Luận văn Nghiên cứu một thiên hà thấu kính hấp dẫn có độ dịch chuyển đỏ z = 0.7 sử dụng dữ liệu alma

1.19) and gas emission ladders (Figure 1.20) are important data for the interpretation of these observations. Figure 1.18 From left to right and up-down: Very Large Array (VLA, radio); ALMA (mm/submm); Plateau de Bure (PdBI, IRAM, mm); Pico Veleta (IRAM, mm); Herschel (IR); Very large telescope (VLT, visible, NIR, NUV), Hubble Space Telescope (HST, visible, NIR, NUV); Galex (UV); Chandra (X-ray). 1.3.3 Observing the dust [23] Starburst galaxies are galaxies undergoing an exceptionally high rate of star formation (SFR). The Spectral energy distribution (SED, Figure 1.19) of Far Infrared emission (FIR) measures the heating generated by star formation 24 from the black body radiation of the dust and approximately measures the star formation rate at all redshifts. However a small change in shape, measured by an increase of the ratio of the 8 micron luminosity to the total FIR luminosity, reveals the presence of a separate family of starbursts, relatively more compact and forming more stars, believed to be triggered by mergers as opposed to galaxies forming stars via steadier processes. Starburstiness, RSB , defined as the relative value of the SFR, is also a measure of the mass doubling timescale. IR8 , relative value of the 8 micron emission is correlated to RSB for both local and distant galaxies. For galaxies that are spatially resolved, compactness and IR8 are correlated, high IR8 values being associated with large compactness. Large RSB , large IR8 , compact galaxies also stand out in the SFR vs M(stars) diagram. Figure 1.19 SEDs observed in the local Universe for Main Sequence (left) and Starburst (right) galaxies. 1.3.4 Observing the gas [24] CO is the main tracer of cold gas thanks to its high moment of inertia in spite of CO/H2~10-4. The signal increases with temperature and density. Quasar hosts reach the higher level of excitation (high SFR, compact emission region). The FIR luminosity increases with the CO luminosity for local as well as distant galaxies. Hyper-SB galaxies, quasar hosts and powerful radio galaxies show the most extreme gas properties in terms of gas excitation, star formation efficiency and compact, although complex, gas morphologies, suggesting compact, hyper SBs simultaneous with AGN accretion. FWHM CO line widths show little correlation with both CO and FIR luminosities. Only local line widths are corrected for inclination, which is unknown for distant galaxies. When the gas and/or dust emissions can be spatially resolved, they often display clumpy and turbulent morphologies at the ~10 kpc scale but also, sometime, give evidence for rotating discs. Z=0 Z=0 Polycyclic Aromatic Hydrocarbons Hot dust Cold dust 25 Figure 1.20 Excitation of rotational CO emission levels (Carilli & Walter 2013) as a function of J for various types of galaxies (left), for various densities (centre left) and for various temperatures (centre-right). Right: dust (FIR) luminosity vs gas (CO) luminosity for various types of galaxies. 1.3.5 Observing the central black hole Black hole masses are measured from the kinematics of the surrounding ionized gas (luminosity and FWHM of Balmer HI line) or variability of X-ray emission. They are strongly correlated to the velocity dispersion of the gas in the host galaxy. The M-σ diagram relates the mass MBH of the black hole, MBH ~ 0.2% M(bulge), to the velocity dispersion σ. At high redshifts, the lack of knowledge of inclination of the disc with respect to the plane of the sky is an important factor of error on the velocity dispersion. Figure 1.21 Up-left: X-ray variability vs SMBH mass ([24]). Down-left: SMBH mass vs line width. Right: dependence of the SMBH mass on velocity dispersion [25]. 1.3.6 Galaxy evolution The mass fraction between molecular gas and stars in massive disc galaxies (Figure 1.22 left) increases by an order of magnitude from z=0 to z~2. Hence, the peak of cosmic star formation corresponds to the epoch when 26 typical star forming galaxies were dominated by cool gas, not by stars (Figure 1.22 centre). The star formation rate density (Figure 1.23) peaked ~3.5 Gyr after the Big Bang, at z~1.9, and declined exponentially at later times with a time scale of ~3.9 Gyr. Half of the stellar mass observed today was formed before z=1.3. About one quarter formed before the peak and another quarter after z=0.7. Less than 1% of today’s stars formed during the epoch of re- ionization. The co-moving rates of star formation and central black hole accretion show similar rise and fall, giving evidence for co-evolution of