674:
2851:
1932:
1944:. Proper distance is also equal to the locally measured distance in the comoving frame for nearby objects. To measure the proper distance between two distant objects, one imagines that one has many comoving observers in a straight line between the two objects, so that all of the observers are close to each other, and form a chain between the two distant objects. All of these observers must have the same cosmological time. Each observer measures their distance to the nearest observer in the chain, and the length of the chain, the sum of distances between nearby observers, is the total proper distance.
1740:
2348:. The cosmological definitions used to define the velocities of distant objects are coordinate-dependent ā there is no general coordinate-independent definition of velocity between distant objects in general relativity. How best to describe and popularize that expansion of the universe is (or at least was) very likely proceeding – at the greatest scale – at above the speed of light, has caused a minor amount of controversy. One viewpoint is presented in Davis and Lineweaver, 2004.
1481:
2911:
1966:, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance. Comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a universe empty of mass, where both sorts of distance can be measured. When the density of mass in the
577:
54:
2875:
1735:{\displaystyle \chi ={\begin{cases}|\kappa |^{-1/2}\sinh ^{-1}{\sqrt {|\kappa |}}r,&{\text{if }}\kappa <0\ {\text{(a negatively curved āhyperbolicā universe)}}\\r,&{\text{if }}\kappa =0\ {\text{(a spatially flat universe)}}\\|\kappa |^{-1/2}\sin ^{-1}{\sqrt {|\kappa |}}r,&{\text{if }}\kappa >0\ {\text{(a positively curved āsphericalā universe)}}\end{cases}}}
2899:
589:
2887:
1990:. In this case, for two events which are simultaneous according to the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events, which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a
2863:
1434:
686:
Although general relativity allows the formulation of the laws of physics using arbitrary coordinates, some coordinate choices are more natural or easier to work with. Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers
2356:
Within small distances and short trips, the expansion of the universe during the trip can be ignored. This is because the travel time between any two points for a non-relativistic moving particle will just be the proper distance (that is, the comoving distance measured using the scale factor of the
1935:
The evolution of the universe and its horizons in proper distances. The x-axis is distance, in billions of light years; the left-hand y-axis is time, in billions of years since the Big Bang; the right-hand y-axis is the scale factor. This is the same model as in the earlier figure, with dark energy
677:
The evolution of the universe and its horizons in comoving distances. The x-axis is distance, in billions of light years; the left-hand y-axis is time, in billions of years since the Big Bang; the right-hand y-axis is the scale factor. This model of the universe includes dark energy which causes an
1744:
Most textbooks and research papers define the comoving distance between comoving observers to be a fixed unchanging quantity independent of time, while calling the dynamic, changing distance between them "proper distance". On this usage, comoving and proper distances are numerically equal at the
795:
Space in comoving coordinates is usually referred to as being "static", as most bodies on the scale of galaxies or larger are approximately comoving, and comoving bodies have static, unchanging comoving coordinates. So for a given pair of comoving galaxies, while the proper distance between them
725:
Most large lumps of matter, such as galaxies, are nearly comoving, so that their peculiar velocities (owing to gravitational attraction) are small compared to their Hubble-flow velocity seen by observers in moderately nearby galaxies, (i.e. as seen from galaxies just outside the
1205:
2357:
universe at the time of the trip rather than the scale factor "now") between those points divided by the velocity of the particle. If the particle is moving at a relativistic velocity, the usual relativistic corrections for time dilation must be made.
2340:. In general relativity no coordinate system on a large region of curved spacetime is "inertial", but in the local neighborhood of any point in curved spacetime we can define a "local inertial frame" in which the local speed of light is
908:
652:
factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster).
819:. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula (derived using the
665:. Comoving distance and proper distance are defined to be equal at the present time. At other times, the Universe's expansion results in the proper distance changing, while the comoving distance remains constant.
2068:
2252:
2189:
1072:
in accordance with special relativity. For a derivation see "Appendix A: Standard general relativistic definitions of expansion and horizons" from Davis & Lineweaver 2004. In particular, see
2383:
1967:
1199:
820:
178:
734:
2322:
2283:
2126:
2095:
1859:
2005:
of proper distance with respect to cosmological time) and calls this a "velocity", then the resulting "velocities" of galaxies or quasars can be above the speed of light,
1429:{\displaystyle ds^{2}=-c^{2}\,d\tau ^{2}=-c^{2}\,dt^{2}+a(t)^{2}\left({\frac {dr^{2}}{1-\kappa r^{2}}}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)\right).}
1042:
996:
549:
1108:
2936:
770:
1763:
1476:
1176:
1156:
1128:
1917:
1888:
1792:
1812:
1456:
1196:
1070:
826:
2450:
Davis, T. M.; Lineweaver, C. H. (2004). "Expanding
Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe".
