2350:
1569:. Since this increases with the volume of the universe, the expansion pressure is effectively constant, independent of the scale of the universe, while the other terms decrease with time. Thus, as the density of other forms of matter ā dust and radiation ā drops to very low concentrations, the cosmological constant (or "dark energy") term will eventually dominate the energy density of the Universe. Recent measurements of the change in Hubble constant with time, based on observations of distant
2422:
2386:
2410:
2362:
2398:
2374:
79:, most of the energy was in the form of radiation, and that radiation was the dominant influence on the expansion of the universe. Later, with cooling from the expansion the roles of matter and radiation changed and the universe entered a matter-dominated era. Recent results suggest that we have already entered an era dominated by
1246:
is increasing with time. In contrast, the Hubble parameter seems to be decreasing with time, meaning that if we were to look at some fixed distance d and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.
1892:
p. 44: "Taken together, all the current data provide strong evidence for the existence of dark energy; they constrain the fraction of critical density contributed by dark energy, 0.76 Ā± 0.02, and the equation-of-state parameter,
1772:
106:
Some insight into the expansion can be obtained from a
Newtonian expansion model which leads to a simplified version of the Friedmann equation. It relates the proper distance (which can change over time, unlike the
1054:
1909:ā¼ 10 Gyr. These results are robust ā data from any one method can be removed without compromising the constraints ā and they are not substantially weakened by dropping the assumption of spatial flatness."
735:
1577:
1470:
1363:
1251:
898:
138:
565:
960:
1642:
1425:
1119:
1310:
1528:
247:
86:
Using the dimensionless scale factor to characterize the expansion of the universe, the effective energy densities of radiation and matter scale differently. This leads to a
1168:
826:
431:
1244:
1206:
473:
621:
1558:, the Hubble parameter is also thought to be constant, so the expansion law of the dark-energy-dominated era also holds for the inflationary prequel of the big bang.
1826:
1672:
769:
523:
353:
326:
186:
135:
382:
279:
641:
585:
496:
299:
159:
58:
137:
which is constant and set to today's distance) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting
2051:
1684:
1565:
is given the symbol Ī, and, considered as a source term in the
Einstein field equation, can be viewed as equivalent to a "mass" of empty space, or
2349:
2256:
2120:
2034:
1945:
1463:
1886:). During the radiation-dominated era, a(t) ā t; during the matter-dominated era, a(t) ā t; and for the dark-energy-dominated era, assuming
1554:. The dark-energy-dominated era began after the matter-dominated era, i.e. when the Universe was about 9.8 billion years old. In the era of
1208:
is increasing over time. This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy
2139:
1781:, and only holds for a positive sign of the cosmological constant, which is the case according to the currently accepted value of the
965:
1458:
to radiation until the universe was about 378,000 years old (redshift 1100). This second moment in time (close to the time of
667:
2340:
1447:
2442:
831:
532:
2022:
1964:
Frieman, Joshua A.; Turner, Michael S.; Huterer, Dragan (2008-01-01). "Dark Energy and the
Accelerating Universe".
1321:
1459:
31:
1254:
which is used to model the expanding universe, if at present time we receive light from a distant object with a
903:
83:, but examination of the roles of matter and radiation are most important for understanding the early universe.
2268:"First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters"
1590:
1439:, the energy density of matter exceeded both the energy density of radiation and the vacuum energy density.
2233:
2215:
Zelik, M and
Gregory, S: "Introductory Astronomy & Astrophysics", page 497. Thompson Learning, Inc. 1998
1937:
1376:
1059:
1882:ā² 0.5. The evolution of the scale factor is controlled by the dominant energy form: a(t) ā t (for constant
1265:
2297:
2159:
2112:
2026:
1841:
2176:
1483:
1782:
1562:
1555:
1332:
1129:
194:
2289:
2267:
2073:
1983:
2302:
1135:
777:
387:
2414:
1790:
1581:
1474:
1367:
1211:
1173:
651:
526:
69:
2402:
2390:
2315:
2279:
2089:
2063:
1999:
1973:
1778:
1539:
1348:
647:
436:
2252:
2116:
2030:
1941:
650:, which are presented in the case of a locally isotropic, locally homogeneous universe by the
590:
108:
2366:
2307:
2081:
2077:
1991:
1987:
1931:
1846:
2135:
1995:
1804:
1650:
1546:
is proposed as the last of the three phases of the known universe, the other two being the
747:
501:
331:
304:
164:
113:
2143:
1675:
646:
The evolution of the scale factor is a dynamical question, determined by the equations of
358:
255:
1573:, show this acceleration in expansion rate, indicating the presence of such dark energy.
