<p>Most of this answer is based on <a href="https://ui.adsabs.harvard.edu/abs/2003ApJS..148..213P/abstract" rel="nofollow noreferrer">First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications For Inflation</a>.  As I am not an expert in this area, I am more transcribing the computation rather than interpreting it.</p>
<p>The tensor/scalar ratio is given by:
<span class="math-container">$$r \equiv \frac{\Delta^{2}_{h}(k_{0})}{\Delta^{2}_{\mathcal{R}}(k_{0})}$$</span></p>
<p>Where the scalar power spectrum is:
<span class="math-container">$$\Delta^{2}_{\mathcal{R}}(k) \equiv \frac{k^{3}}{(2\pi^{2})\langle|\mathcal{R}_{k}|^{2}\rangle}$$</span></p>
<p>and the tensor power spectrum is:
<span class="math-container">$$\Delta^{2}_{h}(k) \equiv \frac{k^{3}}{(2\pi^{2})\langle|h_{+k}|^{2}+|h_{\times k}|^{2}\rangle}$$</span></p>
<p><span class="math-container">$\mathcal{R}$</span> is the &quot;curvature perturbation in the comoving gauge&quot;</p>
<p><span class="math-container">$h_{+k}$</span> and <span class="math-container">$h_{\times k}$</span> are the &quot;two polarization states of the primordial tensor perturbation&quot;</p>
<p>The spectra are parameterized by:</p>
<p><span class="math-container">$$\Delta^{2}_{\mathcal{R}}(k) = \Delta^{2}_{\mathcal{R}}(k_{0}) \left(\frac{k}{k_{0}}\right)^{n_{s}(k_{0})-1+\frac{1}{2}\frac{dn_{s}}{d\ln k}\ln(\frac{k}{k_{0}})}$$</span></p>
<p><span class="math-container">$$\Delta^{2}_{h}(k) = \Delta^{2}_{h}(k_{0}) \left(\frac{k}{k_{0}}\right)^{n_{s}(k_{0})+\frac{1}{2}\frac{dn_{t}}{d\ln k}\ln(\frac{k}{k_{0}})}$$</span></p>
<p><span class="math-container">$\Delta^{2}_{\mathcal{R}}(k_{0})$</span> and <span class="math-container">$\Delta^{2}_{h}(k_{0})$</span> are normalization constants and <span class="math-container">$k_{0}$</span> is a &quot;pivot wavenumber.&quot;</p>

CC BY-SA 4.0
2022-01-15T13:30:30.620Z
2022-01-15T13:30:30.620Z
GrapefruitIsAwesome
<p>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.</p>

<p>I know that r is <strong>defined</strong> by the ration of <strong>amplitude</strong> of primordial tensor and scalar perturbation. But what we in fact <strong>measure</strong> 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.</p>

CC BY-SA 4.0
2020-03-05T01:12:31.683Z
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?
https://arxiv.org/pdf/1405.5538.pdf
Dean Pitts
CC BY-SA 4.0
2020-03-05T01:21:46.573Z
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
Dean Pitts
<cosmology><data-analysis><cosmic-microwave-background>
How to determine scalar-to-tensor ratio r from CMB polarization spectrum?
Ziang Yan
# How to determine scalar-to-tensor ratio r from CMB polarization spectrum?

(Asked by: username_0 on 2014-10-26)

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.

**Tags:** cosmology, data-analysis, cosmic-microwave-background

> **Comments:**
> **username_1**: 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? https://arxiv.org/pdf/1405.5538.pdf  
> **username_1**: 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  

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## 1 answers

### Answer by username_2 on 2022-01-15. Score: 1

Most of this answer is based on [First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications For Inflation](https://ui.adsabs.harvard.edu/abs/2003ApJS..148..213P/abstract). 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."


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