619:
796:
would have been smaller in the past and will become larger in the future due to the expansion of space, the comoving distance between them remains
1962:
between the two points, observers situated between the two points would have different cosmological ages when the geodesic path crossed their own
737:
Comoving coordinates separate the exactly proportional expansion in a
Friedmannian universe in spatial comoving coordinates from the scale factor
788:. The comoving spatial coordinates tell where an event occurs while cosmological time tells when an event occurs. Together, they form a complete
2850:
2795:
2716:
2689:
2586:
2552:
2518:
2427:
353:
2544:
673:
1202:
where the metric takes the form (in reduced-circumference polar coordinates, which only works half-way around a spherical universe):
2773:
1940:
Cosmological time is identical to locally measured time for an observer at a fixed comoving spatial position, that is, in the local
562:
1931:
2020:
612:
195:
557:
271:
2001:
If one divides a change in proper distance by the interval of cosmological time where the change was measured (or takes the
1745:
current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted
2194:
2131:
645:
200:
123:
1955:) that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or
1947:
It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to
2941:
2931:
2333:
1987:
1974:'), then the cosmological coordinate system used to write this metric becomes a non-inertial coordinate system in the
699:
605:
581:
128:
1817:
2841:
2787:
2578:
1995:
347:
327:
135:
80:
2372:
1994:
in flat
Minkowski spacetime), or the coordinate distance between the events in the inertial frame where they are
948:, this expression gives the correct distance that would be measured by a hypothetical tape measure at fixed time
662:
173:
2009:. Such superluminal expansion is not in conflict with special or general relativity nor the definitions used in
919:
804:
342:
107:
807:
which explains how constant comoving distances are reconciled with proper distances that increase with time.
710:. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local
310:
190:
2399:
515:
317:
259:
2510:
2469:
528:
500:
322:
2300:
2261:
2104:
2073:
1496:
2903:
460:
420:
390:
337:
293:
281:
185:
75:
815:
Comoving distance is the distance between two points measured along a path defined at the present
2891:
2879:
2606:
2485:
2459:
2010:
1952:
711:
541:
480:
450:
415:
385:
332:
276:
45:
31:
2344:
and in which massive objects such as stars and galaxies always have a local speed smaller than
1052:
of the light particles, an observer in an inertial frame always measures the speed of light as
2791:
2769:
2712:
2685:
2679:
2582:
2559:
2548:
2514:
2423:
2377:
1983:
816:
789:
733:
719:
658:
510:
2706:
2639:
2593:
2525:
1005:
959:
2477:
2366:
740:
593:
405:
395:
380:
231:
100:
1748:
1461:
1161:
1141:
1113:
1079:
2761:
2758:
Gravitation and
Cosmology: Principles and Applications of the General Theory of Relativity
1975:
1893:
1864:
1768:
520:
455:
440:
425:
410:
400:
264:
2098:
692:
161:
2473:
952:, i.e. the "proper distance" (as defined below) after accounting for the time-dependent
2915:
2855:
1971:
1941:
1797:
1441:
1181:
1055:
938:
698:
A comoving observer is the only observer who will perceive the universe, including the
505:
465:
2097:
is the recession velocity due to the expansion of the universe (the velocity given by
2925:
2388:
1948:
1049:
945:
702:, to be isotropic. Non-comoving observers will see regions of the sky systematically
679:
490:
475:
375:
2733:
2489:
2328:. Even in special relativity the coordinate speed of light is only guaranteed to be
1110:
is defined as a quantity with the dimension of distance while the radial coordinate
2910:
2867:
1978:
of special relativity where surfaces of constant
Minkowski proper-time Ļ appear as
727:
715:
495:
470:
445:
430:
286:
2833:
691:. Such observers are called "comoving" observers because they move along with the
2813:
2657:
2572:
2538:
2504:
1923:
is just the distance that would be measured by rulers between them at that time.
718:. The velocity of an observer relative to the local comoving frame is called the
17:
2783:
785:
243:
236:
657:
roughly corresponds to where a distant object would be at a specific moment of
2827:
2255:
2002:
1963:
998:
in the integrand. By "comoving speed of light", we mean the velocity of light
703:
485:
2765:
2017:
in this sense; the total velocity of any object can be expressed as the sum
1979:
1956:
1890:
is the scale factor (e.g. Davis & Lineweaver 2004). The proper distance
707:
435:
2464:
1076:. 16ā22 in the referenced 2004 paper [note: in that paper the scale factor
2605:
Hogg, David W. (1999-05-11). "Distance measures in cosmology". p. 4.
