Review Questions

1. In curved spacetime, what is the role of the angle χ (known as the "development angle") and what does it quantify (see Figure 24.1)?

2. Given the static distance element, dD, what does the product a(t)dD represent? What is the quantity a(t) called and what does it parameterize?  

3. A higher order term to the linear Hubble-Lemaître law is quantified as the "deceleration parameter", q0 .  What does the deceleration parameter measure?

4. What is the sign of the unitless curvature constant for an "open", "flat", and "closed" universe.  

5. In words, what dynamic does Einstein's first field equation describe? What dynamic does his second field equation describe? What are the three components to the pressure and energy-density "fluid"?  

6. What physics is described by the continuity equation and how does each of the pressure and energy-density terms depend on the sound speed of the energy-density "fluid"?

7. What are the four present-epoch cosmological parameters, Ω? Write out the expression for each.

8. Consider Equations 24.51 through 21.54, which give a compact form of the Einstein field equations.  What is the interpretation of the function E(t) and E'(t)?  Write out these two functions and, for each of the four cosmological parameters, state what power of the expansion factor, a0/a(t), it scales with.

9. What defines a Friedmann-Einstein cosmological model?  In this model, what is the critical density? What does it mean if the Universe is at the critical density?  If it is below the critical density? If it is higher than the critical density? What parameter values have historically been adopted for an "open" Friedmann-Einstein Universe? What parameter values define an Einstein-de Sitter Universe?

10.  Describe how the Robertson-Walker metric is partitioned into temporal, and linear and transverse spatial elements. Further, describe how the spatial elements are written as both proper and co-moving elements. What is the interpretation of the co-moving spatial elements?

11. What is the approximate value of the Hubble-Lemaître constant? What is the Hubble tension? How significant is it? What might be some resolutions?

12. What is the ΛCDM model? Briefly describe what is known about dark matter and dark energy? What are the approximate values of ΩM and ΩΛ?   What is the "737 Cosmology" and what is its utility?

13. In the 737 Cosmology, describe the evolution of the cosmological parameters.  Over what range of expansion parameter was the Universe radiation dominated?  Matter dominated? Dark energy dominated?  

Problems

1. Under construction