Primodial power spectrum of scalar perturbations

[thecommexokid]

For scalar modes \mathcal{R}_k originating in the Bunch-Davies vacuum at the onset of inflation, I have the following equation for their primordial power spectrum:
P_{\mathcal{R}}(k)=\frac{4\pi}{\epsilon(\eta_k)}\bigg( \frac{H(\eta_k)}{2\pi} \bigg)^2,
where:

• c = G = ħ = 1,
• k is the comoving wavenumber of the mode,
• ε is the slow-roll parameter,
• H is the Hubble parameter, and
• η_k is the (conformal) time when the mode \mathcal{R}_k exited the horizon.

In my notes, I have this equation labeled as "Well-known result:", but now I am struggling to even find a source for it. Can anyone offer any assistance in finding a reference for this? My goal is to have some understanding of its derivation.

 

[Chronos]

Pardon my curiosity, but, this formula appears to ignore tensor modes. Would these not be important?

 

[bapowell]

For scalar modes \mathcal{R}_k originating in the Bunch-Davies vacuum at the onset of inflation, I have the following equation for their primordial power spectrum:
P_{\mathcal{R}}(k)=\frac{4\pi}{\epsilon(\eta_k)}\bigg( \frac{H(\eta_k)}{2\pi} \bigg)^2,
where:

• c = G = ħ = 1,
• k is the comoving wavenumber of the mode,
• ε is the slow-roll parameter,
• H is the Hubble parameter, and
• η_k is the (conformal) time when the mode \mathcal{R}_k exited the horizon.

In my notes, I have this equation labeled as "Well-known result:", but now I am struggling to even find a source for it. Can anyone offer any assistance in finding a reference for this? My goal is to have some understanding of its derivation.

This result is obtained by solving the mode equation of inflaton fluctuations in the limit of slow roll evolution. After relating the inflaton fluctuation to the curvature perturbation (an association that depends explicitly on a choice of gauge), the power spectrum of the curvature perturbations can be readily obtained. A good review that is available free online is that by Liddle and Lyth: http://arxiv.org/pdf/astro-ph/9303019.pdf.

I am happy to help with questions on the particulars of the calculation.

Chronos: this formula is the amplitude of the scalar perturbations only. Indeed, tensors are important and need to be included when calculating temperature and polarization anisotropies.