arXiv:1503.02812v1 [astro-ph.CO] 10 Mar 2015 as,freape h tr iil otenkdeeaea are tru stars. eye the the naked of representative of the means distribution to no by visible selection stars special very the example, for cause, of out says: random he ”at then stars and M space”, Type in and stars ”dwarf” A the when and Type happens ”giant” chooses what considering one of by existence begins Eddington Russell. the stars. and proposed Hertzsprung had by that They introduced type) Hertzsprung-Russel been spectral recently the vs. had magnitude in (absolute stars other diagram of among (HR) distribution type” the spectral topics with associated ”Phenomena universe, book his In contribution Eddington’s 1.1. ac- into take to should leads observer space in the count. scattered that are effects and have interesting luminosities bodies celestial of that range vi- fact a astronomer observations The mind instruments. affect an in with problems of had made similar task) Bruno but the era, observations, least telescopic sual at the (or Before life difficult. the make that h ih ftercnrlsa nypol i h hr dialog reflect third may the (in they book poorly 3) his only and star of away, central far their too of too be light be may the they may 1) they planets: rea- 2) those some all faint, out see filled cannot pointed we cosmos he why the systems, sons planetary of the and vision for stars problems his with to discussed leads he Universe When the observer. in cos- position and that fixed understood our philosophical already a Bruno Giordano but thinker, mological astronomer, an not Although Introduction 1. srnm Astrophysics & Astronomy ue1,2021 11, June esyitninly”u ftesasi pc” be- space”, in stars the of ”out intentionally say We effects selection some of aware faintly was Bruno Thus eew td hssaitclba ncsooy clarifyin cosmology, in bias derived statistical (1920) Methods. this Malmquist study sky”. we the Here ”from gathered or space” pc.W lodsussm ocpulapcsta explain that aspects conceptual form some Results. his discuss that also 1936 We in showed space. Malmquist way the to analogously Aims. Accepted / U Received Astronomy, and Physics of Department Observatory, Tuorla aei nepnigFidanuies.Ti shlstu f true holds is This universe. Friedmann expanding an in made eest h itiuino bevd(norce o K-e for (uncorrected observed of distribution the to refers e words. Key fteifiieuies n worlds, and universe infinite the Of n11,Edntndrvdafruafrtedffrnebetwe difference the for formula a derived Eddington 1914, In digo 11)dsussi h chapter the in discusses (1914) Eddington digo-amus isi omlgclcontext cosmological a in bias Eddington-Malmquist h amus oml o h nrni difference intrinsic the for formula Malmquist The tla oeet n h tutr fthe of structure the and movements Stellar edrvdteMlqitrlto ihnagnrlcosmolog general a within relation Malmquist the derived We ehd:saitcl–Glxe:dsacsadrdhfs– redshifts and distances Galaxies: – statistical Methods: aucitn.25489˙final no. manuscript 1584). .Teerikorpi P. ABSTRACT e n xadn rvoswork. previous expanding and g etor ffect h h iprin1 dispersion the eur iprino lot3mg hl usl a de- had 1 Russell would of while mag type value mag, 11 mean spectral 3 of a almost M rived gap of the wide dispersion for the a dwarfs require way and this mean giants in between brighter explain to a that but shows dispersion, dis- same normal value the a is with sample magnitude-limited tribution a in stars among ecnldsta h eeto ffc,wihh a just h had he which giants. and effect, dwarfs selection explain cannot the discovered, that concludes he ls ihaGusa uioiyfnto L)( (LF) function luminosity Gaussian a with class uino bouemagnitudes absolute of bution luminosities”. implying intrinsic without the re- selection, in might