Topics in Galaxy Formation (4) Fluctuations in the Cosmic Microwave - - PowerPoint PPT Presentation

Topics in Galaxy Formation

(4) Fluctuations in the Cosmic Microwave Background Radiation

• The Ionisation of the Intergalactic Gas Through the Epoch of Recombination
• The Sachs-Wolfe Effect
• Intermediate Angular Scales
• Small Angular Scales
• The Results of WMAP and the SDSS
• Comparison with Other Estimates

1

The Ionisation of the Intergalactic Gas Through the Epoch of Recombination

The optical depth of the intergalactic gas increases rapidly with redshift once the gas becomes fully ionised. Temperature fluctuations which originate at redshifts greater than the redshift of recombination are damped out by Thomson scattering. The fluctuations we observe originate in a rather narrow redshift range about that at which the optical depth of the intergalactic gas is unity. At the epoch of recombination, the plasma was 50% ionised when the temperature of the background radiation was about 4,000 K. Photons emitted in the recombination of hydrogen atoms must have energies hν ≥ hνα, where να is the frequency of the Lyman-α transition which has wavelength 121.6 nm. These photons can either reionise

• ther hydrogen atoms directly, or else raise them to an excited state H∗, from which the

electron can be ejected by the much more plentiful soft photons in the black-body spectrum. The Lyman-α photons are destroyed by the two-photon process in which two photons are liberated from the 2s state of hydrogen in a rare quadrupole transition with spontaneous transition probability w = 8.23 sec−1.

2

The Probability Distribution of Last Scattering

Jones and Wyse (1985) provided a convenient analytic expression for the degree of ionisation through the critical redshift range: x = 2.4 × 10−3(Ω0h2)1/2 ΩBh2

• z

1000

12.75

. (1) Ω0 is the density parameter for the Universe as a whole and ΩB the density parameter

• f baryons. The optical depth of the intergalactic gas at redshifts z ∼ 1000 is

τ = 0.37

• z

1000

14.25

. (2) Because of the enormously strong dependence upon redshift, the optical depth of the intergalactic gas is always unity very close to a redshift of 1070, independent of the exact values of Ω0, ΩB and h. This probability distribution for the range of redshifts from which the photons of the background radiation we observe today were last scattered is given by p(z) dz = e−τ dτ dz dz , (3) which can be closely approximated by a Gaussian distribution with mean redshift 1070 and standard deviation σ = 80 in redshift.

3

Fluctuations at the Last Scattering Layer

The physical scale at the present epoch corresponding to the thickness of the last scattering layer is given by dr = c H0 dz z3/2Ω1/2 . (4) If we take the thickness of the last scattering layer to correspond to a redshift interval ∆z = 120 at z = 1070, this is equivalent to a physical scale of 10(Ω0h2)−1/2 = 25 Mpc at the present epoch. The mass contained within this scale is M ≈ 3 × 1014(Ω0h2)1/2 M⊙ ∼ 1014 M⊙, corresponding roughly to the mass of a cluster of galaxies. The comoving scale of d = 10(Ω0h2)−1/2 Mpc corresponds to a proper distance d/(1 + z) at redshift z and hence to an angular size θ = d(1 + z) D = 10(Ω0h2)−1/2 DMpc = 6Ω1/2 arcmin , (5) since D = 2c/H0Ω0, if Ω0z ≫ 1.

4

Perturbations on the Last Scattering Layer

The diagram shows schematically the size of various small perturbations compared with the thickness of the last scattering layer. On very large scales, the perturbations are very much larger than the thickness of the layer. On scales less than clusters of galaxies, many perturbations overlap, reducing the amplitude of the perturbations.

5

Large Angular Scales - the Sachs-Wolfe Effect

On the very largest scales, the dominant source of intensity fluctuations results from the fact that the photons we observe have to climb out of the gravitational potential wells associated with perturbations which are very much greater in size than the thickness of the last scattering layer. On the scales of interest, the fluctuations at the epoch of recombination far exceed the horizon scale and so the perturbations would represent a change of the gravitational potential of everything within the horizon. More properly, we should describe these perturbations as metric perturbations. These ‘super-horizon’ perturbations raise the thorny question of the choice of gauge to be used in relativistic perturbation theory. A general relativistic treatment, first performed by Sachs and Wolfe (1967), is needed. The result is ∆T/T = (1/3)∆φ/c2, recalling that ∆φ is a negative quantity.

6

The Sachs-Wolfe Effect The Coles-Lucchin Argument

Coles and Lucchin (1995) rationalised how the Sachs–Wolfe answer can be found. In addition to the Newtonian gravitational redshift, because of the perturbation of the metric, the cosmic time, and hence the scale factor R, at which the fluctuations are

• bserved, are shifted to slightly earlier cosmic times. Temperature and scale factor

change as ∆T/T = −∆R/R. For all the standard models in the matter-dominated phase R ∝ t2/3 and so the increment of cosmic time changes as ∆R/R = (2/3)∆t/t. But ∆ν/ν = −∆t/t is just the Newtonian gravitational redshift, with net result that there is a positive contribution to ∆T/T of −(2/3)∆φ/c2. The net temperature fluctuation is ∆T/T = 1

3∆φ/c2.

