Why does this redshift versus lighttravel time equation work?(adsbygoogle = window.adsbygoogle || []).push({});

z=-ln(1-t)/sqrt(1-t^2)

where (z) is the cosmological redshift, and (t) is the look-back time where the present equals 0 and the origin of time equals 1.

Ned Wright's cosmology calculator with the default inputs [H_o=69.6, OmegaM=.286, OmegaVac=.714] yields very close values to this equation, especially in the z<2 range that we actually have observational data for. To convert (t) to Gigayears as the calculator outputs, just multiply (t) by the age of the universe, 13.721 Gyr in the default calculator setup.

We know (z) is related to scale (a), by

a=1/1+z

So the rest of the values output by the calculator-- co-moving radial distance, angular size distance, and luminosity distance-- are also in agreement with what you would get with the alternate equation after integration over time, multiplication etc.

Why does this work? It doesn't consider mass density, radiation, dark energy, or any free parameter. In the supernova paper, Perlmutter's team notes that the "empty universe" model of Milne generates a nearly best fit graph to their data set, although no equations are included. Is this equation a expression of an emtpy FLRW model? Simply inputing OmegaM=0 and OmegaVac=0 into Ned Wright's calculator does not yield the same relation.

I have a pet theory as to why the equation works, but I won't force it on anybody right now. Mostly I'm eager to see someone else compare the values generated by the equation and the calculator, and get your 2 cents on how close they seem. Coincidence? Could such a shorthand equation be useful to ballpark numbers, even without a explanation?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Why does this redshift v. light travel time equation work?

Loading...

Similar Threads - does redshift light | Date |
---|---|

I How does flux density scale with redshift? | Sep 3, 2017 |

B Why does expansion redshift light if... | Feb 19, 2017 |

Cosmological redshift: Where does the energy go? | Nov 2, 2014 |

Why does redshift indicate that space-time is expanding | Feb 19, 2011 |

How much does dust affect light from high redshift quasars? | Feb 28, 2009 |

**Physics Forums - The Fusion of Science and Community**