Verifying the Identity of High-Redshift Massive Galaxies Through the Clustering of Lower Mass Galaxies Around Them
aa r X i v : . [ a s t r o - ph ] J u l Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 31 October 2018 (MN L A TEX style file v2.2)
Verifying the Identity of High-Redshift Massive GalaxiesThrough the Clustering of Lower Mass Galaxies AroundThem
Joseph A. Mu˜noz ⋆ and Abraham Loeb † Harvard-Smithsonian Center for Astrophysics, 60 Garden St., MS 10, Cambridge, MA 02138, USA
31 October 2018
ABSTRACT
Massive high-redshift galaxies form in over-dense regions where the probability offorming other galaxies is also strongly enhanced. Given an observed flux of a galaxy, theinferred mass of its host halo tends to be larger as its inferred redshift increases. As themass and redshift of a galaxy halo increase, the expected clustering of other galaxiesaround it gets stronger. It is therefore possible to verify the high-redshift identityof a galaxy (prior to an unambiguous spectral identification) from the clustering ofother galaxies around it. We illustrate this method for the massive galaxy suggestedby Mobasher et al. (2005) to be at redshift z ∼ .
5. If this galaxy were to exist at z ∼ .
5, there should have been a mean of ∼
10 galaxies larger than a hundredth ofits mass and having z-band magnitudes less than ∼
25 detected as i-dropouts in theHUDF. We calculate an approximate probability distribution for neighbor galaxies anddetermine that there is less than a ∼ .
3% chance of detecting no massive neighborgalaxies. The lack of other massive z ∼ . Key words:
Cosmology: Theory – Galaxies: High-redshift
The purported detection of a very massive galaxy, HUDF-JD2, at a redshift of z ∼ . z ∼ z ∼ .
5, the estimated mass of the halo contain-ing this galaxy is M JD ∼ × M ⊙ for a reasonable starformation efficiency, f ⋆ , of 10%. In the concordance cosmo-logical model, the expected number of such galaxies in theHubble Ultra Deep Field (HUDF) field-of-view is less than10 − (Barkana & Loeb 2006). Since the expected probabil-ity of finding such a galaxy in the HUDF is extremely low,its redshift identification is of great importance for testingthe standard cosmological model.Here we point out that, despite the low average abun- ⋆ E-mail: [email protected] † E-mail: [email protected] dance of galaxies like JD2 (due to its large mass and highredshift), such a galaxy cannot exist alone. The large-scaleover-density implied by its existence naturally results inneighboring halos over and above what would be expectedfrom random fluctuations in the average galaxy population.The surrounding over-dense region behaves as if it is partof a closed universe, in which the formation of all galaxiesoccurs earlier. We approach this problem analytically in this
Paper . In addition to the deeper fundamental understandinggained from such a treatment, the extreme rarity of objectslike JD2 make a statistical analysis using numerical simula-tions difficult. In §
2, we show how the excursion set formal-ism can be used to calculate an approximate probability dis-tribution for the number of neighbors around massive galax-ies. We then calculate, in particular, the expected clusteringof bright galaxies around the Mobasher et al. (2005) galaxyin §
3, and explore the dependence of our results on the starformation efficiency, duty cycle, and power-spectrum nor-malization in §
4. In §
5, we generalize these results to otherhalo masses and redshifts. Finally, § c (cid:13) Mu˜noz & Loeb for the universe with the
WMAP3 cosmological parameters(Spergel et al. 2007).
We assume the simple model for Lyman-break Galaxies(LBGs) considered by Stark et al. (2007), which associatesLBGs with merger-activated star formation in dark-matterhalos and includes suppression of the star formation ef-ficiency in low-mass halos by supernova feedback. In themodel, the star formation duty cycle, ǫ DC , gives the frac-tion of halos that contain active star formation. This frac-tion has recently been calibrated by the measured luminosityfunction of LBGs at z ∼ − σ er-rors of ǫ DC = 0 . +0 . − . (Stark et al. 2007). Here we adopt aconservative value of ǫ DC = 0 .
