Warum fliegen Photonen immer mit Lichtgeschwindigkeit
07.03.2016 um 00:17Trotzdem war die Rekombinationsfläche keine "14 mya ly" und auch keine "14 mio ly" von uns entfernt als die CMB sich auf den Weg zu uns gemacht hat. Also wo ist deine Rechnung?
Input:
(* Syntax: Mathematica ||| lcdm.yukterez.net *)
kg = 1; m = 1; sek = 1; K = 1; (* Units *)
set = {"GlobalAdaptive", "MaxErrorIncreases" -> 100, Method -> "GaussKronrodRule"}; (* Integration Rule *)
n = 100; (* Recursion Depth *)
tE = 300 Gyr; (* Eventhorizon Limit *)
c = 299792458 m/sek; (* Lightspeed *)
ca = 1; (* Perturbation Velocity *)
G = 667384*^-16 m^3 kg^-1 sek^-2; (* Newton Constant *)
Gyr = 10^7*36525*24*3600*sek; (* Billion Year Scale *)
Glyr = Gyr*c; (* Billion Lightyear Scale *)
Mpc = 30856775777948584200000 m; (* Megaparsec *)
kB = 13806488*^-30 kg m^2/sek^2/K; (* Boltzmann *)
h = 662606957*^-42 kg m^2/sek; (* Planck *)
ρR = 8 π^5 kB^4 T^4/15/c^5/h^3; (* Radiation Density *)
ρΛ = ρc[H] ΩΛ; (* Dark Energy Density *)
T = 2725/1000 K; (* CMB Temperature *)
ρc[H_] := 3 H^2/8/π/G; (* Critical Density *)
H0 = 67150 m/Mpc/sek; (* Hubble Constant *)
ΩR = ρR/ρc[H]; ΩM = 317/1000; ΩΛ = 683/1000 - ΩR; ΩT = ΩR + ΩM + ΩΛ; ΩK = 1 - ΩT; (* Density Parameters *)
aE[t_] := Power[(Sqrt[ΩM/ΩΛ] Sinh[(3 H0 Sqrt[ΩΛ])/2 t])^2, (3)^-1]; (* Solving Region *)
w[a_, w0_] := (1 + w0) (Sqrt[1 + (ΩΛ^-1 - 1) a^-3] - (ΩΛ^-1 -1) a^-3 Tanh[1/Sqrt[1 + (ΩΛ^-1 - 1) a^-3]]^-1)^2 (1/Sqrt[ΩΛ] - (ΩΛ^-1 - 1) Tanh[Sqrt[ΩΛ]]^-1)^-2 -1;
F[a_, w0_] := Sqrt[ΩR a^-4 + ΩM a^-3 + ΩK a^-2 + ΩΛ a^(-3 (w[a, w0] + 1))]; (* Density Function by Scalefactor*)
φ[z_, w0_] := Sqrt[ΩR (z + 1)^4 + ΩM (z + 1)^3 + ΩK (z + 1)^2 + ΩΛ ((1 + z)^(3 (w[1/(z + 1), w0] + 1))) ]; (* Density Function by Redshift *)
H[a_, w0_] := H0 F[a, w0]; (* Hubble Parameter by Scalefactor *)
ε[z_, w0_] := H0 φ[z, w0]; (* Hubble Parameter by Redshift *)
int[f_, {x_, xmin_, xmax_}] := Quiet[NIntegrate[f, {x, xmin, xmax}, Method -> set, MaxRecursion -> n]];
ta[A_, w0_] := int[1/a/ H[a, w0], {a, 0, A}]; (* Time by Scalefactor *)
α[τ_, w0_] := Quiet[A /.FindRoot[ta[A, w0] - τ, {A, 1}]] (* Scalefactor by Time *)
tz[Z_, w0_] := int[1/(1 + z)/ ε[z, w0], {z, Z, \[Infinity]}]; (* Time by Redshift *)
χ[τ_, w0_] := Z /. Quiet[FindRoot[tz[Z, w0] - τ, {Z, 0}]] (* Redshift by Time *)
rH[τ_, w0_] := c/H[α[τ, w0], w0]; (* Hubble Radius *)
lC[τ_, w0_] := int[-c α[τ, w0]/a^2/H[a, w0], {a, 1, α[τ, w0]}]; (* Light Cone of t0 *)
Lc[τ_, t_, w0_] := int[-c α[τ, w0]/a^2/H[a, w0], {a, α[t, w0], α[τ, w0]}]; (* Light Cone of t *)
