Complete Cosmology

The Concordance Model of Cosmology

Core ideas

The concordance model, $\Lambda$CDM, describes a nearly flat universe with radiation, baryons, cold dark matter, dark energy, and nearly scale-invariant primordial perturbations. It is not a list of facts: it is a parameterized model connecting expansion, nucleosynthesis, CMB anisotropies, and galaxy clustering.

For review, be able to state the components of $\Lambda$CDM, define density parameters, explain flatness, and identify the key observational pillars. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\Omega_i=\frac{\rho_i}{\rho_{\rm crit}},\qquad \rho_{\rm crit}=\frac{3H_0^2}{8\pi G},\qquad \sum_i\Omega_i\simeq1$$

Worked example

Consider a flat $\Lambda$CDM universe with $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$, $\Omega_m = 0.3$, and $\Omega_\Lambda = 0.7$.

1. Compute the critical density: $$\rho_{\rm crit} = \frac{3H_0^2}{8\pi G} = \frac{3(70\ \mathrm{km\,s^{-1}\,Mpc^{-1}})^2}{8\pi (6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}})} \approx 9.2\times10^{-27}\ \mathrm{kg\,m^{-3}}.$$
2. The present matter density is $\rho_m = \Omega_m \rho_{\rm crit} \approx 0.3 \times 9.2\times10^{-27} \approx 2.8\times10^{-27}\ \mathrm{kg\,m^{-3}}$.
3. The cosmological constant energy density is $\rho_\Lambda = \Omega_\Lambda \rho_{\rm crit} \approx 6.4\times10^{-27}\ \mathrm{kg\,m^{-3}}$, which corresponds to $\Lambda = 8\pi G \rho_\Lambda \approx 1.2\times10^{-52}\ \mathrm{m^{-2}}$.

Because $\Omega_m + \Omega_\Lambda = 1$, the curvature parameter $\Omega_k \simeq 0$, consistent with a spatially flat geometry.

Problems with Solutions

Problem 1. Calculate $\rho_{\rm crit}$ for $H_0 = 67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$.

Solution. Convert $H_0$ to SI: $67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}} = 2.184\times10^{-18}\ \mathrm{s^{-1}}$. Then

$$\rho_{\rm crit} = \frac{3(2.184\times10^{-18})^2}{8\pi(6.674\times10^{-11})} \approx 8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}.$$

Problem 2. A model has $\Omega_m = 0.25$, $\Omega_\Lambda = 0.70$, and $\Omega_k = 0.05$. Verify that $\sum_i \Omega_i = 1$ and state whether the universe is open or closed.

Solution. The sum is $0.25 + 0.70 + 0.05 = 1.00$. Because $\Omega_k > 0$, the curvature term is negative in the Friedmann equation and the spatial geometry is open (negative curvature).

Problem 3. At redshift $z=1$, by what factor is the matter density higher than today, assuming matter domination?

Solution. In a matter-dominated era, $\rho_m \propto a^{-3}$ and $a = (1+z)^{-1}$. At $z=1$, $a=1/2$, so $\rho_m(z=1) = \rho_{m,0}(1+z)^3 = \rho_{m,0}\, 2^3 = 8\rho_{m,0}$. The density is 8 times larger.

Section summary. Modern cosmology is organized around the predictive $\Lambda$CDM model.

The Expanding Universe

Core ideas

Expansion is encoded in the scale factor $a(t)$. Cosmological redshift measures how wavelengths stretch with the universe. Distances are subtle because emission time, observation time, and geometry differ; comoving, angular-diameter, and luminosity distances answer different observational questions.

For review, be able to convert between scale factor and redshift, define Hubble rate, distinguish common cosmological distances, and interpret lookback time. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$1+z=\frac{a_0}{a(t)},\qquad H=\frac{\dot a}{a},\qquad d_L=(1+z)^2d_A$$

Worked example

A Type Ia supernova is observed at redshift $z = 0.5$ in a flat universe. Its observed flux yields a luminosity distance $d_L = 2.8\ \mathrm{Gpc}$.

1. The angular-diameter distance is $$d_A = \frac{d_L}{(1+z)^2} = \frac{2.8\ \mathrm{Gpc}}{(1.5)^2} \approx 1.24\ \mathrm{Gpc}.$$
2. A physical diameter $D = 10\ \mathrm{kpc}$ at this redshift subtends an angle $$\theta = \frac{D}{d_A} = \frac{10\ \mathrm{kpc}}{1.24\times10^6\ \mathrm{kpc}} \approx 8.1\times10^{-6}\ \mathrm{rad} \approx 1.7''.$$
3. The scale factor at emission was $a = (1+z)^{-1} = 2/3$, meaning the universe was $2/3$ its present size when the light was emitted.

Problems with Solutions

Problem 1. What is the scale factor corresponding to $z = 3$?

Solution. $a = (1+z)^{-1} = 1/4 = 0.25$.

Problem 2. A galaxy at $z = 2$ has a physical separation of $100\ \mathrm{kpc}$ from a quasar. What is their comoving separation?

