Jatila van der Veen
Physics Department, University of California, Santa Barbara
jatila@cfi.ucsb.edu

Philip Lubin,
Physics Department, University of California, Santa Barbara
lubin@cfi.ucsb.edu

Paolo Natoli, UCSB Physics Department, on leave from
Università di Roma "Tor Vergoata" Vialle della Ricerca Scientifica, 1
I-00133 Roma, Italy
Natoli@cfi.ucsb.edu

Recommended preparation: RAAP image processing activities "COBE for (Big) Kids: Discovering how the results of the Cosmic Background Exploreccccr Satellite constrain models of structure formation in the early universe" and "If it’s not DARK it doesn’t MATTER: searching for MACHOs in the galactic halo".
 
 

Introduction

In the early 1960’s Arno Penzias and Robert Wilson, then at Bell Labs, noticed a small discrepancy in their microwave instruments that indicated an excess of radiation coming in from space. Not content to ignore it, they soon made one of the profound discoveries of the 20^th century: they had found the embers of the early univese. This radiation, which is nearly constant in every direction, in all seasons, and at all times of day, is now called the Cosmic Microwave Background Radiation and is almost universally accepted to be evidence of a hot dense beginning to our universe. When viewed in the light of Edwin Hubble’s discovery of the redshifts of galaxies some 30 years earlier, this microwave background could be nothing other than the highly redshifted and cooled relic radiation from a very hot infant universe, now seen at a black body temperature of around 3 Kelvin. . (This corresponds to a black body radiation curve which peaks between about 0.5 and 0.1 cm, or about 60 and 300 GHz.)

The CMB photons are the oldest observable radiation, and come to us from the time when matter and radiation first separated, when the universe was cool enough to become transparent. This era is sometimes referred to as the Recombination Era, for it is believed that this was the first time the universe was cool enough for protons to capture an electron, and the first neutral hydrogen could form, some 300,000 years after the Big Bang.

For nearly three decades after its discovery, the only variation in the CMB that was found was the so called "dipole anisotropy", which revealed a slightly warmer temperature in the direction of the Virgo Supercluster, and a slightly cooler temperature in the opposite direction. (That is approximately 11h Right Ascension, -7^o Declination on the Celestial Sphere.) This bipolar temperature variation is thought to be due to our motion relative to the CMB photons, and is well explained as a "Doppler shift" from Special Relativity. Once this motion was corrected for, the CMB remained isotropic to one part in 10^4.
 
 

 
 

Throughout the 1980’s, several groups around the world used balloon-borne and ground-based microwave antennas, radiometers, and bolometers to search for this elusive structure. Their measurements showed that any structure in the CMB was at or below the one part in 10⁵ level. The effect of these measurements was to pressure the theoretical understanding towards more exotic explanations, such as inflation, since the structure , not yet seen, was uncomfortably small.

In 1992, data from the Cosmic Background Explorer (COBE) satellite, launched by NASA in 1989, showed evidence for minute temperature variations (anisotropy) in the CMB at a level of just one part in 10⁵ at angular scales of greater than 10 degrees 

The COBEdata could not be used to measure anisotropies at anglular scales below 7 degrees, the size of the beam.

At about the same time (even prior in one case) data from the South Pole and balloon borne experiments showed anisotropies at one degree scales at a similar amplitude level. As we will see, it is particularly important to measure at angular scales smaller than 1 degree, since this is the scale below which matter has been in "causal" contact. By definition, two points are in causal contact if there has been enough time since the beginning of the Universe for light to travel from one point to the other. Above this size scale, there had not been enough time for one region to influence another. This is a critical scale and, not unexpectedly, the anisotropies show a marked transition at this angular scale.

Although COBE represented a major breakthrough in our understanding of cosmology, the anisotropies it mapped are at an angular scale that is much larger than we can study with optical astronomy. Even the largest structures we can observe today, such as the Great Wall of galaxies discovered by Margaret Geller and John Huchra of Harvard in the early 199 0’s, are small by comparison.

Still, questions such as: "What is the actual density of the universe?", "What fraction of all the matter in the universe is "baryonic?", and "What is the correct value of the Hubble parameter?" remain unanswered. More accurate measurements of the small-scale fluctuations in the CMB should help us answer these and other questions about the physical processes which operated in the very earliest stages of the universe, and which have profound implications for our understanding of fundamental particles, forces, and symmetries in Nature.

