ROLE OF GALAXY MERGERS IN COSMIC STAR FORMATION HISTORY

Quantitative galaxy morphologies are always measured within well-defined apertures in order to probe the same physical regions of galaxies; these apertures are optimized to include as much galaxy flux as possible while minimizing the effect of noise plus SB dimming. When comparing low- to high-redshift galaxy morphologies, SB dimming (SB ∝(1 + z)⁴) can artificially decrease the galaxy size. To avoid such an effect, the galaxy size can be measured through the dimensionless parameter:

$$\begin{equation} \eta (r) =I(r)/{\langle }I(r){\rangle }, \end{equation} \tag{ 1 }$$

where I(r) is the SB at radius r and 〈I(r)〉 is the mean SB within radius r (Petrosian 1976). In the standard approach (Conselice et al. 2000, 2003), the Petrosian radius R_p is defined at η(r)= 0.2 and the galaxy aperture radius is defined to be 1.5 R_p. The SB I(r) can be measured within either circular apertures or elliptical apertures. For inclined noninteracting galaxies, the mean galaxy ellipticity and position angle trace the true galaxy light well. The R_p defined within an elliptical aperture is generally larger than that defined within a circular aperture and can be two times larger for extremely inclined systems. However, we have found that interacting galaxies usually have large variations in their position angles and ellipticities. The measured galaxy size for them depends on the adopted position angle and ellipticity. Because of this uncertainty for the interacting systems, we adopt circular apertures universally for all galaxies in this paper. For most local LIRG systems, we found that the difference in R_p between circular apertures and elliptical apertures with mean ellipticity and position angle is within 20%. Therefore, using circular apertures does not introduce large errors.

The concentration parameter measures how compact the light distribution is. It has been shown to correlate well with the Hubble type (Kent 1985; Bershady et al. 2000). Early-type galaxies generally have more compact light distributions than do late-type ones. With the curve of growth measured within the galaxy aperture, the concentration can be defined as the ratio of the two radii enclosing two fixed fractions of the light. Following Kent (1985) and Bershady et al. (2000), we have used the following definition in this paper:

$$\begin{equation} C=5\log {\frac{R_{80}}{R_{20}}}, \end{equation} \tag{ 2 }$$

where R₈₀ and R₂₀ are the radii that enclose 80% and 20% of the total light, respectively.

The asymmetry parameter (A) was first introduced by Abraham et al. (1996) and subsequently developed by Conselice et al. (2000, 2003). It quantitatively describes the level of the galaxy asymmetry and provides a direct measurement of the importance of asymmetric substructures in galaxy morphologies, such as merging companions and tidal tails, which are used to identify mergers in visual classification. Accurate measurements of the asymmetry parameter are thus critical to understand the role of galaxy mergers in galaxy evolution. The galaxy asymmetry is defined as

$$\begin{equation} A_{\rm galaxy+noise} = \frac{{\sum }|I_{\rm galaxy+noise} - I_{\rm galaxy+noise}^{180^{\circ }}|}{{\sum }I_{\rm galaxy+noise}}, \end{equation} \tag{ 3 }$$

where A_galaxy+noise is the asymmetry of the galaxy signal plus the noise, I_galaxy+noise is the image of galaxy signal plus the noise, and $I_{\rm galaxy+noise}^{180^{\circ }}$ is I_galaxy+noise with the image rotated by 180° around a rotation center. The true pure galaxy asymmetry is then given by

$$\begin{equation} A_{\rm galaxy} = {\rm min}(A_{\rm galaxy+noise}) - A_{\rm noise}^{\rm corr}, \end{equation} \tag{ 4 }$$

where min(A_galaxy+noise) is the minimum of the galaxy signal plus noise asymmetry, and A^corr_noise is the noise correction obtained by rotating a background image around a center and normalizing by the galaxy brightness:

$$\begin{equation} A_{\rm noise}^{\rm corr} = \frac{{\sum }|B - B^{180^{\circ }}|}{{\sum }I_{\rm galaxy+noise}}, \end{equation} \tag{ 5 }$$

where B is the background image without any object.

The overall measurements of galaxy concentration and asymmetry are composed of the following steps:

• 1.  
  Guess the initial center.
• 2.  
  Measure the R_p and concentration.
• 3.  
  Search for the rotation center that gives the minimum of A_galaxy+noise within a 1.5 R_p aperture.
• 4.  
  Use the new rotation center to remeasure the R_p and concentration.
• 5.  
  Use the above rotation center and new R_p to search for a new rotation center that gives the minimum of A_galaxy+noise within a 1.5 R_p aperture.
• 6.  
  Correct for the noise A^corr_noise.

The galaxy centers can be easily located for noninteracting galaxies but not for interacting ones. Here, we adopt the rotation center giving the minimum of A_galaxy+noise to measure the galaxy size as is done in steps 1–3. Note that the asymmetry rotation center in step 5 is usually not very different from that in step 3, as the "walking around" method invented by Conselice et al. (2000) is generally robust to find the minimum A_galaxy+noise. For discussions about the galaxy size definition, the algorithm searching for the minimum A_galaxy+noise, and the dependence of A_galaxy+noise on the resolution and correlated noise, see Conselice et al. (2000, 2003).

The technical details about our new A^corr_noise are given in Appendix A. Here, we present a summary of the procedure to determine this parameter. A set of 1000 randomly produced noise asymmetry measurements is carried out by putting circular regions in the background image around the target. The value of this distribution at the 15% probability low-end tail is used as A^corr_noise, the median of true noise corrections. The error (∼3σ) in the final measured galaxy asymmetry is taken to be two times the standard deviation of these randomly produced noise asymmetries. In reality, some galaxies are always present in the field of targets. To account for this problem, a success rate is defined for the 1000 circular region placements as the fraction of circular regions containing no galaxy signal indicated by the SExtractor segmentation image. Circular regions containing any galaxy signal are not used. More sets of 1000 placements are generated until 1000 successful measurements are reached. If the success rate for one set of 1000 placements is lower than 50%, the circular region size for the following set is taken to be 80% of this set. Then, the measured noise asymmetry is rescaled to that with the original size by assuming that the noise asymmetry is proportional to the aperture area.