14, in accordance with our as-sumed star-formation efficiency of f ⋆ = 10% (which matchesthe fraction of Ω b in stars today). The remaining fraction,1 − ǫ DC , of halos at z ∼ z = 3, there is, as yet,no analytical method to predict this extra bias at higherredshift. Thus, our calculations for the number of neighbor-ing LBGs around massive galaxies at high redshift are lowerlimits that could be modified in the future with a betterunderstanding of the evolution of the “temporal” bias withmass and redshift.According to the excursion set prescription (Zentner2007), if the linear density fluctuations in the universe areextrapolated to their values today and smoothed on a co-moving scale R , a point whose over-density exceeds a criticalvalue of δ c ( z ) ≈ . D ( z = 0) /D ( z ), where D ( z ) is the lin-ear growth factor at redshift z , belongs to a collapsed objectwith a mass M = (4 / π ρ crit R if R is the largest scale forwhich the criterion is met, where ρ crit is the critical densityof the universe today. The critical value of the over-density,extrapolated to today from z = 6 .
5, is δ c ( z = 6 . ∼ .
6. Fora Gaussian random field of initial density perturbations, asindicated by
WMAP3 measurements of cosmic microwavebackground (Spergel et al. 2007), the probability distribu-tion of the extrapolated and smoothed over-density, δ R , isalso a Gaussian: Q ( δ R , S ( R )) dδ = 1 p π S ( R ) exp „ − δ R S ( R ) « dδ, (1)with zero mean and a variance given by: S ( R ) = Z k max dk π k P ( k ) , (2)where P ( k ) is the linear power-spectrum of density fluctu-ations today as a function of wave-number k , and k max =1 /R . Since equation (2) is a monotonically decreasing func-tion of R (or M ), the smoothing scale can be uniquely spec-ified by the variance of the over-density field smoothed on that scale. The critical threshold for collapse introduces asmall correction to the probability distribution such thatthe distribution of δ R becomes: Q ( δ R , S ( R )) = Q ( δ R , S ( R )) − Q (2 δ c − δ R , S ( R )) . (3)The conditional probability distribution of δ on a scalespecified by S given a value of δ on a scale larger scalespecified by S < S is: Q ( δ , S | δ , S ) = Q ( δ − δ , S − S ) . (4)Using Bayes Theroem, the conditional probability of δ ona scale S given δ on a smaller scale specified by S > S is: Q ( δ , S | δ , S ) dδ ∝ Q ( δ , S | δ , S ) Q ( δ , S ) dδ (5)with a constant of proportionality such that the integral ofequation (5) is unity. Setting δ = δ c ( z ) and S = S ( M ),where M is the host halo mass of a detected massive galaxy,gives the probability distribution of δ = δ R on any comov-ing scale R > R , due to the presence of that galaxy, where R is the radius corresponding to M .However, the excursion set formalism calculates the col-lapse of objects (a nonlinear effect) by considering the be-havior of the linear over-density field extrapolated to thepresent day. This method functions entirely in Lagrangiancoordinates (which move with the flow) and does not takeinto account how the over-density field changes in the quasi-linear regime. Obviously, when a region collapses, matter ispulled in from the surrounding region to fill the void. Thus,if we assume the existence of JD2 at z = 6 .
5, the materialin the rest of the HUDF at that redshift would have startedoutside the region earlier in the universe’s history.We denote the Lagrangian radius of a region as R L .Early in the history of the universe, before the region beginsto collapse, this radius is equal to the radius of the region inEulerian coordinates (which do not move with the flow). Asthe region collapses, the Eulerian radius shrinks, while R L remains unchanged. We denote the final Eulerian radius ofthe region at the redshift at which it is observed as R E .The extent of the collapse depends on the magnitudeof the over-density in the presence of the massive galaxywhose probability distribution we have just calculated inLagrangian coordinates. The more over-dense the region,the larger it would have to be initially to collapse to thesame value of R E . Similarly, a lower value of the over-densitywould mean that the material inside R E came from a rel-atively smaller Lagrangian size. We would like a mapping,then, between the comoving Eulerian size of a viewed re-gion (such as the HUDF), and the comoving Lagrangiansize of the region from where the same material originatedin the early universe. This can be obtained via the sphericalcollapse model (Mo & White 1996). A spherically symmet-ric perturbation of Lagrangian radius R L and over-density δ L > R E at redshift z given by: R E = 310 1 − cos θδ L D ( z = 0) D ( z ) R L (6)11 + z = 3 × /
20 ( θ − sin θ ) / δ L (7)For a fixed value of R E , there is a one-to-one relationshipbetween R L and the value of δ L that collapses R L to R E . c (cid:13) , 000–000 lustering Around Early Galaxies Figure 1.
