4.4 PECULIAR VELOCITIES (peculiar in the sense of not associated to the Hubble flow rather than odd) The expansion of the Universe also stretches the de Broglie wavelength of freely moving massive particles in proportion to the scale factor. Recalling that λ=h/p where p is the momentum of the particle, we infer that p red shifts as a 1 : This formula can also be derived by simple special relativistic kinematics, which is valid in an infinitesimal region around a comoving observer. Consider a particle at time t having peculiar velocity vp(t) relative to a comoving observer. The particle will, at time t + δt, have crossed a proper distance vp(t)δt, and pass a second comoving observer whose velocity relative to the first one is O O Hubble law The second observer will attribute to the particle the velocity Taylor expansion to first order in δv
Rewriting the previous equation as and noting that the relativistic momentum expression implies one concludes that the proper momentum p(t) measured by the comoving observer the particle passes at time t decays with increasing scale factor as If the particle is non-relativistic, then vp~p and the peculiar velocity itself decreases as a 1. Thus, the expansion of the universe eventually brings massive particles at rest in the comoving frame.
An immediate consequence of this fact is that, since the kinetic temperature Tm of a non-relativistic non-interacting gas is proportional to the peculiar velocity squared, this will vary with cosmic time as The expansion of the Universe also makes the radiation blackbody temperature vary as if cosmic gas and radiation are decoupled, the former will cool faster than the latter.
Matter and Radiation Temperatures Today Recombination
4.5 GRAVITY WHERE THE ACTION IS The laws of Newtonian mechanics can be formulated in terms of a variational principle called the principle of least action. Consider the simple case of a particle of mass m moving in 1D in a potential Φ(x). The equations of motion are summarized by the Lagrangian Newton s law can be expressed as Lagrange s equation
Consider the possible paths between an event (xa,ta) and an event (xb,tb.) For each path construct a real number called its action: Φ(x)=0 The action is an example of a functional a map from functions [in this case x(t) s] to real numbers. Among all the curves connecting A and B, those that minimize the action δs=0 satisfy Lagrange s equation A particle obeying Newton s law follow a path of extremal or least action. The motion of a free test particles in curved space-time can be summarized by a similar variational principle the principle of extremal proper time: A geodesic is an extremum of the action on the set of curves.
The proper time along a timeline world line between events A and B is World lines that minimize the action between A and B are those that satisfy Lagrange s equations σ=parameter x i =x i (σ) geodesic equation is the relativistic analog of
All freely falling particles follow geodesic paths in curved space-time. Distribution of matter (more generally, stress-energy) determines space-time curvature. Space-time tells matter how to move. (Along geodesic paths) Matter tells spacetime how to curve. (Field equations) Compare to equivalent description of Newtonian gravity: Gravitational force tells matter how to accelerate. (F = mia) Matter tells gravity how to exert force. (F = GMmg/r 2 ) mi=mg
4.6 COSMIC DYNAMICS FRW To solve for a(t) and k we must substitute this metric into Einstein s field equations, differential equations relating the metric functions gij to the density and pressure of matter the GR analogs of the Newtonian Poisson equation Einstein tensor stress-energy tensor Here the Riemann tensor Rij and the Ricci (curvature) scalar R = g ik Rik are functions of the metric tensor gij and its first two derivatives, and give the curvature of space. Tij is the stress energy tensor it measures the relevant properties of all matter in the Universe. As the stress energy tensor is assumed to be symmetric:
there are potentially 10 Einstein equations (the number of independent components of a 4 x 4 symmetric matrix). energy density energy flux shear stress pressure momentum density momentum flux If the metric has additional symmetries, the number of independent Einstein eqs. may be much less. A homogeneous and isotropic perfect fluid is characterized by a density ρ and pressure p. Such a fluid has a stress energy tensor given by
If the metric is of FRW type, there are only 2 distinct equations: Relates the curvature of the Universe to its matterenergy content and expansion rate. Gravity pushes instead of pulls if pressure is negative and p< ρ/3. Friedmann equation acceleration equation
newly created
The cosmological debate (SS vs. BB) acquired religious and political aspects. Pope Pious XII announced in 1952 that big-bang cosmology affirmed the notion of a transcendental creator and was in harmony with Christian dogma. Steady-state theory, denying any beginning or end to time, was in some minds loosely associated with atheism. Gamow suggested steadystate theory be part of the USSR Communist Party line, although in fact Soviet astronomers rejected both steady-state and big-bang cosmologies as "idealistic" and unsound. Hoyle himself associated steady state theory with personal freedom and anti-communism!
The universe's ostensible fine tuning is the result of selection bias: i.e., only in a universe capable of eventually supporting life will there be living beings capable of observing any such fine tuning, while a universe less compatible with life will go unbeheld. WAP states that the Universe must have the age and the fundamental physical constants necessary to accommodate conscious life. As a result, it is unremarkable that the universe's fundamental constants happen to fall within the narrow range thought to be compatible with life...
Universe s Fine Tuning ΩRAD ΩM ΩΛ Selection Effect: AP? e
A=5 5 Li3 A=8 8 Be4 UNSTABLE! STABLE ISOTOPES 7 Li, 9 Be
2γ The net energy release of the process is 7.275 MeV. Because the triple-alpha process is unlikely, it requires a long period of time to produce carbon. One consequence of this is that no Carbon was produced in the Big Bang because within minutes after the Big Bang, the temperature fell below that necessary for nuclear fusion. Ordinarily, the probability of the triple alpha process would be extremely small. Hoyle s resonance