BeskrivningFriedmann universes.svg	English: The age and ultimate fate of the universe can be determined by measuring the Hubble constant today and extrapolating with the observed value of the deceleration parameter, uniquely characterized by values of density parameters (ΩM for matter and ΩΛ for dark energy). A "closed universe" with ΩM > 1 and ΩΛ = 0 comes to an end in a Big Crunch and is considerably younger than its Hubble age. An "open universe" with ΩM ≤ 1 and ΩΛ = 0 expands forever and has an age that is closer to its Hubble age. For the accelerating universe with nonzero ΩΛ that we inhabit, the age of the universe is coincidentally very close to the Hubble age. Intended as a replacement for Universe.svg and Universos.gif.
Datum	23 september 2009
Källa	Eget arbete
Skapare	BenRG
SVG utveckling InfoField	Källkoden till denna SVG är giltig.  Den här Det diagram skapades med Other tools   This diagram uses embedded text that can be easily translated using a text editor. Den här filen ersätter filen File:Universe.svg. Notera att ersatta bilder i normalta fall inte ska raderas. Deutsch ∙ English ∙ español ∙ فارسی ∙ français ∙ magyar ∙ Bahasa Indonesia ∙ italiano ∙ 日本語 ∙ 한국어 ∙ македонски ∙ മലയാളം ∙ Nederlands ∙ polski ∙ prūsiskan ∙ português do Brasil ∙ русский ∙ slovenščina ∙ svenska ∙ 中文（简体） ∙ 中文（繁體） ∙ +/−

This diagram uses the following exact solutions to the Friedmann equations:

${\displaystyle {\begin{cases}a(t)=H_{0}t&\Omega _{M}=\Omega _{\Lambda }=0\\{\begin{cases}a(q)={\tfrac {\Omega _{M}}{2(1-\Omega _{M})}}(\cosh q-1)\\t(q)={\tfrac {\Omega _{M}}{2H_{0}(1-\Omega _{M})^{3/2}}}(\sinh q-q)\end{cases}}&0<\Omega _{M}<1,\ \Omega _{\Lambda }=0\\a(t)=\left({\tfrac {3}{2}}H_{0}t\right)^{2/3}&\Omega _{M}=1,\ \Omega _{\Lambda }=0\\{\begin{cases}a(q)={\tfrac {\Omega _{M}}{2(\Omega _{M}-1)}}(1-\cos q)\\t(q)={\tfrac {\Omega _{M}}{2H_{0}(\Omega _{M}-1)^{3/2}}}(q-\sin q)\end{cases}}&\Omega _{M}>1,\ \Omega _{\Lambda }=0\\a(t)=\left({\tfrac {\Omega _{M}}{\Omega _{\Lambda }}}\sinh ^{2}\left({\tfrac {3}{2}}{\sqrt {\Omega _{\Lambda }}}H_{0}t\right)\right)^{1/3}&0<\Omega _{M}<1,\ \Omega _{\Lambda }=1-\Omega _{M}\end{cases}}}$

Some of the shown models are implemented as an animation at Cosmos-animation.

use strict;
use Svg;
use Math::Trig qw(sinh cosh acos asinh acosh pi);

sub ScaleFunc {
	my ($H0, $M0, $with_lambda) = @_;
	if ($M0 == 1) {
		my $q0 = 2/(3*$H0);
		return sub { my ($q) = @_; ($q - $q0, (1.5 * $H0 * $q) ** (2/3)) };
	}
	if ($with_lambda) {
		my $L0 = 1 - $M0;
		# assume 0 < $M0 < 1
		my $a = ($M0/$L0) ** (1/3);
		my $b = 1.5 * $H0 * sqrt($L0);
		my $q0 = asinh(sqrt($L0/$M0)) / $b;
		return sub { my ($q) = @_; ($q - $q0, $a * (sinh($b * $q) ** (2/3))) }
	} else {
		# \Omega_{\Lambda_0} = 0
		my $k0 = 1 - $M0;
		if ($M0 == 0) {
			return sub { my ($q) = @_; ($q - 1/$H0, $q * $H0) }
		} else {
			my $a = $M0 / (2 * abs($k0));
			my $b = 1 / ($H0 * sqrt(abs($k0)));
			my $c = $a * $b;
			if ($M0 > 1) {
				my $d = $a * (2 / ($H0 * $M0) - acos(2/$M0 - 1) * $b);
				return sub { my ($q) = @_; ($c * ($q - sin($q)) + $d, $a * (1 - cos($q))) }
			} else {
				# 0 < M < 1
				my $d = $a * (acosh(2/$M0 - 1) * $b - 2 / ($H0 * $M0));
				return sub { my ($q) = @_; ($c * (sinh($q) - $q) + $d, $a * (cosh($q) - 1)) }
			}
		}
	}
}

