At a team meeting, I once presented a slide jokingly noting that “z is an abomination, it is neither multiplicative, additive or a shift”. Of course, redshift's saving grace is that a human's computational ability is sufficient to convert to the inverse scalefactor, add unity and you get , where is the cosmological scalefactor with the common convention that the present-day value .
Using the logarithmic shift , the relationship is evidently . Spacing in logarithm of the scalefactor has desirable properties when considering galaxy populations or cosmology (Table 1). Figure 2 shows the separation in line-of-sight comoving distance () versus redshift and zeta for two different cosmologies. The black lines show
in each plot. These are inversely proportional to (e.g. Hogg 1999) and , respectively. Notably varies less, particular at . This is a desirable property since large-scale structure is evaluated using comoving distances. Spacing in corresponds to constant velocity and approximately constant comoving distance.
The turnover in demonstrates the onset of dark energy dominating the dynamics for the `737 cosmology'. This is evident even without the comparison to the Einstein-de-Sitter (EdS) cosmology because for a non-accelerating universe ( ), is constant. For the EdS model, and so that
, | (15) |
Also shown in Figure 2, with vertical lines, are the points at which the universe halves its age (737 cosmology), with increasing and . For , the last half of cosmic time covers only a small fraction of the plot (), whereas for , the spacing is approximately logarithmic in time. For an EdS model, it would be equally spaced in because . For the 737 cosmology, an increase in of corresponds to halving the age of the universe across the epochs shown. A generic plot related to galaxy evolution shows the cosmic star-formation rate (SFR) density, logarithmically scaled, versus but often scaled linearly in (Madau & Dickinson, 2014; Hopkins & Beacom, 2006). This a recognition of the aesthetic of separation.
note | |||
0.1 | 0.105 | 0.905 | present-day galaxy properties |
0.5 | 0.649 | 0.607 | transition to cosmic acceleration |
1.0 | 1.72 | 0.368 | peak of cosmic SFR density |
1.5 | 3.48 | 0.223 | |
2.0 | 6.39 | 0.135 | end of reionization |
2.5 | 11.2 | 0.0821 | |
3.0 | 19.1 | 0.0498 | first stars |
7.0 | 1096 | 0.000912 | matter-radiation decoupling |