The cosmic microwave background radiation
Laszlo DobosDept. of Physics of Complex Systems
[email protected] 5.60
May 18, 2018.
Origin of the cosmic microwave radiationPhotons in the plasma are scattered constantly
380 thousand years after the Big Bang: recombinationI temperature falls (well) below the ionization energy of HI around 3000 K (determine from the Saha equation)I protons combine with electron into neutral H atomsI the Universe becomes gradually transparent for thermal
photon
Surface of the last scatteringI photons are scattered for the very last time in the plasmaI average free path becomes larger than size of the horizonI free streaming, in a mere 13.8 bn years, they reach usI today its temperature is redshifted to 2.7 K, microwave
Spectrum of the cosmic microwave backgroundI same as at the time of last scattering but redshiftedI Planck curve but varies from direction to direction
Surface of last scattering
The farthest surface we can ever observe via EM radiation
I temperature anisotropies in the order of δT/T ∼ 10−5
I temperature fluctuations follow density fluctuations
I even here where we are, was plasma at early times
I structure visible around us must come from densityfluctuations of the early plasma
COBE - dipole
COBE
WMAP
Map of the cosmic microwave radiation
Source: Planck Consortium (2013)
Acoustic oscillations and the surface of last scatteringBefore photon decoupling
I fluctuations inside the horizon oscillate
I amplitude of a plane wave changes with time
I early universe: no crosstalk between wave numbers
Surface of the last scattering
I imprint of the oscillating modes at decoupling
I each mode”catches” decoupling at different phase
I imprint of each mode with corresponding amplitude
I density is from the combination of all modes
I temperature depends on density only
I adiabatic modes
After decoupling
I photo pressure disappears
I fluctuations are affected by gravity only
I linear grows on large, non-linear growth on small scales
Amplitude of adiabatic modes
Modes with maximum amplitudes
When is the amplitude of a mode with wave number k maximal?
I if it had enough time to fully
I it had exactly enough time to compress fully
I 1/4 period or 3/4 period
I wavelength equal to the size of the acoustic horizon
k−1 = vs · t∗
I and all the harmonics of them
Amplitude of other wavelengths depend on the phase they werecaught in recombination.
What do we see from the early fluctuations?
Sachs–Wolfe effect (primordial)
I fluctuations just before decoupling with different amplitudes
I when plasma denser, a bit hotter but also deeper gravitationalpotential
I photons have to”climb out” of potential wll
I lose energy, photons from denser regions will appear colder
I denser regions will appear slightly colder in CMB
Projection effects
I fluctuations are treated as plane waves
I surface of last scattering appears as surface of a sphere
I how do we see plane waves intersected by a sphere?
Projection of a plane wave
ϑ
ϑ´
Map of the cosmic microwave background
Source: Planck Consortium (2013)
Power spectrum of the CMB
Express temperature fluctuations by spherical harmonics
T (θ, φ)
T0=∞∑l=0
l∑m=−l
a(lm)Y(lm)(θ, φ)
Power spectrum is averaging by directions
Cl =1
2l + 1
l∑m=−l
|alm|2
Power spectrum as measured by Planck
2 10 500
1000
2000
3000
4000
5000
6000
D `[µ
K2 ]
90 18
500 1000 1500 2000 2500
Multipole moment, `
1 0.2 0.1 0.07Angular scale
Source: Planck Consortium (2013)
Peaks of the power spectrumFirst acoustic peak
I wave number which had enough time to reach maximalamplitude (1/4 period) by t∗
I wavelength equal to the size of the acoustic horizon rs at t∗I its redshift z can be measured from temperature of CMB
I compare rs with DA(z)-vel ⇒ Ω = 1
Second acoustic peak
I wavelength reaching 3/4 period by t∗I baryons fell into the potential formed by dark matter
I
I a foton – barion interaction1 depends in wavelength offluctuation
I second peak has smaller amplitude as first one
I measures the amount of baryonic matter1baryon drag
Other peaks and the plateauThird acoustic peak
I sensitive to the baryon-dark matter ratio
Higher harmonics
I with decreasing amplitude
I due to Silk damping
Plateau at large angles (small `-s)
I we would not expect any correlations
I similar to horizon problem
I evidence for cosmic inflation inflation
I measurements with large error (Poisson noise)
The problem of cosmic variance
I CMB can only be measure from a single point of the U
