1766:
1506:
380:
735:
67:. Later, as radiation redshifts away, these become mass perturbations, and they start to grow linearly. Only long after that, starting with small mass scales and advancing over time to larger mass scales, do the perturbations actually collapse to form (for example) galaxies or clusters of galaxies, in so-called hierarchical structure formation (see
139:
507:
1761:{\displaystyle f(\delta _{M}>\delta _{c})=\int _{\delta _{c}}^{\infty }d\delta _{M}~{\frac {1}{{\sqrt {2\pi }}\sigma (M)}}\exp \left(-{\frac {1}{2}}{\frac {\delta _{M}^{2}}{\sigma ^{2}(M)}}\right)={\frac {1}{2}}\operatorname {erfc} \left({\frac {1}{\sqrt {2}}}{\frac {\delta _{c}}{\sigma (M)}}\right)~.}
948:
1950:
1087:
In other words, fluctuations are small at some early cosmological time, and grow until they cross a threshold ending in gravitational collapse into a halo. These perturbations are modeled linearly, even though the eventual collapse is itself a non-linear process.
2247:
765:
Qualitatively, the prediction is that the mass distribution is a power law for small masses, with an exponential cutoff above some characteristic mass that increases with time. Such functions had previously been noted by
Schechter as observed
375:{\displaystyle dn\equiv N(M)dM={\frac {1}{\sqrt {\pi }}}\left(1+{\frac {n}{3}}\right){\frac {\bar {\rho }}{M^{2}}}\left({\frac {M}{M^{*}}}\right)^{\left(3+n\right)/6}\exp \left(-\left({\frac {M}{M^{*}}}\right)^{\left(3+n\right)/3}\right)dM}
730:{\displaystyle M^{*}=\left({\frac {{\bar {\rho }}^{1-{\frac {n}{3}}}}{2\sigma ^{2}}}\right)^{\frac {3}{3+n}}=\left({\frac {{\bar {\rho }}_{0}^{1-{\frac {n}{3}}}}{2\sigma _{0}^{2}}}\right)^{\frac {3}{3+n}}\cdot {\frac {R_{0}^{2}}{R^{2}}}}
78:
is related to the fraction of volume samples in which the smoothed initial density fluctuations are above some density threshold. This yields a formula for the mass function (distribution of masses) of objects at any given time.
813:
1829:
2302:; at some cosmological time reaches gravitational collapse after the universe has expanded by a factor of 1/δ since that time. Using this, the normal distribution of the fluctuations, written in terms of the
2155:
2150:
1362:
2152:
One can make an ad-hoc argument and say that negative perturbations are not contributing in this scheme so that we are mistakenly leaving out half of the mass. And so, the Press-Schechter ansatz is
2082:
2021:
1298:
762:
is the scale of the universe at that time. Parameters with subscript 0 are at the time of the initial creation of the fluctuations (or any later time before the gravitational collapse).
1140:
1082:
1047:
1178:
444:
473:
758:
is the standard deviation per unit volume of the fluctuation from which the object was formed had gravitationally collapsed, at the time of the gravitational collapse, and
1481:
1824:
1446:
1797:
1391:
1207:
2360:
2300:
2278:
1970:
1247:
979:
756:
2340:
500:
1012:
134:
1417:
806:
2320:
1501:
1227:
402:
105:
770:, and are now known as Schechter luminosity functions. The Press-Schechter formalism provided the first quantitative model for how such functions might arise.
