Elements of observational cosmology 28.05.2009 Anisotropies in the CMB: what they are and what has...

Elements of observational cosmology

28.05.2009 Anisotropies in the CMB: what they are and what has created them?

CMB: 5 years of WMAP

Cosmic microwave background radiation

Cosmic microwave background radiation (CMB, CMBR) – diffuse background radiation observed at cm – mm wavelenghts

1965, Penzias & Wilson, Bell Telephone Laboratories (1965), cm.

1989 – COBE 2002 – WMAP 2009 - Planck

Cosmic microwave background radiation

T = 2.725 +/- 0.002 K The most ideal black

body in the Universe Reileigh-Jeans tail in

the cm range, the Wien function and maximum in mm.

Fluctuations ~10^{-5}

CMB: isotropy

T = 2.725 +/- 0.002 K – practically uniform all over the sky

CMB: dipole ~10-4 : dipole anisotropy (one part

of the sky is 1/1000 “warmer” than the opposite)

the dipole amplitude is 3.353+/-0.024 mK

interpretation: we are moving with respect to the CMB rest frame, i.e. mainly the “proper motion” of our galaxy

T = T0(1+(v/c)cosθ) θ – an angle with respect to

the direction of the max intensity of CMB (galactic coordinates l = 264.25o i b = 48.22o, where Hydra/Centaurus galaxy cluster is); v – the velocity of Earth with respect to CMB, ~350 km/s

The most widely used statistical tool used to analyze fluctuations in the CMB is power

spectrum of its temperature fluctuations

Temperature fluctuations: ΔT/T(Θ).

In case of the adiabatic origin of the CMB fluctuations they are related to the density field fluctuations by:

TT

=13

BB

Power spectrum is defined as the Fourier transform of the autocorrelation function of fluctuations (i.e. an excess probability over random that two points at angular distance Θ have a difference in temperature ΔT).

CMB power spectrum

Since we measure CMB on all the sky, we have to operate in the space of spherical coordinates. Then, it is convenient to decompose ΔT/T into shperical harmonics:

TT

n =∑l=0

∞∑m=−l

lalmY lm n

In such a case the correlation function of the fluctuations can be expressed by a series of the Legendre polynomials:=⟨ T

Tn T

T n ' ⟩= 1

4 ∑l2l1C l P l n⋅n '

CMB power spectrum

=⟨ TT

n TT

n ' ⟩= 14 ∑l

2l1C l P l n⋅n '

C l=⟨a lm2 ⟩

And it can be shown that C

l is an

angular power spectrum of the CMB fluctuations:

CMB power spectrum can be devided into three “interesting ranges, which display the

effect of different physical processes

large scale plateau (super-horizon

scales)

acoustic oscillations (in the primordial plasma

before recombination on

the scales comparable to the horizon size then)

Damping tail

Classification of the CMB Anisotropies (from: Tegmark 1995)

The origin of the CMB power spectrum

Before z~1000 CMB photons were (1+z) times hotter and able to ionize H atoms.

“Baryon-photon fluid”. After the recombination

epoch at z=1000, CMB photons stopped to react with matter (at least roughly speaking).

Differences of matter density then were imprinted in the temperature fluctuations and can be observed now

from: Wayne Hu

The origin of the CMB power spectrum?

These perturbations have given rise to today's structures through the gravitational instability mechanism. The response of the fluid to the gravitational potential fluctuations allows us to measure the properties of the fluid in an expanding universe known to be filled with dark matter, which allows us to extract basic cosmological parameters, as well as those of the seed perturbations, which can be used to pin down the nature of large scale structure formation in the universe.


-> CMB maps -> power spectrum.

All sky fluctuations -> angular distrubution in the multipole space l (proportional to the inverse angle) instead of usual Fourier variable k.


Acoustic oscillations

The strongest and the most important structures in the spectrum result from acoustic oscillations in the photon-baryon fluid just before recombination.


Acoustic oscillations Radiation pressure

from the photons resists the gravitational compression of the fluid into potential wells and sets up acoustic oscillations in the fluid

(Springs represent photon pressure and balls represent the effective mass of the fluid.)


