Observational Cosmology
The Cosmic Microwave Background Part I: CMB Theory
Kaustuv Basu
Course website: http://www.astro.uni-bonn.de/~kbasu/ObsCosmo
Observational Cosmology K. Basu: CMB theory and experiments Outline of the CMB Lectures
CMB theory CMB experiments
➡ Discovery of the CMB ➡ Cosmological parameter estimation ➡ Thermal spectrum of the CMB ➡ Planck results - what’s new? ➡ Temperature anisotropies and the angular power spectrum ➡ CMB Polarization results (the BICEP2 story) ➡ CMB polarization - its origin and power spectra ➡ Basics of CMB analysis: map making and foreground ➡ CMB secondary anisotropies subtraction
Observational Cosmology K. Basu: CMB theory and experiments 2 CMB theory overview
The thermal (blackbody) spectrum
Temperature anisotropies
Polarization anisotropies
Secondary anisotropies
Observational Cosmology K. Basu: CMB theory and experiments 3 Microwave Background Radiation
~ 1%
~ 400 photons per cm3 ~1 eV/cm3
• CMB dominates the radiation content of the universe
• It contains nearly 93% of the radiation energy density and 99% of all the photons (and gives rise to the high baryon-to-photon ratio!)
Observational Cosmology K. Basu: CMB theory and experiments 4 Discovery of the CMB
McKellar (1940)
• In 1940, McKellar discovers CN molecules in interstellar space from their absorption spectra (one of the first IS-molecules)
• From the excitation ratios, he infers the “rotational temperature of interstellar space” to be 2° K (1941, PASP 53, 233)
• In his 1950 book, the Nobel prize winning spectroscopist Herzberg remarks: “From the intensity ratio of the lines with K=0 and K=1 a rotational temperature of 2.3° K follows, which has of course only a very restricted meaning.”
Observational Cosmology K. Basu: CMB theory and experiments 5 Discovery of the CMB
•After the “α-β-γ paper”, Alpher & Herman (1948) predict 5 K radiation background as by-product of their theory of the nucleosynthesis in the early universe (with no suggestion of its detectability).
• Shmaonov (1957) measures an uniform noise temperature of 4±3 K at λ=3.2 cm.
• Doroshkevich & Novikov (1964) emphasize the detectability of this radiation, predict that the spectrum of the relict radiation will be a blackbody, and also mention that the twenty- foot horn reflector at the Bell Laboratories will be the best instrument for detecting it!
No Nobel prize for these guys!
Observational Cosmology K. Basu: CMB theory and experiments 6 Discovery of the CMB
• Originally wanted to measure Galactic emission at λ=7.3 cm
• Found a direction- independent noise (3.5±1.0 K) that they could not get rid of, despite drastic measures
• So they talked with colleagues..
• Explanation of this “excess noise” was given in a companion paper by Robert Dicke and collaborators (no Nobel prize for Dicke either, not to mention Gamow!)
Observational Cosmology K. Basu: CMB theory and experiments 7 The Last Scattering Surface
All photons have travelled roughly the same distance since recombination. We can think of the CMB being emitted from inside of a spherical surface, we’re at the center. (This surface has a thickness)
Observational Cosmology K. Basu: CMB theory and experiments 8 Thickness of the last scattering “surface”
The visibility function is defined as the probability density that a photon is last scattered at redshift z: g(z) ~ exp(-τ) dτ/dz
Probability distribution is well described by Gaussian with mean z ~ 1100 and standard deviation δz ~ 80.
Observational Cosmology K. Basu: CMB theory and experiments 9 The last scattering “sphere”
anisotropy amplitude ΔT/T ~ 10−5
Observational Cosmology K. Basu: CMB theory and experiments 10 Horizon scale at Last Scattering
The particle horizon length at the time of last scattering (i.e. the distance light could travel since big bang) is give by
The factor 2c/H0R0 is approximately the comoving distance to the LSS (Ωm=1 EdS universe).
This tells us that scales larger than 1.7◦ in the sky were not in causal contact at the time of last scattering. However, the fact that we measure the same mean temperature across the entire sky suggests that all scales were once in causal contact - this led to the idea of Inflation.
