On the abundance of gravitational arcs produced by submillimeter galaxies at radio and submm wavelengths
aa r X i v : . [ a s t r o - ph . C O ] O c t Astronomy&Astrophysicsmanuscript no. radio June 24, 2018(DOI: will be inserted by hand later)
On the abundance of gravitational arcs produced bysubmillimeter galaxies at radio and submm wavelengths
C. Fedeli , , ⋆ and A. Berciano Alba , ⋆⋆ Dipartimento di Astronomia, Universit`a di Bologna, Via Ranzani 1, I-40127 Bologna, Italy INAF-Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna, Italy INFN, Sezione di Bologna, Viale Berti Pichat 6 /
2, I-40127 Bologna, Italy Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA, Dwingeloo, The Netherlands Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands
Astronomy & Astrophysics, submitted
Abstract.
We predict the abundance of giant gravitational arcs produced by submillimeter galaxies (SMGs) lensed by fore-ground galaxy clusters, both at radio and submm wavelengths. The galaxy cluster population is modeled in a realistic waywith the use of semi-analytic merger trees, while the density profiles of individual deflectors take into account ellipticity andsubstructures. The adopted typical size of the radio and submm emitting regions of SMGs is based on current radio / CO obser-vations and the FIR-radio correlation. The source redshift distribution has been modeled using three di ff erent functions (basedon spectroscopic / photometric redshift measurements and a simple evolutionary model) to quantify the e ff ect of a high redshifttail on the number of arcs. The source number counts are compatible with currently available observations, and were suitablydistorted to take into account the lensing magnification bias. We present tables and plots for the numbers of radio and submmarcs produced by SMGs as a function of surface brightness, useful for the planning of future surveys aimed at arc statisticsstudies. They show that e.g., the detection of several hundred submm arcs on the whole sky with a signal-to-noise ratio of atleast 5 requires a sensitivity of 1 mJy arcsec − at 850 µ m. Approximately the same number of radio arcs should be detectedwith the same signal-to-noise ratio with a surface brightness threshold of 20 µ Jy arcsec − at 1 .
1. Introduction
The e ff ect of gravitational lensing constitutes a unique researchtool in many astrophysical fields, since it allows one to investi-gate the structures of both the lenses (e.g., galaxies and galaxyclusters) and the lensed background sources, as well as to probethe three-dimensional mass distribution of the Universe. Inparticular, one of the most spectacular phenomena associatedwith the gravitational deflection of light are the giant arcs ob-served in galaxy clusters, which are caused by extended back-ground sources lying in the regions where the lensing mag-nification produced by the cluster is strongest. The e ff ectivesource magnification can easily reach ∼
30 in these cases, pro-viding the opportunity to detect and spatially resolve the mor-phologies and internal dynamics of high redshift backgroundsources at a level of detail far greater than otherwise possi-ble. An illustration of this powerful technique was presented inSwinbank et al. (2007), where a magnification factor of 16 bythe cluster RCS 0224-002 allowed them to study the star for-mation activity, mass and feedback processes of a Lyman break ⋆ E-mail: [email protected] ⋆⋆ E-mail: [email protected] galaxy at z ∼
5, something that (without the help of lensing)would not be possible beyond z ∼ / submm wavebands is expected to be muchlarger than in the optical (Blain 1996, 1997): Due to the spec-tral shape of the thermal dust emission, the observed submmflux density of dusty galaxies with a given luminosity remains C. Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths approximately constant in the redshift range 1 . z . ff ect, together with the steep slope ofthe observed submm number counts, produces a strong magni-fication bias that makes submm galaxies (hereafter SMGs) anideal source population for the production of lensed arcs.The SMGs were first detected about a decade ago withSCUBA (Smail et al. 1997; Hughes et al. 1998; Barger et al.1998; Eales et al. 1999). The current observational evidenceindicates that these objects are high-redshift dust obscuredgalaxies, in which the rest frame FIR peak of emission isobserved in the submm band. Their FIR luminosities, in therange 10 − L ⊙ , are ∼
100 times higher than what is ob-served in local spirals. Their energy output seems to be dom-inated by star formation processes induced by galaxy inter-actions / mergers, although a good fraction ( ∼ − ∼
30 clusters observed with SCUBA (Smail et al.2002; Chapman et al. 2002; Cowie et al. 2002; Knudsen et al.2008), only 4 multiply imaged SMGs have been reportedto date (Borys et al. 2004; Kneib et al. 2004; Knudsen et al.2008). This extremely low detection rate is due to three ma-jor limitations of current 850 µ m surveys: (i) very small skycoverage ( ∼ ∼ ffi cient resolution ( ∼ ′′ ) to resolve extended lensed struc-tures. Current e ff orts to increase the surveyed area at 850 µ minclude the SASS , the SCLS and the all sky survey that willbe carried out with the HFI bolometer on board of the Planck satellite , but their resolutions will still be insu ffi cient to iden-tify lensed arcs. The only instrument that can currently providesub-arcsecond resolution in submm (at 890 µ m) is the SMA .However, the tight correlation between radio syn-chrotron and FIR emission observed in star-forming galaxies(van der Kruit 1973; Helou et al. 1985), provides an alternativeway to obtain high resolution images of SMGs. Commonlyreferred to as ”the FIR / radio correlation”, it covers aboutfive orders of magnitude in luminosity (Condon 1992; Garrett2002) out to z ∼ Submillimeter Common User Bolometer Array (Holland et al.1999), which used to be mounted at the James Clerk MaxwellTelescope (JCMT) located in Hawaii SCUBA-2 All Sky Survey, a ∼ × square degree survey witha 5 σ depth of 150 mJy and 15 ′′ resolution SCUBA-2 Cosmology Legacy Survey, a ∼
35 square degree sur-vey with a 5 σ depth of 3.5 mJy and 15 ′′ resolution σ depth of ∼
350 mJy and ∼ ′′ resolution The Submillimeter Array in Hawaii massive star formation: While young massive stars produce UVradiation that is re-emitted in the FIR by the surrounding dust,old massive stars explode as supernovae, producing electronsthat are accelerated by the galactic magnetic field and gener-ate the observed radio synchrotron emission (Harwit & Pacini1975; Helou et al. 1985). Therefore, given the possible com-mon physical origin of both emissions, radio interferometricobservations can be used as a high-resolution proxy for the rest-frame FIR emission of high- z galaxies observed in the submm.The advent of ALMA will open a new window intomm / submm astronomy at sub-arcsecond resolution and sub-mJy sensitivity, allowing the detection of resolved gravitationalarcs produced by SMGs. Although its small instantaneous fieldof view (FOV) severely limits ALMA’s survey capability, a 25meter submm telescope (CCAT ) is going to be built on a highpeak in the Atacama region to provide wide field images ( ∼ ) with a resolution of ∼ . ′′ at 350 µ m. With the com-bined capabilities of both instruments, arc statistics studies inthe submm might be possible.At the same time, radio interferometry is also experienc-ing major technological improvements. In particular, the VLA and MERLIN are currently undergoing major upgrades whichwill boost their sensitivities by factors of 10 −
30 and dramat-ically improve their mapping capabilities. The new versions ofthese arrays ( e -MERLIN and EVLA) will be fully operationalin 2010 and 2012 respectively. In order to assess the prospectsfor the study of gravitationally lensed arcs at submm and radiowavelengths, in this work we report detailed theoretical predic-tions about the abundance of arcs produced by the SMG popu-lation at 850 µ m and 1.4 GHz.The paper is organized as follows. In Section 2, we in-troduce all the relevant quantities that are necessary to derivethe total number of arcs detectable on the sky. In Section 3we present and discuss all the observational information aboutSMGs that is required for the subsequent strong lensing analy-sis: morphology, redshift distribution and number counts. A de-scription of our cluster population model and the way in whichthe abundance of large arcs is computed is given in Section4. The derived arc redshift distributions and number of arcsare presented in Section 5, while in Section 6 we discuss howthese relate to previous findings in the literature. A summaryand conclusions are presented in Section 7.The adopted cosmology corresponds to the standard Λ CDM model with cosmological parameters inferred from theWMAP-5 data release in conjunction with type Ia supernovaeand baryon acoustic oscillation datasets (Dunkley et al. 2009;Komatsu et al. 2009), namely Ω m , = . Ω Λ , = . σ = .