Acoustic oscillations The shorter the wavelength of the

potential fluctuation, the faster the fluid oscillates such that at last scattering the phase of the oscillation reached scales with the wavelength.

Since regions of compression (maxima) represent hot regions and refaction (minima) cold regions, there will be a harmonic series of peaks in wavelength associated with the acoustic oscillations. It is the features of these peaks that provides us with an array of cosmological tests.

The time evolution of a single wavelength of the potential fluctuation (Fourier amplitude time evolution on right).

Since the photons must climb out of the potential wells at last scattering, the effective temperature, Theta+Psi, is reduced at long wavelengths to 1/3 Psi. This effect is commonly known as the Sachs-Wolfe effect.


Acustic oscillations as baryonic weight Baryons increase the

effective mass of the fluid. This changes the balance between pressure and gravity in the fluid. Gravitational infall now leads to greater compression of the fluid in the potential well. Just like a mass on a spring, gravity shifts the zero point of the oscillator.


Acustic oscillations as baryonic weight This increases the amplitude

of the oscillation, since the initial conditions now represent a larger displacement from the zero point. More importantly it changes the absolute value (rms, dotted lines) of the maxima vs. minima of the effective temperature fluctuation. Compressions are enhanced over rarefactions of the fluid inside potential wells. Thus the relative heights of the peaks present one way of measuring the baryon content of the universe. In the scale-invariant adiabatic model, this is how the anisotropy depends on the baryon content:

Hu & White 1996


Acustic oscillations as baryonic weight

Change in peak heights as the baryon content is raised (h=0.5 -> Omega_b h^2 = 0.00125-0.125)

Baryons increase the amplitude of the oscillations as well as cause an alternation in the odd and even peak heights.

Hu & White 1996


Doppler effect Since the turning

points of the oscillation are at the extrema, the fluid oscillates 90 degrees out of phase with the temperature. Its line-of-sight motion relative to the observer causes a Doppler shift.


Doppler effects and baryons If the baryons are neglected,

the Doppler oscillations have the same amplitude as the temperature oscillations. However, as we have seen, baryons increase both the amplitude of the oscillation and the rms magnitude of the compressional peaks strongly. Furthermore, by conservation of energy, as the effective mass of the oscillator goes up, the velocity goes down as the square root of the mass. Together, these factors imply that in the real universe which has a significant baryon content, the Doppler effect is subdominant.

Relative Suppression by Baryons


Doppler effects and baryons An analytic

decomposition of the anisotropy shows this effect clearly.

The total contribution includes damping processses and ISW effects.

The Doppler effect plays a more significant role if the universe is reionized. Hu & White 1997


Adiabatic fluctuations Realistic models involve time-dependent potentials giving rise to

forced oscillations in the photon-baryon fluid. This allows to separate two competing models for structure formation cleanly from the CMB: adiabatic and isocurvature models.

Adiabatic fluctuation, of which inflation is the unique causal generator, begin with gravitational potential wells established in the very early universe.

On the other hand isocurvature models, begin with no potential fluctuations. They are only generated as matter is moved around inside the causal horizon. Cosmological defects provide isocurvature conditions, but there are other more phenomenologically based models.


Adiabatic fluctuations The behavior of the potential wells, in

particular the feedback influence of density perturbations generated by the acoustic oscillations on the potential wells themselves, serve to distinguish the two general cases: Adiabatic case Isocurvature case


Adiabatic fluctuations Adiabatic fluctuations: as the

fluid begins to compress, photon pressure resists the increase in the density perturbation, thereby allowing the gravitational potential to decay. Left in a highly compressed state, the fluid then oscillates as a cosine wave with enhanced amplitude.


Adiabatic fluctuations Isocurvature case: Radiation

fluctuations are here set up to eliminate the potential initially. Photon pressure resists the implied rarefaction of the fluid. The fluid then falls back into the well, lending its self-gravity to enhance the depth. At the compressional maxima, photon pressure again causes potential decay leaving the fluid in a highly compressed state. However, here it drives a sine oscillation.