Inflationary theories suggests that the Universe went through a period of very fast expansion, which would have stretched a small, causally connected patch of the Universe into a region of size comparable to the size of the observable Universe today.
Observational Cosmology K. Basu: CMB theory and experiments 11 Recombination temperature & redshift
The temperature of last scattering depends very weakly on the cosmological parameters (weaker than logarithmically).
It is mainly determined by the ionization potential of hydrogen and the baryon-to-photon ratio.
-3 Ne ~ 500 cm (roughly same as Galactic HII regions)
Te = Tr = 2970 K = 0.26 eV zr = 1090
Why this is so low compared to ionization potential of hydrogen, 13.6 eV?
Observational Cosmology K. Basu: CMB theory and experiments 12 Recombination temperature & redshift
This ratio implies that recombination occurs at substantially lower temperature than the binding energy B = 13.6 eV of the Hydrogen atom, at T ≈ 0.3 eV 3 only, or at a redshift of z✳ ≈ 10 .
This follows from Saha’s ionization equilibrium equation, which in terms of ionization fraction X (since ne = np) and photon-to-baryon ratio η reads as
Observational Cosmology K. Basu: CMB theory and experiments 13 Recombination temperature & redshift
Once Q13.6 eV/Tb reaches a certain level, recombination proceeds rapidly. If we define the moment of recombination when X = 1/2, we get
Observational Cosmology K. Basu: CMB theory and experiments 14 Hydrogen recombination in Saha equilibrium vs. more precise numerical calculation by the RECFAST code. The features seen before hydrogen recombination are due to helium recombination.
The relative rapidity of the recombination process makes its redshift practically independent on the standard cosmological parameters.
Observational Cosmology K. Basu: CMB theory and experiments 15 Epoch of decoupling
The photon Thomson scattering rate is controlled by how many electrons are available. The scattering rate Γ is thus a function of ionization fraction:
For free streaming of electrons, this scattering rate should at least be of the order of Hubble time, H(z). Now recombination and photon decoupling takes place in the matter dominated era, so from Friedmann equation:
Setting Γ ~ H (condition for photon free streaming), we get
Actually, when Γ ~ H, the system is not in equilibrium (Saha equation not valid)! A more precise calculation yields
Observational Cosmology K. Basu: CMB theory and experiments 16 The thermal spectrum of the CMB
Observational Cosmology K. Basu: CMB theory and experiments 17 The CMB blackbody
2006 Nobel Prize in Physics!
The blackbody shape is preserved at all redshifts if expansion is purely adiabatic. The frequency shifts by 1/a, and energy density by 1/a4.
Observational Cosmology K. Basu: CMB theory and experiments 18 Measurement of TCMB
Credit: D. Samtleben Ground and balloon based experiments have been measuring CMB temperature for decades with increasing precision
but it was realized that one has to go to the stable thermal environment of outer space to get a really accurate measurement (and observe in the Wien part).
Measured blackbody spectrum of the CMB, with fit to various data Credit: Ned Wright
Observational Cosmology K. Basu: CMB theory and experiments 19 COBE
Launched on Nov. 1989 on a Delta rocket.
DIRBE: Measured the absolute sky brightness in the 1-240 μm wavelength range, to search for the Infrared Background
FIRAS: Measured the spectrum of the CMB, finding it to be an almost perfect blackbody with T0 = 2.725 ± 0.002 K
DMR: Found “anisotropies” in the CMB for the first time, at a level of 1 part in 105
2006 Nobel prize in physics
Credit: NASA
Observational Cosmology K. Basu: CMB theory and experiments 20 FIRAS on COBE
• One input is either the sky or a Far Infra-Red Absolute blackbody, other is a pretty good blackbody Spectrophotometer • Zero output when the two inputs are equal A diferential polarizing • Internal reference kept at T0 (“cold Michelson interferometer load”), to minimize non-linear response of the detectors Credit: NASA • Residual is the measurement!