of division groups two real mode any the double nearness if the de- for asks ”from been or he sult had brightness motion), for parallaxes proper either the (large chosen which been for had stars rived selection the a of As possibility effect. the ponders then Eddington vision, where enaslt magnitude, absolute mean hi itne i at rprmtos,poie htit distribution that the provided knows from one motions), derived and proper be Gaussian fact, may is (in stars lumi- distances A-type the of their how derived (LF) investigated formula function general He distribu- more nosity 1922). the spatial (1920, of homogeneous Malmquist case by special a a for is It valid tion. is (1) Equation formula Malmquist General 1.2. class candle. a standard Such Gaussian sample. a magnitude-limited called a be as may sky the from n antdsu oalmtn antd.Oersl was result One magnitude. limiting a to up magnitudes ent h iesoeo h isrelation. bias the of scope wide the M M l sas ai ntepeec fetnto nEuclidean in extinction of presence the in valid also is ula eea eainfrti ieec nEcienspace. Euclidean in difference this for relation general a iest fTru I-10 Piikki¨o, Finland FIN-21500 Turku, of niversity i rblmti n nt-admgiue when magnitudes finite-band and bolometric or i ntecretntto digo’ oml swritten is formula Eddington’s notation current the In hnh hw nafwsrksta o omldistri- normal a for that strokes few a in shows he Then di- giants and dwarfs the of reality the discussing When m z = − dpnetetnto)aprn magnitudes. apparent extinction) -dependent M h M M M 0 omlg:dsac scale distance Cosmology: − 0 ntema bouemgiue fsas”in stars of magnitudes absolute mean the en 0 i clfaeok nldn remn’ model, Friedmann’s including framework, ical − = stema bouemgiuewe stellar a when magnitude absolute mean the is 0 . 1 69 − . 382 σ /k / M 2 ( √ 2 σ ln d ihti oml thn,Eddington hand, at formula this With . 2 M 2 k d , a m hsntto) h rqec of frequency the notation), (his ) ( . m 4mgfralseta ye.Thus types. spectral all for mag 14 ) σ sas ai o observations for valid also is M h iprin ssampled is dispersion) the = M ihtemean the with a
( c m S 2021 ESO fappar- of ) a M ( m M 0 ) the = 0 and (1) M 2 P. Teerikorpi: Eddington-Malmquist bias in a cosmological context the Malmquist formula for the mean value of M of a sam- An early work on the Malmquist bias in cosmology was ple gathered through the apparent magnitude ”window” made by Bigot & Triay (1990), kindly communicated by 1 m 2 dm: them to us after the paper T98 was published. The present ± discussion should facilitate access to their technical treat- d ln a(m) M = M σ2 . (2) ment, where one result is the Malmquist integral relation, h im 0 − M dm Eq.(3), and where they conclude that the constant correc- The a(m) term in Eq.(2) depends on the spatial distribu- tion (Eq. (1)) is no longer valid for distant objects. tion of stars. It has a simple constant form, when the num- Here we derive the general formula using the cosmolog- −α ical route, but analogously to the way Malmquist (1936) ber density reads r , where r is the distance: M m = 2 h i remarkably showed that Eq.(2) is valid not only in trans- M0 (3 α)0.461σM . With α = 0, the Eddington Eq. (1) is obtained.− − A strongly thinning density with α = 3 is re- parent Euclidean space, but also in the presence of inter- stellar extinction. We first assume fully transparent space quired for no bias, M m = M0. Then the large volume at large distances, whichh i contributes high-luminosity stars to and start with the differential bias (Eq. (2)), whose deriva- the sample, is fully compensated for by the lower number tion illustrates well the classical and cosmological aspects density of stars. of the bias and from which it is easy to derive the integral bias.