Solution. Comoving distance is independent of expansion, defined as $r = d_{\rm phys}/a = d_{\rm phys}(1+z)$. Thus $r = 100\ \mathrm{kpc} \times 3 = 300\ \mathrm{kpc}$.

Problem 3. In a matter-dominated flat universe, the Hubble parameter scales as $H(z) = H_0(1+z)^{3/2}$. Compute $H$ at $z = 1000$ for $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$.

Solution. $H(1000) = 70 \times (1001)^{3/2} \approx 70 \times 3.17\times10^4 \approx 2.2\times10^6\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$.

Section summary. Expansion turns cosmic time into observable redshift and distance relations.

The Fundamental Equations of Cosmology

Core ideas

The Friedmann equations follow from applying Einstein’s equation to a homogeneous and isotropic universe. Fluids dilute according to their equation of state: radiation as $a^{-4}$, matter as $a^{-3}$, and a cosmological constant as constant density.

For review, be able to derive density scaling, use Friedmann acceleration, compare radiation-, matter-, and dark-energy-dominated eras. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$H^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3},\qquad \dot\rho+3H(\rho+p/c^2)=0$$

Worked example

Consider a spatially flat ($k=0$), matter-dominated universe with $\Lambda = 0$. The Friedmann equation reduces to

$$H^2 = \frac{8\pi G}{3}\rho_m.$$

Because $\rho_m = \rho_{m,0} a^{-3}$, we write $H = \dot a/a$ and obtain

$$\dot a^2 = \frac{8\pi G\rho_{m,0}}{3} \frac{1}{a}.$$

Let $a \propto t^{2/3}$. Then $\dot a = \frac{2}{3} t^{-1/3} a_0^{2/3}$ and

$$H = \frac{\dot a}{a} = \frac{2}{3t}.$$

The age of such a universe is $t_0 = 2/(3H_0)$. For $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}} \approx 2.27\times10^{-18}\ \mathrm{s^{-1}}$,

$$t_0 = \frac{2}{3 \times 2.27\times10^{-18}} \approx 9.4\times10^{17}\ \mathrm{s} \approx 9.3\ \mathrm{Gyr}.$$

Problems with Solutions

Problem 1. Show that radiation density scales as $\rho_r \propto a^{-4}$ using the continuity equation $\dot\rho + 3H(\rho + p/c^2) = 0$ with $p = \rho c^2/3$.

Solution. Substitute $p = \rho c^2/3$:

$$\dot\rho + 3H\left(\rho + \frac{\rho}{3}\right) = \dot\rho + 4H\rho = 0.$$

Since $H = \dot a/a$, this becomes $d\rho/\rho = -4\, da/a$, integrating to $\rho \propto a^{-4}$.

Problem 2. A flat universe has $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$. Write the Hubble parameter $H(z)$ and evaluate it at $z = 1$. Solution.

$$H(z) = H_0\sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda}.$$

At $z=1$: $H(1) = H_0\sqrt{0.3 \times 8 + 0.7} = H_0\sqrt{3.1} \approx 1.76\,H_0 \approx 123\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$.

Problem 3. In a matter-dominated universe with small curvature, the Friedmann equation is $H^2 = H_0^2[\Omega_m a^{-3} + (1-\Omega_m)a^{-2}]$. At what scale factor does the curvature term equal the matter term?

Solution. Set $\Omega_m a^{-3} = (1-\Omega_m)a^{-2}$, giving $a = \Omega_m/(1-\Omega_m)$. For $\Omega_m = 0.3$, $a = 0.3/0.7 \approx 0.43$.

Section summary. Friedmann dynamics connects cosmic contents to expansion.

The Origin of Species

Core ideas

The early universe was hot and dense, so particle abundances were set by equilibrium, freeze-out, decays, and nuclear reactions. Big-bang nucleosynthesis predicts light elements; recombination releases the CMB; baryogenesis and dark matter freeze-out are deeper origin questions.

For review, be able to explain thermal equilibrium, freeze-out, nucleosynthesis, recombination, and why relic abundances probe early physics. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\Gamma(T)\sim H(T)\quad\mathrm{at\ freeze\!-\!out},\qquad T\propto a^{-1}$$

Worked example

Estimate the temperature of the universe at photon decoupling (recombination). The ionization energy of hydrogen is $13.6\ \mathrm{eV}$. Recombination occurs roughly when the typical photon energy is a few times below the ionization energy, i.e., $T_{\rm rec} \sim 0.3\ \mathrm{eV}/k_B \approx 3500\ \mathrm{K}$.

Using $T \propto a^{-1}$ and $T_0 = 2.725\ \mathrm{K}$ today, the corresponding redshift is

$$1+z_{\rm rec} = \frac{T_{\rm rec}}{T_0} \approx \frac{3500}{2.725} \approx 1284,$$

so $z_{\rm rec} \approx 1100$, which matches detailed calculations. At this time, the photon mean free path became larger than the Hubble radius and the CMB was released.

