Bose-Einstein spectrum 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(h ν /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, both radiative Compton and bremsstrahlung processes permit the generation of photons needed to ensure true thermal equilibrium. 22 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Bose-Einstein spectrum The claim that thermal equilibrium is established at the high redshift of ~2 × 10 6 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 an epoch corresponding to a redshift of ~2 × 10 6 , 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. 23 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 ine ffj cient ( Γ > H − 1 ) , so we get y << 1. Lectures 2+4 (K. Basu): CMB theory and experiments 24 Observational Cosmology
μ - and y -type distortions Ref: Khatri & Sunyaev, arXiv:1203.2601 25 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Current limits on Spectral Distortions • Energy added after z~2x10 6 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 (z < 10 5 ) show up as “y distortion”, where y ~ σ T n e kT e (e.g. the Sunyaev-Zel’dovich e fg ect). 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 << 10 5 ), e.g. free-free distortions, are also tightly constrained. COBE measurement: Y fg < 1.9 x 10 -5 (95% CL) 26 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
μ - and y -type distortions Sensitivity of the proposed Pixie satellite Kogut et al. (2011), arXiv:1105.2044 27 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Future missions for spectral measurement Pixie satellite concept Cosmic Origins Explorer (PRISM, CoRE+) 28 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Recombination stages Sunyaev & Chluba (2009) 29 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Spectral features during recombination 30 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Temperature anisotropies Lectures 2+4 (K. Basu): CMB theory and experiments 31 Observational Cosmology
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. 32 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 ? 33 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Doppler e fg ects on CMB (~ β 2 e fg ects) Planck collaboration (2013) 34 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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! 35 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
DMR on COBE Di fg erential Microwave Radiometer • Di fg erential radiometers measured at frequencies 31.5, 53 and 90 GHz, over a 4-year period • Comparative measurements of the sky o fg er far greater sensitivity than absolute measurements Credit: NASA 36 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
COBE DMR Measurements Credit: Archeops team Credit: NASA 37 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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. Thus after 7 years, the data could still Credit: NASA be added and noise lowered (of course, the improvement will be marginal). 38 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
WMAP results after 1st year Internal Linear Combination map (Credit: WMAP Science Team) 39 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Planck results after 1.5 years (next lecture) 40 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 41 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Analogy: Fourier series Sum sine waves of di fg erent frequencies to approximate any function. Each has a coe ffj cient, or amplitude. 42 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Spherical harmonics 43 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Spherical harmonics l~180º/ θ relates to the angular size of the pattern, whereas m relates to the orientation of the pattern. Thus 〈 |a lm | 2 〉 is independent of m. The mean 〈 a lm 〉 is zero, but we want to calculate its variance: (Reason is the S-W e fg ect!) Lectures 2+4 (K. Basu): CMB theory and experiments 44 Observational Cosmology
Visualizing the multipoles Lectures 2+4 (K. Basu): CMB theory and experiments 45 Observational Cosmology
Visualizing the multipoles (real sky!) 46 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Power at di fg erent scales Credit: Wayne Hu 47 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
CMB power spectrum How to constrain cosmology from this measurement? Use spherical harmonics in place of sine waves: Calculate coe ffj cients, a lm , and then the statistical average: Amplitude of fluctuations on each scale ⎯ that’s what we plot. (TT power spectrum) 48 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Make your own CMB experiment! • Design experiment to measure • Find component amplitudes • Plot against l (where l is inverse of angular scale, l ~ π / θ ) 49 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Generating theoretical C l ?? OUTPUT INPUT Physics Favorite cosmological model: powerful Ω m , Ω Λ , σ 8 , H 0 , .. cosmological codes (CMBFast or CAMB) Fit to data 50 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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: For an idealized full-sky observation, the variance of each measured C l is: How well we can estimate an average value from a sample depends on how many points we have on the sample. This is called the cosmic variance, and it is an unavoidable source of uncertainty when constraining models! Lectures 2+4 (K. Basu): CMB theory and experiments 51 Observational Cosmology
