Appendix

Calculation of the local curvature of the fitted and theoretical spectral forms

Linear spectral forms

With the fitted polynomial coefficients $a_0,...\, a_3$, we can write a linear form of the fit as:

$$S(\nu) = C_0 \exp(\tau)\,,$$ (A1)

with $C_0 = \exp(a_0)$ and

$$\tau = \sum^3_{i=1} a_i (\ln \nu)^i \, .$$

And the derivatives used in (14) are given by the following formulae:

$$\frac{{\rm d}S}{{\rm d}\nu} = C_0 \exp(\tau) \frac{{\rm d}\tau}{{\rm d}\nu}$$ (A2)

$$\frac{{\rm d}^2S}{{\rm d}{\nu}^2} = C_0 \left[\exp(\tau) \fr... ...d}{\nu}^2} + \exp(2\tau) \frac{{\rm d}\tau}{{\rm d}\nu} \right]$$ (A3)

$$\frac{{\rm d}\tau}{{\rm d}\nu} = \frac{1}{\nu} [a_1 + 2a_2 \ln\nu + 3a_3(\ln\nu)^2]$$ (A4)

$$\frac{{\rm d}^2\tau}{{\rm d}{\nu}^2} = -\frac{1}{\nu^2} [3a_3(\ln\nu)^2 + (2a_2\, -\, 6a_3)\ln\nu + (a_1\, -\, 2a_2)].$$ (A5)

The power-law fit is given by:

$$S(\nu) = C_1 \left(\frac{\nu}{\nu_1}\right)^{\alpha_{\rm t}}... ...\nu_1}\right)^{\alpha_{\rm o}-\alpha_{\rm t}}\right]\right\}\,,$$ (A6)

with the corresponding derivatives:

$$\frac{{\rm d}S}{{\rm d}\nu} = C_1\left[\nu^{\alpha_{\rm t}} ... ...d}\nu} + \alpha_{\rm t}\nu^{(\alpha_{\rm t}-1)} f_{\nu}\right]$$ (A7)

$$\begin{eqnarray} \frac{{\rm d}^2S}{{\rm d}{\nu}^2} & = & C_1\Bigg[\nu^{\alpha_{\... ...uad\quad\quad\quad + (\alpha_{\rm t}-1)\alpha_{\rm t}f_{\nu}\Bigg]\end{eqnarray}$$ (A8)

$$f_{\nu} = 1 - \exp\left[-\left(\frac{\nu}{\nu_1}\right)^{\lambda}\right]\,,\quad \lambda=\alpha_{\rm o}-\alpha_{\rm t}$$ (A9)

$$\frac{{\rm d}f_{\nu}}{{\rm d}\nu} = \frac{\lambda\nu^{\lambd... ...}} \,\exp\left[-\left(\frac{\nu}{\nu_1}\right)^{\lambda}\right]$$ (A10)

$$\frac{{\rm d}^2f_{\nu}}{{\rm d}\nu^2}\! =\! \frac{\lambda \n... ...\!\!\left[-\!\left(\frac{\nu}{\nu_1}\right)^{\lambda}\!\right].$$ (A11)

Logarithmic spectral forms

Following the same considerations, the logarithmic form for the polynomial fit and its derivatives are:

$$S_{\rm log}(\nu) = \sum^3_{i=0} a_i(\ln\nu)^i$$ (A12)

$$\frac{{\rm d}S_{\rm log}}{{\rm d}(\ln\nu)} = a_1+2a_2\nu+3a_3\nu^2$$ (A13)

$$\frac{{\rm d}^2S_{\rm log}}{{\rm d}(\ln\nu)^2} = 2a_2+6a_3\nu.$$ (A14)

And for the power-law fit:

$$S_{\rm log}(\nu) = \ln C_1 + \alpha_{\rm t}(\ln\nu - \ln\nu_1) + \ln f_{\nu}$$ (A15)

$$\frac{{\rm d}\,f_{\nu}}{{\rm d}(\ln\nu)} = \lambda \left(\fr... ...da}\,\exp\left[-\left(\frac{\nu}{\nu_1}\right)^{\lambda}\right]$$ (A16)

$$\frac{{\rm d}^2f}{{\rm d}(\ln\nu)^2} = \lambda^2 \left(\frac... ...,\exp\!\!\left[-\left(\frac{\nu}{\nu_1}\right)^{\lambda}\right]$$ (A17)

$$\frac{{\rm d}S_{\rm log}}{{\rm d}(\ln\nu)} = \frac{1}{f_{\nu}}\frac{{\rm d}f_{\nu}}{{\rm d}(\ln\nu)} + \alpha_{\rm t}$$ (A18)

$$\frac{{\rm d}^2S_{\rm log}}{{\rm d}(\ln\nu)^2} = \frac{1}{f^... ... \left(\frac{{\rm d}f_{\nu}}{{\rm d}(\ln\nu)}\right)^2 \right].$$ (A19)