Problems with Solutions

Problem 1. The CMB temperature today is $T_0 = 2.725\ \mathrm{K}$. What was the photon temperature at $z = 100$?

Solution. $T(z) = T_0(1+z) = 2.725 \times 101 \approx 275\ \mathrm{K}$.

Problem 2. The number density of CMB photons today is $n_\gamma \approx 410\ \mathrm{cm^{-3}}$. What was it at $z = 1100$?

Solution. Photons dilute as $a^{-3}$ and are conserved (after decoupling), so $n_\gamma(z) = n_{\gamma,0}(1+z)^3$. Thus $n_\gamma(1100) \approx 410 \times (1101)^3 \approx 5.5\times10^{11}\ \mathrm{cm^{-3}}$.

Problem 3. Big-bang nucleosynthesis occurs at $T \sim 80\ \mathrm{keV}$. Convert this to kelvin and estimate the corresponding redshift.

Solution. $T = 80\times10^3\ \mathrm{eV}/(8.617\times10^{-5}\ \mathrm{eV\,K^{-1}}) \approx 9.3\times10^8\ \mathrm{K}$. The redshift is $z \approx T/T_0 \approx 3.4\times10^8$.

Section summary. Cosmic species are fossils of early-universe reaction rates.

The Inhomogeneous Universe: Matter and Radiation

Core ideas

Small perturbations grow into structure. Density contrast, velocity, gravitational potential, and radiation perturbations evolve differently before and after horizon entry. Photon-baryon acoustic oscillations leave signatures in the CMB and matter power spectrum.

For review, be able to define density contrast, explain horizon entry, describe acoustic oscillations, and read a matter power spectrum qualitatively. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\delta(\bm x,t)=\frac{\rho(\bm x,t)-\bar\rho(t)}{\bar\rho(t)},\qquad P(k)=\langle|\delta_{\bm k}|^2\rangle$$

Worked example

In the linear regime, a matter overdensity has $\delta = 0.01$ inside a region of comoving radius $R = 10\ \mathrm{Mpc}$. The mean matter density today is $\bar\rho_m \approx \Omega_m \rho_{\rm crit} \approx 2.5\times10^{-27}\ \mathrm{kg\,m^{-3}}$.

1. The local density is $\rho = \bar\rho(1+\delta) = 1.01\,\bar\rho \approx 2.53\times10^{-27}\ \mathrm{kg\,m^{-3}}$.
2. The excess mass in the region is $$\Delta M = \delta\,\bar\rho \times \frac{4\pi}{3}R^3 = 0.01 \times 2.5\times10^{-27}\ \frac{4\pi}{3} (10\times3.086\times10^{22}\ \mathrm{m})^3 \approx 3.1\times10^{44}\ \mathrm{kg} \approx 1.5\times10^{14}\ M_\odot.$$
3. If the power spectrum at this scale is $P(k) \approx 10^4\ (\mathrm{Mpc}/h)^3$, the variance of the Fourier mode is $\langle|\delta_{\bm k}|^2\rangle = P(k) \approx 10^4\ (\mathrm{Mpc}/h)^3$.

Problems with Solutions

Problem 1. Define the density contrast $\delta$ and explain what $\delta = -0.5$ means physically.

Solution. $\delta = (\rho - \bar\rho)/\bar\rho$. A value of $-0.5$ means the local density is half the cosmic mean; the region is an underdense void.

Problem 2. The matter power spectrum has dimensions of (length)$^3$. Why?

Solution. In three dimensions, the Fourier transform of $\delta(\bm x)$ uses $d^3x$, so $|\delta_{\bm k}|^2$ is dimensionless and $P(k) = \langle|\delta_{\bm k}|^2\rangle$ must have dimensions of $k^{-3}$, i.e., volume.

Problem 3. At what comoving scale does a mode enter the horizon during matter domination?

Solution. The comoving horizon is $\chi_H \approx 2c/H_0\sqrt{\Omega_m}(1+z)^{-1/2}$. At matter-radiation equality ($z_{\rm eq} \approx 3400$), the horizon scale is roughly $\lambda_{\rm eq} \approx 100\ \mathrm{Mpc}$ comoving. Modes with $k > 2\pi/\lambda_{\rm eq}$ entered earlier during radiation domination.

Section summary. Structure formation starts from small coupled matter-radiation perturbations.

The Inhomogeneous Universe: Gravity

Core ideas

Gravity turns perturbations into growing structure. In Newtonian gauge, potentials source motion and lensing; in the subhorizon matter era, density perturbations obey a simple growth equation. Relativistic gauge issues matter on large scales.

