe reached GUT symmetry breaking energies, the symmetry did not break right away. Instead, the Universe entered a state similar to that of a supercooled liquid. The physical description of this is that the Universe entered a state of false vacuum. That is, the nominal vacuum was not the true ground state. This false vacuum has the curious property of a constant (and VERY large) energy density.
The expected value of the energy density is ~10e95 erg/cm3. As the energy density is constant, the thermodynamic implication is that it leads to a negative pressure. Thus, if a bubble of true vacuum nucleates it will have P=0 which is much greater than P=-10e95. So the bubble expands drastically, driving the inflation, and returning the Universe to it's true ground state.
In order to see how this works mathematically, one must couch the expression for the second derivative of the scale-factor in general relativistic terms:
Where P is the false vacuum pressure, and u is the false vacuum energy density (which is much greater than the radation energy density). So this reduces to the following:
This integrates to the following expression for the scale factor:
Here Ri is the scale when inflation starts, and tau is
If t(end of inflation) ~ 10e-32 sec, this implies Rf ~ 10e50 Ri.
And after inflation we go back to standard Big Bang evolution.
The horizon scale just before inflation is c ti, with ti~10e-34 sec. This means the horizon scale is about 3x10e-24 cm. After inflation, this causally connected volume corresponds to ~10e26 cm.
Our present horizon scale is about 2x10e28 cm. At tf (end of inflation), this scale corresponded to only about 200 cm or so. In other words, our entire observable Universe only amounts to about 10e-24 of the the causally connected region that underwent inflation. Thus the horizon problem is solved.
One can express the density parameter as follows:
Which will drive Omega to within one part in 10e50 of 1.
We don't know if this idea is correct. We do know that we should be able to test the physics of the strong-electroweak symmetry breaking in the next decade or so. That should tell us if this idea is correct, and will undoubtedly open deeper mysteries for us to think about.
One of those is the cause of cosmic acceleration. Another is the first symmetry breaking, between gravity and everything else. To understand the nature of that problem let's jump back to before the inflationary epoch, and consider just how hard we can push the clock backwards. The answer to that question is that we can go back to the point that requires a theory of Quantum Gravity in order to understand the dynamics.
Consider a Black Hole of some mass M. It has a singularity inside its event horizon. The maximum uncertainty in the position of the singularity can be no more than the size of the Schwarzchild radius. The Heisenberg Uncertainty principle relates the uncertainty in position to the uncertainty in momentum. If the Black Hole mass is small, then the uncertainty in position must also be small, and thus the uncertainty in momentum will be large. One can express the uncertainty in momentum as c times the uncertainty in energy. And one can further express energy as either kinetic or gravitational energy. QM meets GR when these are equal. Some algebra:
This defines a quantity known as The Planck Mass. The Planck mass is not a very exceptionaly mass at first glance. But note that the result was derived in the context of the mass of a black hole. Now consider the resulting Schwarzschild radius for such a Black Hole:
This defines a quantity known as The Planck Length. Finally, one can derive the time it takes a photon to travel one Planck length:
This defines a quantity known as The Planck Time. When the Universe is one Planck time old, it has a horizon length of one Planck length. We have no means of describing the properties of the Universe at these times, but the expectation is that behavior is governed by a single force law at these energies (the "Theory of Everything").
