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Home , Cosmic variance, Diffusion damping

Understanding the Cosmic Microwave Background and its relevance in

Author: David Fern´andezBonet Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗

Advisor: Alberto Manrique

Abstract: The Cosmic Microwave Background (hereafter CMB) allows the accurate determi- nation of cosmological parameters upon a careful study of its . In this work, physical phenomena that give rise to such anisotropies are discussed, along with an exhaus- tive analysis on the impact of baryon and density over the power spectrum’s shape by using an online simulator. Finally, a sky map, illustrating the spatial distribution of temperature anisotropies, is generated.

I. INTRODUCTION give a qualitative idea of the scale and intensity of CMB anisotropies (section IV). We end up with the conclusions and baryons were strongly coupled via Thom- (section V). son scattering in the early . When this cooled down to about 3000 K, the ionisation rate quickly II. ANGULAR POWER SPECTRUM decreased, recombined and photons could freely propagate. In other words, the Universe became trans- parent. For historical reasons, this period is called re- The quantitative analysis of the statistical properties combination. The observed radiation that constitutes of the temperature anisotropies requires computing the the Cosmic Microwave Background (CMB) is, in fact, angular correlation function of the temperature contrast made up by photons decoupled at recombination. ∆T = T − hT i, or rather, its spherical harmonics trans- The CMB radiation follows the blackbody spectrum form. It yields information about the magnitude of the up to a high degree of precision. Today, its associated perturbations and their angular scale. temperature is near 2.7 K and it can be detected in First, it is necessary to define the normalized temper- any direction. This fact strongly implies that the en- ature Θ in the directionn ˆ on the celestial sphere as the deviation ∆T from the average: Θ(ˆn) = ∆T . The multi- tire was once in thermal equilibrium, hT i and thus in causal contact. However, extremely accurate pole decomposition of this temperature field in terms of measurements of the CMB done with COBE revealed spherical harmonics Ylm is given by the following expres- that there were small temperature fluctuations, the so sion [2]: called anisotropies[1]. They were imprinted by tiny den- sity perturbations during the epoch that matter and ra- Z Θ = Θ(ˆn)Y ∗ (ˆn)dΩ , (1) diation were coupled. It is believed with a high degree of lm lm certainty that cosmic structure arises from those density where Ω is the solid angle and integration is done over the perturbations. entire sphere. The asterisk denotes the complex conju- When recombination occurs, photons can travel freely gate. The order m describes the angular orientation of a carrying valuable information about the physics of the fluctuation mode. The degree (or multipole) l describes Universe just before decoupling. Such information can its characteristic angular frequency, such that a given be condensed in the angular power spectrum, a powerful multipole corresponds to an angular scale θ ∼ 180o/l. tool that is notably useful to study the CMB anisotropies. The reciprocal of l is proportional to the so-called fluc- Since the shape of the power spectrum depends on global tuation wavelength. properties of the Universe, cosmological parameters can If the inflationary model is correct, the coefficients Θlm be estimated with great accuracy using this tool. can be assumed to be uncorrelated random Gaussian In this work, the angular power spectrum (section variables with zero mean and a that is inde- II) and physical phenomena associated with anisotropies pendent of orientation (statistical isotropy). In this way, (section III) are discussed. Using CAMB’s web interface ∗ hΘ Θl0m0 i = Clδll0 δmm0 . The angular power spectrum (an online simulator of CMB anisotropies), graphical re- lm Cl is then defined by: sults are obtained showing some insight on the power spectrum’s shape and its relationship to cosmological pa- l rameters, in particular, the baryonic and cold dark mat- 1 X ∗ C = Θ Θ 0 0 . (2) ter densities (section IV). Also, a sky map is generated to l 2l + 1 lm l m m=−l Only 2l + 1 modes can be used in order to determine Cl. Therefore, when determining the angular power spec- ∗Electronic address: [email protected] trum, there will be an intrinsic statistical UNDERSTANDING THE CMB David Fern´andezBonet

