Hello J, "The Edges of Our Universe" is a nice and very accesible paper by Toby Ord that discusses different natural "edges" to the universe that arise in General Relativity (and in particular the ΛCDM model) and some futurist implications.
https://arxiv.org/abs/2104.01191 Anyway, the tail end of the paper encourages the reader to do some actual calculations in Excel, providing the necessary starting point. I decided to whip this up in J and wanted to share: Bonus question: For you j903ers, does compensated summation in S improve comparison with Table 3 in the paper? Extra Bonus question: Can you tacit-ify S? === LCDM.ijs === NB. Small program to calculate various functiosn for the ΛCDM model of our NB. universe. Entirely culled from Toby Ord's excellent and accessible NB. "The Edges of Our Universe". NB. NB. Note that all distances are measured in comoving coordinates. NB. a(t): The scale factor. Unitless NB. NB. There is no closed-form expression for a(t) which makes it difficult to NB. calculate. Fortunately, however, a(t) is bijective, and it's inverse, t(a), NB. has a closed-form expression. Thus, here we treat a as an independent NB. variable and calculate time t below. NB. NB. Note that a(t) grows exponentially in t. NB. NB. x a y: Generates a logarithmic division of the range 10^(-x) to 10^x where NB. succesive values have a ratio of x. NB. (a y) -: 1.005 a y a=: 1.005&$: : (10&^@i:@(] j. <.@<:@((% 10&^.)~ 2&*))) NB. z(a): The redshift NB. a_now NB. z(a) = ------ - 1 NB. a(t) NB. NB. Note: a_now := 1 z=: _1 + % NB. Integral: v is the function to integrate and u is the measure. NB. NB. / b NB. u S v x < -- > | v(x) du(x) NB. / a NB. NB. where a = {.x, b = {:x, and the partition is given by u(x) S=: 2 : '+/\@(+/\inv@u * v)' NB. Time t(a): Inverse function of a(t). Units: years. NB. NB. / a a' da' NB. t(a) = 1/H | ---------------------------- NB. / 0 √(R + M a' + K a'^2 + Λ a^4) NB. NB. H =: H_0 = 1/14.4 Gy^_1 Hubble parameter NB. R =: Ω_{R,0} = 9.8e_5 Radiation density NB. M =: Ω_{M,0} = 0.308 Matter density NB. K =: Ω_{K,0} = 0 Curvature of space NB. Λ =: Ω_{Λ,0} = 0.692 Dark matter density t=: ] S (14.4e9&*@(*: %&%: 9.8e_5 0.308 0 0 0.692&p.)) NB. d_γ(t): Light travel distance since Big Bang to time t. Units: light-years NB. Equivalent to r_o: Radius of observable universe. NB. NB. / t c NB. d_γ(t) = | ------ dt' NB. / 0 a(t') NB. NB. c = 1 ly/y d=: t S % ro=: d NB. D_γ: Asymptotic limit of d_γ(t) NB. Note that a grows logarithmically with time, so a parameter of 100 NB. here gives roughly the value of d at 2^10^100 years. D=: {: d a 100 NB. r_a: Radius of the affectable universe ra=: D - d NB. Convenience verbs and modifiers T=: [: (t ,. z ,. ] ,. d ,. ra) a F=: ("_1)(#~`)(`:6) ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