1.3. About terminology 2.1. Bolometric magnitude The Eddington-Malmquist (or Malmquist) bias refers to the major difference in sampling luminous objects ”from We begin with the necessary formulae using the bolometric space” versus ”from sky”. The Malmquist relation is the magnitude. Instead of the classical distance, we use the red- general Eq. (2), while the Eddington formula is the special shift z as the parameter indicating the distance. Then the case, Eq. (1). observed apparent (bolometric) magnitude m is related to Butkevich et al. (2005) termed the bias in Eq.(2) differ- the absolute (bolometric) magnitude M as M = m µ(z), where µ(z) is the Friedmann model-dependent distance− ential, while integral bias was used to denote Malmquist’s 1 other formula, modulus of an object at redshift z. We consider a class of objects with a Gaussian LF Φ(M) 2 d ln A(mlim) for the bolometric magnitudes. Then the number of objects M int = M0 σM , (3) 1 h i − dmlim in the sky observed with the apparent magnitude m 2 dm (differential counts) may be obtained using the analogue± of where M is the mean for the whole magnitude-limited h iint the equation of von Seeliger (1898), now summing over the sample and A(m) is the cumulative distribution up to the redshift: magnitude limit m . ∞ lim ω dV When standard candle data are inspected as M vs. m a(m)dm = Φ(m µ(z))ρ(z) dzdm, (4) or M vs. r (r = distance), respective biasesh of Typei 1 4π Z0 − dz h i and Type 2 appear (as reviewed by Teerikorpi 1997, espe- where ρ(z) gives the co-moving spatial number density of cially Table I therein), which were also called classical and objects, possibly varying as a function of redshift, V (z) is distance-dependent by Sandage (1994). the co-moving volume up to redshift z, and ω is the solid Type 1 relates to the bias treated here, that is, how dV angle covered by the region under survey ( dz is the co- M m differs from M0. Type 2 refers to the magnitude cut- moving volume derivative). h i off effect when for instance a Hubble diagram is inspected By derivation, one obtains another needed expression as m versus log z. Often the Type 2 aspect is also called, a ∞ da(m) ω dΦ(m µ(z)) dV little misleadingly, the Malmquist bias. = − ρ(z) dz. (5) Another parameter is the Eddington bias, which de- dm 4π Z0 dm dz notes the influence of random measurement errors on de- Using Eq.(4), the average value of the absolute magnitude rived distribution functions (Eddington 1913, 1940). The 1 of the objects observed at m 2 dm reads Eddington bias was discussed by Teerikorpi (2004), who ∞ ± also considered how it works in concert with the Malmquist ω dV M ma(m)= (m µ(z))Φ(m µ(z))ρ(z) dz.(6) bias, Eq.(2). h i 4π Z0 − − dz We derive in Sect. 2 the Malmquist relation in a general cosmological context, first using the bolometric magnitude Inserting the Gaussian LF, Eq. (5) becomes ∞ and then for a finite-band magnitude. In Sect. 3, the result da(m) ω m µ(z) M dV = − − 0 Φ(m µ(z))ρ(z) dz is illustrated and compared with our earlier studies. Section dm −4π Z σ2 − dz 4 contains concluding remarks. 0 M 1 = 2 ( M ma(m) M0a(m)),(7) −σM h i − 2. Malmquist equation in a cosmological context from which one finally obtains We have previously discussed cosmological Malmquist bias 2 d ln a(mb) in the Hubble diagram at high redshifts (Teerikorpi 1998, Mb mb = M0b σM , (8) 2003; or T98, T03). This was made by calculating the be- h i − dmb haviour of the average log z for a Gaussian standard 1 h im One may write µ(z) = 5 log rlum(z)/10pc. The luminosity candle, taking into account the different foreground and distance rlum(z) is obtained using the well-known Mattig equa- background volumes as given by Friedmann models. The tion and its generalizations (e.g., Baryshev & Teerikorpi 2012), Malmquist formula was not directly considered. once the values of the Friedmann model parameters are fixed. P. Teerikorpi: Eddington-Malmquist bias in a cosmological context 3 where b means that the magnitudes are bolometric. (e.g., in the cosmological context) one considers the dis- Equation (8) is the same as the classical Malmquist for- tance modulus µ as a variable instead of z. Then mula. Below we discuss some further