# Latest cosmological parameters

I'm looking for the latest values (with uncertainties) of the four main cosmological density parameters $\Omega_i$ : \begin{align}\tag{1} \Omega_{\text{mat}} &={} ?, &\Omega_{\text{rad}} &={} ?, &\Omega_{\Lambda} &={} ?, &\Omega_{k} &={} ?. \end{align} I know that $\Omega_{\text{mat}} \approx 0.30$, $\Omega_{\text{rad}} \approx 0.00$, $\Omega_{\Lambda} \approx 0.70$ and $\Omega_{k} \approx 0.00$, but I would like to have more precise values (with uncertainties, if possible). Take note that these parameters are constrained by the following relation : $$\tag{2} \Omega_{\text{mat}} + \Omega_{\text{rad}} + \Omega_{\Lambda} + \Omega_{k} \equiv 1.$$

Of course, I checked Wikipedia but I don't trust it very much :

https://en.wikipedia.org/wiki/Lambda-CDM_model

I've also checked on arXiv. For example :

but I don't find clear final and consensual values in this paper.

• What makes you think that you will find consensual answers? Also note that the sum of the density parameters does not necessarily equal unity. Dec 14 '16 at 10:53
• @RobJeffries, what do you mean the sum of the densities does not necessarily equal unity ? Unless there's another form of energy or the Friedmann-Lemaître equations aren't good, they should add to 1.
– Cham
Dec 14 '16 at 14:24
• They sum to 1 in a flat universe. Dec 14 '16 at 17:47
• Whether or not they sum to 1 is an experimental result, not a given. All we know for sure now is that they seem to be pretty close to 1.00 May 29 '18 at 12:36

Cosmological parameters are measured in a variety of ways, and their values will depend on which measurements you trust the most. The paper you link to (Planck Collaboration et al. 2016) with the 2015 results from the Planck observations of the cosmic microwave background is probably the one that most people will accept, but even in that paper you will find different values, depending on which observables you combine.

You will find the values in their Table 4. I think that most people use the values in the column called "TT+lowP+lensing" (e.g. Geil et al. 2016, Ricotti et al. 2016, and Liu et a. 2016), which is the "conservative" choice. However, you'll also find some (e.g. Chevallard & Charlot 2016 and Silk 2016) who use the values in the last column, called "TT,TE,EE+lowP+lensing+ext". These values takes into account external data (baryonic acoustic oscillations and supernovae data), which reduce the uncertainties, arguably to unnaturally small values. The TT, TE, TT, and lowP refer to the polarization maps used, and "lensing" to the weak gravitational lensing measurements by Planck.

## Standard cosmological parameters

The table below is a modified version from the Planck paper where I show only the most used parameters:

Here, $n_s$ is the slope of the primordial power spectrum, $H_0$ is the Hubble constant in km s–1 Mpc–1, $\Omega_\Lambda$ and $\Omega_m$ are the density parameters of dark energy and total (dark+baryonic) matter, $\Omega_\mathrm{b}h^2$ and $\Omega_\mathrm{c}h^2$ are the density parameters of baryonic and dark matter, multiplied by the factor $h \equiv H_0/100$ (squared), $\sigma_8$ is the matter density fluctuations on scales of 8 (comoving) Mpc, $z_\mathrm{re}$ is the redshift at which the Universe was reionized (assuming instant reionization), and the last row shows the inferred age of the Universe in billion years.

## Curvature density

The constraints on the curvature parameter $\Omega_K$ is given in Table 5, which has somewhat different combinations of data. All in all, Planck constrains the curvature to $|\Omega_K| < 0.005$, but you will rarely offend anyone by simply setting the curvature to zero.