black holes and their host galaxies. The detection of CO, [CII] and dust out to z~7 when the Universe was less than 1 Gyr old and when there had been little time to enrich the ISM with C and O reveals the coeval formation of massive galaxies and SMBHs in extreme starbursts at such early times. Metallicity is strongly correlated with the dust to gas ratio for local galaxies and decreases with redshift. Star formation between Big Bang and z~2.5 (2.5 Gyr later) was sufficient to enrich the Universe as a whole to 1% of solar metallicity. The rise of the mean metallicity of the Universe to ~1 ‰ solar 1 Gyr after Big Bang was accompanied by the production of fewer than 10 ionizing photons per baryon, implying ~25% escape probability from galaxy to IGM, a high value compared to what is observed at z<~3. Figure 1.22 Left: gas to stars mass ratio vs redshift. Centre: SFR density vs redshift. Right: stellar mass density vs redshift. [26] Does the SMBH grow by accreting material from the host galaxy or from mergers? To answer this question (Figure 1.23 right) one assumes that all SMBH grow through accretion (AGN), observes high z AGNs and deduces their mass and mass accretion rate from their intrinsic luminosity, integrates mass accretion rate over time for the whole population and extrapolates the mass that the SMBH would have at z=0, compares with masses of SMBH observed in local galaxies. A good match implies that growth is mostly though accretion (not mergers) in luminous AGN phase (most massive built first and in quasar phase); as time goes, growing SMBHs become increasingly obscured. Growth is correlated with star formation in the host galaxy. 27 Figure 1.23 Left: SFR density (orange) and luminosity density (blue) vs redshift. Centre: black hole accretion vs redshift from IR (blue) and X-ray (green) measurements. Right: SMBH growth rate vs mass (Credit: [28]). 1.4. Radio interferometry and data reduction 1.4.1 Generalities [4] Radio interferometry (Figure 1.24) correlates the signals of two antennas separated by a baseline λb, with the length b of vector b measured in units of wavelengths. It has coordinates (u,v) and the (u,v) plane is called the Fourier plane. A pair of antennas gives a pair of points in the Fourier plane: b and –b. Neglecting the PSF of each individual antenna, i.e. assuming a pencil beam with no side lobes, pointing to a direction ξ (unit vector) in the sky plane, the delay between the two signals is λb.ξ, and the time dependence of their sum reads exp(iωt)+exp(iω[t+λb.ξ])=exp(iωt){1+exp(2iπb.ξ)} with ωλ=2π. It has the form of a rapidly oscillating term modulated by a signal B{1+exp(2iπb.ξ)} where B is the signal amplitude. The complex quantity V=Bexp(2iπb.ξ) is called the visibility. It is easily generalized to an extended source as V(b)=∫∫B(ξ)exp(2iπb.ξ)dΩ where dΩ is the solid angle element. The visibility is the Fourier transform of the source brightness measured in the sky plane as ξ=(l,m), while the visibility is measured in the Fourier plane as a function of the baseline b=(u,v). Note that V(–b)=V*(b): by introducing an additional delay in one of the signal one can measure the real and imaginary components of the visibility. In practice this is done online by the correlator. What is measured by the interferometer is therefore the visibility, which is the Fourier transform in the Fourier (u,v) plane (baseline b) of the source brightness B(ξ) in the sky plane. If the visibility were measured everywhere in the Fourier plane, one would obtain the source brightness by a simple Fourier transform. But in practice, the Fourier plane is only explored in a limited number of points, bk, with visibility Vk=|Vk|e±iφk. The map I(ξ)=∑k 2|Vk| 28 cos(2πbkξ–φk ) is called the dirty map. The visibility measured with baseline bi for a point source in direction ξ0 reads V(bi)~exp(2iπbi ξ0 ). The dirty map, which in this case is called the dirty beam, is therefore B(ξ)~2∑icos{2πbi.