817 and H = h
100 km s − Mpc − , with h = .
2. Strong lensing statistics
The choice of the best parameters to be used in order to char-acterize the morphological properties of long and thin gravita- The Atacama Large Millimeter Array in Chile Cornell Caltech Atacama Telescope The Very Large Array in New Mexico The UK Multi-Element Radio Linked Interferometer Network. Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 3 tional arcs is still a matter of debate. In this work, we adoptedthe quite popular choice of the length-to-width radio d , whichhas to be larger than a certain threshold d (usually 7 . z s , thee ffi ciency of the galaxy cluster population to produce arcs withlength-to-width ratio d ≥ d , is parametrized by the opticaldepth τ d ( z s ) = π D Z z s Z + ∞ n ( M , z ) σ d ( M , z ) dMdz , (1)where D s is the angular diameter distance to the source redshiftand n ( M , z ) is the total number of clusters present in the unitredshift around z with mass in the unit interval around M . Thecross section σ d ( M , z ) is the area of the region on the sourceplane where a source has to lie in order to produce (at least) onegravitational arc with d ≥ d , for a single cluster with mass M at redshift z . This depends in general on the cluster structure,the source properties, and the redshifts of both the cluster andthe source. Since in realistic situations sources are distributedat di ff erent redshifts, we can calculate the average optical depthby integrating τ d ( z s ) over the source redshift distribution p ( z s ),¯ τ d ( z s ) = Z z s p ( ξ ) τ d ( ξ ) d ξ. (2)In this way, the total number of arcs with length-to-widthratio d ≥ d can be calculated as (Bartelmann et al. 1998) N d = π N ¯ τ d , (3)where N is the observed surface density of sources, and ¯ τ d = ¯ τ d ( + ∞ ) is the total average optical depth (i.e., the average op-tical depth with the integral extending to all possible sourceredshifts). Therefore, the number of arcs produced by a givenpopulation of background sources can be calculated by provid-ing the following observational constraints: (i) the characteris-tic source shape and size, which is necessary to compute thecluster cross sections σ d ( M , z ), (ii) the source redshift distri-bution p ( z s ), which is necessary to evaluate the optical depth¯ τ d , and (iii) the cumulative source number counts N .
3. Characteristics of the SMG population
In strong lensing statistics studies, it is customary to character-ize the size of elliptical background sources using the equiv-alent e ff ective radius R e ≡ √ ab , which is the radius of a cir-cle that has the same area of the elliptical source with semi-major axis a and semi-minor axis b . In addition, the orien-tation of sources is randomly chosen, and to account for thedi ff erent source shapes the value of the axis ratio b / a is con-sidered to vary within a certain interval. The typical valuesof these parameters used in optical ray-tracing simulations are R e = . ′′ and b / a randomly varying in the interval [0 . , Fig. 1. E ff ective radii and axis ratios derived for the 1 . µ m counterpartof GN20 detected with the SMA, whose size was derived byfitting a Gaussian (triangle) and an elliptical disk (square) tothe data (Younger et al. 2008). Error bars are computed throughstandard error propagation.Due to the low resolution of submm single-dish observa-tions (e.g., FWHM ∼ ′′ for SCUBA at 850 µ m), current esti-mates of the typical size of SMGs are based on continuum ra-dio (Chapman et al. 2004; Biggs & Ivison 2008) and millime-ter (e.g. Tacconi et al. 2006) interferometric observations ofsmall source samples. In particular, Biggs & Ivison (BI08 here-after) combined 1 . ff ective radius and axis ratio for eachof the 12 radio sources reported in BI08 (black circles). Notethat, although the b / a interval [0 . ,
1] used in optical lensingsimulations contains 11 out of the 12 radio sources, there areseveral error bars that extend below its lower limit. In addition,the optical e ff ective radius of 0 . ′′ is not suitable to describethem. Since the source sample is too small (and the error barsrather large) to derive a reliable size distribution , we decidedto use an e ff ective radius close to the median of the sample.As a result, the size of the 1.4 GHz radio emission produced bythe SMG population has been characterized in the following by b / a randomly varying within [0 . ,
1] and R e = . ′′ .Since SMGs seem to follow the FIR-radio correlation (e.g.Kov´acs et al. 2006), their emission at both 1.4 GHz and 850 µ mis expected to be associated with massive star formation. Thismeans that, as a first approximation, the same morphological C. Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths
Fig. 2.
Arc width probability distributions derived from a set of sources at z s = MareNostrum simulation at z = .
3. The two panels illustrate the di ff erence between selecting arcswith a length-to-width ratio d ≥ d = . d ≥ d =
10 (right). The black histograms were produced using the typicalsource size of SMGs at 1.4 GHz and 850 µ m assumed in this work ( R e = . ′′ and b / a randomly varying in the interval [0 . , R e = . ′′ and b / a ∈ [0 . , e -MERLIN radio interferometer at 1 . µ m emission of a SMG (GN20)for the first time (see Figure 1).If we wish to characterize gravitational arcs via theirlength-to-width ratio and make comparisons between obser-vations and theoretical predictions in an unbiased way, it iscrucial to resolve their width. To address if the resolution pro-vided by radio and submm instruments could be an issue forarc statistics studies, we investigated the width distribution ofarcs produced by a population of sources that is being lensedby a galaxy cluster. In particular, we used the most massivelensing cluster at z = . MareNostrum cosmologicalsimulation (Gottl¨ober & Yepes 2007), a large n -body and gas-dynamical run, whose lensing properties recently have beenstudied by Fedeli et al. (2009, in preparation). The mass dis-tribution of this cluster was projected along three orthogonaldirections, for which we derived deflection angle maps by stan-dard ray-tracing techniques (Bartelmann & Weiss 1994). Then,a set of sources at z s = ffi ciency peaks for lenses at z ∼ .
3) with R e = . ′′ and axis ratios randomly varying in the interval [0 . ,
1] waslensed through the three projections. As usual in this proce-dure, the sources are preferentially placed near the lensingcaustics following an iterative procedure to enhance the prob-ability of the production of large arcs. The bias introduced by this artificial increase of sources is corrected for by assigninga weight ≤ d ≥ d with d = . d =
10 (right). As expected, reducingthe source equivalent size produces a decrease in the width oflensed images. Note also that the behavior of the distributionsis practically independent of the minimum length-to-width ra-tio used to select the arcs. The most important feature, however,is that both distributions drop to zero for widths below ∼ . ′′ ,meaning that virtually no radio / submm (or optical) arcs havewidths smaller than that value. At 1.4 GHz, ∼ . ′′ resolu-tion is already accessible with MERLIN / e -MERLIN ( ∼ . ′′ ).Therefore, resolving the width of long and thin images for arcstatistics studies is in principle already possible at radio wave-lengths. However, until the advent of ALMA, the ∼ . ′′ reso-lution of the SMA at 950 µ m will only be able to resolve a verysmall fraction of arcs produced by the most extended SMGs.Finally, we would like to stress two points related to thechoice of source morphological parameters presented in thissection. First, the most luminous SMGs seem to be the result ofmerger processes (e.g., Greve et al. 2005; Tacconi et al. 2006), . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 5 Fig. 3.
