Abstract

 

Throughout history, cosmologists have constructed theories that are, essentially, the building blocks of the universe. Over this paper, I will describe the theory of inflation and the problems that were solved by its creation. I will also dive into the cosmic microwave background (CMB) and its history, as well as what it tells us about the formation of the early universe and how we know it today. Using python, I also generated and examined simulated data to tell us more about the distribution of the universe and factors that affect its analysis.

 

Introduction

 

In the world of cosmology, the great majority of cosmologists support the theory of the Big Bang – the idea that the universe underwent exponentially large expansion within a split second. Despite the research studies conducted in recent years surrounding the Big Bang, cosmologists are still searching for ways to prove the theory. Such proofs have been analyzed through observations created around the cosmic microwave background – background radiation leftover from the Big Bang – that didn’t take shape until the last couple of decades. Through the development of new technologies such as the COBE satellite 1  to analyze this background radiation, cosmologists are a step closer to proving the Big Bang theory and now have a deep understanding of how the universe was formed and what initial conditions impacted its creation, such as the theory of inflation.

 

Inflation

 

In modern cosmology, three main problems have persisted across decades in the search for theorizing the creation of the universe: the flatness problem, the monopole problem, and the horizon problem. However, the theory of inflation is one that aims to solve these three governing problems. Coined by cosmologist Alan Guth in 19812 , inflation is defined as a period of time between 10^-33 and 10^-32 seconds after the Big Bang singularity, or the initial “creation” of the universe where the universe underwent very rapid exponential expansion. This expansion saw the length of the universe expanding at a linear factor of at least 10^26, with the volume – the amount of space in the universe – expanding at a factor of at least 10^78. This inflationary epoch, as it is described, is a theory that is becoming increasingly popular among cosmologists due to its properties that can theoretically resolve these three governing problems.

 

The Flatness Problem

 

The flatness problem, also known as one of Dicke’s coincidences, was proposed by American astronomer and physicist Robert Dicke in 1969. The problem surrounds the understanding of the Friedmann equations, a set of equations in cosmology by cosmologist Alexander Friedmann that relates the scale factor of expansion of the universe, its density, and its curvature3.

 

According to the Friedmann equations, the universe is theorized to be almost perfectly flat in the present day, but this would propose that the universe would have to be even flatter in the past due to the idea of expansion. This flatness has to do with the density of matter in the universe and its relation to its “critical density”. Critical density describes the average density of 43matter required for the universe to be at its limit to eventually prevent any further expansion, and when at this critical density the universe is said to be flat4. The Friedmann equations follow the formula ρc=3H2/8πG, where ρc is the critical density required for a flat universe, H is the Hubble Constant, and G is the gravitational constant5. Since the values of H and G are constants, the value of ρc is derived to be equal to approximately 10^-26 km m^-36. When incorporating the actual density of the universe into this equation, the equation then follows that ρ/ρc=3H2/8πG, where ρ/ρc – the ratio of the actual density over the critical density – can be represented by the greek letter Ω5. 

 

Friedmann theorized that if Ω < 1, then the universe would be “open” and infinitely expand, following a negative curvature (like the shape of a saddle). If Ω > 1, Friedmann theorized that the universe would be “closed” and eventually collapse in on itself, following positive curvature (like the shape of a sphere). In reality, the value of Ω is almost exactly the value of 1, meaning that the universe is almost exactly flat7. The flatness problem, however, arises when comparing the current value of Ω to the value of Ω when the universe was initially expanding.

 

At the beginning of the universe’s creation immediately after the Big Bang, it can be theorized that the shape of the universe could have been any shape, but the rapid exponential expansion from the inflationary period flattened it out suddenly, thus solving the flatness problem. The period of inflation made this Ω value so close to 1 that the universe is almost perfectly flat, despite it being inferred to be a different value from Ω in the past when it was initially formed.

 

The Horizon Problem 

 

The second issue that arises with the Big Bang model is the horizon problem. This discrepancy centers around the temperature of the universe when involving the Cosmic Microwave Background (CMB). The CMB can be described as the background microwave radiation left over from the Big Bang, and this radiation is a consistent 2.7 Kelvin throughout, achieving an equilibrium8. However, according to the Big Bang model, since some regions in the universe today are so far apart from each other, CMB photons can’t make contact with each other. This means that this temperature equilibrium shouldn’t theoretically be achieved as it is in our modern model of the universe; the distance between these regions in lightyears is so great that it exceeds the actual age of the universe. 