Observational Cosmology K. Basu: CMB theory and experiments 21 FIRAS Measurements
Fundamental FIRAS measurement is the plot at the bottom: the diference between the CMB and the best-fitting blackbody. The top plot shows this residual added to the theoretical blackbody spectrum at the best fitting cold load temperature.
The three curves in the lower panel represents three likely non-blackbody spectra:
Red and blue curves show efect of hot electrons adding energy before and after recombination (roughly), the grey curve shows efect of a non-perfect blackbody as calibrator (less than 10-4) Credit: Ned Wright
Observational Cosmology K. Basu: CMB theory and experiments 22 Thermalization of the CMB
Process that changes photon energy, not number: Compton scattering: e + γ = e + γ
Processes that creates photons: Radiative (double) Compton scattering: e + γ = e + γ + γ Bremsstrahlung: e + Z = e + Z + γ
At an early enough epoch, timescale of the radiative Compton scattering rate must be shorter than the expansion timescale. They are equal at z ~ 2x106 , or roughly two months after the big bang.
The universe reaches thermal equilibrium by this time through scattering and photon-generating processes. Thermal equilibrium generates a blackbody radiation field. Any energy injection before this time cannot leave any spectral signature on the CMB blackbody.
For pure adiabatic expansion of the universe afterwards, a blackbody spectrum — once established — should be maintained.
Observational Cosmology K. Basu: CMB theory and experiments 23 Thermalization of the CMB
Kinetic equilibrium can be established by any process with a timescale less than H −1
Under KE, as opposed to thermal equilibrium, the spectrum is Bose-Einstein: n = [ exp(hν/kT + μ) − 1] −1
Clearly, μ plays a only small role at high frequencies, but the discrepancy becomes larger as the frequency drops.
True thermal equilibrium requires the creation and destruction of photons as well as energy redistribution by scattering (see, for instance, Kompaneets, 1957). In the early Universe, radiative Compton scattering permits the generation of photons needed to ensure true thermal equilibrium. The bremsstrahlung process is less important, due to the high photon-to-baryon ratio.
The efciency of these two processes keep on dropping with time, and after about z ~ 2 x 106 they cannot keep the universe in thermal equilibrium anymore.
Observational Cosmology K. Basu: CMB theory and experiments 24 Bose-Einstein spectrum
The claim that thermal equilibrium is established at the high redshift of ~2 × 106 is equivalent to the claim that μ is driven essentially to zero by that redshift. However, if energy is added to the CMB radiation field after redshift of ~2×106, there may still be time to reintroduce kinetic equilibrium, but not full thermal equilibrium. In that case, the spectrum would be a Bose-Einstein spectrum (y > 1).
Ref: Khatri & Sunyaev, arXiv:1203.2601
Observational Cosmology K. Basu: CMB theory and experiments 25 μ- and y-type distortions
At a later epoch (up to recombination and during reionization) any energy input will create a y-type distortion. Here Comptonization is inefcient ( Γ > H−1), so we get y << 1. This can be created at low redshifts down to z = 0.
Ref: Khatri & Sunyaev, arXiv:1203.2601
Observational Cosmology K. Basu: CMB theory and experiments 26 μ- and y-type distortions Ref: Khatri & Sunyaev, arXiv:1203.2601 Sunyaev,arXiv:1203.2601 & Khatri Ref:
Observational Cosmology K. Basu: CMB theory and experiments 27 Current limits on Spectral Distortions
• Energy added after z~2x106 will show up as spectral distortions. Departure from a Planck spectrum at fixed T is known as “μ distortion” (B-E distribution). μ distortion is easier to detect at wavelengths λ >10 cm. COBE measurement: |μ| < 9 x 10-5 (95% CL)
• The amount of inverse Compton scattering at later epochs 5 (z < 10 ) show up as “y distortion”, where y ~ σT ne kTe (e.g. the Sunyaev-Zel’dovich efect). This rules out a uniform intergalactic plasma as the source for X-ray background. COBE measurement: y < 1.2 x 10-5 (95% CL)
• Energy injection at much later epochs (z << 105), e.g. free-free distortions, are also tightly constrained. -5 COBE measurement: Yf < 1.9 x 10 (95% CL)
Observational Cosmology K. Basu: CMB theory and experiments 28 Future missions for spectral measurement
Pixie satellite concept
Cosmic Origins Explorer (PRISM, CoRE+)
Observational Cosmology K. Basu: CMB theory and experiments 29 μ- and y-type distortions
Sensitivity of the proposed Pixie satellite Kogut et al. (2011), arXiv:1105.2044
Observational Cosmology K. Basu: CMB theory and experiments 30 Recombination stages Sunyaev & Chluba (2009) Chluba & Sunyaev
Observational Cosmology K. Basu: CMB theory and experiments 31 Spectral features during recombination
Observational Cosmology K. Basu: CMB theory and experiments 32 Spectral features during recombination From Chluba & Sunyaev (2006)
Frequency dependent modulation in the CMB temperature from the H I and He II recombination epoch, after subtracting the mean recombination spectrum. This signal is unpolarized and same in all directions on the sky.
