# Understanding The Fluctuations In The CMB Maps

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If I'm understanding this correctly the fluctuations in CMB are a result of the "last scatter" of photons when electrons joined together with nuclei that before this formed plasma. CMB is among other things used to see the early matter density fluctuations of the early universe, right?

So, how is this seen? Are the warmer parts in the maps where the matter is most dense? And are these spots warmer because more photons are coming from that spot, or because of some other reason?

The anisotropies in the CMB are caused by four effects; three at the surface of last scattering (SoLS), and one after:

1. Temperature differences
• Denser regions will be more compressed and thus hotter, on average. Hence, an overdensity will result in a hotter spot, with a fractional fluctuation $$Delta T/T_0$$.
2. Gravitational redshift
• Photons climbing up (or falling down) the gravitational potential $$phi$$ of their SoLS will lose (or gain) energy. Hence, an overdensity will result in a colder spot, and an underdensity in a hotter spot. This effect is also called the non-integrated Sachs-Wolfe effect and changes the temperature by $$Delta T = phi/3c^2$$, where $$c$$ is the speed of light.
3. Doppler shift
• The SoLS has a bulk motion $$mathbf{v}$$. Generally, overdensities have gas infalling, and hence result in a colder spot. In the direction $$hat{mathbf{n}}$$, the temperature difference will be $$Delta T = T_0 mathbf{v}/c$$.
4. The integrated Sachs-Wolfe effect
• If a potential well is getting shallower in time (which happens during dark-energy domination), photons receive a net blueshift in crossing the well and the CMB appears hotter. This gives a temperature difference of $$Delta T = 2Deltaphi/c^2$$, where $$Deltaphi$$ is the change in the potential while a photon traversed the potential well.

The total temperature fluctuation is the sum of all these terms. In general, the gravitational redshift will dominate over the temperature differences, which in turn dominates over the Doppler shift.

A review of the CMB fluctuations is given by Challinor (2013).

## 15.3: Observations of Variations in the CMB

Some students just went to a public lecture where a cosmologist showed a map of the CMB. The students are discussing what the colors in the map mean.

• Krystal: I have no idea what that map was showing. It just looks like Swiss cheese to me.
• Lenny: Well, I know it was microwave light not regular light, so the different colors could stand for something other than color. I just don&rsquot know what.
• Marianne: I think it was a map of where the galaxies are. Like each one of those spots was a galaxy .
• Nasir: It doesn&rsquot look like galaxies to me.

When scientists look at the spatial distribution of the CMB in detail, they notice that superimposed on the nearly uniform background are tiny (about 10 &ndash 100 parts per million) variations in temperature. This departure from uniformity, called anisotropy, was first observed by the Differential Microwave Radiometer (DMR) instrument on the COBE satellite in 1991 (Figure 15.6). This was important because for the first time it showed that the microwave background was not completely even. Since COBE&rsquos early result, fluctuations in the CMB have been measured with increasing precision by dozens of instruments.

Figure 15.2 such that the red spots are slightly hotter than the blue spots by about 1/10,000th of a degree Kelvin . Animated Figure 15.7 shows a map of the CMB as observed by COBE and then later by the WMAP satellite at better resolution . It is possible to distinguish smaller angular sizes in the WMAP data . The red spots in the map correspond to slightly hotter regions and the blue spots correspond to cooler regions. The contrast between the hotter and colder regions is greatly exaggerated in order to see the effect their typical temperature differences are about 1/10, 000th of a degree!

Animated Figure 15.7 CMB anisotropy was first observed by the COBE satellite in 1991 at large angular scales (initial image in animation). Since then, it has been observed with better resolution by dozens of other instruments, including the WMAP satellite (final image in animation). The contrast has been enhanced compared to the CMB map in Figure 15.2 such that the red spots are slightly hotter than the blue spots by about 1/10,000th of a degree Kelvin. The WMAP image has better angular resolution than the COBE/DMR image. The difference between DMR and WMAP is similar to the difference between a near-sighted person without their glasses and a person who does not need glasses trying to recognize faces in a crowded room. The person who does not need glasses can see more detail than the person who does. Credit: NASA/WMAP Science Team

In both COBE and WMAP data, microwave signals from our Galaxy and other galaxies have been removed from the map, leaving only an image of the CMB. This is possible because the CMB has a unique blackbody spectral signature. Other sources of microwave emission, such as dust in our galaxy, have different spectral signatures. By observing the sky at several different frequencies, we are able to isolate and then subtract the sources other than the CMB. This process is illustrated in Animated Figure 15.8, with data from the Planck satellite.

