File:Friedmann universes.svg
Summary
| Description |
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.
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| Date | ||||
| Source | Own work | |||
| Author | BenRG | |||
| SVG development |  This diagram was created with an unknown SVG tool. Category:Translation possible - SVGThis diagram uses embedded text that can be easily translated using a text editor.
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Formulas
This diagram uses the following exact solutions to the Friedmann equations:
See also
Some of the shown models are implemented as an animation at Cosmos-animation.
Perl code
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');
Licensing
 |
I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. |