Observational Cosmology K. Basu: CMB theory and experiments 33 Temperature anisotropies
Observational Cosmology K. Basu: CMB theory and experiments 34 Amplitude of temp. anisotropies
CMB is primarily a uniform glow across the sky!
Turning up the contrast, dipole pattern becomes prominent at a level of 10-3. This is from the motion of the Sun relative to the CMB.
Enhancing the contrast further (at the level of 10-5, and after subtracting the dipole, temperature anisotropies appear.
Observational Cosmology K. Basu: CMB theory and experiments 35 The CMB dipole
I’(ν’)=(1+(v/c) cos θ)3 I(ν) ν’=(1+(v/c) cos θ) ν T(θ)=T (1+(v/c) cos θ)
• Measured velocity: 390±30 km/s • After subtracting out the rotation and revolution of the Earth, the velocity of the Sun in the Galaxy and the motion of the Milky Way in the Local Group one finds: v = 627 ± 22 km/s • Towards Hydra-Centaurus, l=276±3° b=30±3°
Can we measure an intrinsic CMB dipole ?
Observational Cosmology K. Basu: CMB theory and experiments 36 Modulation and aberration of anisotropies Both these are level ~β efects. Modulation is the same as dipole, but working on ΔT instead of T0. Aberration is a relativistic shift in position.
Planck collaboration (2013)
Observational Cosmology K. Basu: CMB theory and experiments 37 Anisotropy amplitude
From the fact that non-linear structures exist today in the Universe, the linear growth theory predicts that density perturbations at z = 1100 (the time of CMB release) must have been of the order of
After the CMB was found in 1965, fluctuations were sought at the relative level of 10-3, but they were not found. Eventually they were found at a level of 10-5.
The reason is that already before the CMB release the DM perturbations started growing independently. While the radiation-Baryon fluid oscillated and therefore didn’t grow in amplitude, the DM perturbations continued to grow. Before the DM dominated the mass (i.e. z~3300) this growth was slow (logarithmic), while once DM dominate the mass the growth was linear. Since DM has no coupling to the electromagnetic spectrum, nor to the baryons, this growth happened without pumping the perturbations in the CMB to equal levels.
In fact, this can be seen as a proof of the existence of such a DM as a non- interacting form of matter!
Observational Cosmology K. Basu: CMB theory and experiments 38 Relikt-1
Launched July 1983. Single frequency (37 GHz) radiometer
Strukov et al. (1992)
Analysis seriously delayed by the breakup of the USSR!
A 1992 paper reported a temperature decrement of -71 ± 43 μK at large angles at 90% confidence, including systematics.
Observational Cosmology K. Basu: CMB theory and experiments 39 DMR on COBE
Diferential Microwave Radiometer
• Diferential radiometers measured at frequencies 31.5, 53 and 90 GHz, over a 4-year period • Comparative measurements of the sky ofer far greater sensitivity than absolute measurements Credit: NASA
Observational Cosmology K. Basu: CMB theory and experiments 40 COBE DMR Measurements
Credit: Archeops team
Credit: NASA 2006 Nobel Prize in Physics for George Smoot.