 

When viewing the CMB, the distance between the farthest objects in the universe is around 46 billion light years apart, but the age of the universe is only around 13.8 billion years old; for these two objects in space to reach an equilibrium with each other would be impossible9. Essentially, the time it takes for the CMB to transfer from one side of the universe to another is impossible as light cannot move faster, meaning that both regions couldn’t have made contact, and thus couldn’t reach the same equilibrium temperature of 2.7 Kelvin.

 

According to the inflationary model, the universe was initially extremely small and then it suddenly expanded during the inflationary epoch period. The regions of the universe that seem ever so distant apart today would have been in contact with each other in the early universe, and during inflation these regions grew exponentially far from each other; solving the suspicion that this equilibrium would have never been achieved.

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The Monopole Problem 

 

The final issue associated with the Big Bang model was the monopole problem that is based upon the Grand Unified Theory. The Grand Unified Theory (GUT) is a model that describes the four fundamental forces: strong, weak, electromagnetic, and gravity forces. Three of these four fundamental forces – strong, weak, and electromagnetic – can be categorized as one unifying theoretical framework, known as the GUT scale10. This is because during the early existence of the universe when it was small and compact, it was extremely hot and dense, causing all three of these forces to be detected as one unifying force. As time went on and the universe cooled down, this drastic difference in temperature caused the GUT forces to divide throughout the universe in the form of magnetic monopoles. Magnetic monopoles are hypothetical particles with a magnetic charge with only one isolated magnetic pole, causing them to have different magnetic forces than other traditional magnetic particles. However, these monopoles have never been directly observed due to their small density, leading to the rise of the monopole problem11.

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The issue of detecting these monopoles can be solved with the theory of inflation, ultimately describing how the inflationary period made these monopoles diluted. As inflation occurred and the universe suddenly expanded, this expansion decreased the presence of magnetic monopoles in the universe, making them increasingly harder to observe. The density of them in the universe is so small because of inflation, making it not surprising that they are difficult to see.

 

The Cosmic Microwave Background

 

The Cosmic Microwave Background (CMB) is the background microwave radiation in the universe. After the Big Bang, the universe underwent massive inflation, expansion, and cooling. In essence, the CMB is able to allow astronomers and cosmologists alike to gather and analyze data on the primordial universe, or specifically around 13.7 billion years ago. At the time of the formation of the universe with the Big Bang, the temperature release was initially drastic, being estimated at around 1000 trillion degrees Celsius right after its occurrence. However, after around 380,000 years after the Big Bang, the universe had cooled to around 3000 Kelvin, or 2700 degrees Celsius12. According to cosmologists, This recombination occurred because the universe had cooled to a point where electrons were able to combine with protons to form hydrogen atoms; the temperature was too low for them to separate again. Paired with a phenomena called decoupling – a period where different particles fall out of equilibrium with each other – radiation was released into the universe, and such radiation is what we experience today with the radiation of the CMB13. Although the CMB’s formation was such a long time ago, its radiation still floats through space, but instead at a temperature of around 2.7 Kelvin, or -270 degrees Celsius. This temperature is so low that it can’t be detected by the naked eye, but advancements in cosmology as of recent have made this detection possible.

 

Discovered in June 1963 by Robert Wilson and Arno Penzias in Holmdel, New Jersey, the CMB was first detected as a “noise”, like static, that interrupted their observations at their previous organization Bells Telephone Laboratories. After initial confusion as to what could have caused the interruption, Wilson and Penzias looked into it further. Around the same time of their discovery of the “noise”, other notable physicist Robert Dicke of Princeton University theorized that if the Big Bang theory was the reason for the creation of the universe, then a background radiation of around 3 degrees Kelvin must exist in the universe14. 

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Wilson and Penzias took this information for what they could and began to look for any evidence that could support Dicke’s theory. After later analyzing in their laboratory, Wilson and Penzias consolidated their findings with Dicke to establish that there was a low-background radiation in the universe present at around 3 degrees Kelvin, just as Dicke theorized. Shortly after discovering this enigma, the physicists officially published their findings – that there was indeed a radiation signal present in space that they named the cosmic microwave background. They declared that the CMB would be able to provide proof of the preexisting but vaguely supported Big Bang Theory, a theory that describes how the universe was created with an extremely outwards explosion of energy that expanded and grew from a single point. 

