Model Time

t

$a\left(t\right)$

$\dot{a}\left(t\right)$

$\ddot{a}\left(t\right)$

Redshift

$z=\frac{1}{a\left(t\right)}-1$

Interpretation

$t\lesssim -4$

$Re\left[a\left(t\right)\right]\approx 1$

$Im\left[a\left(t\right)\right]\approx 0$

Far negative time.

$-4\lesssim t<0$

$Complex\left[a\left(t\right)\right]$

wave packet

$Complex\left[\dot{a}\left(t\right)\right]$

wave packet

$Complex\left[\ddot{a}\left(t\right)\right]$

wave packet

Oscillation of scale factor in a localized interval of near negative time.

$t=0$

0

1

$\infty$

$\infty$

Big Bang, $\ddot{a}\left(t\right)$ explodes.

$0<t\lesssim 0.064$

Early accelerating expansion. Scale factor monotonically increasing with the arrow of time.

$t\approx 0.064$

≈+0.10

Max ≈ +1.71

0

≈9.03

End of the Dark Ages, reionization and large-scale structuration of the universe.

$0.064\lesssim t\lesssim 0.162$

Decelerating expansion. Matter dominated.

$t\approx 0.162$

≈+0.26

≈+1.47

Min ≈ −3.25

≈2.87

Deceleration extremum. Matter dominated.

$0.162\lesssim t\lesssim 0.668$

Decelerating expansion. Matter dominated.

$t\approx 0.668$

≈+0.73

Min ≈ +0.71

0

≈0.36

End of deceleration.

$0.668\lesssim t<1$

Late accelerating expansion. “Dark Energy” dominated.

$t=1$

1

1

2

0

Time now. Late accelerating expansion. “Dark Energy” dominated.

$1<t\lesssim 4$

Near positive time. Late accelerating expansion. “Dark Energy” dominated.

$t=4.0708$

≈8 × 10¹⁸⁴

$a\left(t\right)$ ≈ Number of Planck volumes in observable universe

$t\gtrsim 4$

Far positive time. Not physical.