aspects of the result. ω dV dz(µ) F (µ)= ρ(z(µ)) (z(µ)) , (13) 4π dz dµ 2.2. K-correction K(z) and extinction E(z) ∞ The above derivation, with the bolometric magnitude a(m)= Φ(m µ)F (µ)dµ = (Φ ⋆ F )(m), and (14) in cosmology, corresponds to the case of transparent Z−∞ − Euclidean space in Malmquist’s original study, where the apparent magnitude could be bolometric or finite-band (no 2 d ln (Φ ⋆ F )(m) redshift). ∆Mm = σ . (15) − dm In practice, a finite-band magnitude mi is measured that gives rise to a redshift-dependent K-effect Ki(z). In Here we have the essential mathematical reason for the that case, one replaces M = m µ(z) by Mi,c = mi wide scope of the Malmquist relation because it is based on K (z) µ(z), and the end result− is similar to Eq.(8), now− i − the simple Gaussian convolution of the function F, which for the K-corrected Mi-magnitude Mi,c and the observed carries all cosmological factors (geometry, luminosity dis- mi magnitude: tance, possible number density evolution), and the distance modulus µ may be viewed as a dummy integration variable. 2 d ln a(mi) Mi,c mi = Mi0 σM . (9) With the K-effect (Sect.2.2), the new variable is constructed h i − dmi from µ(z)+ K(z), which is normally a monotonically in- We emphasize a subtlety in the magnitudes in Eq.(9). The creasing function. 2 difference Mi,c mi Mi0 indicates how much the mean value of theh intrinsici − (K-corrected) absolute magnitude of the objects at the observed apperent magnitude mi differs 3. Discussion from the actual mean Mi0 of the Gaussian LF. In the right- To illustrate the result in a concrete way, it is instructive side expression, the distribution a(mi) is that of the ob- to make numerical calculations of the left and right sides of served (uncorrected) apparent magnitude. the Malmquist equation in Friedmann space. Adding a z-dependent extinction E(z) to the model (Mi,c = mi Ki(z) E(z) µ(z)) leads to the same result. The symbol− m may− designate− either a bolometric or (as 3.1. Illustrations of the result in practice) a finite-band magnitude in the remaining text. For example, consider a class of Gaussian standard can- dles with M0 = 23 mag and σM = 0.4 mag. Bolometric 2.3. Integral relation magnitudes are− first assumed and the Einstein-de Sitter model with H0 = 50 km/s/Mpc is used (the exact value In the integral bias the relevant variable is the limiting mag- of H is not relevant). Then the distance modulus is µ = nitude m up to which the sample is complete in the 0 lim 5 log(1 + z (1 + z)1/2)+45.4 and the co-moving volume inspected region of the sky (in the derivation of the differ- −dV 1/2 2 5/2 ential bias it is not required that the sample is complete in derivative is dz ((1 + z) 1) /(1 + z) . Using these relations∝ in Eqs.− (4) and (6), with ρ(z) = this sense). Then up to mlim, the number of objects is constant (no number evolution), we calculate the logarith- mlim mic distribution of apparent magnitudes a(m) as shown in A(mlim)= a(m)dm, (10) Z−∞ Fig.1 (an arbitrary zero-point) and the average M at differ- ent observed m, or M m. The slope of log a(m) is indicated and the mean absolute magnitude for the whole sample is for a few apparenth magnitudes.i Note the expected classical mlim slope 0.6 at bright magnitudes (low redshifts). M intA(mlim)= M ma(m)dm. (11) Calculation shows that the expression M0 h i Z−∞ h i 2 − (d log a(m)/dm)/0.6 1.382σM indeed reproduces the numerically calculated× M (see the upper part of Fig.1). From what was discussed above, we know what M m h im is (i.e., Eq.(9)) both classically and cosmologicallyh fori a Of course, this agreement is just as expected from the Gaussian LF, and inserting this into Eq.(11) results in derived Malmquist relation. However, in presenting Fig. 1, we wish to underline several aspects. In the distribution 2 dA(mlim) a(m), m is the observed, uncorrected apparent magnitude. M intA(mlim)= M0A(mlim) σM , (12) h i − dmlim The difference M m M0 for intrinsic absolute magni- tudes can be derivedh i without− detailed information on the from which follows Eq.