The polarization of the Cosmic Background Radiation (CBR) is an important source of cosmological information. However, the signal from the polarized component of the CBR should be only a few microkelvin strong, and has so far eluded detection. The experiment described below will attempt to measure this signal.
In order to give the reader some understanding of not only the experiment itself, but also the phenomenon it will investigate, this paper is divided into two parts. The first part is a qualitative discussion of the nature and origin of the CBR polarization, which indicates the sorts of signals that we can reasonably expect. The second part is a description of the telescope and experimental techniques which will be used to detect these signals.
A Brief Introduction to CBR and its Polarization
So far no one has detected the polarization of the CBR. We therefore provide this general introduction to CBR anisotropies (1) to demonstrate that the existence of polarization is a robust prediction given the generally accepted picture of the CBR, (2) to justify our observational methods and (3) to illustrate how the polarization of the CBR can provide interesting information about the structure and evolution of the early universe. This discussion is quite qualitative and pedagogical. More in-depth and quantitative discussions of the CBR anisotropies can be found in the references cited below. Starting with the basics, the first section of this part describes how we know the CBR comes from a plasma that filled the entire universe in the distant past. The next section shows how the light emitted from such a plasma carries information about the material properties of that plasma. In particular we will show that the polarization traces variations in the bulk flow of the plasma. This will enable us to relate the variations in the CBR to the dynamics of the early universe and to make some rough predictions about the size and scale of the polarization, as discussed in the final section.
What is the CBR, and where does it come from?
The Cosmic Background Radiation, as its name suggests, is a form of electromagnetic radiation that comes at us from every point in the sky and seems to fill all of space. This radiation can be distinguished from the light that comes from other sources, such as stars, dust, etc. by its distinctive 2.7 Kelvin blackbody spectrum. (A blackbody spectrum is the spectrum of radiation emitted by material which absorbs any light incident upon it, the shape of this spectrum depends only on the temperature of the material.)
From its ubiquity and its spectrum we can deduce the origin of the CBR. The blackbody spectrum of the CBR suggests that it comes from some material which strongly scatters light. The only known objects in the universe which interact strongly with electromagnetic radiation are the charged electrons and nuclei of ordinary atoms (Other, more exotic, forms of matter which might fill the universe, are generally believed not to interact with electromagnetic radiation).
A plasma, consisting of free charged particles, can generate blackbody radiation. For example, the light emitted from our sun has a blackbody component. However, at low temperatures, like 2.7 Kelvin, the electrons and nuclei are not free as in a plasma, but are bound together into neutral molecules (most of which is hydrogen).Since these molecules do not have a net charge, they do not interact much with electromagnetic fields and therefore cannot produce blackbody radiation.
Thus we have a problem reconciling the blackbody shape of the spectrum, which indicates that it originated from a plasma, and the low source temperature the spectrum suggests. We can resolve this apparent paradox by considering another cosmological phenomenon, universal expansion. The light we receive from any galaxy is redshifted by a factor roughly proportional to its distance from us. We can interpret these redshifts as the result of a relative velocity between us and the other galaxies. Hence, each galaxy is moving away from us with a speed dependent on its distance from us. Assuming that we do not inhabit a special place in the universe, this means any two galaxies are moving apart at a rate that depends on their separation. Thus it is said that the universe is expanding (see Peebles 1993 for a much more detailed discussion of universal expansion).
Since the galaxies seem to be moving apart, in the past they must have been packed more closely together. Similarly, the photons that make up the CBR were also packed closer together in the past, and so in the CBR would be more intense. In addition, the distance between the crests of an electromagnetic wave has grown with time just like the distance between galaxies or photons. Therefore in the past the CBR photons also had smaller wavelengths and higher energies. For these reasons, the spectrum of the CBR has changed with time. It turns out that the spectrum of the CBR always remains that of a blackbody, but the temperature indicated by the spectrum decreases with time (Peebles 1993). Hence, in the past the temperature of the CBR was higher, and if we go far enough back in time the CBR becomes so hot that its photons can ionize hydrogen. This yields the plasma which we need to generate the CBR. Notice that this plasma fills the entire universe at early times.
The CBR therefore originates from the hot, ionized phase of the early universe. This phase of the universe existed until about 300,000 years after the big bang. At this time, the plasma had expanded and cooled down enough for the electrons and nuclei to combine to form neutral atoms (which cannot strongly scatter light) and the universe became transparent to the CBR photons. This period is known as decoupling. Following decoupling, the photons cooled and free streamed until they encounter our detectors today. The CBR therefore gives us a picture of the universe when it was only 300,000 years old (less than 1% of its current age)!
