29. Cosmology Goals :

1. 29. Cosmology Goals: 1. Examine a simple cosmological model to gain insights into the parameters of interest. 2. Note how the observed cosmological parameters depend upon the type of universe considered: flat, closed, or open. 3. Add complications to the simple model to make it more similar to reality, then examine what observations tell us about the applicability of the adopted models.

2. Observational Background The rectangular area below lies upwards from the Galactic plane, thereby sampling the universe.

3. A close-up of the region of sky where the Hubble Space Telescope Ultra Deep Field image lies.

5. Every “fuzzy” object in the image is a distant galaxy. Foreground stars in our own Galaxy have associated diffraction spikes because they are point-like images. Star

6. Redshift-distance relation (again).

7. The inverse of the Hubble constant (which is the slope of the Hubble relation) has units of time and is called the Hubble Time. It is an estimate of the age of the universe (backwards extrapolation), provided that the expansion began at some point in the past and has been continuing at the same rate ever since. for H0 = 71 km/s/Mpc, and 15 billion years for H0 = 65 km/s/Mpc. The Milky Way’s globular clusters are all less than 14 billion years old.

8. Schematics of the meanings for “homogeneous” and “isotropic.”

9. A Simple Pressureless “Dust” Model of the Universe Chapter 29 begins with the presentation of a simple model to illustrate the basic cosmological parameters, an expanding universe filled with pressureless “dust” of uniform density, ρ(t), with an arbitrary point as the origin. Such a universe is isotropic and homogeneous, and expands about an arbitrary point chosen as the origin. Let r(t) be the radius of a thin spherical shell of mass m at time t. The shell expands along with the universe with recessional velocity v(t) = dr(t)/dt. As the shell expands, its kinetic energy K decreases and its gravitational potential U increases, but the total energy E remains the same. The total energy of the shell can be expressed in terms of two constants, k and , such that E = −½mkc22. The constant k has units of (length)−2, while  (“varpi”) can be thought of as the present radius of the shell, in other words: r(t0) = .

10. The pressureless “dust” model of the universe.

11. Conservation of energy for the mass shell implies that: where Mr is the mass interior to the shell: Although the radius and density of the shell constantly change, the combination r3(t)ρ(t) remains constant because the total mass of dust in the shell does not change as the universe model expands. Removal of m and substituting for Mr leads to: Note that k > 0 represents a closed universe, k < 0 an open universe, and k = 0 a flat universe.

12. The cosmological principle requires that the expansion is the same for all shells, which implies that the radius of a particular shell identified by  is:

13. Here r(t) is the co-ordinate distance while  is the co-moving co-ordinate. R(t) is a dimensionless scale factor. Thus, R(t0) = 1 corresponds to r(t0) = . The scale factor is equal to Remitted/Robserved. which from previously means that the scale factor and redshift are related by: The comment that r3(t)ρ(t) remains constant means that the product R3ρ remains constant for all shells, and, since R(t0) = 1, implies: where ρ0 is the density of the dust-filled universe at present. From above we know that:

14. The evolution of this pressureless “dust” universe is described by the time behaviour of the scale factor R(t). The Hubble parameter, H(t), can be expressed in terms of the scale factor as: But v(t) is the time derivative of r(t): Therefore: The conservation of energy equation therefore becomes: or:

15. The left side of the equation applies to all shells while the right side of the equation contains only constants. It can be rewritten as: For a flat, closed universe, k = 0, so the density ρ is the critical density ρc required for closure: We can evaluate the parameter for our present universe using: That yields the present value for the critical density: about 11h2 hydrogen atoms per cubic metre.

16. Protons and neutrons are baryons, so hydrogen ions and element nuclei in stars are a form of baryonic (quark-based) matter. Electrons are leptons, but when bound to atomic nuclei are counted as a component of baryonic matter, along with mesons. Non-baryonic matter, such as so-called dark matter, is generally of unknown composition. One of the questions of cosmology is how much of the universe is baryonic and how much non-baryonic. The ratio of the measured density of the universe to the critical density is denoted Ω, the density parameter: with a present value of:

17. Some estimates of interest:

19. From the previous equation for density ratios: or: From previous relations it follows that: And, at t = t0: If Ω0 > 1, then k > 0 and the universe is closed. But if Ω0 < 1, then k < 0 and the universe is open. Most cosmologists appear to want Ω0 = 1 (k = 0) in order to have a flat universe.

20. Equating some of the previously-derived equations yields: With the previous two equations one obtains: And: According to the equations, at very early times as R → 0 and z → ∞ the sign of Ω− 1 does not change and Ω → 1 no matter what its present value. By inference, the very early universe was essentially flat, regardless of its present nature.

21. Thoughts on an analogy: baking raisin bread is often used to picture an expanding universe. No matter which raisin represents the Sun, all raisins appear to increase their distances with time, at a rate proportional to their distance. But where is the edge?

22. The expansion of a flat (Ω0 = 1), one-component universe of pressureless dust as function of time is found by further combination of the equations: Taking the square root, rearranging, and integrating yields: Solving gives: and: or: for Ω0 = 1.

23. tH = 1/H0 is referred to as the Hubble time. If Ω0 ≠ 1 the density is not equal to the critical density. For Ω0 > 1 the universe is closed and the solutions become: For Ω0 < 1 the universe is open and the solutions become:

24. The variable x parameterizes the solutions. The behaviour of a closed universe for Ω0 = 2 is shown in the appended figure. The “bounce” that occurs after contraction of the universe is a mathematical artifact and does not necessarily imply an oscillating universe. Recall the hyperbolic trigonometric functions: so Ropen increases monotonically with t. The appended figure displays the solution for Ω0 = 0.5. When Ω0 ≤ 1 the model universe continues to expand forever.

