Mattig's relation and dynamical distance indicators

0.1 MB·

by  Pekka Teerikorpi

Astronomische Nachrichten, 22 January 2016
Mattig’s relation and dynamical distance indicators
P. Teerikorpi1 and Yu.V. Baryshev2
1 Tuorla Observatory,Department of Physics and Astronomy,University of Turku,21500 Piikkio¨, Finland
2 AstronomyDepartment,SaintPetersburgStateUniversity,StaryjPeterhoff,198504,StPetersburg,Russia
The dates of receipt and acceptance should be inserted later
6 Key words Cosmology: distance scale – Cosmology: theory
1
We discuss how the redshift (Mattig) method in Friedmann cosmology relates to dynamical distance indi-
0
2 cators based on Newton’s gravity (Teerikorpi 2011). It belongs to the class of indicators where the relevant
length inside the system is the distance itself (in this case the proper metric distance). As the Friedmann
n
model has Newtonian analogy, its use to infer distances has instructive similarities to classical dynamical
a
distance indicators. In view of the theoretical exact linear distance-velocity law, we emphasize that it is
J
conceptuallycorrecttoderivethecosmological distanceviatheroute:redshift(primarily observed)→space
1
expansionvelocity(notdirectlyobserved)→metricdistance(physicallengthin”cm”).Importantproperties
2
of theproper metric distance are summarized.

]
c Copyrightlinewillbeprovidedbythepublisher
q
-
r
g 1 Introduction on one of the bodies. Idealized examples could be the
[ Earth-MoonandSun-Earthpairs.Assuming a circular