2823:
2808:
2611:
2393:
1991:
1959:
903:{\displaystyle \chi =\int _{t_{e}}^{t}c\;{\frac {\mathrm {d} t'}{a(t')}}}
781:
688:
633:
168:
70:
63:
678:
accelerating expansion after a certain point in time, and results in an
53:
2336:; in a non-inertial frame the coordinate speed may be different from
2481:
2862:
2380:, for the apparent faster-than-light movement of distant galaxies.
1930:
732:
672:
2834:
An explanation from the Atlas of the
Universe website of distance
929:
is the time of emission of the photons detected by the observer,
30:"Physical distance" redirects here. For the general concept, see
2293:
if the light is emitted towards our position at the origin and +
784:
according to a clock of a comoving observer and is a measure of
2128:
is the "peculiar velocity" measured by local observers (with
2063:{\displaystyle v_{\text{tot}}=v_{\text{rec}}+v_{\text{pec}}}
1728:
2646:
by V. F. Mukhanov, along with the accompanying discussion.
648:
used by cosmologists to define distances between objects.
1590:(a negatively curved ‘hyperbolic’ universe)
2818:
2540:
Measuring the
Universe: The Cosmological Distance Ladder
1722:(a positively curved ‘spherical’ universe)
2819:
iCosmos: Cosmology
Calculator (With Graph Generation )
2684:. Springer Science & Business Media. p. 219.
1198:
in the commonly used comoving coordinate system for a
2839:
2452:
2396:, for the link between comoving distance to redshift.
2303:
2264:
2197:
2134:
2107:
2076:
2023:
1896:
1867:
1820:
1800:
1771:
1751:
1484:
1464:
1444:
1208:
1184:
1164:
1144:
1116:
1082:
1058:
1008:
962:
829:
743:
2824:
General method, including locally inhomogeneous case
2247:{\displaystyle v_{\text{pec}}=a(t){\dot {\chi }}(t)}
2184:{\displaystyle v_{\text{rec}}={\dot {a}}(t)\chi (t)}
1178:must be distinguished from the coordinate distance
2316:
2277:
2246:
2183:
2120:
2089:
2062:
2013:. Even light itself does not have a "velocity" of
1911:
1882:
1853:
1806:
1786:
1757:
1734:
1470:
1450:
1428:
1190:
1170:
1150:
1122:
1102:
1064:
1036:
990:
902:
764:
792:, giving both the location and time of an event.
2297:if emitted away from us) but the total velocity
644:(or physical distance) are two closely related
1438:In this case the comoving coordinate distance
613:
8:
2445:
2443:
2441:
2439:
2369:for comparison with other distance measures.
2708:An Introduction to the Science of Cosmology
2571:LachiĆØze-Rey, Marc; Gunzig, Edgard (1999).
2384:FriedmannāLemaĆ®treāRobertsonāWalker metric
861:
821:FriedmannāLemaĆ®treāRobertsonāWalker metric
620:
606:
220:
94:
52:
36:
2610:
2463:
2378:Faster-than-light Ā§ Cosmic expansion
2308:
2302:
2269:
2263:
2224:
2223:
2202:
2196:
2149:
2148:
2139:
2133:
2112:
2106:
2081:
2075:
2054:
2041:
2028:
2022:
1895:
1866:
1819:
1799:
1770:
1750:
1720:
1703:
1688:
1680:
1678:
1666:
1652:
1645:
1640:
1631:
1622:
1605:
1588:
1571:
1556:
1548:
1546:
1534:
1520:
1513:
1508:
1499:
1491:
1483:
1463:
1443:
1407:
1399:
1387:
1374:
1356:
1340:
1319:
1309:
1298:
1276:
1268:
1262:
1246:
1238:
1232:
1216:
1207:
1183:
1163:
1158:for the comoving distance. However, this
1143:
1115:
1081:
1057:
1012:
1007:
966:
961:
865:
862:
852:
845:
840:
828:
803:The expanding Universe has an increasing
780:coordinate is the elapsed time since the
742:
730:local to the observed "lump of matter").
772:This example is for the ΛCDM model.
661:, which can change over time due to the
2937:Coordinate charts in general relativity
2846:
2410:
251:
223:
115:
44:
2681:Relativity and the Nature of Spacetime
2526:Extract of page 37 (see equation 2.39)
1044:] which is time-dependent even though
2574:The Cosmological Background Radiation
2545:Springer Science & Business Media
811:Comoving distance and proper distance
700:cosmic microwave background radiation
7:
2705:Raine, Derek; Thomas, E. G. (2001).
