# 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

• Ch. 12 in Schutz with input from elsewhere
•
• Aside: Boltzmann equation for neutrino (or photon) transport: LHS is just the geodesic equation

## Cosmological Principle

• The universe is homogeneous and isotropic (at large enough scales)
• The properties of the universe are the same for all observers
• "The universe plays fair"
• Not proven, but observationally true for scales greater than few $\times 10^{7-8}~\mathrm{lyr}$ (this value might need to be updated)
• Guided the theoretical developments in cosmology
• Planck satellite sees two violations: hemispheric asymmetry and a "cold spot"

## Hubble Expansion

• Every galaxy is traveling (on average) away from all other galaxies
• There is no "center"
• The velocity relative to the observer is proportional to the distance, $v = H d$
• Hubble's parameter, $H$, it's value at zero redshift (now) is Hubble's constant, $H_0$ $$H_0 \approx 2.3 \times 10^{-18}~\mathrm{s}^{-1} \approx 71~\mathrm{km}~\mathrm{s}^{-1}~\mathrm{Mpc}$$
• Current anomaly: Hubble's constant from gravitational lensing disagrees with the value from CMB

## Preferred Cosmological Reference Frame

• SR: All inertial reference frames are equivalent
• GR: There are no global inertial frames, only local ones
• Cosmology: choose a time coordinate such that hypersurfaces are homogeneous and isotropic
• Thus cosmology has a preferred frame

## Inaccessible Regions of Spacetime

• Spacetime inside an event horizon is inaccessible to all outside observers
• Part of spacetime is inaccessible to us as well, "particle horizon"

## Cosmological Metrics

• We want to ensure that
• Hypersurfaces of constant time are homogeneous and isotropic
• Mean rest frame of galaxies agrees with our definition of simultaneity
• Comoving coordinates: galaxy locations are fixed, and the expansion of the universe expands the space between them
• Metric along hypersurface of constant time $$d \ell^2(t) = h_{ij}(t) dx^{i} dy^{j} \Rightarrow d \ell^2(t) = R^2(t) h_{ij} dx^{i} dy^{j}$$
• Generalize to time coordinate $$ds^2 = -dt^2 + g_{0i} dt dx^{i} + R^2(t) h_{ij} dx^{i} dy^{j}$$
• We need $g_{00}=-1$ so coordinate time is proper time and $g_{0i}$ must vanish so that our basis vectors remain orthogonal

## Cosmological Metrics II

• Deduce form of $h_{ij}$. Must be spherically symmetric and isotropic. The usual form $$d \ell^2 = e^{2 \Lambda(r)} dr^2 + r^2 d \Omega^2$$ is isotropic about the origin, but we want the curvature to be isotropic (this process is related to that used to construct the Schwarzchild metric in "isotropic coordinates")
• The way to do this is to use the metric $$d \ell^2 = \frac{dr^2}{1 -k r^2} + r^2 d \Omega^2$$ with $k$ a "curvature parameter" so that the full metric is $$ds^2 = -dt^2 + R^2(t) \left[ \frac{dr^2}{1 -k r^2} + r^2 d \Omega^2\right]$$
• The curvature in this (the Robertson-Walker) metric depends only on $k$, $R$ and $R$'s first and second time derivatives
• WLOG, set $k=-1,0,+1$ (open, flat, closed)

## Cosmological Redshift

• Distance measurements change from the expansion, so redshift is sometimes a more useful distance measurement
• Our metric is isotropic and homogeneous, so we can put the origin anywhere
• Coordinate time is proper time so light travels along a null trajectory $$0 = - dt^2 + R^2(t) d \chi^2$$
• The metric is independent of $\chi$ so $p_{\chi}$ is conserved, and thus $p^{0} \propto R(t)^{-1}$
• The (cosmological part of the) redshift is given by $$1 + z_{\mathrm{cosmo}} = R(t_0)/R(t)$$
• Full redshift is given by $1+z_{\mathrm{full}} = (1 + z_{\mathrm{cosmo}})(1 + z_{\mathrm{motion}})$

## Expansion with the RW metric

• The Hubble parameter is $$H(t)=\frac{\dot{R}(t)}{R(t)}$$ and $R(t)$ is the "scale factor of the universe"
• Alternatively, $$R(t) = R_0 \exp \left[\int_{t_0}^{t} H(t^{\prime}) dt^{\prime}\right]$$ and $$1+z(t) = \exp \left[- \int_{t_0}^{t} H(t^{\prime}) dt^{\prime}\right]$$ thus (why?) $$H(t) = -\frac{\dot{z}}{1+z}$$