Left panel . Redshift distribution of SMGs derived from the spectroscopic sample of Chapman et al. (2005) (cyan his-togram) and the photometric sample of Aretxaga et al. (2007) (dark-grey histogram). The curves SPZ and PHZ correspond tothe best fits provided by Eq. (4) to the spectroscopic and photometric data, respectively. The curve CHM corresponds to the bestGaussian fit to the simple evolutionary model for SMGs used in CH05 (as quoted in CH05), normalized to the redshift interval[0 , + ∞ ]. Note that the binning adopted here is di ff erent from the one used in CH05, so the ”redshift desert” in the redshift interval z = . − . Right panel . Thecumulative distributions corresponding to the fits reported on the left panel, calculated by using Eq. (5).hence it is unlikely that their true shape is elliptical, as we as-sumed. However, if a merging source is lensed as an arc at aparticular wavelength, irregularities in its shape and internalstructure will not significantly change the global morphologicalproperties of the arc, like the length-to-width ratio. What canhappen is that the length-to-width ratio of an arc changes withwavelength because the emitting region of the source at thosewavelengths have di ff erent sizes. For some of the wavelengthsthe source might even look like a group of small isolated emit-ting regions instead of a continuous one, which means that inthe image plane it will be observed as a group of disconnectedmultiple images instead of a full arc. A very illustrative exam-ple of this scenario is SMM J04542-0301, an elongated regionof submm emission which seems to be associated with a mergerat z = . µ Jy, the typical size derived from this sample might bedi ff erent from the one that could be derived from fainter SMGs.Note, however, that the change of the cluster cross section withsource size has a very small slope for R e between 0.2 ′′ and 1.5 ′′ (Fedeli et al. 2006). Therefore, deviations from R e = . ′′ within this interval (which is two times larger than the inter-val that contains the sizes measured by BI08 and Tacconi et al. (2006), see Figure 6 of BI08) will have a negligible e ff ect onthe derived number of arcs. A key point in trying to estimate the abundance of strong lens-ing features that are produced by the galaxy cluster populationis the redshift distribution of background sources. Distributionspeaked at higher redshift, or with a substantial high- z tail, willhave in general more potential lenses at their disposal, andhence will produce larger arc abundances as compared to low- z -dominated distributions. In addition, the lensing e ffi ciency forindividual deflectors is also an increasing function of the sourceredshift.The most robust estimate of the redshift distribution ofSMGs to date is based on the ∼
15 arcmin SCUBA surveycarried out by Chapman et al. (2005) (CH05 hereafter). Radioobservations were used to pinpoint the precise location of thesubmm detections, allowing the identification of optical coun-terparts that could provide precise spectroscopic redshifts. Thefinal sample is composed of 73 SMGs with 850 µ m flux den-sities > . > µ Jy.On the other hand, the SCUBA Half-Degree ExtragalacticSurvey (SHADES, Mortier et al. 2005; van Kampen et al.
C. Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths ) 850 µ m survey to date .From their catalog of 120 SMGs, 69 have robust radio counter-parts with S µ m ≥ S . ≥ µ Jy. Photometricredshifts for this sub-sample were calculated by Aretxaga et al.(2007) (AR07 hereafter) by fitting Spectral Energy Distribution(SED) templates to the available photometry at 850 µ m and1 . µ m (additional photom-etry at millimeter wavelengths was also used for 13 out of the69 sources). The histogram of the resulting photometric red-shift distribution, together with the spectroscopic one reportedby CH05, are shown in Figure 3. The accuracy on the photo-metric redshifts derived by AR07 is ∆ z ∼ .
65. Note that the re-quirement for a radio counterpart biases these two redshift dis-tributions against SMGs with z >
3, due to the less favorable K-correction in the radio compared with submm. Using a simpleevolutionary model, CH05 estimated that the fraction of SMGs( S µ m > z ∼ . z ∼ ∼ ff ect of this high- z tail on thepredicted number of arcs, we used three di ff erent analytic ex-pressions to characterize the redshift distribution of SMGs inour calculations (see Figure 3). One of them (CHM) corre-sponds to the best Gaussian fit to the distribution predicted bythe CH05 evolutionary model, as quoted in CH05. The othertwo (SPZ and PHZ) were obtained by fitting the CH05 andAR07 histograms with the following analytic function, usuallyadopted for optical strong lensing studies (Smail et al. 1995), p ( z s ) = β z Γ (3 /β ) z exp − z s z ! β , (4)where Γ ( x ) is the complete Euler-gamma function evaluated at x , z is a free parameter that broadly selects the position of thepeak, and β is another free parameter that defines the extensionof the high-redshift tail. Note that, unlike the histograms, thisfunction drops to zero at low redshift. However, the contribu-tion to the global lensing optical depth coming from sourcesat z . ffi ciency (see also the subsequent discussion inSection 5). The resultant best-fitting parameters of these threefunctions are summarized in Table 1.The SPZ curve constitutes a good representation of CH05data not corrected for spectroscopic incompleteness . ThePHZ curve, on the other hand, does not describe the AR07 dis- The largest SMG survey to date (0.7 deg ) has being carried outat 1 . Due to the lack of strong spectral features falling into the obser-vational windows, it is not possible to measure the spectroscopic z ofsources in the range z = . − . Table 1.
Parameters of the redshift distributions presented inFigures 3 and 4.
Nickname z (1)0 β (1) rms (2) z (3)p SPZ 1 .
99 2 . − . .
51 13 . − . − − .
30 2 . (4) − − .
64 2 . (4) − − .
81 2 . (1) best fit parameters of Eq. (4). (2) width of the best Gaussian fit. (3) peak position of the distribution. (4) the parameters associated with thesedistributions, provided by M. Swinbank,were misquoted in SW08 and CH05. tribution very well, failing to reproduce its z ∼ − . z part of the photometrichistogram in Figure 3, the function needs to raise very steeplyand hence, by construction, it must also drop steeply at high- z . Despite the function in Eq. (4) not being a good choice forfitting the photometric data, we nevertheless included the PHZcurve in our calculations because it highlights the consequenceof choosing a distribution that is truncated at z ∼
3. Finally,the CHM curve allows us to predict the number of expectedarcs if the CH05 histogram is corrected for spectroscopic in-completeness and high-z SMGs without radio counterparts (ra-dio incompleteness). We stress that at this stage we are not in-terested in using the best possible representation for the truesource redshift distribution, but only to adopt a few motivatedchoices that broadly cover the range of realistic possibilities,in order to check the corresponding e ff ect on the abundance ofarcs.To show more clearly the di ff erent behavior in the high-redshift tail of our three choices, we present their cumulativedistributions in the right panel of Figure 3, namely P ( z ) = Z z p ( ξ ) d ξ. (5)In particular, when P ( z ) ≃ z ∼ .
5, we still have P ( z ) ≃ . ∼
20% of the SMGs stillcan be found at z & . ∆ z p = . . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 7 Fig. 4.
Left panel . Histogram of the redshift distribution of SMGs derived by Chapman et al. (2005), corrected for spectroscopicincompleteness. This correction was implemented by interpolating the CH05 distribution in the region of the redshift desert(M. Swinbank, private communication). The SPZC line indicates the best Gaussian fit to the histogram. The blue solid linecorresponds to the redshift distribution of SMGs with S µ m > S . > µ Jy predicted by the semi-analyticmodel presented in Swinbank et al. (2008). The SPZ curve presented in Figure 3 has being included for comparison.
Right panel .The same histogram, also corrected for radio incompleteness using CH05 evolutionary model. The blue solid line corresponds tothe redshift distribution of SMGs with S µ m > ∆ z ∼ .