2352:Short distances vs. long distances
956:via the inverse scale factor term
866:
348:2dF Galaxy Redshift Survey ("2dF")
25:
2644:Physical Foundations of Cosmology
2324:is generally different from
563:Timeline of cosmological theories
328:Cosmic Background Explorer (COBE)
2909:
2897:
2885:
2873:
2861:
2849:
2780:Principles of Physical Cosmology
587:
576:
575:
2814:Ned Wright's cosmology tutorial
343:Sloan Digital Sky Survey (SDSS)
196:Future of an expanding universe
2809:Distance measures in cosmology
2422:. Cambridge University Press.
2317:{\displaystyle v_{\text{tot}}}
2278:{\displaystyle v_{\text{pec}}}
2254:, the dots indicating a first
2241:
2235:
2220:
2214:
2178:
2172:
2166:
2160:
2121:{\displaystyle v_{\text{pec}}}
2090:{\displaystyle v_{\text{rec}}}
1906:
1900:
1877:
1871:
1854:{\displaystyle d(t)=a(t)\chi }
1845:
1839:
1830:
1824:
1781:
1775:
1689:
1681:
1641:
1632:
1557:
1549:
1509:
1500:
1295:
1288:
1138:Many textbooks use the symbol
1097:
1086:
1031:
1020:
985:
974:
894:
883:
753:
747:
682:beyond which we can never see.
558:History of the Big Bang theory
354:Wilkinson Microwave Anisotropy
1:
1919:between two galaxies at time
687:who perceive the universe as
550:Discovery of cosmic microwave
201:Ultimate fate of the universe
2732:J. Baez and E. Bunn (2006).
1988:inertial frame of reference
1986:from the perspective of an
1927:Uses of the proper distance
1624:(a spatially flat universe)
318:Black Hole Initiative (BHI)
2958:
2788:Princeton University Press
2736:. University of California
2658:"Homogeneity and Isotropy"
2579:Cambridge University Press
1970:is set to zero (an empty '
81:Chronology of the universe
29:
2711:. CRC Press. p. 94.
2678:Petkov, Vesselin (2009).
2627:Gravitation and Cosmology
2506:Introduction to Cosmology
2373:Expansion of the universe
1048:, at any point along the
933:is the present time, and
663:expansion of the universe
174:Expansion of the universe
2577:(illustrated ed.).
2543:(illustrated ed.).
2418:Huterer, Dragan (2023).
338:Planck space observatory
124:Gravitational wave (GWB)
1037:{\displaystyle c/a(t')}
991:{\displaystyle 1/a(t')}
954:comoving speed of light
191:Inhomogeneous cosmology
27:Measurement of distance
2656:Wright, E. L. (2009).
2537:Webb, Stephen (1999).
2318:
2279:
2248:
2185:
2122:
2091:
2064:
1937:
1913:
1884:
1855:
1808:
1788:
1765:, the proper distance
1759:
1736:
1472:
1452:
1430:
1192:
1172:
1152:
1124:
1104:
1066:
1038:
1002:comoving coordinates [
992:
904:
773:
766:
765:{\displaystyle a(t)~.}
683:
2511:John Wiley & Sons
2420:A Course in Cosmology
2400:Shape of the universe
2319:
2280:
2249:
2186:
2123:
2092:
2065:
1936:and an event horizon.
1934:
1914:
1885:
1856:
1809:
1794:at an arbitrary time
1789:
1760:
1758:{\displaystyle \chi }
1737:
1473:
1471:{\displaystyle \chi }
1453:
1431:
1193:
1173:
1171:{\displaystyle \chi }
1153:
1151:{\displaystyle \chi }
1125:
1123:{\displaystyle \chi }
1105:
1103:{\displaystyle R(t')}
1067:
1039:
993:
905:
767:
736:
676:
282:Large-scale structure
260:Shape of the universe
2503:Roos, Matts (2015).
2301:
2262:
2195:
2132:
2105:
2074:
2021:
1912:{\displaystyle d(t)}
1894:
1883:{\displaystyle a(t)}
1865:
1818:
1814:is simply given by
1798:
1787:{\displaystyle d(t)}
1769:
1749:
1482:
1462:
1442:
1206:
1182:
1162:
1142:
1114:
1080:
1056:
1006:
960:
827:
741:
669:Comoving coordinates
594:Astronomy portal
552:background radiation
529:List of cosmologists
2942:Physical quantities
2638:See the diagram on
2560:Extract of page 263
2474:2004PASA...21...97D
1976:Minkowski spacetime
1130:is dimensionless.]
