Issue |
A&A
Volume 680, December 2023
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Article Number | L5 | |
Number of page(s) | 5 | |
Section | Letters to the Editor | |
DOI | https://doi.org/10.1051/0004-6361/202348202 | |
Published online | 11 December 2023 |
Letter to the Editor
Compact to extended Lyman-α emitters in MAGPI: Strong blue-peak emission at z ≳ 3
1
School of Mathematical and Physical Sciences, Macquarie University, NSW, 2109
Australia
e-mail: tamal.mukherjee@hdr.mq.edu.au
2
Macquarie University Astrophysics and Space Technologies Research Centre, Sydney, NSW, 2109
Australia
3
Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218 Hawthorn, 3122 VIC, Australia
4
Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT, 2611
Australia
5
International Centre for Radio Astronomy Research (ICRAR), Curtin University, Bentley, WA, Australia
6
International Centre for Radio Astronomy Research (ICRAR), The University of Western Australia, Crawley, WA, 6009
Australia
7
School of Physics, University of New South Wales, Sydney, NSW, 2052
Australia
8
Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW, 2006
Australia
9
ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO-3D), Australia
Received:
9
October
2023
Accepted:
9
November
2023
We report the discovery of three double-peaked Lyman-α emitters (LAEs) exhibiting strong blue-peak emission at 2.9 ≲ z ≲ 4.8 in the VLT/MUSE data obtained as part of the Middle Ages Galaxy Properties with Integral Field Spectroscopy (MAGPI) survey. These strong blue-peak systems provide a unique window into the scattering of Lyα photons by neutral hydrogen (H I), suggesting gas inflows along the line of sight and low H I column density. Two of them at z = 2.9 and z = 3.6, are spatially extended halos with their core regions clearly exhibiting stronger emission at the blue peak than at the red peak. However, spatial variations in the peak ratio and peak separation are evident over 25 × 26 kpc (z = 2.9) and 19 × 28 kpc (z = 3.6) regions in these extended halos. Notably, these systems do not fall in the regime of Lyα blobs or nebulae. To the best of our knowledge, such a Lyα halo with a dominant blue core has not been observed previously. In contrast, the LAE at z ∼ 4.8 is a compact system spanning a 9 × 9 kpc region and stands as the highest-redshift strong blue-peak emitter ever detected. The peak separation of the bright cores in these three systems ranges from Δpeak ∼ 370 to 660 km s−1. The observed overall trend of decreasing peak separation with increasing radius is supposed to be controlled by H I column density and gas covering fraction. Based on various estimations, in contrast to the compact LAE, our halos are found to be good candidates for Lyman-continuum (LyC) leakers. These findings shed light on the complex interplay between Lyα emission, gas kinematics, and ionizing radiation properties, offering valuable insights into the evolution and nature of high-redshift galaxies.
Key words: galaxies: evolution / galaxies: high-redshift / cosmology: observations
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Investigations of the circumgalactic medium (CGM) have significantly added to our understanding of the mechanisms driving galaxy formation and evolution. Interactions involving gas exchanges between galaxies and the surrounding intergalactic environment shape the evolution of galaxies. Lyman-α (Lyα) emission is a key tracer of CGM gas around high-redshift star-forming galaxies. Extended Lyα emission has been observed in various galactic environments: (i) Lyα halos around individual galaxies (Kunth et al. 2003; Steidel et al. 2011; Matsuda et al. 2012; Hayes et al. 2013; Leclercq et al. 2017) and (ii) Lyα blobs (LABs) associated with multiple sources (Francis et al. 1996; Fynbo et al. 1999; Matsuda et al. 2011; Cai et al. 2017). The origin of this diffuse Lyα emission in these halos remains a subject of ongoing debate, with several possibilities proposed, including scattering from star-forming regions, Lyα fluorescence, gravitational cooling radiation from accretion and emission from satellite galaxies (Haiman et al. 2000; Zheng et al. 2011; Byrohl et al. 2021).