25, which is the field-to-field variation between theseven sub-fields in the CH05 sample due to cosmic variance.Therefore, we can consider SPZ as a good representation ofthe current observations, despite the fact that it comes from ahistogram that was not corrected for spectroscopic incomplete-ness.After the computations of the number of arcs were com-pleted, we became aware of the fact that the parameters quotedin CH05 for the best Gaussian fit to their simple evolutionarymodel (CHM, see Table 1) were incorrect (M. Swinbank, pri-vate communication). As it is shown in the right panel of Figure4, the CHM distribution has a higher- z tail as compared to thecorrect Gaussian fit (CHMC) and the prediction of the semi-analytical model (SWS). Since the true high- z tail of the red-shift distribution of SMGs is expected to be in between thecases considered in our calculations (SPZ, PHZ and CHM),and (as it will be discussed in Section 5.2) the final e ff ect ofthe source redshift distribution on the number of arcs is smallgiven the many uncertainties involved, we considered it unnec-essary to repeat the calculations for CHMC. The final ingredient needed to estimate the number of arcs pro-duced by SMGs is the observed surface density of this sourcepopulation, both at 1 . µ m. Let n ( S ) be the dif- ferential number counts, defined as the surface density of un-lensed galaxies per unit flux density S . Integrating n ( S ) overall fluxes above a given threshold, we obtain the respective cu-mulative number counts N ( S ) ≡ Z + ∞ S n ( ξ ) d ξ. (6)Let µ be the lensing-induced magnification of images on thelens plane, and µ + ≡ | µ | . If P ( µ + | d ) is the magnification prob-ability distribution for sources that are imaged into arcs with d ≥ d , then the magnified di ff erential number counts can becalculated as (Bartelmann & Schneider 2001) n ( S ) = Z + ∞ n S µ + ! P ( µ + | d ) µ + d µ + . (7)Hence, the magnified cumulative number counts read as N ( S ) ≡ Z + ∞ S n ( ξ ) d ξ = Z + ∞ N S µ + ! P ( µ + | d ) µ + d µ + . (8)As can be seen in Eq. (8), the lensing magnification bias hasa twofold e ff ect. On one side, sources that would be too faintto be detected without the action of lensing are amplified, andhence the respective images are brought above the detectionthreshold. On the other side, the unit solid angle is stretchedby lensing magnification, implying that the number density of C. Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths sources is decreased. Which one of these two e ff ects wins de-pends on the local slope of the unmagnified cumulative numbercounts. In particular, if N ( S ) ∝ S − α and α >
1, the numberdensity of sources will be increased, while if α < N ( S ) the flux density is derived by integrating the sur-face brightness over the area of the source, for the magnifiedcounts N ( S ) the integral is performed over the area of the re-sulting arc.Since our main motivation was to provide predictions forthe abundance of giant arcs to be detected in surveys carriedout with future instruments, we needed to provide the predictednumber of arcs as a function of the surface brightness, insteadof flux density. The reason is that we are working under theassumption that arcs are resolved structures, and therefore theyare observed as extended objects. Under these circumstances,the flux integrated over the resolution element of the instru-ment (seeing, PSF, pixel or beam) is no longer the total fluxof the source (as in the case of unresolved sources), and itmay therefore be below the limiting flux although the arc asa whole is not. In other words, arc detectability under these cir-cumstances is not limited by the flux density but rather by thesurface brightness.In order to take this into account, we had to convert theobserved number counts as a function of flux density into num-ber counts as a function of surface brightness. Assuming thatthe size of sources is given by R e , and that the surface bright-ness is constant across it, then N ( B ) = N ( S /π R ). Note that,since the surface brightness is not a ff ected by lensing, the mag-nification bias will manifest itself only through the solid anglestretching. Therefore, the cumulative magnified number counts(as a function of surface brightness) can be written as N ( B ) = N ( B ) Z + ∞ P ( µ + | d ) µ + d µ + . (9)Among other things, this implies that the magnification biaswill always decrease the cumulative number counts, irrespec-tive of the shape of the unmagnified ones.In the following we used the magnification distributiongiven by Fedeli et al. (2008), which is represented by the su-perposition of two Gaussians. In particular, we adopted the P ( µ + | d ) function for d =
10, but the result is virtually thesame also for the case d = .
5. Note however, that this (con-ditional) magnification distribution was computed for a back-ground population of sources that have di ff erent morphologiesthan SMGs (see Section 3.1). In principle, the bimodality ofthe magnification distribution is expected to be preserved be-cause it only depends on the caustic structure (Li et al. 2005),but it can be a ff ected by the source morphology in two op-posite ways. On one hand, since SMGs are smaller than inFedeli et al. (2008), we expect large arcs to form closer to thecritical curves, and therefore to have larger magnifications onaverage. On the other hand, the fact that SMGs are more elon-gated will favor the formation of large arcs in regions of lowermagnification. Given the uncertainties in other parts of the cal-culation, we consider that the use of a magnification distribu-tion derived for optical sources will have a marginal e ff ect on Table 2.
Parameters for the 850 µ m di ff erential number counts. Name S ∗ (mJy) n (4)0 , ∗ α β DB (1) . + . − . ±
48 2 . + . − . . + . − . SB (2) . ± .
08 1039 ± − . ± . − SM (3) − (1) best fit double power-law function, Eq. (10). (2) best fit Schechter function, Eq. (11). (3) shallowest Schechter function consistent with the data. (4) expressed in deg − for DB and in deg − mJy − for SB and SM. the derived number of arcs produced by SMGs. For a compre-hensive review of the many e ff ects that could a ff ect the estima-tion of arc abundances by galaxy clusters, see the discussion inFedeli et al. (2008). For the latest and most complete estimate of the submm num-ber counts at 850 µ m we refer to Knudsen et al. (2008) (here-after KN08), who carried out a combined analysis of the countsderived from the Leiden SCUBA Lens Survey (LSLS) and theSHADES survey. With an area of 720 arcmin , the SHADESsurvey is the largest blank-field submm survey completed todate, and therefore the least a ff ected by cosmic variance. Itprovides the best constraints for the submm number counts inthe flux density range 2 −
15 mJy (Coppin et al. 2006). On theother hand, the LSLS survey targeted 12 galaxy cluster fieldswhich cover a total area of 71 . in the image plane. Itprovides the deepest constraints at the faint end of the submmcounts (0 .
11 mJy, after correcting for the lensing magnifica-tion). In their analysis, KN08 used two functions to character-ize the combined di ff erential number counts from both surveys:A double power-law, n ( S ) = n , ∗ / S ∗ ( S / S ∗ ) α + ( S / S ∗ ) β (10)and a Schechter function (Schechter 1976), n ( S ) = n , ∗ SS ∗ ! α + e − S / S ∗ . (11)Moreover, when fitting the observed cumulative numbercounts, they added the supplementary constraint that the inte-grated light well below 0 . ff er by afactor of ∼ . . . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 9 Fig. 6.
The cumulative number counts of SMGs at 850 µ m (top panels) and 1.4 GHz (bottom panels), as a function of fluxdensity (left panels) and surface brightness (right panels). The conversion from flux density into surface brightness was doneassuming that the emission at both submm and radio wavelengths is homogeneous and have R e = . ′′ . Thin lines correspond to S . = S µ m /
50, while thick lines assume S . = S µ m / d ≥ d =
10. SM indicates the shallowest Schechter function that is consistent with thedata, while DB represents the best fit double power-law (see the text for more details).by SMGs at 850 µ m for the two following cases: (i) the shal-lowest Schechter function consistent with the combined LSLSand SHADES data (also shown in Figure 5) and (ii) the bestfit double power law function, hereafter refered to as SM andDB, respectively. The first one provides the minimum expectednumber of arcs consistent with observations, whereas the sec- ond one gives the number of arcs predicted by the best fit to thedata (see Table 2).The cumulative number counts derived for these two casesas a function of flux density are shown in the top left panelof Figure 6, including the corresponding counts corrected formagnification bias using Eq. (8). In the same Figure, the topright panel shows the cumulative number counts as function Fig. 5.
Comparison between observed and predicted cumula-tive number counts. The red and black solid curves correspondsto the best-fit double power-law (DB) and the best-fit Schechterfunction (SB) derived by Knudsen et al. (2008) for the com-bined 850 µ m cumulative number counts from the LSLS survey(Knudsen et al. 2008) and the SHADES survey (Coppin et al.2006). The black dashed line indicates the shallowest Schechterfunction consistent with the data of these two surveys (SM).The blue solid line indicates the cumulative number counts pre-dicted by the semi-analytic model presented in Swinbank et al.(2008) for SMGs with S µ m > S µ m > S . > µ Jy, and SMGswith S µ m > S . > . µ Jy are indicated bythe cyan solid and dashed lines, respectively. Note that the cyandashed line and the blue solid line are almost indistinguishable.of surface brightness. The corresponding counts corrected formagnification bias (which will be used to compute the numberof arcs) were derived using Eq. (9).