For review, be able to use Poisson’s equation for cosmological perturbations, solve qualitative growth, and distinguish physical perturbations from gauge artifacts. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\ddot\delta+2H\dot\delta-4\pi G\bar\rho_m\delta=0,\qquad \nabla^2\Phi=4\pi Ga^2\bar\rho_m\delta$$

Worked example

In a matter-dominated Einstein—de Sitter universe ($a \propto t^{2/3}$), the linear growth equation is

$$\ddot\delta + 2H\dot\delta - 4\pi G\bar\rho_m\delta = 0.$$

Because $H = 2/(3t)$ and $4\pi G\bar\rho_m = 2/(3t^2)$ (from the Friedmann equation), the equation becomes

$$\ddot\delta + \frac{4}{3t}\dot\delta - \frac{2}{3t^2}\delta = 0.$$

Trying a power law $\delta \propto t^n$ gives $n(n-1) + \frac{4}{3}n - \frac{2}{3} = 0$, or $3n^2 + n - 2 = 0$, with solutions $n = 2/3$ (growing mode) and $n = -1$ (decaying mode). Thus the growing mode is $\delta_+(t) \propto a(t)$, and a perturbation that was $\delta = 10^{-5}$ at recombination ($z \approx 1100$) grows to $\delta_0 = 10^{-5}(1+z)^{-1} \approx 10^{-2}$ today.

Problems with Solutions

Problem 1. Show that in a matter-dominated universe, $\delta_+ \propto a$ is a solution of the growth equation.

Solution. With $a \propto t^{2/3}$, $\delta = a$, $\dot\delta = \dot a = Ha$, $\ddot\delta = \dot H a + H\dot a = (\dot H + H^2)a$. Using $\dot H = -4\pi G\bar\rho_m/3 = -H^2/2$ (EdS) gives $\ddot\delta = H^2 a/2$. Substituting into the growth equation:

$$\frac{H^2 a}{2} + 2H(Ha) - 4\pi G\bar\rho_m a = \frac{H^2 a}{2} + 2H^2 a - \frac{3H^2 a}{2} = 0.$$

Problem 2. A density contrast $\delta = 0.02$ exists on a scale of $5\ \mathrm{Mpc}/h$. Use Poisson’s equation to estimate the gravitational potential $\Phi$.

Solution. $\nabla^2\Phi = 4\pi G a^2 \bar\rho_m \delta$. Approximating $\nabla^2 \sim -k^2$ with $k \sim 2\pi/(5\ \mathrm{Mpc}/h) \approx 1.26\,h\ \mathrm{Mpc^{-1}}$, and using $4\pi G a^2\bar\rho_m \approx (3/2)\Omega_m H_0^2 a^{-1}$ today ($a=1$), we get $|\Phi| \sim (3/2)\Omega_m H_0^2 \delta/k^2 \sim 10^{-5}$, consistent with CMB potential wells.

Problem 3. What is the physical difference between the growing and decaying modes?

Solution. The growing mode ($\delta \propto a$) becomes more important with time and seeds structure. The decaying mode ($\delta \propto t^{-1}$ or $a^{-3/2}$ in EdS) redshifts away and is negligible after a short period unless artificially excited by initial conditions.

Section summary. Gravity amplifies primordial perturbations into cosmic structure.

Initial Conditions

Core ideas

Inflation explains why initial perturbations are nearly adiabatic, Gaussian, and scale invariant. The curvature perturbation is the central variable because it remains conserved on superhorizon scales for adiabatic modes. Its power spectrum seeds both CMB anisotropy and structure.

For review, be able to define adiabatic perturbations, read the scalar power spectrum, explain spectral tilt, and connect inflation to horizon and flatness problems. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$P_{\mathcal R}(k)=A_s\left(\frac{k}{k_*}\right)^{n_s-1},\qquad n_s\approx1$$

Worked example

Planck 2018 reports $A_s \approx 2.1\times10^{-9}$ and $n_s \approx 0.965$ at $k_* = 0.05\ \mathrm{Mpc^{-1}}$. Evaluate the primordial curvature power spectrum at $k = 0.002\ \mathrm{Mpc^{-1}}$.

Using

$$P_{\mathcal R}(k) = A_s\left(\frac{k}{k_*}\right)^{n_s-1},$$

we find

$$P_{\mathcal R}(0.002) = 2.1\times10^{-9}\left(\frac{0.002}{0.05}\right)^{-0.035} = 2.1\times10^{-9} \times (0.04)^{-0.035}.$$

Since $\ln(0.04) \approx -3.22$, $(0.04)^{-0.035} = e^{0.113} \approx 1.12$. Thus $P_{\mathcal R}(0.002) \approx 2.35\times10^{-9}$. The slight scale dependence means there is about $12\%$ more power on larger scales (smaller $k$) than at the pivot.

Problems with Solutions

Problem 1. Evaluate $P_{\mathcal R}$ at $k = 0.1\ \mathrm{Mpc^{-1}}$ for $A_s = 2.1\times10^{-9}$, $n_s = 0.965$, $k_* = 0.05\ \mathrm{Mpc^{-1}}$.

Solution. $P_{\mathcal R}(0.1) = 2.1\times10^{-9}(0.1/0.05)^{-0.035} = 2.1\times10^{-9} \times 2^{-0.035} \approx 2.1\times10^{-9} \times 0.976 \approx 2.05\times10^{-9}$.

Problem 2. A model has $n_s = 1$ exactly. What does this imply about the power spectrum?

Solution. $P_{\mathcal R}(k) = A_s$, independent of scale. This is the Harrison—Zel’dovich scale-invariant spectrum.