857:
294:Structure formation
186:Friedmann equations
76:Age of the universe
40:Part of a series on
2932:Physical cosmology
2594:Extract of page 11
2465:astro-ph/0310808v2
2314:
2275:
2244:
2181:
2118:
2087:
2060:
2011:physical cosmology
1953:special relativity
1938:
1909:
1880:
1851:
1804:
1784:
1755:
1732:
1727:
1468:
1448:
1426:
1188:
1168:
1148:
1120:
1100:
1062:
1034:
988:
946:integral over time
900:
836:
774:
762:
712:frame of reference
684:
634:standard cosmology
333:Dark Energy Survey
277:Large quasar group
46:Physical cosmology
32:Distance (physics)
2796:978-0-691-01933-8
2718:978-0-7503-0405-4
2691:978-3-642-01962-3
2625:Steven Weinberg,
2588:978-0-521-57437-2
2581:. pp. 9ā12.
2554:978-1-85233-106-1
2520:978-1-118-92329-0
2429:978-1-316-51359-0
2311:
2272:
2232:
2205:
2157:
2142:
2115:
2084:
2057:
2044:
2031:
1984:Minkowski diagram
1807:{\displaystyle t}
1723:
1719:
1706:
1693:
1625:
1621:
1608:
1591:
1587:
1574:
1561:
1451:{\displaystyle r}
1347:
1191:{\displaystyle r}
1065:{\displaystyle c}
944:Despite being an
898:
817:cosmological time
790:coordinate system
786:cosmological time
758:
722:of the observer.
720:peculiar velocity
659:cosmological time
650:Comoving distance
646:distance measures
638:comoving distance
630:
629:
301:
300:
143:
142:
18:Light travel time
16:(Redirected from
2949:
2914:
2913:
2902:
2901:
2900:
2890:
2889:
2888:
2878:
2877:
2876:
2866:
2865:
2854:
2853:
2845:
2784:P. J. E. Peebles
2746:
2745:
2743:
2741:
2729:
2723:
2722:
2702:
2696:
2695:
2675:
2669:
2668:
2666:
2664:
2653:
2647:
2636:
2630:
2623:
2617:
2616:
2614:
2612:astro-ph/9905116
2602:
2596:
2592:
2568:
2562:
2558:
2534:
2528:
2524:
2509:(4th ed.).
2500:
2494:
2493:
2467:
2447:
2434:
2433:
2415:
2367:Distance measure
2323:
2321:
2320:
2315:
2313:
2312:
2309:
2284:
2282:
2281:
2276:
2274:
2273:
2270:
2258:), so for light
2253:
2251:
2250:
2245:
2234:
2233:
2225:
2207:
2206:
2203:
2190:
2188:
2187:
2182:
2159:
2158:
2150:
2144:
2143:
2140:
2127:
2125:
2124:
2119:
2117:
2116:
2113:
2096:
2094:
2093:
2088:
2086:
2085:
2082:
2069:
2067:
2066:
2061:
2059:
2058:
2055:
2046:
2045:
2042:
2033:
2032:
2029:
1918:
1916:
1915:
1910:
1889:
1887:
1886:
1881:
1860:
1858:
1857:
1852:
1813:
1811:
1810:
1805:
1793:
1791:
1790:
1785:
1764:
1762:
1761:
1756:
1741:
1739:
1738:
1733:
1731:
1730:
1724:
1721:
1717:
1707:
1704:
1694:
1692:
1684:
1679:
1674:
1673:
1661:
1660:
1656:
1644:
1635:
1626:
1623:
1619:
1609:
1606:
1592:
1589:
1585:
1575:
1572:
1562:
1560:
1552:
1547:
1542:
1541:
1529:
1528:
1524:
1512:
1503:
1477:
1475:
1474:
1469:
1457:
1455:
1454:
1449:
1435:
1433:
1432:
1427:
1422:
1418:
1417:
1413:
1412:
1411:
1392:
1391:
1379:
1378:
1361:
1360:
1348:
1346:
1345:
1344:
1325:
1324:
1323:
1310:
1303:
1302:
1281:
1280:
1267:
1266:
1251:
1250:
1237:
1236:
1221:
1220:
1197:
1195:
1194:
1189:
1177:
1175:
1174:
1169:
1157:
1155:
1154:
1149:
1129:
1127:
1126:
1121:
1109:
1107:
1106:
1101:
1096:
1071:
1069:
1068:
1063:
1043:
1041:
1040:
1035:
1030:
1016:
997:
995:
994:
989:
984:
970:
918:′) is the
909:
907:
906:
901:
899:
897:
893:
878:
877:
869:
863:
856:
851:
850:
849:
771:
769:
768:
763:
756:
622:
615:
608:
592:
591:
590:
579:
578:
272:Galaxy formation
232:Lambda-CDM model
221:
213:Components
95:
56:
37:
21:
2957:
2956:
2952:
2951:
2950:
2948:
2947:
2946:
2922:
2921:
2920:
2908:
2898:
2896:
2886:
2884:
2874:
2872:
2860:
2848:
2840:
2805:
2762:Steven Weinberg
2754:
2752:Further reading
2749:
2739:
2737:
2734:"Preliminaries"
2731:
2730:
2726:
2719:
2704:
2703:
2699:
2692:
2677:
2676:
2672:
2662:
2660:
2655:
2654:
2650:
2637:
2633:
2624:
2620:
2604:
2603:
2599:
2589:
2570:
2569:
2565:
2555:
2547:. p. 263.
