Model Universe Including Pressure

Transcription

1 Model Universe Including Pressure The conservation of mass within the expanding shell was described by R 3 ( t ) ρ ( t ) = ρ 0 We now assume an Universe filled with a fluid (dust) of uniform density ρ, pressure P and temperature T r is the radius of the comoving spherical surface The first law of thermodynamics can be applied to the fluid within the expanding sphere (conservation of internal energy U) du = dq dw Because the entire Universe has the same temperature and there cannot be any heat flow dq Because the expansion of the Universe is adiabatic, any change in internal energy has to be due to work by the fluid

2 Model Universe Including Pressure We obtain du = dw = P dv 4 = P π 3 d ( r 3 ) Introducing the internal energy density yields 4 u = U π r 3 3 d ( r 3 u ) = P d ( r 3 ) 1

3 Model Universe Including Pressure Using u = ρ c 2 yields d ( r 3 ρ ) = P c 2 d ( r 3 ) Finally, using r = R ϖ we obtain the Fluid Equation d ( R 3 ρ ) = P c 2 d ( R 3 )

4 Model Universe Including Pressure Using the Fluid Equation together with the Friedmann Equation we can obtain the Acceleration Equation d 2 R 4 3 P = π G ρ + R 2 3 c 2 This equation is a valid solution of Einstein s field equations (including relativistic effects) for a spherically symmetric distribution of matter It was proven by G.D. Birkhoff in 1923 (Birkhoff s theorem) The acceleration only depends on the density, the pressure, and the radius To solve for these three variables, we need an independent equation of state that links the variables

5 Model Universe Including Pressure The state equation can be written generally as (w a constant) P = w u = w ρ c 2 For pressureless dust we have For the blackbody we had found which gives w m = 0 P rad = u rad / 3 w rad = 1/3 Using this result in the fluid equation yields R 3(1+w) ρ = ρ 0

6 Cosmology: Cosmic Microwave Background

7 The Idea of a Big Bang George Gamow and Ralph Alpher proposed in 1948 the idea of an initial hot dense Universe to explain the abundance of elements Fred Hoyle, the proponent of a steady-state Universe, referred to their work publicly as a Big Bang idea The mean free path of photons in a hot and dense would have been short so that the Universe would have been in a thermodynamic equilibrium (besides expansion) The blackbody spectrum describes the radiation in such an environment Normalized abundance of elements in the Sun s photosphere

8 Cosmic Background Radiation George Gamow and Ralph Alpher calculated a value of 5 K for the remainder of the radiation in the present day Universe It turns out that the present blackbody temperature T 0 and the primordial temperature are related R T = T 0 The fusion of hydrogen atoms requires T = 10 9 K and ρ = 10-2 kg m -3 Combining with our earlier results R 3 ( t ) ρ ( t ) = ρ 0 and yields for the scale factor This results to ρ b 0 = kg m -3 R = ( ρ b 0 / ρ b ) ⅓ = T 0 = R T ( R ) = 3.47 K

9 Cosmic Microwave Background Using Wien s Law, we can obtain the peak wavelength of the blackbody spectrum λ max = m K T 0 = m In 1965 Arno Penzias and Robert Wilson detected a background radio signal when using a horn antenna for communication with the Telstar satellite Their background radiation corresponded to a 2.7 K blackbody with a peak wavelength of λ max = 1.06 mm The present day value for the Cosmic Microwave Background (CMB) from the WMAP satellite measurements is T 0 = ± K

10 Cosmic Microwave Background Cosmic Microwave spectrum obtained with the COBE satellite

11 Cosmic Microwave Background

12 Cosmic Microwave Background The Cosmic Microwave Background originates from the Big Bang when the Universe was a singularity We expect that the radiation is isotropic and that any observer measures the same spectrum A Doppler shift emerges from the observer s peculiar velocity v relative to the Hubble flow This shift in wavelength can be expressed as a change of temperature ( 1 β 2 ) ½ T moving = T rest 1 β cos θ T rest ( 1 + β cos θ ) The second term on the rhs is called the dipole anisotropy of the CMB β = v / c

13 Dipolar Anisotropy and Resolution Temperature resolution dependence from COBE data At around 2.73 K the microwave sky looks very homogeneous At a resolution of 3 mk the dipolar anisotropy due to the motion of the Earth towards the center of the local galaxy cluster with respect to the Hubble flow becomes apparent After subtracting the dipolar anisotropy, the inhomogeneity of the microwave structure is visible at a resolution of K

14 The Sunyaev-Zel dovich Effect When low-energy photons of the CMB pass through hot ionized intracluster gas of about 10 8 K of a galaxy cluster they can undergo inverse Compton scattering The low-energy photons gain energy in the process, leading to an increase of frequency of the amount 4 k T e Δ f = f m e c 2 The resulting distortion of the CMB spectrum is referred to as the Sunyaev-Zel dovich effect The effect can be used to calculate a decrease in temperature Δ T = T 0 2 k T e m e c 2 τ τ optical depth of intracluster gas

15 The Sunyaev-Zel dovich Effect Frequency shift of CMB photons, distorting the blackbody spectrum

16 The Sunyaev-Zel dovich Effect Temperature changes superimposed on images of galaxy clusters showing the Sunyaev-Zel dovich effect