What generates variations in the CBR?
Since the CBR gives us a picture of the universe roughly 300,000 years after the big bang, the variations in both the temperature and the polarization of the CBR reflect the structure of the early universe. In order to clarify this relationship, we will here describe generally how the variations in the emitted light from a plasma reflect the material properties of that plasma.
The variations in the temperature of the radiation reflect the variations in the density of the plasma. This should make some amount of sense, since when a plasma is compressed, its temperature increases. However, the simple relation between temperature and density is complicated by gravitational redshifts (The photons leaving a dense region have to climb out of a potential well and hence suffer a red shift and so the temperature indicated by the observed radiation may be different from that of the actual plasma--- see Hu, Sugiyama and Silk 1997).
The polarization of the radiation, on the other hand, traces the velocity of the plasma. To understand why this is so, we must first understand how the scattering of light by charged particles can generate polarization. A charged particle moves in response to electromagnetic fields. Thus when light shines upon such a particle, the electromagnetic wave causes it to oscillate in the plane perpendicular to the direction of motion of the incident photon. This oscillating charge will generate new electromagnetic waves (i.e.\ emit light). Since the light emitted by a moving charge is polarized along the axis of motion of the charge (see the description of dipole radiation in textbooks such as Griffiths 1989 or Jackson 1975), the light emitted by the particle will tend to be polarized in the plane of the particle's motion (see Figure 1). Hence, even if the light incident on a charged particle is unpolarized, the scattered light can have a polarized component.
Figure1: Light from moving charges. Figure (a) shows the classical pattern of radiation emitted by an electron moving along a single axis. Note how the light is always polarized along the axis of motion (only the electric field of the electromagnetic wave is shown). Figure (b) shows the radiation emitted by an electron moving in a single plane. This field is simply the sum of the field shown in (a) and a similar field rotated 90 degrees to reflect the fact the electron now moves in two directions. Note now that the light is polarized in the plane of motion. The electrons will move in this way when either (a) polarized or (b) unpolarized light shines on them from above (or below).
The charged particles in a plasma have light shining upon them from every direction. If the (unpolarized) incident light on a charged particle is equally bright in all directions, the particle will be pushed about every which way. Since the charge moves equally in all planes, the scattered light has no average polarization. However, if the incident light is brighter along one axis than another (i.e.\ the intensity of the incident light has a quadrupole moment), then the charged particle moves more along one plane than another, and the emitted radiation is linearly polarized (see Figure 2). Thus a quadrupole moment in the intensity (i.e. temperature) of the incident radiation generates a polarized component in the scattered radiation}. (Other patterns in the temperature of the incident radiation (dipole, hexapole, etc) do not define a single plane where the amplitude of the particle's motion is the greatest, and hence cannot give rise to polarization-- See Hu and White 1997 and Melchiorri and Vittorio 1996).
Figure 2: Polarization from anisotropy. The charge at the origin sees the light coming in along one axis is brighter than that coming in along the other axis. This causes the particle to oscillate more in one plane than another, so that the scattered light along the third axis has a polarized component.
This might suggest that the polarization will just follow the temperature variations, and thus traces the density of the plasma. However, since the electrons in the plasma scatter light so easily, the photons cannot travel very far between scattering events; and the incident light on an electron comes from its nearest neighbors. On these small scales, the same scattering with electrons which generates the polarization tends to heat up cold regions and cool down hot regions, erasing the temperature variations. Therefore such variations are not the source of significant polarization.
However, local quadrupole moments can be generated by the bulk motion of the fluid. Say the plasma is flowing radially towards some point with some divergence. If we switch to the frame of the plasma at some point, we see the fluid moving in toward the selected point. However, since the relative velocities along a radial line are different from those along a azimuthal ring, the magnitude of this relative velocity has a quadrupole moment. Since the light emitted by a fluid moving toward you is brighter than the light emitted by the same fluid moving away from you (the photons are not emitted isotropically in this frame), there is a quadrupole moment in the intensity of the radiation incident upon the bit of plasma (see Figure) and the light scattered by the charged particles in this region will have a polarized component. { Therefore variations in the velocity field of the plasma can give rise to polarized emission from the plasma} (Coulson, Crittendon and Turok 1994).