2536:
2535:
2531:
2521:
2502:
2501:
2497:
2482:10.1071/AS03040
2449:
2448:
2437:
2430:
2417:
2416:
2412:
2408:
2363:
2354:
2304:
2299:
2298:
2265:
2260:
2259:
2198:
2193:
2192:
2135:
2130:
2129:
2108:
2103:
2102:
2077:
2072:
2071:
2050:
2037:
2024:
2019:
2018:
1929:
1892:
1891:
1863:
1862:
1816:
1815:
1796:
1795:
1767:
1766:
1747:
1746:
1726:
1725:
1701:
1662:
1639:
1628:
1627:
1603:
1594:
1593:
1569:
1530:
1507:
1492:
1480:
1479:
1460:
1459:
1440:
1439:
1403:
1383:
1370:
1366:
1362:
1352:
1336:
1326:
1315:
1311:
1308:
1304:
1294:
1272:
1258:
1242:
1228:
1212:
1204:
1203:
1180:
1179:
1160:
1159:
1140:
1139:
1136:
1112:
1111:
1089:
1078:
1077:
1054:
1053:
1023:
1004:
1003:
977:
958:
957:
928:
886:
879:
870:
864:
841:
825:
824:
813:
739:
738:
671:
655:Proper distance
642:proper distance
626:
588:
586:
568:
567:
554:
551:
544:
542:Subject history
534:
533:
525:
370:
362:
361:
358:
355:
313:
303:
302:
265:Galaxy filament
218:
206:
205:
157:
152:Expansion
145:
144:
129:Microwave (CMB)
108:Nucleosynthesis
92:
35:
28:
23:
22:
15:
12:
11:
5:
2955:
2953:
2945:
2944:
2939:
2934:
2924:
2923:
2919:
2918:
2906:
2894:
2882:
2870:
2858:
2838:
2837:
2831:
2821:
2816:
2811:
2804:
2803:External links
2801:
2800:
2799:
2777:
2753:
2750:
2748:
2747:
2724:
2717:
2697:
2690:
2670:
2648:
2631:
2629:(1972), p. 415
2618:
2597:
2587:
2563:
2553:
2529:
2519:
2513:. p. 37.
2495:
2435:
2428:
2409:
2407:
2404:
2403:
2402:
2397:
2391:
2386:
2381:
2375:
2370:
2362:
2359:
2353:
2350:
2334:inertial frame
2307:
2268:
2243:
2240:
2237:
2231:
2228:
2222:
2219:
2216:
2213:
2210:
2201:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2156:
2153:
2147:
2138:
2111:
2080:
2053:
2049:
2040:
2036:
2027:
1972:Milne universe
1942:comoving frame
1928:
1925:
1908:
1905:
1902:
1899:
1879:
1876:
1873:
1870:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1803:
1783:
1780:
1777:
1774:
1754:
1729:
1716:
1713:
1710:
1702:
1700:
1697:
1691:
1687:
1683:
1677:
1672:
1669:
1665:
1659:
1655:
1651:
1648:
1643:
1638:
1634:
1630:
1629:
1618:
1615:
1612:
1604:
1602:
1599:
1596:
1595:
1584:
1581:
1578:
1570:
1568:
1565:
1559:
1555:
1551:
1545:
1540:
1537:
1533:
1527:
1523:
1519:
1516:
1511:
1506:
1502:
1498:
1497:
1495:
1490:
1487:
1467:
1458:is related to
1447:
1425:
1421:
1416:
1410:
1406:
1402:
1398:
1395:
1390:
1386:
1382:
1377:
1373:
1369:
1365:
1359:
1355:
1351:
1343:
1339:
1335:
1332:
1329:
1322:
1318:
1314:
1307:
1301:
1297:
1293:
1290:
1287:
1284:
1279:
1275:
1271:
1265:
1261:
1257:
1254:
1249:
1245:
1241:
1235:
1231:
1227:
1224:
1219:
1215:
1211:
1187:
1167:
1147:
1135:
1132:
1119:
1099:
1095:
1092:
1088:
1085:
1061:
1033:
1029:
1026:
1022:
1019:
1015:
1011:
987:
983:
980:
976:
973:
969:
965:
939:speed of light
926:
896:
892:
889:
885:
882:
876:
873:
868:
860:
855:
848:
844:
839:
835:
832:
812:
809:
800:at all times.