Two satellites currently under construction, one by NASA , and the other by the European Space Agency (ESA), which are scheduled for launch early in the next century, should help us answer these questions. Called MAP and Planck respectively, these two satellites will map the entire microwave sky at angular scales of a few arc minutes to one degree, thus increasing our ability to measure the temperature fluctuations that were present in the universe at the recombination time. When the data are analyzed, we may be able to resolve many of the unanswered questions in cosmology today.

A qualitative look at two cosmological models

Even before the data arrive from MAP and Planck, cosmologists are modeling the spectrum of the possible anisotropies. Already, using the data we have so far, we can place constraints on the various cosmological parameters, such as the expansion rate of the Universe, density of matter, and what portions of our classical understanding of physics can and cannot be applied to conditions in the early Universe.

The Standard Cosmological Model, often called the Big Bang Theory, states that the Universe began from an initially hot, dense state, from which it expanded and is still expanding today. The observational pillars of this model are the observed red shifts of galaxies, the cosmic microwave background radiation, and the observed abundances of hydrogen, deuterium, and helium. In addition, we observe that the Universe is homogeneous and isotropic at sufficiently large scales.

One consequence of this model is the so-called Hubble Law, which provides an explanation for red shifts of galaxies, namely that the velocity of a galaxy as seen from Earth is a linear function of its distance: . Strictly speaking, this is an approximation, and holds only for velocities much less than the speed of light.

One important parameter in the context of this model is W₀ ("Omega naught") ,

the ratio of the total density of all matter in the Universe to the so-called critical density:

r_total / r_critical = W_o (1)

Universes with W_o > 1 are said to be CLOSED, implying that there is sufficient matter to cause the Universe to eventually collapse back on itself ("Big Crunch"). Universes with W_o < 1 are said to be OPEN, implying that the Universe will expand forever. Universes with W_o = 1 are called FLAT, implying that the curvature of space is zero when the total density of all matter equals the critical density of the Universe. This is equivalent to saying that the total kinetic energy of expansion exactly balances the gravitational potential energy of all the matter in the Universe, thus producing an expansion rate that asymptotically approaches zero.

Within the Standard Cosmological Model, there are basically two different theories which describe the growth of perturbations in the CMB, which lead to different expectations in the spatial distribution of CMB anisotropies and their spectra: random density fluctuations and topological defects in space-time itself.

The Inflationary Model is based on a random distribution of density perturbations in the earliest moments of the Universe. A very rapid expansion known as inflation suddenly stretched out the Universe by a factor of 10²⁰ or so. Inflation may have started somewhere around 10^-35 of a second after the Big Bang, and lasted perhaps 10^-33 of a second (no one is really sure), but it had the effect of smoothing out the amplitudes of the original density perturbations. This model explains why the original density perturbations would have left such small scale anisotropies in the CMB.

During the first few minutes of the Universe, all the primordial hydrogen and helium nuclei were formed, but for the next 300,000 years the density and temperature were so great that the Universe was opaque to all forms of electromagnetic radiation. This era is often called the "tight coupling era" because baryons and photons were tightly coupled by Compton scattering and electromagnetic interactions in a "photon-baryon fluid" (PB fluid).

The remnants of the original density fluctuations from the Big Bang provided the mechanism for gravity-driven oscillations in the PB fluid. Competition between local gravitational collapse due to excess mass density, and adiabatic expansion due to radiation pressure caused oscillations in the PB fluid, which in turn sent out acoustic waves that propagated at the local speed of sound.

At recombination, some 300,000 years after the Big Bang, matter and radiation essentially decoupled from each other, but the density contrasts remained embedded in the young Universe. It is believed that a large portion of the mass of the Universe is locked up in "dark matter", so named because it is only observed by its gravitational interactions and not by any electromagnetic interactions. Photons don’t scatter off dark matter like they do off protons and electrons, hence any dark matter formed early on would have been free to collapse gravitationally at a much earlier era.

After decoupling there was much less interaction between the photons and the now neutral matter. The baryons collected in the gravitational potential wells left by the dark matter, later growing into the structures we find today. The photons were free to propagate on their own through the Universe, imprinted with the signature of the density perturbations just prior to decoupling.

Recombination was not instantaneous, but is thought to have taken place over a time interval corresponding to a "delta z" of ~ 100, or roughly 10% of the red shift of the LSS itself. This had the effect of damping out the smaller wavelength (higher l – see below) peaks in the spectrum.

