2024-01-10 Cosmology Nomograms

pre { font-family: "Major Mono Display", monospace; white-space: pre-wrap; } /* keeps wrapping within container. Remember to use
where necessary. It will wrap and respect [newline] characters. */ Cosmology Ruler Bookmark"Dole, 2024) and Paper-and-pencil cosmological calculator (Pilipenko, 2013)

Separated by a number of years, these two authors have come up with rather similar ideas about how people can rapidly get an approximate answer to the fairly involved question of what an observed redshift means in terms of when the redshift was imposed on the light signal, and thus limits on when the light source was active.

The impression is sometimes given that a redshift can be directly recalculated into the age of the event(s) concerned (before today) and the time after the Big Bang at which these events occurred.

That's not the case. In fact you need to integrate a model of the cosmological universe, with matter density (and it's changes, and the inferred degree of deceleration ; "dark matter" and "bright" matter), the increasing amount of positive acelleration from "dark energy" in the recent parts of those events, and the Hubble constant at this time (which also changes with the inferred matter density changes). All of which is a fairly complex calculation, which no reasonable person can be expected to do in their head, on demand.

Both of these authors have come up with ideas about how to get reasonably accurate answers to such questions in real time. To those of my generation who grew up with (just) slide rules and log tables, not calculators, their solutions will be familiar, but to generations who grew up with digital computers everywhere, these analogue computers might seem a little odd. But when you're looking for an answer accurate to 2 or 3 significant figures, this sort of solution can give you that answer faster than you can enter the parameters into a calculator - assuming you have already set up the calculation method after looking up the procedure.

Nomograms are a way of recording the interrelations between several parameters which are linked by one or more equations. A simple graph (mathematical sense, not computing/ networks sense) is a relationship between two parameters, typically expressed as "x" (the independent variable) and "y" a "function of x" (or "f(x)" in a more recent notation) which you'd read off the other Cartesian axis. Or if you used polar coordinates, you'd link the parameters by a "radius" and an angle compared to the axis. Different expressions of the same underlying relationship. Well, nomograms are a way of interlinking three or more variables, in a way that can't really be done without a 3-d printer.

A nomogram is a way of linking several variables into one expression or an equation. You set out a graduated line to reperesent one variable - on which the user chooses a point for their value for that parameter (the graduation also implise a range of validity for the underlying equations, which are frequently approximate solutions). A second line (graduated differently, and not necessarily straight or parallel to the first, and also incorporating limits of validity) represents a second parameter of the equation, and typically the user then projects a straight line from those two points to intersect a third line on a graduation - which is the solution to the equation represented by the lines, thir angles, and the scales. You can generalise the system to more complex networks of lines and scales, allowing you to handle 4-parameter systems and higher.

The reader may recognise the slide rule as a particular form using several parallel lines, scales linearly, logarithmically, reciprocally, and with various trigonometrical functions. For several centuries, they were the scientist's analogue calculator of choice, for calculations accurate to 3 or possibly 4 significant figures. For greater accuracy, you'd need to use tables of functions, which would go up to 8 significant figures (by which point, they were bookshelves, not tables). My generation always carried a set of 4-figure logarithmic and trigonometrical tables with our science and maths text books.

All of which is background you can no-longer assume the present generation will understand. They need this to understand how to use either the nomograms or the "bookmarks".

Both presenters provide several ranges of z (redshift) to allow for use in the nearby universe (z less than 1), the intermediate universe (z between 1 and 30) or the very early universe (z greater than 10). There is recent "news" about a galaxy being found at redshift of 11.6 ; it's the current record holder, but it will be overtaken eventually.

I'm not going to reproduce the nomograms here. They're intended for a page printout sitting in your desk-tidy. Get the paper linked to above, and print the relevant pages. Better if you've got a laser printer and card or plastic stock to make them more hard-wearing, but they're not high tech.

The bookmark you might be able to use on-screen. It includes it's own scale, but if you "roll your own" cosmoligy thoughts, you'll probably find it a handy reference. One side covers z from 0 to 30 (the modern universe, linearly) and the other side covers z from 0.1 to 1000 (the early universe, logarithmically). In addition to the obvious z, age, and lookback time (how old the universe was when the redshift event happened), there is also a scale of angle per kiloparsec (which I think you'd use for planning observation campaigns to have a good probability of finding a good number of examples of [whatever you're hypothesising].

This image is scaled for 100dots per cm, and should fit onto a 30cm-wide printing area. Or you can scale it to fit your screen.

1	6	7
2	5	8
3	4	9