761:
755:
752:
749:
746:
716:comoving frame
670:
667:
628:
627:
625:
624:
617:
610:
602:
599:
598:
597:
596:
584:
570:
569:
566:
565:
560:
555:
548:
545:
540:
539:
536:
535:
532:
531:
524:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
443:
438:
433:
428:
423:
418:
413:
408:
403:
398:
393:
388:
383:
378:
372:
371:
368:
367:
364:
363:
360:
359:
352:
350:
345:
340:
335:
330:
325:
320:
314:
309:
308:
305:
304:
299:
298:
297:
296:
284:
279:
274:
262:
254:
253:
249:
248:
247:
246:
234:
226:
225:
219:
212:
211:
208:
207:
204:
203:
198:
193:
188:
176:
171:
158:
151:
150:
147:
146:
141:
140:
139:
138:
136:Neutrino (CNB)
126:
118:
117:
113:
112:
111:
110:
93:
91:Early universe
90:
89:
86:
85:
84:
83:
78:
73:
58:
57:
49:
48:
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2954:
2943:
2940:
2938:
2935:
2933:
2930:
2929:
2927:
2917:
2912:
2907:
2905:
2895:
2893:
2883:
2881:
2871:
2869:
2864:
2859:
2857:
2852:
2847:
2843:
2835:
2832:
2829:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2806:
2802:
2797:
2793:
2789:
2785:
2781:
2778:
2775:
2774:0-471-92567-5
2771:
2768:(July 1972).
2767:
2763:
2759:
2756:
2755:
2751:
2735:
2728:
2725:
2720:
2714:
2710:
2709:
2701:
2698:
2693:
2687:
2683:
2682:
2674:
2671:
2659:
2652:
2649:
2645:
2641:
2635:
2632:
2628:
2622:
2619:
2613:
2608:
2601:
2598:
2595:
2590:
2584:
2580:
2576:
2575:
2567:
2564:
2561:
2556:
2550:
2546:
2542:
2541:
2533:
2530:
2527:
2522:
2516:
2512:
2508:
2507:
2499:
2496:
2491:
2487:
2483:
2479:
2475:
2471:
2466:
2461:
2458:(1): 97ā109.
2457:
2453:
2446:
2444:
2442:
2440:
2436:
2431:
2425:
2421:
2414:
2411:
2405:
2401:
2398:
2395:
2392:
2390:
2389:Proper length
2387:
2385:
2382:
2379:
2376:
2374:
2371:
2368:
2365:
2364:
2360:
2358:
2351:
2349:
2347:
2343:
2339:
2335:
2331:
2327:
2305:
2296:
2292:
2288:
2266:
2257:
2238:
2229:
2226:
2217:
2211:
2208:
2199:
2175:
2169:
2163:
2154:
2151:
2145:
2136:
2109:
2100:
2078:
2051:
2047:
2038:
2034:
2025:
2016:
2012:
2008:
2004:
1999:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1958:
1954:
1950:
1949:proper length
1945:
1943:
1933:
1926:
1924:
1922:
1903:
1897:
1874:
1868:
1848:
1842:
1836:
1833:
1827:
1821:
1801:
1778:
1772:
1752:
1742:
1714:
1711:
1708:
1698:
1695:
1685:
1675:
1670:
1667:
1663:
1657:
1653:
1649:
1646:
1636:
1616:
1613:
1610:
1600:
1597:
1582:
1579:
1576:
1566:
1563:
1553:
1543:
1538:
1535:
1531:
1525:
1521:
1517:
1514:
1504:
1493:
1488:
1485:
1465:
1445:
1436:
1423:
1419:
1414:
1408:
1404:
1400:
1396:
1393:
1388:
1384:
1380:
1375:
1371:
1367:
1363:
1357:
1353:
1349:
1341:
1337:
1333:
1330:
1327:
1320:
1316:
1312:
1305:
1299:
1291:
1285:
1282:
1277:
1273:
1269:
1263:
1259:
1255:
1252:
1247:
1243:
1239:
1233:
1229:
1225:
1222:
1217:
1213:
1209:
1201:
1200:FLRW universe
1185:
1165:
1145:
1133:
1131:
1117:
1093:
1090:
1083:
1075:
1059:
1051:
1050:null geodesic
1047:
1027:
1024:
1017:
1013:
1009:
1001:
981:
978:
971:
967:
963:
955:
951:
947:
942:
940:
936:
932:
925:
921:
917:
913:
890:
887:
880:
874:
871:
858:
853:
846:
842:
837:
833:
830:
822:
818:
810:
808:
806:
801:
799:
793:
791:
787:
783:
779:
778:comoving time
759:
750:
744:
735:
731:
729:
723:
721:
717:
713:
709:
705:
701:
696:
694:
690:
681:
680:event horizon