1122:
658:
2293:
2426:
2354:
1451:
1443:
1340:
626:
570:
481:
284:
144:
43:
17:
2436:
2093:
1767:{\displaystyle H_{0}={\sqrt {8\pi G\rho _{\mathrm {full} }/3}}={\sqrt {\Lambda /3}}.}
1455:
35:
2319:
2052:"Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pdf)"
2003:
2421:
2378:
2334:
Relation of the scale factor with the cosmological constant and the Hubble constant
2019:
Gravity from the Ground Up: An
Introductory Guide to Gravity and General Relativity
2085:
1777:
This exponential dependence on time makes the spacetime geometry identical to the
744:. The Hubble parameter varies with time, not with space, with the Hubble constant
2333:
2246:
1566:
1466:
were last scattered, is often mistaken as marking the end of the radiation era.
1436:
1336:
1262:, then the scale factor at the time the object originally emitted that light is
80:
1870:
p. 6: "The
Universe has gone through three distinct eras: radiation-dominated,
1576:
For a dark-energy-dominated universe, the evolution of the scale factor in the
741:
1362:
For a radiation-dominated universe the evolution of the scale factor in the
1570:
1344:
1901:
is constant. This implies that the
Universe began accelerating at redshift
1797:
and the age of the universe is of the order of 13.8 billion years, or
2284:
1851:
1469:
For a matter-dominated universe the evolution of the scale factor in the
1356:
1255:
76:
1056:, and substituting the above definition of the Hubble parameter gives
1352:
1347:(referring generally to the constituents of the universe which moved
2373:
2311:
2068:
94:
at a later time and, since about 4 billion years ago, a subsequent
2206:
Ryden, Barbara, "Introduction to
Cosmology", 2006, eqn. 6.33, 6.41
2188:
Ryden, Barbara, "Introduction to
Cosmology", 2006, eqn. 5.25, 6.41
1978:
1049:{\displaystyle {\dot {d}}(t)={\frac {d(t){\dot {a}}(t)}{a(t)}}}
1132:, which means that the second derivative of the scale factor
2224:
Ryden, Barbara, "Introduction to
Cosmology", 2006, eqn. 6.33
2136:
Is the universe expanding faster than the speed of light?
1170:
is positive, or equivalently that the first derivative
2338:
1807:
1687:
1653:
1593:
1486:
1379:
1268:
1214:
1176:
1138:
1062:
968:
906:
834:
780:
750:
730:{\displaystyle H(t)\equiv {{\dot {a}}(t) \over a(t)}}
670:
629:
593:
573:
535:
504:
484:
439:
390:
361:
355:, usually also referred to as comoving distance, and
334:
307:
287:
258:
197:
167:
147:
116:
46:
1820:
1766:
1666:
1636:
1522:
1419:
1304:
1238:
1200:
1162:
1113:
1048:
954:
892:
820:
763:
729:
635:
615:
579:
559:
517:
490:
467:
425:
376:
347:
320:
293:
273:
241:
180:
153:
129:
52:
1435:Between about 47,000 years and 9.8 billion years
90:in the very early universe but a transition to a
893:{\displaystyle {\dot {d}}(t)=d_{0}{\dot {a}}(t)}
1832:(The Hubble time is 13.79 billion years).
560:{\displaystyle 13.799\pm 0.021\,\mathrm {Gyr} }
1130:the expansion of the universe is accelerating
8:
2107:Jones, Mark H.; Robert J. Lambourne (2004).
1897:ā ā1 Ā± 0.1 (stat) Ā±0.1 (sys), assuming that
1547:
1446:was about 47,000 years old (redshift 3600),
1966:Annual Review of Astronomy and Astrophysics
498:counted from the birth of the universe and
1578:FriedmannāLemaĆ®treāRobertsonāWalker metric
1471:FriedmannāLemaĆ®treāRobertsonāWalker metric
1462:), at which the photons which compose the
1364:FriedmannāLemaĆ®treāRobertsonāWalker metric
1252:FriedmannāLemaĆ®treāRobertsonāWalker metric
955:{\displaystyle d_{0}={\frac {d(t)}{a(t)}}}
384:is the scale factor. Thus, by definition,
2301:
2283:
2251:. Cambridge: Cambridge University Press.