Figure 3: Velocities and polarization. In (a) we see a plasma flowing into a point. We switch to the reference frame of the fluid at a given point in the plasma in (b). In this frame we see the fluid moving in towards this bit of fluid. These relative velocities have a quadrupole moment. Consequently, there is a quadrupole moment in the intensity of the light that is emitted towards the bit of plasma. These variations in the intensity of the incident light give rise to polarization as shown in Figure 2 above.
Where do the variations in the CBR come from? The above discussion shows how the variations in the material properties of a plasma give rise to variations in the temperature and polarization of the light emitted by that plasma. Now let us turn our attention to the particular primordial plasma which gave rise to the CBR. From observations of the variations in the CBR we can infer the dynamics of this plasma. However, for the purposes of this writing we will instead use the dynamics of the plasma to predict the polarization of the CBR.
The inhomogeneities this primordial plasma evolved in response to pressure, gravity and universal expansion; therefore a full treatment of the dynamics is very complicated and well beyond the scope of this paper. Furthermore, the behavior of the plasma depended on a host of poorly constrained cosmological parameters. This is, of course, why experiments such as this, which can shed some light on these dynamics, are interesting. However, it also means that we still have only a rough idea of what the plasma was doing before it decoupled. Therefore in this section we will use some basic principles to understand the gross features of the plasma's dynamics. We will then couple this understanding with observations of the temperature variations
to come up with some general predictions regarding the polarization.
First of all, it is important to realize that the density and velocity of the plasma could have varied on many different scales (figure 4). There could have been regions the size of our galaxy that were on average hotter than other such regions, and there could also have been regions the size of the entire observable universe that were on average hotter than other similar sized regions. This could make the analysis of the motions of the plasma very complicated. However, since the variations in the CBR have been found to be very small (about a part in 100,000), the variations in the material properties of the plasma were only small perturbations. In this sort of situation, the variations on a given scale evolve independently of those on all other scales (This is a consequence of the equations of motion being approximately linear). Therefore we can consider the variations on each scale separately, in effect, pretending that the universe only varied on that scale (see Hu, Sugiyama and Silk 1997).
Figure 4: Variations on multiple scales. In this simple case, the value of the function has variations on both a small scale and a large scale, in a similar way, the early universe had variations on many different scales.
Of course, we expect variations on different scales to have evolved differently. One of the most important and basic factors responsible for such differences arises from the fact that the speed of light is finite. Because of this fact, any observer in the universe can only see things as far away as the speed of light times the age of the universe. The size of this observable region is known as the horizon scale. Note that the horizon scale depends on the age of the universe and grows with time.
The size of the scale of the variations relative to the horizon scale has dramatic effects on the evolution of the variations. If we have variations in the density of a plasma on a scale much greater than the horizon scale, the density of the plasma does not change, much over the horizon scale, so the universe looks roughly homogeneous to any charged particle in the plasma, and so the plasma has no inclination to move. On the other hand, if the scale of the density variations is much smaller than the horizon scale, then the charged particles can receive photons from regions of different densities, and thus they can feel forces pushing them towards overdense regions (gravity) or towards under dense regions (pressure). Therefore, variations on scales greater than the horizon scale will not change much with time, while variations on scales smaller than the horizon scale can evolve as the plasma moves about in response to pressure and gravitational forces.
The CBR gives us a picture of the plasma at decoupling, thus the important horizon scale to the CBR variations is the horizon scale at decoupling. Since the horizon scale grows with time, this scale is smaller than the horizon scale now, which means that the CBR can carry information about variations in the plasma which were on scales larger than the horizon scale. Variations on these scales could not change appreciably from their initial state and are therefore extremely interesting for investigating inflation and other features of the extremely early universe. However, since the plasma did not have much inclination to move on these scales the polarization is expected to be extremely small.
We also have information about variations in the plasma which were smaller than the horizon size at decoupling. On these scales the plasma was able to flow around in response to gravitational and pressure forces before the photons decoupled from the plasma. These scales can therefore be informative regarding the kinematic properties of the plasma (mass density, sound speed, etc.), which depend on its composition. These variations consequently carry information about the content of our universe.