The upper panel shows the probability distribution ofthe over-density δ (extrapolated to z = 0) due to cosmic vari-ance (Eq. 8) in a spherical region that collapsed to the size ofthe HUDF assuming the existence of a 2 × M ⊙ halo con-taining JD2 at z = 6 .
5. The central panel shows, for each valueof the over-density, the resulting number of LBGs in halos withmass above 2 × M ⊙ expected in this region. The final prob-ability distribution of the number of LBGs in the region above2 × M ⊙ is plotted in the bottom panel, taking into accountboth cosmic variance and Poisson fluctuations. In the presence of a massive galaxy, the probability dis-tribution of δ (in the Lagrangian sphere that collapsed to R E ) can be computed by considering the possible histories ofthe region R E having collapsed from different possible R L ’sweighted by the probability of the corresponding value of δ L in each R L . These weights are given by equation (5). The re-sulting probability distribution of δ (dropping the subscriptL) “seen” in a fixed R E can be expressed generally as: dP ( δ | M ) dδ ∝ Q ( δ, R L ( δ, R E ) | δ c ( z ) , R ( M )) , (8)where again the constant of proportionality is set so that R ( dP ( δ | M ) /dδ ) dδ = 1.Now that we know the distribution of over-densities inwhich the massive galaxy sits, we can easily calculate theexpected number of neighbor galaxies seen in R E in thepresence of each value for the over-density, ¯ N ( δ | M ), andthus, the probability distribution of ¯ N . Barkana & Loeb(2004) calculate the mass function of objects in a regionof fixed over-density by combining the Sheth-Tormen andPress-Schechter prescriptions in the regimes for which eachbest fits results from numerical simulations. Given the rela-tionship between δ and R L for a given R E , the over-densitydetermines the Lagrangian scale size. We find¯ N ( δ | M ) = ( V ( δ ) − V ( M )) ǫ DC Z m max m min dn bias dm ( m, z, δ ) dm (9) dn bias dm ( m, z, δ ) = dn ST dm ( m, z ) f PS ( δ c ( z ) − δ, S ( m ) − S ( δ )) f PS ( δ c ( z ) , S ( m )) , (10)where dn ST /dm is the Sheth-Tormen mass function, V ( δ ) isthe volume enclosed by the Lagrangian radius specified by δ , and f PS ( δ c ( z ) , S ) dS = δ c ( z ) S / Q ( δ c ( z ) , S ) dS (11)is the mass fraction at z contained in halos with mass in therange corresponding to ( S, S + dS ). The mass limit m max isthe mass enclosed by a sphere of today’s critical density withradius R L corresponding to δ . We ignore the probability that R L , while containing an over-density δ < δ c ( z ), might bepart of a larger collapsed region with δ > δ c ( z ). This is agood assumption given the unlikely occurrence of δ = δ c ( z )on the scale of M in the first place.The resulting probability distribution of ¯ N , due to cos-mic variance, is given by: P ( ¯ N | M ) d ¯ N = dP ( δ ( ¯ N ) | M ) dδ dδd ¯ N , (12)where δ ( ¯ N ) is the inverse of equation (9) and dδ/d ¯ N is itsderivative. Poisson fluctuations contribute additional vari-ation in the actual number, N , of neighbor galaxies. Theprobability, P ( N | M ), of each discrete value of N in thepresence of a mass M galaxy at high redshift is obtainedby convolving a discretized version of equation (12) with thePoisson distribution in the following way: P ( N | M ) = ∞ X ¯ N =0 ˜ P ( ¯ N | M ) P Poisson ( N, ¯ N ) , (13)where P Poisson ( k, λ ) = λ k e − λ k ! , ˜ P ( ¯ N = 0 | M ) = Z . P ( ¯ N | M ) d ¯ N d ¯ N , and for ¯
N > P ( ¯ N | M ) = Z ¯ N +0 . N − . P ( ¯ N ′ | M ) d ¯ N ′ d ¯ N ′ . (14)Equation (13) can be compared to galaxy counts in surveys.In particular, if no galaxies in halos above some minimummass, m min , are seen as neighbors to another galaxy in ahalo of mass M , then the quantity 1 − P ( N = 0 | M ) isapproximately the confidence by which we can rule out ei-ther the existence of a halo of mass M or the cosmologicalmodel. We show results for neighbor i-dropouts around JD2 in theHUDF. If JD2 is indeed a massive galaxy at z ∼ .
