Question

3. The Cosmological constant When formulating general relativity, Einstein believed that the Universe was static. In order to arrange a static Universe, he proposed a change to the equations, something he would later famously call his "greatest blunder". That was the introduction of a cosmological constant, A, to the Friedmann equation (1) (a) (3 points) By referring to the Friedmann equation 1), derive an equation for a static Universe when k -0 (b) (8 points) Last week you derived an equation that describes the acceleration (i.e., ä/a) of a flat, dust-dominated Universe. Now repeat the derivation, but including the cosmological constant. Under what conditions will Л contribute to the (positive) acceleration that is observed for our Universe? (c) (4 points) Is the universe described in part (a) achievable? Give a qualitative (or mathematical) argument for your answer. Explain in words why this cannot be the correct description of our Universe (d) (3 points) In cosmology, we often define the density parameter, Ω(t)-a(1)/Pe(1), where pe is the critical density defined as 3H2 pelt) and H (1) = à/a is Hubble's parameter. By referring to the Predmann equation (1), explain physically this choice of definition for pe Total: 18 points]

Answer 1

a(ts (at) 匕 C1 a Ct)3 3 univers e to be tedici you set G P2423 3C2 3

En 2. a then th e unive丫se with matte r density 12 02 3 No, it is not achieva ble (스 is miss 1 na n stati cos m of '0 9 1(2/ Constant /\ \ s 2- f23 hich Hzen the universe Lor il expa nd forever modets the ecpanding universe has EO D) Tt we debined citical densihy 2 Critical 3 H then we see that , the tot density Sin +8rtSa equals the citiced dengity theno meaning the universe is flat.By the eautvalence o expahsioh rate this also means that the universe expan critically ,the cticaly densitu is therchose the

the de nei ty at which the universe ex pa nds Criicady given the value at e xc