Obviously, since the plasma moves around in response to the forces on it, significant polarization can be generated on these scales. The movement of the plasma also has a pronounced effect on the temperature variations. This flow of plasma into and out of dense regions causes the size of the variations to increase (this is the result of some rather complex interactions between the forces acting on the plasma, and is beyond the scope of this writing, see Hu and Sugiyama 1995 and Kodoma and Sasaki 1984 for information on this subject). Furthermore, the competition between gravitational forces ( pulling material into dense regions) and pressure (which pushes it back out) gives rise to oscillations in the plasma. In an oscillating fluid, there is a simple relation between the variations in the velocity and the density of the fluid (analogous to the simple relation between the position and velocity of an oscillating mass on a spring). Consequently there is also a simple relation between the polarization and temperature variations of the CBR on these scales: On these scales the polarization will be on the order of 10% the size of the temperature variations. (They are not of comparable size because only the portion of the velocity field that gives rise to local quadrupole moments can generate polarization.)
From these basic considerations, we can understand a very basic feature of the variations in the CBR. Variations in the plasma larger than the horizon scale at decoupling will generate radiation with little polarization and some amount of temperature variations. On the other hand, variations on scales smaller than the horizon size at decoupling will generate radiation with some polarization and a larger amount of temperature variations.
It is of course necessary to relate the length scales of the variations in the decoupling plasma to the angular scales of the CBR anisotropies we see on the sky. The CBR photons seen today have been free streaming since decoupling, so they originate from a spherical shell of the decoupling plasma centered on us. Since the universe has expanded while the photons were in transit, the physical radius of this shell at decoupling (which would allow us to set angular scales equal to some physical distance scales) depends on how much universe has expanded since decoupling (Figure 5). This makes the CBR sensitive to the total amount of universal expansion. This is interesting, but also a bit of a problem, since this cosmological parameter is still not well measured.
Figure 5: From variations in the plasma to variations in the CBR. In (a) we see the decoupling plasma, with the shell that will give rise to the CBR photons we receive today. As the photons decouple from the plasma and move towards us (and in all other directions), the universe expands, as shown in (b). Thus the apparent size of the variations on the sky we see when these photons reach us today (c) depends on the expansion rate of the universe.
Nevertheless, the close relationship between temperature and polarization variations allows us to make some fairly rugged predictions about the size and scale of the polarization of the CBR. Numerous experiments have measured temperature variations on a variety of scales. The measurements indicate an increase in the size of the variations at angular scales of about a degree (see Figure 6). This suggests that a degree on the sky subtends roughly the horizon scale at decoupling. Therefore we expect polarization to be strongest on sub-degree scales. Furthermore, since the observed temperature variations are on the order of tens of microkelvin on these scales, we can expect the polarization to be at most a few microkelvin.
PostScript Plot a PostScript Plot b
Figure 6: Observations of CBR anisotropies. The variations in the (a) temperature and (b) polarization of the CBR across the sky are visualized using Cl plots. In these plots we decompose the variations on the sky into spherical harmonics, with multiple number l. In general, variations of angular size q have approximately l=p/q. The rms variations per logarithmic interval in l are displayed versus l. The curves are theoretical predictions based on different cosmological models, which were generated using the program CMBFAST written by Seljak and Zaldariagga. The dots on the upper plot represent the temperature variations observed by a variety of experiments. Each observation measures the rms temperature variation within a range of l to be within a ceratin error bar, and so defines a box on the Cl plot. We display a fixed number of dots within this box for each observation, so the density of points gives an impression of what the actual temperature variations are likely to be. The theoretical curves (and to some extent, the experimental data) show the basic features described in the text. On large angular scales (low l) there is some temperature variations and very little polarization, while on smaller angular scales (higher l) there is an increase in the size of the temperature variations and the polarization becomes significant. The series of peaks and troughs in the curves reflect the oscillatory nature of the plasma's motion mentioned in the text. On smaller angular scales the variations die away due to the finite distance between photon-electron scattering events in the plasma.
A Description of the Telescope that will look for Polarization The preceding discussion indicates that we can reasonably expect that the polarization will be (1) strongest on sub-degree scales and (2) on the order of a few microkelvin strong at those scales. Now we shall describe the experimental techniques which should allow us to detect such a signal. First we describe the frequency range and angular resolution of our telescope, which should maximize the contribution of the CBR to the polarized signal sent into our detector. Next we describe the correlation receiver that allows us to detect the small amount of polarization. Then we describe where on the sky we will look for the polarized signal.
