# Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

Use down and up arrows to proceed to the next or previous slide.

## Outline

• Midterm Overview and Ch. 23-25 in Guidry
• Big bang

## Distances in the RW metric

• Presume one galaxy is at $(t,0,0,0)$ and another at $(t,r,0,0)$ at some fixed coordinate time $t$
• Our metric is $$ds^2 = -dt^2 + a^2(t) \left[ \frac{dr^2}{1 -k r^2} + r^2 d \Omega^2\right]$$ thus the proper distance is $$\ell = a(t) \int_0^{r} \frac{dr}{\sqrt{1-kr^2}}$$
• For a flat, $k=0$, universe, $$r=\frac{\ell}{a}$$
• For a closed universe, $k=1$, $$r = \sin\left(\frac{\ell}{a}\right)$$
• For an open $k=-1$ universe, $$r=\mathrm{sinh} \left(\frac{\ell}{a}\right)$$

## Friedman Equations, Revisited

• The Friedman equations (using $a(t)$ for the scale factor and putting in some factors of $G$) are $$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G \rho}{3} - \frac{k}{a^2}$$ and $$\frac{2 \ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{k}{a^2} = - 8 \pi G p$$
• By differentiating the first equation with respect to time and eliminating $\ddot{a}$, one gets $$\frac{\dot{\rho}}{\rho} + 3 \left( 1 + \frac{p}{\rho} \right) \frac{\dot{a}}{a} = 0$$
• This is just conservation of mass and energy, and can also be derived from $${T^{\mu \nu}}_{;\nu} = 0$$
• In order to solve, we need an equation of state, $p=p(\rho)$

## Friedman Equations, Revisited II

• Matter is dilute, so only take terms linear in $\rho$, then $p=w \rho$ with sound speed $c_s=\sqrt{w}$
• For nonrelativistic gas, $w \ll 1$ ($w=0$ is matter dominated) and for relativistic gas, $w=1/3$ (radiation dominated)
• Cosmological constant has $w=-1$ (vacuum dominated)
Single-component cosmologies from Guidry

## Back to the Big Bang

• For radiation (matter) dominated, we have $$\frac{\dot{\rho}}{\rho} + 4(3) \frac{\dot{a}}{a}=0$$ and $$a(t) \sim t^{1/2(2/3)}$$
• Observations suggest that $$\frac{\rho_b}{\rho_{\gamma}} \sim 10^{3-4}$$ thus currently matter dominated
• At earlier times, the temperature increases, but the energy density in photons increases faster than that due to matter, thus universe was previously radiation dominated
• Extrapolating further back in time, we get $T \rightarrow \infty$ and $a \rightarrow 0$
• This is the most basic argument for the big bang

## Big Bang Thermodynamics

• Thermodynamics of ultrarelativistic fermions and bosons $T \gg m$ (and $\mu=0$) $$n = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{p^2~dp}{e^{p/T}\pm1}$$ using upper sign for fermions and lower sign for bosons and $$\varepsilon = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{E p^2~dp}{e^{p/T}\pm1}$$
• These integrals give $$n=\frac{g_i \zeta(3) T^3}{\pi^2} \left( \frac{7}{8} \mp \frac{1}{8} \right); \qquad \varepsilon=\frac{g_i \pi^2 T^4}{90} \left( \frac{15}{16} \mp \frac{1}{16} \right)$$
• Thus the energy density for relativistic particles can be written as $$\varepsilon = \frac{g_{\mathrm{eff}} \pi^2 T^4}{30}$$ where $g_{\mathrm{eff}} = \sum g_b + \frac{7}{8} \sum g_f$

 Temperature Additional Particles 4 $g_{\mathrm{eff}}$ $T • Where$T_C$corresponds to the confinement-deconfinement transition between quarks and hadrons • Number of degrees of freedom for$T>m_{W,Z}$depends on unknown particle physics (like Higgs) • This assumes equilibrium • Because$s \sim \sum_i n_i$, entropy is approximately constant ## Photon opacity in the Early Universe • When electrons are present, Thomson scattering means matter is opaque to radiation • Cooling eventually allows the formation of neutral hydrogen atoms and photons decouple from matter • For historical reasons, this is called "recombination" • This happens at$z=1100$, when$T \sim 4000~\mathrm{K}$, at$t=4\times 10^{5}~\mathrm{yr}$• This results in the cosmic microwave background • Time between recombination and the formation of the first stars is sometimes called the "Dark Ages" ## Olber's paradox • In flat space, surface brightness is independent of distance • Surface brightness is flux density per unit angular area $$S = m + 2.5 \mathrm{log}_{10} A$$ where$m$is the apparent magnitude and$A$is the visual area • As distance increases,$A$decreases but$m$increases to match • This is not obvious when gravitational lensing is involved, but still holds from Liouville's theorem: phase space distribution is constant along photon rays • Surface is a better indicator for large or close objects • Static infinite universe would be bright rather than dark • Speed of light is finite and thus the volume is finite • Big bang says universe was hotter (and thus brighter in the past), but the expansion leads to redshift ## Out of Equilibrium • Reaction rates scale as $$\Gamma \sim n \left< v \sigma \right>$$ where$n$is the number of scatterers per unit volume,$v$is the relative velocity and$\sigma$is the cross section • Expect thermal equilibrium as long as $$\Gamma \gt \frac{\dot{a}}{a} = H \approx \frac{1}{2t}$$ • Weak interactions decouple when$T \sim 1~\mathrm{MeV}$which happens at$1~\mathrm{sec}$(showing this will be HW) ## Inflation • Dark energy of the magnitude required to match observations today is negligable in early universe • Early evolution thus dominated by matter$p \sim 0$or radiation$p=\rho/3$which leads to a decelerating expansion ($\ddot{a}<0$) • Yet expansion is currently increasing with time • A cosmological constant cannot fix this • Inflation postulates an early period where ($\ddot{a}>0\$) to explain these observations
• Universe begins in a false vacuum and undergoes a phase transition to the true vacuum and releases energy into the plasma
• First-order phase transition would cause larger inhomogeneities, thus believed to be second order

## Group Work

• Group 1: Spokesperson: Tuhin Das Leonard Mostella Ibrahim Mirza Group 2: Spokesperson: Satyajit Roy Erich Bermel Group 3: Spokesperson: Jason Forson Andrew Tarrence
• In class, I stated that as the temperature increases, the energy density in photons increases faster than that due to matter. Argue that this is true from basic stat. mech.
• When the temperature drops below the strange quark mass, the number of effective degrees of freedom drops by 21/2. How is this number obtained? See, e.g. the PDG's big bang cosmology review.