Figure 5 also shows the cumulative submm number counts pre-dicted by the SW08 model (blue solid line) compared withthe results from di ff erent 850 µ m SCUBA surveys. Note that,although the model tends to over-predict the counts at faintfluxes compared with the best fits provided by KN08 (red andblack solid lines), it is consistent with the observational er-rors. The cyan solid line indicates the predicted counts forSMGs with radio counterparts assuming S . > µ Jy.The fact that its shape is di ff erent from the shape of the bluesolid curve is because current observations only detect radioemission from ∼
60% of the observed SMGs. However, if weallow the sensitivity threshold to go down to the µ Jy level ex-pected for e -MERLIN, the SW08 model indicates that it would be possible to detect all the radio counterparts of SMGs with S µ m > / radio correlation (e.g.Kov´acs et al. 2006), the 1 . µ m number counts introduced in theprevious section (DB and SM). As shown in Figure 7 of CH05,the ratio between the 850 µ m flux density and the 1 . S µ m / S . ratio between 50 and100. Therefore, we decided to use these two scaling factors toderive first order upper and lower limits of the radio numbercounts of SMGs. The resultant cumulative radio number countsare shown in the lower panels of Figure 6. Note that the valuesof 50 and 100 chosen for the submm / radio flux density ratio aremeant to be indicative, since there are still many sources thatdisplay a ratio below 50 or above 100. The reader interestedin results given by di ff erent values of this ratio can scale thecurves appropriately in the upper panels of Figure 6. Also, ex-act numerical values can be made available by the authors uponrequest.
4. Strong lensing optical depth
To compute the total optical depth for lensed SMGs, we con-structed a synthetic cluster population composed of q = , . × ] M ⊙ h − at z =
0. Notethat it is not necessary to extract these masses according tothe cluster mass function, since this is already taken into ac-count in Eq. (1) by weighting the cross sections with the func-tion n ( M , z ). The structure of each cluster is modeled using theNFW density profile (Navarro et al. 1995, 1996, 1997), whichconstitutes a good representation of average dark-matter halosover a wide range of masses, redshifts and cosmologies in nu-merical n -body simulations (Dolag et al. 2004). Several studiesof strong lensing and X-ray luminous clusters also show thatthese are well fitted by an NFW profile (Schmidt & Allen 2007;Oguri et al. 2009). This profile also has the advantage that itslensing properties can be described analytically (Bartelmann1996).To account for the asymmetries of real galaxy clusters, thehalos are assumed to have elliptically distorted lensing poten-tials. However, instead of considering a single ellipticity valueto describe all the synthetic cluster lenses, we derived an el-lipticity distribution from a set of numerical clusters extractedfrom the MareNostrum simulation (Gottl¨ober & Yepes 2007).The strong lensing analysis required to generate this ellipticitydistribution was taken from Fedeli et al. (2009, in preparation),as described in Section 3.1. For each simulated cluster, the lens-ing analysis along three orthogonal projections was performed,computing the cross sections for arcs with d ≥ d = . z s =
2. For each of these projections, we foundthe ellipticity e of the NFW lens whose cross section is closest . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 11 Fig. 7.
The distribution of NFW lens ellipticities fitting thecross sections of a sample of numerical clusters. The red dashedline represents the best fit log-normal distribution, whose me-dian and dispersion are labeled in the top-right corner of theplot.to the cross section of the numerical cluster, i.e., we found theellipticity that minimizes the quantity r ( e ) = (cid:12)(cid:12)(cid:12) σ (n)7 . − σ . ( e ) (cid:12)(cid:12)(cid:12) , (12)where σ (n)7 . is the cross section of the numerical lens, and σ . ( e )is that of the NFW lens for a given potential ellipticity e .Figure 7 shows the distribution of the ellipticities that min-imize the quantity r ( e ) in Eq. (12) for all clusters in our nu-merical analysis. The dashed red line is derived by fitting thedistribution with a log-normal function of the kind p ( e ) de = √ πσ e exp " − [ln( e ) − ln( e )] σ d ln( e ) , (13)where the best-fit parameters are e = .
31 and σ e = .
23. Theellipticity values used to characterize the potential of the syn-thetic NFW cluster lenses were then randomly extracted fromthe above distribution.Elliptical NFW profiles are a good representation of realis-tic cluster lenses only when the clusters do not undergo majormerger events (Meneghetti et al. 2003). Since the merger ac-tivity of galaxy clusters is known to have a significant e ff ecton the statistics of giant arcs (Torri et al. 2004; Fedeli et al.2006), it has to be taken into account in the construction ofthe synthetic cluster population. For this reason, we used theexcursion set formalism developed by Lacey & Cole (1993) Also referred to as the ”extended Press & Schechter (1974) for-malism” (see also Bond et al. 1991; Somerville & Kolatt 1999) to con-struct a backward merger tree for each model cluster at z =
0, assuming that each merger is binary (see discussion inFedeli & Bartelmann 2007). When a merger happens, the eventis modeled assuming that the two merging halos (also describedas elliptical NFW density profiles) approach each other at aconstant speed. The duration of the merger is set by the dy-namical timescales of the two halos (see Fedeli & Bartelmann(2007) and Fedeli et al. (2008) for a detailed description of themodeling procedure).With the synthetic cluster population constructed in thisway, the total average optical depth was derived by comput-ing individual cross sections with the semi-analytic algorithmdeveloped by Fedeli et al. (2006), especially designed to esti-mate the strong lensing cross sections of individual lenses ina fast and reliable way. The optical depth for a discrete set oflenses can be recast as τ d ( z s ) = π D Z z s q − X i = ¯ σ d , i ( z ) Z M i + M i n ( M , z ) dM dz , (14)where the masses M i (1 ≤ i , q ) have to be sorted from the low-est to the highest at each redshift step, and the quantity ¯ σ d , i ( z )is defined as¯ σ d , i ( z ) ≡ (cid:2) σ d ( M i , z ) + σ d ( M i + , z ) (cid:3) . (15)This e ff ectively means that, for all the clusters with mass be-tween M i and M i + , we assume the average cross section of themodel dark-matter halos with masses M i and M i + . The algo-rithm of Fedeli et al. (2006) for computing strong lensing crosssections consists of first assuming sources as point-like circles,and then introducing the e ff ect of source ellipticity according toKeeton (2001). The source finite size is taken into account byconvolving the local lensing properties over the typical sourcesize.The total average optical depth is calculated by integratingEq. (14) over the source redshift distribution. E ff ectively, the p ( z s ) weighting is avoided since we assigned individual sourceredshifts (randomly extracted from the distribution p ( z s )) ateach of the q halos in the cluster sample and evolved theirmerger trees back in time until the respective source redshift.Given the large number of synthetic dark-matter halos used,this approach allows one to omit p ( z s ) in Eq. (2) when the red-shift integral is discretized.Despite the fact that the ellipticity distribution used tomodel the synthetic cluster population was derived for d ≥ d = .
5, the best fit parameters of Eq. (13) can be used to computethe cross sections for arcs also with d =
10 without compro-mising the results. The reason is that, although there might bea mild dependence of the distribution of lensing ellipticities on d , it is the overall caustic structure that defines the abundanceof arcs above a certain d , regardless of its precise value. Infact, the criterion used to determine the ellipticity distribution isbased on the similarity between the cross sections of the NFWlens and the numerical lens, which is an indirect way of com-paring the overall caustic structures produced by both kinds of lenses. To verify this argument, the ellipticity distribution wasre-computed using a criterion that is directly related to the caus-tic structure, that is, by defining the best fitting ellipticity as theone that minimizes the modified Hausdor ff distance betweenthe critical lines of the numerical and NFW lenses. The ellip-ticity distribution obtained in this way is very similar to the onedepicted in Figure 7 ( e = . rms = . e = .
31, see Figure7) is fully consistent with the one obtained by Meneghetti et al.(2003) comparing the deflection angle maps ( e ∼ .
