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

• Back to Cosmology by Weinberg
• Ryden, "Introduction to Cosmology"

Structure Formation

• Universe homogeneous over 100 Mpc scales
• Strong fluctuations over smaller scales
• Clusters, superclusters and voids: large-scale structure
• Superclusters are typically flattened, and just beginning to collapse (virialize)
2dF Galaxy Redshift survey; Coless et al. (2001)

Gravitational Instability

• Spatially averaged energy density $$\bar{\varepsilon}(t) = \frac{1}{V} \int_V \varepsilon(\vec{r},t) d^3 r$$
• Define a dimensionless density fluctuation $$\delta(\vec{r},t) \equiv \frac{\varepsilon(\vec{r},t) - \bar{\varepsilon}(t)}{\bar{\varepsilon}(t)}$$ where $\delta_{\mathrm{min}} = -1$ .
• Start with simplified case: static homogeneous matter-only spherical universe with uniform mass density, $\bar{\rho}$
• Density is $\bar{\rho}(1+\delta)$

Density Fluctuations

• Gravitational acceleration at the surface of the sphere is $$\ddot{R} = -\frac{G \Delta M}{R^2} = - \frac{G}{R^2} \left( \frac{4 \pi}{3} R^3 \bar{\rho} \delta \right)$$ thus $$\frac{\ddot{R}}{R} = -\frac{4 \pi G \bar{\rho}}{3} \delta(t)$$ and a positive $\delta$ causes collapse
• Use mass conservation, $$M = \frac{4 \pi}{3} \bar{\rho} \left[ 1 + \delta(t) \right] R(t)^3$$ to obtain $$R(t) = R_0 \left[ 1 + \delta(t) \right]^{-1/3}$$ with $R_0^3 = [3 M/(4 \pi \bar{\rho})]$

Density Fluctuations II

• Use small $\delta$ expansion $$R(t) = R(0)[1-\delta(t)/3]$$ thus $$\ddot{\delta} = 4 \pi G \bar{\rho} \delta$$ which has solution $$\delta(t) = A_+ e^{t/t_{\mathrm{dyn}}} + A_- e^{-t/t_{\mathrm{dyn}}}$$ with $$t_{\mathrm{dyn}}^{-2} = 4 \pi G \bar{\rho}$$ which is independent of the size of the fluctuation, depending only on the average density
• Something must stop this exponential growth

Density Fluctuations III

• Exponential growth halted by pressure, and pressure perturbations travel at the speed of sound $$t_{\mathrm{press}} \sim \frac{R}{c_s}$$ thus to be stabilized by collapse we must have $$t_{\mathrm{press}} < t_{\mathrm{dyn}}$$ Thus, for stability, the region must be smaller than the Jeans length $$\lambda_J \sim c_s t_{\mathrm{dyn}} = c_s \left(\frac{c^2}{G \bar{\varepsilon}}\right)$$
• We've ignored some numerical factors, the more precise result is $$\lambda_J = c_s \left(\frac{\pi c^2}{G \bar{\varepsilon}}\right)$$
• E.g Earth's atmopshere, the Jeans length is 10^5 km, so structures smaller than this are stable

Cosmological Fluctuations

• Characteristic time is the Hubble time $$H^{-1} = \left(\frac{3 c^2} {8 \pi G \bar{\varepsilon}}\right)^{1/2}$$ and comparing this with the dynamical timescale one finds that $$H^{-1} = \left( \frac{3}{2}\right)^{1/2} t_{\mathrm{dyn}}$$ and $$\lambda_J = 2 \pi \left( \frac{3}{2}\right)^{3/2} \frac{c_s}{H}$$
• Now, take the radiation-dominated case, where $c_s \sim c/\sqrt{3}$, then $$\lambda_J \approx 3 c/H$$ i.e. the "Hubble distance"

Contribution of matter

• Matter typically has $c_s \ll 1$
• when photons and baryons are in equilibrium, before $z\approx 1100$, the Jeans length of the universe was large
• After decoupling, then $$c_{s,\mathrm{matter}} = c \left( \frac{k T}{m c^2}\right)^{1/2}$$ where $T_{\mathrm{dec}} \approx 0.26~\mathrm{eV}$ and $m \approx 1.22 m_p = 1140~\mathrm{MeV}$, thus $$c_{s,\mathrm{matter}} = c \approx 10^{-5}$$ and the new Jeans mass is $$10^{5}~\mathrm{M}_{\odot}$$

Include Expansion of Universe

• Overdense regions have to compete with the expansion of the universe
• Mass conservation equation is now $$M = \frac{4 \pi}{3} \bar{\rho}(t) [ 1+ \delta(t) ] R(t)^3$$ where $\bar{\rho}(t) \propto a(t)^{-3}$ is now time-dependent
• This implies $$R(t) \propto a(t) \left[1+\delta(t)\right]^{-1/3}$$ or taking two time derivatives $$\frac{\ddot{R}}{R} = \frac{\ddot{a}}{a} - \frac{\ddot{\delta}}{3} - \frac{2 \dot{a} \dot{\delta}}{3 a}$$
• Then $$\ddot{\delta} + 2 H \dot{\delta} = 4 \pi G \bar{\rho} \delta$$ where now we have an additional "Hubble friction" term which slows perturbations ($H = \dot{a}/a$)