475:
is the mean (baryonic and dark) matter density of the universe at the time the fluctuation from which the object was formed had gravitationally collapsed, and
2482:
Sheth, R. K., & Tormen, G. (1999). Large-scale bias and the peak background split. Monthly
Notices of the Royal Astronomical Society, 308(1), 119-126
63:
cosmological models, perturbations on all scales are imprinted on the universe at very early times, for example by quantum fluctuations during an
943:{\displaystyle M{\frac {dn}{dM}}={\frac {1}{\sqrt {\pi }}}{\frac {\bar {\rho }}{M}}\left({\frac {M}{M^{*}}}\right)^{1/2}e^{-M/M^{*}}}
2466:
2433:
1945:{\displaystyle \sigma (M)^{2}={\frac {\Sigma ^{2}}{M^{2}}}={\frac {V\cdot \sigma ^{2}}{M^{2}}}={\frac {\sigma ^{2}}{M\cdot \rho }}}
2087:
2505:
1303:
767:
2371:
2510:
2242:{\displaystyle F(>M)=\operatorname {erfc} \left({\frac {1}{\sqrt {2}}}{\frac {\delta _{M}}{\sigma (M)}}\right)~,}
2026:
1978:
1252:
1094:
1054:
778:
1019:
64:
2415:
1145:
2500:
74:
Press and
Schechter observed that the fraction of mass in collapsed objects more massive than some mass
407:
449:
1826:
depend on redshift, so the above probability does as well. The variance given in the 1974 paper is
982:
1451:
1802:
1424:
68:
24:
1773:
1367:
1183:
2462:
2429:
48:
2345:
2285:
2254:
1955:
1232:
964:
741:
2454:
2421:
782:
44:
36:
2325:
478:
988:
110:
32:
2402:
Formation of
Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation
1396:
788:
2305:
1486:
1212:
387:
90:
40:
1051:
Halos are spherical, virialized overdensities with a density above a critical density
985:
and the variance of this distribution is scale-dependent, given by the power spectrum
2494:
2401:
87:
The Press–Schechter formalism predicts that the number of objects with mass between
39:) of a certain mass within a given volume of the Universe. It was described in an
2390:
60:
2370:
A number of generalizations of the Press–Schechter formula exist, such as the
1483:
we can directly compute the probability of halos forming with masses at least
404:
is the index of the power spectrum of the fluctuations in the early universe
1249:
is convolved with a top-hat window function). The sphere radius is of order
2458:
2481:
28:
957:
The Press–Schechter formalism is derived through three key assumptions:
2417:
The
Encyclopedia of Cosmology, Volume 1: Galaxy Formation and Evolution
2391:
Dark Matter Halos, Mass
Functions, and Cosmology: a Theorist’s View
502:
is a cut-off mass below which structures will form. Its value is:
2425:
1972:
is the mass standard deviation in the volume of the fluctuation.
1016:
Matter perturbations grow linearly with the growth function
2145:{\textstyle f(\delta _{M}>\delta _{c})={\frac {1}{2}}~.}
1357:{\displaystyle \delta _{M}({\vec {x}})\geq \delta _{c}~,}
2023:
we expect all matter to be contained in halos such that
1448:
are
Gaussian distributed with an average 0 and variance
2090:
2029:
2348:
2328:
2308:
2288:
2257:
2158:
1981:
1958:
1832:
1805:
1776:
1509:
1489:
1454:
1427:
1399:
1370:
1306:
1255:
1235:
1215:
1186:
1148:
1097:
1057:
1022:
991:
967:
816:
791:
744:
510:
481:
452:
410:
390:
142:
113:
93:
781:
of 1), is very close to the spectrum of the current
2354:
2334:
2314:
2294:
2272:
2251:the fraction of matter contained in halos of mass
2241:
2144:
2076:
2015:
1964:
1944:
1818:
1791:
1760:
1495:
1475:
1440:
1411:
1385:
1356:
1292:
1241:
1221:
1201:
1172:
1134:
1076:
1041:
1006:
973:
942:
800:
750:
729:
494:
467:
438:
396:
374:
128:
99:
808:has a simpler form. Written in mass-free units:
2084:However, the above equation gives us the limit
1975:Note, that in the limit of large perturbations
2077:{\textstyle f(\delta _{M}>\delta _{c})=1~.}
27:for predicting the number of objects (such as
8:
2016:{\displaystyle \sigma (M)\gg \delta _{M}~,}
1293:{\displaystyle M\sim {\bar {\rho }}R^{3}~.}
2347:
2327:
2307:
2287:
2256:
2206:
2200:
2188:
2157:
2126:
2114:
2101:
2089:
2053:
2040:
2028:
2001:
1980:
1957:
1923:
1917:
1906:
1895:
1882:
1871:
1861:
1855:
1846:
1831:
1810:
1804:
1775:
1725:
1719:
1707:
1686:
1660:
1649:
1644:
1638:
1628:
1589:
1583:
1574:
1561:
1554:
1549:
1533:
1520:
1508:
1488:
1453:
1432:
1426:
1398:
1372:
1371:
1369:
1342:
1321:
1320:
1311:
1305:
1278:
1263:
1262:
1254:
1234:
1214:
1188:
1187:
1185:
1156:
1155:
1147:
1135:{\displaystyle \delta _{M}({\vec {x}})~,}
1112:
1111:
1102:
1096:
1068:
1056:
1033:
1021:
990:
966:
961:Matter in the Universe has perturbations
932:
923:
916:
902:
898:
886:
877:
857:
855:
843:
820:
815:
790:
773:The case of a scale-free power spectrum,
743:
719:
709:
704:
698:
676:
663:
658:
638:
631:
626:
615:
614:
611:
584:
571:
551:
544:
533:
532:
529:
515:
509:
486:
480:
454:
453:
451:
430:
409:
389:
351:
331:
319:
310:
281:
261:
249:
240:
227:
213:
211:
196:
173:
141:
112:
92:
1091:We introduce the smoothed density field
2383:
1077:{\displaystyle \delta \geq \delta _{c}}
7:
1042:{\displaystyle \delta \propto D_{+}}
2362:gives the Press-Schechter formula.