Both the Inflationary and Topological Defect Models predict that in the earliest moments of the Universe matter and radiation existed in one state, and the four Fundamental forces we observe today were united into one Superforce. Somewhere in the tiniest fraction of a second after the Big Bang, the Universe passed through a succession of transitions in which first the strong nuclear force differentiated, then the weak nuclear force, and finally electromagnetism.

According to the Topological Defect Models, these transitions which broke the original symmetry of the Universe left defects in the fabric of spacetime itself, which could have taken various forms, such as strings, knots, domain walls, and other discontinuities. These regions of discontinuity in spacetime became regions of varying gravitational potential where matter later collected to form the large scale structures we find in the Universe. The mathematics behind the Topological Defect Model are very complicated and indeed not well understood at this time.

The predicted spectrum of anisotropies for each model should look radically different. The Inflationary Model predicts a CMB spectrum that resembles that of a complex sound wave that damps out at high frequencies. The Topological Defect Model predicts a CMB spectrum that rises slowly to one peak, and slowly damps out. In the following figure, the spectrum labeled "Strings" is based on the Topological Defect Model, while the other three spectra are based on Standard Models in which the cosmological parameters are varied, as discussed

below. 
 
 

The Problem Of The Critical Density Of The Universe And The Search For Dark Matter

Will the universe expand forever, or eventually collapse in a "big crunch"? The answer to this question depends on the density of matter in the universe. In a spatially "flat" Universe there should be just enough matter so that the gravitational potential energy of all the matter just balances the kinetic energy of expansion, and the velocity of expansion will eventually approach zero. In other words,the expansion velocity of the universe at any radius is just equal to the escape velocity of any chunk of matter at that radius.

We can use this assumption to get an idea of how the critical density and the Hubble constant are related:

We define the gravitational potential of any test mass at infinity, relative to a large mass, as 0. In our flat universe model, the kinetic energy of expansion of this test mass also approaches 0 as it approaches infinity. We define the initial kinetic energy of expansion and initial gravitational potential energy at some initial radius r to be K_o and U_o, respectively.

and

where G is the Universal Gravitational Constant, which has the value 6.67 x 10 ^-11 Newton-meter²-kg^-2 in SI units.

By Conservation of Energy,

                   which leads to .

Isolating v_o, we can calculate the escape velocity necessary for mass m to escape the gravitational pull of M and just barely make it to "infinity" , asymptotically approaching zero kinetic energy:

Now, consider a sufficiently large, spherical region of the universe in which the density is r . The mass M of this region is

Substituting this expression for the mass M into our equation for escape velocity, we get

Now recall Hubble’s Law, which relates the recession velocity of any galaxy to its distance from us, r:

where H_o is the Hubble parameter, in units of km/sec/Megaparsec. Whether v (r) is equal to the escape velocity necessary for the universe to escape its own gravitational influence (so to speak), depends on whether the average density, r , is equal to the critical density, r_{c ,} necessary to balance the expansion at some time in the future.  To estimate the critical density, we equate v(r) and v_o , giving us

which, after a bit of algebraic massaging, leads to an expression for the critical density of the universe in terms of H_o and the universal gravitational constant, G:

If we take the current moderate estimate of H_o= 65 km/sec/Mpc, we arrive at a value of r_c of approximately 8 x 10^-30 gr/cm³. If we assume "mass follows light", that is, all the radiation we observe, at all frequencies, is an indication of all the matter in the observable universe, then there is insufficient mass to close the universe, by at least an order of magnitude.

This observation has led to the predictions of the so-called dark matter within and between the galaxies.

Dark Matter: The Evidence

Over the past few decades, several observations have led to the conclusion that a large portion of the mass of the universe must be in some form that we cannot detect electromagnetically. These observations include the velocities of galaxies in clusters, the rotation rates of spiral galaxies, the mass:luminosity ratios of galaxies, and extended halos observed around elliptical galaxies.

The line-of-sight velocities of galaxies can be computed from the Doppler shifts in their spectra. Galaxies in large clusters are observed to have rather high random velocities, but do not seem to be flying apart. From the dimensions of a cluster and the observed velocities, we can estimate the amount of mass that must be contained within the cluster to keep it gravitationally bound. It turns out that the gravitational mass per galaxy that is necessary to keep clusters together is several times larger than the mass per galaxy which is measured by independent means.