Complex radiative transfer and resonant scattering of Lyα photons have given rise to diverse spectral profiles, with the double-peaked profile being particularly intriguing (Gronke et al. 2016; Verhamme et al. 2017). The escape of ionizing photons (> 13.6 eV) into the intergalactic medium (IGM) played a pivotal role in cosmic reionization, with faint, low-mass, early star-forming galaxies (Bunker et al. 2010; Finkelstein et al. 2015) contributing significantly. However, the interplay between the emission and escape of ionizing radiation, particularly Lyman-continuum (LyC) photons (λ < 912 Å), remains a puzzle. This enigma is partly due to the inherent opacity of the neutral IGM (Madau 1995; Inoue et al. 2014). This opacity of the IGM to LyC radiation beyond z ≳ 4.5 makes direct observation of LyC emission challenging (Vanzella et al. 2018). Double-peaked profiles serve as a probe to indirectly infer characteristics related to LyC leakage at higher redshifts. The peak separation, correlated with the neutral hydrogen (H I) column density (NHI), plays a pivotal role in determining the LyC escape fraction, , in high-redshift galaxies (Verhamme et al. 2015; Kakiichi & Gronke 2021). Furthermore, observations and simulations show a relation between
and Lyα equivalent widths (EWs; Verhamme et al. 2017; Maji et al. 2022).
The majority of double-peaked profiles observed in Lyα emitters (LAEs; Shapley et al. 2003; Izotov et al. 2018a; Kerutt et al. 2022) and LABs (e.g., Matsuda et al. 2006) predominantly exhibit a dominant red peak, indicating gas outflows. A dominant red peak is obvious as intergalactic absorption tends to extinguish the blue peak. However, some z > 6 LAEs display double-peaked Lyα profiles where the blue peak has a high probability of being scattered by the increasingly neutral IGM at higher redshifts (Hu et al. 2016; Hayes et al. 2021; Endsley et al. 2022). Despite the common association of strong LAEs with low-redshift LyC leakers (Bian & Fan 2020), the opposite is not necessarily true. An interesting distinction arises: Lyα double-peaked profiles with a stronger and broader blue peak than the red peak – implying gas inflows (Blaizot et al. 2023) – have been detected in two Lyα nebulae at z ∼ 3 (Vanzella et al. 2017), two LABs (Ao et al. 2020; Li et al. 2022) and, only in the outskirts of z ∼ 2 Lyα halos (Erb et al. 2018); Erb23. Besides these extended halos and luminous blobs, where two peaks typically originate from different portions within an extended halo or spatially separated regions within the blob, only two compact and stronger blue-peak LAEs have been reported to date (Furtak et al. 2022; Marques-Chaves et al. 2022), with the authors claiming that both blue and red peaks originate from the LAE itself.
In this Letter, we report the discovery of three strong “blue-peak” LAEs, spanning redshifts 2.9 ≲ z≲ 4.8 and obtained as part of the Middle Ages Galaxy Properties with Integral Field Spectroscopy (MAGPI1) survey (Foster et al. 2021). The structure of this Letter is as follows. In Sect. 2, we provide details of observations and data reduction including MAGPI survey Lyα identification. In Sect. 3, we present our results, followed by a discussion in Sect. 4. Throughout this Letter, we assume a standard flat ΛCDM cosmology with parameters H0 = 70 km s−1 Mpc−1, Ωm = 0.3 and ΩΛ = 0.7.
2. Observations and data reduction
The MAGPI survey is an ongoing Large Program (Program ID: 1104.B-0526) on the VLT/MUSE, targeting 56 fields (with stellar masses > 7 × 1010 M⊙). As of September 2023, 48 fields have been completed. The primary objective of this survey is to conduct a detailed spatially resolved spectroscopic analysis of stars and ionized gas within 0.25 < z < 0.35 galaxies (see Foster et al. 2021). Data are taken using the MUSE Wide Field Mode (1′×1′) with a spatial sampling rate of 0.2″ pixel−1 and an average image quality of 0.65″ FWHM. Each field is observed in six observing blocks, each comprising 2 × 1320 s exposures, resulting in a total integration time of 4.4 h. The survey primarily employs the nominal mode, providing a wavelength coverage ranging from 4700 Å to 9350 Å, with a dispersion of 1.25 Å. Ground-layer adaptive optics (GLAO) is used to correct atmospheric seeing effects, resulting in a 270 Å wide gap between 5780 Å and 6050 Å due to the GALACSI laser notch filter. Deeper MAGPI data resulted in the detection of both foreground sources within the local Universe and distant background sources, including LAEs up to z ∼ 6.4.