Local LIRGs and ultra-luminous infrared galaxies (ULIRGs) are generally galaxies undergoing mergers with morphological signatures of tidal tails, multiple nuclei, and highly asymmetric features (Sanders & Mirabel 1996). To account for the redshift dependence of galaxy merger morphologies due to SB dimming, we measured the morphologies of local LIRGs and of local LIRGs redshifted to different redshifts. We retrieved Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS)/WFC-F435W images of 88 local LIRGs observed in Program ID-10592 (PI: Aaron Evans). This local LIRG sample is a complete subsample of the IRAS Revised Bright Galaxy Sample (RBGS; i.e., f_60 μm > 5.24 Jy; Sanders et al. 2003) above L_IR = 10^11.4 L_☉. The sample size is large enough for statistically valid comparisons. The distance of this sample covers the range from 35 to ∼350 Mpc. The median distance of 135 Mpc corresponds to a physical resolution of 80 pc. These galaxies are also bright enough to be useful when redshifted to higher redshifts. These local LIRGs are currently experiencing merging processes, spanning a wide range of merging stages from well-separated galaxy pairs to the final merging remnants.

To obtain quantitative morphology measurements, foreground stars and background galaxies should be removed from the images first. Given the importance of the background region in accurate morphology measurements (see Appendix A), we carried out a careful removal of contaminators, especially given that some of the LIRGs have tens of foreground stars. We used SExtractor (Bertin & Arnouts 1996) to obtain the segmentation map of each image. To make sure the extended low-SB emission was included in the object, we first rebinned the image by a factor of 4 × 4 and then adjusted the parameters in SExtractor so the extended emission was detected. This segmentation map was then resampled to the original resolution by assuming that the original pixels belonging to a rebinned pixel conserve the segmentation value. We then cleaned the neighborhood of the target by replacing pixels of nontarget object pixels randomly with values of background pixels. As the neighborhood is dominated by the background pixels (greater than 95%), such replacement does not introduce any significant correlation in the final cleaned neighborhood background. Sometimes a target was identified as several different objects by SExtractor. We visually inspected each image to make sure such substructures were still included in the target during the neighborhood clean. Sometimes foreground stars lie within the target region and cannot be separated by SExtractor. We used the IRAF package IMEDIT to remove these contaminators further.

As the local LIRG sample is a flux-limited sample instead of a volume-limited sample, we have applied weights of $\frac{1}{V_{\max }}$/$\sum \frac{1}{V_{\max }}$ to each object to obtain an equivalent volume-limited asymmetry distribution, where V_max is measured at the redshift where the object with a given IR luminosity has f_60 μm = 5.24 Jy.

3.2.1. HST Images and Morphology Measurements

3.2.2. Spitzer MIPS and IRAC Data

3.2.3. Redshifts and Morphology-Redshift Catalog

The redshift-morphology catalog was further limited to objects with reliable asymmetry measurements, defined as those with R_gal= 1.5R_p > 15 pixels (003 pixel⁻¹) and error(A) < 0.1 in a given band. The accuracy of the galaxy asymmetry depends on the resolution and S/N. For a galaxy with small size, the resolution is not high enough to resolve galaxy structures and the asymmetry becomes artificially small (Conselice et al. 2000). For galaxies with low S/Ns, the asymmetry uncertainty is large due to the noise. For the study in this paper, we used an asymmetry uncertainty cut (error(A) < 0.1 at approximately 3σ level) instead of an S/N cut, as our revised asymmetry method can characterize the uncertainty due to complicated background structures. Figure 2 shows the advantage of the asymmetry uncertainty cut over the S/N cut. While both S/N and 〈S/N〉 are tightly correlated with the asymmetry uncertainty at high S/N (e.g., 〈S/N〉> 3), the scatter of the correlation becomes large at 〈S/N〉< 3. A high-S/N cut (e.g., 〈S/N〉> 3) will exclude many objects with relatively small asymmetry uncertainty (error(A) < 0.1) and a low-S/N cut (e.g., 〈S/N〉> 1) will introduce objects with large asymmetry uncertainty (error(A) > 0.1). Our error(A) < 0.1 criterion only excludes objects with large asymmetry uncertainties. Most objects with error(A) > 0.1 have relatively inhomogeneous backgrounds. All objects with error(A) > 0.3 are at the edge of the field where the exposure is relatively shallow and shows a large gradient toward the edge.

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We now have a redshift-morphology catalog with redshift information and reliable morphology measurements. The redshift evolution of galaxy morphologies will be characterized in a subsample of this catalog that is complete to a certain value. Figure 3 shows the completeness cut at 50%, 70%, and 90% of the redshift-morphology catalog in R_gal versus magnitude in three bands, where the completeness is defined as the ratio of the number of galaxies with redshift measurements and secure morphologies to the total number of galaxies in an R_gal-magnitude bin. The absolute values at different redshift bins are given in Table 1. Here, we do not account for incompleteness of the HST/ACS catalog, as its incompleteness at m_z< 25 is negligible.

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Figure 4 shows the absolute magnitude, galaxy size, SB and IR luminosity as a function of redshift for all m_z< 25 and MIPS-detected objects with redshift measurements. The absolute rest-frame B-band magnitude is measured through the version 4.1.4 KCORRECT code (Blanton et al. 2003; Blanton & Roweis 2007) using four ACS bands. For the first three quantities, the 70% complete redshift-morphology cut is drawn in each redshift bin and the 50% complete IR flux limit is drawn for the IR luminosity. As shown in the figure, most objects with m_z < 25 and z< 1.2 are included in the final redshift-morphology catalog. This is because the incompleteness of the redshift-morphology catalog is mainly caused by the redshift incompleteness instead of the criterion of secure morphologies.

