I have heard that the actual vacuum energy density which is up to 120 orders smaller than the predicted QED value can be measured in experiments or cosmological observations?
What are these experiments?
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It's not so much measured as inferred.
The cosmological constant problem is that the Planck density $\frac{c^5}{G^2\hbar}$ is much larger than the empirical value of $\rho_\Lambda:=\frac{c^2\Lambda}{8\pi G}$, with $\Lambda\approx1.1\times10^{-52}\operatorname{m}^{-2}$ the cosmological constant. So your question is how we measure $\Lambda$, or equivalently $\Omega_\Lambda:=\frac{3c^2\Lambda}{H_0^2}$, with $H_0\approx2.2\times10^{-18}\operatorname{s}^{-1}$ the Hubble constant.
There's some controversy over the empirical value of $H_0$, as we can measure it in different ways in slight disagreement: you can compare Cepheid variables, or infer historic expansion from CMB radiation.
Meanwhile, $\Omega_\Lambda\approx0.69$ is estimated from $1-\Omega_m-\Omega_k$ (see e.g. slides 23-29 here). While $\Omega_m$, which includes dark matter, is estimable from how fast stars are at the edges of galaxies, $\Omega_k$ is empirically negligible according to CMB anisotropies.