(3). Friedmann model in question. This is also true for some It is interesting to note that the Malmquist differential number evolution (ρ(z)). As the slope of log a(m) decreases and integral relations also apply to the extreme spatial dis- starting from 0.6, the Malmquist bias decreases for this tribution z = constant. This is considered in Appendix A. model as well. Figure 2 presents similar calculations, but now assum- 2.4. Malmquist relation via convolution ing finite-band apparent magnitudes subject to a K-effect. For simple illustration, the K-correction is taken to be The Malmquist relation can also be considered via convo- 2 lution. Namely, the distribution a(m) results from a con- For a class of identical objects (M = M0), Φ ∝ the Dirac volution of a Gaussian function Φ and a function F when δ-function, and the convolution results in a(m) ∝ F (m − M0). 4 P. Teerikorpi: Eddington-Malmquist bias in a cosmological context
Hubble log z vs. m diagram. A uniform spatial distribution was assumed with no comoving number evolution. We conclude that the curves of Mattig (1958) for Friedmann models need to be corrected for a non-constant Malmquist bias in the log z vs. m Hubble diagram (T98; also Bigot & Triay 1990). It was found that at bright m (low z) the Malmquist shift is close to classical, as expected. Then it generally decreases (in absolute value) towards fainter magnitudes. The same can now be seen from the Malmquist relation. When changing the model for example by adding the cosmological constant Λ, one simply asks how the slope of the counts a(m) changes. While the comoving volume derivative V ′(z) becomes steeper with increasing z when a positive Λ is added, (tending to increase the slope), the Fig. 1. log a(m) vs. bolometric magnitude m (Eq.(4)) for luminosity distance also increases (diminishing the slope). a standard candle with M = 23.0 and σ = 0.4 mag 0 − M The rapidly increasing volume is more important; it re- (the E-deS model, see text). The upper figures above the sults in steeper a(m), which means a larger Malmquist bias, curve are the mean values M m calculated from Eq. (6) that is, closer to the classical one. For example, referring to for m = 8, 16 and 22 mag,h respectively.i The lower figures Fig.1, at m = 22 mag the pure flat Λ model, ΩΛ = 1, would (∆) are the Malmquist bias values as calculated from the lead to the bias 0.15 instead of 0.08 for the E-deS flat slopes shown below the curve. model. Thus the− Λ-model leads to− a weaker m-dependence of the bias, as also derived in T98 and T03. For the K-effect, it was pointed out in T98 that if K(z) increases with z so that the objects become fainter quicker than when they are only due to the bolometric factor, then the backside volume effectively decreases and the trend in the Malmquist bias away from the classical case increases, as also seen here in Figs. 1 and 2. The K-effect can be important and increase the deviation of the bias from the classical constant value.
3.3. Practical note That we can derive the difference of the intrinsic magni- tudes M and M in principle from minimal information h im 0 (σM ) and raw data (a(m)) does not mean that we may gen- erally forget factors such as the K-correction when applying Fig. 2. log a(m) vs. finite-band apparent magnitude m for this result. a Gaussian standard candle. Here the K-effect needed in For example, we consider a Gaussian standard candle the calculations of a(m) and the average absolute mag- in a test of the Friedmann model, assuming that we have nitude M corresponds to the correction K(z)=2.5z. h im been able to determine a(m) and know σM . Other parameters have the same values as in Fig.1. This At the observed m the average M m is predicted to be and Fig.1 also show roughly where the redshift distribution dln a(m) h i M σ2 . This value is compared with the average peaks around m = 22 mag. 0 − M dm M m,data derived from the K-corrected apparent magni- htudesi of the objects at the observed (uncorrected) mag- nitude m. Each object has a known redshift, so one may K(z)=2.5z mag, roughly like for elliptical galaxies in op- calculate for each its K-corrected absolute magnitude for a tical wavebands (Coleman et al. 1980), making them ap- given Friedmann model. Therefore the test requires know- parently fainter than would be the bolometric expectation