(ξ– ξ0)}. It is maximal at ξ=ξ0, which is fortunate, but will only look as a decent PSF if the (u,v) coverage is good enough, covering a broad range of distances and directions. Indeed, if the baselines were all parallel to a same direction χ, the dirty beam would be elongated along that direction, which would be unsatisfactory. Note that we introduce here the concepts of beam and PSF for the interferometer as a whole, defined as the image of a point source; they have nothing to do with the beam (usually called primary beam) or PSF of each single dish that defines the field of view. In order to get the best possible dirty beam, one needs to use a pattern of antennas that optimizes the (u,v) coverage, which is normally systematically done. In order to increase the density of measurements in the Fourier plane, one may make observations using different antenna patterns (multi-configuration) and/or let the Earth rotation do this for us (super-synthesis). Figure 1.24 Schematic interference between signals received by two antennas (left) and signal treatment (right). In the radio interferometry jargon, one says that to “clean” the dirty map, one needs to de-convolve the effect of the PSF. This is called “de- convolution”. It means to produce a map that would be obtained with a well behaved PSF, which one calls the “clean beam”. In practice there exist several codes that allow for de-convolving the dirty map and are commonly used by radio astronomers such as CASA or GILDAS. Once data have been acquired by an interferometer, their reduction proceeds in two phases: calibration transforming from raw data to visibilities, and imaging/de-convolution transforming from dirty map to clean map. The calibration of ALMA data is done in CASA that produces as output a data set or (u,v) table, which contains “calibrated” visibilities. Imaging and de-convolution are done either with GILDAS/MAPPING or with CASA using as input the calibrated (u,v) 29 visibility table and giving as main output a set of (l,m,v) spectral cubes. Spectral cubes, or data cubes, consist of two sky coordinates, l and m, and one frequency, v, (that can be used to calculate a Doppler shift and, therefore, a velocity) at which the brightness is measured. 1.4.2 Reducing the ALMA data of the CO emission of RX J1131 We use ALMA observations, project number 2013.1.01207.S (PI: Paraficz Danuta), collected on July 19th 2015 using the standard 12-m array. The number of antennas was 37, the shortest and longest baselines were 27.5 m and 1.6 km respectively, which gives an angular resolution of 0.3 arcsec and a maximum recoverable scale of 16 arcsec. The antenna configuration and (u,v)-coverage are shown in Figure 1.25. Observations were carried out in Band 4 in two execution blocks, the total integration time spent on source was 75 minutes. The available bandwidth was divided into four spectral windows: three for the continuum, 128 channels and 2 GHz bandwidth each, centred on 137.2, 149.1 and 151.0 GHz. The fourth spectral window was centred on the red-shifted CO(2–1) line (νrest=230.538 GHz): 480 channels, 3.9 MHz channel width; together they add up to 1.875 GHz total bandwidth. Precipitable water vapour varied between 1.0 and 1.4 mm, which implies good observation conditions for this frequency band. The phase calibrator quasar J1130-1449 was ~ 2.5o away from the target and was typically observed every 7 minutes. Quasar J1058+0133 was used for bandpass calibration while Titan and Ganymede were used for flux calibration. Figure 1.25 Left: antenna configuration. Right: (u,v)-coverage. The data were reduced (calibration and imaging) using the Common Astronomy Software Application package (CASA; [30]). The procedure is described in the scripts available on the ALMA Science Archive ( Imaging was performed using standard 30 CLEAN method, with Brigg weighting (robust=0.5) applied to the calibrated visibilities. Continuum emission was presented in subsection 1.2.3.1 and is illustrated in Figure 1.26. It is seen to closely match the HST optical images with the exception of image D which is not detected; emission from the lens galaxy is observed at the same location as in the visible. Figure 1.26 Left: continuum brightness distribution inside the region (x2+y2)1/2< 3 arcsec from the lens galaxy. The Gaussian fit corresponds to a noise σ of 12.8 μJy beam-1. Right: continuum map, contours step of 5σ, starting at 3σ, beam size 0.34×0.27 arcsec2 shown in the lower left corner. Red crosses indicate positions of the HST quasar images and of the lens galaxy. For the CO(2-1) line emission, we first used the data reduced by ALMA staff but we realized that they used too short a frequency interval to define the continuum subtraction and we redid the imaging with proper continuum subtraction. The beam size is 380×290 mas2 with position angle PA=66o and the noise rms level is 0.382 mJy beam–1 per channel. Note that the beam size obtained by P18 is larger, 440×360 mas2, because they use natural rather than robust weighting; on the other hand, their noise level is accordingly smaller. We do not claim that our choice of robust weighting is better than the choice of natural weighting adopted by P18; our aim is instead to understand the effect of a different data reduction on the results obtained. The data are presented in the