MUSE data reduction was performed using pymusepipe2, a Python wrapper for the MUSE reduction pipeline (Weilbacher et al. 2020). A comprehensive discussion of the data reduction procedures will be presented in the first data release (Mendel et al., in prep.). LSDCat (Herenz & Wisotzki 2017) was used for the identification of faint sources – particularly LAEs – accompanied by both automated and visual inspection. This extensive search led to the detection of 360 new LAEs distributed across 35 MAGPI fields (Mukherjee et al., in prep.). Among these, a preliminary examination of LAE profiles revealed the presence of 35 double-peaked profiles, of which 3 distinct double-peak LAEs show strong blue-peak emissions that are presented in this Letter.
3. Results
We find dominant blue-peak emission features in three newly discovered double-peak LAEs at redshifts of z = 2.9, z = 3.6, and z = 4.8, corresponding to MAGPI field IDs 1534, 2302, and 1208, respectively (see Table 1). Alongside strong Lyα emissions, the MUSE data also reveal various faint rest-frame UV emission and absorption lines: (i) For LAE at z = 2.9, we identify emission lines such as C IVλ1548, 1550, and C III] λ1908, as well as multiple absorption lines from both high- (Si IV and C IV) and low-ionization (Si II and O I) transitions. (ii) In the case of z = 3.6 LAE, we tentatively observe C IVλ1548, 1550 and C III] λ1908 emissions and absorption lines from C IV and low-ionization Si II transitions. (iii) For z = 4.8 LAE, we detect Si IVλ1393, 1402, and C IVλ1548, 1550 emission lines, as well as absorption lines from high (Si IV and C IV) and low ionization (Si II and O I) transitions. For the former two, C III] λ1908 lines are either too faint or have insufficient S/N to derive an accurate systemic redshift (Bacon et al. 2021) while for the latter, C III] is outside the spectral window. Following Verhamme et al. (2018), an approximate systemic redshift can be derived by taking the average of the mean wavelengths of the two asymmetric Gaussians fitted to the two Lyα peaks, which gives the following values: zsys ∼ 2.9484, zsys ∼ 3.6136 and zsys ∼ 4.788.
Properties of strong blue-peak LAEs.
For our detections, we use the Python/MPDAF package (Bacon et al. 2016) designed for MUSE data cube analysis. We generate continuum-subtracted Lyα narrow-band (NB) images from the MUSE data cube. We perform double-skewed Gaussian fitting to 1D Lyα emission profiles, utilizing the Python/pyplatefit package (Bacon et al. 2023). Lyα NB images along with corresponding 1D profiles are shown in Fig. 1. To quantify the strength of the blue peak, we measure blue-to-total flux ratios (Fblue/Ftotal; where Ftotal = Fblue + Fred). Hence, Fblue/Ftotal > 0.5 implies a stronger blue peak. We also estimate other parameters including peak separations (Δpeak), Lyα luminosities, total rest-frame equivalent widths (EW0), and the full width at half maximum (FWHM) of the blue and red peaks. We determine the LyC escape fraction () based on the empirical relationship between
and Δpeak from Izotov et al. (2018b). These results, along with respective source coordinates (right ascension (RA) and declination (Dec) of the bright cores), are summarised in Table 1. We also estimate LyC escape using C IVλ1550/C III] λ1908 following the work of Schaerer et al. (2022). To investigate spatial variations in 1D profiles of our sources, we generate pixel-by-pixel maps color-coded according to Fblue/Ftotal and Δpeak on the Lyα NB images (see Fig. 2).