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Two main results described in the Appendices are important for the study of morphology evolution. First, as shown in Appendix A, we found that the underlying structure of the sky background is important in accurate asymmetry measurements and we propose a revised asymmetry measurement approach. Second, by redshifting local LIRGs and low-z (z = 0.2–0.4) GOODS galaxies to higher redshifts, asymmetry deficits due to SB dimming for these two galaxy populations are derived as a function of redshift. As shown in Appendix B, originally more asymmetric objects show larger asymmetry deficits. Such dependence indicates that to recover the intrinsic distribution of a galaxy population at a given redshift, a low-z galaxy population with intrinsically similar asymmetry should be used to construct the asymmetry corrections. Such a low-z galaxy population can be defined through comparing the observed asymmetries of redshifted low-z galaxies to those of the galaxy population at a given redshift, as the observed asymmetries still on average correlate with original asymmetries even given the larger asymmetry deficits for originally more asymmetric objects (see Figure 1).

Figure 5 shows the observed asymmetry of local LIRGs, GOODS LIRGs, and GOODS field galaxies at different redshifts compared to the observed asymmetry of redshifted local LIRGs and redshifted low-z (z = 0.2–0.4) GOODS galaxies. The GOODS LIRGS are the objects at L_IR > 2.5 ×10¹¹ L_☉ and satisfying the 70% completeness cut of the redshift-morphology catalog at the corresponding redshifts (see Figure 3 and Table 1). The GOODS field galaxies within a redshift interval are all galaxies satisfying the 70% completeness cut of the highest redshift bin. For the redshifted local LIRGs and low-z GOODS galaxies, the observed asymmetry becomes progressively smaller at higher redshift as caused by SB dimming (see the Appendix for discussion).

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Figure 5 shows a trend of the observed asymmetry for different subsets of the galaxy population within a given redshift bin: A(redshifted z = 0.2–0.4 GOODS galaxies) ≈A(GOODS galaxies) <A(GOODS LIRGs) <A(redshifted local LIRGs). This observed asymmetry trend and Figure 1 (i.e., that the observed asymmetry on average correlates with intrinsic asymmetry and that the asymmetry correction also increases with intrinsic asymmetry) indicate that the asymmetry corrections based on the asymmetry deficits of redshifted low-z GOODS galaxies should give a reasonable estimate of the true corrections for GOODS field galaxies. The corrections for GOODS LIRGs could be as small as the values based on redshifted low-z GOODS galaxies or as large as those based on redshifted local LIRGs. We therefore show these two cases as lower and upper limits. Note that GOODS LIRGs with L_IR > 10¹¹ L_☉ follow the same relation, although they are slightly more symmetric than LIRGs with L_IR > 2.5 ×10¹¹ L_☉. As the asymmetry deficit at a given redshift shows a large variation for different low-z objects (see Figures 22 and 23), the whole distribution of the asymmetry deficit is used to assign a probability for the final corrected asymmetry of a galaxy at a given redshift. For field galaxies, such an asymmetry deficit distribution is further constructed as a function of the galaxy B-band brightness (see Figure 23).

We now correct the observed asymmetry of GOODS LIRGs using asymmetry deficits determined from redshifted low-z GOODS galaxies and local LIRGs, which give conservative lower and upper limit estimates, respectively, as discussed above. With these corrections, the merger fraction of GOODS LIRGs as a function of redshift can be derived. Note that the asymmetry criterion for mergers is usually defined as A > 0.35 (e.g., Conselice et al. 2003). Given a systematic shift of 0.05 between our asymmetry method and that in the literature (see Appendix A), we adopted A > 0.3 to be consistent.⁶ Figure 6 shows our result for LIRGs at L_IR > 2.5 × 10¹¹ L_☉ and LIRGs at L_IR > 10¹¹ L_☉. The shaded areas are enclosed by lower and upper limits to the merger fractions. The figure shows that LIRGs are dominated by mergers up to z = 1.2, with merging fractions ∼50% in all redshift bins. Consistent with what has been found in the above section, high-redshift LIRGs show slightly lower merger fractions than local LIRGs.

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As a test of the conclusions from the asymmetry calculations, we also visually classified the IRAS RBGS (Sanders et al. 2003) as a function of the IR luminosity. For L_IR< 10^11.5 L_☉, we used the Digital Sky Survey for galaxies at distance <60 Mpc, where the spatial resolution is <400 pc. At 10^11.5 L_☉ <  L_IR< 10¹²L_☉, HST images were used. The merger fractions are 12%, 41%, 80%, and 95%, respectively, in IR-luminosity bins log(L_IR/L_☉) of [10.5, 10.99], [11.0, 11.49], [11.5, 12.0], and [12.0, 12.49], where the fraction for the last bin was taken from Sanders & Mirabel (1996). At L_IR< 10¹² L_☉, our numbers are consistent with Sanders & Mirabel (1996), lying between their pure merger fraction and merger+close-pair fraction, as some galaxies in close pairs have disturbed morphologies and are thus identified as mergers by us. Multiplying these fractions with the local IR LFs, we obtained merger fractions of 45% and 80% at L_IR > 10¹¹ L_☉ and L_IR > 2.5 × 10¹¹ L_☉ as shown in Figure 6. These values are nearly identical to the asymmetry-identified ones.

Our result of high merger fractions for high-redshift LIRGs is consistent with the UDF result (Shi et al. 2006) where no corrections are applied for the observed asymmetry given the much deeper exposure in the UDF. Such consistency further indicates that the adopted correction is valid for the high-redshift GOODS LIRGs.

