a(t) for concordance model, with approximations & errors shown
Comparison between true a(t) in black and three simple approximations for the three principle eras: radiation (red), matter (purple), dark energy (blue). The approximations are derived assuming a single component, as shown in sec 6c ii as well as in the Toolbox. Fractional errors are shown as dashed lines. Errors are below 10% except for times between 10-6 and 10-6 tH,o or roughly 14 kyr to 1.4 Myr (unfortunately, spanning recombination). The matter era fit is improved by adding 13,600 yr to the time to compensate for the error stemming from integration over the radiation era (shown as the second purple dashed line). Although it is customary to push the radiation approximation well into the first second, in practice the earliest times are complicated by the presence of more relativistic components (parameterized by g*, which increases from 1 to 130 for the standard model), as well as inflation at very early times.
The ranges over which the approximations work within 10% are:
| radiation era: | log(a) < -3.8, | t < 15 kyr |
| matter era: | 0.002 < a < 0.8, | 1 Myr < t < 10 Gyr |
| dark energy era: | a > 0.8, | t > 10 Gyr. |
Figure made for this website.