primordial gravitational waves

After reading some literatures

I still have some questions that I don't understand.

According to cosmic inflationary, we know that Primordial Gravitational Waves(PGWs) could be produced in the very early universe. And the appearance of PGWs is the result of the metric's tensor perturbation.

But Why are there perturbations, such as scalar perturbation and tensor perturbation ?

We know that tensor-to-scalar ratio $r=\frac{A_T}{A_S}$ is one important quantity. The relation between $r$ and inflaton (the scalar field) $\phi$ is that $$r=\frac{8}{m_{Pl}^2}\left(\frac{\dot \phi}{H}\right)^2$$

So if we can detect the value of $r$, we can know something about Inflation and PGWs.

The second question is that how do astronomers measure $r$ through direct way (laser interferometer) or indirect way (CMB polarization)?

Finally, I don't konw why someone likes to place $h_0^2$ before PGWs' energy density $\Omega_{gw}$, i.e. $\Omega_{hw}h_0^2$, see (Luuk Wagenaar 2016 Inflation, quantum fluctuations and gravitational waves ).

What does that $h_0^2$ mean ？

• Thought I came across a piece of information, perhaps relevant to this question, that might just be useful? Please see Gravitational-wave sensitivity curves, by C J Moore, et al. Specifically, page 7, eq. 28 which defines Hubble's constant as: $$H_0 = h_{100}\times100kms^{-1}Mpc^{-1}$$ Where $h_{100}$ is Reduced Hubble's Parameter. This is apparently used in plotting a graph (page 26) of the product of Energy Density ($\Omega_{GW}$) and Reduced Hubble Parameter squared against the Frequency of Gravitational Waves. – Dhruv Saxena May 19 '17 at 4:22