5, thenthere should be many massive galaxies visible around it. Fig-ure 1 shows the distribution of δ given by equation (8) insidethe Lagrangian patch that collapses to an angular Euleriansize of ∼ ′′ , enclosing an area on the sky roughly equiv-alent to the HUDF field-of-view. This angular scale corre-sponds to a comoving size of ∼ . z = 6 .
5, which c (cid:13)000
5, which c (cid:13)000 , 000–000 Mu˜noz & Loeb
Figure 2.
The dependence of the number of neighbor galaxieson the star formation efficiency and the duty cycle assuming afixed value of 2 × M ⊙ for the stellar mass of JD2. The uppertwo panels show the mean number of neighbor LBGs at z = 6 . . × M JD = 2 × M ⊙ inthe HUDF due to the presence of JD2 and the probability ofdetecting none of these objects as a function of the duty cycle.The assumed one-to-one relationship between the star formationefficiency and the duty cycle given in equation (15) is plotted inthe third panel, while the lower panel shows how the host halomass of JD2 depends on the duty cycle through its relationship tothe star formation efficiency. The vertical dashed line denotes thebest-fit value of the duty cycle given by Stark et al. (2007), whilethe dotted lines indicate the 1 − σ bounds. Since the validity ofequation (15) cannot be verified beyond the range of the 2 − σ contour, we truncate the plot near its lower boundary at ǫ DC of ∼ . assuming an extrapolated over-density of δ = 6 .
5, encloseswithin a spherical radius a mass of ∼ M ⊙ . Also shownare the expected number of neighbors inside that radius,given each value of the over-density, and the resulting prob-ability distribution of the number of neighbors that includesboth cosmic variance and Poisson fluctuations.While the HUDF should be sensitive to LBGs in hostsas small as ∼ × M ⊙ , the number of random galax-ies expected in such hosts without correlations from JD2 iscomparable to the number of excess neighbors that resultfrom these correlations. This creates some ambiguity in de-tecting the excess over the background. On the other hand,while the mean number of uncorrelated LBGs in halos aslarge as ∼ × M ⊙ is orders of magnitude smaller thanthe excess due to JD2, the probability of detecting no suchneighbor is not small enough for a lack of a detection to bemeaningful.Thus, we consider neighbor LBGs in halos with massesabove 2 × M ⊙ . For a duty-cycle ǫ DC = 0 .
14 and starformation efficiency f ⋆ = 0 .
1, these galaxies have lumi-nosities at 1500 ˚A above ∼ × ergs s − Hz − or z-bandmagnitudes at z ∼ . ∼
25. The mean abun-
Figure 3.
The dependence of the number of neighbor galaxies onthe normalization of the matter power spectrum σ . The upperpanel shows the mean number of neighbor LBGs at z = 6 . . × M JD in the HUDF due to thepresence of JD2, while the lower panel indicates the probabilityof detecting no such objects. The vertical dashed line denotes theWMAP value used in the rest of the Paper. dance of such galaxies in the absence of JD2 is much lessthan unity, and indeed, no such objects have been detectedin the HUDF (Bouwens et al. 2006, 2007). Figure 1 showsthat there should be a mean of ∼
10 very bright i-dropoutLBGs in hosts with masses larger than within an angularLagrangian radius of ∼ ′′ of JD2, where this is a lowerlimit due to the fact that LBGs should be more clusteredthan halos (Scannapieco & Thacker 2003). The probability,given by equation (13), of detecting no such galaxies in thisregion is P ( N = 0 | M JD ) ∼ × − . Thus, we can rule outJD2 at redshift z ∼ . z ∼
2, as allowed by spectral fits, or there is a problem withour assumed cosmological model.