 

With this newly found information, they were then able to send out more research expeditions and projects to further explore the CMB to hopefully gather information about it and its importance to the universe. Of which was the CMB’s impact on proving the theory of inflation. Scientists were able to hypothesize that slight fluctuations were visible in the radiation of the CMB, and that these fluctuations were created as a result of the inflationary period. Essentially, if there were fluctuations in the CMB, then something must have caused these fluctuations – the CMB.

 

COBE Discoveries

 

A number of these research expeditions were conducted by NASA’s Cosmic Microwave Background Explorer (COBE) satellite, of which Wilson, Penzias, and Dicke collaborated with in order to gather further evidence on the CMB. COBE was initially launched on November 18, 1989 and, according to NASA: “carried three instruments, a Far Infrared Absolute Spectrophotometer (FIRAS) to compare the spectrum of the cosmic microwave background radiation with a precise blackbody, a Differential Microwave Radiometer (DMR) to map the cosmic radiation precisely, and a Diffuse Infrared Background Experiment (DIRBE) to search for the cosmic infrared background radiation”1. These instruments, however, were terminated in December 23, 1993 and the operation of the COBE satellite was replaced and continued by the NASA WMAP, or the Wilkinson Microwave Anisotropy Probe, mission from 2001-2010 that also worked to conduct information on the CMB15. The WMAP was also later replaced by the European Space Agency’s Planck satellite that also worked to achieve gathering information on the CMB. Because of the efforts of satellites such as the COBE, WMAP, and Planck, astronomers and cosmologists have gained extensive knowledge on the start of the universe and the factors that contributed to its formation. During the time that the COBE satellite was in commission, it had made two key discoveries that supported the theory of the Big Bang and the CMB.

 

Black-body Spectrum

 

First, it declared that the CMB had a near-perfect blackbody spectrum, or a spectrum with frequencies measured from thermal electromagnetic radiation within a surrounding body in thermodynamic equilibrium that is emitted by a black body, or an idealized physical body that absorbs all incident electromagnetic radiation. Essentially, this is radiation at thermal equilibrium that can be used to measure the relationship between an object’s temperature and the wavelength of the electromagnetic radiation that it emits. When displayed on a spectrum, this blackbody radiation is continuous, meaning that it gives off some light at all wavelengths, and also peaks at a specific wavelength. In terms of the CMB, this means that its blackbody spectrum is nearly perfect and that it would absorb almost all the radiation that falls on it; it would not reflect or transmit any radiation16.

 

Temperature Anisotropies

 

Second, the COBE declared that the CMB has faint anisotropies, or small fluctuations in temperature from the blackbody radiation left over from the Big Bang. As measured by the COBE, these temperature fluctuations were also around 2.725 Kelvin, the approximate temperature of the CMB. Astronomers were able to analyze these anisotropic fluctuations in order to reveal components of the universe that initially remained hidden in the past. They concluded that in the cooler areas of the anisotropies there were presence of galaxy clusters, and in the warmer areas there were no clusters present17. These anisotropies can also be classified into two categories based on the circumstances on which they occurred: primary anisotropies and secondary anisotropies.

 

Primary anisotropies can be described as fluctuations that have occurred before or during the recombination event. On the other hand, secondary anisotropies are fluctuations that have occurred after the recombination period where photons and other interactions like background radiation, hot gasses, or gravitational potentials have influenced the fluctuations18. Ultimately, astronomers were able to use these temperature fluctuations of the CMB to determine the approximate location of preexisting galaxy clusters, leading to greater understanding of how the universe is arranged.

 

Polarization Anisotropies

 

Polarization is a term that surrounds the properties of light and how it is oriented. When CMB photons in the early stages of the universe were reflected off each other and other particles perpendicular to the direction of propagation these photons and light waves became polarized19. The CMB polarizes in two different spectrums called E-modes and B-modes. These polar spectrums are able to support the theory of inflation when further analyzed. E-modes – an analogy for electric field (E-field) – are waves that were polarized at 90 degrees to propagation, while B-modes – an analogy for magnetic field (B-field) were polarized at 45 degrees to propagation. 

In particular, cosmologists have used B-modes specifically in order to provide evidence for the theory of inflation. B-modes can be created by two main mechanisms: the gravitational lensing of E-modes and the generation of gravitational waves arising from inflation20. As a result, cosmologists have predicted there to be a presence of two types of B-modes, those generated during the inflationary period, and those generated by gravitational lensing later in the universe’s existence. The theory of inflation, being initially difficult to approve its existence, was able to be supported with the detection of B-modes. Although hard to detect and have never been observed, cosmologists concluded that inflation would have released massive gravitational waves that polarized waves from the CMB and created B-modes. Hence, if B-modes are observed, then the theory of inflation can be confirmed.

