Cheat sheetsCosmology Cheat Sheet
Cosmology Cheat Sheet
Reference tables, formulas, and small calculators. A grab-bag of reference material I find myself looking up often.
Planck 2018 Best Fit
Best-fit ΛCDM parameters from Planck 2018
68% confidence limits for the base ΛCDM model from the TT,TE,EE+lowE+lensing+BAO likelihood. Adapted from Wikipedia.
| Parameter | Symbol | Value |
|---|---|---|
| Baryon density | $\Omega_b h^2$ | 0.02242 ± 0.00014 |
| Cold dark matter density | $\Omega_c h^2$ | 0.11933 ± 0.00091 |
| 100 × approximation to $r_s/D_A$ (CosmoMC) | $100\,\theta_{\rm MC}$ | 1.04101 ± 0.00029 |
| Reionization optical depth | $\tau$ | 0.0561 ± 0.0071 |
| Curvature power spectrum amplitude | $\ln(10^{10} A_s)$ | 3.047 ± 0.014 |
| Scalar spectral index | $n_s$ | 0.9665 ± 0.0038 |
| Hubble constant (km s$^{-1}$ Mpc$^{-1}$) | $H_0$ | 67.66 ± 0.42 |
| Dark energy density | $\Omega_\Lambda$ | 0.6889 ± 0.0056 |
| Matter density | $\Omega_m$ | 0.3111 ± 0.0056 |
| Density fluctuations at $8 h^{-1}$ Mpc | $S_8 = \sigma_8(\Omega_m/0.3)^{1/2}$ | 0.825 ± 0.011 |
| Reionization redshift | $z_{\rm re}$ | 7.82 ± 0.71 |
| Age of the Universe (Gyr) | $t_0$ | 13.787 ± 0.020 |
| Decoupling redshift | $z_*$ | 1089.80 ± 0.21 |
| Comoving sound horizon at $z_*$ (Mpc) | $r_*$ | 144.57 ± 0.22 |
| 100 × angular scale of sound horizon at last scatter | $100\,\theta_*$ | 1.04119 ± 0.00029 |
| Baryon-drag redshift | $z_{\rm drag}$ | 1060.01 ± 0.29 |
| Comoving sound horizon at $z_{\rm drag}$ (Mpc) | $r_{\rm drag}$ | 147.21 ± 0.23 |
Timeline
Important events in cosmic history
| Event | Time $t$ | Redshift $z$ | Temperature $T$ |
|---|---|---|---|
| EW phase transition | 20 ps | $10^{15}$ | 100 GeV |
| QCD phase transition | 30 µs | $10^{12}$ | 150 MeV |
| Neutrino decoupling | 1 s | $6 \times 10^9$ | 1 MeV |
| $e^+ e^-$ annihilation | 6 s | $2 \times 10^9$ | 500 keV |
| BBN | 3 min | $4 \times 10^8$ | 100 keV |
| Matter-radiation eq. | 60 kyr | 3400 | 0.75 eV |
| Recombination | 260–380 kyr | 1100–1400 | 0.26–0.33 eV |
| Photon decoupling | 380 kyr | 1000–1200 | 0.23–0.28 eV |
| Reionization | 100–400 Myr | 11–30 | 2.6–7.0 meV |
| Matter-Λ equality | 9 Gyr | 0.4 | 0.33 meV |
| Present | 13.8 Gyr | 0 | 0.24 meV |
Dynamics
Scalings and Friedmann formulas
Scalings in Each Era
| Radiation domination | Matter domination | $\Lambda$ domination | |
|---|---|---|---|
| $a(t)$ | $t^{1/2}$ | $t^{2/3}$ | $\exp(H_0 t)$ |
| $\eta(a)$ | $\propto a$ | $\propto a^{1/2}$ | $-1/(aH)$ |
| $H(a)$ | $\propto a^{-2}$ | $\propto a^{-3/2}$ | const. |
In Λ domination $\eta = \int dt/a$ asymptotes to a finite future conformal time; the form above is the leading $a \to \infty$ behavior.