eH[τ_, w0_] := α[τ, w0] int[c/(α[time, w0]), {time, τ, tE}]; (* Event Horizon *)
pH[τ_, w0_] := int[-α[τ, w0] c/a^2/H[a, w0], {a, α[τ, w0], 0}]; (* Particle Horizon *)
g[τ_, w0_] := tc /. Quiet[FindRoot[pH[tc, w0]/c - τ, {tc, τ}]]; (* Conformal Time *)
ωR[τ_, w0_] := ΩR α[τ, w0]^-4/ρc[H[α[τ, w0]]]; (* Radiation Evolution *)
ωM[τ_, w0_] := ΩM α[τ, w0]^-3/ρc[H[α[τ, w0]]]; (* Matter Evolution *)
ωK[τ_, w0_] := ΩK α[τ, w0]^-2/ρc[H[α[τ, w0]]]; (* Curvature Evolution *)
ωΛ[τ_, w0_] := ΩΛ α[τ, w0]^(-3 (w[α[τ, w0], w0] + 1))/ρc[H[α[τ, w]]]; (* Dark Energy Evolution *)
t0[w_] := ta[1, w0]/Gyr; (* Age of the Universe, now *)
"t0 in Gyr" -> t0[-1]
z1 = 1089; (* Redshift *)
t1 = tz[z, -1]; "Age at Emission" -> t1/Gyr "Gyr"
tr = tz[0, -1] - tz[z, -1]; "Light Travel Time" -> tr/Gyr "Gyr"
lC1 = lC[t, -1]; "Distance at Emission" -> lC1/Glyr "Glyr"
lC2 = lC[t, -1] (z1 + 1); "Distance at Absorption" -> lC2/Glyr "Glyr"
v1 = ε[z, -1] lC1; "Recessional Velocity at Emission" -> v1/c "c"
v2 = H0 lC2; "Recessional Velocity at Absorption" -> v2/c "c"
Output:
"Age at Emission" -> 0.000403849665 "Gyr"
"Light Travel Time" -> 13.8192005139 "Gyr"
"Distance at Emission" -> 0.0416130189885 "Glyr"
"Distance at Absorption" -> 45.3581906975 "Glyr"
"Recessional Velocity at Emission" -> 63.1225507272 "c"
"Recessional Velocity at Absorption" -> 3.11497927827 "c"
Mit Betonung auf
"Distance at Emission" -> 0.0416130189885 "Glyr"
, Hilbert Jämes
Input:
(* Syntax: Mathematica ||| lcdm.yukterez.net *)
kg = 1; m = 1; sek = 1; K = 1; (* Units *)
set = {"GlobalAdaptive", "MaxErrorIncreases" -> 100, Method -> "GaussKronrodRule"}; (* Integration Rule *)
n = 100; (* Recursion Depth *)
tE = 300 Gyr; (* Eventhorizon Limit *)
c = 299792458 m/sek; (* Lightspeed *)
ca = 1; (* Perturbation Velocity *)
G = 667384*^-16 m^3 kg^-1 sek^-2; (* Newton Constant *)
Gyr = 10^7*36525*24*3600*sek; (* Billion Year Scale *)
Glyr = Gyr*c; (* Billion Lightyear Scale *)
Mpc = 30856775777948584200000 m; (* Megaparsec *)
kB = 13806488*^-30 kg m^2/sek^2/K; (* Boltzmann *)
h = 662606957*^-42 kg m^2/sek; (* Planck *)
ρR = 8 π^5 kB^4 T^4/15/c^5/h^3; (* Radiation Density *)
ρΛ = ρc[H] ΩΛ; (* Dark Energy Density *)
T = 2725/1000 K; (* CMB Temperature *)
ρc[H_] := 3 H^2/8/π/G; (* Critical Density *)
H0 = 67150 m/Mpc/sek; (* Hubble Constant *)
ΩR = ρR/ρc[H]; ΩM = 317/1000; ΩΛ = 683/1000 - ΩR; ΩT = ΩR + ΩM + ΩΛ; ΩK = 1 - ΩT; (* Density Parameters *)
aE[t_] := Power[(Sqrt[ΩM/ΩΛ] Sinh[(3 H0 Sqrt[ΩΛ])/2 t])^2, (3)^-1]; (* Solving Region *)