Animated Figure 15.8 Microwave radiation from our Galaxy and other galaxies has been removed from the data in order to make a map of only the CMB. Credit: ESA and the Planck Collaboration

Another source of microwave light that can be found in CMB maps is a signature from the interaction of CMB photons with galaxy clusters, known as the Sunyaev-Zel&rsquodovich (SZ) effect. Because of observations of the SZ effect, we know the CMB must be cosmological, that is, coming from all over the Universe, not just a nearby source (such as our Galaxy). Light from the CMB is distorted in the vicinity of galaxy clusters, which means it must be coming from places beyond the galaxy clusters (which are themselves far away) in order to be affected. The SZ effect can be seen at great distances, because it does not follow an inverse square law. This makes it ideal for finding galaxy clusters. Several groups are involved in efforts to measure the SZ effect and use it to survey galaxy clusters, as described in Going Further 15.1: The Sunyaev-Zel&rsquodovich (SZ) Effect.

Note: The Sunyaev-zel&rsquodovich Effect

The Sunyaev-Zel&rsquodovich effect is caused by interactions of photons with charged particles. We have already discussed on several occasions how photons can be scattered by charged particles, typically electrons. Sometimes this scattering merely changes the direction of the photons without changing their energy. That is what happens to sunlight as it passes through Earth&rsquos atmosphere, and the preferential scattering of short wavelengths over longer causes the sky to appear blue on a clear day.

However, it is also possible for the scattering of light to transfer energy to or from the photons. If the population of scattering electrons has on average more energy than the population of photons, then energy is transferred to the photons, which appear hotter as a result. On the other hand, if the typical energy of the electrons is lower than the energy of the photons, then the light will transfer energy to the electrons. The light cools at it looses energy to the electrons.

The microwave background began as light with a typical temperature around 3000K. Since the Universe has grown by a factor of about a thousand since the CMB was created, the CMB light now is characterized by a temperature of around 3K, and of course, it was cooling from 3000K to 3K over its entire journey. However, some of the microwave photons have encountered galaxy clusters as they traveled through the cosmos. These photons have had their temperature modified by the scattering described previously.

In addition to containing galaxies, galaxy clusters also contain immense amounts of hot gas. The gas, which actually dominates the total baryonic mass of the clusters, has a temperature around 10 million kelvin, much higher than that of the CMB. This is because it sits deep in the gravitational well of the cluster. When CMB photons encounter the hot gas they gain energy from its particles because the CMB is so much cooler than the typical electron in the cluster gas. As a result, the photons passing through a cluster have a slightly increased temperature than CMB photons that don&rsquot pass through a cluster. The spectrum is distorted and shifted to higher energy. The CMB photons mark the positions of the clusters on the sky because they cause the clusters to stand out as small spots in the microwave background. At higher wavelengths than the peak of the CMB, these spots are hotter, and at longer wavelengths (lower energies), these spots are colder, because the photons that would normally be present have been shifted to a different part of the spectrum. An example of an SZ cluster observation is shown in Figure B.15.1.

FIGURE B.15.1: SZ EFFECT. Galaxy cluster Abell 2744 as seen by the South Pole Telescope (SPT). Light from the CMB interacts with the hot gas in the cluster, causing a cold spot at the wavelengths that SPT observes. This is called the Sunyaev-Zel&rsquodovich (SZ) effect. Credit: SPT Collaboration.

The SZ effect is important at high-l values, or small angular scales, because galaxy clusters are not very large on the sky. The small SZ temperature fluctuations are not caused by any primordial cosmological parameters, they are modifications of the original CMB spectrum by intervening mass concentrations as its photons travel to us. These fluctuations must be accounted for in the cosmological analysis of the CMB since they introduce a spurious signal (or noise, if you prefer) on the true cosmological perturbations.