Adiabatic fluctuations Because the two general

classes of perturbations drive different harmonics (cosine vs. sine), they can be cleanly distinguished by taking the ratios of the locations of the peaks

However, isocurvature models (e.g. cosmic strings) can also have strong effects after horizon crossing which can wash out these signals (see Magueijo et al. 1995 ). Likewise adiabatic models could be reionized and lose their peak structure.

Hu & White (1996)


Adiabatic fluctuations These feedback effects

increase with the radiation-matter density ratio allowing a measure of the matter-radiation equality epoch. Ignoring diffusion damping, we see that the peak heights in the adiabatic model rise by a factor of approximately 25 (20 in the presence of realistic neutrino content) through the equality scale: leading to a probe analogous to the shape parameter of large scale structure but in anticorrelation with the latter effect. The ability to measure the equality epoch allows for a sensitive probe of the total matter content of the universe. Hu & White (1997)


Adiabatic fluctuations The matter-radiation

ratio is raised while keeping all other parameters fixed (Omega0h2= 0.1-1)

With highly sensitive measurements there is even the possibility of measuring properties of the radiation such as the neutrino mass

Hu & White (1996)


Diffusion damping In reality, the coupling

between the baryons (electrons) and the photons is imperfect since the photons possess a mean free path to Compton scattering. As the photons random walk through the baryons, hot and cold regions are mixed. Fluctuations damp nearly exponentially as the diffusion length overtakes the wavelength.


Diffusion damping At last scattering, the

ionization fraction decreases due to recombination, thus increasing the mean free path of the photons. The effective diffusion scale becomes, by definition, the thickness of the last scattering surface providing a cut off in the anisotropy spectrum.

Hu & White (1997)


Diffusion damping Since this scale is

sensitive to the baryon content and the thermal history while being almost entirely independent of the model for structure formation, it provides a valuable tool for measuring these cosmological quantities as well as the curvature of the universe..

Hu & White (1997)


Projection problems A spatial temperature

fluctuation on the last scattering surface appears to us as an anisotropy on the sky. The conversion from physical scale into angular scale depends on the curvature of the universe and the distance to the last scattering surface. The former can alternately be thought of as gravitational lensing from the background curvature rather than curvature fluctuation. Consider first the case of positive curvature:


Projection problems Closed Universe: Photons free

stream to the observer on geodesics analogous to lines of longitude to the pole. Thus the same angular scale represents a smaller physical scale in a closed universe.


Projection problems The opposite effect is true of

open universes: Acoustic features appear at

harmonics of a fixed physical scale at last scattering, the sound horizon. Measurement of the angular scale that such features subtend on the sky can thus measure the curvature of the universe through an angular size distance test.


Projection problems Decreasing the distance to the

last scattering surface decreases the physical scale associated with a given angular scale. We shall see that sources in the foreground of last scattering such as the ISW effect are affected by this property. However for effects that arise purely from the last scattering surface, such as the acoustic features, the presence of curvature merely scales the features in angular or multipole l space

Angular diameter distance scaling with curvature and lambda (Omega_K=1-Omega_0-Omega_Lambda, fixed Omega_0h^2 and Omega_Bh^2), Hu & White 1997


Projection problems With the matter-radiation and

baryon-photon ratios fixed, acoustic features just scale with the projection or angular size distance. Geodesic deviation brings the features to higher multipoles (smaller angles) in an open universe.


Projection problems In a cosmological constant

(Lambda) universe the rapid expansion causes a decrease in the comoving distance to last scattering bringing the features to slightly larger angles. The degeneracy between these two effects is broken at larger angles by the late Integral Sachs-Wolf (ISW) effect.


Projection problems The spacing between the

peaks provides the most robust test of the curvature. In models where the variation in the gravitational potential is slow compared with the natural frequency of the oscillator, the natural period of the oscillation sets the separation betwen the peaks.

Unfortunately uncertainties in the Hubble constant h and to a lesser extent the baryon-photon ratio muddy the situation a bit.

Hu & White (1996)


Projection problems Here we have taken extremely

conservative uncertainties in the baryon content and realistic uncertainties in the Hubble constant. Note that since we here allow the matter content Omega_0 h^2 to vary with Omega_0 in the Lambda universe, the scaling of the peaks does not take on the simple form described above.