The radiation density is a bit more convoluted. It has a contribution from both photons and neutrinos, and their densities are related as $$\rho_\nu = N_\mathrm{eff} \frac{7}{8} \left(\frac{4}{11}\right)^{4/3}\rho_\gamma,$$ where $N_\mathrm{eff} = 3.046$ is the effective number of neutrino species. Following the procedure by Pulsar in this answer, but with updated parameters (i.e. the $N_\mathrm{eff}$ given above, and the average CMB temperature of $T_0 = 2.722\pm0.027$ (Eq. 83a)), I get that $$\begin{array}{rcl} \Omega_\mathrm{rad}h^2 & = & \Omega_\nu h^2+ \Omega_\gamma h^2 \\ & = & (1.7018 + 4.6213) \times 10^{-5} \\ & = & 4.1620\times10^{-5}, \end{array}$$ that is, with $h = 0.6781$, $$\Omega_\mathrm{rad} = 9.0513\times10^{-5}.$$

## Recap

So, answering your question is a little difficult, as there's no single answer, and since the curvature is given with 95% confidence ("2$\sigma$"), rather than 68% ("1$\sigma$"). For the Hubble constant and the matter and dark energy, I'd recommend $H_0 = 67.81\pm0.92 \,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, $\Omega_\mathrm{m} = 0.308\pm0.012$ and $\Omega_\Lambda = 0.692\pm0.012$.

For curvature, I would use 0 (especially because your calculations probably would need to switch between ordinary and hyperbolic trigonometry depending on the sign), but if you do want to include uncertainty, you can say $\Omega_K=0\pm0.005$ (95%). Or, you could simply use $\Omega_K = 1 - \Omega_\mathrm{m} - \Omega_\Lambda - \Omega_\mathrm{rad}$ and propagate the errors to get $\Omega_K = 0\pm0.017$, which will give you a more conservative value.

For radiation, to propagate uncertainties you would have to know the covariance matrix of the input parameters, but since the total error is dominated by that of the Hubble constant, standard error propagation $(\sigma_{\Omega_\mathrm{rad}}/\Omega_\mathrm{rad} \simeq 2 \sigma_h/h)$ yields a value of $\Omega_\mathrm{rad} = (9.0513\pm0.2456)\times10^{-5}$.

So, to be explicit my recommendation is: $$\{\Omega_\mathrm{m}, \Omega_\Lambda, \Omega_\mathrm{rad}, \Omega_K\} = \{ 0.308, 0.692, 9.05\times10^{-5},0\} \pm \{0.012, 0.012, 2.46\times10^{-6}, 0 \}.$$ But I think the most important thing is to state where you take the parameters from. People rarely state why the choose a particular set of parameters, and although Planck gives very small error bars, other probes give error bars so small that they're basically mutually incompatible. That's why you can still easily get away with $\{\Omega_\mathrm{m}, \Omega_\Lambda, \Omega_\mathrm{rad}, \Omega_K\} = \{ 0.3,0.7,0,0\} \pm \{0,0,0,0\}$.

• Thanks. Please, could you edit the answer to add a list of the four values I was asking for, with their uncertainties ? It is still not clear what "final" values I should take, as the most "accurate", "reliable" or "consensual" ones.
– Cham
Dec 13 '16 at 22:15
• Also, I thought that the most reliable value of Hubble's constant was $H_0 \approx 71,9 \; (\text{km}/\text{s})/\text{Mpc}$. What should be the best value (i.e. the most reliable one) ?
– Cham
Dec 13 '16 at 22:29
• @Cham: See my update.
– pela
Dec 13 '16 at 23:08
• A fine answer to an unanswerable question. Note that $h= H_0/100$. Dec 14 '16 at 7:21
• Thanks, @Rob. And Cham, I don't know for which purpose you need the parameters, but I think the most important thing is to state where you take the parameters from. People rarely state why the choose a particular set of parameters, and although Planck gives very small error bars, other probes gives error bars so small that they're basically mutually incompatible. That's why you can still easily get away with {Ωm, ΩΛ, Ωrad, Ωk} = {0.3, 0.7, 0, 0} ± {0, 0, 0, 0}.
– pela
Dec 14 '16 at 9:15