675:
668:
666:
664:
660:
656:
651:
647:
643:
639:
635:
623:
618:
616:
611:
609:
604:
603:
601:
600:
595:
585:
583:
574:
573:
572:
571:
564:
561:
559:
556:
553:
547:
546:
543:
538:
537:
530:
527:
526:
522:
519:
517:
514:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
452:
449:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
373:
366:
365:
357:
351:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
315:
312:
307:
306:
295:
292:
288:
285:
283:
280:
278:
275:
273:
270:
266:
263:
261:
258:
257:
256:
255:
250:
245:
242:
238:
235:
233:
230:
229:
228:
227:
222:
216:
210:
209:
202:
199:
197:
194:
192:
189:
187:
184:
180:
177:
175:
172:
170:
167:
163:
160:
159:
155:
149:
148:
137:
134:
130:
127:
125:
122:
121:
120:
119:
114:
109:
106:
102:
99:
98:
97:
96:
88:
87:
82:
79:
77:
74:
72:
69:
65:
62:
61:
60:
59:
55:
51:
50:
47:
43:
39:
38:
33:
19:
2904:Solar System
2786:. Publisher:
2779:
2764:. Publisher:
2757:
2738:. Retrieved
2727:
2707:
2700:
2680:
2673:
2661:. Retrieved
2651:
2643:
2634:
2626:
2621:
2600:
2573:
2566:
2539:
2532:
2505:
2498:
2455:
2451:
2419:
2413:
2355:
2345:
2341:
2337:
2329:
2325:
2294:
2290:
2286:
2285:is equal to
2099:Hubble's law
2014:
2006:
2000:
1996:simultaneous
1946:
1939:
1920:
1743:
1437:
1137:
1073:
1045:
999:
953:
949:
943:
934:
930:
923:
920:scale factor
915:
911:
814:
805:scale factor
802:
797:
794:
777:
775:
724:
704:blue-shifted
697:
685:
654:
649:
641:
637:
631:
356:Probe (WMAP)
290:
287:Reionization
268:
240:
214:
182:
165:
162:Hubble's law
153:
132:
104:
67:
2892:Outer space
2880:Spaceflight
2740:28 February
2663:28 February
1968:FLRW metric
1964:world lines
1134:Definitions
941:in vacuum.
714:called the
708:red-shifted
693:Hubble flow
311:Experiments
244:Dark matter
237:Dark energy
179:FLRW metric
116:Backgrounds
2926:Categories
2828:Fortran 77
2406:References
2256:derivative
2003:derivative
1980:hyperbolas
391:Copernicus
369:Scientists
224:Components
2766:Wiley-VCH
2230:˙
2227:χ
2170:χ
2155:˙
1957:spacelike
1849:χ
1753:χ
1709:κ
1686:κ
1676:
1668:−
1647:−
1637:κ
1611:κ
1577:κ
1554:κ
1544:
1536:−
1515:−
1505:κ
1486:χ
1466:χ
1405:ϕ
1397:θ
1394:
1372:θ
1334:κ
1331:−
1256:−
1244:τ
1226:−
1166:χ
1146:χ
1118:χ
838:∫
831:χ
689:isotropic
521:Zeldovich
421:Friedmann
396:de Sitter
323:BOOMERanG
252:Structure
217:Structure
101:Inflation
2830:software
2790:(1993).
2490:13068122
2394:Redshift
2361:See also
1992:geodesic
1960:geodesic
1705:if
1607:if
1573:if
1094:′
1028:′
982:′
891:′
875:′
798:constant
782:Big Bang
582:Category
501:Suntzeff
461:LemaƮtre
411:Einstein
376:Aaronson
169:Redshift
71:Universe
64:Big Bang
2916:Science
2856:Physics
2842:Portals
2470:Bibcode
1982:in the
1046:locally
1000:through
937:is the
506:Sunyaev
491:Schmidt
481:Penzias
476:Penrose
451:Huygens
441:Hawking
426:Galileo
2794:
2772:
2715:
2688:
2585:
2551:
2517:
2488:
2426:
2332:in an
2101:) and
2070:where
1861:where
1718:
1620:
1586:
910:where
757:
580:
516:Wilson
511:Tolman
471:Newton
466:Mather
456:Kepler
446:Hubble
406:Ehlers
386:Alpher
381:AlfvƩn
289:
267:
239:
181:
164:
156:Future
131:
103:
66:
2868:Stars
2640:p. 28
2607:arXiv
2486:S2CID
2460:arXiv
728:group
496:Smoot
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