Redshift and Cosmological Distances

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David_Sims
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Joined: Sun Apr 07, 2024 9:39 pm

Redshift and Cosmological Distances

Post by David_Sims » Mon Jan 20, 2025 1:44 am

Relationship between
• Red Shift
• Light Travel Distance
• Comoving Distance
• Proper Distance at the Time of Light Emission
• Age of the Universe at the Time of Light Emission


Constants and assumptions

Mass density parameter, Ωm = 0.311016
Dark energy density parameter, ΩΛ = 0.688900
Radiation density parameter, Ωr = 0.000084
Spacetime curvature parameter, Ωk = 0 ← flat spacetime

Speed of light, c = 299792458 m/s
Hubble constant, H = 67.4 km/sec/Mpc

Number of kilometers in a megaparsec, Q = 3.0856775815e+19
Number of meters in a light year, L = 9.4607304726e+15

A distance scale factor, B = cQ/H
B = 1.3724968349e+26 meters

The distance scale factor in light years
B/L = 14,507,302,991 light years

Light travel distance, t, in light years

t = (B/L) · ∫(0,z) { 1 / [ (1+x) √[ Ωm(1+x)³ + Ωr(1+x)⁴ + ΩΛ ] ] } dx

The comoving distance, d, in light years

d = (B/L) · ∫(0,z) { 1 / [ √[ Ωm(1+x)³ + Ωr(1+x)⁴ + ΩΛ ] ] } dx

Note that the above relationships are true only when the curvature of spacetime is zero — i.e., when Ωk=0.

When Ωk≠0, a further step (not shown here) is required.

The proper distance at the time of light emission, D, in light years

D = d/(z+1)

The age of the universe at the time of observation

T = (B/L) (1 year/lightyear) · ∫(0,∞) { 1 / [ (1+x) √[ Ωm(1+x)³ + Ωr(1+x)⁴ + ΩΛ ] ] } dx

The age of the universe at the time of light emission

T’ = T−t

Note that the proper distance at the time of light emission has a maximum, Dmax = 5,869,055,037 light years, at redshift z=1.59218, at which the corresponding light travel distance, t, is 9,772,747,726 light years, the corresponding comoving distance, d, is 15,213,647,086 light years, and the corresponding age of universe at light emission, T', is 4,067,852,739 years.
.

Example problem #1

The redshift
z = 0.0574869863 ← input

The light travel distance

t = (14,507,302,991 light years) · ∫(0 , 0.0574869863) { 1 / [ (1+x) √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] ] } dx

t = 800,000,000 light years

The comoving distance

d = (14,507,302,991 light years) · ∫(0 , 0.0574869863) { 1 / [ √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] } dx

d = 822,674,744 light years

The proper distance at the time of light emission

D = 822,674,744 light years / (1 + 0.0574869863)
D = 777,952,594 light years

The age of the universe at the time of observation

T = (14,507,302,991 years) · ∫(0,∞) { 1 / [ (1+x) √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] ] } dx

T = 13,840,600,465 years

The age of the universe at the time of light emission

T’ = 13,840,600,465 years − 800,000,000 years
T’ = 13,040,600,465 years
.

Example problem #2

The redshift
z = 12.7 ← input

The light travel distance

t = (14,507,302,991 light years) · ∫(0,12.7) { 1 / [ (1+x) √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] ] } dx

t = 13,500,400,949 light years

The comoving distance

d = (14,507,302,991 light years) · ∫(0,12.7) { 1 / [ √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] } dx

d = 33,178,822,601 light years

The proper distance at the time of light emission

D = 33,178,822,601 light years / (1 + 12.7)
D = 2,421,811,868 light years

The age of the universe at the time of observation

T = (14,507,302,991 years) · ∫(0,∞) { 1 / [ (1+x) √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] ] } dx

T = 13,840,600,465 years

The age of the universe at the time of light emission

T’ = 13,840,600,465 years − 13,500,400,949 years
T’ = 340,199,516 years
.

Example problem #3

The redshift
z = 0.7579951971 ← input

The light travel distance

t = (14,507,302,991 light years) · ∫(0 , 0.7579951971) { 1 / [ (1+x) √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] ] } dx

t = 6,835,778,094 light years

The comoving distance

d = (14,507,302,991 light years) · ∫(0 , 0.7579951971) { 1 / [ √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] } dx

d = 9,000,000,000 light years

The proper distance at the time of light emission

D = 9,000,000,000 light years / (1 + 0.7579951971)
D = 5,119,467,911 light years

The age of the universe at the time of observation

T = (14,507,302,991 years) · ∫(0,∞) { 1 / [ (1+x) √[ 0.311016 (1+x)³ + 0.000084 (1+x)⁴ + 0.6889 ] ] } dx

T = 13,840,600,465 years

The age of the universe at the time of light emission

T’ = 13,840,600,465 years − 6,835,778,094 years
T’ = 7,004,822,371 years
.

TABLE

z = redshift
D = proper distance at the time of light emission (MLY)
t = light travel distance (MLY)
d = comoving distance (MLY)
T’ = age of the universe at the time of light emission (MY)

z, D, t, d, T’

0 , 0 , 0 , 0 , 13841
1 , 5559 , 7966 , 11118 , 5874
2 , 5793 , 10550 , 17380 , 3289
3 , 5323 , 11688 , 21295 , 2152
4 , 4801 , 12296 , 24006 , 1543
5 , 4336 , 12665 , 26018 , 1175
6 , 3940 , 12907 , 27586 , 932
7 , 3606 , 13077 , 28851 , 763
8 , 3322 , 13200 , 29900 , 639
9 , 3078 , 13294 , 30788 , 546
10 , 2868 , 13367 , 31552 , 473
11 , 2684 , 13425 , 32218 , 415
12 , 2523 , 13472 , 32806 , 368
13 , 2380 , 13511 , 33329 , 329
14 , 2253 , 13543 , 33800 , 296
15 , 2139 , 13571 , 34225 , 269
16 , 2036 , 13594 , 34613 , 245
17 , 1942 , 13615 , 34967 , 225
18 , 1857 , 13632 , 35294 , 207
19 , 1779 , 13648 , 35595 , 192
20 , 1708 , 13661 , 35874 , 178
1088 , 41.75 , 13840 , 45467 , 0.380 ← CMBR
∞ , 0 , 13841 , 46405 , 0 ← Big Bang

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