 

Power Spectrum

 

These anisotropies that cosmologists discovered can be displayed on a series of graphs after applying different equations. Data can be plotted on a graph with with the x-axis showing different angular scales from a viewer – in this case the Planck satellite – going from 90 to 0 degrees, and the y-axis showing the temperature fluctuations of the anisotropies, or essentially the power signal of the CMB, thus capturing the spatial wavelength of fluctuations in temperature of the CMB across the universe. Also plotted on the x-axis are the multiple moment values, 𝓁, that are inversely proportion values to the angular scale. To find the multipole moment, one is able to use the equation 𝓁=180/θ. The red plot points on the spectrum in Figure 1 capture these fluctuations, showing a range of fluctuations at a particular angular scale/multipole moment. A green line of best fit can also be then generated through it, capturing the most widely accepted portrayal of the estimation of these fluctuations. Essentially, one can see the relative strength of temperature fluctuations at different angular measurements in the universe.

 

As seen in the power spectrum, the most fluctuations occur in the one degree to 0.07 degree range. In the 90 degree to one degree range, there are less fluctuations in temperature, as there is only a measured around 1000 micro-Kelvin. The temperatures then “peak” at the one degree value, reaching a temperature with a high of around 5500-5600 micro-Kelvin. After this one degree measurement, the temperature then steeply decreases and fluctuates more leading up to the 0.07 degree value, with two more major peaks happening between these values.

 

By analyzing the trends in the power spectrum, one can make out key themes in the universe’s history. It is theorized that the first “peak” in the power spectrum was caused by sound waves in the early universe compressing and creating great amounts of temperature. This compression was caused by the period of inflation, as during its occurrence the universe expanded and all matter was greatly affected21. This is also the period of time where photons were polarized into E-modes and these anisotropies were created. This peak also is also able to illustrate a general idea of the geometry of the universe. After analyzing WMAP data, scientists were able to determine that at this peak at multiple moment 𝓁=100, the critical density of the universe is remarkably close to zero, meaning that it is flat; this too supports the flatness problem mentioned earlier. After this large first peak, there are smaller corresponding peaks along the power spectrum that capture periods in the universe where soundwaves had compressed similarly, but not to the same extreme. The CMB power spectrum is able to give a holistic idea of how the temperature of the universe is structured and distributed in space.

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Figure 1: CMB Temperature Power Spectrum22

 

Similar to temperature anisotropies, a power spectra can also be generated for E-mode and B-mode anisotropies, as featured in Figure 2. The E-modes power spectra is constructed similarly to the temperature power spectra in that the x-axis and y-axis are representative of the same measurements and units. The x-axis follows with the monopole moment, and the y-axis shows E-mode fluctuations.

 

The spectrum is closely related to the temperature power spectra, as where in the regions where there are dips in the E-mode spectra, there is a “peak” in temperature. Compared to the temperature power spectra, the peaks on this spectra correlate to the dips in the E-mode power spectra. Both of the power spectrums show structure formation and are complementary to each other; where there are trends in fluctuations in one, there are correlating fluctuations in the other. Also visualized on the B-modes power spectra on the same graph, the fluctuations appear in different arrangements than E-modes. In the universe, B-modes are formed from inflationary waves, and the fluctuations created by these waves in B-modes are also represented.

 

Furthermore, there are different “r” values represented throughout the spectrum. This “r” value is represented as the tensor to scalar ratio, and on the spectrum finds itself at a value of r=0.3 and r=0.01, with them representing the B-mode power spectrums at two different tensor to scalar ratio values. By analyzing these different “r” values, astronomers get a general idea of how to scale inflation as well as different amplitudes of B-mode power signals.

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Figure 2: CMB E-mode and B-mode Anisotropy Power Spectrum23

 

Data Analysis

 

Using code and data compiled from an online github resource centered around CMB analysis, I used a theoretical CMB temperature power spectrum to generate a map of the CMB24. I first used simulated data to generate a temperature anisotropy heat map in python that highlights fluctuation in temperature throughout the CMB. I then used a similar methodology that shows a point source map of the CMB to realize noise from the galaxy. I then generated full sky maps  that emphasize noise fluctuations from the instrument that impact raw measurements of the CMB. It is crucial to simulate CMB measurements with noise as a way to enhance analysis techniques for removing  systematic noise such as point sources from the galaxy and detector noise from the telescope (like the Atacama Cosmology Telescope), and atmospheric noise that skews our measurements of the CMB.  