The rotation velocitiesof stars in the spiral arms of nearby galaxies can also be measured from their Doppler shifts. It is observed that the rotation velocities of galaxies do not follow the predicted "Keplerian" pattern, that is, the velocity of stars in a spiral galaxy does not decrease steadily with distance from the nucleus, as would be expected if most of the mass of the galaxy were contained within the nucleus. Rather, the rotation velocities decrease in the inner few kiloparsecs of spiral galaxies, but remain fairly constant in the outer reaches of spiral galaxies, indicating that there must be a tremendous amount of unseen mass out there.

From the mass:luminosityrelationship for stars, based on well-known theories of stellar evolution and structure, we can infer the mass of a star in terms of solar masses from its luminosity - the light output at all wavelengths. If we normalize the mass:luminosity relationship so that M¤ :L¤ = 1 (that is, the mass:luminosity ratio of the Sun is equal to 1), then measure the masses of clusters gravitationally, we find that large clusters of galaxies have mass:luminosity ratios of 200 - 400. Clearly this indicates that there is more mass in these rich clusters than can be seen from their light output.

Observations of elliptical galaxies reveal the presence of faint shells that extend out two or three times farther than the bulk of the starlight, forming dark halos around the luminous regions of the galaxies.

Dark Matter: Cold Or Hot, Normal Or Exotic?

When we speak of "hot" or "cold" dark matter, we refer to the possible scenarios at the time in the universe when matter and radiation first separated, and matter dominated over radiation. Cold dark matter (CDM) refers to particles whose random motions were very slow compared to the speed of light, and whose energies were very much less than their rest masses. Hot dark matter (HDM) refers to particles such as the hypothetical massive neutrino (with a mass of about 50 electron volts, or 10^-4 of the mass of the electron) which had relativistic velocities just prior to the era of matter domination.

The simplest possibility for the dark matter component of the universe is a very large number of small bodies, like Jupiter, that never grew large enough to "shine", and that are too small for us to detect. If one of these bodies should pass across our line of sight when we are observing a more distant star, a temporary brightening of the starlight should be observed, due to the bending and focusing of the light as it passes by the dark body. This is called gravitational lensing .

Several groups around the world are searching for gravitational lenses in our galactic halo and bulge, but so far very few have been detected. These objects have been playfully called MACHOs, standing for Massive Compact Halo Objects.

The other possibility is that the dark matter is in some exotic form which is not baryonic - that is, not made up of the protons and neutrons of ordinary matter. Since these particles do not interact easily with ordinary matter, they have been given the name Weakly Interacting Massive Particles, or WIMPs.l

The most popular exotic matter candidates that have been proposed are axions and light supersymmetric particles, or LSP’s, thought to be relics of the very early universe, left over from the highest energy era, when the four fundamental forces of nature are believed to have been united.

Cosmic Strings and Textures: GUT Theorists’ Answer to Inflation

If the Inflationary Cosmologists are correct, the fluctuations in the temperature of the early universe should take the form of random Gaussian noise. According to the Topological Defect Model, however, that the origin of the CMB anisotropies comes from the defects in spacetime caused by the breaking of fundamental symmetries in Nature as the universe cooled from an initially hot, uniform state at energies over 10¹⁶ GeV per particle. Theories which relate the four fundamental forces we observe today (gravity, electromagnetic, strong and weak nuclear) to one Superforce at high energies are called Grand Unified Theories (GUTs) or the Theory of Everything (TOEs). If the GUT model is correct, then the anisotropies in the CMB should show very different characteristics when examined at the resolution of the MAP and PLANCK experiments. Instead of being random hot and cold spots, the CMB maps should have line-like discontinuities in temperature, and the angular power spectrum should look smooth instead of lumpy at higher harmonics, greater than about l = 200 or so.

Understanding the Spectrum of CMB Anisotropies

The problem of predicting the CMB spectrum from a chosen set of initial conditions for the perturbations involves solving the first order General Relativity field equations (which may or may not include a cosmological constant - L - see description below) for the various components of the cosmic fluid. These include dark matter and neutrinos as well as photons and baryonic matter. These are very complicated equations, but can be viewed as generalizations of the classical fluid dynamic equations which express the continuity of flow and the conservation of energy and momentum under the influence of gravity.

The PB fluid is visualized more or less as a plasma of relativistic particles, in which the speed of sound is given as
  where R is the radius of the Universe at time t.
As R approaches 0, i.e., the farther back you go in time, 

Each little pocket of matter can be thought of as behaving like a tiny harmonic oscillator.The baryons contribute mass which causes the collapse, and the photons contribute the resistance to collapse in each pocket of material. What results is something like a damped harmonic oscillator, in which density (instead of displacement) is the variable, the expansion rate of the Universe (H) provides the damping term, and gravity provides the driving force for each oscillating pocket of PB fluid.