![]() |
Fig. 1. Spatial and spectral profiles: shown are continuum-subtracted Lyα NB images smoothed with a 2D Gaussian Kernel of σ = 1 (top row) and a 1D Lyα emission profile fitted with a double-skewed Gaussian (bottom row). The 1D spectra of the cores for the sources at z = 2.9, 3.6, and 4.8 are extracted using 0.8″, 0.8″, and 0.6″ apertures (solid white circles in the NB images), respectively. These are shown in black solid lines in the bottom row. Dashed gray lines for the sources at z = 2.9 and z = 3.6 show the scaled Lyα spectra of the entire halo. The z = 4.8 LAE, which is a compact source, is covered in its entirety within the 0.6″ aperture. White dashed contours in the NBs indicate the position of the continuum. For the z = 3.6 LAE, three distinct regions are labeled as A, B, and C in the NB image. The colorbars are in units of erg s−1cm−2 pix−2 where 1 pix = 0.2″. |
![]() |
Fig. 2. Flux-ratio and peak-separation maps: Top row: pixel by pixel map color-coded according to blue-to-total flux ratio. Bottom row: peak separation map. These are arranged from left to right in increasing order of redshift (z = 2.9, 3.6, and 4.8). Peak separation maps are scaled within the range of 200 − 600 km s−1. |
Our double-peaked Lyα spectral profiles are unique, showing prominent and stronger blue peaks than the red peaks. We observe a variety of spatial distributions of Lyα emissions, where the continuum-subtracted Lyα NB images reveal two extended Lyα halos and one compact, point-like LAE (Fig. 1, top row). Arranged by increasing redshift, these LAEs at z = 2.9, z = 3.6, and z = 4.8 span regions measuring 24.8 × 26.4 kpc, 19.2 × 28.2 kpc, and 8.7 × 8.7 kpc, respectively. Notably, the extended halos at z = 2.9 and z = 3.6 exhibit spatially varying spectra, featuring varying Fblue/Ftotal and Δpeak in different spatial locations. In contrast, the compact LAE predominantly displays an overall blue-peak-dominated spectral profile, with mild variations in the outskirts. Below we break down our spectral and spatial analysis for each of the three cases:
(i) z = 2.9 LAE (field ID 1534) shows a compact “continuum-like” bright core and an extended halo (Fig. 1, left panel in top row). Such spatial profiles are characterized by a “two-component model” (see Wisotzki et al. 2016; Leclercq et al. 2017). Two “tail-like” regions are identified within the halo. A faint continuum is also detected at the source location. Due to source crowding, faint continuum emissions from neighboring sources were also detected. To address this, we subtracted the continuum emissions from neighboring sources to obtain pure emission lines of our source. Subsequently, we spectrally collapsed this processed data cube over the Lyα wavelength range in order to construct a continuum-subtracted Lyα NB image showing an extensive Lyα halo spanning 24.8 × 26.4 kpc. The halo displays spatially varying double-peaked spectra, with the brightest core featuring the strongest blue peak with Fblue/Ftotal = 0.74 (EWblue, 0 = 46.3 ± 3.7 Å and EWred, 0 = 14.4 ± 1.7 Å). Figure 1 (bottom row, left panel) presents a continuum-subtracted 1D profile of the core extracted adopting a circular aperture of 0.8″ radius. The overall 1D profile of the halo reveals a symmetric double peak with a slightly stronger blue peak. This spatial variation is evident in the Fblue/Ftotal map (Fig. 2, top row, left panel), showing that the blue peak dominates in the bright core and in its associated tail in the upper part of the halo, while the red peak dominates in the lower regions and in the outskirts. There is a smooth gradient from blue-dominated regions to red-dominated regions, with a boundary of Fblue = Fred in the central region of the halo. We also observe a significant spatial variation in Lyα peak separation (see Fig. 2, bottom row, left panel), ranging from ∼200 to 750 km s−1 with larger separations (≳480 km s−1) in the central region and smaller ones (≲400 km s−1) in the bright core and outskirts. We measure a C IV/C III] ratio of 0.83 for this source.