It is then a straightforward calculation to show that, for the Ω0 = 1 model, the temperature fluctuations depend upon angular scale as ∆T T ≈ 1

3

∆φ c2 ∝ θ(1−n)/2 . (6)

7

The Power Spectrum of the Fluctuations in the Cosmic Microwave Background Radiation

For the preferred Harrison-Zeldovich spectrum n = 1, we expect the power spectrum to be independent of angle on large angular scales. The flatness of the power spectrum on large angular scales was discovered by COBE and fully confirmed by the power spectrum

• btained by the Wilkinson Microwave

Anisotropy Probe (WMAP). The detailed shape on large angular scales depends upon the choice of cosmological model.

8

Intermediate Angular Scales

In the case of Cold Dark Matter, all scales are unstable and grow according to the standard formula from the time they enter the horizon.

• The proper horizon scale at the epoch of recombination is

rH = 3ct = 2c H0Ω1/2 (1 + z)−1.5 = 5.8 × 1021(Ω0h2)−1/2 m , (7) corresponding to a comoving scale of 200(Ω0h2)−1/2 Mpc.

• We also need the Jean’s length of the photon-dominated plasma. Using the

concordance values of the cosmic parameters, we find ΩBh2 = 2.4 × 10−2 so that the inertia in the baryonic matter is more or less the same as the inertial mass in the radiation at the epoch of recombination. Therefore, the appropriate sound speed to use is very close to c/ √ 6 and λs = c √ 6t = 7 × 1020 (Ω0h2) m , (8) This scale corresponds to a comoving length scale of 32.5(ΩBh2)−1 ≈ 200 Mpc.

9

Intermediate Angular Scales

• We can compared this with the sound horizon on the last scattering surface

λs = cst (9) where t = 2 3H0Ω1/2 (1 + z)−1.5 is the age of the Universe at that time. Not surprisingly, this is of exactly the same

• rder as the Jeans’ length at that time. The importance of this result that this

corresponds to the maximum wavelength which the sound waves can have on the last scattering surface.

• This scale corresponds to an observed angular scale of about 1◦ on the sky.
• Note the important point that the sound horizon depends upon both the energy

density in the cosmic background radiation and the baryonic density.

10

Intermediate Angular Scales

The first acoustic peak is associated with perturbations on the scale of the sound horizon at the epoch of recombination. The amplitudes of the acoustic waves at the last scattering layer depend upon the phase difference from the time they came through the horizon to last scattering layer, that is, they depend upon

• dφ =
• ω dt .

(10) Let us label the wavenumber of the first acoustic peak k1. Oscillations which are nπ out

• f phase with the first acoustic peak also correspond to maxima in the temperature

power spectrum at the epoch of recombination. There is, however, an important difference between the even and odd harmonics of k1. The odd harmonics correspond to the maximum compression of the waves and so to increases in the temperature, whereas the even harmonics correspond to rarefactions of the acoustic waves and so to temperature minima. The perturbations with phase differences π(n + 1

2) relative to

that of the first acoustic peak have zero amplitude at the last scattering layer and correspond to the minima in the power spectra.

11

Intermediate Angular Scales

To find the acoustic peaks, we need to find the wavelengths corresponding to frequencies ωtrec = nπ . (11) Adopting the short wavelength dispersion relation , ω2 = c2

sk2 − 4πG̺B = c2 s(k2 − k2 J) ≈ c2 sk2 ,

(12) the condition becomes cskntrec = nπ kn = nπ λs = nk1 . (13) Thus, the acoustic peaks are expected to be evenly spaced in wavenumber. The separation between the acoustic peaks thus provides us with further information about various combinations of cosmological parameters.

12

Intermediate Angular Scales

The next task is to determine the amplitudes of the acoustic peaks in the power

• spectrum. The complication is that the acoustic oscillations take place in the presence
• f growing density perturbations in the dark matter, which have greater amplitude than

those in the acoustic oscillations. Therefore, in dark matter scenarios, the acoustic waves are driven by the larger density perturbations in the dark matter with the same wavelength, that is, the perturbations are forced oscillations. In a simple approximation, growth rate of the oscillation is driven by the growing amplitude of the dark matter perturbations: d2∆B dt2 = ∆D4πGρD − ∆Bk2c2

s .

