I am trying to understand how the width of a supernova light curve depends on the redshift of its component frequencies.(adsbygoogle = window.adsbygoogle || []).push({});

Let us make the simple assumption that the light curve is Gaussian. The inverse Fourier transform of a Gaussian is given by:

$$\large e^{-\alpha t^2}=\int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha}}e^{-\frac{(\pi f)^2}{\alpha}}e^{2\pi ift}\ df$$

Now if all the components of the light curve are redshifted by a factor [itex]k[/itex] then I think the right-hand side of the above equation becomes:

$$\large \int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha}}e^{-\frac{(\pi f)^2}{\alpha}}e^{2\pi ikft}\ df$$

I now change variables in the integral using:

$$f'=kf$$

The above integral becomes the inverse Fourier transform of a modified Gaussian curve:

$$\large \int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha k^2}}e^{-\frac{(\pi f')^2}{\alpha k^2}}e^{2\pi if't}\ df'$$

Thus it seems that if the components are redshifted by a factor [itex]k[/itex] the light curve transforms in the following way:

$$\large e^{-\alpha t^2} \rightarrow e^{-\alpha k^2t^2}$$

Is this correct?

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# Redshift of supernova light curve

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