• Full result is $$\ddot{\delta} + 2 H \dot{\delta} = \frac{4 \pi G \bar{\varepsilon}_m}{c^2} \delta$$ where $\bar{\varepsilon}_m$ is the energy density contribution from matter with $c_s \ll 1$
• Rewriting in terms of $$\Omega_m = \frac{\bar{\varepsilon}_m}{{\varepsilon}_m} = \frac{8 \pi G \bar{\varepsilon}_m}{3 c^2 H^2}$$ then $$\ddot{\delta} + 2 H(t) \dot{\delta} - \frac{3}{2} \Omega_m H(t)^2 \delta = 0$$ so density perturbations wait until universe is matter dominated. Radiation-matter equality at $z\sim 3230$
• To see this, presume $\Omega_m \approx 0$, and notes that $H \sim 1/(2t)$, then $$\ddot{\delta} + \frac{\dot{\delta}}{t} \approx 0$$ which means density fluctuations grow logarithmically

Dark Energy and Matter-Dominated Regime

• When universe is dominated by dark energy, the Hubble parameter is constant, thus $$\ddot{\delta} + 2 H_{\Lambda} \dot{\delta} \approx 0$$ which has solution $$\delta(t) \sim C_1 + C_2 e^{-2 H_{\Lambda} t}$$ so fluctuations are constant while the energy density drops exponentially $\bar{\varepsilon}_m \propto e^{-3 H_{\Lambda} t}$
• Matter-dominated, then $H \sim 2/(3t)$ and $\Omega_m = 1$, then power-law solution $$\delta(t) \approx D_1 t^{2/3} + D_2 t^{-1}$$ and as $t$ becomes large the growing term dominates, violating the simple linear model

Matter-Dominated Regime and Fourier Transform

• Universe becomes matter-dominated, density perturbations increase
• Regions with $\delta=1$ collapse and virializes
• If baryonic matter is able to cool efficiently, then it converges to the center and forms stars
• Density perturbations in dark matter started before recombination, at $z \approx 3570$, baryons fall into these potential wells after recombination at $z \approx 1100$
• Fluctuations in momentum space (complex) $$\delta(\vec{r}) = \frac{V}{(2 \pi)^3} \int \delta_{\vec{k}} e^{-i \vec{k}\cdot\vec{r}} d^3~k$$ and its inverse $$\delta_\vec{k} = V \int \delta(\vec{r}) e^{i \vec{k}\cdot\vec{r}} d^3~r$$

Power Spectrum

• Fluctuations in momentum space obey $$\ddot{\delta}_{\vec{k}} + 2 H \dot{\delta}_{\vec{k}} - \frac{3}{2} \Omega_m H^2 {\delta}_{\vec{k}} = 0$$ as long as $a(t) 2 \pi/k$ is large compared to Jeans length
• The quantity $a(t) 2 \pi/k$ must also be small compared to $c/H$ so that waves are causally connected to each other
• The mean square amplitude of the fluctuations in momentum space define the power spectrum $$P(k) = \left< \left| \delta_{\vec{k}}\right|^2 \right>$$ where the absolute value signs indicate real part and the averaging is over wavenumbers $\vec{k}$

Power Spectrum, Correlation Functions

• The N-point correlation function (in coordinate space) is $$\xi_N \equiv \left< \delta(\vec{x}_1) \delta(\vec{x}_2) \ldots \delta(\vec{x}_N)\right>$$ where if the universe is homogeneous and isotropic then these correlation functions depend only on relative distances
• Thus, the two-point correlation function is the Fourier transform of the power spectrum
• Most cosmological models predict that the density fluctuations created by inflation are Gaussian fields
• Gaussian fields are those where the magnitude and phase of the Fourier transform of the density fluctuations are normally distributed
• Also, $P(k) \sim k^n$ with $n \sim 1$ (Harrison-Zel'dovich)

Power Spectrum II

• Turns out that $n=1$ is the only power law which prevents divergence of fluctuations on large and small scales
• $n$ is the "scalar spectral index" measured taking value of $0.9667 \pm 0.0040$
• $n=1$ also implies the spectrum is scale-invariant
• Deviations from power-law measured by the "running of the spectral index" or $dn/d~\mathrm{ln}~k = -0.023\pm 0.011$

Group Work

• Group 1: Spokesperson: Erich Bermel Andrew Tarrence Ibrahim Mirza Group 2: Spokesperson: Tuhin Das Satyajit Roy Group 3: Spokesperson: Leonard Mostella Jason Forson
• Google the CMB power spectrum and describe as many of the features as you can. Also, it is frequently plotted as a function of mulitpolarity, $\ell$. What does $\ell$ refer to and how is it related to the observations?