(14) The sound speed is given by cs = c √ 3

• 4̺rad

4̺rad + 3̺B

1/2

= c

• 3(1 + R)

, (15)

13

Intermediate Angular Scales

The temperature fluctuations are related to the density perturbations by the standard adiabatic relation Θ0 = δT T = 1

3

δρ ρ = 1

3∆B ,

(16) These are known as monopole perturbations. The motion of the perturbations also give rise to temperature fluctuations because d∆/dt = −∇ · δv. The associated Doppler effect is Θ1 = δT/T0 = δv cos θ/c. These are referred to as the dipole terms in the temperature fluctuation spectrum. We have not yet taken account of the gravitational redshifting of the temperature perturbations since they take place within the gravitational potential well of amplitude Ψ. Therefore, the temperature fluctuation observed by the distant observer is Θ0(t) + Ψ/c2, recalling that Ψ is a negative quantity. As a result, the temperature fluctuations are Θ0(t) + Ψ c2 =

• Θ0(0) + (1 + R)

c2 Ψ

• cos ωt − R

c2Ψ . (17) The inclusion of the gravitational redshift effect enables us to relate the acoustic perturbations to the Sachs-Wolfe oscillations.

14

Intermediate Angular Scales

In the limit R → 0, the monopole and dipole temperature fluctuations are of the same

• amplitude. However, when the inertia of the baryons can no longer be neglected, the

monopole contribution becomes significantly greater than the dipole term. At maximum compression, kλs = π, the amplitude of the observed temperature fluctuation is (1 + 6R) times that of the Sachs–Wolfe effect. Furthermore, the amplitudes of the oscillations are asymmetric if R = 0, the temperature excursions varying between −(Ψ/c2)(1 + 6R) for kλs = (2n + 1)π and (Ψ/c2) for kλs = 2nπ. These results can account for the some of the prominent features of the temperature fluctuation spectrum. The temperature perturbations associated with the acoustic peaks are much larger than the Sachs–Wolfe fluctuations. The asymmetry between the even and odd peaks in the fluctuation spectrum is associated with the extra compression at the bottom of the gravitational potential wells when account is taken of the inertia of the perturbations associated with the baryonic matter.

15

Small Angular Scales

• Silk Damping scale results in the suppression of high wave number modes on

scales less than about 8 Mpc at the present epoch.

• The superposition of perturbations damps out the perturbations within the last

scattering layer.

• The Sunyaev-Zeldovich effect associated with hot intergalactic gas in clusters of

galaxies creates additional small scale perturbations.

16

The WMAP Power Spectrum

Many of the features of the above analysis can be observed in the WMAP power spectrum.

• The location of the maximum of the

first peak in the power spectrum.

• The asymmetry between the first,

second and third peaks.

• The flatness of the spectrum at low

values of l.

• The polarisation and the large

signal at very small values of l.

17

Parameter Estimation using WMAP and SDSS

Max Tegmark and his colleagues have used the WMAP power-spectrum and polarisation to make parameter estimates. The yellow areas show probability distributions using WMAP alone; the red areas include the power spectrum of galaxies from the Sloan Digital Sky Survey.

18

Parameter Estimation using WMAP and SDSS

Parameter Status WMAP alone WMAP + SDSS ωB = ΩBh2 Not optional 0.0245+0.0050

−0.0019

0.0232+0.0013

−0.0010

ωD = ΩDh2 Not optional 0.115+0.020

−0.021

0.1222+0.0090

−0.0082

ΩΛ Not optional 0.75+0.10

−0.10

0.699+0.042

−0.045

w τ Not optional 0.21+0.24

−0.11

0.124+0.083

−0.057

Ωk∗ As Not optional 0.98+0.56

−0.21

0.81+0.15

−0.09

ns 1.02+0.16

−0.06

0.977+0.039

−0.025

α r nt b Not optional No constraint 1.009+0.073

−0.083

fν = ρν/ρD

19

Concordance Values of the Cosmological Parameters

Parameter Definition Value H0 Hubble’s constant 72 km s−1 Mpc−1 Ωk space curvature ΩΛ dark energy density parameter 0.72 Ω0 = ΩB + ΩD total matter density parameter 0.28 ΩB baryon density parameter 0.047 ΩD dark matter density parameter 0.233 ns scalar spectral index 1 As amplitude of scalar power-spectrum 0.89 τ reionisation optical depth 0.17

20

Independent Estimates of Cosmological Parameters

• Hubble Space Telescope estimate of Hubble’s constant h = 72 ± 7 km s−1

Mpc−1.

• Estimates of ΩΛ from Type1A supernovae, ΩΛ ≈ 0.7.
• Average Mass Density in the Universe from Infall into Superclusters: Ωm = 0.3 if

h = 0.7.

• Synthesis of the light elements: ωb = 0.022 ± 0.002.
• Nucleocosmochronology: The best estimate of the age of the Galaxy is

Tgal = 12 ± 2 billion years.

• Ages of Globular Clusters T ≈ 13 billion years.

21

The Modifi ed Initial Power Spectrum

Max Tegmark and his colleagues have shown how many other pieces of data are consistent with this picture. Note:

• Overlap of WMAP

and SDSS power spectra.

• Statistics of

gravitational lensing.

• Power spectrum of

neutral hydrogen clouds.

22

The Acoustic Oscillations in the Galaxy Distribution

AAT 2dF galaxy survey SDSS galaxy survey

23