The damping scale also provides an angular size distance test for the curvature. Because it is sensitive to the Compton mean free path at last scattering, uncertainties in the baryon content introduce a larger uncertainty in pinning down the curvature:

Of course, by combining these scales one can build a measurement of the curvature that is robust to changes in the other cosmological parameters.

Hu & White (1996)


Integral Sachs-Wolf effect Between z=1000 and today, photons free-stream to the

observer assuming that reionization did not take place too early. However, gravitational effects still play a role in modifying the anisotropy spectrum. The simplest example is the gravitational redshift effects from potential perturbations. The gravitational redshift felt by the photons climbing out of the potential well at last scattering has already been discussed and encorporated in the effective temperature of the acoustic oscillations. However, if the depth of the potential wells between the last scattering surface and the observer change as the photons cross it, the differential redshifts will induce further anisotropies in the CMB.


Integral Sachs-Wolf effect There are two pieces to the

gravitational redshift that leads to a doubling of the effect. If the depth of the potential well changes as the photon crosses it, the blueshift from falling in and the redshift from climbing out no longer cancel leading to a residual temperature shift. The second effect is general-relativistic in nature. A potential fluctuation represents a fluctuation in the space-time curvature. Heuristically, the wavelength of the photon is "stretched" along with the fabric of space-time. If the potential well decays, the photons will be blueshifted as the space-time fabric "contracts" further anisotropies in the CMB.


Integral Sachs-Wolf effect The wavelength of a

photon stretches with the space-time fabric in a potential well leading to redshifts and blueshifts as the potential varies.


Accurate theoretical anisotropy power spectra for popular models have long been available (e.g. see the Berkeley Archive) and codes to calculate them are now fast and public (e.g. Seljak & Zaldarriaga 1996: CMBFast). Here we concentrate on the gross features of the spectrum which are likely to occur in most models.

The physical scales associated with the acoustic (A), matter-radiation equality (eq), damping (D),and late ISW (KLambda) anisotropy formation mechanisms are imprinted on the anisotropy power spectrum.

A scale-invariant adiabatic example. Power spectrum of temperature anisotropies in multipole space (l is proportional to the inverse of the angle). Independent of these scalar effects, there may be tensor contributions below the acoustic scale in l .


Accuracy of satellites CMB anisotropy experiments should be able to reach the required angular

resolution, sensitivity, sky coverage and reduction of systematic effects to probe the acoustic features in the spectrum. These include ground, long-duration balloon, and in particular two satellite experiments.

With this kind of sensitivity, most of the cosmological parameters of a near-scale invariant adiabatic model can be fit to the several percent level. Assumptions about the underlying model may be relaxed as information in the higher peaks is uncovered to provide more of a true measurement of the parameters.

Whether these goals can actually be achieved depends on the level of foreground contamination (see e.g. Tegmark & Efstathiou (1995)) and systematic effects. Galactic foregrounds will limit the area of clean sky available, increasing the error bars as the inverse square root of the fraction of sky, and foreground subtraction will increase the noise. Systematic effects (sidelobes, calibration, striping in the map from 1/f noise, etc.) may also degrade the sensitivity.

We are also here assuming that uncertain secondary anisotropies such as non-linear effects from reionization and gravitational lensing create only a small perturbation to the primary spectrum. This may not be a good assumption beyond l = 1000 .


Accuracy of satellites

PLANCKWMAP


Re-ionisation We know that the universe has been

ionized at least until z = 5 from the null detection of the Gunn-Peterson effect. Hence after standard recombination at z = 1000 , the universe must have underwent reionization. However, the amount of rescattering of CMB photons is negligible unless the ionization persists through redshifts of several tens. This is unlikely in models without excessive small scale

power and in such models, e.g. scale invariant adiabatic ones, degree scale anisotropy detections are already severely constraining the amount of reionization allowable. Still, it is useful to recall the basic physical effects associated with reionization.


Re-ionisation Rescattering Damping: Rescattering

damps fluctuations in the same manner as diffusion. Scattering eliminates anisotropies leaving them only in the unscattered fraction exp(-optical depth). Since outside the horizon, streaming has not yet converted temperature fluctuations to anisotropies, power is only lost below the horizon at the rescattering epoch.