Cosmic and sample variance • Cosmic variance : on scale l , there are only 2l+1 independent modes • Averaging over l in bands of Δ l ≈ 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/ √ f sky and the resulting sample variance is called variance (f ~0.65 for the PLANCK satellite) 52 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
The origin of temperature anisotropies 53 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 Please don’t confuse between the “creation” of the CMB photons, and their “release” from the last scattering surface! CMB photons are created at much earlier e p o c h t h r o u g h m a t t e r / a n t i - m a t t e r annihilation, and thus, were formed as g a m m a r a y s ( n o w c o o l e d d o w n t o microwave) 54 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Primordial temperature anisotropies Creation of anisotropies 55 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Sources of Δ T Max Tegmark (astro-ph/9511148) 56 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Power spectrum: primary anisotropies Acoustic peaks Sachs-Wolfe plateau Damping tail 57 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 a fg ected the CMB photons: • Perturbations in the gravitational potential (Sachs-Wolfe e fg ect): 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 • 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 58 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Sachs-Wolfe e fg ect Δν / ν ~ Δ T/T ~ Φ /c 2 As we explain below, additional e fg ect of time dilation 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 e fg ect are due to two competing e fg ects: (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. 59 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Sachs-Wolfe e fg ect The second contribution (redshift due to time delay) is more tricky. 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. Lectures 2+4 (K. Basu): CMB theory and experiments 60 Observational Cosmology
Sachs-Wolfe e fg ect For power-law index of unity for the primary density perturbations, n s =1 (Zel’dovich, Harrison ~1970), the Sachs-Wolfe e fg ect produces a flat power spectrum: C l SW ~ 1/ l (l+1) . (For Sachs-Wolfe e fg ect) Lectures 2+4 (K. Basu): CMB theory and experiments 61 Observational Cosmology
Measuring C l 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, una fg ected 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! 62 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 63 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Acoustic oscillations Equations of motion for self-gravitating non-relativistic gas: Continuity eqn. From these three we get the perturbation equation Euler eqn. Poisson eqn. For radiation-dominated era, set p= ρ c 2 /3 We want to solve for this δ . 64 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Acoustic oscillations (perturbation equation) 65 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 a distinct wavy pattern in the power spectrum of the CMB. Lectures 2+4 (K. Basu): CMB theory and experiments 66 Observational Cosmology
Angular variations Density fluctuation on the sky from a single k mode, and how it appears to an observer at di fg erent 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 67 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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: harmonics 2nd harmonic: mode compresses and rarifies by recombination 3rd harmonic: mode compresses, rarifies, compresses ➡ 2nd, 3rd, .. peaks 68 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 69 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Doppler shifts Times in between maximum compression/rarefaction, modes reached maximum velocity This produced temperature enhancements via the Doppler e fg ect (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) 70 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Damping and di fg usion • Photon di fg usion (Silk damping) suppresses fluctuations in the baryon- photon plasma • 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 This is same as a low-resolution comparable instrument blurs all the details! to the oscillation scales Power falls off 71 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Power spectrum summary Acoustic peaks Sachs-Wolfe plateau Damping tail ISW rise 72 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Which way the peaks move? Credit: Wayne Hu 73 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 74 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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 75 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
DM in the power spectrum Cold dark matter Baryons Credit: Max Tegmark 76 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
E fg ect of curvature Ω k 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, Ω Λ ) also causes ISW e fg ect on large scales, by altering the growth of structures in the path of CMB photons. 77 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