We discuss briefly possible causes for the low merger fractions found in other studies: Bridge et al. (2007) identified mergers based on asymmetry parameters; Zheng et al. (2004) and Bell et al. (2005) visually classified mergers; and Lotz et al. (2008a) used Gini-M₂₀ to identify mergers. Bridge et al. (2007) quantified the merger fraction of LIRGs in the Spitzer First Look Survey (FLS), where the imaging is shallower than GOODS. They did asymmetry corrections according to Conselice et al. (2005), which we have found underestimate the correction for LIRGs (see Section 6). This undercorrection probably accounts for the low merger values (∼10%–15%) they derive. Zheng et al. (2004) obtained a visually classified merger fraction of 16% for LIRGs at 0.4 < z<1.2 in the CFRS field. Figure 20 warns of the bias of visual classification, as merging galaxies can look like normal galaxies with no or some weak asymmetric structures due to SB dimming. Note that Figure 20 is constructed for the GOODS field. At the depth of the images in the CFRS field, more asymmetric structures should be lost and high-redshift LIRGs will be artificially more symmetric. A similar bias possibly exists for the visual classification used in Bell et al. (2005). The discrepancy between Gini-M₂₀ and asymmetry may be caused in part by the different timescales over which galaxies can be identified as mergers. For gas-rich galaxy mergers, the timescale for asymmetry is several times longer than that for Gini-M₂₀ (Lotz et al. 2008b).

However, we have compared the performance of CA and Gini-M₂₀ on a local sample of galaxies and confirmed that to first order they give similar results for merger fractions. Since Gini-M₂₀ is less affected by the ratio of S/N, it is not clear what the implications of our analysis would be for morphology studies using it at z ∼ 1. Although Lotz et al. (2008a) discuss these issues, a more detailed investigation would be desirable.

Figure 7 shows the merger fraction for galaxies at M_B < −19.75 and M_B < −18.94 − 1.3z, compared to other works, where the asymmetry correction is based on the asymmetry deficits found for redshifted low-z GOODS galaxies. Our work shows a weak redshift evolution of galaxy merger fractions, f_merger ∝(1 + z)^0.5±0.3 for M_B < −19.75 and f_merger ∝ (1 + z)^0.9±0.3 for M_B < −18.94 − 1.3z. This result is consistent with the morphological study of Lotz et al. (2008a). Again, however, our merger fraction (20%–30%) is several times higher than other works except for Shi et al. (2006). This could be caused by the fact that the visual classification and asymmetric classification suffers from strong redshift dependence as discussed above. Only Shi et al. (2006) used images deep enough to detect asymmetric features as faint as for nearby galaxies and thus their result based on uncorrected asymmetry is consistent with our result.

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We notice that our relatively high merger fraction (20%) at z = 0.2–0.4 may require a rapid evolution to that at z = 0. However, the current understanding of the galaxy merger fraction at z < 0.2 is much less constrained mainly due to the small volume and use of shallow images with poor spatial resolution. For example, De Propris et al. (2007) measured a merger fraction with asymmetry for a complete galaxy sample from the Millennium Galaxy Catalog (MGC) and found a merger fraction of 2%–4% depending on the definition of possible contaminators. However, the poor spatial resolution (1.3 arcsec) and shallow exposure (sky noise = 26 mag arcsec⁻²) of the MGC survey (Cross et al. 2004), provide rest-frame B-band image quality for galaxies at z = 0.1 only comparable to z = 1 galaxies observed in the GOODS survey, implying a possibly significant underestimate of the merger fraction due to SB effects. The image quality is even worse for the SDSS and 2dFGRS images. Current HST data do not provide a statistically large complete galaxy sample at z < 0.2. The fraction of mergers identified as galaxy pairs is also not well constrained, from 1% (De Propris et al. 2007) to 5% (Lin et al. 2008) for the same set of data but with different methods, which again suggests that more thorough studies are required to have good constraints on galaxy mergers at z < 0.2.

4.4.1. Methodology for Constructing IR Luminosity Function

The IR LFs of galaxy mergers are derived broadly following the works of Le Floc'h et al. (2005) and Shi et al. (2008). The 1/V_max method (Schmidt 1968) is applied to galaxy mergers with m_z< 25, f_24 μm > 60 μJy and known redshifts. The comoving number density in a given luminosity bin can be written as

$$\begin{equation} \Phi ({\rm Log}L_{\rm IR})d({\rm Log}L_{\rm IR}) = {\sum }\omega /V_{\max }, \end{equation} \tag{ 6 }$$

where ω is the weight to correct for incompleteness and V_max is the maximum volume for an object to be included in the sample. V_max is given by

$$\begin{equation} V_{\max } = \int _{z_{\rm low}}^{z_{\rm high}} \Omega \frac{dV}{dz} dz, \end{equation} \tag{ 7 }$$

where [z_low, z_high] is the redshift over which the object can be detected and Ω is the survey solid angle (730 arcmin²). Note that the field edge excluded for the morphology study is a small fraction (1%) of the total area. While z_low is always fixed to the low end of a redshift interval, the maximum redshift, z_high, is defined as

$$\begin{equation} z_{\rm high} = {\rm min}(z_{\rm bin}^{\rm high}, z^{\rm limit}_{\rm IR}, z^{\rm limit}_{m_{z}}), \end{equation} \tag{ 8 }$$

where z^high_bin is the high end of a redshift interval and z^limit_IR is the limiting redshift at which the observed IR flux reaches the limiting flux of 60 μJy. The K-correction for 24 μm flux is based on the star-forming templates of Rieke et al. (2009). $z^{\rm limit}_{m_{z}}$ is the limiting redshift where the observed z-band magnitude reaches m^limit_z = 25. The K-correction in the z band is based on the ACS photometry and KCORRECT code (Blanton et al. 2003; Blanton & Roweis 2007).

