# 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

• Cosmology by Weinberg
• Cyburt et al. (2016) for BBN

## Distance Indicators

• Distance measurements are key in determining the Hubble parameter and its dependence on redshift
• Is the Hubble parameter larger or smaller at earlier times? How would you determine this?
• For nearby objects
• Trigonometric parallax
• Photometric parallax
• Red clump stars
• RR Lyrae stars
• Eclipsing binaries
• Cepheid variables
• Confusion between "peculiar motion" and Hubble expansion

## Distance Indicators I - Tully-Fisher

 Phenomenological correlation between angular velocity of stars in a galaxy around center and the luminosity of that galaxy Standard correlation includes only stars Slightly different correlations in different bands Found tighter correlation when luminosity includes stars and gas Mass $\propto v^{3-4}$ Originally applied to spiral galaxies, applied to ellipticals in "Faber-Jackson" relation Also evidence for dark matter Tully and Fisher (1977), this figure from Karachentsev et al. (2002)

## Distance Indicators II - Type Ia supernova

 White dwarf always has a mass near the Chandrasekhar limit Luminosity correlated with rise and decline time of the emitted light Emitted light is from the decay of Nickel-56 "Phillips relationship" In detail, $$M_{\mathrm{max}}(B) = -21.7 + 2.7 \Delta m_{15}(B)$$ Calibrate correlation with other distance measurements "Standardizable candle" From Phillips

## More General Cosmological Models

• The Friedmann equations give $$\rho = \frac{3 H_0^2}{8 \pi G} \left[ \Omega_{\Lambda} + \Omega_M \left( \frac{a_0}{a}\right)^3 + \Omega_R \left(\frac{a_0}{a}\right)^4\right]$$ with $$\rho_{V0} = \frac{3 H_0^2 \Omega_{\Lambda}}{8 \pi G}, \quad \rho_{M0} = \frac{3 H_0^2 \Omega_{M}}{8 \pi G}, \quad \rho_{R0} = \frac{3 H_0^2 \Omega_{R}}{8 \pi G},$$ and $$\Omega_{\Lambda} + \Omega_M + \Omega_R = 1-\Omega_K$$ and $$\Omega_K = -\frac{K}{a_0^2 H_0^2}$$
• Using the Friedmann equation $$\dot{a}^2 + K = \frac{8 \pi G \rho a^2}{3}$$

## More General Cosmological Models II

• ...and defining $x \equiv a/a_0$, we get $$dt = \frac{dx}{H_0 x \sqrt{\Omega_{\Lambda} + \Omega_K x^{-2} + \Omega_M x^{-3}+ \Omega_R x^{-4}}}$$
• Define $t=0$ to be at $z=\infty$ and we have $a/a_0 = 1/(1+z)$ so $$dt = \frac{-dz}{H_0 (1+z) \sqrt{\Omega_{\Lambda} + \Omega_K (1+z)^{2} + \Omega_M (1+z)^{3}+ \Omega_R (1+z)^{4}}}$$ thus the age of the universe is $$t = \frac{1}{H_0} \int_0^{1} \frac{dx}{x \sqrt{\Omega_{\Lambda} + \Omega_K x^{-2} + \Omega_M x^{-3}+ \Omega_R x^{-4}}}$$
• This can be integrated in the approximation that $$\Omega_R \approx \Omega_K = 0$$

## Cosmic Microwave Background

• Energy density in radiation is $$U = \frac{8 \pi^5 k_B^4 T^4}{15 h^3 c^3}$$ which gives $$\Omega_{\gamma} \equiv \frac{\rho_{\gamma 0}}{\rho_{0,\mathrm{crit}}} = 2.47 \times 10^{-5} h^{-2}$$ where $$H = 100~h~\mathrm{km}/\mathrm{Mpc}/\mathrm{s}$$ which naturally implies that $\Omega_{\gamma}$ is too small enough to be ignored in the current epoch.

## Cosmic Microwave Background

• Neutrinos increase this energy density by about a factor of two
• The present number density of photons is $$n_{\gamma} = \frac{30 \zeta(3) a_B T^3}{\pi^4 k_B} = \frac{410~\mathrm{photons}}{\mathrm{cm}^3}$$
• Compare to baryons, $$n_B = \frac{3 \Omega_B H_0^2}{8 \pi G m_N} = 1.1 \times 10^{-5} \Omega_B h^2 ~\mathrm{nucleons}/\mathrm{cm}^3$$
• Current values of baryon-to-photon ratio actually a few $\times 10^{-10}$

## Recombination

• Matter and radiation decoupled when electrons and protons combine at $z=1100$ and $T=4000~\mathrm{K}$ and $t=360,000~\mathrm{yr}$.
• Rough estimate from Saha equation for $$p + e \leftrightarrow H + \gamma$$ is $$\frac{n_p n_e}{n_{\mathrm{H}}} = \left( \frac{m_e k_B T}{2 \pi \hbar^2} \right) \exp \left( - \frac{E_I}{k_B T}\right)$$ where $E_I$ is the ionization energy $13.6~\mathrm{eV}$ of hydrogen.
• In reality, excited states (2p, 2s) make a contribution
• Deviations from equilibrium impact these results at the 10% level