ing K(z) and the expression for the luminosity distance. at increasing redshifts. Again, here a(m) is the raw distri- Referring to Fig. 2, one might have derived the slope 0.46 at m = 16 and hence the bias 0.17 mag. Then bution of observed magnitudes, without the K-correction. − the Friedmann model is correct, which gives M 16,data = 23.17 mag, as derived from the K-corrected apparenth i mag- 3.2. Comparison with the bias analysis in T98 −nitudes of the objects at the uncorrected m = 16 mag. With all the data, one requires that the difference We have explained what happens to the Malmquist bias in cosmology in terms of the luminosity distance and the d ln a(m) ∆M = M M σ2 (16) corresponding comoving volume (T98). As the cosmological m h im,data − 0 − M dm distances and volumes are related in a way different from does not depend on m. In addition, ∆Mm 0 for the the classical distances, the bias is generally not constant in 3 h i ≈ Friedmann models, but depends on apparent magnitude. correct Hubble constant and M0. 3 In the cited analysis, instead of M m, we calculated the The same is valid in stellar statistics if the Malmquist rela- quantity log z , which is directlyh suitablei to analyse the tion is to be used to derive the mean absolute magnitude of a h im P. Teerikorpi: Eddington-Malmquist bias in a cosmological context 5
4. Concluding remarks Malmquist (1936) reported that his relation is also valid when light extinction is added to the static Euclidean space he considered. His study inspired the present work, which shows the scope of the Malmquist relation from classical situations to Friedmann cosmological models. The cosmological factors (luminosity distance, comov- ing volume derivative, and number evolution) are all re- flected in the slope of the (log) counts vs. observed apparent magnitude. The K-effect for finite-band magnitude and also possible z-dependent extinction are automatically included in the right side of the Malmquist relation. We emphasized conceptual aspects of the Malmquist relation in view of its important role in stellar statistics. However, prospects for its practical use for high-luminosity Fig. A.1. Differential Malmquist relation as applied to the objects in extragalactic astronomy are not so immediate. case of a cluster at distance modulus µ = 10. The LF is First, a constant Gaussian LF is rare or absent for ob- Gaussian with M0 = 5 and σ = 1 mag. The slope of jects found at low and high redshifts. Second, it requires the ln a(m) curve at any− m, multiplied by σ2, gives the dln a(m) many data to dermine dm with good accuracy. In ad- difference m0 m. dition, the Type 1 Malmquist analysis is too simplistic if − the objects are not detected on the basis of non-variable brightness, but there is a chain of measurement and lumi- to a Gaussian function with the mean m0 = M0 + µ(z0) nosity inference as for Ia supernovae. and is observed up to the sample limit mlim. The second The Type 2 approach is often applied in luminosity- term in the right side of Eq.(9) for any m
Appendix A: Case of a cluster (z = const.) References In the integrations of Sect.2 (e.g., in Eq. 4) integration can Baryshev, Yu., & Teerikorpi P. 2012, Fundamental Questions of be restricted to a finite z range (where the objects in ques- Practical Cosmology (Springer, Berlin) tion exist). The density law ρ(z) takes care of this. Bigot, G. & Triay, R. 1990, Phys.Lett.A 150, 227 Equation (9) also applies for a δ-function-like ρ(z), for Butkevich, A., Berdyugin, A., & Teerikorpi, P. 2005, MNRAS 362, 321 instance, for a cluster at z = z0. Then a(m) is proportional Coleman G.D., Wu C.-C., & Weedman D.W. 1980, ApJSS 43, 393 Eddington, A.S. 1913, MNRAS 73, 359 stellar class. hMidata − M0 is derived from the histogram of the Eddington, A.S. 1914, Stellar Movements and the Structure of the apparent magnitudes. Then to infer M0, hMidata must be com- Universe (Macmillan, London) puted, which requires distances and extinctions for each sample Eddington, A.S. 1940, MNRAS 100, 35 star. Hendry, M.A., & Simmons, J.F.L. 1994, ApJ 435, 515 6 P. Teerikorpi: Eddington-Malmquist bias in a cosmological context
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