(ii) We find an extended halo around the z = 3.6 LAE (field ID 2302), showing three distinct bright regions labeled as A, B and C (Fig. 1, top row middle panel), all displaying double-peak emission. Remarkably, these bright regions are spatially separated, a unique distribution seen in Lyα halos. Due to source faintness and low S/N, a continuum light was not immediately visible. However, applying a median filter to the data cube revealed faint continuum light at the location of the brightest core. Region A, the brightest core, displays prominent, strong blue-peak emission with Fblue/Ftotal = 0.60 (EWblue, 0 = 47.2 ± 13.0 Å and EWred, 0 = 37.8 ± 11.8 Å) and is encircled with an aperture of 0.8″ radius. The entire halo shows a symmetric double-peaked profile with almost equal peak strengths. The flux-ratio map illustrates (see Fig. 2, top row middle panel) the dominance of the blue component mainly in the bright core (A) and in the lower diffuse halo (C), with the red peak becoming more prominent as we move away from the core and the other bright regions. We further observe larger peak separations (∼400−600 km s−1) in region A and at the center of the halo, and regions B and C exhibit peak separations of ∼390 km s−1, while comparatively lower peak separation (≲370 km s−1) is seen in other regions within the halo and in the outskirts (Fig. 2, bottom row, middle panel). We find a C IV/C III] ratio of 1.1.
(iii) The LAE at z = 4.8 (field ID 1208) shows a compact spatial profile (Fig. 1, top row, right panel) resembling that of “point-source LAEs” as studied by Bacon et al. (2015). Continuum light is detected at the location of Lyα emission in this source. The Lyα profile shows a broader and stronger blue-peak emission. The flux-ratio map (Fig. 2, top row, right panel) confirms the prevalence of the blue component across most of the Lyα-emitting region, with slight red-peak dominance in the outskirts. Similar to previous cases, we find larger peak separations (≳600 km s−1) in the central region and smaller values (≲450 km s−1) in the outskirts.
4. Discussion
In this Letter, we report three new double-peaked LAEs with dominant blue-peak emissions. These are ∼9% of the sample of double-peaked LAEs found in the MAGPI survey, which is consistent with the results obtained in cosmological zoom-in simulations (Blaizot et al. 2023). Lyα emission profiles with dominant blue peaks have previously been observed in different spatially separated regions within LABs or luminous Lyα nebulae. By comparison with the spatial size of LABs and nebulae (Vanzella et al. 2017; Li et al. 2022), we find that our extended LAEs are not in the regime of LABs or nebulae. However, we classify z = 2.9 and z = 3.6 LAEs as extended Lyα halos (see discussions in Wisotzki et al. 2016; Leclercq et al. 2017). The source at z = 2.9 shows a similar spatial distribution to that described by Wisotzki et al. (2016) and Leclercq et al. (2017), featuring a continuum-like bright core surrounded by an extended halo. On the other hand, the extended source at z = 3.6 reveals a more complex Lyα spatial distribution, displaying three bright regions within the halo. This complexity suggests complex kinematics involving gas inflows and outflows in this galaxy (Vanzella et al. 2017; Ao et al. 2020; Li et al. 2022; Erb et al. 2018). This could also indicate a merger system, where two progenitors are merging into a single descendent (e.g., Tilvi et al. 2011). A faint continuum light from these two sources is also detected in the MUSE data. However, lack of deep imaging data limits our ability to perform robust continuum subtraction and radial profile estimation.
Erb et al. (2018, 2023) report complex kinematics in double-peaked extended Lyα halos at z ∼ 2, with the authors observing a dominant red peak in the central bright region with large peak separations, and a dominant blue peak at the edge of the halos where peak separations become narrower. For both of our extended halos, we find peak separation distribution similar to those described by Erb et al. (2018, 2023), but an inverse (core-to-halo) distribution for flux ratios. In both of our extended halos, the bright cores powering the Lyα emission are dominated by blue peaks, while red peaks dominate as we move away from the bright cores towards the edges. This means that we have discovered the first two Lyα halos at z = 2.9 and z = 3.6 with dominant blue cores.
In contrast, we find an overall blue-peak-dominated Lyα spectral profile in the compact LAE at z = 4.8. Furtak et al. (2022) discovered the first such compact lensed object at z = 3.2. This was followed by a similar detection in a compact UV-bright star-forming galaxy at z = 3.6 (Marques-Chaves et al. 2022). Furtak et al. (2022) further find that both the blue and red peaks originate from the compact LAE itself. Similar to their flux ratio map, we observe that the entire Lyα-emitting region is dominated by the blue peak, while the red peak becomes stronger at the edges.
