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 ≥ hν α , where ν α is the frequency of the Lyman- α transition which has wavelength 121.6 nm. These photons can either reionise other 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 (Ω 0 h 2 ) 1 / 2 � 12 . 75 z � . (1) Ω B h 2 1000 Ω 0 is the density parameter for the Universe as a whole and Ω B the density parameter of baryons. The optical depth of the intergalactic gas at redshifts z ∼ 1000 is � 14 . 25 z � τ = 0 . 37 . (2) 1000 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 Ω 0 , Ω B 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 ) d z = e − τ d τ d z d z , (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 c d z d r = . (4) z 3 / 2 Ω 1 / 2 H 0 0 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(Ω 0 h 2 ) − 1 / 2 = 25 Mpc at the present epoch. The mass contained within this scale is M ≈ 3 × 10 14 (Ω 0 h 2 ) 1 / 2 M ⊙ ∼ 10 14 M ⊙ , corresponding roughly to the mass of a cluster of galaxies. The comoving scale of d = 10(Ω 0 h 2 ) − 1 / 2 Mpc corresponds to a proper distance d/ (1 + z ) at redshift z and hence to an angular size = 10(Ω 0 h 2 ) − 1 / 2 θ = d (1 + z ) = 6Ω 1 / 2 arcmin , (5) 0 D D Mpc since D = 2 c/H 0 Ω 0 , if Ω 0 z ≫ 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/c 2 , 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 observed, 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 ∝ t 2 / 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/c 2 . The net temperature fluctuation is ∆ T/T = 1 3 ∆ φ/c 2 . It is then a straightforward calculation to show that, for the Ω 0 = 1 model, the temperature fluctuations depend upon angular scale as ∆ T ∆ φ c 2 ∝ θ (1 − n ) / 2 . ≈ 1 (6) 3 T 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 obtained 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 2 c (1 + z ) − 1 . 5 = 5 . 8 × 10 21 (Ω 0 h 2 ) − 1 / 2 m , r H = 3 ct = (7) H 0 Ω 1 / 2 0 corresponding to a comoving scale of 200(Ω 0 h 2 ) − 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 Ω B h 2 = 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 6 t = 7 × 10 20 c λ s = √ (Ω 0 h 2 ) m , (8) This scale corresponds to a comoving length scale of 32 . 5(Ω B h 2 ) − 1 ≈ 200 Mpc. 9
Intermediate Angular Scales • We can compared this with the sound horizon on the last scattering surface λ s = c s t (9) where 2 (1 + z ) − 1 . 5 t = 3 H 0 Ω 1 / 2 0 is the age of the Universe at that time. Not surprisingly, this is of exactly the same order 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 � � d φ = ω d t . (10) Let us label the wavenumber of the first acoustic peak k 1 . Oscillations which are nπ out of 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 k 1 . 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 ωt rec = nπ . (11) Adopting the short wavelength dispersion relation , ω 2 = c 2 s k 2 , s k 2 − 4 πG̺ B = c 2 s ( k 2 − k 2 J ) ≈ c 2 (12) the condition becomes k n = nπ c s k n t rec = nπ = nk 1 . (13) λ s 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
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