I Toward the spectral index in slow roll

[ergospherical]

Getting a bit confused over a manipulation; I have a power spectrum $P_{R}(k)$ in terms of $V$ and $\epsilon_V$ (to be evaluated at $k=aH$) for the curvature perturbation $R$ and from this need to get a spectral index, i.e. apply $d/d(\ln{k})$. So we need to transform this operator into "something" $\times d/d\phi$. From slow roll we get $3H \dot{\phi} = -V'$ and $H^2 = V/(3M_{pl}^2)$, combining to$$\frac{\dot{\phi}}{H} = -\frac{M_{pl}^2 V'}{V}$$Then from the LHS I need to convert into something related to $d\phi/dk$. We know $k=aH$ so could try something like$$\frac{\dot{\phi}}{H} = a \frac{d\phi}{da} = a \frac{d\phi}{dk} \frac{dk}{da}$$but from there on it's not clear to me how to manipulate $\frac{dk}{da}$ into something without more time derivatives? The answer should be$$\frac{d}{d\ln k} = - M_{pl}^2 \frac{V'}{V} \frac{d}{d\phi}$$i.e. suggesting that $$\frac{a}{k} = \frac{da}{dk}$$ but why is that the case, as $H = H(t)$ is not fixed?

 

[ergospherical]

Closing old (unanswered) threads. It's been a couple months since I've needed this, but I figured out the general procedure eventually. Let's see if I remember. The relation between the spectral index and the power spectrum is$$n_s - 1 = \frac{d \log P_R}{d\log k} \bigg{|}_{k=aH}$$and the play is to break this into:
$$n_s -1 = \frac{d \log P_R}{dN} \frac{dN}{d\log k} \bigg{|}_{k=aH}$$The quantity $N := \log a$ is the number of $e$-foldings of inflation. For the comoving curvature perturbation $R$, the power spectrum goes like $P_R \sim H^2 / (M_{\mathrm{pl}}^2 \epsilon)$ evaluated at horizon crossing, so
$$\frac{d \log P_R}{dN} = 2 \frac{d\log H}{dN} - \frac{d\log \epsilon}{dN} = -2\epsilon -2(\epsilon - \eta)$$
where the slow roll parameters $\epsilon := -d\log H/dN$ and $\eta := -d\log H_{,\phi}/dN$ respectively. As for the second factor: at the horizon crossing $k=aH$, or $$\log k = \log a + \log H$$Then$$\frac{dN}{d\log k} = \left( \frac{d\log k}{dN} \right)^{-1} = \left(1 + \frac{d\log H}{dN}\right)^{-1} \approx 1+\epsilon$$Therefore to linear order in the slow roll parameters $n_s - 1 = (2\eta - 4\epsilon) \big{|}_{k=aH}$. Further, given that the potential slow roll parameters $\epsilon_V$ and $\eta_V$ (defined in terms of $V(\phi)$) are related to the actual slow roll parameters via $\epsilon \approx \epsilon_V$ and $\eta \approx \eta_V - \epsilon_V$, we get $$n_s - 1 = (2\eta_V - 6\epsilon_V) \big{|}_{k=aH}$$which relates the spectral index (observable!) to the potential slow roll parameters that you can calculate directly from the form of the potential $V(\phi)$.