 

Heat Map

 

For instance using this data, an angular power spectra can be generated that shows temperature anisotropies in the CMB, as shown in Figure 3. Similar to the temperature power spectra mentioned earlier, this spectra shows temperature fluctuations in the CMB as measured by the telescopes but displays them on a heat map. The heat map presents the common multipole moment 𝓁 at varying values across the space in the universe and displays the temperature as it appears at such differing multipoles. As portrayed in the map, the temperature fluctuates between -400 micro-Kelvin and 400 micro-Kelvin, but shows an almost equal balance between temperatures greater than 0 or less than 0. This represents the variations that the CMB has when analyzing different angles of it through instruments; it is not uniform throughout, but shows fluctuations between large temperature values.

 

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Figure 3: CMB Temperature Anisotropy Heat Map 

 

Point Source Map

 

Using a similar methodology as the heat source map generated from simulated CMB values taken, a point source map can also be created. Point sources in the CMB essentially come from astrophysical objects in the universe such as Active Galactic Nuclei (AGN) or Dust Star Forming Galaxies (DSFGs). When put on a map, DSFGs grow brighter due to their higher frequencies, while AGNs are the opposite. Using both a poisson and exponential distribution for the pre-registered data values, a sky map can be generated that presents the point sources of such formations in the universe at different angles, as shown in Figure 4. Although very hard to see initially, the “specks” in the map represent clusters with varying temperatures and locations at different angles.

 

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Figure 3: CMB Point Source Map

 

The Sunyaev-Zel'dovich effect, or SZ effect for short, is a distortion that occurs when CMB photons collide with high-energy cluster electrons and receive an energy boost, allowing easier detection of disturbances in density in the sky. Compiling the SZ effect with pre-registered data from the telescopes from before, a new point source map can then be generated. Points on the map, similar to the other point source map, capture clusters of formations and galaxies. The effect makes it easier to view these galaxies than before, as well as trace their masses. The SZ map follows in Figure 5.

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Figure 5: CMB Sunyaev-Zel'dovich Effect Point Source Map

 

Full Sky Map

 

Compiling the CMB anisotropy map, the point source map, and the SZ map, a new map can be generated that captures the full sky at different angles and temperatures using a gaussian distribution.. Similar to the original CMB map with simulated data, the map is fairly balanced and uniform, seen in Figure 6.

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Figure 6: CMB Full Sky Map

 

This map, however, is one that doesn’t account for noise that would naturally occur in an astronomical instrument. This noise is composed of white noise that is created from a typical gaussian distribution, atmospheric noise that occurs at large angular scales, and 1/f noise – noise that is created from a signal such that the energy of the map is inversely proportional to the frequency of the signal. Depending on how much noise is present in detectors in ground instruments, the heat map may appear to be less or more cohesive. For instance when there is no 1/f noise but there is both white noise and atmospheric noise present, the heat map will be generated as follows in Figure 7. This map is much more distorted and difficult to analyze than the original full sky map with no noise.

 

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Figure 7: CMB Full Sky Map with White Noise and Atmospheric Noise

 

On the other hand, you can “whiten” or reduce this noise by using a high pass filter or square filter. By using python, you can manipulate the data to portray the effects that the filter has on the sky map. For instance, when setting a value N that is set to “mask” the sky is lower, then there is less noise, and vice versa for when the value is larger. As seen in Figure 8 and 9, a lower N value would result in a clearer image, but a higher N value would result in a more blurry image. 

      

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Figure 8: CMB Full Sky Map with Low N               Figure 9: CMB Full Sky Map with High N 

 

This data representation is extremely helpful to visualize how small obstacles – like noise in instruments – in the data collection process can make a huge impact on the overall readings and analyzing of data.

 

Conclusion

 

The theories surrounding the creation of the universe are ideas that are still being proven and disproven today. The Big Bang theory and the theory of inflation are two governing ideas that cosmologists have developed in recent decades to prove the formation of the universe as we know it. Different components analyzed from the CMB have played a part in our understanding, such as anisotropies, polarization, and E-modes and B-modes. Upon running further analysis on data gathered from telescopes that analyzed the CMB, one can see how each component of the CMB plays a part in presenting information about the formation of our universe and how it came to be. With the use of data collected from the cosmic microwave background, the Big  bang theory and inflation have become ever so close to becoming the basis for our understanding. 