However, the SZ effect is interesting in its own right because it provides a way to understand how galaxy clusters have formed and evolved over the history of the Universe. Because the CMB comes to us from essentially the beginning of the Universe, it probes as far as can be probed by photons. What&rsquos more, the SZ effect depends only on the angular size of the clusters and the temperature of their gas, it is not affected by the 1/r 2 dimming law that affects many astronomical measurements. Thus SZ provides a powerful means to study structures anywhere along the line of sight. One of the many telescopes studying the SZ effect is the South Pole Telescope. Animated Figure B.15.2 shows this telescope being built.

The South Pole Telescope (SPT) is one of several telescopes studying the SZ effect, which can be used to find galaxy clusters. Credit: SPT Collaboration.

In the next activity, you will examine several CMB maps to get a sense of the temperature and angular scales of variations in the CMB.

## Unexpected topology of the temperature fluctuations in the cosmic microwave background

In: Astronomy and astrophysics , Vol. 627, No. July 2019, A163, 01.07.2019.

Research output : Contribution to journal › Article › Academic › peer-review

T1 - Unexpected topology of the temperature fluctuations in the cosmic microwave background

N2 - We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 핊2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of "masks" is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3σ and 4σ at scales of 3-7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck's predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power- spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super- horizon scales involved, may motivate the study of primordial non- Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.

AB - We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 핊2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of "masks" is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3σ and 4σ at scales of 3-7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck's predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power- spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super- horizon scales involved, may motivate the study of primordial non- Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.

## Cosmic Background Explorer

COBE Spacecraft Construction: In 1989, the Cosmic Background Explorer (COBE) spacecraft was launched into an Earth orbit to make a full sky map of the Cosmic Microwave Background (CMB) radiation. COBE found very subtle irregularities in the otherwise very uniform CMB, findings that are considered important evidence in support of the Big Bang theory.
Credit: NASA / COBE Science Team

The Cosmic Background Explorer (COBE) mission was led by John Mather. The COBE group was trying to figure out how to do measurements of the cosmic microwave background radiation. Even though the particular science I had been doing at Chicago was different, the techniques we had been using for infrared experiments were exactly what was needed to build the COBE. John developed a “theory of operation” for the detectors we were going to use on the COBE. This is basically a mathematical description of how the device should work. At the time, John’s wife was a ballet teacher in New York, and he would sit during her classes with a little programmable calculator (this was before there were laptops) to try to figure out the problem. This was a really different way of doing things, because people usually started by thinking about what detectors were going to be used, and what the different types of detectors were capable of doing, and where to buy them, and so on. But John was trying to develop a theoretical basis, so that you would understand how the performance of the detector could change if you modified the device in certain ways.

COBE All-Sky Map: The final COBE all-sky map of the apparent fluctuations in the cosmic background, thought to be the seeds of the large scale structure of the Universe.
Credit: NASA

## Planck's most detailed map ever reveals an almost perfect Universe

The anisotropies of the Cosmic microwave background (CMB) as observed by Planck. The CMB is a snapshot of the oldest light in our Universe, imprinted on the sky when the Universe was just 380 000 years old. It shows tiny temperature fluctuations that correspond to regions of slightly different densities, representing the seeds of all future structure: the stars and galaxies of today. Credit: ESA

(Phys.org) —Acquired by ESA's Planck space telescope, the most detailed map ever created of the cosmic microwave background – the relic radiation from the Big Bang – was released today revealing the existence of features that challenge the foundations of our current understanding of the Universe.

The image is based on the initial 15.5 months of data from Planck and is the mission's first all-sky picture of the oldest light in our Universe, imprinted on the sky when it was just 380 000 years old.

At that time, the young Universe was filled with a hot dense soup of interacting protons, electrons and photons at about 2700ºC. When the protons and electrons joined to form hydrogen atoms, the light was set free. As the Universe has expanded, this light today has been stretched out to microwave wavelengths, equivalent to a temperature of just 2.7 degrees above absolute zero.

This 'cosmic microwave background' – CMB – shows tiny temperature fluctuations that correspond to regions of slightly different densities at very early times, representing the seeds of all future structure: the stars and galaxies of today.

According to the standard model of cosmology, the fluctuations arose immediately after the Big Bang and were stretched to cosmologically large scales during a brief period of accelerated expansion known as inflation.

