Say I have a spectrum of a galaxy at a redshift $z$, in flux density units of erg/s/cm^2/Angstrom. I'd like to recover the spectrum (in the same flux density units) at z=0, i.e. at its rest wavelengths. How do I do this?

I saw this paper (Distance measures in cosmology, Hogg (2000)) but am still confused, because of their use of luminosity and luminosity densities, and in my class I only have fluxes.

  • $\begingroup$ Since these are differential quantities, a careful application of the chain rule should be all that's necessary for a basic answer. Since I am not careful, I'll defer to someone who is ;-) $\endgroup$ – uhoh Apr 16 '19 at 1:26

You are familiar with the relationship between flux and luminosity, right? This is usually taken as the definition of luminosity distance in extragalactic astronomy, $$ F = \frac{L}{4\pi D_L^2}.$$ The quickest way to convert this into a relationship with spectral variables is to use differential 1-forms. So, $L \rightarrow L_\lambda \,\mathrm{d}\lambda_e$ and $F \rightarrow F_\lambda \,\mathrm{d}\lambda_o$, with $\lambda_e$ the photon wavelength at emission (also called the rest frame wavelength) and $\lambda_o$ the observed wavelength at $z=0$ (also called the comoving wavelength). Thus \begin{align} F_\lambda \,\mathrm{d}\lambda_o & = \frac{L_\lambda \,\mathrm{d}\lambda_e}{4\pi D_L^2} \Rightarrow\\ F_\lambda & = \frac{L_\lambda}{4\pi D_L^2} \times \frac{\mathrm{d}\lambda_e}{\mathrm{d}\lambda_o}. \end{align} Now you just need to take your derivative using the definition of redshift, $\lambda_o = (1+z)\lambda_e$, and use some way of calculating the luminosity distance in your cosmology.

Formally, a lot of people like to use $\nu F_\nu = \lambda F_\lambda$ as though it were a total flux, but it's not. The derivation will work, but only because redshifting scales all frequencies/wavelengths by a single constant. $\nu F_\nu$ isn't a measure of the flux under any part of the spectrum, it's a spectral flux using a different coordinate system. Consider the following \begin{align} F & = F_\nu \,\mathrm{d}\nu \\ &= \nu F_\nu \frac{\mathrm{d}\nu}{\nu} \\ &= F_{\ln \nu} \,\mathrm{d}\ln \nu. \end{align} In other words, $\nu F_\nu$ is still a spectral flux. If $F_\nu$ is in $\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{Hz}^{-1}$ then $\nu F_\nu$ will be in $\mathrm{ergs}\,\mathrm{cm}^{-2}\,(e\text{-fold})^{-1}$. The $e\text{-fold}$ is a pseudo-unit, like radians, dex, octave, decibel, or magnitudes. In principle it can be omitted, just like how angular frequencies can be in radians per second, or just $\mathrm{s}^{-1}$.

Fun math exercise: show that $\ln(10) \nu F_\nu$ has units of $\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{dex}^{-1}$.


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