2109:An Introduction to Galaxies and Cosmology
2067:
1977:
1959:
1957:
1812:
1806:
1751:
1746:
1733:
1717:
1716:
1701:
1692:
1686:
1658:
1652:
1622:
1592:
1551:
1510:
1506:
1485:
1416:
1403:
1399:
1378:
1284:
1267:
1216:
1215:
1213:
1178:
1177:
1175:
1140:
1139:
1137:
1064:
1063:
1061:
1009:
1008:
993:
970:
969:
967:
920:
911:
905:
870:
869:
863:
836:
835:
833:
800:
779:
755:
749:
690:
689:
686:
669:
628:
604:
592:
572:
546:
545:
534:
509:
503:
483:
450:
438:
414:
395:
389:
360:
339:
333:
312:
306:
286:
257:
238:
229:
196:
172:
166:
161:to their distance at some reference time
146:
121:
115:
45:
1637:{\displaystyle a(t)\propto \exp(H_{0}t)}
478:The scale factor is dimensionless, with
2345:
1922:
1863:
2111:. Cambridge University Press. p.
1996:10.1146/annurev.astro.46.060407.145243
1420:{\displaystyle a(t)\propto t^{1/2}.\,}
1114:{\displaystyle {\dot {d}}(t)=H(t)d(t)}
328:is the distance at the reference time
1890:= ā1, asymptotically a(t) ā exp(Ht)."
1464:cosmic microwave background radiation
1305:{\displaystyle a(t)={\frac {1}{1+z}}}
7:
2266:Spergel, D. N.; et al. (2003).
2248:Structure formation in the universe
1523:{\displaystyle a(t)\propto t^{2/3}}
27:Expansion of the universe parameter
1878:ā³ 0.5; and dark-energy dominated,
1748:
1727:
1724:
1721:
1718:
553:
550:
547:
68:, this is a key parameter of the
25:
2173:Physical Foundations of Cosmology
1874:ā³ 3000; matter-dominated, 3000 ā³
1454:, although the universe remained
242:{\displaystyle d(t)=a(t)d_{0},\,}
2420:
2408:
2396:
2384:
2372:
2360:
2348:
2272:Astrophysical Journal Supplement
1335:, and until about 47,000 years
740:where the dot represents a time
281:is the proper distance at epoch
2234:The Nobel Prize in Physics 2011
1580:is easily obtained solving the
1473:is easily obtained solving the
1128:Current evidence suggests that
1631:
1615:
1603:
1597:
1496:
1490:
1389:
1383:
1278:
1272:
1233:
1227:
1195:
1189:
1163:{\displaystyle {\ddot {a}}(t)}
1157:
1151:
1108:
1102:
1096:
1090:
1081:
1075:
1040:
1034:
1026:
1020:
1005:
999:
987:
981:
946:
940:
932:
926:
887:
881:
853:
847:
821:{\displaystyle d(t)=d_{0}a(t)}
815:
809:
790:
784:
721:
715:
707:
701:
680:
674:
610:
597:
456:
443:
426:{\displaystyle d_{0}=d(t_{0})}
420:
407:
371:
365:
268:
262:
222:
216:
207:
201:
1:
2050:Planck Collaboration (2016).
1239:{\displaystyle {\dot {d}}(t)}
1201:{\displaystyle {\dot {a}}(t)}
66:RobertsonāWalker scale factor
2056:Astronomy & Astrophysics
567:giving the current value of
2245:Padmanabhan, Thanu (1993).
2086:10.1051/0004-6361/201525830
1789:The current density of the
1785:, Ī, that is approximately
962:, so combining these gives
774:From the previous equation
188:. The formula for this is:
75:In the early stages of the
2459:
2197:Padmanabhan (1993), p. 64.
2142:November 28, 2010, at the
2023:Cambridge University Press
1322:Chronology of the universe
1319:
468:{\displaystyle a(t_{0})=1}
1544:dark-energy-dominated era
1534:Dark-energy-dominated era
771:being its current value.
96:dark-energy-dominated era
32:expansion of the universe
2236:. Retrieved 18 May 2017.
2171:Mukhanov, V. F. (2005),
2017:Schutz, Bernard (2003).
1930:Steven Weinberg (2008).
