Exploring the Early Universe: Concepts, Simulations, and Observations

1. 30. The Early Universe Goals: 1. Introduce the various concepts associated with studies of the early universe, despite questions about applicability. 2. Discuss the various simulations computed as models of the universe, asking the question “can cosmology be an observational science?” 3. Present the various analyses of the cosmic microwave background as well as more recent related observations.

2. Your instructor is uncomfortable with much of this chapter of the textbook, for various reasons. Comments in the text such as: “Although the foregoing argument is only little better than a dimensional analysis…” “The moral is that specific predictions based on grand unified theories should be viewed with some caution.” “This is the earliest time that can be probed experimentally.” leave the impression of considerable uncertainty in what is being presented. So much is presented as background only.

3. Your instructor Many cosmologists

4. Subatomic Physics Background This section is best left as a reading exercise, although it is relevant to many of the ideas about the early universe.

5. There is perhaps undue concern about the existence of dark matter, either in “hot” form (relativistic particles, HDM) or “cold” form (CDM). The latter could simply consist of a high proportion of low mass stellar objects of low luminosity, brown dwarfs or stellar-mass black holes (MACHOs − massive compact halo objects) in the Galactic halo, although searches in the ’90s found far fewer than predicted. Alternate candidates include WIMPs (weakly interacting massive particles) that interact with matter only through their gravitational effects. HDM is not favoured because of the difficulties of forming structure in the early universe, so most current models include both cold dark matter and a cosmological constant Λ, i.e. ΛCDM models. There is no guarantee that the real universe matches expectations from such models. Modellers can model just about anything, without regard to reality.

6. The Planck time is the only combination of fundamental constants with units of time, and is defined using Planck’s constant, the gravitational constant, and the speed of light: Another parameter arises from Heisenberg’s uncertainty principle, which is panned by Marmet. A black hole represents the most compact region within which mass can be contained, as defined by the Schwarzschild radius RS. So, for an uncertainty in position given by: the corresponding uncertainty in momentum is:

7. which implies a very large uncertainty in momentum for low-mass primordial black holes formed in the early universe. At the relativistic limit, E pc, so the uncertainty in energy is: which describes the approximate kinetic energy of such a black hole. The gravitational potential energy of such a hole is given by: The Planck mass corresponds to the limit where the sum of kinetic and potential energies is identically zero, i.e.:

8. and is therefore defined as: The Planck length is generated from the Schwarzschild radius: by neglecting the factor of “2” and inserting the Planck mass into the relation: although it is noted that such arguments are little more than dimensional analyses.

9. Various epochs in the early universe are characterized by transitions from one basic parameter to another.

10. It is an article of faith for physicists that before the Planck time the four fundamental forces of nature (gravity, electromagnetism, strong and weak nuclear forces) were merged, and separated through spontaneous symmetry breaking at that instant. Some physicists do not consider gravity as a force, e.g. Roy Bishop.

11. Problems with the Standard Model of the Big Bang Why is the CBR so smooth? This is known as the horizon problem, arising from the time when light decoupled from matter.

12. Why is the universe so nearly flat? This is known as the flatness problem, arising from the relationship considered earlier (example problems): If, at the time of decoupling when [zWMAP] = 1089, the density parameter had been Ω = 0.9991 rather than 1, then we would have Ω0 = 0.5 today. If, on the other hand, the density parameter had been as small as Ω = 0.5 at that instant, then it would not be long before the density of the universe had decreased to a point where stars and galaxies could not be formed. It seems that a fine-tuning is required.

13. Why are there no magnetic monopoles? This is known as the monopole problem, Magnetic monopoles are a single magnetic charge, i.e. an isolated magnetic pole. It implies a lack of defects in the early universe in which magnetic monopoles might have been formed. Inflation. This concept dates from 1980, when Alan Guth proposed that in the first fraction of a second of the universe it spontaneously expanded at superluminal speed in such a fashion as to smooth out any initial irregularities in the distribution of matter. The concept does solve many of the problems described previously, but introduces one of its own, namely how to explain the introduction of a non-physical “fudge factor” to resolve problems arising from a physical analysis of the early universe.

14. An example of the concept of inflation in cosmology.

15. How the “scale factor” changes with time

16. Virtual particles and vacuum energy are used as possible explanations for the concept of inflation. See the discussion of the Casimir effect and virtual particles. Much of the discussion is governed by order of magnitude estimates, but the end result is the same: inflation does solve both the horizon problem and the flatness problem. See plots in Fig. 30.4.

18. Matter-Antimatter Asymmetry Why is the universe dominated by matter instead of antimatter? Existing antiparticles are explained by high-energy collisions with normal matter, e.g. pair production (proton-antiproton pair) through the collision of two energetic protons or via rare ultra-high-energy cosmic rays. The discrepancy implies that the formation of matter particles was slightly more likely than the formation of antimatter particles in the early universe.

19. The Origin of Structure Presumably stars and galaxies formed from the existence of density inhomogeneities in the early universe that reached the Jean’s mass for a star or galaxy.