Problem 3. Inflation solves the horizon problem because a mode that is inside the horizon today was inside the inflationary horizon before inflation ended. Estimate the number of e-folds needed if the current horizon is $14\ \mathrm{Gpc}$ and the Hubble radius at the end of inflation was $10^{-28}\ \mathrm{m}$.

Solution. The ratio of physical scales is $\sim 14\ \mathrm{Gpc}/10^{-28}\ \mathrm{m} \approx 1.4\times10^{52}$. The required e-folds are $N \sim \ln(1.4\times10^{52}) \approx 120$, but because expansion after inflation also stretches scales, only $\sim 50$—$60$ e-folds of inflation are typically needed.

Section summary. Initial conditions are encoded in the primordial curvature power spectrum.

Growth of Structure: Linear Theory

Core ideas

Linear theory describes perturbations while $|\delta|\ll1$. Matter perturbations grow after matter-radiation equality, baryon acoustic features survive statistically, and transfer functions encode the processing of primordial fluctuations.

For review, be able to use growth factors, transfer functions, equality scale, and linear bias at a conceptual and formula level. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$P_m(k,z)=D^2(z)T^2(k)P_{\rm prim}(k),\qquad f=\frac{d\ln D}{d\ln a}$$

Worked example

Suppose the primordial power spectrum is $P_{\rm prim}(k) = A k^{n_s}$ with $A = 2\times10^{-9}$ at a reference scale, and the transfer function at $k = 0.1\ h\ \mathrm{Mpc^{-1}}$ is $T(k) \approx 0.9$. The linear growth factor today is normalized to $D(0)=1$, and at $z=1$ it is $D(1) \approx 0.55$ for $\Lambda$CDM.

The matter power spectrum at $z=1$ is

$$P_m(k,z=1) = D^2(1)\,T^2(k)\,P_{\rm prim}(k) = (0.55)^2 \times (0.9)^2 \times P_{\rm prim}(k) \approx 0.245\,P_{\rm prim}(k).$$

If $P_{\rm prim}(0.1) \approx 2\times10^3\ (\mathrm{Mpc}/h)^3$ (after normalization to match observed large-scale amplitude), then $P_m \approx 490\ (\mathrm{Mpc}/h)^3$. The growth rate is

$$f = \frac{d\ln D}{d\ln a} \approx \Omega_m(z)^{0.55} \approx 0.89$$

at $z=1$ for $\Omega_m = 0.3$.

Problems with Solutions

Problem 1. If $D(z=0)=1$ and $D(z=1)=0.55$, by what factor have perturbations grown between $z=1$ and today?

Solution. The ratio is $D(0)/D(1) = 1/0.55 \approx 1.82$. Perturbations grew by about $82\%$.

Problem 2. A galaxy survey measures $\delta_g = 0.1$ in a region. If linear bias is $b = 1.5$, what is the underlying matter overdensity?

Solution. Using $\delta_g = b\,\delta_m$, we get $\delta_m = \delta_g/b = 0.1/1.5 \approx 0.067$.

Problem 3. The comoving wavenumber entering the horizon at matter-radiation equality is $k_{\rm eq} \approx 0.01\ (\Omega_m h^2)\ \mathrm{Mpc^{-1}}$. Evaluate $k_{\rm eq}$ for $\Omega_m h^2 = 0.14$.

Solution. $k_{\rm eq} \approx 0.01 \times 0.14\ \mathrm{Mpc^{-1}} = 1.4\times10^{-3}\ \mathrm{Mpc^{-1}}$, corresponding to a comoving scale $\lambda_{\rm eq} \approx 2\pi/k_{\rm eq} \approx 4500\ \mathrm{Mpc}$. (More precise calculations give $k_{\rm eq} \sim 0.015\,h\ \mathrm{Mpc^{-1}}$, i.e., $\sim 400\ \mathrm{Mpc}/h$, but this illustrates the scaling.)

Section summary. Linear theory links primordial fluctuations to large-scale structure.

The Cosmic Microwave Background

Core ideas

The CMB is radiation from last scattering. Its temperature anisotropies come from acoustic oscillations, gravitational redshift, Doppler motion, diffusion damping, and projection effects. Peak positions measure geometry; peak heights measure contents.

For review, be able to interpret the CMB angular power spectrum, name the Sachs-Wolfe and acoustic effects, and connect peaks to parameters. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\frac{\Delta T}{T}(\hat{\bm n})=\sum_{\ell m}a_{\ell m}Y_{\ell m}(\hat{\bm n}),\qquad C_\ell=\langle|a_{\ell m}|^2\rangle$$

Worked example

The first acoustic peak of the CMB temperature power spectrum is observed at $\ell_1 \approx 220$. The angular diameter distance to last scattering in a flat $\Lambda$CDM model is $d_A \approx 14\ \mathrm{Gpc}$. The physical sound horizon at recombination is $r_s \approx 150\ \mathrm{Mpc}$.

