Question

It has been speculated that the present-day acceleration of the universe is due to the existence of a false vacuum, which will eventually decay. Suppose that the energy density of the false vacuum is ??? = 0.69??,? = 3360 MeV · m?³, and that the current energy density of matter is ??,? = 0.31??,? = 1510 MeV · m?³. (a) Starting from the solution of the Friedmann equation for a universe containing only ? + matter (see eq. 5.101 from Ryden) H?t = 2 / (3????,?) ln [ (a/a???)³?² + ?(1 + (a/a???)³) ], (where a??? = (??,? / ???,?)¹?³ = (0.31 / 0.69)¹?³ = 0.766, is the scale factor at matter-? equality) derive an expression for a(t) in the case of a ? a???, that is, for strong ? domination. (b) Use the previous result to find the Hubble parameter H(t) = à(t)/a(t) in this limit, and then find from it the corresponding Hubble time. Now suppose that the false vacuum is fated to decay instantaneously to radiation at a time t_f = 50t?. (Assume, for simplicity, that the radiation takes the form of blackbody photons.) (c) To what temperature will the universe be reheated at t = t_f, and what will be the energy density of radiation? (d) What will be the scale factor at t = t_f? (e) What will be the energy density of matter at t = t_f?

          It has been speculated that the present-day acceleration of the universe is due to the existence of a false vacuum, which will eventually decay. Suppose that the energy density of the false vacuum is ??? = 0.69??,? = 3360 MeV · m?³, and that the current energy density of matter is ??,? = 0.31??,? = 1510 MeV · m?³.

(a) Starting from the solution of the Friedmann equation for a universe containing only ? + matter (see eq. 5.101 from Ryden)

H?t = 2 / (3????,?) ln [ (a/a???)³?² + ?(1 + (a/a???)³) ],

(where a??? = (??,? / ???,?)¹?³ = (0.31 / 0.69)¹?³ = 0.766, is the scale factor at matter-? equality)

derive an expression for a(t) in the case of a ? a???, that is, for strong ? domination.
(b) Use the previous result to find the Hubble parameter H(t) = à(t)/a(t) in this limit, and then find from it the corresponding Hubble time.

Now suppose that the false vacuum is fated to decay instantaneously to radiation at a time t_f = 50t?. (Assume, for simplicity, that the radiation takes the form of blackbody photons.)

(c) To what temperature will the universe be reheated at t = t_f, and what will be the energy density of radiation?
(d) What will be the scale factor at t = t_f?
(e) What will be the energy density of matter at t = t_f?

It has been speculated that the present-day acceleration of the universe is due to the existence of a false vacuum, which will eventually decay. Suppose that the energy density of the false vacuum is ??? = 0.69??,? = 3360 MeV · m?³, and that the current energy density of matter is ??,? = 0.31??,? = 1510 MeV · m?³.

(a) Starting from the solution of the Friedmann equation for a universe containing only ? + matter (see eq. 5.101 from Ryden)

H?t = 2 / (3????,?) ln [ (a/a???)³?² + ?(1 + (a/a???)³) ],

(where a??? = (??,? / ???,?)¹?³ = (0.31 / 0.69)¹?³ = 0.766, is the scale factor at matter-? equality)

derive an expression for a(t) in the case of a ? a???, that is, for strong ? domination.
(b) Use the previous result to find the Hubble parameter H(t) = à(t)/a(t) in this limit, and then find from it the corresponding Hubble time.

Now suppose that the false vacuum is fated to decay instantaneously to radiation at a time tf = 50t?. (Assume, for simplicity, that the radiation takes the form of blackbody photons.)

(c) To what temperature will the universe be reheated at t = tf, and what will be the energy density of radiation?
(d) What will be the scale factor at t = tf?
(e) What will be the energy density of matter at t = tf?

Added by Irene B.

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