Observational Cosmology K. Basu: CMB theory and experiments 41 WMAP: 2001-2010
Note the same dual receivers as COBE. This design, plus the very stable conditions at the L2, minimizes the “1/f noise” in amplifiers and receivers.
Credit: NASA Thus after 7 years, the data could still be added and noise lowered (of course, the improvement gradually diminishes).
Observational Cosmology K. Basu: CMB theory and experiments 42 WMAP results after 1st year
Internal Linear Combination map
(Credit: WMAP Science Team)
Observational Cosmology K. Basu: CMB theory and experiments 43 Planck 2015 results
Observational Cosmology K. Basu: CMB theory and experiments 44 CMB temperature anisotropies
• The basic observable is the CMB intensity as a function of frequency and direction on the sky. Since the CMB spectrum is an extremely good black body with a fairly constant temperature across the sky, we generally describe this observable in terms of a temperature fluctuation
• The equivalent of the Fourier expansion on a sphere is achieved by expanding the temperature fluctuations in spherical harmonics
Observational Cosmology K. Basu: CMB theory and experiments 45 Analogy: Fourier series
Sum sine waves of diferent frequencies to approximate any function.
Each has a coefcient, or amplitude.
Observational Cosmology K. Basu: CMB theory and experiments 46 Spherical harmonics
Observational Cosmology K. Basu: CMB theory and experiments 47 Visualizing the multipoles
Observational Cosmology K. Basu: CMB theory and experiments 48 Visualizing the multipoles (real sky!)
Observational Cosmology K. Basu: CMB theory and experiments 49 Power at diferent scales
Credit: Wayne Hu
Observational Cosmology K. Basu: CMB theory and experiments 50 Spherical harmonics
l~180º/θ relates to the angular size of the pattern, whereas m relates to 2 the direction on the sky. Thus〈|alm| 〉is independent of m (since CMB is isotropic).
The mean〈alm〉is zero (Gaussian random field), but we can calculate its variance:
(Reason is the S-W efect!)
Observational Cosmology K. Basu: CMB theory and experiments 51 CMB power spectrum
How to constrain cosmology from this measurement?
Use spherical harmonics in place of sine waves:
Calculate coefcients, alm, and then the statistical average:
Amplitude of fluctuations on each scale ⎯ that’s what we plot. (TT power spectrum)
Observational Cosmology K. Basu: CMB theory and experiments 52 Cosmic variance
We only have one Universe, so we are intrinsically limited to the number of independent m-modes we can measure − there are only (2l + 1) of these for each multipole.
We obtained the following expression for the power spectrum:
2 We can see (2l+1)Cl follows a χ distribution. It has (2l+1) degrees of freedom, thus its variance is simply 2(2l+1). Therefore we get:
How well we can estimate an average value from a sample depends on how many points we have on the sample. This cosmic variance is an unavoidable source of uncertainty Observationalwhen constraining Cosmology models! K. Basu: CMB theory and experiments 53 Cosmic and sample variance
• Cosmic variance: on scale l, there are only 2l+1 independent modes
• Averaging over l in bands of Δl ≈1 makes the error scale as l-1
• If the fraction of sky covered is f, then the errors are increased by a factor 1/√fsky and the resulting variance is called sample variance (f ~0.65 for the PLANCK satellite)
Observational Cosmology K. Basu: CMB theory and experiments 54 The origin of temperature anisotropies
Observational Cosmology K. Basu: CMB theory and experiments 55 Primordial temperature anisotropies
At recombination, when the CMB was released, structure had started to form
This created the “hot” and “cold” spots in the CMB
These were the seeds of structure we see today
Should not confuse between the “creation” of the CMB photons before the blackbody surface (z >106), and their “release” from the last scattering surface .