(See Max Tegmark’s illuminating paper "Doppler Peaks and All That: CMB Anisotropies and What They can Tell Us", in which he explains all the math.. )

When the equations are solved over a range of spatial scales (or wavenumbers), the solution takes the form of a series of sines and cosines (Fourier series), which lead to a particular spectrum of density (and hence temperature) fluctuations.

The elegance of this model is that the details of the CMB spectrum depend heavily on the gravity-driven acoustic oscillations. The amplitudes of the peaks depend on the ratio of baryons (inertia) to photons (restoring force), as well as the percent of dark matter. The baryon density (W_b) itself depends on the rate at which space is stretched as the Universe expands – the Hubble parameter.

The interdependence of W_b and H₀ is often expressed in the parameter W_bh², where h = H₀/100.

The initial conditions of the Universe determine the phase shift of the peaks in the spectrum. Thus it should be possible to extract a wealth of cosmic information from the spectrum of CMB anisotropies! For an excellent visualization of how the CMB spectrum is affected by the various cosmic parameters, see Wayne Hu’s home page.

All of the above description is a simplified version of how the model predicts the spectrum of CMB anisotropies we should measure, given a certain set of initial conditions. The problem of experimentally deriving the CMB spectrum from observations of microKelvin fluctuations in the background temperature of the Universe is a separate issue! This problem is similar to trying to decipher the sound signals from an orchestra playing random tunes from a great distance away, when there is a lot of noise contamination from nearby traffic and shouting children that you must first interpret and remove. In measuring the CMB one has to remove the signal contamination which comes from looking through our galaxy, intergalactic dust, stray radiation, and instrument noise.

Let us digress for a moment to the case of a vibrating guitar string, with a characteristic length, density, diameter, and tension. If the string is plucked, a pleasant sound will be produced, which is a combination of the normal modes of vibration of the string. The spectrum of the sound wave will tell you the relative power in each mode, and the actual wave form can be approximated by adding a series of sine waves with frequencies that are integral multiples of the fundamental (longest wavelength). The more terms you include in the series, the more closely you will approach the actual sound.

Now consider a circular drumhead with a characteristic diameter, thickness, density, and tension. If the drumhead is struck, it will vibrate in a characteristic manner which is a combination of its normal modes of vibration. Because it is a 2-dimensional surface however, its normal modes include vibrations with nodes that form concentric circles as well as nodes that are radii. The superposition of these 2-dimensional normal modes produces the sound you hear, which can be modeled with a series of time-varying Bessel functions in radius and azimuthal angle.

Finally, imagine a giant circular balloon filled with water, supported by strings which allow it to vibrate freely if tapped, but not roll away. You tap the balloon in several places, which causes pressure waves to travel through the balloon, and also around its surface. After a few seconds of oscillation, the entire balloon is suddenly frozen, so that the lumps in its surface at that instant are frozen into it forever. The variations in the surface height of the spherical balloon can be described with a Spherical Harmonic Series as a function of two angles, one measured between "equator and poles" so to speak ("small circles"), and the other around the equator with respect to some arbitrary reference ("great circles").

Since we are really looking out toward the LSS at the inside surface of an imaginary sphere, the spectrum of anisotropies is best described with a spherical harmonic expansion in temperature:

where T is the measured sky temperature, in microKelvins, as a function of two orthogonal angles, q and j , and the are the spherical harmonics.
(The are products of a function of cosine (theta ) and [(cos (mphi) + isin (mphi)]. For a full treatment, see any good book on partial differential equations!)

"l" represents the number of nodes between equator and poles, while "m" represents the number of longitudinal nodes. m varies from – l to + l , with l > 0, and for each l the sum is taken over all possible m’s. The a_l,m are the expansion coefficients, which are like the individual amplitudes in a Fourier series.

We define a new quantity, C_l , which is the squared average of the expansion coefficients for any l, averaged over all possible m’s:

The angular power spectrum, defined as  , gives the relative strength of the temperature variation we can measure for any order l, averaged over all possible m’s for that value of l.

The l = 0 term is often called the monopole term, and corresponds to the surface of a completely smooth and featureless sphere. In terms of the CMB, the monopole term is the 2.726 Kelvin microwave background, which is uniform out to a few milliKelvin.

