ASTR 340 Assignment 7: Standard Models of the Universe and Cosmological Constants - Prof. , Assignments of Astronomy

Download ASTR 340 Assignment 7: Standard Models of the Universe and Cosmological Constants - Prof. and more Assignments Astronomy in PDF only on Docsity! ASTR 340 – Fall 2006 Prof. Ostriker ASSIGNMENT No. 7 DUE: Thursday, Nov. 9 READING: Finish Chapter 11 of the text. Read Chapter 12 of the text. 1. Age of the Universe in Standard Models Using diagrams as needed, discuss how the present age of the Universe t0 relates to the Hubble time, tH = 1/H0, in the three kinds of Standard Models (i.e. Λ = 0) of the evolution of the Universe. 2. Deceleration parameter What is the deceleration parameter? How is it related to the density parameter ΩM in Standard Models? Describe how ΩM and q are obtained from observations, and how this information can be used to test whether the “standard model” approach is correct. 3. Generalized Friedmann equation The generalized version of the Friedmann equation, allowing for a cosmological constant Λ, is H2R2 = H20R 2 0 [ ΩM ( R0 R ) + ΩΛ ( R R0 )2 + Ωk ] , where ΩM = ρ08πG/(3H 2 0 ), ΩΛ = Λ/(3H 2 0 ), and Ωk = −kc 2/(R0H0) 2. The value HR = Ṙ is the rate of change of the cosmic scale factor R, but can also be thought of as the expansion speed V of a subvolume of the Universe of radius R. (Note that eq. 11.28 of the text has a typo.) (a) Explain which term in the generalized Friedmann equation is most important when the Universe is very young, which is most important when it is very old (provided that V is always positive), and why. (b) Are there, in principle, solutions of the generalized Friedmann equation in which V is not always positive? Explain. 4. de Sitter Universe What is the de Sitter model of the Universe? What is the value of the Hubble parameter in the de Sitter model? How does the de Sitter solution relate to the future of our Universe, if the cosmological constant is not zero? –1–