CMB photons are created at much earlier epoch through matter/anti-matter annihilation, and thus, were formed as gamma rays (now cooled down to microwave)
Observational Cosmology K. Basu: CMB theory and experiments 56 PrimordialCreation temperature of anisotropies anisotropies
Observational Cosmology K. Basu: CMB theory and experiments 57 Sources of ΔT Max Tegmark (astro-ph/9511148) Tegmark Max
Observational Cosmology K. Basu: CMB theory and experiments 58 Power spectrum: primary anisotropies
Acoustic peaks
Sachs-Wolfe plateau Damping tail
Observational Cosmology K. Basu: CMB theory and experiments 59 Sources of primary anisotropies
Quantum density fluctuations in the dark matter were amplified by inflation. Gravitational potential wells (or “hills”) developed, baryons fell in (or moved away).
Various related physical processes afected the CMB photons:
• Perturbations in the gravitational potential (Sachs-Wolfe efect): photons that last scattered within high-density regions have to climb out of potential wells and are thus redshifted
• Intrinsic adiabatic perturbations (the acoustic peaks): in high-density regions, the coupling of matter and radiation will also compress the radiation, giving a higher temperature. This creates sound-like waves.
• Velocity perturbations (the Doppler troughs): photons last-scattered by matter with a non-zero velocity along the line-of-sight will receive a Doppler shift
Observational Cosmology K. Basu: CMB theory and experiments 60 Sachs-Wolfe efect
Δν/ν ~ Δ T/T ~ Φ/c2
Additional efect of time dilation (see next slide) while the potential evolves leads to a factor of 1/3 (e.g. White & Hu 1997):
The temperature fluctuations due to the so-called Sachs-Wolfe efect are due to two competing efects: (1) the redshift experienced by the photon as it climbs out of the potential well toward us and (2) the delay in the release of the radiation, leading to less cosmological redshift compared to the average CMB radiation.
Observational Cosmology K. Basu: CMB theory and experiments 61 Sachs-Wolfe efect
The second contribution (redshift due to time delay) is easy to compute. Because of general relativity, the proper time goes slower inside the potential well than outside. The cooling of the gas in this potential well thus also goes slower, and it therefore reaches 3000 K at a later time relative to the average Universe.
Observational Cosmology K. Basu: CMB theory and experiments 62 Sachs-Wolfe efect
For power-law index of unity primary density perturbations, ns=1 (Zel’dovich, Harrison ~1970), the Sachs-Wolfe efect produces a flat SW power spectrum: Cl ~ 1/ l (l+1).
(For Sachs-Wolfe efect)
Observational Cosmology K. Basu: CMB theory and experiments 63 Measuring Cl at low multipoles (l≤100)
The horizon scale at the surface of last scattering (z ~ 1100) corresponds roughly to 2°. At scales larger than this (l ≥ 100), we thus see the power spectrum imprinted during the inflationary epoch, unafected by later, causal, physical processes.
Due to the limit of cosmic variance, the measurements by COBE some ~25 years ago was already of adequate accuracy!
Observational Cosmology K. Basu: CMB theory and experiments 64 Acoustic oscillations
• Baryons fall into dark matter potential wells: Photon baryon fluid heats up
• Radiation pressure from photons resists collapse, overcomes gravity, expands: Photon-baryon fluid cools down
• Oscillating cycles exist on all scales. Sound waves stop oscillating at recombination when photons and baryons decouple.
Credit: Wayne Hu
Springs: Balls: photon baryon pressure mass
Observational Cosmology K. Basu: CMB theory and experiments 65 Acoustic oscillations
Equations of motion for self-gravitating non-relativistic gas (here photon-baryon fluid):
Continuity eqn.
From these three we get the perturbation equation Euler eqn. for the photon gas
Poisson eqn.
For radiation-dominated era, set p=ρc2/3
We want to solve for this δ.
Observational Cosmology K. Basu: CMB theory and experiments 66 Acoustic oscillations
(perturbation equation)
Observational Cosmology K. Basu: CMB theory and experiments 67 Acoustic oscillations
The last equation is a standing wave with an interesting property: its phase is fixed by the factor in parenthesis. This means that for every wave number k we know what the phase of the oscillating standing wave is at the time of the CMB release. For some modes this phase may be π/2, in which case the density fluctuation has disappeared by the time of CMB release, but the motion is maximum. For others the density fluctuation may be near maximum (phase 0 or π). This gives the distinct wavy pattern in the power spectrum of the CMB.