q 1 A. Sachs-Wolfe effect proportional to 2l+1 . This is known as the cosmic vari- ance, which takes the exact form: For large scales (l < 100), the main contributor in ∆C r 2 l = . (3) shaping the power spectrum is the so called Sachs-Wolfe Cl 2l + 1 effect. It may be divided into the ordinary Sachs-Wolfe As it can be seen in Fig.(1), low multipole values are effect and the integrated Sachs-Wolfe effect (ISW). The associated with a high degree of intrinsic uncertainty. It former one refers to the energy shift that CMB photons is important to note that the full uncertainty in the power experience when exposed to the gravitational effect of in a given multipole degrades from instrumental noise, fi- density perturbations at the time of recombination. The nite beam resolution, and observing over a finite fraction energy gained or lost, depending on the sign of the grav- of the full sky. Finer technology is needed in order to itational potential ψ, will induce a temperature contrast obtain reliable data for l > 2000. which can be observed in the CMB [3]. The quantity that is usually plotted against the ISW is the same physical effect, but in this case the multipole l, sometimes termed the TT (temperature- time elapsed becomes relevant. It describes the energy temperature correlation) spectrum, is shift that photons undergo after recombination when they travel through large massive structures until they reach the earth. This situation is quantitatively different 2 2 l(l + 1) from the ordinary Sachs-Wolfe effect since the potential ∆T ≡ T Cl , (4) CMB 2π well presented to the is localized in a single region where TCMB = 2.7K is the blackbody temperature of along the path. The blueshift when entering into the po- the CMB. tential well would normally be compensated by the red- shift when leaving, but large structures evolve with time, meaning that the process is not symmetric. In order to compute the temperature due to this effect, an integral over the time variation of the gravitational potential must be done. Thus, the name ”integrated” Sachs-Wolfe effect. This kind of effect is mostly relevant at large scales (or low l) because at smaller scales, photons must traverse many density perturbations and the fluctuations tend to cancel out. The large-scale region in the power spectrum is known as the ”Sachs-Wolfe Plateau” (not noticeable in Fig. (1) owing to the logarithmic scale) for its shape.

B. Acoustic peaks

There are two main factors that cause acoustic peaks: gravity and radiation pressure. While the gravitational pull caused by ordinary and dark matter tends to gen- FIG. 1: The TT power spectrum. The red curve represents erate high-density baryon zones and enhances collapse, the best-fit cosmological model to the WMAP data (includ- photonic pressure has the opposite effect. This leads to ing ). It also includes high-l data from other acoustic oscillations in all scales, that is, the photon- experiments. Note the multipole scale on the bottom and the baryon fluid has compressing and expanding regions angular scale on the top. From [2]. which means that baryon density oscillates constituting sound waves. At trec, these fluctuations are captured in the CMB and can be detected as anisotropies on the sky. Hot spots in a sky map represent high-density re- III. PHYSICAL PROCESSES gions that are compressing whereas cold spots represent low-density expanding regions. When discussing the mechanisms of generation of the The first peak of the power spectrum is the peak with primordial CMB anisotropies, the three most important lowest l (or highest wavelength) and it corresponds to a physical processes are: the gravitational shift in the fre- maximum compression phase at decoupling, thus leading quency that photons experience (the Sachs–Wolfe effect), to a high temperature contrast in the CMB. The second the acoustic oscillations propagating through plasma due peak is, in turn, associated to regions that first com- to gravity and radiation pressure and the temperature pressed and then reached maximum rarefaction (maxi- contrast damping on small scales caused by photon dif- mum expansion) due to radiation pressure. Generalising, fusion [4]. odd peaks are related to the compression phase whereas

