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Planck 2018 Best Fit Cosmology
Adapted from Wikipedia
| Parameter | Symbol | TT,TE,EE+lowE +lensing 68% limits | TT,TE,EE+lowE +lensing+BAO 68% limits |
|---|
| Baryon density | $\Omega_b h^2$ | 0.02237±0.00015 | 0.02242±0.00014 |
| Cold dark matter density | $\Omega_c h^2$ | 0.1200±0.0012 | 0.11933±0.00091 |
| 100x approximation to rs / DA (CosmoMC) | $100\theta_{MC}$ | 1.04092±0.00031 | 1.04101±0.00029 |
| Thomson scattering optical depth due to reionization | $\tau$ | 0.0544±0.0073 | 0.0561±0.0071 |
| Power spectrum of curvature perturbations | $\ln(10^{10}A_s)$ | 3.044±0.014 | 3.047±0.014 |
| Scalar spectral index | $n_s$ | 0.9649±0.0042 | 0.9665±0.0038 |
| Hubble’s constant (km s$^{-1}$ Mpc$^{-1}$) | $H_0$ | 67.36±0.54 | 67.66±0.42 |
| Dark energy density | $\Omega_\Lambda$ | 0.6847±0.0073 | 0.6889±0.0056 |
| Matter density | $\Omega_m$ | 0.3153±0.0073 | 0.3111±0.0056 |
| Density fluctuations at 8h$^{-1}$ Mpc | $S_8 = \sigma_8(\Omega_m/0.3)^{0.5}$ | 0.832±0.013 | 0.825±0.011 |
| Redshift of reionization | $z_{re}$ | 7.67±0.73 | 7.82±0.71 |
| Age of the Universe (Gy) | $t_0$ | 13.797±0.023 | 13.787±0.020 |
| Redshift at decoupling | $z_*$ | 1089.92±0.25 | 1089.80±0.21 |
| Comoving size of the sound horizon at z = z* (Mpc) | $r_*$ | 144.43±0.26 | 144.57±0.22 |
| 100× angular scale of sound horizon at last-scattering | $100\theta_*$ | 1.04110±0.00031 | 1.04119±0.00029 |
| Redshift with baryon-drag optical depth = 1 | $z_{\rm drag}$ | 1059.94±0.30 | 1060.01±0.29 |
| Comoving size of the sound horizon at z = z_drag | $r_{\rm drag}$ | 147.09±0.26 | 147.21±0.23 |
| Legend: | | | |
- 68% limits: Parameter 68% confidence limits for the base ΛCDM model
- TT, TE, EE: Planck Cosmic microwave background (CMB) power spectra; TT represents temperature power spectrum, TE is temperature-polarization cross spectrum, and EE is polarisation power spectrum
- lowE: Planck polarization data in the low-ℓ likelihood
- lensing: CMB lensing reconstruction
- BAO: Baryon acoustic oscillations, JLA: Joint Light-curve Analysis (of supernovae), H0: Hubble constant
Important Events in Cosmology
| Event | time $t$ | Redshift $z$ | Temperature $T$ |
|---|
| EW Phase Transition | 20 ps | $10^{15}$ | 100 GeV |
| QCD Phase Transition | 30 $\mu$s | $10^{12}$ | 150 MeV |
| Neutrino Decoupling | 1 s | $6\times 10^9$ | 1 MeV |
| $e^+ e^-$ annihilation | 6 s | $2\times 10^9$ | |
| BBN | 3 min | $4 \times 10^8$ | 100 keV |
| Equality | 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 |
| DM-$\Lambda$ Equality | 9 Gyr | 0.4 | 0.33 meV |
| Present | 13.8 Gyr | 0 | 0.24 meV |
Scalings
| Radiation Domination | Matter Domination | $\Lambda$ Domination | |
|---|
| $a(t)$ | $t^{1/2}$ | $t^{2/3}$ | $\exp(H_0 t)$ | |
| $\eta(a)$ | $a$ | $a^{1/2}$ | | |
| $H(a)$ | $a^{-2}$ | $a^{-3/2}$ | const. | |
| | | | |
$$
\begin{align*}
\frac{dz}{dt} &= H(z) (1+z)\\
\frac{d}{d\eta} &= a^2 H \frac{d}{da}\\
\frac{H^2}{H_0^2}&= \Omega_{r,0}\alpha^{-4} + \Omega_{0,m}\alpha^{-3} + \Omega_{0,k}\alpha^{-2}+\Omega_{0,\Lambda} \\
\end{align*}
$$
Particle Physics
Mandelstam Variable Identities
Identities for $2\to 2$ scattering where the incoming particles have momenta and masses $p_1, p_2$ and $m_1, m_2'$ respectively. The outgoing particles have momenta and masses $k_1, k_2$ and $m_1', m_2'$.
$$
\begin{align*}
p_1 \cdot p_2 & = s - m_1^2 - m_2^2 \\
k_1 \cdot k_2 & = s - m_1'^2 - m_2'^2 \\
p_1 \cdot k_1 & = m_1^2 + m_1'^2 - t \\
p_2 \cdot k_2 & = m_2^2 + m_2'^2 - t \\
p_1 \cdot k_2 &= m_1^2 - m_2'^2 - u \\
p_2 \cdot k_1 &= m_2^2 - m_1'^2 - u
\end{align*}
$$
### Particle Masses and Degrees of Freedom
| Type | | mass | spin | $g$ |
|---|
| quarks | $t, \overline{t}$ | 173 GeV | $1/2$ | $2\cdot 2\cdot 3=12$ |
| $b, \overline{b}$ | 4 GeV | | |
| $c, \overline{c}$ | 1 GeV | | |
| $s, \overline{s}$ | 100 MeV | | |
| $d, \overline{d}$ | 5 MeV | | |
| $u, \overline{u}$ | 2 MeV | | |
| gluons | $g_i$ | 0 | 1 | $8 \cdot 2=16$ |
| leptons | $\tau^\pm$ | 1777 MeV | $1/2$ | $2\cdot 2 =4$ |
| $\mu^\pm$ | 106 MeV | | |
| $e^\pm$ | 511 keV | | |
| $\nu_\tau, \overline{\nu}_\tau$ | < 0.6 eV | $1/2$ | $2\cdot 1 = 2$ |
| $\nu_\mu, \overline{\nu}_\mu$ | < 0.6 eV | | |
| $\nu_e, \overline{\nu}_e$ | < 0.6 eV | | |
| gauge bosons | $W^+$ | 80 GeV | 1 | 3 |
| $W^-$ | 80 GeV | | |
| $Z^0$ | 91 GeV | | |
| $\gamma$ | 0 | | |
| Higgs boson | $H^0$ | 125 GeV | 0 | 1 |
| Therefore, $g_b=28$ and $g_f=90$ so $g_*=g_b +\frac{7}{8}g_f=106.75$. | | | | |