Overall, we observe that Lyα peak separation decreases with increasing radius, which is consistent with the findings of Erb et al. (2018, 2023). This trend is primarily driven by NHI, where greater NHI values result in larger peak separations. Lyα peak separation can be influenced by the cloud covering fraction (CF) within a clumpy outflow, where higher CF mimics higher NHI (Gronke et al. 2016) and impacts the velocity shift of escaping photons. However, this strongly depends on other parameters, such as velocity and gas temperature (e.g., Kakiichi & Gronke 2021; Li et al. 2022).
Verhamme et al. (2017) find that strong LyC emitters are strong LAEs with high-rest-frame EWs (≳70 Å) and large Lyα escape fractions. Simulations suggest a connection between double-peak separation and NHI, indicating that for low NHI, a narrow peak separation (≲400 km s−1) serves as a strong indicator of LyC escape (Verhamme et al. 2015; Kakiichi & Gronke 2021). These predictions are confirmed observationally for low-redshift LyC leakers (Verhamme et al. 2017; Izotov et al. 2021; Flury et al. 2022). Further, a high C IV/C III] ratio (≳0.75) has been proposed as a good tracer of LyC leakage in low-redshift galaxies (Schaerer et al. 2022). We measure high-rest-frame EWs in Lyα, relatively narrow peak separations (see Table 1), and C IV/C III] > 0.75 for the two extended LAEs, making them good candidates for LyC leakers. In contrast, we find a relatively large peak separation and a very low rest-frame EW for the compact LAE, implying a relatively high NHI in this system and almost no escape of LyC photons.
Radiative transfer simulations, incorporating shells or clumpy, multi-phase models with varying outflow velocities typically focus on double-peaked profiles of Lyα with stronger red peaks (Gronke & Dijkstra 2016; Erb et al. 2023). Cosmological hydrodynamics simulations suggest that red-dominated lines preferentially arise in face-on directions, while blue-dominated lines are seen in the edge-on directions (Blaizot et al. 2023). A Lyα line with a stronger blue peak than the red peak usually implies inflows of CGM gas along the line of sight during the accretion phase (Vanzella et al. 2017; Ao et al. 2020; Blaizot et al. 2023). In this context, our new discoveries suggest inflowing gas systems or edge-on morphologies, warranting further investigation in order to understand the complex gas kinematics and the environments of such rare LAEs. These investigations will help us to constrain the mechanisms that regulate their formation and evolution.
Acknowledgments
We wish to thank the ESO staff, and in particular the staff at Paranal Observatory, for carrying out the MAGPI observations. MAGPI targets were selected from GAMA. GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. GAMA was funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. GAMA photometry is based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 179.A-2004, ID 177.A-3016. The MAGPI team acknowledges support from the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.
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All Tables
All Figures
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Fig. 1. Spatial and spectral profiles: shown are continuum-subtracted Lyα NB images smoothed with a 2D Gaussian Kernel of σ = 1 (top row) and a 1D Lyα emission profile fitted with a double-skewed Gaussian (bottom row). The 1D spectra of the cores for the sources at z = 2.9, 3.6, and 4.8 are extracted using 0.8″, 0.8″, and 0.6″ apertures (solid white circles in the NB images), respectively. These are shown in black solid lines in the bottom row. Dashed gray lines for the sources at z = 2.9 and z = 3.6 show the scaled Lyα spectra of the entire halo. The z = 4.8 LAE, which is a compact source, is covered in its entirety within the 0.6″ aperture. White dashed contours in the NBs indicate the position of the continuum. For the z = 3.6 LAE, three distinct regions are labeled as A, B, and C in the NB image. The colorbars are in units of erg s−1cm−2 pix−2 where 1 pix = 0.2″. |
In the text |
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Fig. 2. Flux-ratio and peak-separation maps: Top row: pixel by pixel map color-coded according to blue-to-total flux ratio. Bottom row: peak separation map. These are arranged from left to right in increasing order of redshift (z = 2.9, 3.6, and 4.8). Peak separation maps are scaled within the range of 200 − 600 km s−1. |
In the text |
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