The angular scale of the sound horizon is

$$\theta_s = \frac{r_s}{d_A} \approx \frac{150}{14\times10^3} \approx 1.07\times10^{-2}\ \mathrm{rad} \approx 0.61^\circ.$$

The corresponding multipole is

$$\ell \approx \frac{\pi}{\theta_s} \approx \frac{\pi}{1.07\times10^{-2}} \approx 293.$$

A more accurate treatment including projection effects, early integrated Sachs—Wolfe, and the precise sound speed shifts this to $\ell_1 \approx 220$, but the estimate captures the basic geometry.

Problems with Solutions

Problem 1. The temperature anisotropy $\Delta T/T \sim 10^{-5}$ corresponds to what temperature fluctuation in $\mu\mathrm{K}$ at $T_0 = 2.725\ \mathrm{K}$?

Solution. $\Delta T = 10^{-5} \times 2.725\ \mathrm{K} = 2.725\times10^{-5}\ \mathrm{K} = 27.25\ \mu\mathrm{K}$.

Problem 2. Estimate the angular separation on the sky corresponding to $\ell = 1000$.

Solution. $\theta \approx \pi/\ell \approx \pi/1000 \approx 3.14\times10^{-3}\ \mathrm{rad} \approx 0.18^\circ \approx 10.8'$.

Problem 3. The $a_{\ell m}$ coefficients for $\ell = 2$ are measured as $a_{20}=1.0\times10^{-5}$, $a_{21}=0.5\times10^{-5}$, $a_{2,-1}=-0.3\times10^{-5}$, etc. What is the ensemble average $\langle|a_{2m}|^2\rangle$ if all five modes have the same variance?

Solution. For a given $\ell$, $\langle|a_{\ell m}|^2\rangle = C_\ell$. If the variance is the same for all $m$, then $C_2 = \langle|a_{2m}|^2\rangle$. If the measured values are one realization, the estimator is $\hat C_2 = (2\ell+1)^{-1}\sum_m |a_{2m}|^2$. With hypothetical equal variance $|a_{2m}|^2 = 10^{-10}$ for all five modes, $\hat C_2 = 10^{-10}$.

Section summary. The CMB is a precision image of early-universe perturbations.

The Polarized CMB

Core ideas

CMB polarization is produced by Thomson scattering of radiation with a quadrupole anisotropy. Scalar perturbations generate E-modes; tensor perturbations can generate primordial B-modes. Lensing converts some E-mode power into B-modes.

For review, be able to distinguish E and B polarization, explain Thomson scattering origin, and state why B-modes are important. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\mathrm{scalar}\Rightarrow E,\qquad \mathrm{tensor/lensing}\Rightarrow B,\qquad C_\ell^{EE},C_\ell^{BB}$$

Worked example

Thomson scattering of CMB photons off free electrons produces linear polarization with a characteristic $E$-mode pattern. The polarization amplitude is roughly $10\%$ of the temperature quadrupole anisotropy at last scattering. If the temperature quadrupole is $\Delta T_\ell \sim 20\ \mu\mathrm{K}$ at the relevant angular scales, the polarization amplitude is

$$P \sim 0.1 \times \frac{20\ \mu\mathrm{K}}{2.725\ \mathrm{K}} \sim 7\times10^{-7}.$$

In terms of $C_\ell$, the first peak of the $EE$ power spectrum is $C_\ell^{EE} \approx 0.1\, C_\ell^{TT}$ near $\ell \sim 300$. For scalar perturbations, the polarization pattern is purely $E$-mode; primordial $B$-modes from tensor perturbations are predicted at a much lower level, with $r = C_\ell^{BB}/C_\ell^{TT} \sim 0.01$ or less.

Problems with Solutions

Problem 1. Why does Thomson scattering produce polarization only when the radiation has a quadrupole anisotropy?

Solution. In the electron rest frame, Thomson scattering is symmetric for unpolarized radiation only if the incoming intensity is isotropic. A dipole anisotropy produces no net polarization when averaged over all scattering directions. A quadrupole intensity pattern creates a linear polarization because scattering perpendicular to the hot and cold directions differs, leaving a net linear polarization in the scattered radiation.

Problem 2. If the tensor-to-scalar ratio is $r = 0.05$ and $C_\ell^{TT} \approx 5\times10^{-10}$ at $\ell = 80$, estimate $C_\ell^{BB}$ from primordial tensors at that scale.

Solution. Roughly, $C_\ell^{BB} \approx (r/10)\,C_\ell^{TT}$ near the reionization bump. For $r=0.05$, $C_\ell^{BB} \approx 0.005 \times 5\times10^{-10} = 2.5\times10^{-12}$. This is well below current detection limits ($r \lesssim 0.03$ from BICEP/Keck).

Problem 3. Lensing converts $E$-modes to $B$-modes. If the lensing potential has power $C_\ell^{\phi\phi} \approx 10^{-7}$ at $\ell \sim 100$, why is the lensing $B$-mode considered a foreground for primordial tensor searches?