When compared to the best fit of observations to the standard model of cosmology, Planck’s high-precision capabilities reveal that the fluctuations in the cosmic microwave background at large scales are not as strong as expected. The graphic shows a map derived from the difference between the two, which is representative of what the anomalies could look like. Credit: ESA

Planck was designed to map these fluctuations across the whole sky with greater resolution and sensitivity than ever before. By analysing the nature and distribution of the seeds in Planck's CMB image, we can determine the composition and evolution of the Universe from its birth to the present day.

Overall, the information extracted from Planck's new map provides an excellent confirmation of the standard model of cosmology at an unprecedented accuracy, setting a new benchmark in our manifest of the contents of the Universe.

But because precision of Planck's map is so high, it also made it possible to reveal some peculiar unexplained features that may well require new physics to be understood.

"The extraordinary quality of Planck's portrait of the infant Universe allows us to peel back its layers to the very foundations, revealing that our blueprint of the cosmos is far from complete. Such discoveries were made possible by the unique technologies developed for that purpose by European industry," says Jean-Jacques Dordain, ESA's Director General.

"Since the release of Planck's first all-sky image in 2010, we have been carefully extracting and analysing all of the foreground emissions that lie between us and the Universe's first light, revealing the cosmic microwave background in the greatest detail yet," adds George Efstathiou of the University of Cambridge, UK.

One of the most surprising findings is that the fluctuations in the CMB temperatures at large angular scales do not match those predicted by the standard model – their signals are not as strong as expected from the smaller scale structure revealed by Planck.

Another is an asymmetry in the average temperatures on opposite hemispheres of the sky. This runs counter to the prediction made by the standard model that the Universe should be broadly similar in any direction we look.

Furthermore, a cold spot extends over a patch of sky that is much larger than expected.

The asymmetry and the cold spot had already been hinted at with Planck's predecessor, NASA's WMAP mission, but were largely ignored because of lingering doubts about their cosmic origin.

"The fact that Planck has made such a significant detection of these anomalies erases any doubts about their reality it can no longer be said that they are artefacts of the measurements. They are real and we have to look for a credible explanation," says Paolo Natoli of the University of Ferrara, Italy.

"Imagine investigating the foundations of a house and finding that parts of them are weak. You might not know whether the weaknesses will eventually topple the house, but you'd probably start looking for ways to reinforce it pretty quickly all the same," adds François Bouchet of the Institut d'Astrophysique de Paris.

One way to explain the anomalies is to propose that the Universe is in fact not the same in all directions on a larger scale than we can observe. In this scenario, the light rays from the CMB may have taken a more complicated route through the Universe than previously understood, resulting in some of the unusual patterns observed today.

"Our ultimate goal would be to construct a new model that predicts the anomalies and links them together. But these are early days so far, we don't know whether this is possible and what type of new physics might be needed. And that's exciting," says Professor Efstathiou.

Beyond the anomalies, however, the Planck data conform spectacularly well to the expectations of a rather simple model of the Universe, allowing scientists to extract the most refined values yet for its ingredients.

Normal matter that makes up stars and galaxies contributes just 4.9% of the mass/energy density of the Universe. Dark matter, which has thus far only been detected indirectly by its gravitational influence, makes up 26.8%, nearly a fifth more than the previous estimate.

Conversely, dark energy, a mysterious force thought to be responsible for accelerating the expansion of the Universe, accounts for less than previously thought.

Finally, the Planck data also set a new value for the rate at which the Universe is expanding today, known as the Hubble constant. At 67.15 kilometres per second per megaparsec, this is significantly less than the current standard value in astronomy. The data imply that the age of the Universe is 13.82 billion years.

"With the most accurate and detailed maps of the microwave sky ever made, Planck is painting a new picture of the Universe that is pushing us to the limits of understanding current cosmological theories," says Jan Tauber, ESA's Planck Project Scientist.

"We see an almost perfect fit to the standard model of cosmology, but with intriguing features that force us to rethink some of our basic assumptions.

"This is the beginning of a new journey and we expect that our continued analysis of Planck data will help shed light on this conundrum."