The incompleteness correction ω includes corrections for the IR detection, IR objects associated with optical counterparts at m_z < 25, the redshift measurement success rate, and the criterion of secure morphologies. The incompleteness of the IR detections as a function of the 24 μm flux density is given in Papovich et al. (2004). Our sample is limited to f_24 μm = 60 μJy at which the incompleteness is ∼50%. There is no incompleteness correction for IR objects associated with optical counterparts with m_z< 25. This is because m_z< 25 is deep enough to detect nearly all optical counterparts of the IR objects brighter than f_24 μm = 60 μJy, given the rough correlation between the IR luminosity and optical luminosity in Le Floc'h et al. (2005). The incompleteness correction for redshift measurements is defined as the ratio of the number of all objects to that of objects with redshift measurements within a three-dimensional magnitude–color–color space m_z–(B − i)–(V − z). Such corrections can account for redshift measurement success as a function of the galaxy brightness and color. For the whole ACS photometry sample at m_z< 25, the success rate for redshift measurements is 68%. The final correction is for galaxies with secure morphologies, i.e., galaxy radius R_gal > 15 pixel and asymmetry uncertainty error(A) < 0.1. Such a correction is defined as the ratio of the number of all objects to that of objects with secure morphologies within a given luminosity bin and a given redshift bin. Note that these corrections are small (less than 1.2) except for the highest luminosity bin in 0.4 <z< 0.6 (a correction factor of 1.5). This indicates that the final IR LF of galaxy mergers is mainly based on objects with secure morphologies.

4.4.2. IR Luminosity Function of Galaxy Mergers

Figure 8 shows the IR LFs of galaxy mergers within different redshift bins. As shown in the figure, our IR LFs of field galaxies match those of Le Floc'h et al. (2005) well. The 1σ uncertainties of the IR LFs of galaxy mergers include the Poisson noise, the uncertainty in the 24 μm-to-L_IR conversion and cosmic variance. Since our field size is almost as large as that in Le Floc'h et al. (2005) and 24 μm flux density is used to derive the total IR luminosity in both works, for the uncertainty in the 24 μm-to-L_IR conversion and cosmic variance, we simply adopted the result obtained by Le Floc'h et al. (2005), an average of 0.2 dex upper side uncertainty and 0.1 dex lower side uncertainty. The final uncertainty including the Poisson noise is plotted in Figure 8 and listed in Table 2. The solid lines are the fit to the IR LFs of field galaxies while the dotted lines are the fit to the IR LFs of galaxy mergers. The curve for the fit is the Schetcher function. Although the Schetcher function fails to fit the local field galaxy IR LF (Pérez-González et al. 2005), our limited available data points do not allow us to use a formula with more parameters, such as a double power law. The slope of the Schetcher function is fixed as the average value of the fit to the IR LFs in the first two redshift bins. The fitted result is listed in Table 3.

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Table 2. Infrared Luminosity Function of Merger Galaxies

	Φ (h370 Mpc−3 Log L−1☉)
Log(LIR (h−270 L☉))	0.4 <z< 0.6	0.6 <z< 0.8	0.8 <z< 1.0	1.0 <z< 1.2
10.75a	(4.60+2.27−1.33) ×10−4	...	...	...
11.25a	(3.05+1.55−0.96) ×10−4	(3.64+1.77−1.01) ×10−4	(5.66+2.69−1.45) ×10−4	(6.26+3.28−2.13) ×10−4
11.75a	(4.23+3.11−2.61) ×10−5	(1.83+0.93−0.58) ×10−4	(4.40+2.10−1.15) ×10−4	(5.58+2.63−1.41) ×10−4
12.25a	...	(1.56+1.38−1.23) ×10−5	(1.07+0.56−0.37) ×10−4	(1.95+0.96−0.56) ×10−4
12.75a	...	...	...	(2.90+1.85−1.44) ×10−5
10.75b	(4.60+2.27−1.33) ×10−4	...	...	...
11.25b	(3.06+1.55−0.96) ×10−4	(3.54+1.72−0.99) ×10−4	(5.14+2.44−1.33) ×10−4	(5.85+3.11−2.06) ×10−4
11.75b	(4.22+3.10−2.61) ×10−5	(1.40+0.73−0.48) ×10−4	(2.74+1.33−0.76) ×10−4	(2.73+1.32−0.75) ×10−4
12.25b	...	(1.12+1.12−1.03) ×10−5	(7.11+3.94−2.74) ×10−5	(1.11+0.57−0.36) ×10−4
12.75b	...	...	...	(1.73+1.27−1.06) ×10−5

Notes. ^aThe galaxy asymmetries are corrected based on the redshifted local LIRGs. ^bThe galaxy asymmetries are corrected based on the redshifted low-z (z= 0.2–0.4) GOODS field galaxies.

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Table 3. Parameters of Schetcher Function Fit to IR LFs of Galaxy Mergers

Redshift	Log(Φ*) (h370 Mpc−3 Log L−1☉)	Log(L*) (h−270 L☉)	α
0.4 < za<0.6	−3.63+0.06−0.05	11.37+0.08−0.03	−1.12+0.00−0.00
0.6 < za<0.8	−3.70+0.06−0.10	11.75+0.07−0.00	−1.12+0.00−0.00
0.8 < za<1.0	−3.59+0.09−0.03	12.04+0.17−0.00	−1.12+0.00−0.00
1.0 < za<1.2	−3.62+0.02−0.03	12.29+0.04−0.06	−1.12+0.00−0.00
0.4 < zb<0.6	−3.63+0.06−0.05	11.37+0.08−0.04	−1.12+0.00−0.00
0.6 < zb<0.8	−3.72+0.07−0.11	11.70+0.07−0.01	−1.12+0.00−0.00
0.8 < zb<1.0	−3.67+0.04−0.04	11.98+0.13−0.04	−1.12+0.00−0.00
1.0 < zb<1.2	−3.80+0.02−0.08	12.26+0.03−0.07	−1.12+0.00−0.00

Notes. ^aThe galaxy asymmetries are corrected based on the redshifted local LIRGs. ^bThe galaxy asymmetries are corrected based on the redshifted low-z (z= 0.2–0.4) GOODS field galaxies.