Treball de Fi de Grau 2 Barcelona, June 2020 UNDERSTANDING THE CMB David Fern´andezBonet even peaks are related to the rarefaction phase. Since IV. ANISOTROPIES AND COSMOLOGY odd peaks are associated with how much the plasma com- presses, they are enhanced if the concentration of baryons The influence of cosmological parameters over is increased [5]. This idea will be further developed in anisotropies and ultimately, over the power spectrum, section IV. will be discussed in this section. Variations of these pa- Assuming adiabatic perturbations, the wavelength of rameters lead to shape, position and height changes on the first peak can be found using the relation treccs = λ1, the peaks. a˙ where cs is the sound speed in the photon-baryon fluid. The Hubble rate is defined as H ≡ a (where a is the −1 The angular size of this wavelength viewed from earth is scale factor) and its current value is H0 = 100h km s Mpc−1, where h = 0.678±0.009 is the Hubble parameter [7]. It is linked to the contents of the Universe through the equation: λ1 π θ1 ' , l1 ' , (5) DA(trec) θ1 8πG Λ κ H2 = ρ + − ; 1 = Ω + Ω + Ω , (6) 3 3 a2 m Λ κ where the angular diameter distance DA depends on the assumed cosmology of the universe and the scale factor. where the matter-energy density ρ, the cosmological The position of the first peak computed via Eq. (5) as- constant Λ and curvature κ are introduced. The o 2 suming a flat universe yields θ1 ≈ 1 [6]. It agrees with right hand equation is the result of dividing by H , experimental data and provides strong evidence for null thus a dimensionless form is obtained. Ωm includes curvature. About one degree is, in fact, the angular scale baryons (ordinary matter made by , neutrons of the at time trec (the distance sound and electrons, denoted by Ωb) and also dark matter can travel before recombination). Given the high degree (we will assume the cold type, denoted by Ωcd). ΩΛ of isotropy that the CMB exhibits on all scales and the and Ωκ are respectively related to and the fact that no causal process could explain that isotropy curvature of the universe. High quality experimental level beyond 1o, this naturally leads to the ”horizon prob- data allows the determination of these parameters: 2 2 lem”. Cosmic inflation offers a remarkable answer to that Ωbh = 0.02226 ± 0.00023, Ωcdh = 0.1186 ± 0.0020, problem. ΩΛ = 0.692 ± 0.012 [7]. In a flat universe model Ωk = 0.

The CAMB or Code for Anisotropies in the Microwave Background is a Fortran 90 and Python application which computes CMB spectra given several cosmolog- C. Silk damping ical parameters inputs. It was developed by Antony Lewis and Anthony Challinor in collaboration with Silk damping, also referred as diffusion damping, is NASA’s LAMBDA team (https://lambda.gsfc.nasa. a physical process that made the CMB more uniform. gov/toolbox/tb_camb_form.cfm). Its latest version During recombination, diffusing photons travelled from (April, 2014) was released alongside a web interface regions with high temperature to regions with lower tem- that optimizes the computing process and minimizes perature, thus reducing anisotropies. The strength of the user mandatory requirements. Whereas this is this effect depends on the distance photons travel before the tool used to obtain sky map and power spectra being scattered, that is, its diffusion length. A higher data, gnuplot is the software that allowed its graphical diffusion length will ultimately lead to a more damped representation. More details about CAMB code can be spectra. This characteristic length, in turn, behaves as found at http://camb.info/. −1 λD ∝ (xenb) , where xe is the electronic ionisation frac- tion (nearly 1 at recombination) and nb is the baryon The effect of slight baryon density variations can be number density. It is important to note that the more seen in Fig. (2). Baryon loading enhances odd peaks ionised baryons are, the shorter the photons could travel the same way adding mass in a spring enhances the dis- before encountering one and being scattered. tance travelled by such mass [8]. Because radiation pres- h 2i sure must overcome deeper potential wells, even peaks The damping factor, defined as D ∼ exp − λD  λ are lowered. It can be said that the oscillation presents (where λ is the wavelength of the acoustic mode be- asymmetry in the sense that the extrema that constitute ing suppressed) is a reliable indicator on how ”damped” compressions are increased over those that represent ex- the fluctuations are, with maximum damping at D = 0. pansion or rarefactions. Because the exact form of higher λD  Therefore, the higher the value λ , the lower the order peaks depends on several other parameters, the pre- damping factor. Since λ ∝ l−1, the high-multipole part viously mentioned effect can be seen in a concise way in of the power spectrum is severely damped. In particular, the height difference between the first and second peak. Silk damping exponentially decreases anisotropies in the As the baryon number density is increased, the distance CMB on a scale l > 800 known as Silk scale. becomes higher.

