How does one measure the curvature $k$ in FLRW metric?

Question

How does one measure the curvature parameter $k$ in the FLRW metric? $$ds^2=-c^2dt^2+a^2(t)[\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\phi^2]$$ In particular, what is the convenient equation (involving $k$) that is/can be used to measure $k$?

EDIT I'm looking for an answer that will explain the measurement of curvature $k$ with the same clarity as the measurement of spring constant $\kappa$ from the one-dimensional simple harmonic equation $F=-\kappa x$ i.e., having measured the applied force $F$ (can be done with a spring balance may be) and the corresponding displacement $x$ (by a meter rule), one can measure $\kappa$. Similarly, if the equation involving curvature $k$ contains non-trivial physical quantities (such as the components of Riemann curvature tensor etc), I would like to know how each of them is measured.

Answer 1

This is a very hard question to answer in detail as it requires several pages of mathematics to derive the required formulas (there is no easy fit like $F=-kx$ as you suggested)

I will not derive the formula (it can be found in e.g. Dodelson) but after some work you obtain:

$$\Delta(m-M) = 5\log\left\{ \left( \frac{c}{H_0}\sqrt{\frac{k}{\Omega_{total}-1}}(1+z) \right)S_k\left(\sqrt{\frac{\Omega_{total}-1}{k}}\int_0^z \frac{dz'}{(1+z')\sqrt{(1+z')(1-\Omega_\Lambda) + \Omega_\Lambda/(1+z')^2}} \right)\right\} \\ - 5\log\left(\frac{1}{2}((1+z)^2-1)\right)$$

Where $\Delta(m-M) = (m-M)_{'real'\ universe} - (m-M)_{empty\ universe}$. M can be obtained by using standard candles such as supernovae type Ia, it is than easy calculate $(m-M)_{empty\ universe}$ via an equation similar to the one above and $m_{'real'\ universe}$ is simply the magnitude that we measure. Therefore $\Delta(m-M)$

$S_k(...) = sinh(...), sin(...) or 1$ depending on the value of $k$

$\Omega_\Lambda, \Omega_{total}=\Omega_\lambda+\Omega_{matter}, H_0$ and k remain unknown.

The next step is to measure a lot of standard candles at various redshifts z and plot their $\Delta(m-M)$ relation as a function of z. This should obey the relation above. All that remains to be done is to run a fitting script that fits the above function to $\Delta(m-M)$ for various values of $k, \Omega_\Lambda,...$ the best fit gives us the observed cosmology.

In the figure below you can see such a fit from a project I made previous semester where we had to calculate k for some dataset.

Obviously the fits are hard to make due to degeneracies in the fit and uncertainty plots can be made as like this one:

My results for the above fit were: $\Omega_{matter} = 0.286 \pm 0.031$, $\Omega_{\Lambda} = 0.721 \pm 0.025$, $H_0(km/s/MPc) = 70.3 \pm 2.58$ which is consistent with k = 0.

I hope this helped :)

Answer 2

You simply measure the ratio of the circumference of a circle to its radius.

Take a spatial submanifold, and for convenience we'll take $a=1$ (the units of the radial distance can always be chosen to make $a=1$ as any chosen time). Then the spatial metric becomes:

$$ d\ell^2=\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\phi^2 $$

Draw a circle with yourself at the origin. To get the circumference of the circle we integrate around the equatorial angle $\phi$ while keeping $r$ fixed and $\theta$ fixed at $\pi/2$. Since $dr = d\theta = 0$ the metric becomes:

$$ d\ell^2=r^2\sin^2\theta d\phi^2 $$

The circumference is then:

$$ C = \int_0^{2\pi}\,rd\phi = 2\pi r $$

which shouldn't surprise us unduly :-)

Now we take a measuring tape and measure out the distance to the circle. In this process we are keeping $\theta$ and $\phi$ fixed so $d\theta = d\phi = 0$ so our metric becomes:

$$ d\ell^2=\frac{dr^2}{1-kr^2} $$

So the distance we measure is:

$$ R = \int_0^r\,\frac{dr}{\sqrt{1-kr^2}} $$

The integral depends on the sign of $k$. For positive $k$ (closed universe) we get:

$$ R = \frac{\sin^{-1}(\sqrt{k}\,r)}{\sqrt{k}} $$

and for negative $k$ (open universe) we get:

$$ R = \frac{\sinh^{-1}(\sqrt{|k|}\,r)}{\sqrt{|k|}} $$

To find $k$ simply substitute $r = C/2\pi$, where $C$ is our experimentally measured circumference, and solve the resulting equation for $k$.