Homogeneous Universe

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Problem 1: homogeniety vs isotropy

Most cosmological models are based on the assumption that the Universe is spatially homogeneous and isotropic. Give examples to show that the two properties do not automatically follow one from the other.

Problem 2: global isotropy

Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?

Problem 3: examples

What three-dimensional geometrical objects are both homogeneous and isotropic?

Problem 4: the Big Bang "explosion"

Why the notion of Big Bang regarding the early evolution of the Universe should not be treated too literally?

Problem 5: Galilean invariance of the Hubble law

Show that the Hubble's law is invariant with respect to Galilean transformations.

Problem 6: Hubble law from homogeniety and isotropy

Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.

Problem 7: preservation of homogeniety

Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.

Problem 8: stationary model of the Universe

In the 1940-ties Bondi, Gold and Hoyle proposed a stationary model of the Universe basing on the generalized cosmological principle, according to which there is no privileged position either in space or in time. The model describes a Universe, in which all global properties and characteristics (density, Hubble parameter and others) remain constant in time. Estimate the rate of matter creation in this model.

Problem 9: Hubble flow and peculiar velocities

Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{km/s}.$ Estimate how distant a galaxy should be for its peculiar velocity to be negligible compared to the velocity of Hubble flow $V_H=H_{0}R$.

Problem 10: the age of the Universe

Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).

Problem 11: Olbers paradox resolved

Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.