19 68ApJ. . .152L.115G The AstrophysicalJournal,Vol.152,May1968 as aproofforthevalidityof“bigbang”cosmologicaltheory.Actually,weknow t,T may stillbedetectableataverylowtemperature. that, iftheUniverseexpandedfromaninitialhigh-temperaturestate,radiation temperature woulddecreaseduringtheexpansionandacosmicequilibriumradiation however, forfrequencieshigherthanthefrequencyatwhichspectrumhasamaxi- responding toablack-bodyspectrumatT=3°K.Noexactmeasurementsareavailable, mum, ifitreallycorrespondstoa3°Kblackbody.Becauseofthisremaininguncer- tainty and,aboveall,becauseofthefar-reachingconsequencescommoninterpre- originally dismissedbecausethesummedfluxofallknownradiogalaxiesfallsshort tation, itappearsdesirabletoexplorealternativeexplanations.Theideathattheback- ground mayresultfromthesuperpositionofunresolvedextragalacticradiosourceswas sources withdifferentpropertiesexistthathavesofargonelargelyundetectedexcept the observationsbyalargefactor.Thequestionremains,however,whetherradio plore whatthepropertiesofsuchsourceswouldhavetobeweretheyaccountfor for thesummationoffluxfrommanythem.Itispurposethispapertoex- this microwaveradiation. Robertson andWalkerexpression by asourceattimet=hwithfrequencyvv\willreachtheobserverorigin where R{t)isascalefactorofthemetricandKcurvatureindex.Aphotonemitted with afrequencyv=v$time/.WecanthendefineDopplerratio garding theevolutionofsources.Wecanthen take K=0andR(t)e,whereT cause thisallowsthespecificationofallnecessary quantitieswithoutassumptionsre- is theHubbletime,anditpossibletoshowthat the co-movingradialcoordinateof source whosephotonwereceiveattimet=towith aredshiftzcanbewritten(Metzner 0 and Morrison1959) © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The discoveryofthemicrowave-radiationbackgroundhasbeengenerallyregarded The accumulatedmeasurementsfitthelow-energysideofcurveverywell,cor- If weassumeahomogeneousandisotropicuniverse,itslineelementisgivenbythe We shallmakethecalculationsforcaseofasteady-state Euclideanuniversebe- CAN THEOBSERVEDMICROWAVEBACKGROUNDBEDUETOA * Onleaveofabsencefromthe LaboratoriodiAstrofísica,Frascati,Italy. Comell-Sydney UniversityAstronomyCenter,SpaceSciencesBuilding, ds* = SUPERPOSITION OFSOURCES? Thomas GoldandFrancoPacini* cW -R\t) Received February29,1968 r =cTe-^iz-1). (3) Ithaca, NewYork 2 [dr +r{dJdsin6d<¡>)] vi R{t,) vq R(h) = L115 2 (1 +Kr/4) (i) (2) 19 68ApJ. . .152L.115G k I(v) ground frequency intervalv<>
We wish to point out explicitly two important facts contained in the above expres- sions (10) and (11): {a) The radiation background due to sources having an intrinsic power-law spectrum in the frequency interval (í'o,*'i) is of the type I(v) oc for v < ^o, i.e., it does not depend upon the spectral index of the sources, (b) For spectra S(v) = Av~k with & > 1, the spectral index of the background is the same as that of the sources in the frequency interval (po^i); for spectra S(v) = Avk with ¿ > — 1, this is true only for frequencies v > such that {vi/v)k~z 1. The observed microwave background at low frequencies has a slope k = 2. If we want to explain this radiation as the superposition of unresolved sources, we must require that the individual sources have a slope of about 2. A decisive test would be the investigation of the spectral behavior of the background at still lower frequencies—should the black- body interpretation be correct, no change in the slope would be found, while, if the back- ground is due to unresolved sources, the slope would become k = 3. Unfortunately measurements at frequencies below about 1 GHz cannot be very accurate because the radiation from the Galaxy becomes dominant and it is therefore difficult to perform this decisive test. On the other hand, the measurements show that for frequencies larger than about 10 GHz the energy spectrum of the background does not increase as fast as v2. It can, however, be seen from expression (10) that a similar behavior can be expected also in the case of superposition of individual sources, provided that their intrinsic spectrum has a low-frequency cutoff at a critical frequency not much larger than a few times 10 GHz. If we have a spectral index k = 2, the requirements upon the number density of sources can be expressed as follows :
= 3 X . (12)
-63 Since reff = 3° K, we obtain pA ~ 10 (in cgs units). On the other hand, the observed degree of homogeneity imposes some lower limits on the number density of unresolved sources. In order to estimate this lower limit, we note that most of the background in the observed frequency interval is due to relatively close sources; we can then neglect for our purposes the redshift factors and consider the Universe as a static sphere of radius R = cT = 1028 cm having the observer in the center. The energy received in the solid angle du/^ir from sources within r and r + Jr is given by
The background flux which reaches the observer within a solid angle Jco/áx is then
I(14) where R is the radius of the observable Universe. The fluctuations in the number of emitting sources will be of the order of [(Jco/é^éTr^pJr]172, and they will give rise to fluctuations in the background of the order of dco _p ^\1/2 (15) An Rq 47t/ ’ where R$ is the minimum distance at which we can find a source. By taking into account the probability of finding a source within a certain solid angle, we must require ^tRq3p = 47t/Jco. The ratio a between fluctuations and background can be related to the space density of the sources through the relation 1 47T 1 1 (16) P smx dw R* az •
© American Astronomical Society • Provided by the NASA Astrophysics Data System L118 THOMAS GOLD AND FRANCO PACINI
The présent limits on small-scale variations of the background are therefore consistent with this model if the space density of the sources is not much smaller than that of galaxies. We wish, however, to point out that such an estimate is very pessimistic, be- cause individual sources can indeed be detected standing against the background and therefore some quoted values for the homogeneity are not meaningful for our purposes. In general, the ratio between the wth-most luminous source and background would be In n 2/3 (17) T ~ (36tp)^R * A comparison of the observed distribution function of the temperature (sources in- cluded) over the sky for any given observing beam width will allow one to see more clearly what space density and average strength of sources would be required. One can then decide whether such sources could exist without having been recognized» Finally, we note that Heeschen (1968) has recently reported remarkable radio emis- sion at centimeter wavelengths of two otherwise normal-looking elliptical galaxies, NGC 1052 and NGC 4278. The radio emission from these galaxies in the meter wave- lengths is too weak to be detected with present sensitivities. If such emission in the centimeter range is common among other galaxies, it may indeed contribute significantly to the microwave background.
We would like to thank Dr. C. Hazard for useful discussions. This work was sup- ported by the Office of Naval Research under contract N00014-67-A-0077-0007.
REFERENCES Heeschen, D. S. 1968, Ap. J. {Letters), 151, L135. Metzner, A. W. K., and Morrison, P: 1959, M.N.R.A.S., 119, 6.
Copyright 1968. The University of Chicago. Printed in U.S.A.
© American Astronomical Society • Provided by the NASA Astrophysics Data System