1801:. The Hubble constant,
1674:in the exponential, the
1366:is obtained solving the
616:{\displaystyle a(t_{0})}
2078:2016A&A...594A..13P
1988:2008ARA&A..46..385F
1938:Oxford University Press
1548:radiation-dominated era
1327:Radiation-dominated era
88:radiation-dominated era
18:Scale factor (universe)
2138:(see final paragraph)
1842:Cosmological principle
1822:
1768:
1668:
1647:Here, the coefficient
1638:
1524:
1450:density surpassed the
1421:
1339:, the dynamics of the
1306:
1240:
1202:
1164:
1115:
1050:
956:
894:
822:
765:
731:
637:
617:
581:
561:
519:
492:
469:
427:
378:
349:
322:
295:
275:
243:
182:
155:
141:at any arbitrary time
131:
54:
2154:Davies, Paul (1992),
1823:
1821:{\displaystyle H_{0}}
1783:cosmological constant
1769:
1669:
1667:{\displaystyle H_{0}}
1639:
1563:cosmological constant
1525:
1422:
1320:Further information:
1307:
1241:
1203:
1165:
1116:
1051:
957:
895:
823:
766:
764:{\displaystyle H_{0}}
732:
638:
618:
582:
562:
520:
518:{\displaystyle t_{0}}
493:
470:
428:
379:
350:
348:{\displaystyle t_{0}}
323:
321:{\displaystyle d_{0}}
296:
276:
244:
183:
181:{\displaystyle t_{0}}
156:
132:
130:{\displaystyle d_{C}}
55:
34:is parametrized by a
1830:ā70.88 km s Mpc
1805:
1685:
1651:
1591:
1552:matter-dominated era
1484:
1431:Matter-dominated era
1377:
1266:
1212:
1174:
1136:
1060:
966:
904:
832:
778:
748:
668:
627:
591:
571:
533:
502:
482:
437:
388:
377:{\displaystyle a(t)}
359:
332:
305:
285:
274:{\displaystyle d(t)}
256:
195:
165:
145:
114:
92:matter-dominated era
60:. Also known as the
44:
2294:2003ApJS..148..175S
1793:is of the order of
1791:observable universe
1582:Friedmann equations
1475:Friedmann equations
1368:Friedmann equations
652:Friedmann equations
527:age of the universe
525:set to the present
70:Friedmann equations
62:cosmic scale factor
2443:Physical cosmology
1818:
1779:de Sitter universe
1764:
1664:
1634:
1540:physical cosmology
1520:
1437:after the Big Bang
1417:
1337:after the Big Bang
1302:
1236:
1198:
1160:
1111:
1046:
952:
890:
818:
761:
727:
648:general relativity
633:
613:
577:
557:
515:
488:
465:
423:
374:
345:
318:
291:
271:
239:
178:
151:
127:
50:
2258:978-0-521-42486-8
2122:978-0-521-83738-5
2036:978-0-521-45506-0
1947:978-0-19-852682-7
1759:
1741:
1300:
1250:According to the
1224:
1186:
1148:
1072:
1044:
1017:
978:
950:
878:
844:
828:one can see that
725:
698:
636:{\displaystyle 1}
580:{\displaystyle a}
491:{\displaystyle t}
294:{\displaystyle t}
154:{\displaystyle t}
109:comoving distance
64:or sometimes the
53:{\displaystyle a}
16:(Redirected from
2450:
2425:
2424:
2413:
2412:
2411:
2401:
2400:
2399:
2389:
2388:
2387:
2377:
2376:
2365:
2364:
2363:
2353:
2352:
2344:
2323:
2305:
2287:
2285:astro-ph/0302209
2262:
2237:
2231:
2225:
2222:
2216:
2213:
2207:
2204:
2198:
2195:
2189:
2186:
2180:
2169:
2163:
2152:
2146:
2133:
2127:
2126:
2104:
2098:
2097:
2071:
2047:
2041:
2040:
2014:
2008:
2007:
1981:
1961:
1952:
1951:
1927:
1910:
1868:
1847:Lambda-CDM model
1831:
1827:
1825:
1824:
1819:
1817:
1816:
1800:
1796:
1788:
1773:
1771:
1770:
1765:
1760:
1755:
1747:
1742:
1737:
1732:
1731:
1730:
1702:
1697:
1696:
1673:
1671:
1670:
1665:
1663:
1662:
1643:
1641:
1640:
1635:
1627:
1626:
1556:cosmic inflation
1529:
1527:
1526:
1521:
1519:
1518:
1514:
1452:radiation energy
1426:
1424:
1423:
1418:
1412:
1411:
1407:
1349:relativistically
1311:
1309:
1308:
1303:
1301:
1299:
1285:
1245:
1243:
1242:
1237:
1226:
1225:
1217:
1207:
1205:
1204:
1199:
1188:
1187:
1179:
1169:
1167:
1166:
1161:
1150:
1149:
1141:
1120:
1118:
1117:
1112:
1074:
1073:
1065:
1055:
1053:
1052:
1047:
1045:
1043:
1029:
1019:
1018:
1010:
994:
980:
979:
971:
961:
959:
958:
953:
951:
949:
935:
921:
916:
915:
900:, and also that
899:
897:
896:
891:
880:
879:
871:
868:
867:
846:
845:
837:
827:
825:
824:
819:
805:
804:
770:
768:
767:
762:
760:
759:
736:
734:
733:
728:
726:
724:
710:
700:
699:
691:
687:
659:Hubble parameter
642:
640:
639:
634:
622:
620:
619:
614:
609:
608:
586:
584:
583:
578:
566:
564:
563:
558:
556:
524:
522:
521:
516:
514:
513:
497:
495:
494:
489:
474:
472:
471:
466:
455:
454:
432:
430:
429:
424:
419:
418:
400:
399:
383:
381:
380:
375:
354:
352:
351:
346:
344:
343:
327:
325:
324:
319:
317:
316:
300:
298:
297:
292:
280:
278:
277:
272:
248:
246:
245:
240:
234:
233:
187:
185:
184:
179:
177:
176:
160:
158:
157:
152:
136:
134:
133:
128:
126:
125:
59:
57:
56:
51:
21:
2458:
2457:
2453:
2452:
2451:
2449:
2448:
2447:
2433:
2432:
2431:
2419:
2409:
2407:
2397:
2395:
2385:
2383:
2371:
2361:
2359:
2347:
2339:
2330:
2303:10.1.1.985.6441
2265:
2259:
2244:
2241:
2240:
2232:
2228:
2223:
2219:
2214:
2210:
2205:
2201:
2196:
2192:
2187:
2183:
2170:
2166:
2156:The New Physics
2153:
2149:
2144:Wayback Machine
2134:
2130:
2123:
2106:
2105:
2101:
2049:
2048:
2044:
2037:
2016:
2015:
2011:
1963:
1962:
1955:
1948:
1929:
1928:
1924:
1919:
1914:
1913:
1891:
1869:
1865:
1860:
1838:
1829:
1808:
1803:
1802:
1798:
1794:
1786:
1712:
1688:
1683:
1682:
1676:Hubble constant
1654:
1649:
1648:
1618:
1589:
1588:
1536:
1502:
1482:
1481:
1456:optically thick
1433:
1395:
1375:
1374:
1329:
1324:
1318:
1289:
1264:
1263:
1210:
1209:
1172:
1171:
1134:
1133:
1058:
1057:
1030:
995:
964:
963:
936:
922:
907:
902:
901:
859:
830:
829:
796:
776:
775:
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661:is defined as:
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1972:(1): 385ā432.
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1905:ā¼ 0.4 and age
1862:
1861:
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2403:Outer space
2391:Spaceflight
1567:dark energy
1448:massāenergy
81:dark energy
2069:1502.01589
2025:. p.
1917:References
1571:supernovae
1316:Chronology
742:derivative
2367:Astronomy
2298:CiteSeerX
2094:119262962
1979:0803.0982
1933:Cosmology
1787:2 Ā· 10 s.
1749:Λ
1714:ρ
1707:π
1613:
1607:∝
1500:∝
1442:When the
1393:∝
1357:neutrinos
1345:radiation
1333:Inflation
1222:˙
1184:˙
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976:˙
876:˙
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696:˙
684:≡
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2437:Category
2320:10794058
2140:Archived
2004:15117520
1852:Redshift
1836:See also
1550:and the
1256:redshift
77:Big Bang
2427:Science
2355:Physics
2341:Portals
2290:Bibcode
2074:Bibcode
2062:: A13.
1984:Bibcode
1353:photons
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2379:Stars
2316:S2CID
2280:arXiv
2177:p. 58
2090:S2CID
2064:arXiv
2000:S2CID
1974:arXiv
1858:Notes
1828:, is
1678:, is
543:0.021
2253:ISBN
2117:ISBN
2031:ISBN
1942:ISBN
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