Newtonianaccelerationr¨ofaparticleinthe forcefield orbit, the radial acceleration leads to d = GM, with
1 V2
v of a spherical mass M the orbital speed from the period T as V = 2πd. This
1 1 r¨ T
2 =− × . (1) leaves M ∝d3 so one needs an auxiliary relation M ∝
5 r2 G M dn, where n 6= 3. For the Earth-Moon pair one writes
6 offers knownwaysto infer the distance d ofa gravitat-
GM = gr2 (so n = 0). For the Sun the gravitational
5 ing system. Such dynamical distance indicators often E E
redshift z gives (T2011) GM =z c2θ d (n=1).
0 belong to ”physical” (Sandage et al. 2006) or ”one- g sun g sun
. Such cases involved circular orbits. In the Galaxy–
1 step” (Jackson 2007) methods, in contrast to relative
0 distance indicators such as standard candles and rods. M31 pair and in the flow of dwarf galaxies away from
6 the Local Group the motion is (nearly) radial and one
The latter give luminosity and angular size distances.
1 cannot directly infer the acceleration in Eq.(1). How-
The distance type fromphysicalindicatorsdepends on
:
v themethod;essentialisthatthedistanceisdirectlyob- ever, with data on the mass and ”bang time” one can
i derive,byTolman-Bondi-typecalculations,theexpected
X tained in cm, with no midway distance as a reference.
Note thatr inEq.(1)canbe either a lengthwithin distance-velocity relation that gives the distance if the
r
radial velocity is measured (e.g., Gromov et al. 2001).
a a remote system whose distance is wanted or it can be
the distance itself from the observer to the system. A supreme distance indicator of this category with
radial motion is the Friedmann universe itself. Its spe-
cial features are discussed below.
1.1 When r is within a distant system
When the accelerationcan be related to an observable
2 The Friedmann universe
velocity quantity V, Teerikorpi(2011;T2011)grouped
dynamical distance indicators according to how the
M/d degeneracy in the equation M =c θdV2 is over- The spheres cut from this world model are known to
1
expandinaNewtonianway(McCrea&Milne1934).In
comeifonecanexpressM/dinanotherformcontaining
terms of internal hyperspherical coordinates, with the
knownquantitiesandtheunknownd(θistheanglethe
size r is observed at). T2011 discussed cases M ∝ dn scale factor a chosen to be dimensionless and the co-
movingradialcoordinater sothatthemetricdistance
where n = 0,2 or 3, and the distance d can be deter- c
is d=ar =r at the present time (a=1):
mined. c c
1 1 a¨r
=− × c , (2)
1.2 When r is the desired distance d from us (ar )2 G M
c eff
InsomecasesthelengthinEq.(1)isthedistancedfrom where the effective mass M = 4π(ar )3(ρ −2ρ )
eff 3 c m Λ
ustoanobject,asinabinarysystemwiththeobserver (ρ is the density of the gravitating matter and ρ is
m Λ
Copyrightlinewillbeprovidedbythepublisher
2 P.Teerikorpi and Yu.V.Baryshev:Mattig’s relation and dynamical distance indicators
thatoftheΛterm).WenotethatEq.(2)isexactinthe AfterHarrison’s(1993)work,itisnaturaltounder-
Friedmann model, not an approximation valid, say, at line the exact link between the expansion velocity and
shortdistances(Baryshev2008;Baryshev&Teerikorpi the metric distance. It is also instructive and logical
2012). when consideringthe expansionlaw asa distance indi-
Herethedynamicalequationis,ofcourse,Einstein’s cator, to derive first the present moment V(z) for the
fieldequation.Withthesymmetriesunderlyingtheworld redshift z and then divide it by H0. The mathematics
model it leads to Friedmann’s equations (and Eq.(2)). is essentially as in Mattig’s original way.
The puzzling similarity of Eq.(2) to Eq.(1) has led to In general, the light emitted from a galaxy having
discussions on its meaning (e.g., Layzer 1954) and if the redshift z, has gone through a comoving radial co-
it could throw light on conceptual problems in Fried- ordinateintervalr beforehavingreachedus.Thenone
c
manncosmology(e.g.,Baryshev2008).Herewesimply can write for the present rate of increase of the metric
askiftheanalogyextendstohowoneobtainsthe(well- distance between us and the galaxy, using adr =cdt:
c
known) distance indicator in the Friedmann world. dar a0 da
c
V(z)=( ) =(a˙) c . (5)
Wecannotmeasuretheaccelerationofspaceexpan- dt 0 0 Z aa˙
a0/(1+z)
sion a¨r between us and the remote object. However,
c
One may express the derivative a˙ in terms of a using
we can replace a¨ in Eq.(2) by −qa˙2/a, where q is the
the first Friedmann equation. Changing the variable
decelerationparameter,constantonallscalesr atthe
c
from a to z, noting how ρ and ρ depend on a, and
epoch t (fixed a(t)), due to the system’s regularity. So m Λ
dividing by H gives the known formula
0
G M
(a˙r )2 = × eff , (3) V(z)
c
q ar d(z) =
c
H
0
whichresemblestheepressionsoftenappearingforclas- c z dx
sical dynamical distance indicators (T2011). Writing = (6)
V = a˙r and d = ar , we see also here the M/d de- H0 Z0 (Ωm(1+x)3+ΩΛ+ω(1+x)2)12
c c
generacy. It can be overcome, as Meff ∝ d3, so n = 3. whereω =1−Ωm−ΩΛ(=0forflatspace).Onlyforthe
Inserting the expression for M into Eq.(3) one gets flat pure Λ model the expansion velocity is the ”clas-
eff
γ sical Doppler” V(z)=cz. Eq.(6) can be generalisedto
V =a˙r =( )1/2×ar =H(t)d, (4)
c c includeenergytransferbetweenthemattercomponents
q
(Teerikorpi et al 2003; Gromov et al 2004).
thelineardistance-velocitylawintheFriedmannworld.
Here γ = 4πG(2ρ −ρ ) is constant for a fixed scale
3 Λ m
factor.It is this law that offers a route to the distance. 4 Concluding remarks
In a landmark study, Harrison (1993) underlined
that the relation V = Hd is exact. V is the rate at We showed the steps from redshift to metric distance
which the (proper) metric distance d between two ob- in order to sum up the idiosyncracies of the redshift
jects at rest in space grows. The Hubble constant H method where the system is the universe as described
0
fixesthis directlyunobservablelawinitscurrentstate. by the Friedmann model. This machinery as a dynam-
ical distance indicator involves tightly packed all the
TheexpansionrateV cannotbedirectlymeasured.
ingredients of cosmological physics, such as the Cos-
In classicalindicators if the radialvelocity is needed it
