# How to determine scalar-to-tensor ratio r from CMB polarization spectrum?

CMB polarization spectrum can tell us about the primordial scalar and tensor perturbation. By analyze B and E mode angular spectrum power spectrum and temperature power spectrum we can determine the scalar-to-tensor ratio r, as many articles implies.

I know that r is defined by the ration of amplitude of primordial tensor and scalar perturbation. But what we in fact measure is the power spectra. So I wonder how can we get the value of r from the spectra we measure thus we can study the physics of primordial perturbations and inflation.

• Introduction The possible detection of tensor perturbations in the cosmic microwave background (CMB) by the BICEP2 experiment suggests that inflation occurred at a high energy scale Ei = 2 × 1016 GeV, just two orders of magnitude below the reduced Planck scale MPl =1/√8πG = 2.4×1018 GeV This is what I have read in this. How Many e-Folds Should We Expect from High-Scale Inflation? arxiv.org/pdf/1405.5538.pdf Mar 5, 2020 at 1:12
• VI. CONCLUSIONS The recent BICEP2 discovery, if verified, suggests that high-scale cosmic inflation is the correct theory of the very early Universe. With characteristic energy of order 1016 GeV, observational signatures of inflation open the door to physics on the threshold of the Planck scale. Many models of inflation are currently being investigated for their ability to fit precision CMB observations. The current success of relatively simple models of inflation, driven by a single scalar field with a potential and a canonical kinetic term Mar 5, 2020 at 1:21

Most of this answer is based on First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications For Inflation. As I am not an expert in this area, I am more transcribing the computation rather than interpreting it.

The tensor/scalar ratio is given by: $$r \equiv \frac{\Delta^{2}_{h}(k_{0})}{\Delta^{2}_{\mathcal{R}}(k_{0})}$$

Where the scalar power spectrum is: $$\Delta^{2}_{\mathcal{R}}(k) \equiv \frac{k^{3}}{(2\pi^{2})\langle|\mathcal{R}_{k}|^{2}\rangle}$$

and the tensor power spectrum is: $$\Delta^{2}_{h}(k) \equiv \frac{k^{3}}{(2\pi^{2})\langle|h_{+k}|^{2}+|h_{\times k}|^{2}\rangle}$$

$$\mathcal{R}$$ is the "curvature perturbation in the comoving gauge"

$$h_{+k}$$ and $$h_{\times k}$$ are the "two polarization states of the primordial tensor perturbation"

The spectra are parameterized by:

$$\Delta^{2}_{\mathcal{R}}(k) = \Delta^{2}_{\mathcal{R}}(k_{0}) \left(\frac{k}{k_{0}}\right)^{n_{s}(k_{0})-1+\frac{1}{2}\frac{dn_{s}}{d\ln k}\ln(\frac{k}{k_{0}})}$$

$$\Delta^{2}_{h}(k) = \Delta^{2}_{h}(k_{0}) \left(\frac{k}{k_{0}}\right)^{n_{s}(k_{0})+\frac{1}{2}\frac{dn_{t}}{d\ln k}\ln(\frac{k}{k_{0}})}$$

$$\Delta^{2}_{\mathcal{R}}(k_{0})$$ and $$\Delta^{2}_{h}(k_{0})$$ are normalization constants and $$k_{0}$$ is a "pivot wavenumber."