20. The textbook adopts the approach that many such inhomogeneities developed during the brief “inflationary” epoch, when baryonic matter was coupled with photons, the expansion being so rapid that the sizes of overdense and underdense regions vastly exceeded the particle horizon, inhibiting particle motion (Fig. 30.6). In overdense regions ρ' > ρ, so the Hubble flow in such regions is described by: whereas: for the flat region of the surrounding universe. The equations can be combined to yield:

21. So that density fluctuations at that stage can be written as: During the radiation era: and the scale factor varied as: so the density fluctuations then must correlate with time and initial values in the radiation era as:

22. During the matter era, however: while the scale factor in a flat early universe varied as: so the density fluctuations must correlate with time and initial values during the matter era as: At some later time in the radiation era the particle horizon expanded to include the entire region of enhanced adiabatic density perturbation, making all parts of the region causally connected. The fate of such fluctuations then depended upon the relative values of their masses with respect to the Jean’s mass.

23. For a static (non-expanding) medium, the minimum mass required for the density fluctuation δρ/ρ to increase with time is the Jean’s mass: The same expression is valid in an expanding universe, although with different consequences for values falling below the above minimum. The expression can be rewritten using the speed of sound: and the adiabatic expression for pressure as a function of density: where C is a constant, and the ideal gas law.

24. Namely: with γ = 5/3 for an ideal monotonic gas. The expression for the Jean’s mass then becomes: The two density terms are different. Prior to recombination the numerator value of ρ corresponded to the baryonic mass density. During that era the temperatures of matter and radiation were equal, so:

25. The value of ρ in the denominator, however, is dominated by photons, so: and, prior to recombination: so the speed of sound was: The Jean’s mass at that time was therefore proportional to 1/T3 until recombination.

26. The supposed variation of Jean’s mass with decreasing temperature T of the universe is depicted in Fig. 30.7.

28. In the Big Bang model of cosmology, the formation of galaxies like the Milky Way occurred only after several previous stages in which the first stars were formed and galaxy mergers occurred.

29. Until recombination occurred, adiabatic density fluctuations in the early universe underwent acoustic oscillations much like pulsations in stars, with the sound waves traversing the medium at ~58% of light speed. Such oscillations produced regions of compression and rarefaction that were imprinted on the CMB when recombination occurred. Smaller adiabatic fluctuations did not survive that phase. The minimum mass required to survive damping by photon leakage can be estimated, and works out to be about 7.7 × 1013M, roughly the dimensions of a cD galaxy or a rich cluster of galaxies. Following recombination, less massive density fluctuations existed from previously ”frozen” isothermal fluctuations unaffected by the dissipation effects of acoustic oscillations. The Jean’s mass at that time was about 1.6 × 106M, roughly the dimensions of a globular cluster.

30. The Gunn-Peterson trough in the Lyman-α forest for high redshift quasars, indicating less ionized H from z ~ 5-6.

31. It has also been suggested that galaxy formation is biased towards regions of overdensity in the early universe. Former faculty member Michael West was one of the first to suggest the point. It would explain the separation of galaxy strands from galactic voids in the observable universe, which has been recognized for many years.

32. A Simple Model of Acoustic Oscillations Consider a simple cylinder of cross-sectional area A and length 2L filled with gas (below). Let a movable piston be located in its centre. The equilibrium values of pressure and density are P0 and ρ0, so that the mass of the piston is m = 2LAρ0.

33. If the piston is displaced one way or the other, the density on one side will change by an amount Δρ accompanied by a pressure change of: where we can approximate: and: It follows that: Examine a displacement x of the piston, where the total mass of gas on either side does not change.

34. i.e.: likewise: So: Therefore, to first order:

35. The resulting equation of motion with Newton’s second law becomes: or: which, when simplified, becomes: The result is simple harmonic motion of the piston: with angular frequency:

36. If a gravitational field is now added in one direction, say the positive x-axis: The resulting equation of motion is: To solve this equation, set: So: which gives rise to simple harmonic motion with angular frequency ω = vs/L about y = 0, where the equilibrium position is:

37. It means that the compressions in region 2 along the direction of the gravitational field are of greater magnitude than the rarefactions in region 2. The equilibrium density in region 2 (i.e. when y = 0) is: The implication is that local concentrations of matter should fall inward rather than expand outward with the Hubble flow, i.e. collapsing initial configurations were preferred to expanding ones in the early universe. It is now possible to examine the angular power spectrum of the CMB and interpret the peaks and troughs in terms of fluctuations in the early universe.

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

42. Given that

46. Sources: Wilkinson Microwave Anisotropy Probe (WMAP) Arcminute Cosmology Bolometer Array Receiver (ACBAR) BOOMERanG experiment (Balloon Observations Of Millimetric Extragalactic Radiation ANd Geophysics) Cosmic Background Imager (CBI) Very Small Array (VSA) 65 experiments in total

49. The first peak is attributed to the compression of a large region (large L) that reached maximum compression at the time of decoupling. The first trough is attributed to a smaller region (smaller L) that began oscillating earlier, when it was sub-horizon sized. It could oscillate faster so it arrived with δT = 0 at the time of decoupling. The second peak (second harmonic) is attributed to oscillation of a still smaller region that reached maximum rarefaction at the time of decoupling. The relative heights of the first and second peaks is used to estimate the density of baryonic matter in the universe. The third peak (third harmonic) is assumed to arise from an oscillation reaching second compression at the time of decoupling. The superhorizon is attributed to a vast fluctuation that remained over the particle horizon until recombination.

50. Comparisons with model universes.