Observational Cosmology K. Basu: CMB theory and experiments 68 Angular variations
Density fluctuation on the sky from a single k mode, and how it appears to an observer at diferent times:
These frames show one superhorizon temperature mode just after decoupling, with representative photons last scattering and heading toward the observer at the center. (From left to right) Just after decoupling; the observer’s particle horizon when only the temperature monopole can be detected; some time later when the quadrupole is detected; later still when the 12-pole is detected; and today, a very high, well-aligned multipole from just this single mode in k space is detected.
Animation by Wayne Hu Observational Cosmology K. Basu: CMB theory and experiments 69 Acoustic peaks
Oscillations took place on all scales. We see temperature features from modes which had reached the extrema
• Maximally compressed regions were hotter than the average Recombination happened later, corresponding photons experience less red-shifting by Hubble expansion: HOT SPOT
• Maximally rarified regions were cooler than the average Recombination happened earlier, corresponding photons experience more red-shifting by Hubble expansion: COLD SPOT 1st peak Harmonic sequence, like waves in pipes or strings:
2nd harmonic: mode compresses and rarifies by harmonics recombination 3rd harmonic: mode compresses, rarifies, compresses
➡ 2nd, 3rd, .. peaks
Observational Cosmology K. Basu: CMB theory and experiments 70 Harmonic sequence
Credit: Wayne Hu
Modes with half the 1 wavelengths osccilate twice as fast (ν = c/λ). 2 3 Peaks are equally spaced in
Observational Cosmology K. Basu: CMB theory and experiments 71 Doppler shifts
Times in between maximum compression/rarefaction, modes reached maximum velocity
This produced temperature enhancements via the Doppler efect (non-zero velocity along the line of sight)
This contributes power in between the peaks
➡ Power spectrum does not go to zero (it does at very high l-s)
Observational Cosmology K. Basu: CMB theory and experiments 72 Damping and difusion
• Photon difusion suppresses fluctuations in the baryon-photon plasma (Silk damping)
• Recombination does not happen instantaneously and photons execute a random walk during it. Perturbations with wavelengths which are shorter than the photon mean free path are damped (the hot and cold parts mix up) Thickness of the LSS is In the power spectrum scales below 3 comparable ~0.2° (l > ~10 ) are damped. to the oscillation scales
Power falls off
Observational Cosmology K. Basu: CMB theory and experiments 73 Power spectrum summary
Acoustic peaks
Sachs-Wolfe plateau Damping tail ISW rise
Observational Cosmology K. Basu: CMB theory and experiments 74 Which way the peaks move?
Credit: Wayne Hu
Observational Cosmology K. Basu: CMB theory and experiments 75 Baryon loading
The presence of more baryons add inertia, and increases the amplitude of the oscillations (baryons drag the fluid into potential wells).
Perturbations are then compressed more before radiation pressure can revert the motion.
This causes a breaking of symmetry in the oscillations, enhancing only the compressional phase (i.e. every odd-numbered peak).
This can be used to measure the abundance of cosmic baryons. Credit: Wayne Hu
Observational Cosmology K. Basu: CMB theory and experiments 76 Baryons in the power spectrum
Credit: Wayne Hu
Power spectrum shows baryon enhance every other peak, which helps to distinguish baryons from cold dark matter
Observational Cosmology K. Basu: CMB theory and experiments 77 DM in the power spectrum
Cold dark matter Baryons
Credit: Max Tegmark
Observational Cosmology K. Basu: CMB theory and experiments 78 Efect of curvature
Ωk does not change the amplitude of the power spectrum, rather it shifts the peaks sideways. This follows from the conversion of the physical scales (on the LSS) to angular scales (that we observe), which depends on the geometry.
Curvature (cosmological constant, ΩΛ) also causes the ISW efect on large scales, by altering the growth of structures in the path of CMB photons.
Observational Cosmology K. Basu: CMB theory and experiments 79 CMB parameter cheat sheet
Observational Cosmology K. Basu: CMB theory and experiments 80 Questions?
Observational Cosmology K. Basu: CMB theory and experiments 81