Treball de Fi de Grau 3 Barcelona, June 2020 UNDERSTANDING THE CMB David Fern´andezBonet

There are two more issues to be considered: oscilla- because the main source generating the potential is not tion’s period and damping. Fig. (2) shows how less radiation. Because dark matter reduces the radiation- baryons imply lower oscillation periods, as well as more dominated time, an increase in such kind of matter will damping. The former fact can be explained again using reduce the driving effect thus reducing peak’s amplitude. the spring analogy: adding mass will lower the frequency The regions on the power spectrum that are more sen- of the oscillations, thus increasing the period. Interest- sitive to this effect are the high-multipole regions, since ingly, temperature contrast enhancement in odd peaks lower wavelengths oscillate earlier and have more expo- and lower oscillation frequency have the same cause. Ad- sure to the radiation era. ditional damping is explained by the photonic diffusion Because the first and second peaks are highly de- length dependence on the number density of baryons: pendent on gravitational effects caused by photons and −1 λD ∝ (xenb) . A lower nb will induce an increased baryons, the third peak is the most reliable indicator damping effect by enlarging the diffusion length. of dark matter, or, more formally, the dark matter to The fact that baryons are related to at least the three radiation ratio. A third peak large enough to be higher distinct considered cases taking part in shaping the power than the second peak is a strong sign pointing out dark spectrum (distance between 1st and 2nd peaks, oscilla- matter domination [9]. tion period and damping) leads to high precision mea- surements of Ωb.

FIG. 3: Angular power spectrum for different cold dark mat- ter densities Ωcd. Baryon density and curvature are fixed, FIG. 2: Angular power spectrum for different baryon densities 2 such that Ωbh = 0.02, Ωκ = 0. Dark matter has an over- Ωb. Cold dark matter density and curvature are fixed, such 2 all suppressing effect on the peaks. For high values of dark that Ωcdh = 0.11, Ωκ = 0. Baryonic matter enhances odd matter density the third peak becomes larger than the second peaks and increase the distance between first and second peak. one.

The most obvious effect of introducing cold dark mat- In order to obtain sky maps, CAMB uses the Hierar- ter density variations is to reduce or to enhance the over- chical Equal Area isoLatitude Pixelisation (or HEALPix) all amplitude of the peaks as it can be seen in Fig. (3). software. Its function is to map a 2-sphere (the surface The amount of dark matter plays a relevant role in de- of a sphere in a three-dimensional space) and to partition termining at which time the universe transitioned from the resulting region. This last process is called pixelas- a radiation-dominated era to a matter-dominated era. ing, and it consists in dividing the sphere into 12 base The higher the dark matter density, the earlier it will pixels and increase the resolution by the division of each happen. Gravitational potential wells cannot be consid- pixel into four new ones. ered as fixed at the radiation era. Indeed, the potential The output generated by HEALPix is a .fits file (flex- decays in such a manner that drives the amplitude of the ible image transport system). Since they have a large oscillations up. This happens because the main source of combination of variants, reading this kind of files can be the potential is radiation itself. The complete decay hap- complicated if the exact process of their creation is un- pens at maximum compression, when pressure eventually known. After several tries, the software that best fitted stops radiation from any further compression. Because the purpose of depicting a sky map turned out to be the the baryon-photon fluid do not experience a gravitational fv software, a FITS file viewer and editor developed at potential anymore, it bounces back with an enhanced am- the High Energy Astrophysics Science Archive Research plitude. This is the so called driving effect, and it does Center (HEASARC) at NASA/GSFC. Given that target not take place when the universe is matter-dominated users of this software are cosmologists and NASA em-