3, usinglenses placed at z ∼ . z s =
5. Results
The redshift distribution of arcs with d ≥ d (arc redshift dis-tribution hereafter) is expected to provide information aboutthe redshift distribution of the background source populationthat is being lensed. However, it will also be distorted by thefact that the abundance of massive and compact galaxy clustersevolves with redshift, and that the lensing e ffi ciency dependson the relative distances of sources and lens with respect to theobserver.To assess the potential of this approach to gather informa-tion about the intrinsic redshift distribution of SMGs, we de-rived the arc redshift distribution associated with each of thethree source redshift distributions used as inputs in our calcu-lations (see Figure 3). This was done by computing, for eachinput distribution, the average optical depth ¯ τ d ( z s ) for severaldi ff erent values of z s . That is, we excluded in the computationof the average optical depth those model clusters (and theirrespective chain of progenitors) whose associated source red-shifts were > z s . The resultant cumulative arc redshift distribu-tions, obtained after normalizing the optical depth for each z s to the total average optical depth ¯ τ d ( + ∞ ) = ¯ τ d , are shown inthe right panel of Figure 8. The lines correspond to the best fitfor each distribution provided by the simple function¯ τ d ( z s )¯ τ d ( + ∞ ) = − exp − z s z s , ∗ ! γ , (16)where z s , ∗ indicates where the transition between the extrema 0and 1 occur, and γ indicates how sharp this transition is. Thebest-fit parameters for each of the three input source redshiftdistributions are summarized in Table 3. The correspondingdi ff erential arc redshift distributions are shown in the left panelof Figure 8. This parameter constitutes one of the best ways to quan-tify the morphological di ff erence between two sets of points(Dubuisson & Jain 1994). See also Rzepecki et al. 2007 for a di ff er-ent application to gravitational lensing. Table 3.
Parameters of the arc redshift distributions shown inFigure 8. p ( z s ) z s , ∗ γ SPZ 2 .
64 4 . .
31 8 . .
53 3 . Once more, these arc redshift distributions have beenshown for arcs with length-to-width ratio larger than d = d = . d =
10, the resulting arc redshift dis-tribution also does not change significantly between the twochoices.A comparison between Figure 8 and Figure 3 shows that, asexpected, the arc redshift distributions reflect the general prop-erties of the source redshift distributions used as input, but thereare also some noteworthy di ff erences between them. For in-stance, the arc redshift distribution associated with CHM tendsto zero at very low redshift, unlike in the case of the originalCHM distribution. The reason is that low redshift sources donot produce many arcs, because (i) they have very few potentiallenses at their disposal, and (ii) the lensing e ffi ciency of thoselenses is very low due to geometric suppression. This results ina lack of low redshift arcs in the distribution, which shifts itspeak to higher redshifts compared with the CHM peak (fromz p = . p ∼ . z p = .
76 to z p & z s . ffi ciency and the one due to the cuto ff of the in-put distribution to allow a significant shift in its peak, and theonly possible consequence for the distribution is to shrink andincrease the peak height in order to preserve the normalization.In general, it is apparent that the di ff erences between di ff er-ent source redshift distributions are somewhat enhanced whenit comes to the arc redshift distribution. Therefore, assumingthat redshift information is available for arcs, this approach canin principle be used to obtain some information about the gen-eral characteristics of the source redshift distribution, althoughit will probably not allow one to distinguish between redshiftdistributions that are very similar. In this section we present and discuss the main results of thiswork: the predicted number of arcs produced by SMGs at radioand submm wavelengths. To that end, we computed the total . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 13
Fig. 8.
Left panel . Di ff erential redshift distributions of SMGs producing large gravitational arcs, corresponding to the cumulativedistributions reported in the right panel. Right panel . The cumulative redshift distributions of SMGs producing radio arcs with d ≥ d =
10 for input source redshift distributions SPZ , PHZ and CHM (see the text for more details). Lines are the best fitfunctions given in Eq. (16) and Table 3, according to the labels in the plot.average optical depth for each of the three source redshift distri-butions presented in Section 3.2 (PHZ, SPZ and CHM), and forarcs with length-to-width ratio higher than both d = . d =
10. These quantities were then multiplied by the magni-fied cumulative source number counts presented in Section 3.3(SM and DB), to obtain the arc number counts as function ofsurface brightness. The results, extrapolated to the whole sky,are shown in Figure 9. A detailed list with the predicted numberof submm and radio arcs for di ff erent sensitivities is presentedin Tables 4 and 6, respectively.Note that the arc number counts given by the source redshiftdistributions SPZ and PHZ are almost indistinguishable on thescale of Figure 9, irrespective of the length-to-width threshold d adopted. As expected, SPZ produces more large arcs thanPHZ because it peaks at higher redshift, but only by a factor of ∼ ∼ d = . d =
10 (as wasalso expected). However, the ratio between the number of arcswith d ≥ d = . d ≥ d =
10 is not exactly equal to theratio in the respective optical depths, since the magnificationdistributions for the two kinds of arcs are also di ff erent (see thediscussion in Fedeli et al. 2008). Finally, when it comes to comparing the results from thetwo adopted source number counts (DB and SM), we see thatthe di ff erence in the predicted number of arcs is negligible forsubmm surface brightness limits greater than 5 mJy arcsec − .However, at B µ m = . − , the function DB pre-dicts 2 times more arcs than SM, and the di ff erence becomes afactor of 5 for 0 . − . In the radio domain, the di ff er-ence between DB and SM is negligible for B . = µ Jyarcsec − , a factor ∼ µ Jy arcsec − , a factor ∼ µ Jyarcsec − and a factor ∼ µ Jy arcsec − . Therefore, the un-certainty in the predicted number of arcs is clearly dominatedby the uncertainty of the source number counts at the faint sur-face brightness end.Considering an all-sky submm survey with enough resolu-tion to resolve individual arcs with d ≥ d = . ∼ . ′′ ),and a sensitivity of B µ m = − , our calculationspredict between ∼
500 (PHZ SM) and ∼
600 (SPZ DB) arcswith a signal-to-noise ratio (SNR) larger than 5. In the case of d ≥ d =
10, the expected number of arcs would be between200 and 250. If the sensitivity is reduced to 0 . − ,these predictions can vary between ∼ ∼ d = .
5, and between 1400 and 3600 for d = B . = . − would detect between 8 and 50arcs for d = . d =
10) at SNR ≥
5, with the main uncertainty given by the S µ m / S . ratio used to obtain the radio number counts by scaling the ob-served submm counts. If the limiting radio surface brightnessis reduced to ∼ µ Jy arcsec − , the predicted number of arcscould be increased to ∼ − d = . −
600 for
Fig. 9.
The total number of arcs with d ≥ d = . d ≥ d =
10 (right panels) that are predicted to be observed inthe whole sky above the surface brightness reported on the abscissa. Results for each of the three input source redshift distributionsas well as both source number counts adopted in this work are shown, according to the labels. The two top panels refer to surfacebrightness at 850 µ m, while the bottom panels refer to 1 . S . = S µ m / S . = S µ m / ff erence in scale on the abscissa between top and bottompanels. d = ≥
5, it would be necessary to go asdeep as S . = µ Jy arcsec − . The largest number of arcs isgiven by the CHM source redshift distribution, which is abouta factor of 2 larger than the number of arcs predicted by SPZand PHZ. It is plausible that future radio and submillimeter surveysof galaxy clusters would focus on the most massive objects,since the center of attention of many multiwavelength studiesis on X-ray bright clusters. To roughly evaluate the e ff ect ofthis kind of selection, we re-computed the optical depths byincluding only those clusters in our synthetic population withmass M ≥ × M ⊙ h − . The resulting arc number counts . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 15 Fig. 10.
