1. Paper-and-pencil cosmological calculator
S.V. Pilipenko
Astrospace Center of the Lebedev Physical Institute, 117997 Moscow, Profsojuznaja 84/32
Moscow Institute of Physics and Technology, 141700 Dolgoprudnyj, Russia, Institutskij per. 9
A cosmological calculator is usually a program which computes relations between redshift,
distance, physical and angular sizes, luminosity and apparent magnitude for some cosmological
model characterized by a convenient set of parameters such as the Hubble constant at present,
H0 , the dimensionless matter and dark energy densities Ωm and ΩΛ . Several on-line calculators
arXiv:1303.5961v1 [astro-ph.CO] 24 Mar 2013
are available at NASA NED web site (http://ned.ipac.caltech.edu/help/cosmology calc.html).
Although these calculators are ready to use, there are situations when there is no the Internet
or a PC at hand. Online calculators also may be not enough vivid to use them in education
since they usually calculate the answer only for a single input redshift. One solution to these
two problems is a calculator based on the nomogram method.
The proposed paper-and-pencil calculator is designed for the ΛCDM cosmological model with
recent cosmological parameters from the Planck mission: H0 = 67.15 km/s/Mpc, ΩΛ = 0.683
and Ωm = 0.317 [1]. The calculator contains the following quantities:
• z — redshift;
• H — current value of the Hubble constant, km/s/Mpc;
• r comov — comoving distance, Mpc;
• dm — distance modulus;
• age — age of the Universe, Gyr;
• time — lookback time, Gyr;
• size 1” — physical size of an object which is seen as an 1” arc on the sky, kpc;
• angle 1kpc — angular size of a rod with physical size 1 kpc, arcsec.
In order to use the calculator, one needs to find a known value on a respective vertical scale.
All the other values are situated on the same horizontal level. For the comfort of using a ruler,
the redshift scale is repeated twice. The calculators are available for three redshift intervals:
z < 20, z < 1, z < 0.1. The space between major (labelled) tics on each vertical scale is always
divided into ten equally-spaced intervals of the denoted value.
The code used to produce these calculators is public available and can be found at
http://code.google.com/p/cosmonom/
References
[1] Planck collaboration XVI, 2013, submitted to A&A, arXiv:astro-ph/1303.5076

2. H0 = 67.15, OLambda = 0.683, Om = 0.317
z H r_comov dm age time size 1" angle 1kpc z
0.0 0. 35. 0.0 0.0 10.00 0.0
13.0 1.0 1.00
2.0 0.50
12.0 2.0
40. 0.30
1000. 11.0 3.0 4.0
80. 10.0 4.0 0.20
42. 9.0 5.0 6.0
0.5 2000. 0.5
8.0 0.15
6.0
100.
7.0 7.0
3000. 8.0
44. 6.0
1.0 8.0 1.0
4000. 5.0
9.0
8.7
4.0 0.11
10.0 8.7
5000.
2.0 200. 46. 2.0
3.0
11.0
6000.
300. 8.0
3.0 3.0
2.0
7000. 12.0
400. 12.2
4.0 4.0
48. 12.4
0.15
12.6
5.0 8000. 5.0
600.
1.0 12.8 6.0
6.0 6.0
800. 0.8 13.0
9000.
8.0 1000. 0.20 8.0
0.6 13.2
50.
10.0 10.0
13.4 4.0
0.4
12.0 10000. 12.0
2000.
14.0 0.30 14.0
16.0 16.0
3000. 13.6
18.0 0.2 18.0
20.0 20.0

3. H0 = 67.15, OLambda = 0.683, Om = 0.317
z H r_comov dm age time size 1" angle 1kpc z
0.000 0. 25. 0.0 0.000
100.00
13.8
10. 30. 20.00
31.
0.005 20. 0.1 10.00 0.005
32.
30. 0.1
40. 33. 13.7
0.010 0.2 5.00 0.010
50. 4.00
60. 34. 0.2 0.3
70. 13.6 3.00
80. 0.4
0.020 90. 0.3 0.020
100. 35. 13.5 0.5 2.00
68.
0.4
0.030 13.4 0.030
1.50
36. 0.5
13.3
0.040 0.040
200. 13.2
1.0 1.00
13.1
69. 37.
0.060 13.0 0.060
0.80
300.
1.0
1.5
0.080 0.080
70. 38. 0.60
400.
0.100 12.5 0.100
2.0
71. 1.5
39.
72. 600. 12.0 0.40
2.0
73. 3.0
74. 800. 0.30
0.200 40. 0.200
75.
76. 11.0
1000. 3.0 4.0 0.25
77.
78.
0.300 79. 41. 0.300
80. 10.0 5.0 0.20
4.0
1500.
0.400 0.400
42. 9.0 6.0
5.0
0.500 2000. 0.500
90.
0.15
0.600 8.0 0.600
6.0 7.0
2500. 43.
0.700 100. 0.700
7.0
0.800 7.0 0.800
110. 3000.
0.900 8.0 0.900
44. 6.0
1.000 120. 1.000

4. H0 = 67.15, OLambda = 0.683, Om = 0.317
z H r_comov dm age time size 1" angle 1kpc z
0.00 0. 25. 0.0 0.00
10. 30. 13.8
31. 15.00
20. 0.1
32.
30. 0.1
40. 33. 13.7 0.2 5.00
0.01 0.01
50. 4.00
60. 34. 0.2 0.3
70. 13.6 3.00
80. 0.4
0.02 90. 0.02
0.3
100. 35. 13.5
0.5 2.00
110.
68.
120.
0.4 0.6
0.03 130. 13.4 0.03
140. 1.50
150. 0.7
36. 0.5
160. 13.3
170. 0.8
0.04 180. 0.04
190. 0.6
13.2 0.9
200.
210.
220. 0.7 1.0 1.00
0.05 0.05
230. 13.1
240. 37. 1.1
69. 0.90
250. 0.8
260. 13.0 1.2
0.06 0.06
270. 0.80
280.
0.9 1.3
290. 12.9
300.
0.07 310. 1.4 0.07
1.0 0.70
320.
12.8
330. 1.5
340.
0.08 350. 1.1 0.08
360. 12.7 1.6
70. 370. 38.
0.60
380. 1.2 1.7
0.09 390. 12.6 0.09
400.
1.8
410.
420. 1.3
12.5
430. 1.9
0.10 0.10