The l = 1 term is called the dipole, and corresponds to a sphere with one part more positive than average and the other more negative. In terms of the CMB "positive" means warmer than background, and "negative" means cooler. The dipole variation of the CMB is approximately +/- .003 Kelvin, relative to the monopole term. (Technically speaking, the origin of the dipole anisotropy in the CMB is not cosmological, but is an effect of our motion relative to the CMB, which causes the sky to appear warmer in the direction of our motion, and cooler in the opposite direction.)

The l = 2 term is the quadrupole, and represents a sphere with two warmer regions around the equator, and a cooler region at the north and south poles. The quadrupole term of the CMB has an amplitude which is two orders of magnitude smaller than the dipole fluctuation.

The angular size  of a temperature fluctuation on the sky depends on the order number "l" approximately as follows:
radians. (5)
Thus, the COBE measurements (with a beam width of 7⁰) correspond to l ~ 20, or somewhat less than 10⁰ of arc on the sky.

The important lesson in all this is that when we plot the CMB power spectrum as a function of order number l , the distribution and amplitude of the peaks in the spectrum depend heavily upon the values of the cosmological parameters such as baryon density vs. radiation density, H_o, whether or not there was cold (non-relativistic) or hot (relativistic) dark matter, and whether the Universe was seeded by these random density/temperature perturbations or by some other, as-yet-poorly understood mechanism.

Hence the reasoning: if we can map the anisotropies and measure the power spectrum of the CMB, we can understand the cosmological parameters that constrain the fundamental physics of the Universe.

Making the models...

The cosmological parameters that we are seeking to measure are:

H₀, the Hubble parameter. This is assumed to be a (constant) expansion rate per Megaparsec (about 3.26 million light years). Independent measurements place H₀ somewhere between 50 and 80 km/sec/Mpc.

W_0, the ratio of the total density of the universe to the critical density.

Omega_b, the ratio of the total density of baryons (normal matter, made up of quarks) to the critical density of the universe.

Omega_cdm, the ratio of the density of cold dark matter to the critical density of the universe. Cold dark matter refers to non-luminous yet gravitationally-interacting matter, whether baryonic or exotic, that is non-relativistic.

Omega_hdm, the ratio of the density of hot (i.e., relativistic) dark matter to the critical density of the universe. Hot dark matter could take the form of massive neutrinos, or some other unknown large particle.

Omega_Lambda , the ratio of the vacuum energy density to the critical density of the universe. Lambda is Einstein’s cosmological constant, some unknown vacuum energy density which contributes to the expansion of the universe .

Recent independent measurements ofhigh red shift Type Ia supernovae http://physics.berkeley.edu/group/supernova/gen.html and high red shift massive galaxy clusters appear to support a non-zero value of Lambda . If true, this would have profound consequences on cosmology and in particular on the CMB anisotropy.

As the universe cooled, charged particles combined to form neutral hydrogen and helium. Helium forms at a higher temperature than hydrogen, thus a certain portion of all baryons became locked up in neutral helium before the time of hydrogen "recombination". The fraction of neutral helium in the primordial universe is also accounted for in running the CMB models, and is thought to have had a value between .23 and .26 of the total baryon density.

It is no trivial matter to solve the equations that govern the spectra of CMB anisotropies! One has to first assume an underlying geometry of space-time according to General Relativity, and then assume a means of perturbing the smoothness of that geometry by temperature and density inhomogeneities. One then has to include the contributions of the various components such as baryons, cold dark matter, hot dark matter, and radiation to the growth of these inhomogeneities, as well as include the effects of collisions between particles and photons and helium fraction.

Here we show the expected spectra for three models: a "Standard Cold Dark Matter" (SCDM) model (blue curve), with Omega_b = .05, Omega_cdm = .95, and Omega_Lambda = 0, giving W_total = 1 for a flat universe, an "Open" model (red curve), with Omega_b = .05, Omega_cdm = .30, and Omega_Lambda = 0, giving a total density of .35 of the critical density, and a flat universe with a positive vacuum energy density (green curve), in which Omega_b = .05, Omega_cdm = .25, and Omega_Lambda= .7. H₀ is the standard 50 km/sec/Mpc.

For comparison, here we show the effect of using H₀ = 75 km/sec/Mpc, while keeping all the other model parameters the same. You can see that changing the value of H₀ has more consequence for temperature anisotropies at length scales much less than 1 degree (higher order than l = 200).