sub SubscriptedText {
	my $text = shift;
	$text->add(shift);
	my $sub = 0;
	for my $t (@_) {
		$sub = !$sub;
		$text->tspan($sub ? (dy => 4, 'font-size' => 12) : (dy => -4))->add($t);
	}
}

my ($image_width,$image_height) = (620,500);
my ($origin_x, $origin_y) = (30.5,450.5);
my $pad_right = 70;
my ($tlo, $thi, $ahi) = (-15,18,2.5);

my $svg = new Svg(width => $image_width, height => $image_height);
#	$svg->rect(width => $image_width, height => $image_height, fill => 'gray');
$svg->defs()->marker(id => 'arrowhead', refX => 0, refY => 3, markerWidth => 10, markerHeight => 6, markerUnits => 'userSpaceOnUse', orient => 'auto')->path(d => 'M 0,0 L 10,3 L 0,6 z');
my $traces = $svg->group(stroke => 'black', 'stroke-width' => 2, fill => 'none');
my $axes = $svg->group(stroke => 'black', 'stroke-width' => 1, fill => 'none');
my $labels = $svg->group('font-family' => 'Nimbus Roman No9 L, Times, serif', 'font-size' => 20, 'text-anchor' => 'middle', stroke => 'none', fill => 'black');
my $H0 = 1 / 13.95;
my $M0 = 0.279;
my ($graphscalex,$graphscaley) = (($image_width-$origin_x-$pad_right)/($thi-$tlo), -$origin_y/$ahi);
my ($graphofsx,$graphofsy) = ($origin_x - $tlo * $graphscalex, $origin_y);
for my $z ([0,0,30,'none'],[$M0,0,3.17,'1,4'],[1,0,26,'2,2'],[6,0,2*pi,'1,3,4,3'],[$M0,1,27,'5,3']) {
	my ($m0,$with_lambda,$max_q,$dashes) = @$z;
	my $f = ScaleFunc($H0,$m0,$with_lambda);
	my (@x,@y);
	for my $i (0..200) {
		($x[$i],$y[$i]) = &$f($i / 200 * $max_q);
	}
	$traces->path('stroke-dasharray' => $dashes, ($m0 == 6 ? () : ('marker-end' => 'url(#arrowhead)')), d => MakePath(\@x, \@y, $graphscalex, $graphscaley, $graphofsx, $graphofsy, 1));
}
$axes->line(x1 => $origin_x, y1 => $image_height-20, x2 => $origin_x, y2 => 20, 'marker-end' => 'url(#arrowhead)');
$axes->line(x1 => 10, y1 => $origin_y, x2 => $image_width - $pad_right + 10, y2 => $origin_y, 'marker-end' => 'url(#arrowhead)');
$labels->text(x => ($origin_x + $image_width) / 2, y => $image_height-10)->add('Billions of years from now');
my $path = '';
for my $gyr (-13.7, -10, -5, 0, 5, 10, 15) {
	my $x = int($gyr * $graphscalex + $graphofsx);
	my $y = $origin_y-5;
	$path .= "M$x.5,${y}l0,10";
	$labels->text(x => $x, y => $origin_y + 20)->add($gyr);
}
$axes->path(d => $path);
$labels->circle(cx => $graphofsx, cy => $graphscaley + $graphofsy, r => 4);
$labels->text(x => $graphofsx-5, y => $graphscaley + $graphofsy, 'text-anchor' => 'end')->add('Now');
$labels->text()->rotate(-90)->translate($origin_x - 8, $origin_y / 2)->add("Average distance between galaxies");
my $trace_labels = $labels->group('font-family' => 'DejaVu Serif, serif', 'font-size' => 16);
SubscriptedText($trace_labels->text(x => 465, y => 30, 'text-anchor' => 'end'), "\x{3A9}", 'M', " = 0.3, \x{3A9}", "\x{39B}", " = 0.7");
SubscriptedText($trace_labels->text(x => 520, y => 50, 'text-anchor' => 'start'), "\x{3A9}", 'M', ' = 0');
SubscriptedText($trace_labels->text(x => 535, y => 70, 'text-anchor' => 'start'), "\x{3A9}", 'M', ' = 0.3');
SubscriptedText($trace_labels->text(x => 540, y => 95, 'text-anchor' => 'start'), "\x{3A9}", 'M', ' = 1');
SubscriptedText($trace_labels->text(x => 540, y => 400, 'text-anchor' => 'start'), "\x{3A9}", 'M', ' = 6');

$svg->write('Friedmann universes.svg');

Jag, upphovsrättsinnehavaren till detta verk, släpper detta verk i public domain. Detta gäller globalt. I vissa länder kan detta inte vara juridiskt möjligt; i så fall: Jag ger härmed envar rätten att använda detta verk för alla ändamål, utan några villkor, förutom villkor som lagen ställer.