Relativity and Cosmology

The thinking around cosmological models has undergone a shift during the last two decades. My prior blog post hinted at the importance of the Hubble radius and the 'emergent spacetime' view that is implied.

Models seem to have moved from the Einstein-de Sitter (matter dominated) view, through the ΛCDM (dark energy dominated) decade to the present 'emergent spacetime' views. The ΛCDM model can actually handle all the views, simply by setting its parameters appropriately. My old Cosmo Calculator^[1] does exactly that and spews out numerous calculated results for a given set of inputs - actually enough information to confuse everyone, except perhaps the experts.

In line with the new thinking, I have teamed up with Marcus from PhysicsForums^[2] to develop a more intuitively simple calculator that may be more appealing at beginners level, while still conforming to the concordant cosmic model of today. It is called CosmoLean and is presently at version A25. 'Lean', because it does not flood the user with results. The main output is a table giving the evolution of some useful parameters over time. Before you dig into it, please read on to get some perspective on it first.

CosmoLean starts with a premise that we can forget about dark energy if we accept that there exists an underlying constant spacetime curvature, but that space is perfectly flat.^[3] Without going into the "why is this so?" type questions, let me just say that it is perfectly in line with all present observations and paints a picture that is compatible with, but conceptually simpler than the dark energy paradigm. In effect it says that from Einstein's 1915 papers^[4] onwards, the laws of gravity had two constants: the local scale gravitational constant (G) and the large scale cosmological constant (Λ), both representing a curvature of spacetime, as applicable to their respective regimes.

Without going into this formidable equation, let's move on to what we need for a new calculator. As in my prior Blog, what we observe today is a (sort of) transient to a future constant Hubble radius (R_H), which only depends on the cosmological constant Λ. How close we are to this future constant (say R_{H_inf}) depends on how much the radiation and matter energies have been diluted by the cosmic expansion. We can completely specify this by three parameters, R_{H_now}, R_{H_inf} and the ratio between matter and radiation energy density. For the latter we can choose the redshift at which they were equal (z_eq) in the past.

R_{H_now} = cT_{H_now} , the speed of light multiplied by the Hubble time. This again is inversely proportional to the Hubble constant (H₀). If we choose units appropriately,^[5] then c=1 and T_H = 1/H₀, which has the value 13.9 Gy, according to today's best observations. The future value R_{H_inf} =16.3 Gly and the past value z_eq ~ 3500 have both been derived from those observations.

To make the cosmological equations^[6] as readable as possible, we have decided to use the symbol Y = T_H = c/R_H for Hubble time. This gets rid of one extra level of subscript. We also use "stretch factor" (S) in place of the usual redshift z, where S = z + 1, a factor that crops up all over the show. Stretch is simply the factor by which wavelengths have increased from the time that light has left an observed source, e.g. a stretch S=2 means wavelengths have been doubled by the cosmic expansion while the photons were in transit.

For the main calculator inputs, we simply have to tell it the values of Ynow, Yinf and S_eq. We may then also specify the output in tabular form in terms of S_upper, S_lower and the output steps in-between, either as step size, or as the number of steps (see the info-tooltip of the live user interface).

To make a connection back to the "old ways" of specifying inputs for cosmo-calculators, we also compute the conventional values and show them at the top-right. The page is pre-populated with default values and there are ample info-tooltips to make it (hopefully) easy to use. So, without further delay, please give it a try and tell us whether you think it is cool, whether it sucks, or anything in-between. It is a work in process, so you may actually still influence the tool.

Some further usage tips will follow...

Click here and take a cosmic dive: TabCosmo6.

Usage tips: see comments #3, #5 below.

-J

[1] http://cr4.globalspec.com/blogentry/20218/Cosmological-Calculator-Update

[2] http://www.physicsforums.com/forumdisplay.php?f=69

[3] It is presently thought that the universe is either spatially flat (or very near to flat), i.e. parallel lines 'here' are still parallel 'there', when considered on a large scale and viewed everywhere at the same cosmic time. However, due to the expansion, light rays that are sent out to be parallel, will not remain parallel over time; hence, spacetime is curved.

[4] Einstein (1915), "Die Feldgleichungen der Gravitation (The Field Equations of Gravitation)"

[5] The conventional Hubble constant is given in units Km/s/Mpc, i.e., a recession speed per distance. With years for time and light-years for distance, speed becomes dimensionless and the speed of light is 1. If we also convert Mega-parsec to billion light years, we get that the present H₀ = 70.36 km/s/Mpc becomes 70.36/978 = 0.0712 Gly^-1. Now if we invert that, we get R_{H_now}= 13.9 Gly and T_{H_now}= 13.9 Gy.

[6] Here are the simplified equations of CosmoLean, for those who can't live without them.