Formula Sheet
$$ \begin{aligned} \frac{dz}{dt} &= -H(z)\,(1+z) \\[4pt] \frac{d}{d\eta} &= a^2 H \,\frac{d}{da} \\[4pt] \frac{H^2}{H_0^2} &= \Omega_{r,0}\,a^{-4} + \Omega_{m,0}\,a^{-3} + \Omega_{k,0}\,a^{-2} + \Omega_{\Lambda,0} \end{aligned} $$Critical density today: $\rho_{\rm crit} = 3 H_0^2 / (8\pi G) \approx 1.878\times 10^{-29}\,h^2\;{\rm g\,cm^{-3}}$.
CMB temperature scaling: $T(z) = T_0\,(1+z)$ with $T_0 = 2.7255\,{\rm K}$.
Particle Physics
Mandelstam identities and Standard Model masses
Mandelstam Variable Identities
For $2 \to 2$ scattering with incoming momenta and masses $(p_1, m_1), (p_2, m_2)$ and outgoing $(k_1, m_1'), (k_2, m_2')$:
$$ \begin{aligned} p_1 \cdot p_2 &= \tfrac{1}{2}\left(s - m_1^2 - m_2^2\right) \\ k_1 \cdot k_2 &= \tfrac{1}{2}\left(s - m_1'^2 - m_2'^2\right) \\ p_1 \cdot k_1 &= \tfrac{1}{2}\left(m_1^2 + m_1'^2 - t\right) \\ p_2 \cdot k_2 &= \tfrac{1}{2}\left(m_2^2 + m_2'^2 - t\right) \\ p_1 \cdot k_2 &= \tfrac{1}{2}\left(m_1^2 + m_2'^2 - u\right) \\ p_2 \cdot k_1 &= \tfrac{1}{2}\left(m_2^2 + m_1'^2 - u\right) \end{aligned} $$with $s + t + u = m_1^2 + m_2^2 + m_1'^2 + m_2'^2$.
Standard Model Masses and Degrees of Freedom
| Type | Mass | Spin | $g$ | |
|---|---|---|---|---|
| Quarks | $t, \bar t$ | 173 GeV | $1/2$ | $2 \cdot 2 \cdot 3 = 12$ |
| $b, \bar b$ | 4 GeV | |||
| $c, \bar c$ | 1 GeV | |||
| $s, \bar s$ | 100 MeV | |||
| $d, \bar d$ | 5 MeV | |||
| $u, \bar u$ | 2 MeV | |||
| Gluons | $g_i$ | 0 | 1 | $8 \cdot 2 = 16$ |
| Charged lep. | $\tau^\pm$ | 1777 MeV | $1/2$ | $2 \cdot 2 = 4$ |
| $\mu^\pm$ | 106 MeV | |||
| $e^\pm$ | 511 keV | |||
| Neutrinos | $\nu_\tau, \bar\nu_\tau$ | < 0.6 eV | $1/2$ | $2 \cdot 1 = 2$ |
| $\nu_\mu, \bar\nu_\mu$ | < 0.6 eV | |||
| $\nu_e, \bar\nu_e$ | < 0.6 eV | |||
| Gauge bosons | $W^+$ | 80 GeV | 1 | 3 |
| $W^-$ | 80 GeV | |||
| $Z^0$ | 91 GeV | |||
| $\gamma$ | 0 | 2 | ||
| Higgs | $H^0$ | 125 GeV | 0 | 1 |
Above all SM thresholds: $g_b = 28$, $g_f = 90$, so $g_* = g_b + \tfrac{7}{8} g_f = 106.75$.
Unit Converter
Convert between natural-unit expressions
Convert between any units. To convert to a dimensionless number,
input GeV^0 as the output unit.
Distance Calculator
Cosmological distances, lookback time, and age at redshift
Computes distances and times from a flat (or curved) FRW model. Defaults to Planck 2018 ΛCDM. $\Omega_k$ is inferred as $1 - \Omega_m - \Omega_\Lambda - \Omega_r$.
The numerical integration uses composite Simpson’s rule on $\int_0^z dz'/E(z')$ with $E(z) = \sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}$. Curvature handling follows Hogg, Distance measures in cosmology (astro-ph/9905116).