 

Acknowledgements

 

I would like to thank physics PhD candidate Sanah Bhimani from Yale University for her great guidance and assistance in the creation of this research paper.

 

References

 

[1] G. F. Smoot. COBE. https://aether.lbl.gov/www/projects/cobe/ (1997).

 

[2] A. H. Guth & W. L. Freedman. Carnegie Observatories Astrophysics Series. Cambridge University Press, 2, 1-3 (2004).

 

[3] D. Sloan. Flatness Problem. http://philosophy-of-cosmology.ox.ac.uk/flatness-problem.html#:~:text=The%20flatness%20problem%20is%20an,energy%20density%20and%20the%20curvature.

 

[4] Swinburne University of Technology. COSMOS - Critical Density. https://astronomy.swin.edu.au/cosmos/C/Critical+Density. 

 

[5] J. A. Peacock. Dynamics of the Expansion. https://ned.ipac.caltech.edu/level5/Peacock/Peacock3_2.html (1999).

 

[6] K. Cahill. Physical Mathematics. Cambridge University Press, 1, 463-468 (2013).

 

[7] D. Darling. Density Parameter https://www.daviddarling.info/encyclopedia/D/density_parameter.html (2016).

 

[8] European Space Agency. Planck and the cosmic microwave background. https://www.esa.int/Science_Exploration/Space_Science/Planck/Planck_and_the_cosmic_microwave_background#TK1a (2007).

 

[9] B. Mattson. Age & Size of the Universe Through the Years. https://imagine.gsfc.nasa.gov/educators/programs/cosmictimes/educators/guide/age_size.html#:~:text=The%20most%20distant%20objects%20in,94%20billion%20light%20years%20across (2017). 

 

[10] D. Darling. Grand Unified Theory. https://www.daviddarling.info/encyclopedia/G/GUT.html (2016). 

 

[11] B. Griswold & E. J. Wollack. WMAP’s Universe - Inflation. https://wmap.gsfc.nasa.gov/universe/bb_cosmo_infl.html#:~:text=a%20uniform%20temperature.-,The%20Monopole%20Problem%3A,abundance%20drops%20to%20undetectable%20levels (2010). 

 

[12] Las Cumbres Observatory. The Early Universe. https://lco.global/spacebook/cosmology/early-universe/. 

 

[13] E. Gawiser & J. Silk. Origin of the Cosmic Background Radiation. ttps://ned.ipac.caltech.edu/level5/Sept05/Gawiser2/Gawiser1.html (2000).

 

[14] A. Chodos & J. Ouellette. This Month in Physics History - June 1963: Discovery of the Cosmic Microwave Background.

https://www.aps.org/publications/apsnews/200207/history.cfm (2002).

 

[15] B. Griswold & E. J. Wollack. Wilkinson Microwave Anisotropy Probe. https://map.gsfc.nasa.gov/ (2012).

 

[16] E. L. Wright. Cosmic Microwave Background. https://www.astro.ucla.edu/~wright/CMB.html (2009).

 

[17] J. G. Bartlett. The standard cosmological model and CMB anisotropies. New Astronomy Reviews, 43, 83-109 (1999).

 

[18] D. D. Reid, D. W. Kittell, E. E. Arsnov, & G. B. Thompson. The picture of our universe: a view from modern cosmology. https://ned.ipac.caltech.edu/level5/Sept02/Reid/Reid5.html (2002).

 

[19] C. Bischoff. CMB Polarization. https://lweb.cfa.harvard.edu/~cbischoff/cmb/#:~:text=CMB%20Polarization%20and%20Inflation&text=Polarization%20describes%20the%20orientation%20of,the%20surface%20of%20last%20scattering (2016). 

 

[20] E. S. Reich. Polarization detected in Big Bang's echo. Nature, 1, 1 (2013).

 

[21] C. Miller. Lecture 21: Cosmic concordance. https://www.astro.umd.edu/~miller/teaching/astr422/lecture21.pdf (2007).

 

[22] European Space Agency. Planck Power Spectrum. https://www.esa.int/ESA_Multimedia/Images/2013/03/Planck_Power_Spectrum (2013).

 

[23] D. Baumann et al. Probing Inflation with CMB Polarization. ResearchGate, 1, 27 (2009).

 

[24] University of Michigan. CMB Analysis Summer School. https://github.com/jeffmcm1977/CMBAnalysis_SummerSchool/blob/master/CMB_School_Part_01.ipynb (2016).