The total number of arcs with d ≥ d = . d ≥ d =
10 (right panels) that are predicted to be observedin the whole sky above the surface brightness reported on the abscissa. Only clusters with mass M ≥ × M ⊙ h − are includedin the calculations. Results for each of the three input source redshift distributions and both source number counts adopted in thiswork are shown, according to the labels. Top panels refer to submillimeter number counts, while bottom panels refer to radionumber counts (notice the di ff erent scale on the horizontal axis). The di ff erence between thin and thick lines is as in Figure 9.are presented in Figure 10, where we used the same scale andline types as in Figure 9 to ease comparison. The correspond-ing numbers of submm and radio arcs predicted for di ff erentsensitivities are listed in Tables 5 and 7, respectively. The mostinteresting feature about these new plots is that the relative dif-ference in the arc number counts produced by the CHM dis-tribution on the one hand, and PHZ and SPZ on the other, issignificantly reduced. This is due to the fact that massive clus- ters are found mainly at low redshift, hence lower-mass higher-redshift lenses, that are accessible only to the CHM distribu-tion, become unimportant.The order-of-magnitude reduction in the abundance of gi-ant arcs when focusing only on clusters with mass M ≥ × M ⊙ h − is in agreement with the fact that the bulk of thelensing signal actually comes from low-mass clusters, since theoptical depth is obviously dominated by the lowest mass ob- jects that are capable of producing a non-vanishing cross sec-tions (see Eq. 1). Note also that, unlike the previous case, PHZproduces slightly more arcs than SPZ. This is due to the factthat we are including in the calculations only low- z clusters,and PHZ actually has more sources with, e.g., z s > d = . B µ m = − , we predict ∼
20 arcs with
SNR ≥ d = − −
90) if B µ m ≥ . − . In the case of a radiosurvey, it would be necessary to push the limiting flux densitydown to ∼ µ Jy arcsec − to detect few hundred arcs (between40 −
200 if d =
6. Comparison with previous work
The probability of strong gravitational lensing due to back-ground sources at radio and submm wavelengths has been apoorly studied issue in the past years. In addition, the fewworks available in the literature usually involve cosmologicalmodels, deflector mass ranges, and modeling approaches forthe source and lens populations that are di ff erent from the onesused in the present work, making the comparison between themdi ffi cult and often not possible. With this note of caution, wetried however to make some of these tentative comparisons inthe following. This required us to repeat the calculations inthe last section using the number counts as a function of fluxdensity instead of surface brightness. The results are shown inFigure 11 only for arcs with d ≥ d = Given the poor resolution of current submm instruments, obser-vational studies of submm arcs have not been possible so far,with only one submm arc candidate reported until now. This arcis supposed to be the brightest region of the extended submmsource SMM J04542 − . − ff erent galaxy evolution models. More re-cent studies (Cooray 1999; Paciga et al. 2009) also have beenfocused on predicting the number of submm lensed sources, butpredictions for the abundance of submm arcs have never beenattempted before.For instance, in the work of Paciga et al. (2009), the au-thors employ the strong lensing analysis of the MillenniumSimulation performed by Hilbert et al. (2007) (see alsoHilbert et al. 2008) in order to compute (i) the average mag-nification of SMGs as a function of flux density, and (ii) thecontribution to the di ff erential number counts given by sources with di ff erent redshifts and magnifications. Hence, their resultscannot be compared with ours in a straightforward way.The only work with which we could try a tentative com-parison is the one by Cooray (1999) (CO99 hereafter), wherethe author provides number counts of (among others) gravi-tationally lensed submm sources as a function of their mag-nification. The clusters were modeled as Singular IsothermalSphere (SIS henceforth) density profiles, which means that theimage magnification equals its length-to-width ratio, as long assources are circular and point-like. Since the sources that weare using are not circular, nor point-like, the following com-parison should be taken with caution. The background submmsources were described by means of the redshift and num-ber distributions observed in the Hubble Deep Field (HDF).Using a Λ CDM cosmology, and considering cluster lenses with M ≥ . × M ⊙ h − , CO99 predicted ∼
500 submm sourcesin the whole sky with lensing magnification larger than 10 and S µ m ≥ S µ m ≥ d ≥ d =
10 in the whole sky.Restricting the cluster mass range to M ≥ × M ⊙ h − re-duces the number of arcs to ∼
100 at most. Matching the massrange of CO99 would only reduce the number of predicted arcseven further, hence being discrepant with CO99 predictions.Assuming that the source population we are considering is thesame, we ascribe at least part of this disagreement to the factthat CO99 considers a very high normalization of the powerspectrum ( σ ≃ . The first observational search for radio arcs in galaxy clustersdates back to Bagchi & Kapahi (1995), where the authors con-sidered a cluster sample of 46 objects with z . . ∼ . . . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 17 Fig. 11.
The number of arcs with d ≥ d =
10 predicted to be observed in the whole sky as a function of the submm fluxdensity at 850 µ m (top two panels) and radio flux density at 1.4 GHz (bottom two). The left panels refer to all clusters in thesample, while the right ones shows results when only clusters with mass M ≥ M ⊙ h − are considered. Di ff erent line styles andcolors refer to di ff erent number count models and source redshift distributions. In the two bottom panels, the thin lines assume S . = S µ m /
50, while the thick lines refer to S . = S µ m / ∼ − . Assuming that cluster galaxies have the same veloc-ity dispersion of the dark-matter particles (Gao et al. 2004;Biviano et al. 2006; Faltenbacher & Diemand 2006, see how- ever Col´ın et al. 2000), and adopting the simulation-calibratedscaling relation of Evrard et al. (2008), this corresponds to amass of at least ∼ × M ⊙ h − . The rest of their clustersshould have a mass & × M ⊙ h − . Assuming a limiting fluxof ∼ . . . M ≥ × M ⊙ h − . Therefore, we find it quite unlikely that the detection claimed by Bagchi & Kapahi (1995) is due to theradio emission from SMGs.The previous considerations suggest the need to go to fluxdensities fainter than 1 mJy in order to detect large radio arcsin galaxy clusters. This kind of implication is also supportedby the results of Cooray (1998), which showed that the radioemission corresponding to large optical arcs in three out of thefour cases he studied is . . A A ff set withrespect with the optical emission or to resolution issues.On the theoretical side, the statistics of radio arcs wasfirst investigated by Wu & Hammer (1993) (WU93 hereafter).Their predictions were made using the evolutionary modelof Dunlop & Peacock (1990) to describe the radio luminosityfunction (dominated by starforming galaxies and AGNs prefer-entially located at z .
1) at 2 . Ω m , =
1. Conversely, we focus on the ra-dio counterparts of SMGs (preferentially located at z ∼
2) at1 . Λ CDM cosmology. As pointed outin Bartelmann et al. (1998), the optical depth for optical arcsproduced by sources at z s ∼ Λ CDM one. Therefore, we can use this prescription for com-parison between both predictions. Note that, since the SED ofSMGs is rather flat at radio wavelengths (spectral index ∼ . . . z . . >
800 km s − , whichcorresponds to a mass & . × M ⊙ h − . After applying thecosmology correction mentioned before, their predicted num-ber of giant radio arcs for the whole sky is (i) ∼ S . & µ Jy, (ii) .
