Astrophysics: Homework Questions and Final Examination

1. Astrophysics ASTR3415: Homework 4, Q.2

2. Astrophysics ASTR3415: Homework 5 Due Wednesday 12/7, 5pm In a model universe with zero cosmological constant, Friedmann’s equation for the evolution of the cosmic scale factor, R(t), may be written as 1. where k is a constant. From this equation, show that the critical matter density is given by Introducing the dimensionless density parameter show that Friedmann’s equation may be re-written as Hence explain why, in this case, the type of model (i.e. open, closed or flat) cannot change as the universe evolves [7] 2. Supernova ‘Velma’, at a redshift ofwas discovered in 1999 by the High-z SN team. Suppose there existed Velman cosmologists who were observing the CMBR when the light we now see from the supernova was emitted. Given that the CMBR is blackbody radiation, with a present-day mean temperature of explain carefully why the Velman cosmologists would have measured the mean temperature of the CMBR to be Approximately what value for the redshift of the CMBR would the Velman cosmologists have measured at that time? (Take the current redshift of the CMBR to be ~1000) [7]

3. The first Friedmann equation, for a matter dominated universe (i.e. ignoring radiation) with a non-zero cosmological constant, can be written as: where and Show that may be written as Hence show that, as stated in the lectures, the Friedmann equation can be re-written as where, for , respectively. Take and , and assume . Using your results for Questions 1 and 2, show that the redshift, , at which satisfies the relation and solve for In the far future, the first Friedmann equation will be dominated by the term. Show that in this case, the scale factor will grow exponentially with time, with solution Problem A2.2 from Liddle (page 135). 3. [4] 4. [8] 5. [5] 6. [9] [10] 7. [Total = 50]

4. Astrophysics ASTR3415: ‘Take Home’ Final Examination Due Wednesday 12/14, 5pm (Note that you will need to consult your notes for some definitions and equations which aren’t given explicitly in the questions. If you’re not sure what I’m referring to in your notes, come ask!) Part 1 Inflation is defined as a period of accelerated expansion of the Universe – i.e. a period in which the scale factor not only increases, but . In certain cosmological models of the early Universe, the scale factor where is a positive constant. What range of values of would correspond to inflationary expansion? [2] Suppose that the Universe is matter dominated, with and . Starting from Friedmann’s equation, and assuming mass conservation, show that the scale factor satisfies the equation: [5] Hence, solve the above differential equation, and choose suitable initial conditions, to show that: [4] Is this model – known as the Einstein de Sitter (EdS) model – an inflationary one? [1] Show that the Hubble constant in the EdS model satisfies [3] 1. 2. 3. 4. 5.

5. Given that the present-day value of the Hubble constant, , determine the present age of the Universe (in years) according to the EdS model. [3] Assuming that the speed of light is constant, and the EdS model is valid throughout the entire history of the Universe *, estimate in this model how far light can have travelled since the Big Bang. [2] Given that the CMBR was emitted at the surface of last scattering, when the Universe was 1000th of its present size, calculate what was the age of the Universe (in years) when the CMBR was emitted, according to the EdS model. [3] How far could light have travelled from the Big Bang to the time at which the CMBR was emitted? (Let this distance be denoted by ) [2] Between the emission of the CMBR and the present-day, the distance has been ‘stretched’ by the expansion of the Universe. What physical size does correspond to today? [2] Assuming that the distance to the CMBR surface of last scattering is given by your answer to Q.9, calculate the angular size subtended today by the physical length . [3] 6. 7. 8. 9. 10. 11. * This assumption is not very good; the early Universe is in fact radiation dominated. We ignore the contribution of radiation here, however, since it greatly simplfies the formulae involved.

6. What is the physical significance of the angular size you have calculated in Q.11? Explain its relevance to the so-called horizon problem, with reference to observations of the temperature distribution on the CMBR. [4] TOTAL = 34 marks It is shown in Liddle A.2 that the relationship between luminosity distance and coordinate distance for a source at redshift z is where is the present-day value of the cosmic scale factor. What are the two physical factors which together result in the appearance of the term in the above equation, which makes distant objects appear more distant than they really are? [2] Consider a flat (i.e. ) cosmological model, but with a non-zero cosmological constant term. In this case it is not possible to derive a simple, analytic expression for the relationship between luminosity distance and redshift. Using Ned Wright’s Cosmology Calculator http://www.astro.ucla.edu/~wright/CosmoCalc.html to do the necessary calculations, make graphs of the quantity versus out to for two different flat models: one with and the other with . (Note that Ned’s page refers to Omega_vac instead of ). [8] 12. Part 2 13. 14.

7. What is the vertical separation of your two graphs at a) b) c) Briefly comment on the relevance of your answers for the rôle of distant supernovae as ‘standard candles’ for measuring the value of [6] TOTAL = 16 marks GRAND TOTAL = 50 marks 15.