So far, we have assumed a star formation efficiency of f ⋆ =10%, a duty cycle of ǫ DC = 0 .
14, and WMAP3 cosmologicalparameters with σ = 0 . ǫ DC and f ⋆ Given a stellar mass of 2 × M ⊙ for JD2(Barkana & Loeb 2006), changes in the star formationefficiency will affect the assumed host halo mass. Moreefficient star formation will, therefore, result in fewerneighbor halos due to the lower corresponding halo mass. c (cid:13) , 000–000 lustering Around Early Galaxies Figure 4.
Results for future detections of massive galaxies atfixed z as a function of host halo mass M . The upper panelshows the mean number of neighbor LBGs with a host halo massabove 0 . × M within the angular Eulerian distance correspond-ing to the HUDF, while the probability of finding no such LBGsin the HUDF is plotted in the lower panel. The dotted, solid,and long-dashed lines denote values for z = 5, 6 .
5, and 8, re-spectively. The vertical dashed line denotes the host halo mass ofJD2.
Meanwhile, changes in the duty cycle will affect the fractionof such halos that are seen as i-dropouts. Yet, these twoparameters are not constrained independently. Figure 4 ofStark et al. (2007) shows very narrow likelihood contours inlog-parameter-space. As a rough approximation, then, weassume that the range allowed by the luminosity functionfitting spans a power-law relationship between the starformation efficiency and the duty cycle. Using the best-fitparameter values and the extremes of the 1 − σ contour,we fit the dependence with a least-squares regression (inlog-space) and find a relationship given by: f ⋆ = A ǫ βDC , (15)where A = 0 .
264 and β = 0 . . M , denoted h N i , is plotted in Figure 2 as a function of ǫ DC . Also shownis the effect on the probability of detecting no such galaxies,given by setting N = 0 in equation (13). As shown, the effectof varying parameters on h N i is only modest. When consid-ering the range of 1 − σ errors on ǫ DC , h N i varies by onlya factor of ∼
2. The correlation between the star formationefficiency and the duty cycle given by equation (15), causesthese parameters to moderate each other’s effect on the ex-pected number of neighbors. While the value of P ( N = 0)varies more significantly, our choice of parameters through-out the rest of the paper gives conservative values for both h N i and P ( N = 0). The best-fit value of ǫ DC = 0 .
25 derivedby Stark et al. (2007) results in an even smaller probabilityof detecting no bright neighbor dropouts if JD2 is at z = 6 . Figure 5.
Results for future detections of massive galaxies as afunction of redshift z . The upper panel shows the mean numberof neighbor LBGs with a host halo mass above 0 . × M inthe HUDF, while the probability of finding no such LBGs in theHUDF is plotted in the lower panel. The solid, dotted, and long-dashed lines denote values for M = 2 × M ⊙ , 10 M ⊙ , and2 × M ⊙ , respectively. The vertical short-dashed line denotes z = 6 .