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Figure 8 also shows the fraction of galaxy mergers as a function of the IR luminosity in different redshift bins. At all redshifts, the merger fraction increases with the IR luminosity, a trend similar to that for local galaxies (Sanders & Mirabel 1996).

4.4.3. B-Band Luminosity Function of Galaxy Mergers

The rest-frame B-band LFs of galaxy mergers are also derived as shown in Figure 9. The result of the merger B-band LF is listed in Table 4. Again, Schetcher functions are used to fit the data points where the slope is fixed as the average value of the fit to the LF in the first three redshift bins. The fitted result is listed in Table 5. For the B-band LFs, there are enough data points to break the luminosity–density degeneracy. The result is shown in Figure 10. For the general field B-band LF, Φ* ∝ (1 + z)^0.1±0.5 and ΔM*_B ∝ (−0.4 ± 0.3)Δz, which are generally consistent with the literature data (see Faber et al. 2007, and references therein). For the merger B-band LF, Φ* ∝ (1 + z)^0.5±0.5 and ΔM*_B ∝ (−0.4 ± 0.3)Δz.

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Table 4. B-Band Luminosity Function of Merger Galaxies

	Φ (h370 Mpc−3 Log L−1☉)
MB	0.2 <z< 0.4	0.4 <z< 0.6	0.6 <z< 0.8	0.8 <z< 1.0	1.0 <z< 1.2
−22.40	...	...	(6.18+2.34−2.34) ×10−5	(3.09+1.34−1.34) ×10−5	(3.93+1.48−1.48) ×10−5
−21.60	(1.28+0.58−0.58) ×10−4	(9.19+3.53−3.53) ×10−5	(1.84+0.53−0.53) ×10−4	(1.71+0.48−0.48) ×10−4	(2.04+0.54−0.54) ×10−4
−20.80	(1.68+0.69−0.69) ×10−4	(3.31+0.93−0.93) ×10−4	(3.61+0.94−0.94) ×10−4	(4.34+1.09−1.09) ×10−4	(2.98+0.76−0.76) ×10−4
−20.00	(3.48+1.14−1.14) ×10−4	(4.87+1.30−1.30) ×10−4	(4.17+1.07−1.07) ×10−4	(5.74+1.41−1.41) ×10−4	(3.37+0.85−0.85) ×10−4
−19.20	(2.04+0.78−0.78) ×10−4	(3.19+0.90−0.90) ×10−4	(3.42+0.90−0.90) ×10−4	(4.42+1.11−1.11) ×10−4	...
−18.40	(2.88+0.99−0.99) ×10−4	(2.48+0.74−0.74) ×10−4	(3.10+0.82−0.82) ×10−4	...	...
−17.60	(1.93+0.75−0.75) ×10−4	(1.92+0.60−0.60) ×10−4	...	...	...
−16.80	(1.07+0.51−0.51) ×10−4	...	...	...	...

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Table 5. Parameters of Schetcher Function Fit to Merger B-band LFs

Redshift	Log(Φ0*) (h370 Mpc−3 M−1B)	M0B*	α
0.2 < z<0.4	−3.52+0.00−0.05	−20.72+−0.03−0.13	−0.53+0.00−0.00
0.4 < z<0.6	−3.43+0.05−0.01	−20.63+−0.11−0.01	−0.53+0.00−0.00
0.6 < z<0.8	−3.39+0.04−0.03	−21.05+−0.08−0.00	−0.53+0.00−0.00
0.8 < z<1.0	−3.29+0.02−0.02	−20.77+−0.01−0.06	−0.53+0.00−0.00
1.0 < z<1.2	−3.47+0.05−0.02	−21.04+−0.02−0.03	−0.53+0.00−0.00

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The breaking of the luminosity–density degeneracy allows us to evaluate the evolution of the pure merger fraction:

$$\begin{equation} n_{\rm galaxy}f_{\rm merger} = n_{\rm merger}. \end{equation} \tag{ 9 }$$

Given n_galaxy ∝ (1 + z)^0.1±0.5 and n_merger ∝ (1 + z)^0.5±0.5 in the B band, we have

$$\begin{equation} f_{\rm merger}^{B} \propto (1+z)^{0.4\pm 0.7}. \end{equation} \tag{ 10 }$$

The weak redshift evolution of the pure merger fraction f^B_merger indicates that the observed weak evolution in a given B-band magnitude is really caused by the weak evolution of the galaxy merger number density relative to the total number density. The merger rate can be estimated through the merger fraction and the timescale of 0.2–1.1 Gyr over which the asymmetry parameter can identify mergers (Lotz et al. 2008b).

While the local LIRGs are dominated by mergers (Sanders & Mirabel 1996), the result for high-redshift LIRGs is controversial, with merger fractions from as low as 10%–20% (Zheng et al. 2004; Bell et al. 2005; Bridge et al. 2007; Lotz et al. 2008a; Melbourne et al. 2008) to ∼50% (Shi et al. 2006). Motivated by the result of Shi et al. (2006) in the UDF field, we re-evaluated the effect of SB dimming on the morphology measurements. We first revised the asymmetry measurement based on simulations. Our new method shows a smaller scatter and does not suffer from a systematic offset compared to the one in the literature. We then obtained redshift-, IR-luminosity- and optical-luminosity-dependent asymmetry corrections by measuring asymmetry deficits of local LIRGs and low-z GOODS galaxies redshifted to different higher redshifts. By applying these corrections, we have found that high-redshift LIRGs are generally highly asymmetric galaxies with implied merger fractions ∼50% up to z= 1.2. For the general galaxy population, we obtained a weak evolution of merger fractions, consistent with results of Bundy et al. (2004), Lin et al. (2008), Lotz et al. (2008a), and de Ravel et al. (2008), but with several times higher merger fractions (20%–30%).