Doppler Effect: Diffusion and rescattering prevents the appearance of large temperature fluctuations. However, the Doppler effect from scattering off electrons caught in the gravitational instability of the baryons can regenerate anisotropies. These contributions are suppressed in the same way as the late ISW effect:


Re-ionisation Photons that last scattered off

opposite sides of the perturbation get Doppler shifted by equal and opposite amounts. Thus for wavelengths far below the thickness of the last scattering surface, Doppler contributions tend to cancel leaving a negligible net effect.


Re-ionisation Non-linear Effects: At very small scales,

higher order contributions are more efficient than the Doppler effect in regenerating anisotropies. These generally make use of combining the Doppler effect with variations in the optical depth. For example, the enhanced baryon density in an overdense region leads to preferential scattering in those regions. If the perturbations are also caught up in a bulk flow, then a Doppler shift arises that can escape cancellation (known as the Vishniac or Ostriker-Vishniac effect):

Similarly, inhomogeneities in the ionization fraction, or contributions from clusters (kinematic Sunyaev-Zel'dovich effect) can give Doppler contributions. Clusters can also produce anisotropic spectral distortions due to the upscattering in frequency of photons by hot electrons (thermal Sunyaev-Zel'dovich effect).

Again, in models without high amplitude small-scale power, it is unlikely that any of these effects will dominate the total anisotropy in the observable regime.

Mikrofalowe promieniowanie tła: przykład opisu zaburzeń (efekt Sunyaeva-Zeldowicza)

Jeśli CMB związane jest z uwolnieniem energii termicznej tuż przed epoką rekombinacji przy z ~1000 i jeśli ilość fotonów została zachowana, to powinno ono było osiągnąć stan równowagi z widmem Bosego-Einsteina

I =2h 3

c2 [exp hk T r−1]−1

gdzie μ – bezwymiarowy potencjał chemiczny

Mikrofalowe promieniowanie tła: opis zaburzeń (przykład)

Interpretacja: rozkład Bosego-Einsteina opisuje stan równowagi fotonów w sytuacji, kiedy mamy “nadmiar” całkowitej energii w stosunku do ilości fotonów, którym jest ona przypisana.

W sytuacji, kiedy promieniowanie ma widmo ciała doskonale czarnego, zarówno gęstość energii jak i gęstość liczbowa fotonów zależą wyłącznie od temperatury T.

W przypadku rozkładu Bosego-Einsteina mamy za to 2 parametry: T

r i μ.

I =2h 3


Mikrofalowe promieniowanie tła: opis zaburzeń

W związku z tym parametr μ może posłużyć do opisu zaburzeń – odchyleń widma od widma ciała doskonale czarnego.

I =2h 3


Mikrofalowe promieniowanie tła: opis zaburzeń: efekt Sunyaeva-Zeldowicza

W przypadku rozpraszania Comptona fotonów CMB na gorących elektronach (w późniejszych epokach) – średnie energie fotonów rosną i widmo zostaje przesunięte do wyższych częstotliwości. Zaburzenie można zapisać jako:

I I

= y x ex

e x−1 [ x e x1e x−1−4 ]

gdzie długość rozpraszania Comptona

I I

= y x ex

e x−1 [ x ex1ex−1−4 ]

x = hν/kT_r, a σT – przekrój na rozpraszanie Thomsona.

Mikrofalowe promieniowanie tła: opis zaburzeń: efekt Sunyaeva-Zeldowicza

Ograniczenia tych parametrów: y <1.5 x 10^{-5} μ < 10^{-4}

W tym wypadku te wartości dają ogranczenia na własności gazu międzygalaktyczngo (bo to do niego należą gorące elektrony, na których CMB może się rozpraszać)

Jest cały szereg innych efektów, powodujących zaburzenia CMB: pierwotne (będące w CMB już od początku) wtórne (których CMB dorobiło się w drodze od

z~1000 do nas)

Elements of observational cosmology 28.05.2009 Anisotropies in the CMB: what they are and what has...

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