5, the redshift of JD2. σ The high over-density in the region surrounding a massivegalaxy at high redshift results from the large density fluctu-ation required to produce such a galaxy. If the over-densitymust reach δ c ( z ) on the scale corresponding to the size ofthe galaxy, then on a smoothing scale only a little larger, theover-density could not have been very low, since the contri-bution from the intervening scales is a Gaussian about zerowith a standard deviation much less than δ c ( z ). However,varying the normalization of the matter power-spectrum willchange the standard deviation of this Gaussian contribution.Reducing the value of σ will decrease the contribution fromthese intervening scales and cause the over-density aroundmassive galaxies to be larger. This will result in more neigh-boring galaxies on average and a lower likelihood of detect-ing none. Conversely, increasing σ will boost the possiblecontribution from intervening scales, shift the probabilitydistribution of δ in regions around massive, high-redshiftgalaxies toward lower values, and result in lower averagenumber of neighbors and a greater chance of detecting none.We explore the dependence of h N i and P ( N = 0) on σ quantitatively in Figure 3. The behavior is just as expected.The dependence is relatively strong given the wide range ofproposed values for σ . However, P ( N = 0) reaches only ∼
1% at σ = 0 . c (cid:13) , 000–000 Mu˜noz & Loeb
As new surveys continue to probe for ever larger galaxies atever higher redshifts (McLure et al. 2006; Rodighiero et al.2007; Wiklind et al. 2007), it is important to generalizeour results to potential future detections of other mas-sive, high-redshift galaxies in the future. We again assume( ǫ DC , f ⋆ , σ ) = (0 . , . , . M , and redshift, z , of the halo hosting the detected galaxyto vary.Figure 4 shows the mean number of neighbor LBGs, h N i , with host halo mass M > . × M that are within anangular Eulerian separation θ = 115 ′′ from a central galaxywith a host halo mass of M situated at a fixed z as a func-tion of M . While the uncorrelated, average abundance oflower mass halos is larger than that of higher mass halos,the plot demonstrates that the average number of neighborsbegins to increase as a function of M , for z = 6 . M ∼ × M ⊙ due to the nonlinear increase inhalo clustering with mass. Below this value, the increasinguncorrelated abundance of objects with lower mass causesthe number of neighbors to increase with decreasing M . At z = 5, h N i decreases as M increases in the entire rangeconsidered. The correlative effects of even a very massivehalo are dominated by the uncorrelated behavior at thisredshift. The mean number of neighbor LBGs within 115 ′′ is greater than unity for all values of M and z considered.The probability of detecting no such neighbors, P ( N =0), is also plotted in Figure 4. Its dependence on M is asexpected; P ( N = 0) increases as h N i decreases.In Figure 5, h N i and P ( N = 0) are plotted as functionsof z at fixed M and θ = 115 ′′ . While the plots of h N i and P ( N = 0) increase and decrease, respectively, with increas-ing z for M = 10 M ⊙ and 2 × M ⊙ due to increasedclustering with the detected halo, this is not indicated for M = 2 × M ⊙ . As shown in Figure 4, d h N i /dM < M = 2 × M ⊙ for all values of z considered indi-cating that clustering is less important for these halos andtheir neighbors. This is also seen in the larger values of h N i for M = 2 × M ⊙ compared to 10 M ⊙ . The increaseis due to an uncorrelated population of halos with mass2 × M ⊙ < M < M ⊙ . For each set of parameters, ahigher value h N i corresponds to a lower value of P ( N = 0),as expected. We have presented a clear signature for the existence of amassive galaxy at high redshift. Due to large correlationsamong massive halos at high redshifts, once such an objectis found, it is unlikely to be alone. On the contrary, the moremassive the halo and the higher its redshift, the greater thenumber of neighbors it has. This signature is helpful in sit-uations where only photometric data is available or whenthe spectroscopic redshift identification is ambiguous. Fora galaxy of a given observed flux, the inferred halo massincreases as its suggested redshift increases. It therefore be-comes easier to rule out a higher-redshift hypothesis, usingdropout techniques to locate the neighboring LBGs that areexpected at a similar redshift. We derived a probability distribution for the number ofneighbors around a massive galaxy at high redshift, whichled to calculations of the mean number of such neighborsand the probability of detecting no such halos.In the case of JD2, the number of neighbor galaxies ob-served in this way can distinguish between the z ∼ . z ∼ z = 6 . ∼
25 is predicted to be ∼
10. Thelack of such bright i-dropouts in the HUDF (Bouwens et al.2006, 2007) implies that JD2 cannot be at such high red-shift with 99 .
7% confidence. Thus, future detections of mas-sive galaxies at high redshift could be verified by looking forbright, massive neighbors.Additional research into exactly how LBGs form indark matter halos and how biased they are compared tothese halos would firm-up the predictions for the probabil-ity distribution of neighbors. Such work would undoubt-edly include various feedback processes and environmen-tal effects on LBG formation. Previous study indicatesthat LBGs are more clustered than dark matter halos(Scannapieco & Thacker 2003), and our results for the num-ber of neighbors are, thus, lower limits on the true values.
We thank Giovanni Fazio, Kamson Lai, Evan Scannapieco,Dan Stark, and Matt McQuinn, for useful discussions. JMacknowledges support from a National Science FoundationGraduate Research Fellowship.
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