General Relativity. Note 14

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Time-space of an isotropic universe. Cosmological redshift. Hubble constant.

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Space−time of an isotropic universe. Cosmological redshift. Hubble constant.

The Friedman's equation for the isotropic universe, ³/_a⁴(a²+a'²)=κε, and the energy conservation equation, dε=−(ε + p)3^da/_a, can be integrated for the matter dominated universe, where the pressure is zero, p=0, and the energy density ε is equal to the mass density μ, ε=μ. The energy conservation gives μa³=Const and the subsequent integration of the Friedmann's equation gives for a closed universe (a²>0) a "Big Bang"→"Big Crunch" scenario: a=a₀(1−cos(η)) , t=a₀(η−sin(η))

For the open isotropic universe the Friedmann's equation provides a "Big Bang"→"Expansion Forever" scenario:

a=a₀(cosh(η)−1) , t=a₀(sinh(η)−η)

For a flat isotropic universe ds²=dt²−b²(t)(dx²+dy²+dz²) the scenario is also "Big Bang"→"Expansion Forever" (see the Exercise): μb³=const, b=const t^2/3.

At early stages with high densities the universe was (probably) rather radiation dominated, p=^ε/₃. This, however, doesn't save us from the singular point at η=0. Indeed, we have (for η<<1) : εa⁴=const , a=const⋅t^1/2

Cosmological redshift. Hubble constant.

In an isotropic universe the radial (dθ=dφ=0) propagation of light (ds²=0) is described by χ=±η+const, from where one can deduce that along the light ray there remains a constant product ωa=const. A light ray with frequency ω₀ emitted at a distance χ and observed at the origin (χ=0) at time η should then have the frequency ω = ω₀ ^a(η−χ)/_a(η) ~= ω₀ (1 − χ ^a'/_a), that is, redshifted, if the universe expands (a'>0). The proper distance l to the source of light is l=χa. Thus the frequency shift z can be written as z=^ω₀−ω/_ω₀=^a'/_a²l = H l, where H is the so called Hubble constant, H=^a'/_a²=¹/_a^da/_dt. The current value of the Hubble constant is H~=¹/_{(13 bil.
years)}.

Inserting ^a'/_a²=H into Friedmann's equations leads to ¹/_a²=H²−^κμ/₃ for a closed universe, and to ¹/_a²=^κμ/₃−H² for an open universe. For the critical density μ_c, such that ^κμ_c/₃=H², the universe is flat.

The current measurements show that the relative density Ω=^μ/_{μ_c} is close to one with an error about few per cent (flatness problem). About 30% of it is "dark matter" and about 70% is "dark energy". The visible matter constitutes only about 3% of the density.

Exercises

Consider a flat (Euclidean) isotropic universe with the metric ds²=dt²−b²(t)(dx²+dy²+dz²):
1. Calculate the Christoffel symbols. [Γ^x_tx=Γ^y_ty=Γ^z_tz=^b'/_b, Γ^t_xx=Γ^t_yy=Γ^t_zz=bb']
2. Calculate the Ricci tensor and the Ricci scalar. [R^t_t=−3^b''/_b, R^x_x=R^y_y=R^z_z=−^b''/_b−2^b'²/_b²]
3. Write down the ^t_t component of the Einstein's equations (with perfect fluid). [3^b'²/_b²=κε]
4. Write down the energy conservation equation ^dV/_V=−^dε/_(ε+p). [3ln(b)=−∫^dε/_(ε+p)]
5. Integrate the equations for a matter dominated universe (p=0, ε=μ). [μb³=const, b=const⋅t^2/3]
6. Integrate the equations for a radiation dominated universe (p=^ε/₃). [εb⁴=const, b=const⋅t^1/2]
Interpret the cosmological red shift ^ω₀−ω/_ω₀=Hl (l is the distance to the "red shifted" galaxy) as a Doppler effect and calculate the velocity with which a galaxy appears to be moving relative to the observer.