## Distance Measurements

• Classical relation between luminosity, flux, and distance $$L = 4 \pi d^2 F$$ thus $L/F$ provides a measurement of $d$. Flux can be easily measured, but $L$ must be inferred.
• For example, if you can measure $T$ for a main sequence star, then you also have information about $L$
• Objects which provide distance measurements are called "standard candles"
• Cosmological distances determined by Type Ia supernovae (sometimes called "standardizable candles" because $L$ is inferred from properties of the Type Ia population)
• The "luminosity distance" is $d_L=[L/(4 \pi F)]^{1/2}$
• Expansion implies $$d_L = R_0 r(1+z) = \left(\frac{z}{H_0}\right) \left[ 1 + \left(1+\frac{\dot{H}_0}{2 H_0^2}\right)z \right] + \ldots$$ Thus measuring luminosity distances gives you information about the expansion of the universe

## Expansion from Supernovae Riess et al. (1998) as in Schutz Supernova distance measurements show the expansion is accelerating Evidence for dark energy or a cosmological constant (still strongest evidence today) Motivates careful understanding of supernovae

## Connecting our Metric to Energy and Pressure

• Energy-momentum conservation implies $$\frac{d}{dt} \left( \rho R^3 \right) = -p \frac{d}{dt} \left( R^3 \right)$$
• "Matter-dominated" (current time), most of the energy density is in cold matter and $p=0$ so $$\frac{d}{dt} \left( \rho R^3 \right) = 0$$
• In "radiation-dominated" era (early universe), $p=\rho/3$, so $$\frac{d}{dt} \left( \rho R^3 \right) = -\frac{\rho}{3} \frac{d}{dt} \left( R^3 \right) \Rightarrow \frac{d}{dt} \left( \rho R^4 \right) = 0$$
• The only nontrivial component of the Einstein tensor is $$G_{tt} = 3 \left(\frac{\dot{R}}{R} \right)^2 + 3 \frac{k}{R^2}$$

## Dark Energy

• Einstein's equations can be modified with a cosmological constant $$G_{tt} + \Lambda g_{tt} = 8 \pi T_{tt}$$
• One can put $\Lambda$ inside the stress-energy tensor $$T^{\alpha \beta}_{\Lambda} = -\frac{\Lambda}{8 \pi}g^{\alpha \beta}$$ and the dark energy density $\rho_{\Lambda} = \Lambda/(8 \pi)$ and the "dark pressure" is negative
• With these definitions $$\frac{\dot{R}^2}{2} = -\frac{k}{2} + \frac{4 \pi R^2}{3} \left(\rho_{\mathrm{matter}} + \rho_{\Lambda}\right)$$
• Observations suggest $k \approx 0$, and matter-dominated implies $\rho_{\mathrm{matter}} R^2$ decreases with time
• The spatial part of Einstein's equations implies $$\frac{\ddot{R}}{R} = - \frac{4 \pi}{3}\left(\rho+3 p\right)$$ so the scale of the universe is affected by pressure as well as energy density

## Results

• One can show then that there must have been a big bang, a earlier time when $R=0$ (and thus the curvature was infinite)
• This is the "cosmological singularity"
• Our expanding universe will never stop
• We proceed towards the future in which, all in compact objects, dwarf stars, planets, and smaller objects at $T=0$
• This is a long time away: we are very close to the static solution where $\rho_{\Lambda} = \rho_{\mathrm{matter}}/2$
• Of course quantum effects can change this picture significantly

## Critical Density and Universe's Parameters

• Rewriting the Friedmann equation $$\frac{3 H^2}{8 \pi} = - \frac{3 k }{8 \pi R^2} + \rho_m + \rho_{\Lambda}$$ and rewriting again $$\rho_H = \rho_k + \rho_m + \rho_{\Lambda}$$ where the Hubble energy density is a critical energy density $$\rho_c = \frac{3}{8 \pi} H_0^2$$ and we measure all densities relative to $\rho_c$ $$1 = \Omega_k + \Omega_m + \Omega_{\Lambda}$$
• The numerical values are $\Omega_{\Lambda}\approx 0.7$, $\Omega_{m}\approx 0.3$, $\Omega_{k}\approx 0$ where $\Omega_{m}=\Omega_b + \Omega_d \approx 0.04 + 0.26$

## Supernova Cosmology Project Amanullah et al. (2010) Combined information from the CMB and BAO gives a complete picture of dark energy vs. dark matter

## Group Work

• Group 1: Spokesperson: Erich Bermel Satyajit Roy Ibrahim Mirza Group 2: Spokesperson: Leonard Mostella Andrew Tarrence Group 3: Spokesperson: Tuhin Das Jason Forson
• Regarding Amanullah et al. (2010), "Spectra and Hubble Space Telescope...":
• What is the connection between Fig. 6, standard candles, and cosmology?
• What is plotted in Fig. 9?
• Explain how this paper is connected to Fig. 10
• What value of $w$ did I assume in the lecture and where did I make that assumption?
• Does Eq. 4 presume anything about the relationship between the uncertainties due to extinction and the uncertainties due to the light curve fitting?