Treball de Fi de Grau 4 Barcelona, June 2020 UNDERSTANDING THE CMB David Fern´andezBonet ployees, its interface is rather complex. Adding the fact out for its clarity, the previously mentioned pro- that the .fits file contained much more than the basic cesses may be identified within the characteristic information to generate a sky map, the filtering process spectrum’s shape and cosmological parameters can was laborious. Eventually, it was possible to plot Fig.(4), be estimated accurately. where one may graphically see the spatial distribution and scale of hot and cold spots. The universe plotted • The reverse process, that is, generating several have Ωκ = 0.2, a non-zero curvature that enlarges the power spectra given different sets of cosmological scale of the anisotropies to make them even more visible. parameters is done using CAMB web interface. In this way, the underlying physics linked with the variation of parameters are exposed and, ulti- mately, a better understanding of them is acquired. While baryon loading increases oscillations’ period, Silk damping and the height of odd peaks, adding cold dark matter has an overall suppressing effect over peaks’ amplitude.

• Given the reduced mandatory extension for this project and the broadness of the studied field, some issues have to be left out. Future work on this topic could include, for example, the effect over the power spectrum of dark energy and curvature vari- FIG. 4: HEALPix generated sky map plotted using fv soft- ware. Red color is associated with cold spots and blue is ations in addition to further research on the Sachs- associated with hot spots. While baryon and dark matter Wolfe plateau’s shape implications. Sky maps cor- 2 2 densities have their usual values Ωbh = 0.02, Ωcdh = 0.11, responding to with different cosmological non-zero curvature is introduced with Ωκ = 0.2, enhancing parameters could be compared and analysed simul- the scale of anisotropies. taneously with its associated power spectra.

V. CONCLUSIONS Acknowledgments

• The CMB constitutes one of the richest sources of Foremost, I would like to thank my advisor Alberto information for cosmologists. There are a consid- Manrique for his support and because he provided valu- erable amount of interesting physical phenomena able resources and perspective when most needed. My associated with CMB’s anisotropies, the most rel- university colleagues fueled the much required motiva- evant of which are the Sachs-Wolfe effect, acous- tion with kind words, and I appreciate them. Lastly, tic oscillations and Silk damping. Upon a careful the thoughtfullness of my family made possible a work- analysis of its power spectrum, a tool that stands friendly environment, which is not to take for granted.

[1] Bennett, C.L., Banday, A.J., Gorski, K.M., et al. 1996, [7] Tanabashi, M. et al. (Particle Data Group), 2018, Phys. ApJLetters, 464, L1. Rev. D 98, 030001. [2] Samtleben, D., S. Staggs, S. & Winstein, B. 2007. Annual [8] Keskitalo, R., 2005, Master’s thesis, University of Helsinki. Review of Nuclear and Particle Science, 57(1): 245–283. Retrieved from: helda.helsinki.fi/handle/10138/20988. [3] Sachs, R.K. & Wolfe, A.M. 1967, Astrophys. J. , 147, 73. [9] Naselsky, D., Nokikov, I. & Nokikov, D. The Physics of [4] Hu, W., Sugiyama, N., & Silk, J. 1997, Nature, 386,37. The Cosmic Microwave Background, (Cambridge univer- [5] Bouchet, F., 2004. Astrophysics and Space Science, 290,69 sity press, New York 2006). [6] Durrer, R., The Cosmic Microwave Background, (Cam- bridge university press, New York 2008).

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