form of a cube of 640×640 pixels, each 70×70 mas2, covering a square of ±22.4 arcsec centred on the continuum emission of the lens galaxy, G, and of 121 Doppler velocity bins, 8.417 km/s each, covering an interval of ±509 km s–1. The rest frequency of the CO(2-1) line emission is 230’538.000 MHz, giving an observed frequency of 139’373.678 MHz for a redshift z=0.6541. The first frequency bin of the data cube is centred on 139’137.963 MHz with channel spacing of 3.907 MHz each, meaning channel 61 for the CO(2-1) line emission: we take its centre as origin of velocity. We originally use coordinates centred on G with the y axis pointing north and the 31 x axis pointing west, position angles being measured on the sky plane counter-clockwise from west. The projected distance from the origin is (x2+y2)1/2. However (see Section 2.2) we shall also use coordinates centred at the best-fit lens centre, 60 mas south and 50 mas east of G, with the y axis pointing 16o east of north and the x axis pointing 16o north of west. Figure 1.27 displays the map of the intensity integrated between –340 and 333 km s–1, the mean velocity map and the Doppler velocity spectrum integrated over a circle of 3 arcsec radius. Also shown in the figure are the maps of the reconstructed source brightness obtained by P18 compared with that of RX J0911 obtained by [3]. A major difference between RX J1131 and RX J0911, much farther away from us, is that the millimetre emission of the former covers the whole caustic while that of the latter covers only the cusp region. A consequence is the importance of the Einstein ring configuration observed in RX J1131, absent from RX J0911. Details of the morpho- kinematics of the CO(2-1) emission are presented in Section 2. Figure 1.27 Upper panels: map of the velocity integrated intensity showing the location of the HST optical images (left, units are Jy beam–1), map of the mean Doppler velocity (centre, units are km s–1) and Doppler velocity spectrum (right). Lower panels: Brightness distribution (left) and maps of the reconstructed source brightness obtained by P18 for RX J1131 (centre) and by [3] for RX J0911 (right); the caustic is shown in the two latter panels. 32 SECTION 2. METHODS 2.1 Gravitational lensing of RX J1131 2.1.1 Generalities [4] A direct consequence of special relativity is that any sensible theory of gravitation must predict that light bends in a gravity field. As a result, light, or generally any electromagnetic radiation, emitted by a distant object and travelling near a very massive object in the foreground will appear to come from a point away from the real source and produce effects of mirage and of light concentration generally referred to as gravitational lensing. As early as 1937, Fritz Zwicky had noted that the effect could allow galaxy clusters to act as gravitational lenses but it was not until 1979 that this effect was confirmed by observation of the so-called “Twin QSO” ([31]). In 1986, [31] and [32] independently discovered the first giant luminous arcs gravitationally lensed by galaxy clusters. Gravitational lenses act equally on all kinds of electromagnetic radiation and lensing has been observed over the whole electro-magnetic spectrum, from radio to X-ray frequencies. Gravitational lensing can be used to study the background source or the foreground lens. One commonly distinguishes between three types of gravitational lensing: microlensing, weak lensing and strong lensing. Microlensing refers to the case of a source passing behind the lens (or a lens passing in front of the source), producing an amplification of its emission at alignment; it has been used to search for Brown Dwarfs, Machos (Massive Astrophysical Compact Halo Objects) and Wimps (Weakly Interacting Massive Particles). It is of some relevance to the case of RX J1131 when stars of the halo of the lens galaxy pass in front of the quasar. Weak lensing corresponds to an extended lens in the foreground, typically a cluster of galaxies and its dark matter content, lensing a number of galaxies in the background; the image of each galaxy is then slightly elongated in a direction perpendicular to the line joining it to the centre of the lensing region and can only be detected on a statistical basis when considering a large number of imaged galaxies. One speaks of strong lensing, as is the case in the present work, when the images can easily be identified as being produced by a same lens and source, in a simple enough configuration (e.g., [33], [34], [35]). The lens may be a star, a galaxy or a cluster of galaxies. The bending is usually small: a galaxy with a mass of 