Advanced User

This page is for users wanting to know how the results of Lightcone are calculated. The calculator uses numerical integration where required, essentially starting from a very early time (soon after the Big Bang) to a rather late time (in the future). It takes inputs from the left-hand side of the input section and mathematically produces all the other values displayed.

Inputs are: the present Hubble radius $R_{0}$, the long term Hubble radius $R_{\infty}$, the redshift for matter-radiation density equality $z_{eq}$ and the total density parameter $\Omega$. The latter equals 1 if the spatial geometry is flat.

Since the factor $z + 1$ occurs so often, an extra parameter (the "stretch factor") $S = z + 1 = 1/a$ is defined, making the equations neater. The Hubble parameter is most conveniently expressed in terms of the $\Omega$'s (rather than in terms of Hubble radii), so we determine:

$\Large\Omega_\Lambda = (R_{0}/R_{\infty})^2$, $\Large\Omega_r = (\Omega-\Omega_\Lambda)/(1+S_{eq})$, $\Large\Omega_m = S_{eq}\Omega_r$

where $\Omega_\Lambda$ is the equivalent energy density parameter of the cosmological constant $\Lambda$, a.k.a "dark energy", $\Omega_r$ is the present radiation energy density parameter, $\Omega_m$ is the present (normal and dark) matter density parameter and $S_{eq}=z_{eq}+1$, the stretch factor at matter-radiation density equality.

The expansion-dependent Hubble parameter $H$ as a function of the present Hubble constant $H_0$, the stretch factor $S$ and the present $\Omega$'s, is given by:

$\Large H = H_0 \sqrt{\Omega_\Lambda + (1-\Omega) S^2 + \Omega_m S^3 (1+S/S_{eq})}$

Hubble radius and Cosmic time (in geometric units, where c=1):

$\Large R(S) = 1/H$, $\Large T(S) = \int_{S}^{\infty}{\frac{dS}{S H}}$

Proper distances 'now', 'then', cosmic event horizon and particle horizon

$\Large D_{now} = \int_{1}^{S}{\frac{dS}{H}}$, $\Large D_{then} = \frac{1}{S} \int_{1}^{S}{\frac{dS}{H}}$, $\Large D_{hor} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}$, $\Large D_{par} = \frac{1}{S}\int_{S}^{\infty}{\frac{dS}{H}}$

The expansion rate as a fractional distance per unit time (at time T):

$\Large\frac{da}{dT} = aH = \frac{a}{R(S)}$

To obtain all the values, it essentially means numerical integration for $S$ from zero to infinity, but practically it has been limited to $10^{-7}< S <10^{7}$ with quasi-logarithmic step sizes, e.g. a small % increase between integration steps.

[Reference] Davis: [url][/url] (2004), Appendix A. All equations converted to Stretch factor S (in place of t and a in Davis).

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