Search Frequencies Determining the frequency to observe at requires quite a few considerations including (1) a peak in the polarization, (2) domination of the CMB polarization over noise polarization at this frequency, and (3) available technology. Our detectors will be looking at 90GHz, a feat tht would have been impossible just a few years ago. We will also have a detector looking at 40 GHz in the neaer future, a much easier enterprise. The benefits of looking at two frequencies increases vastly our ability to view a signal from the period of decoupling
Putting a Fine Point on It: Angular Resolution As we discussed above, the polarized CBR is strongest on sub-degree scales. The optics of this experiment were designed to give us the ability to probe these small angular scales.
Our detectors receive the signal from the sky through a small corrugated feed horn. This horn views a 1.4 meter, off-axis parabolic dish (Figure 7). The dish and horn have been constructed so that the detection system views a region of the sky a fraction of a degree across. This should allow us to probe angular scales where the polarization is expected to be significant.
Figure 7: The Mirror and Base. Our telescope has a 1.4 off-axis parabolic mirror and a small corrugated feed horna(not shown). The ground screens surrounding the telescope keep the telescope from seeing local objects. There will also be an external ground screen beyond that. (not shown here)
Finding the Needle in the Haystack: The Radiometer Even though we maximize the polarized component of the CBR, and minimize the contribution of other sources of polarized signal, the fact remains that the CBR polarization is TINY. We are looking for a few microkelvin difference between the two components of a three Kelvin blackbody signal. Furthermore, this experiment is ground based, so we also have a 30 Kelvin atmosphere to contend with. Therefore our detector must be able to extract the polarization from a huge amount of unpolarized noise.
Figure 8: The radiometer. These figures show the radiometer (a) in principle and (b) in detail. See text for a description of how this device works.
We use a device called a correlation receiver or radiometer to select out the polarized component of the signal from the sky. The workings of this device is shown in Figure. The electric field from the sky that enters the receiver through the feed horn can be decomposed into two components Ea and Eb, as shown. The radiometer takes this field and splits it between two channels. These channels carry a different pair of orthogonal components (1/ 2)^(1/2) (Ea+ Eb) and (1/2)^(1/2) (Ea- Eb). The channels amplify these signals, mix them down, and finally send them into multipliers. These multipliers ideally produce an output voltage proportional to the product of the input fields:(1/ 2)(Ea^2 - Eb^2). Thus the receiver's DC output should be directly proportional to the difference in the power between the two components of the incident field, (i.e. the polarization).
Of course, the multiplier is not ideal and also produces an undesirable output voltage proportional to the total incident power. To deal with this, we insert a phase switch into one of the lines. This device multiplies the field in one of the arms by plus/minus 1, which changes the sign of the polarized signal but leaves the total power signal unchanged. Thus by flipping the phase switch at a certain frequency and locking in on only the signal that varies at that frequency, we can better extract the polarization.
Waiting for the Polarization: Our Obsevation Strategy
Our observing strategy (where we point our telescope on the sky) is strongly influenced by the amount of time we have to wait to see any signal. Even though the radiometer selects out the polarization from the incoming sky signal, this signal will still be noisy because of random fluctuations in the amplifiers. Thus we need to repeatedly measure the polarization on any given point on the sky and take the average of these measurements to find the signal buried in the noise. We therefore have to spend a considerable amount of time focused on one spot in the sky, and can only observe a handful of points within a reasonable amount of time.
Figure 9: The scan pattern.We see where we will look for polarization on the sky. The telescope observes along the ring chopping every few minutes between the two marked spots. At these two spots we measure the difference in the power along the two axis shown. The signal from the two spots should be six hours out of phase and of opposite sign.
We have chosen to observe at a single declination 1 degree from the north pole (Figure 9). If we hold our telescope at a fixed position (with respect to the ground) on this ring, the telescope will see every point on this ring every day as the sky rotates under the beam. The telescope will be able to see on the order of 20 independent spots on this ring.
In fact, we do not look at just one point on the sky, but chop slowly (every few minutes) between two points separated by 90 degrees along the circle. The polarized sky signals detected at these two points should have opposite signs and be six hours out of phase. Thus the two data sets will provide a powerful consistency check on one another.
Conclusions and Outlook
The polarization of the CBR is a fascinating phenomenon and can provide with useful cosmological information. Our telescope should allow us to measure this elusive signal. The telescope is currently being constructed and observations should begin from the roof of the physics building this summer ('99).
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