w[a_, w0_] := (1 + w0) (Sqrt[1 + (ΩΛ^-1 - 1) a^-3] - (ΩΛ^-1 -1) a^-3 Tanh[1/Sqrt[1 + (ΩΛ^-1 - 1) a^-3]]^-1)^2 (1/Sqrt[ΩΛ] - (ΩΛ^-1 - 1) Tanh[Sqrt[ΩΛ]]^-1)^-2 -1;
F[a_, w0_] := Sqrt[ΩR a^-4 + ΩM a^-3 + ΩK a^-2 + ΩΛ a^(-3 (w[a, w0] + 1))]; (* Density Function by Scalefactor*)
φ[z_, w0_] := Sqrt[ΩR (z + 1)^4 + ΩM (z + 1)^3 + ΩK (z + 1)^2 + ΩΛ ((1 + z)^(3 (w[1/(z + 1), w0] + 1))) ]; (* Density Function by Redshift *)
H[a_, w0_] := H0 F[a, w0]; (* Hubble Parameter by Scalefactor *)
ε[z_, w0_] := H0 φ[z, w0]; (* Hubble Parameter by Redshift *)
int[f_, {x_, xmin_, xmax_}] := Quiet[NIntegrate[f, {x, xmin, xmax}, Method -> set, MaxRecursion -> n]];
ta[A_, w0_] := int[1/a/ H[a, w0], {a, 0, A}]; (* Time by Scalefactor *)
α[τ_, w0_] := Quiet[A /.FindRoot[ta[A, w0] - τ, {A, 1}]] (* Scalefactor by Time *)
tz[Z_, w0_] := int[1/(1 + z)/ ε[z, w0], {z, Z, \[Infinity]}]; (* Time by Redshift *)
χ[τ_, w0_] := Z /. Quiet[FindRoot[tz[Z, w0] - τ, {Z, 0}]] (* Redshift by Time *)
rH[τ_, w0_] := c/H[α[τ, w0], w0]; (* Hubble Radius *)
lC[τ_, w0_] := int[-c α[τ, w0]/a^2/H[a, w0], {a, 1, α[τ, w0]}]; (* Light Cone of t0 *)
Lc[τ_, t_, w0_] := int[-c α[τ, w0]/a^2/H[a, w0], {a, α[t, w0], α[τ, w0]}]; (* Light Cone of t *)
eH[τ_, w0_] := α[τ, w0] int[c/(α[time, w0]), {time, τ, tE}]; (* Event Horizon *)
pH[τ_, w0_] := int[-α[τ, w0] c/a^2/H[a, w0], {a, α[τ, w0], 0}]; (* Particle Horizon *)
g[τ_, w0_] := tc /. Quiet[FindRoot[pH[tc, w0]/c - τ, {tc, τ}]]; (* Conformal Time *)
ωR[τ_, w0_] := ΩR α[τ, w0]^-4/ρc[H[α[τ, w0]]]; (* Radiation Evolution *)
ωM[τ_, w0_] := ΩM α[τ, w0]^-3/ρc[H[α[τ, w0]]]; (* Matter Evolution *)
ωK[τ_, w0_] := ΩK α[τ, w0]^-2/ρc[H[α[τ, w0]]]; (* Curvature Evolution *)
ωΛ[τ_, w0_] := ΩΛ α[τ, w0]^(-3 (w[α[τ, w0], w0] + 1))/ρc[H[α[τ, w]]]; (* Dark Energy Evolution *)
t0[w_] := ta[1, w0]/Gyr; (* Age of the Universe, now *)
"t0 in Gyr" -> t0[-1]
z1 = 1089; (* Redshift *)
t1 = tz[z, -1]; "Age at Emission" -> t1/Gyr "Gyr"
tr = tz[0, -1] - tz[z, -1]; "Light Travel Time" -> tr/Gyr "Gyr"
lC1 = lC[t, -1]; "Distance at Emission" -> lC1/Glyr "Glyr"
lC2 = lC[t, -1] (z1 + 1); "Distance at Absorption" -> lC2/Glyr "Glyr"
v1 = ε[z, -1] lC1; "Recessional Velocity at Emission" -> v1/c "c"
v2 = H0 lC2; "Recessional Velocity at Absorption" -> v2/c "c"
Output:
"Age at Emission" -> 0.000403849665 "Gyr"
"Light Travel Time" -> 13.8192005139 "Gyr"
"Distance at Emission" -> 0.0416130189885 "Glyr"
"Distance at Absorption" -> 45.3581906975 "Glyr"
"Recessional Velocity at Emission" -> 63.1225507272 "c"
"Recessional Velocity at Absorption" -> 3.11497927827 "c"
Mit Betonung auf
"Distance at Emission" -> 0.0416130189885 "Glyr"
, Hilbert Jämes