The cosmic microwave background radiation

Laszlo DobosDept. of Physics of Complex Systems

dobos@complex.elte.huE 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)