Solution. Gravitational lensing by large-scale structure recombines $E$-mode polarization into $B$-modes, producing a roughly scale-invariant lensing $B$-mode power spectrum with amplitude $C_\ell^{BB,\rm lens} \sim 10^{-5}\,\mu\mathrm{K}^2$ at $\ell \sim 1000$. This is larger than the primordial tensor $B$-mode for $r \lesssim 0.01$, making it a confusion signal that must be modeled and subtracted.

Section summary. CMB polarization adds geometry and tensor information beyond temperature.

Probes of Structure I: Tracers

Core ideas

Galaxies, quasars, clusters, and the Lyman-alpha forest trace matter imperfectly. Bias, redshift-space distortions, selection functions, and shot noise must be modeled to infer the underlying density field.

For review, be able to define bias, correlation functions, redshift-space distortions, and the relation between tracers and matter. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\delta_g=b\,\delta_m,\qquad \xi(r)=\langle\delta(\bm x)\delta(\bm x+\bm r)\rangle$$

Worked example

A galaxy survey measures the two-point correlation function $\xi_{gg}(r)$. At large scales, the galaxy overdensity is $\delta_g = b\,\delta_m$ with bias $b = 1.8$. The matter power spectrum at $k = 0.1\ h\ \mathrm{Mpc^{-1}}$ is $P_m(k) \approx 2\times10^4\ (h^{-1}\mathrm{Mpc})^3$.

1. The galaxy power spectrum is $P_g(k) = b^2 P_m(k) = (1.8)^2 \times 2\times10^4 \approx 6.5\times10^4\ (h^{-1}\mathrm{Mpc})^3$.
2. The real-space correlation function at separation $r = 10\ h^{-1}\mathrm{Mpc}$ can be estimated by Fourier transforming $P_g(k)$, but a rough estimate is $\xi(r) \sim (b\,\sigma_8)^2 (r/8\ h^{-1}\mathrm{Mpc})^{-1.8}$, giving $\xi(10) \sim 0.5$ for typical values.
3. Shot noise adds a constant $1/\bar n$ to $P_g(k)$. If the number density is $\bar n = 3\times10^{-4}\ (h^{-1}\mathrm{Mpc})^{-3}$, the shot noise is $1/\bar n \approx 3.3\times10^3\ (h^{-1}\mathrm{Mpc})^3$, non-negligible on small scales.

Problems with Solutions

Problem 1. If $\delta_m = 0.02$ in a region and linear bias is $b=1.2$, what is the expected galaxy overdensity?

Solution. $\delta_g = b\,\delta_m = 1.2 \times 0.02 = 0.024$.

Problem 2. Explain why redshift-space distortions cause the clustering of galaxies to appear anisotropic.

Solution. Along the line of sight, peculiar velocities add to the Hubble flow, compressing structures on small scales (fingers of God) and producing a characteristic quadrupole anisotropy on large scales (Kaiser effect). Transverse distances are unaffected, breaking isotropy.

Problem 3. The correlation function is $\xi(r) = \langle\delta(\bm x)\delta(\bm x+\bm r)\rangle$. What is $\xi(0)$ if the field has variance $\sigma^2$?

Solution. $\xi(0) = \langle\delta^2(\bm x)\rangle = \sigma^2$, the variance of the density field.

Section summary. Observed tracers are biased maps of the matter distribution.

Probes of Structure II: Gravitational Lensing

Core ideas

Lensing maps projected mass by measuring deflection, shear, convergence, and magnification. Weak lensing statistically measures large-scale structure; strong lensing probes dense systems; CMB lensing maps matter to high redshift.

For review, be able to define convergence and shear, connect lensing to projected potential, and distinguish weak and strong regimes. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\kappa=\frac{\Sigma}{\Sigma_{\rm crit}},\qquad \Sigma_{\rm crit}=\frac{c^2}{4\pi G}\frac{D_s}{D_lD_{ls}}$$

Worked example

A foreground cluster at $z_l = 0.3$ lenses a background galaxy at $z_s = 1.0$. In a flat $\Lambda$CDM model with $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$, the angular diameter distances are approximately $D_l \approx 1.2\ \mathrm{Gpc}$, $D_s \approx 3.3\ \mathrm{Gpc}$, and $D_{ls} \approx 2.3\ \mathrm{Gpc}$.

The critical surface mass density is

$$\Sigma_{\rm crit} = \frac{c^2}{4\pi G}\frac{D_s}{D_l D_{ls}} = \frac{(3\times10^8)^2}{4\pi(6.674\times10^{-11})} \frac{3.3}{1.2\times2.3\times10^{27}} \approx 1.7\times10^{15}\ \mathrm{M_\odot\,Mpc^{-2}} \approx 2\ \mathrm{g\,cm^{-2}}.$$

If the cluster has a projected mass density $\Sigma = 3\ \mathrm{g\,cm^{-2}}$ at a certain radius, the convergence is $\kappa = \Sigma/\Sigma_{\rm crit} = 1.5$, indicating strong lensing (multiple images possible).