## Vibrations of the Cosmic Drumhead

A multiconnected topology translates into the fact that any object in space may possess several copies of itself in the observable Universe. For an extended object like the region of emission of the CMB radiation we observe (the so-called last scattering surface) it can happen that it intersects with itself along pairs of circles [5]. In this case this is equivalent to say that an observer (located at the center of the last scattering surface) will see the same region of the Universe from different directions. As a consequence the temperature fluctuations will match along the intersection of the last scattering surface with itself as illustrated in the above figure. This CMP map is simulated for a multi-connected flat space - namely a cubic hypertorus whose length is 3.17 times smaller than the diameter of the last scattering surface.

Cosmologists hope to “hear the shape of space”, namely its topology, by analyzing in detail the temperature fluctuations in the cosmic microwave background radiation (CMB). An international team of cosmologists, including researchers from l’Observatoire de Paris, has recently developped a model for the vibrations of the universe. For the first time [1], they have simulated high resolution CMB maps containing the signatures of a wide class of topologies, for comparison with the forthcoming MAP satellite data in early 2003.

In recent years, cosmologists have become interested in the global shape of space [2]. Previously, most of them had neglected the fact that, even if space is flat on a large scale, it can take many different shapes, for instance that of a doughnut-like hypertorus. A space of a given curvature admits a number a topologies. Indeed, 18 flat topologies along with an infinite number of spherical and hyperbolic ones are theoretical candidates to describe the shape of physical space.

Although recent CMB observations constrain the value of space curvature to a very narrow range about zero, they still leave open the question of whether the average curvature is exactly zero (corresponding to a flat universe) slightly positive (spherical universe) or slightly negative (hyperbolic universe), and above all whether the topology is simple (for instance an infinite flat space) or not (for instance a finite flat hypertorus).
In a previous article [3], three authors have proved that the spherical topologies would be more easily detectable observationally than hyperbolic or flat ones. The reason is that, no matter how close space is to perfect flatness, only a finite number of spherical shapes are excluded by observational constraints. Due to the special structure of spherical spaces, topological imprints would be potentially detectable within the observable universe. Thus cosmologists are taking a renewed interest in spherical spaces as possible models for the physical universe. Now the main question is : how to detect the topology of space?

If you sprinkle fine sand uniformly over a drumhead and then make it vibrate, the grains of sand will collect in characteristic spots and figures, called Chladni patterns. These patterns reveal much information about the size and the shape of the drum and the elasticity of its membrane. In particular, the distribution of spots depends not only on the way the drum vibrated initially but also on the global shape of the drum, because the waves will be reflected differently according to whether the edge of the drumhead is a circle, an ellipse, a square, or some other shape.

In cosmology, the early Universe was crossed by real acoustic waves generated soon after Big Bang. Such vibrations left their imprints 300 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations. Thus the CMB temperature fluctuations look like Chladni patterns resulting from a complicated three-dimensional drumhead that vibrated for 300 000 years. They yield a wealth of information about the physical conditions that prevailed in the early Universe, as well as present geometrical properties like space curvature and topology. More precisely, density fluctuations may be expressed as combinations of the vibrational modes of space, just as the vibration of a drumhead may be expressed as a combination of the drumhead’s harmonics.

For the first time, a team of physicists has shown how the shape of a spherical space can be heard in a unique way. They calculated the harmonics (the so-called “eigenmodes of the Laplace operator”) for most of the spherical topologies [4]. Next, starting from a set of initial conditions fixing how the universe originally vibrated (the so-called Harrison-Zeldovich spectrum), they evolved the harmonics forward in time to simulate realistic CMB maps for a number of topologies, including flat and spherical ones [1].

Balloon-borne CMB experiments (Boomerang, DASI, Archeops) have put tight constraints on the curvature of space, but provide too little data to test the topology of the Universe because they cover only a small portion of the sky. The situation is about to change dramatically with the MAP (Microwave Anisotropy Probe) satellite mission.

Launched by NASA in April, 2001, it will provide high resolution maps of CMB fluctuations on the whole sky, excluding the portion obscured by our own Milky Way galaxy. The 6-month MAP data will be released late January or early February 2003. A topological signal as predicted in [5], and simulated in the maps of [1], may be subtly encoded in these data, and may eventually answer the fascinating question whether space is finite.