I for small `-s, statistical sample is very small
I causes significant shot noise
Interaction of the background radiation with theforeground
The background photons right after decoupling
I stream freely in the tenuous neutral universe
First stars and quasars reionize hydrogen
I by this time the universe is even less dense
I CMB photons are scattered but
I not as much that their original pattern could be washed out
Sunyaev–Zel’dovich effect
Hot intracluster medium
I emits light in x-ray
I several millions of Kelvin temperature
I high energy electrons
Inverse Compton scattering
I interaction of high energy electrons with photons
I electrons give energy to photons
I can give a small”kick” from back
Effect on the photons of the CMB
I with the CMB radiation traverses cluster
I a part of the photons gains extra energy
I slightly increases the temperature of the radiation
Szunyajev–Zeldovics-effektus
The integral (late time) Sachs–Wolfe effect2
If a photon
I falls into a potential well ⇒ gains energy
I climbs out of a potential well ⇒ loses energy
I while traversing gravitationally bound systems ∆E = 0
I in the presence of Λ there’s always an effect
CMB photons traverse huge voids and super clusters
I light crossing time is very long
I dark energy and expansion changes the potential well duringthe traversal
I potential gets flatter
I photons might gain/lose some energy during crossing
I hot/cold spots in the CMB pattern
2Called the Rees–Sciama effect when calculated to non-linear order
First evidence for the integral Sachs–Wolfe effect
Have to stack CMD data for lots of voids
Granett, Neyrinck & Szapudi (2008)
Polarization of electromagnetic radiation
Monochromatic electromagnetic plane wave propagating in the zdirection:
Ex = ax(t)ei(ω0t−θx (t)) Ey = ay (t)ei(ω0t−θy (t))
I the CMB is not coherent, nor monochromatic
I such radiation is polarized if the two components correlate
I can be described by the coherence matrix
Iij =
〈ExE∗x 〉
⟨ExE
∗y
⟩〈E ∗x Ey 〉
⟨EyE
∗y
⟩
Stokes-parameterek
Good quantities to measure polarization
I relative intensity in different direction of polarization
I Stokes parameters:
I =⟨E 2x
⟩+⟨E 2y
⟩Q =
⟨E 2x
⟩−⟨E 2y
⟩U = 2Re(
⟨ExE
∗y
⟩)
V = −2Im(⟨ExE
∗y
⟩)
I U and V don’t seem to be easily measurable, but
I = I (0) + I (90)
Q = I (0)− I (90)
U = I (45)− I (135)
V = IR − IL
Stokes parameters
Source of linear polarization
Incident photons are scattered via Thomson scattering
I can cause linear polarization but
I if incoming radiation is isotropic, there is no net polarization
Source of linear polarization
Quadrupole moment of incident radiation can cause net linerpolarization.
Covariance tensor of linear polarization
The Stokes parameters describing linear polarization can be writtenin tensor form:
Pab =1
2
(Q −U−U −Q
)
Polarization of the CMB is measured on the surface of the sphere:
Pab = Pab(θ, φ)
E and B mode
I Similarly to Helmholtz decomposition of the electromagneticfield
I Pab(θ, φ) can be written as the sum of a curl-free and adiv-free term
I these can be written as multipole series
Pab(θ, φ)
T0=∞∑l=2
l∑m=−l
[aE(lm)Y
E(lm)ab(θ, φ) + aB(lm)Y
B(lm)ab(θ, φ)
]I Y E
(lm) and Y B(lm) come from the derivatives of ordinary
spherical harmonics
I The cross-correlation spectrum is defined from the coefficients
CABl =
1
2l + 1
l∑m=−l
aAlmaB∗lm
Quadrupole anisotropy
Three kinds of perturbations can cause quadrupole anisotropy
I m = 0: scalar perturbations : only E mode
I m = ±1: vector perturbations : B mode dominates
I m = ±2: gravitational waves : E and B with similar strength
This is always true locally, for a singla plane wave
I but have to sum over all wave numbers
I what is inherited into the final polarization pattern?
I parity, i.e. E and B modes, don’t mix
I but correlations with the multipole modes of the temperatureare inherited
Why is measuring the polarization important?
B modes originating from the early universe
I vector perturbations decay quickly
I only tensor perturbations can cause B modes
I early time gravity waves
I or later effect from the foreground
Temperature anisotropies are significantly affected by theforeground:
I Sunayev–Zel’dovich effect
I Rees–Schiama effect (integrated Sachs–Wolfe effect)
Polarization is less sensitive to the foreground
I gravitational lensing can cause E → B mixing
I galactic sources can produce B modes
The BB cross-correlation spectrum
The galactic foreground
Source: Planck Konzorcium (2013)