In Sections 4.2 and 4.3, we summarized possible reasons for the lower fractions of morphologically identified mergers in other studies (Le Fèvre et al. 2000; Zheng et al. 2004; Conselice et al. 2003; Cassata et al. 2005; Bell et al. 2005; Bridge et al. 2007; Lotz et al. 2008a; Melbourne et al. 2008).

The low fraction of kinematic galaxy pairs (Le Fèvre et al. 2000; Lin et al. 2008; de Ravel et al. 2008) is at least partly due to the different timescales probed by this method compared to morphology studies. Also, kinematic pairs are exclusively major mergers while a small fraction of highly asymmetric objects may be minor mergers. Our high merger fraction is consistent with the incidence of mergers identified through disturbed velocity fields. Specifically, studies of the velocity fields of ∼60 galaxies at 0.4 <z< 0.75 (Neichel et al. 2008; Yang et al. 2008) find a low fraction of rotationally supported disk galaxies and high fraction (41% ± 7%) of galaxies with complex kinematics. Neichel et al. (2008) also show that the auto classifications (CA and Gini-M₂₀ without corrections) miss half of the galaxies with complex kinematics and misclassify them as normal disk galaxies (also see a similar result from numerical simulations; Lotz et al. 2008b).

We have derived IR LFs of galaxy mergers at different redshifts out to z= 1.2. With these IR LFs, we now can quantitatively evaluate the contribution of galaxy mergers to the cosmic IR energy density out to z∼ 1.2. The open circle in the top panel of Figure 11 shows the cosmic IR energy density estimated by integrating general IR LFs obtained in this work, compared to the work of Pérez-González et al. (2005; gray area). The open star is the evolution of the cosmic IR energy density of galaxy mergers obtained by integrating the IR LFs of galaxy mergers where the upper and lower limits correspond to the result with asymmetry corrections based on redshifted local LIRGs and low-z GOODS field galaxies, respectively. The bottom panel of the figure shows the ratio of the merger IR energy density to the total IR energy density.

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The galaxy mergers-produced IR energy density shows dramatic evolution: Ω^mergers_IR∝ (1+z)^∼5 and (1+z)^∼6 for the result with low and high asymmetry corrections, respectively, compared to the total IR energy density evolution (1+z)^3.0 (Le Floc'h et al. 2005; Pérez-González et al. 2005). At z ≳ 1, the cosmic IR energy density is dominated by galaxy mergers. This result can be expected since LIRGs dominate the IR energy density at z > 0.7 (Le Floc'h et al. 2005; Pérez-González et al. 2005) and LIRGs are dominated by galaxy mergers, as found in this work.

To convert the cosmic energy IR density to the cosmic SFR density, the unobscured star formation as traced by the UV energy density needs to be included. In the local universe, the UV energy density corresponds to about 30% of the cosmic SFR density and decreases to 10% at z = 1.2 (Le Floc'h et al. 2005). The indicated correction from the cosmic IR energy density to the total SFR density for nonmergers should be larger than those for mergers, as the merger IR energy density has contributions from galaxies with on-average higher IR luminosity. This implies that Figure 11 may overestimate the contribution of galaxy mergers to the total cosmic SFR density.

The B-band morphologies of the local LIRGs at L_IR > 10^11.4 L_☉ were measured following the procedure described in Section 2.2. Due to the high resolution and the nature of complicated morphologies, about 10% of the local LIRGs have multiple radii at η(r)= 0.2. For these objects, we used the maximum radius at η(r)= 0.2 that matches the visually identified galaxy size well.

To account for the redshift dependence of LIRG morphologies, we also measured the redshifted B-band morphologies of these LIRGs at redshifts of 0.5, 0.7, 0.9, and 1.1. Since we quantified the GOODS galaxy morphologies using the rest-frame B-band images, we put the local LIRGs into a clean GOODS V-band sky region for local LIRGs redshifted to z = 0.5, i-band sky region for redshifted LIRGs at z = 0.7 and z = 0.9, and z-band sky region for redshifted LIRGs at z = 1.1. The exposure times for GOODS V-, i-, and z-band images are around 5000, 7000, and 20,000 s, respectively. The clean GOODS sky region has a size of 24'' × 24'', which is large enough to cover very extended tidal tails. The GOODS field is too crowded to have a continuous 24'' × 24'' clean region without any intruding object. Our field is obtained by merging four 12'' × 12'' clean regions whose sky fluctuations are consistent with each other within 3%.

To redshift the LIRGs, the pixel size was rebinned through IDL FREBIN.pro which rebins the pixel assuming the flux within a pixel is constant over the pixel area. We did not apply any size evolution, since on average the galaxy size R_p of LIRGs does not change with redshift. The DN per second of the local LIRG B-band images was converted to the GOODS ACS counts at a given band by using their PHOTOFLAM values and then decreased by the square of the luminosity distance and (1 + z), where (1 + z) is the K-correction, since PHOTOFLAM is defined as inverse sensitivity in units of erg cm⁻² s⁻¹ Å⁻¹. The resulting DN per second is then brightened by 0.60z magnitudes, which accounts for the redshift dependence of the rest-frame B-band magnitude of LIRGs as shown in Figure 17.

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During this pixel-rebinned and flux-rescaled process, the original LIRG image noise was correspondingly scaled down. To mimic the galaxy morphologies measured in the real GOODS sky region, the original galaxy background fluctuation should not dominate over the GOODS background in measuring the redshifted local LIRGs. To quantify when the scaled original galaxy background does not affect the redshifted LIRG morphology measurement, we carried out a test to measure the asymmetry of a clean GOODS sky region superposed on a second GOODS sky region scaled down by a given factor. When the second sky region has a scaled-down factor of <0.3, its effect on the asymmetry measurements of the first sky region is negligible. Therefore, we measured all redshifted LIRGs with rescaled original background noise smaller than 0.3 of the GOODS sky noise. At redshifts of 0.5, 0.7, 0.9, and 1.1, the numbers of measurable redshifted LIRGs are 61, 83, 88, and 88 out of a total of 88 local LIRGs. Therefore, the redshifted LIRGs are still representative of the complete local LIRG sample.