Peer reviewed publications and references

[1] A. Riazuelo, J.-P. Uzan, R. Lehoucq and J. Weeks, “Simulating Cosmic microwave background maps in multi-connected universes” (e-print astro-ph/0212223).

[2] J.- P. Luminet: “L’Univers chiffonné”, Fayard, Paris, 2001, 369 p.
R. Lehoucq: “L’univers a-t-il une forme ?”, Flammarion, Paris 2002, 152 p.
J. Weeks : “The Shape of Space”, Dekker, 2nd edition, 2001, 328 p.

[3] J. Weeks, R. Lehoucq and J.-P. Uzan: “Detecting topology in a Nearly Flat Spherical Universe”, (e-print astro-ph/0209389).

[4] R. Lehoucq, J. Weeks, J.-P. Uzan, E. Gausmann and J.-P. Luminet, “Eigenmodes of 3-dimensional spherical spaces and their application to cosmology”, Classical and
Quantum Gravity,(2002) 19, 4683-4708 (e-print gr-qc/0205009).

[5] N. Cornish, D. Spergel and G. Starkman,”Circles in the sky: finding topology with the microwave background radiation”, Classical and Quantum Gravity (1998), 15,
2657-2670 (e-print astro-ph/9801212).

## W ayne h u

The Thomson scattering cross section depends on polarization as (see e.g. [Chandrasekhar] 1960)

where ( ) are the incident (scattered) polarization directions. Heuristically, the incident light sets up oscillations of the target electron in the direction of the electric field vector , i.e. the polarization. The scattered radiation intensity thus peaks in the direction normal to, with polarization parallel to, the incident polarization. More formally, the polarization dependence of the cross section is dictated by electromagnetic gauge invariance and thus follows from very basic principles of fundamental physics.

If the incoming radiation field were isotropic, orthogonal polarization states from incident directions separated by would balance so that the outgoing radiation would remain unpolarized. Conversely, if the incident radiation field possesses a quadrupolar variation in intensity or temperature (which possess intensity peaks at separations), the result is a linear polarization of the scattered radiation (see Fig. 1). A reversal in sign of the temperature fluctuation corresponds to a rotation of the polarization, which reflects the spin-2 nature of polarization.

Fig. 1: Thomson scattering of radiation with a quadrupole anisotropy generates linear polarization. Blue colors (thick lines) represent hot and red colors (thin lines) cold radiation.

In terms of a multipole decomposition of the radiation field into spherical harmonics, , the five quadrupole moments are represented by , . The orthogonality of the spherical harmonics guarantees that no other moment can generate polarization from Thomson scattering. In these spherical coordinates, with the north pole at , we call a N-S (E-W) polarization component Q >0 ( Q <0) and a NE-SW (NW-SE) component U >0 ( U <0). The polarization amplitude and angle clockwise from north are

Alternatively, the Stokes parameters Q and U represent the diagonal and off diagonal components of the symmetric, traceless, intensity matrix in the polarization plane spanned by ( , ),

where are the Pauli matrices and circular polarization is assumed absent.

If Thomson scattering is rapid, then the randomization of photon directions that results destroys any quadrupole anisotropy and polarization. The problem of understanding the polarization pattern of the CMB thus reduces to understanding the quadrupolar temperature fluctuations at last scattering.

Temperature perturbations have 3 geometrically distinct sources: the scalar (compressional), vector (vortical) and tensor (gravitational wave) perturbations. Formally, they form the irreducible basis of the symmetric metric tensor. We shall consider each of these below and show that the scalar, vector, and tensor quadrupole anisotropy correspond to respectively. This leads to different patterns of polarization for the three sources as we shall discuss in ډ.

## Understanding The Fluctuations In The CMB Maps - Astronomy

### Journal Information

International Journal of Astronomy

p-ISSN: 2169-8848 e-ISSN: 2169-8856

### Cosmology with the Cosmic Microwave Background

1 Department of Physics, University of Kashmir, Hazratbal, 190006, Srinagar, J&K, India

2 Inter-University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune, India

Correspondence to: Manzoor A. Malik, Department of Physics, University of Kashmir, Hazratbal, 190006, Srinagar, J&K, India.