Problems with Solutions

Problem 1. Evaluate $\Sigma_{\rm crit}$ for a lens at $z_l = 0.5$ and source at $z_s = 2.0$ using $D_l \approx 1.7\ \mathrm{Gpc}$, $D_s \approx 5.3\ \mathrm{Gpc}$, $D_{ls} \approx 3.9\ \mathrm{Gpc}$.

Solution. $\Sigma_{\rm crit} = \frac{c^2}{4\pi G}\frac{D_s}{D_l D_{ls}} \approx \frac{9\times10^{16}}{4\pi(6.674\times10^{-11})}\frac{5.3}{1.7\times3.9\times10^{27}} \approx 8.5\times10^{15}\ \mathrm{kg\,m^{-2}} \approx 0.85\ \mathrm{g\,cm^{-2}}$.

Problem 2. A circular source has intrinsic radius $R_{\rm src} = 1''$. If the magnification is $\mu = 4$, what is the observed angular area?

Solution. Area magnification is $|\mu|$. The intrinsic area is $\pi (1'')^2$. The observed area is $4\pi (1'')^2 \approx 12.6\ \mathrm{arcsec^2}$. The observed radius is $2''$.

Problem 3. Weak lensing measures shear $\gamma \sim 0.02$ for a population of galaxies. If the mean ellipticity dispersion is $\sigma_\epsilon \approx 0.3$, how many galaxies $N$ are needed to measure the shear with signal-to-noise $S/N = 5$?

Solution. The uncertainty on the mean shear is $\sigma_\epsilon/\sqrt{N}$. Setting $0.02/(0.3/\sqrt{N}) = 5$ gives $\sqrt{N} = 5 \times 0.3 / 0.02 = 75$, so $N \approx 5625$ galaxies.

Section summary. Lensing observes mass through its effect on light paths.

Probes of Structure III: Nonlinear Growth

Core ideas

On small scales, density contrasts become nonlinear. Spherical collapse, halo formation, virialization, N-body simulations, halo mass functions, and baryonic feedback describe the transition from smooth perturbations to galaxies and clusters.

For review, be able to explain nonlinear collapse, virial equilibrium, halo profiles, and why simulations are required. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$2K+U=0,\qquad \rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2}$$

Worked example

A galaxy cluster of mass $M = 10^{15}\ M_\odot$ virializes at $z=0$ with virial radius $R_{\rm vir} \approx 2\ \mathrm{Mpc}$. The circular velocity is

$$v_c^2 = \frac{GM}{R_{\rm vir}} = \frac{(6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}})(2\times10^{45}\ \mathrm{kg})}{2\times3.086\times10^{22}\ \mathrm{m}} \approx 2.2\times10^{12}\ \mathrm{m^2\,s^{-2}},$$

so $v_c \approx 1500\ \mathrm{km\,s^{-1}}$. The virial temperature for an ideal monatomic gas is

$$T_{\rm vir} = \frac{\mu m_p v_c^2}{2k_B} \approx \frac{0.6 \times 1.67\times10^{-27}\ \mathrm{kg} \times 2.2\times10^{12}\ \mathrm{m^2\,s^{-2}}}{2 \times 1.38\times10^{-23}\ \mathrm{J\,K^{-1}}} \approx 8\times10^7\ \mathrm{K} \approx 7\ \mathrm{keV}.$$

This temperature matches the X-ray emission observed from hot intracluster gas. The virial theorem $2K + U = 0$ ensures the cluster is in dynamical equilibrium.

Problems with Solutions

Problem 1. For a self-gravitating system in virial equilibrium, show that the total energy is $E = -K$.

Solution. The virial theorem states $2K + U = 0$, so $U = -2K$. The total energy is $E = K + U = K - 2K = -K$.

Problem 2. An NFW halo has scale density $\rho_s = 10^7\ M_\odot\ \mathrm{kpc^{-3}}$ and scale radius $r_s = 200\ \mathrm{kpc}$. Compute the density at $r = 2r_s$.

Solution. Using $\rho_{\rm NFW}(r) = \rho_s / [(r/r_s)(1+r/r_s)^2]$, at $r=2r_s$:

$$\rho = \frac{10^7}{2 \times (1+2)^2} = \frac{10^7}{18} \approx 5.6\times10^5\ M_\odot\ \mathrm{kpc^{-3}}.$$

Problem 3. In an Einstein—de Sitter universe, spherical collapse turnaround occurs at redshift $z_{\rm ta}$. What is the collapse redshift $z_{\rm coll}$?

Solution. In EdS, parametric time gives $t_{\rm coll} = 2t_{\rm ta}$. Since $t \propto (1+z)^{-3/2}$ during matter domination,

$$(1+z_{\rm coll})^{-3/2} = 2(1+z_{\rm ta})^{-3/2} \quad\Rightarrow\quad 1+z_{\rm coll} = \frac{1+z_{\rm ta}}{2^{2/3}} \approx \frac{1+z_{\rm ta}}{1.587}.$$

For $z_{\rm ta}=1$, $z_{\rm coll} \approx 0.26$.

Section summary. Nonlinear structure requires collapse physics and simulations.