1180:averaged over a sphere with center
1173:{\displaystyle \delta ({\vec {x}})}
1959:
1858:
1562:
14:
2420:. Vol. 2. World Scientific.
953:Assumptions and derivation sketch
439:{\displaystyle P(k)\propto k^{n}}
2404:, W.H. Press, P. Schechter, 1974
2222:
2216:
2171:
2162:
2120:
2094:
2059:
2033:
1991:
1985:
1843:
1836:
1786:
1780:
1741:
1735:
1672:
1666:
1608:
1602:
1539:
1513:
1464:
1458:
1377:
1332:
1326:
1317:
1268:
1193:
1167:
1161:
1152:
1123:
1117:
1108:
1001:
995:
862:
620:
538:
468:{\displaystyle {\bar {\rho }}}
459:
420:
414:
218:
161:
155:
1:
1476:{\displaystyle \sigma (M)~,}
1819:{\displaystyle \delta _{M}}
1441:{\displaystyle \delta _{M}}
783:standard cosmological model
2527:
2372:Sheth–Tormen approximation
1792:{\displaystyle \sigma (R)}
1386:{\displaystyle {\vec {x}}}
1202:{\displaystyle {\vec {x}}}
2282:A fractional fluctuation
21:Press–Schechter formalism
2449:Baumann, Daniel (2022).
2414:Barkana, Rennan (2018).
1229:contained inside (i.e.,
777:=0 (or, equivalently, a
2355:{\displaystyle \sigma }
2295:{\displaystyle \delta }
2273:{\displaystyle >M~.}
1965:{\displaystyle \Sigma }
1242:{\displaystyle \delta }
974:{\displaystyle \delta }
751:{\displaystyle \sigma }
2506:Equations of astronomy
2356:
2336:
2316:
2296:
2274:
2243:
2146:
2078:
2017:
1966:
1946:
1820:
1793:
1762:
1497:
1477:
1442:
1413:
1387:
1358:
1294:
1243:
1223:
1203:
1174:
1136:
1078:
1043:
1008:
975:
944:
802:
752:
731:
496:
469:
440:
398:
376:
130:
101:
2511:Mathematical modeling
2459:10.1017/9781108937092
2357:
2337:
2335:{\displaystyle \rho }
2317:
2297:
2275:
2244:
2147:
2079:
2018:
1967:
1947:
1821:
1794:
1763:
1498:
1478:
1443:
1414:
1388:
1359:
1295:
1244:
1224:
1204:
1175:
1137:
1079:
1044:
1009:
983:Gaussian distribution
976:
945:
803:
779:scalar spectral index
753:
732:
497:
495:{\displaystyle M^{*}}
470:
441:
399:
377:
131:
102:
2346:
2326:
2306:
2286:
2255:
2156:
2088:
2027:
1979:
1956:
1830:
1803:
1774:
1507:
1487:
1452:
1425:
1421:Since perturbations
1397:
1368:
1304:
1253:
1233:
1213:
1184:
1146:
1095:
1055:
1020:
1007:{\displaystyle P(k)}
989:
965:
814:
789:
768:luminosity functions
742:
508:
479:
450:
408:
388:
140:
129:{\displaystyle M+dM}
111:
91:
1654:
1566:
1412:{\displaystyle M~.}
1393:with mass at least
714:
668:
649:
2352:
2332:
2312:
2292:
2270:
2239:
2142:
2074:
2013:
1962:
1942:
1816:
1789:
1758:
1640:
1545:
1493:
1473:
1438:
1409:
1383:
1354:
1290:
1239:
1219:
1199:
1170:
1132:
1074:
1039:
1004:
971:
940:
801:{\displaystyle dn}
798:
748:
727:
700:
654:
613:
492:
465:
436:
394:
372:
126:
97:
69:Physical cosmology
59:In the context of
25:mathematical model
16:Mathematical model
2315:{\displaystyle M}
2266:
2235:
2226:
2198:
2197:
2138:
2134:
2070:
2009:
1940:
1912:
1877:
1754:
1745:
1717:
1716:
1694:
1676:
1636:
1612:
1597:
1582:
1496:{\displaystyle M}
1469:
1405:
1380:
1364:a halo exists at
1350:
1329:
1286:
1271:
1222:{\displaystyle M}
1196:
1164:
1128:
1120:
892:
870:
865:
853:
852:
838:
725:
692:
670:
646:
623:
600:
578:
559:
541:
462:
397:{\displaystyle n}
325:
255:
233:
221:
204:
183:
182:
100:{\displaystyle M}
37:dark matter halos
2518:
2485:
2479:
2473:
2472:
2446:
2440:
2439:
2411:
2405:
2399:
2393:
2388:
2361:
2359:
2358:
2353:
2341:
2339:
2338:
2333:
2321:
2319:
2318:
2313:
2301:
2299:
2298:
2293:
2279:
2277:
2276:
2271:
2264:
2248:
2246:
2245:
2240:
2233:
2232:
2228:
2227:
2225:
2211:
2210:
2201:
2199:
2193:
2189:
2151:
2149:
2148:
2143:
2136:
2135:
2127:
2119:
2118:
2106:
2105:
2083:
2081:
2080:
2075:
2068:
2058:
2057:
2045:
2044:
2022:
2020:
2019:
2014:
2007:
2006:
2005:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1941:
1939:
1928:
1927:
1918:
1913:
1911:
1910:
1901:
1900:
1899:
1883:
1878:
1876:
1875:
1866:
1865:
1856:
1851:
1850:
1825:
1823:
1822:
1817:
1815:
1814:
1798:
1796:
1795:
1790:
1767:
1765:
1764:
1759:
1752:
1751:
1747:
1746:
1744:
1730:
1729:
1720:
1718:
1712:
1708:
1695:
1687:
1682:
1678:
1677:
1675:
1665:
1664:
1653:
1648:
1639:
1637:
1629:
1613:
1611:
1598:
1590:
1584:
1580:
1579:
1578:
1565:
1560:
1559:
1558:
1538:
1537:
1525:
1524:
1502:
1500:
1499:
1494:
1482:
1480:
1479:
1474:
1467:
1447:
1445:
1444:
1439:
1437:
1436:
1418:
1416:
1415:
1410:
1403:
1392:
1390:
1389:
1384:
1382:
1381:
1373:
1363:
1361:
1360:
1355:
1348:
1347:
1346:
1331:
1330:
1322:
1316:
1315:
1299:
1297:
1296:
1291:
1284:
1283:
1282:
1273:
1272:
1264:
1248:
1246:
1245:
1240:
1228:
1226:
1225:
1220:
1208:
1206:
1205:
1200:
1198:
1197:
1189:
1179:
1177:
1176:
1171:
1166:
1165:
1157:
1141:
1139:
1138:
1133:
1126:
1122:
1121:
1113:
1107:
1106:
1083:
1081:
1080:
1075:
1073:
1072:
1048:
1046:
1045:
1040:
1038:
1037:
1013:
1011:
1010:
1005:
980:
978:
977:
972:
949:
947:
946:
941:
939:
938:
937:
936:
927:
911:
910:
906:
897:
893:
891:
890:
878:
871:
866:
858:
856:
854:
848:
844:
839:
837:
829:
821:
807:
805:
804:
799:
785:. In this case,
757:
755:
754:
749:
736:
734:
733:
728:
726:
724:
723:
713:
708:
699:
694:
693:
691:
677:
675:
671:
669:
667:
662:
648:
647:
639:
630:
625:
624:
616:
612:
602:
601:
599:
585:
583:
579:
577:
576:
575:
562:
561:
560:
552:
543:
542:
534:
530:
520:
519:
501:
499:
498:
493:
491:
490:
474:
472:
471:
466:
464:
463:
455:
445:
443:
442:
437:
435:
434:
403:
401:
400:
395:
381:
379:
378:
373:
365:
361:
360:
359:
355:
350:
346:
330:
326:
324:
323:
311:
290:
289:
285:
280:
276:
260:
256:
254:
253:
241:
234:
232:
231:
222:
214:
212:
210:
206:
205:
197:
184:
178:
174:
135:
133:
132:
127:
106:
104:
103:
98:
65:inflationary era
61:cold dark matter
45:William H. Press
2526:
2525:
2521:
2520:
2519:
2517:
2516:
2515:
2491:
2490:
2489:
2488:
2480:
2476:
2469:
2448:
2447:
2443:
2436:
2413:
2412:
2408:
2400:
2396:
2389:
2385:
2380:
2368:
2366:Generalizations
2344:
2343:
2324:
2323:
2304:
2303:
2284:
2283:
2253:
2252:
2212:
2202:
2187:
2183:
2154:
2153:
2110:
2097:
2086:
2085:
2049:
2036:
2025:
2024:
1997:
1977:
1976:
1954:
1953:
1929:
1919:
1902:
1891:
1884:
1867:
1857:
1842:
1828:
1827:
1806:
1801:
1800:
1772:
1771:
1731:
1721:
1706:
1702:
1656:
1655:
1624:
1620:
1588:
1570:
1550:
1529:
1516:
1505:
1504:
1485:
1484:
1450:
1449:
1428:
1423:
1422:
1395:
1394:
1366:
1365:
1338:
1307:
1302:
1301:
1274:
1251:
1250:
1231:
1230:
1211:
1210:
1182:
1181:
1144:
1143:
1098:
1093:
1092:
1064:
1053:
1052:
1029:
1018:
1017:
987:
986:
963:
962:
955:
928:
912:
882:
873:
872:
830:
822:
812:
811:
787:
786:
740:
739:
715:
681:
650:
607:
606:
589:
567:
563:
531:
525:
524:
511:
506:
505:
482:
477:
476:
448:
447:
426:
406:
405:
386:
385:
336:
332:
315:
306:
305:
301:
297:
266:
262:
245:
236:
235:
223:
189:
185:
138:
137:
109:
108:
89:
88:
85:
57:
33:galaxy clusters
17:
12:
11:
5:
2524:
2522:
2514:
2513:
2508:
2503:
2493:
2492:
2487:
2486:
2474:
2467:
2441:
2434:
2406:
2394:
2382:
2381:
2379:
2376:
2367:
2364:
2351:
2331:
2311:
2291:
2269:
2263:
2260:
2238:
2231:
2224:
2221:
2218:
2215:
2209:
2205:
2196:
2192:
2186:
2182:
2179:
2176:
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2012:
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1996:
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722:
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49:Paul Schechter
41:academic paper
15:
13:
10:
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6:
4:
3:
2:
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2468:9781108838078
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2435:9789814656221
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2010:
2002:
1998:
1994:
1988:
1982:
1973:
1936:
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745:
737:
720:
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695:
688:
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672:
664:
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643:
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42:
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34:
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26:
22:
2501:Astrophysics
2477:
2450:
2444:
2426:10.1142/9496
2416:
2409:
2397:
2386:
2369:
2281:
2250:
1974:
1770:Implicitly,
1769:
1420:
1090:
1086:
981:following a
956:
810:
774:
772:
764:
759:
738:
504:
383:
86:
75:
73:
58:
20:
18:
2495:Categories
2378:References
55:Background
2451:Cosmology
2350:σ
2330:ρ
2290:δ
2214:σ
2204:δ
2181:
2112:δ
2099:δ
2051:δ
2038:δ
1999:δ
1995:≫
1983:σ
1960:Σ
1937:ρ
1934:⋅
1921:σ
1893:σ
1889:⋅
1859:Σ
1834:σ
1808:δ
1778:σ
1733:σ
1723:δ
1700:
1658:σ
1642:δ
1626:−
1618:
1600:σ
1595:π
1572:δ
1563:∞
1552:δ
1547:∫
1531:δ
1518:δ
1456:σ
1430:δ
1378:→
1340:δ
1336:≥
1327:→
1309:δ
1269:¯
1266:ρ
1260:∼
1237:δ
1209:and mass
1194:→
1162:→
1150:δ
1142:given by
1118:→
1100:δ
1066:δ
1062:≥
1059:δ
1027:∝
1024:δ
969:δ
934:∗
918:−
888:∗
863:¯
860:ρ
850:π
746:σ
696:⋅
656:σ
636:−
621:¯
618:ρ
569:σ
549:−
539:¯
536:ρ
517:∗
488:∗
460:¯
457:ρ
424:∝
321:∗
303:−
295:
251:∗
219:¯
216:ρ
180:π
150:≡
51:in 1974.
2453:. 2022.
1300:Then if
29:galaxies
2465:
2432:
2342:, and
2265:
2234:
2137:
2069:
2008:
1952:where
1753:
1581:
1468:
1404:
1349:
1285:
1127:
384:where
83:Result
23:is a
2463:ISBN
2430:ISBN
2259:>
2178:erfc
2166:>
2108:>
2047:>
1799:and
1697:erfc
1527:>
136:is:
107:and
47:and
19:The
2455:doi
2422:doi
1615:exp
1503:as
292:exp
71:).
43:by
35:or
2497::
2461:.
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1986:(
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156:(
153:N
147:n
144:d
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121:d
118:+
115:M
95:M
76:M
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