These spectra were generated using a Fortran code by Uros Seljak and Matias Zaldarriaga called CMBFAST which you can download from their webpage and try for youself!

The relative contributions from all the cosmological parameters have different effects on the spectrum of CMB anisotropies. To see animations of how changing these parameters changes the plots of the C_l’s, click HERE

Making Microwave Maps from CMB Spectra

For each model universe, we can predict the statistical distribution on the Last Scattering Surface of small scale temperature variations in the CMB, relative to the accepted value of the average temperature of the universe ("monopole term") of 2.726 K. The next three figures are maps of CMB anisotropies for the Standard Cold Dark Matter (SCDM), Open, and Lambda models that were generated from the spectra for each model, by Paolo Natoli at the University of California, Santa Barbara .

In each map red represents warmer, and blue cooler regions, relative to the background temperature of 2.726 Kelvin.

The first map shows the expected angular distribution of CMB anisotropies in microKelvins, for the Standard Cold Dark Matter Model. The clustering of warm and cool spots occurs at an angular scale of 0.9 – 1.0 degree of arc, corresponding to an l of 200 – the first peak shown in the  spectrum.

In the Open Model (red curve in the first graph) the first peak occurs at l » 500, and you can see the clustering of anisotropies at a finer scale of 0.3 – 0.4 degree. This is a statistical picture of the CMB if the total density of the universe is less than the critical density.

The last map we show is for a flat universe (Omega₀ = 1), but with Lambda = .7 of the critical density.

You can see that the clustering of anisotropies occurs at the .9 – 1.0 degree scales, as in the Standard Model.

In these maps there is no preferred orientation in space. They are statistical representations of temperature fluctuations on the surface of last scattering, projected as if the observer was looking from outside the universe.

The next graph shows all the data that have been collected as of 1998, from a compilation by Max Tegmark, of Princeton University . Each data point represents the total measured power, in microKelvins, of the CMB at a particular spatial frequency or order (l) number.

The vertical lines represent the relative error in each data point, and the initials refer to each experiment. There is an unmistakable clustering of power in the temperature spectrum at the l = 200 region, and a noticeable drop-off at higher orders (smaller angular scales), as predicted by all of the models. The error bars in the higher order measurements, however, are such that it is not yet possible to discriminate between models based on CMB measurements alone. When the MAP and Planck satellites return their data in the first decade of the 21^st Century, it should be possible to better define the CMB spectrum at finer scales.
 
 

A Short Primer on Spherical Harmonics ...just what you always wanted to know!

Just about all models of variations in the cosmic microwave background radiation start with the assumption that they are temperature variations on a sphere which can be modeled with a series of polynomials that depend on altitude and azimuth:

where the a_l,mare constant coefficients chosen so as to satisfy the conditions we expect to find at the boundaries of our system, and the Y_l,mare called Spherical Harmonics.

This sounds complicated, but is really not so bad. Spherical Harmonic Series come from the Legendre Polynomials, which are themselves based on series of cosines, and are solutions to problems that are familiar from potential theory - including electric and gravitational fields.

We can start with a one-dimensional analogy of a vibrating string: If you pluck a guitar string, for example, it will vibrate in very predictable ways called normal modes, which depend mainly on the length, density, and thickness of the string, but also on where you pluck it. The lowest tone (frequency) that is produced by the vibrating string is called the fundamental, and the higher overtones are called harmonics.

If you put a microphone up to the plucked guitar string, and run the output from the mike into a spectrum analyzer, you will see a graph which displays all the frequencies that you are hearing, and their relative amplitudes. This is what we call a power spectrum.

The "l" terms in the Y(l,m) are called the multipoles. Perhaps you are familiar with dipoles, quadrupoles, and octopoles from electromagnetism; the terms refer to the same types of symmetry in the case of potential variations on the surface of a sphere.  l = 0 is called the monopole, l = 1 the dipole, l = 2 the quadrupole, l = 3 the octupole, l = 4 the hexadecapole... The larger the value of l, the smaller the size of the variation in the field under investigation. Among the many uses of the Y(l,m)’s in mathematical physics are modeling the shape of the gravitational field of the Earth and the normal modes of vibration of the Sun (helioseismology).

For a good treatment of spherical harmonics and the 3-D power spectrum of the CMB, go to Max Tegmark's article, Doppler Peaks and all that...