## Dipole Anisotropy

• CMB provides a frame of reference, solar system moves relative to this
• Number of photons is Lorentz invariant, so boost momenta $$| \mathbf{p} | = \left(\frac{1+\beta \cos \theta}{\sqrt{1-\beta^2}}\right) | \mathbf{p}^{\prime} | =$$ and this results in $$T^{\prime} = T \left(\frac{1+\beta \cos \theta}{\sqrt{1-\beta^2}}\right)$$
• Solar system "barycenter" has velocity of 370 km/s (0.1% of $c$) relative to CMB
• Local group of galaxies moves at 630 km/s relative to the CMB

## Sunayev-Zel'dovich effect

• Are CMB photons completely unimpeded by the time they reach the observer?
• No, galaxies contain high-temperature (but low-density) electrons
• Inverse Compton scattering increases the photon energy
• Change in temperature (for $\omega/k_B T_{\gamma} \ll 1$) $$\frac{\Delta T_{\gamma}}{T_{\gamma}} = \frac{\Delta N_{\gamma}}{N_{\gamma}} = - \frac{2 \sigma_T}{m_e c^2} \int~d\ell~n_e(\ell)~k_B T_e(\ell)$$
• Measure $n_e$ from X-ray observations, then use S-Z effect to determine $H_0$

## Big Bang Nucleosynthesis

• Neutron-to-proton ratio nearly equal until 1 second
• Protons slightly favored over neutrons from Saha equation
• Saha equation is relationship between chemical potentials in thermal equilibrium, in this case, $$\mu_n + m_n = \mu_p + m_p + \mu_e$$
• Freeze out (interactions slower than expansion) at $T\sim 0.8~\mathrm{MeV}$
• Competition between fusion and neutron decay leads to 25% deuterium by mass, 75% protons and trace amounts of helium-3, deuterium, lithium-7, etc.
• Uncertainty in neutron lifetime caused difficulties for early models
• These abundances strongly determined by baryon-to-photon ratio

## Big Bang Nucleosynthesis II

 More helium-4 in the universe than can be explained by stellar evolution Deuterium is difficult to create (low binding energy) Observational determinations generally agree with model predictions based on CMB determinations of $\eta$ Except lithium 7, "no solution that is either not tuned or requires substantial departures from standard model physics" Cyburt et al. (2016) Stellar depletion of lithium-7 to explain why observed value is smaller than BBN prediction From NASA

## Big Bang Nucleosynthesis III

• 7 protons for each neutron
• Begin with $$n + p \rightarrow d + \gamma$$ but deuteron binding energy is small so photon breakup is significant until the deuterium bottleneck is resolved
• All neutrons end up in helium-4 $$Y_{\mathrm{helium}-4} = \frac{2(n/p)}{1+(n/p)} \approx 0.25$$
• An accurate fit is $$\eta_{10} = 273.3036 \Omega_B h^2 \left(1 + 7.16958\times10^{-3} Y_p \right) \left(\frac{2.7255~\mathrm{K}}{T_{\gamma}^0}\right)^3$$ where $T_{\gamma}^0$ is the current photon temperature

## Cold Dark Matter

• Cold if (i) non-relativistic, and (at the time of radiation-matter equality) (ii) dissipationless and (iii) collisionless
• Cold dark matter leads to bottom-up structure formation
• Bottom-up: small objects clump first, then merge to form larger object
• What is the opposite of "bottom-up"?
• WIMPs and axions

## WIMP Miracle

• Weak interaction magically provides good dark matter candidates
• Weak scale, $m_W \sim 100~\mathrm{GeV}$
• Weak coupling, $G_F~\sim 10^{-5}~\mathrm{GeV}^{-2}$
• Density of dark matter particles $$\frac{d (na^3)}{dt} = -n^2 a^3 \left<\sigma v\right>$$ or $$n a^3 = \frac{n(t_1) a^3(t_1)}{1+ n(t_1) a^3(t_1) \int_{t_1}^{\infty} \left<\sigma v\right> a^{-3}(t) dt^{\prime}}$$
• Turns out that $$\left<\sigma v\right> \sim G_F^2 m_W^2$$ gives almost exactly the right result

## Group Work

• shuffle list Group 1: Spokesperson: Erich Bermel Tuhin Das Ibrahim Mirza Group 2: Spokesperson: Jason Forson Satyajit Roy Group 3: Spokesperson: Leonard Mostella Andrew Tarrence
• In BBN, at freezeout, the neutron to proton ratio is determined by the Saha equation. Presume that the electron contribution is negligable, thus $\mu_n + m_n = \mu_p + m_p$ and that $n \propto \exp (\mu_i/T)$. What is the neutron to proton ratio at freezeout ($T=0.8~\mathrm{MeV}$) ?
• BBN is delayed until fusion can overcome the "deuterium bottleneck". Cyburt et al. states that this temperature is approximately when $$\eta^{-1} \exp(-E_B/T) = 1$$ where $E_B$ is the binding energy of the deuteron and $\eta$ is the baryon-to-photon ratio. What is $T$?