200 if S . > . S . > a few mJy (such that none should bedetected in surveys in the literature at that time). Consideringonly clusters with mass ≥ × M ⊙ h − (which have a red-shift range comparable to the one used in WU93), our predictednumber of radio arcs with d ≥ d =
10 produced by SMGs inthe whole sky is (i) ∼ − S . > µ Jy, (ii) fewtens if S . > . S . > afew mJy.Our results may thus seem compatible with those of WU99,considering that the number density of SMGs is certainlysmaller than the number density of the entire radio source pop-ulation. This inference is however not conclusive, since WU99are using sources at rather low-redshift. We cannot say whetherconsidering their same redshift distribution and number countswould lead to a discrepant result.In CO99 there is also a study about strongly magnified ra-dio sources, analogous to the submm sources. It is found that ∼
20 sources with amplification larger than 10 should be foundwith S . ≥ µ Jy. For the same parameters, we find inour high-mass cluster study a number ranging from ∼
100 upto ∼ σ . The origin of this dis- crepancy is not clear, although it might be related to the highermass threshold that they adopt, and the fact that the redshiftdistribution considered by CO99 peaks at z ∼ z ∼
2. It should also be noted that CO99 attribute their find-ing many fewer radio arcs than WU93 to the di ff erent numbercount evolution adopted.Our calculations of the abundance of radio and submm arcsare more accurate than the works discussed above for di ff erentreasons, mainly the di ff erent modeling of the cluster popula-tion. To start with, in the literature, investigators often considerall lenses as isolated, spherically symmetric density distribu-tions. Wu & Hammer (1993) investigate the e ff ect of ellipticalmass distributions, but only on the magnification pattern andnot on the e ffi ciency for the production of large arcs. On theother hand, we included the e ff ect of asymmetries, substruc-tures and cluster mergers, that all have been found to be impor-tant to augment arc statistics. Next, we used an NFW densityprofile to model individual lenses, which is a good representa-tion of average dark-matter dominated objects like galaxy clus-ters, while other works have often considered SIS or SIS-likeprofiles, which are more suitable for galaxy lensing. While fora SIS lens model the image magnification equals the length-to-width ratio, it is known to produce fewer gravitational arcs withrespect to the more realistic NFW profile (Meneghetti et al.2003).
7. Summary and conclusions
The advent of the high resolution submm facilities ALMA andCCAT, and the major technological development that radio in-terferometry is currently undergoing (e.g., e -MERLIN, EVLAand SKA) will make possible the study of radio and submmgiant arcs produced by clusters of galaxies. In particular, thestudy of giant arcs produced by submm galaxies (SMGs) seemsparticularly promising for at least two reasons. • It provides the opportunity to detect and spatially resolvethe morphologies and internal dynamics of this populationof dust obscured high-redshift star-forming galaxies, whichis very di ffi cult to study in the optical. • It can provide information about the formation and evolu-tion of the high redshift cluster population, by means of arcstatistics studies.To assess the prospects for these kind of studies, we pro-vided theoretical predictions on the abundance of gravitationalarcs produced by the SMG population at radio (1 . µ m) wavelengths, greatly improving the accu-racy of the results with respect to the first few studies car-ried out a decade (or more) ago. The advantage of radio ob-servations is that the angular resolution and sensitivity pro-vided by interferometers like e -MERLIN and EVLA are al-ready (or will very soon be) at the level required for these kindof studies. However, these frequencies do not benefit from thesame favorable K-correction as mm / submm wavelengths do,which will make the latter a more interesting tool for studyinghigh-redshift clusters as soon as the resolution of sub-mm ob-servations is good enough. The calculation of the number of . Fedeli & A. Berciano Alba: Arcs produced by SMGs at radio and submm wavelengths 19 arcs produced by a background source population requires fourmain ingredients: (i) the source shape and size (ii), the sourceredshift distribution, (iii) the cumulative source number counts,and (iv) a model of the cluster population.The model of the cluster population used in this work wasbased on the extended Press & Schechter (1974) formalism andmade use of an NFW dentity profile to describe each clusterlens. It also included the e ff ect of asymmetries, substructuresand cluster mergers, which have been found to play an impor-tant role in arc lensing statistics.Based on current radio / CO observations and the FIR / radiocorrelation, we have characterized the typical size of the radioand submm emitting regions of SMGs with an e ff ective radius R e = . ′′ , and an axis ratio that varies within the interval b / a ∈ [0 . , ∼ . ′′ resolution.Since the most accurate redshift distribution of SMGs avail-able (Chapman et al. 2005) is based on observations of the ra-dio detected members (which is biased against z ≥ ff erent functions to quantify the e ff ect of a highredshift tail on the predicted number of arcs. The results indi-cate that this e ff ect is less than a factor two if we consider allsimulated clusters during the calculations, and negligible if weonly consider massive clusters.The submm source number counts used in this work cor-respond to the joint fit of the (bright) SHADES survey countsand the (faint) Leiden SCUBA Lens Survey counts presented inKnudsen et al. (2008). To account for the uncertainty in the lowflux end, predictions were made for the best fit to the data, andthe shallowest fit consistent with the data. Note that, althoughonly ∼
60% of the observed SMGs have being detected in ra-dio, the next generation of radio interferometers will be able todetect the radio counterparts of all SMGs with S µ m ≥ S µ m / S . ratio for SMGs at z ∼ − will detect hundreds of arcs with a5 σ significance. In the radio, this number can be achieved witha sensitivity of 10 − µ Jy arcsec − . Obtaining a statisticallysignificant sample of thousands of arcs would require sensitivi-ties of 0 . − in the submm and 1 µ Jy arcsec − in theradio. However, if only massive clusters ( M ≥ × M ⊙ h − )are considered in the calculations, the predicted number of arcsis reduced by about an order of magnitude.Besides the many uncertainties involved in the theoreticalpredictions presented here, the main challenge in designing afuture survey for radio / submm arc statistics studies will be infinding the best compromise between survey area, depth andresolution, three issues that a ff ect the arc detectability in di ff er-ent manners. We believe that this work provides a significantstep forward in this direction. Acknowledgments
We are grateful to M. Bartelmann, A. Blain and L. V. E. Koopmans forreading the manuscript and for many useful comments. We also would like to thank M. Swinbank, for providing us the histograms presentedin Figure 4, their Gaussian fits, and the predictions from his evolution-ary model, and K. K. Kundsen for providing us with the observationaldata presented in Figure 5. We also acknowledge stimulating conver-sation with M. Bonamente, M. Brentjens, M. Joy, A. F. Loenen and I.Prandoni. We wish to thank the anonymous referee for useful remarksthat allowed us to improve the presentation of our work. C. F. acknowl-edges financial contributions from contracts ASI-INAF I / / / / / / References
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Predicted number of radio (1.4 GHz) arcs produced by SMGs for an all-sky survey, using all the synthetic cluster population to compute the optical depths (see Figure9). Values with and without brackets correspond to S µ m / S . =
100 and 50, respectively. B (1)1 . µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − d (2)0 . . . . . . (3)
50 (8) 25 (0) 583 (233) 250 (100) 1333 (575) 575 (250) 8275 (3792) 3600 (1650) 18283 (8275) 7950 (3600) 110017 (50767) 47825 (22075)PHZ DB (3)
50 (8) 17 (0) 550 (225) 233 (92) 1258 (550) 542 (233) 7800 (3575) 3342 (1533) 17242 (7800) 7383 (3342) 103783 (47892) 44450 (20517)CHM DB (3)
100 (17) 42 (8) 1142 (458) 517 (208) 2608 (1142) 1167 (517) 16200 (7425) 7267 (3325) 35808 (16200) 16058 (7267) 215517 (99458) 96633 (44592)SPZ SM (3)
42 (8) 17 (0) 542 (225) 233 (100) 1058 (542) 458 (233) 3567 (2208) 1550 (958) 5608 (3567) 2433 (1567) 14325 (9658) 2394 (4200)PHZ SM (3)
42 (8) 17 (0) 508 (208) 217 (92) 1000 (508) 425 (217) 3367 (2083) 1442 (892) 5292 (3367) 2267 (1442) 13508 (9117) 5783 (3900)CHM SM (3)
83 (8) 42 (8) 1058 (442) 475 (200) 2075 (1058) 933 (475) 6992 (4317) 3133 (1933) 10983 (6992) 492 (3133) 28058 (18925) 12583 (8483) (1) surface brightness limit used to determine the number of arcs. (2) arc length-to-width ratio thresholds. (3) redshift distribution and number count function, as introduced in Figures 3 and 6.
Table 7.
Predicted number of radio (1.4 GHz) arcs produced by SMGs for an all-sky survey, using only clusters with M ≥ × M ⊙ h − to compute the optical depths (seeFigure 10). Values with and without brackets correspond to S µ m / S . =
100 and 50, respectively. B (1)1 . µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − µ Jy arcsec − d (2)0 . . . . . . (3) (3) (3) (3) (3) (3) (1) surface brightness limit used to determine the number of arcs. (2) arc length-to-width ratio thresholds. (3)(3)