mologicalPrinciple,generalrelativity,spaceexpansion
isinferredusingtheDopplereffect.Theexpansionrate
(instead of motion in static space), and the Lemaˆıtre
betweenus andthe objectisalsogotfromthe spectral
redshifteffect.Atthesametime,itisinstructivetosee
lineshiftbutnowusingtheLemaˆıtre(1927)effectthat
that each step has analogies with classical dynamical
gives the exact sense of the cosmologicalredshift.
distance indicators (Eqs.(2)-(5)).
Remarkably, when the density parameters of the
3 From redshift to velocity to distance
Friedmann model Ω and Ω are fixed, a single obser-
m Λ
vationofthe dimensionlessspectralredshiftallowsone
In his classic paper, Mattig (1958) solved the task of
to derive the current space expansion rate (in cm/s)
deriving the formula for the luminosity distance for a
betweenusandtheobjectinquestion.Then,theexact
given redshift. His aim was to express the apparent
magnitude of a standard candle as a function of red- andIfounditextremelyinsufficientthattherelationsz(m)and
shift,andthefocuswasnotonthemetricdistanceand N(m)aregiveninseriesexpansions.Itriedtofindaclosedform
forthe simplestcase, zerocosmological constant andflat space.
theexactlinearexpansionlaw(whichhadnotyetbeen
I succeeded within the two weeks and got through the exami-
adequately discussed at the time).1 nation with a good result.” He also noted: ”the reaction of the
cosmologicalcommunitywasnearlyzero,onlyAllanSandagedis-
1 Wolfgang Mattig derived his resultas student when he had cussed my relations in ApJ 133 (1961). He rendered accessible
togiveatalkoncosmologyforhisgraduateexam.Alettertothe myresults[which]werepublishedinGerman,inanEast-German
present authors reveals that during the short preparation time journal.”TheletterisprintedinBaryshev&Teerikorpi(2002).
”I also studied Heckmann’s book on ’Theorien der Kosmologie’ Forinterestingremarks,seeSandage(1995).
Copyrightlinewillbeprovidedbythepublisher
asna headerwill beprovided by thepublisher 3
distance-velocity relation (with the Hubble constant References
H ) leads to the proper metric distance, as expressed
0
in units of cm.This is the correct,straightforwardand Baryshev,Yu.,2008,inPracticalCosmologyVol.II.,Yu.V.
Baryshev, I.N. Taganov, P. Teerikorpi (eds.), Rus-
simple way to speak about the derivation of the dis-
sian Geographical Society Saint-Petersburg 2008, p. 20
tance within the standard cosmology.
(arXiv:gr-qc/0810.0153)
We find it useful to finish with some notes on the
Baryshev, Yu.& Teerikorpi, P., 2002, Discovery of Cosmic
end result, the metric distance. McVittie (1974) wrote Fractals (World Scientific, Singapore)
that”distanceisameasureofremoteness”.Inhisanal- Baryshev, Yu. & Teerikorpi, P., 2012, Fundamental Ques-
ysis of different distance types that appear in cosmol- tions of Practical Cosmology (Springer, Berlin)
ogyhedidnotpreferanyoneofthemabovetheothers. Eddington, A.S., 1930, MNRAS 90, 668
However,withtheFriedmannmodelasthebasisofcos- Gromov,A.,Baryshev,Yu.,Tuson,D.,Teerikorpi,P.,2001,
Gravitation & Cosmology 7, 140
mology,there are goodreasonsto view the ”tape mea-
Gromov, A.,Baryshev,Yu.,Teerikorpi, P.,2004, A&A415
sure” metric distance d as basic. It comes close to our
813
ordinary physical conception of distance, even though
Jackson, N., 2007, Living Rev. Relativity 10, (2007), 4
it cannot be directly measured.
(http://www.livingreviews.org/lrr-2007-4)
This distance appears in the exact expansion law. Harrison, E.R., 1993, ApJ403, 28
It is also additive, unlike, say, the luminosity distance. Layzer, D., 1954, AJ59, 268
Also, for z =∞ it gives the particle horizon(e.g., 2 c Lemaˆıtre,G.,1927,Ann.Soc.Sci.Brux.47,49(theEnglish
H0 translation in MNRAS91, 483 (1931))
intheEinstein-deSittermodel(Ω =ω =0inEq.(6)).
Λ
Mattig, W., 1958, AN 284, 109
The metric distance is the backbone distance re-
McCrea, W.H.,&Milne, E.A.,1934, Quart.J.Math.5,73
latedinknownwaystotheworkhorsesofpracticalcos-
McVittie, G.C., 1974, QJRAS15, 246
mology:luminosityandangularsizedistances.Thelink Sandage, A., Tammann, G.A., Saha, A., Reindl, B., Mac-
isespeciallysimpleinflatmodels,wheredlum =(1+z)d chetto, F.D., Panagia, N.,2006, ApJ653, 843
and d =(1+z)−1d and the metric distance d is the Sandage, A.R., 1995, in: The Deep Universe, eds. B.
ang
geometric mean of those other distances.2 Binggeli & R.Buser (Berlin, Springer), p.16
Teerikorpi,P.,Gromov,A.,Baryshev,Yu.,2003,A&A407,
Finally,thecurrentmetricdistanceisthebest”mea-
L9
sureofremoteness”characterizingthelocationofadis-
Teerikorpi, P., 2011, A&A 531, A10
tant objectin the galaxyuniverse.So,if its metric dis-
tanceis800Mpc,weatonceknowthatitis1000times
more distant than the Andromeda nebula. Therefore,
the metric distance is also the choice, say, in science
news about very distant galaxies, preferably with the
familiar light-year as unit.
The metric distance in light-years may be so large
thatitsnumericalvalueexceedstheageoftheuniverse
in years. This may disquiet an attentive reader, won-
dering how the light had enough time to cross such
a distance. Then the balloon analogy by Eddington
(1930), illustrating the Hubble law and the changing
metric distance, also offers an intelligible solution to
this ”age paradox”.In fact, the ”light travel distance”
is obtained by weighing the integrand in Eq.(6) by the
redshift factor (1+x)−1.
Acknowledgements We are grateful to Wolfgang
Mattig who kindly informed us on the history of his
1958paperinAstronomischeNachrichten.Wealsothank
the referee for a careful reading of the manuscript.
Yu.B.thankstheSaint-PetersburgUniversityforfinan-
cial support to this work (grant 6.38.18.2014).
2 In non-flat models dlum = (1+z)dext and dang = (1+
z)−1dext, where dext is the external metric distance (so termed
by Baryshev & Teerikorpi 2012), which Mattig (1958) derived
for zero-Λ dust models as his Eq.(12). It coincides with the in-
ternal distance din flat models, being otherwise related to das
dext(z) = RH(kω)−1/2sn((kω)1/2dR(Hz)), where sn = sin when
ω>0(andk=1)andsn=sinhwhenω<0(andk=−1).RH
istheHubbledistancec/H0.
Copyrightlinewillbeprovidedbythepublisher