CMB parameter cheat sheet 78 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
SECONDARY temperature anisotropies 79 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Integrated Sachs-Wolfe e fg ect Temperature anisotropies due to density change and associated gravitational potential (scaler perturbations) at a given point x along the direction n 80 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Integrated Sachs-Wolfe e fg ect • The early ISW e fg ect is caused by the small but non-negligible contribution of photons to the density of the universe • The late ISW e fg ect: • Gravitational blueshift on infall does not cancel redshift on climb-out • Contraction of spatial metric doubles the e fg ect: Δ T/T ~ 2 ΔΦ • E fg ect of potential hills and wells cancel out on small scales 81 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Integrated Sachs-Wolfe e fg ect ISW e fg ect Lectures 2+4 (K. Basu): CMB theory and experiments 82 Observational Cosmology
ISW e fg ect as Dark Energy probe The ISW e fg ect constraints the dynamics of acceleration, be it from dark energy, non-flat geometry, or non-linear growth. Cosmic evolution of dark energy is parametrized by w(a) ≡ p DE / ρ DE For a cosmological constant, w=-1. In general, ρ DE ~ a -3(1+w) In the absence of curvature, measurement of ISW is measurement of DE. 83 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Cosmic variance problem Lectures 2+4 (K. Basu): CMB theory and experiments 84 Observational Cosmology
Cosmic variance problem Problem : Low multipole signals are severely cosmic variance limited Corasaniti, Giannantonio, Melchiorri 2005 Solution : Cross-correlate with other probes of dark energy, with large sky coverage (optical, X-ray or radio surveys of galaxies) 85 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
ISW optical-survey correlation Lectures 2+4 (K. Basu): CMB theory and experiments 86 Observational Cosmology
Δ T from reionization Re-scattering of CMB photons damps anisotropy power ( Δ T 2 ) as e -2 τ , with τ the optical depth to Thomson scattering. For τ = 0.095, this means a 20% reduction from initial power. New perturbations are generated on small scales due to the bulk motion of electrons in over-dense regions (Ostriker-Vishniac e fg ect) 87 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
kSZ e fg ect from reionization kSZ anisotropy at ell ≈ 1000 - 10000 receives a substantial contribution from kSZ anisotropy due to patchy reionization (“patchy kSZ”), which arises again from n e ∗ v pec but, during this epoch, the fluctuations in n e are fractionally large due to the inhomogeneous nature of reionization. The amplitude of the power spectrum of patchy kSZ scales with the duration of the epoch of reionization (EoR), but there is much more information about the bubble size and velocity field during reionization to be obtained if the power spectrum’s shape (and possibly the higher moments of the field) can be measured. CCAT White Paper 88 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Other small e fg ects.. Resonant scattering by atoms and molecules (CMB spectroscopy!) OI 63 µm line Basu, Hernandez-Monteagudo & Sunyaev 2004 89 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Power at small angular scales The signal is actually C l ! Our power spectrum plots boosts the apparent variance at large l by a factor l 2 ! Observations at high- l therefore requires far greater sensitivity. 90 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Local y -distortion: thermal SZ e fg ect 91 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
tSZ power spectrum IR galaxies foreground Ramos-Ceja, Basu et al. 2014 Simulation and observation of high-resolution CMB sky 92 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
tSZ power and foregrounds 93 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
SZ power spectrum SZ power spectrum is a powerful probe of cosmology, primarily through its strong dependance on σ 8 SPT constraints assuming di fg erent tSZ templates 94 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Lensing of the power spectrum 95 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Lensing of the power spectrum Lensing smooths the power spectrum (and E mode polarization) with a width ∆ l~60 This is a small e fg ect, reaching ~10% deep in the damping tail. 96 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
High-ell CMB bandpower measurement Keisler et al. 2011 Lensed CMB 97 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
CMB polarization Lectures 2+4 (K. Basu): CMB theory and experiments 98 Observational Cosmology
How to go further with CMB? Cosmic Variance › We only have one realization (our sky), i.e., one event. › TT at small l (incl. first peak) is now cosmic variance limited. To go further: › TT at large l › Polarization 99 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
Polarization of the CMB 100 Observational Cosmology Lectures 2+4 (K. Basu): CMB theory and experiments
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