26. Because of deceleration, the age of the universe must be less than the Hubble time.

27. Age of the Pressureless “Dust” Universe. Keep in mind that all such mathematical solutions are extrapolations only and do not refer to “the age of the universe.” Recall: and Therefore: for Ω0 = 1 (flat). for Ω0 > 1 (closed). for Ω0 < 1 (open).

28. In the limit as z → ∞ all equations reduce to: where higher order terms are neglected for Ω0 ≠ 1. Because the early universe was flat to a close approximation, highly precise observations are necessary to establish if the universe is flat, closed, or open. The current age of the universe t0 may be found by setting z = 0 in the equation, giving: for Ω0 = 1 (flat). Note that a Hubble constant of H0 = 50 km/s/Mpc yields a Hubble time of tH = 1/H0 = 20  109 years and an age for the present universe of 13.3  109 years, consistent with the derived maximum ages of globular clusters. The solutions for Ω0 ≠ 1 are more complicated, namely:

29. namely: for Ω0 > 1 (closed). for Ω0 < 1 (open). Solutions for the age of the universe in such models are summarized in the following figure.

30. Note that a Hubble constant of H0 = 71 km/s/Mpc produces conflicts with the ages of globular clusters.

31. Lookback Time The lookback timetL is defined as how far back in time one looks when viewing an object of redshift z. That is simply the difference between the present age of the universe and its age at time t(z), i.e.: Solutions depend upon the value of Ω0. The results are: for Ω0 = 1 (flat). Solutions for for Ω0 > 1 (closed) and Ω0 < 1 (open) are more complex. See appended figure for details.

33. Example 29.1.2 If the redshift of quasar SDSS 1030+0524 is z = 6.28, for a flat universe of pressureless dust the lookback time for the quasar is: or 0.6327266. But the age of a flat universe is t0 = 2tH/3. Therefore: Thus, the light from SDSS 1030+0524 was emitted when only ~5% of the history of the universe had unfolded. The universe at that time was also smaller by a factor of ~7 since the scale factor was:

34. The Inclusion of Pressure The inclusion of pressure begins with the original differential equation: with Einstein’s relation Erest = mc2 and the mass density ρ broadened in definition to include relativistic particles of equivalent mass density, where mass now is Erest/c2. It is also useful to describe the conservation of mass using the relation described earlier: Generalization of the first equation is done using the 1st law of thermodynamics, where conservation of internal energy U, work done W, and heat Q is written as:

35. Thermodynamics are used to generalize the pressureless “dust” universe to incorporate pressure-producing components. It is useful to first consider that the entire universe has the same temperature, so there is no heat flow, i.e. dQ = 0. Therefore, any change in internal energy must be produced by work done by the “fluid” component of the universe: With V = 4/3 πr3 one obtains: Define the internal energy per unit volume u as: to obtain:

36. Now substitute for u the equivalent mass density: which generates the relationship: Finally, with: one obtains the fluid equation: In a universe of pressureless dust, P = 0, so R3ρ is constant. With the equations noted previously it is possible to derive the acceleration equation:

37. Acceleration equation: which is an illustration of Birkhoff’s theorem. Note that the effect of pressure (P > 0) is to slow down the expansion. The acceleration equation and the fluid equation contain three unknowns: ρ, P, and R. A solution requires a third relationship for the parameters, in particular the equation of state: where w is a constant of proportionality. Inserting the last into the fluid equation produces: where ρ0 is the present value of the equivalent mass density.

38. A last parameter can be introduced to describe the acceleration of the universal expansion, namely the deceleration parameter q(t): It can also be shown, for a pressureless “dust” universe: Thus, for the present time in such a universe, q0 = 0.5 for a flat universe, q0 > 0.5 for a closed universe, and q0 < 0.5 for an open universe.

39. The Cosmic Microwave Background A new observational parameter was added to the field of cosmology in 1965 with the discovery, by Arno Penzias and Robert Wilson working at the Bell Laboratories in New Jersey, of the 3 K cosmic microwave background radiation, or CMB. The existence of the CMB has subsequently been confirmed by observations with the Cosmic Background Explorer (COBE) satellite. It is usually thought of as the residual glow of the Big Bang, greatly redshifted from 3000 K to 3 K. It may be noted that a variety of alternative explanations have been proposed to explain the glow, but most appear to have weak points. There is even one pertaining to steady-state cosmology, although the predictions of most variants of steady-state cosmology are contradicted by the abundances of helium (He) and the light elements lithium, beryllium, and boron (Li, Be, B).

40. Penzias and Wilson with the radio horn used to discover the 3K microwave background radiation.

41. Origin of the 3K microwave background

42. The 3K background revealed by the COBE satellite, displays a “Doppler shift” of 370.6 ±0.4 km s−1 relative to the Hubble flow, attributed to mass asymmetry in the early universe when matter separated from radiation.

43. The “Doppler effect” seen in COBE measurements of the 3K background.

44. Note that the direction of motion is mainly towards the Virgo cluster.

45. The 3K microwave background matches the radiation from a black body with T = 2.728 K. Note that WMAP finds a best value of [T0]WMAP = 2.725 ± 0.002 K.

46. The 2.728 K background is the constant faint glow from the universe when T = 3000 K, now redshifted by z≈ 1000.

47. The net motion of the Earth relative to the Hubble flow needs to be adjusted for the motion of the Sun about the Galactic centre and the motion of the Galaxy in the Local Group. Both of those adjustments have likely been done erroneously. Thus, the resulting motion of the Local Group relative to the Hubble flow of 627 km s−1 may be slightly in error. No one has yet generated an improved value (likely through lack of knowledge about the LSR velocity). The effect on the temperature of the CMB is given by (Problem 29.21): or:

48. The 3K microwave background with the Doppler shift removed, as recorded by WMAP.