The importance of all this mathematical theory is that there is a powerful theorem that says: If you can find a continuous function f (q ,f ) which can be evaluated over the surface of a sphere, it can be expanded in a uniformly convergent double series of spherical harmonics. In the case of the CMB, we don’t know what that function is a priori; it represents the temperature variations on the surface of last scattering, at the time when matter and radiation first decoupled, which we now see at a redshift of 1000. However, if we can measure the small variations in the CMB today, and determine the coefficients of the spherical harmonic expansion (called the Power Spectrum) which we predict to be a solution of whatever function describes the primordial temperature variations at that time, then we can determine whether our initial model was correct, or - as is usually the case in the scientific process! - refine our model and make more measurements.

Now let’s go to a familiar example in 3 dimensions: an electric field due to some localized distribution of charge. Gauss’ Law for electrostatics tells us that if you find an electric field in space then there must be a source of the field somewhere, because you can measure the variations of the field on any imaginary surface you draw in space and extrapolate back to the source. If you want to discover the source of the field, you merely have to create an imaginary closed volume in that space and measure the total flux of electric field coming through the surface of your imaginary volume, in all directions. If you get a net flux of zero, it means there is no source of the electric field enclosed in your volume; if, on the other hand, you get a non-zero value, then you can immediately point to the source of the field as lying somewhere within your volume.

The same is true for a gravitational potential field. If you measure the variations of the field at a given radius, because you believe you understand something of the nature of how gravitational fields work, you can infer the properties of the source - in this case, a mass distribution. Satellites measure the lumps and bumps on the surface of the Earth, from which we can determine the coefficients of the spherical harmonic expansion which describes the shape of the geoid - the gravitational potential surface of the Earth.

Let’s take a look at how the mathematical model is developed, using the case of an electrostatic field as an example. (Hopefully, the ideas are familiar, even if the math is new!)

We write this in concise mathematical form as:

("Del dot E")

where E is the vector electric field, 4pr is the charge density distributed over the solid angle 4p , and upside-down triangle is the divergence operator, "Del":

Now, as in all potential theory, we let the vector E be represented by the negative gradient of a scalar potential, y , such that

Then we have

which leads to

.
This is often called "Poisson’s equation", after the famous mathematician Poisson. This differential equation of physics appears in the study of many systems, including gravitation.
                The Ѳ operator is just the sum of the second derivatives of the function y (x, y, z) in each direction, and is itself a scalar:
.
Now, we generalize this Poisson equation to the case where the source term is not necessarily a constant, but depends linearly on the function y itself:

where k is a constant. Since the x, y, and z axes are orthogonal, x, y, and z are independent of each other, and this equation can be separated into three equations, each a function of x, y, or z alone:

wherel² + m² + n² = k²(7)

Each of these equations looks suspiciously like the equation of a simple harmonic oscillator, except we have a spatial instead of a temporal variation, and the solutions take the form of sums of sines and cosines.

Now you have probably seen the effect of adding sine waves of different wavelengths on an oscilloscope. If you play a pure tone (from a synthesizer or your computer) into a microphone, and feed the signal into an oscilloscope, you get a pure sine wave. If you play a real instrument, or have a real person sing into the mike, what you see on the screen is a superposition of harmonics: the fundamental, or longest wavelength, and overtones - harmonics - of shorter wavelengths.

To describe the variations in a three dimensional field, you need to add the effects in three dimensions. If we convert from the familiar (x, y, z ) coordinates to the more general spherical coordinates of (r,q ,f ) we transform the simple sines and cosines into a Spherical Harmonic Expansion. In the case of the CMB, we are only interested in the variation of temperature as a function of two angles, q and f , and not of radial distance, because we are measuring the variations on the surface of a sphere that corresponds to a constant redshift of 1000. (We can’t specify the actual distance, because it depends on the rate of expansion of the universe at that time, which we don’t know independently!)

The spherical harmonics are related to the Associated Legendre Polynomials by the equation

                                                                                where the are the Associated Legendre Polynomials

where the are the Legendre Polynomials. (The theory behind all these polynomials can be found in any good book on differential and partial differential equations, particularly one that emphasizes their applications in Physics!)

So, there you have it! Now you can go read all those esoteric papers on small scale CMB anisotropies and structure formation in the Universe, and hopefully know a bit more about what they are talking about!

Click HERE to download a version of CMBFAST that will run under a DOS window as an interactive program. You must unzip the files, and then run CMBFAST. This is an ".exe" file that you have the permission of Uros Seljak and Matias Zaldarriaga to use, but you must reference them!