 Email:

Cosmic Microwave Background is credited with bringing out Cosmology from Speculation to Precision Science. Many of the key cosmological parameters like its age (13.75 Billion Years), its composition (4.6% Baryons, 22.7% Dark Matter, 72.8% Dark Energy) and curvature (flat) are now known to a high degree of accuracy (sub-percent to percent level). It not only provides a more acceptable basis for the Big Bang Model but also gives an observational basis to the formation of structures that we see in the present Universe, ruling out many cosmological models. In this paper, I intend to review as to what we have learnt from the cosmic microwave background and what we expect to learn from it in future, in light of the recent (and upcoming) experimental/observational and theoretical advances.

Keywords: Cosmic Microwave Background, Anisotropies, COBE, WMAP, PLANCK

Cite this paper: Manzoor A. Malik, Cosmology with the Cosmic Microwave Background, International Journal of Astronomy, Vol. 2 No. 2, 2013, pp. 17-22. doi: 10.5923/j.astronomy.20130202.01.

### 2. Discovery and Early History

3K. Again, nobody realized the importance of this work at that point. In 1957, Shamaonov[11] had reported that the absolute effective temperature of the radio emission background (now known as CMB) to be around 4K. Their measurements showed that the radiation intensity was independent of either the time or the direction of observations. This work also went almost unnoticed. In the literature, we find a number of near misses which could all have been treated as a discovery of CMB. In 1964, two radio astronomers, Penzias and Wilson began their search to detect sources of radiation that might potentially harm satellites. To their frustration, they found a uniform signal in the microwave range, seeming to come from all directions. Having no clue what it was, they tried everything to eliminate this “unwanted” background noise but failed. Luckily, Penzias talked to Burke, a scientist at MIT, who advised him to talk to Bob Dicke at Princeton. When the phone call ended, Dicke is reported to have told his colleagues “Boys, we have been scooped”. This was because Bob Dicke and his collaborators (including Dave Wilkinson after whom WMAP was later named) were precisely looking for the signal that was an unwanted noise for Penzias and Wilson and were almost ready with an experiment to find the CMB. CMB had been discovered by Penzias and Wilson! The spectrum of CMB fitted a nearly perfect black body with its temperature at around 3K. The experimental results of Penzias and Wilson[12] detailing the observation and the cosmological interpretation by Dicke, Peebles, Roll and Wilkinson[13] were published side by side in the Astrophysical Journal. For their discovery, Penzias and Wilson, were awarded the 1978 Nobel prize in Physics. Following the serendipitous discovery of the CMB, the next step was to look for the anisotropies in the CMB. Early attempts limited by the low sensitivity of instruments, indicated that CMB was discouragingly smooth. Many experiments were designed to look for the characteristics of CMB but the real breakthrough came with the COBE mission’s confirmation of the perfect black body spectrum of the CMB and the small (1 part in 10 5 ) but significant anisotropies in the cosmic microwave background. John Mather and George Smoot shared the 2006 Nobel prize in Physics for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation respectively.

## W ayne h u

### Key Concepts

• Gravitational waves show a power spectrum with both E and B mode contributions
• Limits on the gravitational wave contribution to the temperature anisotropy imply B-modes < a few tenths of a microKelvin.
• Gravitational waves probe the physics of inflation but will require a thorough understanding of foregrounds and secondary effects for their detection.

If there were only gravitational waves and no density perturbations in the Universe, the CMB temperature, polarization and temperature-polarization cross power spectra would look like:

Notice that the polarization contains power in both the E and B-modes. That we do see acoustic peaks in the spectrum indicates that this scenario cannot actually be true. At most, gravitational waves contribute a fraction of the power in temperature anisotropies. Adding back in the density fluctuations, the power spectrum as a function of the ratio of power in the gravitational wave (tensor, T) versus density (scalar, S) modes becomes:

For realistic values of this ratio or " T/S ", the power in the B-mode corresponds to a tenth of a micro Kelvin signal on scales of l

Needless to say, this signal will be very difficult to detect in the presence of foregrounds and secondary anisotropies that also produce B-modes. The rewards of detecting it are however great. The amplitude and spectrum of the gravitational wave contributions are our best probes of the physics of the inflationary epoch .

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