The galaxy sample that we used is the V-band images of 596 GOODS galaxies at 0.2 <z< 0.4 with secure morphology measurements. For a fiducial galaxy at z = 0.3, the GOODS V-band exposure corresponds to a 10σ rest-frame B-band SB of 25.5 mag arcsec⁻², comparable to the depth of the HST observations of local LIRGs (25.0 mag arcsec⁻²). The algorithm to measure morphologies of redshifted galaxies is almost the same as for redshifted local ULIRGs with two differences. First, we did not require the rescaled original galaxy noise to be <0.3 times the noise in the band where galaxies are redshifted to. Instead, we used the following strategy:

$$\begin{equation} I_{\rm new} = (1-\eta ^{2})^{0.5}B_{\rm new} + I_{\rm old}, \end{equation} \tag{ B1 }$$

where I_new is the final new image, I_old is the rebinned and rescaled original galaxy, B_new is the new background image in the band where the galaxy is redshifted to, and η is the ratio of rescaled and rebinned original galaxy noise to the fluctuation in B_new. This method will make sure that all redshifted galaxies will be used at all redshifts while maintaining the noise level. The main reason that we did not use this method for local LIRGs is that we wanted to quantify the effect of the complicated background structures in deep fields on the asymmetry measurements.

Second, no additional B-band luminosity evolution is applied, although there is a luminosity evolution in the B-band LF (e.g., Faber et al. 2007). This is because, ideally, the asymmetry correction for a distant galaxy should be based on the asymmetry deficit of a redshifted low-z galaxy with intrinsically the same asymmetry and B-band brightness. Applying B-band luminosity evolution will artificially underestimate the effect of SB dimming. To further demonstrate this argument, we measured the asymmetry deficits for redshifted galaxies with B-band luminosity evolution (ΔM_B = −1.1Δz; Faber et al. 2007) and no evolution. This result can be compared to the real result obtained by comparing the asymmetries using GOODS images and UDF images for the same galaxies. The result is shown in Figure 18 for galaxies with 0.4 < z<0.8 within which range the UDF limiting SB in the z-band is comparable to the GOODS v-band limiting SB at z= 0.3. As indicated by the figure, the cases with evolution underestimate the asymmetry correction significantly.

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Figure 19 shows the median asymmetries and the 68% confidence ranges for the local LIRGs and redshifted local LIRGs at z = 0.5, 0.7, 0.9, and 1.1. The exposure time at each redshift is labeled. As shown in the figure, the galaxy appears more symmetric at high redshifts as more low-SB asymmetric structures are embedded in the background fluctuations due to SB dimming. A linear fit to central three points with similar exposure times (∼6000 s) gives δA/δz = −0.38. The asymmetry of the local LIRGs is below the extrapolation of this linear fit as the local LIRGs should be more asymmetric at the exposure time of ∼6000 s. The redshifted LIRGs at z = 1.1 are above the extrapolation of this fit as a deeper exposure at z band detects more faint asymmetric structures. Although the redshift dependence of the galaxy asymmetry has been noticed by Conselice et al. (2000, 2003, 2005), the slope of our dependence is much larger than theirs. Even compared to the test sample of local irregular galaxies (Conselice et al. 2005), our slope is ∼2 times larger. We believed that the reason for such a large difference is the different local galaxy samples used to quantify the redshift dependence of the galaxy asymmetry. While the test galaxy sample in Conselice's works is a reasonable option for study of the general high-redshift galaxy population, it will underestimate the galaxy asymmetry for studies such as quantifying merger fractions. Figure 20 illustrates how tidal tails are progressively lost and galaxies appear more symmetric at higher redshift.

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To further illustrate the dependence of the galaxy asymmetry deficit due to SB dimming on the asymmetry itself, we measured galaxy asymmetries using GOODS z-band images and UDF z-band images for about ∼250 UDF galaxies with high S/N and large size. Figure 1 clearly shows that there is a trend of a larger asymmetry deficits due to SB dimming for more asymmetric galaxies.

Figure 21 shows the asymmetry deficit for redshifted GOODS galaxies at 0.2 < z<0.4. The magnitude limits correspond to the 70% completeness of the redshift-morphology catalog at different redshifts (see Figure 4). Figure 21 further strengthens the conclusion based on local LIRGs that the redshift dependence of galaxy asymmetry is a function of the asymmetry itself. Brighter GOODS galaxies with higher asymmetry at z= 0.3 show a steeper decline, while even the brightest sample shows a slower decline than the local LIRGs that have higher asymmetry.

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Figure 22 shows the probability distributions of the asymmetry deficits for redshifted local LIRGs, where the uncertainty is the Poisson noise. Note that the weight has been applied for this flux-limited sample to obtain a volume-equivalent result (see Section 3.1). The bin size for the asymmetry distribution is 0.05, the mean quadratic asymmetry error. The median value of the distribution in a given redshift bin can be used to correct LIRGs at that redshift. However, as shown in the figure, there is a significant width in the distribution, implying a variation in the asymmetry deficit for an individual galaxy. Such variations are expected given the complicated structures of galaxy morphologies as a function of the SB.

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For general galaxy populations, Figure 23 shows the probability distributions (in absolute object numbers) of the asymmetry deficit for low-redshift (0.2 < z<0.4) GOODS galaxies redshifted to higher redshifts as functions of absolute B-band magnitude. The bin size for the asymmetry distribution is 0.06. At the highest redshift, the distribution for the faintest B-band luminosity bin is not plotted as there is no secure morphology measurement.

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