1011 solar masses will typically produce multiple images separated by only a few arc seconds (e.g., [35], [36], [37]). Strong gravitational lensing has become a textbook topic. In particular, several authors, such as [38] or Saha & Williams (2003), have summarized the main properties in simple terms, underlining the most general qualitative 33 features. While the case of complex lens configurations has been extensively studied (see for example [39]), in particular with the aim of evaluating the mass distribution of baryonic and dark matter in cluster lenses, studies of strongly lensed extended sources are less common. The clearest cases of strong lensing, which are naturally the most studied, are often associated with sources located near the inner caustic of the lens, making the problem highly non linear: when crossing the caustic outward, magnifications become infinite and one switches from a four-image to a two-image configuration. Several authors, such as [38] [40], [41], [42] or [43] have analysed the consequences in the case of extended sources and described their effects in some detail. The most spectacular manifestation of strong lensing is the formation of an Einstein ring, which occurs when source, lens and observer are aligned. The angular size of an Einstein ring is given by the Einstein radius, (see, e.g, [44]) θ=(4GM dLS /[dLdS])1⁄2/c where G is the gravitational constant, M the mass of the lens, dL is the observer-lens distance, dS is the observer-source distance and dLS is the lens-source distance. When axial symmetry is broken, the ring splits in multiple images scattered around the lens. The number and shape of these depend upon the relative positions of the source, lens, and observer, and the shape of the gravitational well of the lensing object (Saha & Williams 2003). There is a relative time delay between different images, corresponding to different light paths. The morphology of multiple images depends on the position of the source with respect to the lens caustic, a curve on which light amplification becomes singular. When crossing the caustic away from the central lens the number of images changes abruptly from 4 to 2 and magnifications change sign, meaning that the images switch from right-handed to left-handed or conversely. The correspondent of the caustic in the image plane is the critical curve, which separates left-handed from right-handed images. 2.1.2 Lens equation: point source [47] Fermat principle states that images form where the gradient of the time delay, τ, cancels. In the approximation of small deflections, which always applies in practice, the time delay can be written as the sum of a geometrical delay and of the gravitational delay proper: τ = τ0[½(i−s)2−ψ]. Here, i and s are the image and source vectors in sky coordinates (in a plane normal to the line of sight), τ0 is a constant time scale and ψ is an effective potential that describes the deflection induced by the lens as a function of the sky coordinates of the image. The effective potential ψ is proportional to the integral of the gravity potential along the line of sight between source and observer. A convenient form, used by many authors, includes an elliptical lens 34 and an external shear, with the axes of the ellipse being taken as coordinate axes without loss of generality: ψ=r0r(1+εcos2φ)½+½γ0r2cos2(φ−φ0) , (1) where (r,φ) are the polar coordinates of the image. The lens term, of strength r0 and aspect ratio [(1+ε)/(1−ε)]½, decreases as 1/r outside the core region. The shear term has a strength γ0 and makes an angle φ0 with the major axis of the lens ellipse. Writing that the gradient of Relation 1 cancels, and calling (rs,φs) the polar coordinates of the source, one obtains the lens equation: rseiφs=reiφ(1−r−1∂ψ/∂r−ir−2∂ψ/∂φ) . (2) There may typically be two or four images depending on the position of the source with respect to the inner caustic of the lens. If the potential is isotropic, the lens equation reduces to rseiφs=eiφ(r−∂ψ/∂r), which has two obvious solutions, one at φ+=φs and the other at φ−=φs+π with r+=∂ψ/∂r+rs and r−=∂ψ/∂r−rs respectively. For rs=0, the alignment is perfect and one obtains an Einstein ring having r=∂ψ/∂r. 2.1.3 Lens equation: extended source [47] To the extent that the source is small and not too close to the lens inner caustic, the image of an extended source is simply obtained by differentiating the lens equation, rseiφs=Dreiφ, where D=Dr+iDi and Dr=1−r−1∂ψ/∂r, Di=−r−2∂ψ/∂φ. In this way, we obtain the relation between a point (rs+drs, φs+dφs) on the source and its image (r+dr, φ+dφ): (drs+irsdφs)eiφs=D(dr+irdφ)eiφ+(∂D/∂r dr+∂D/∂φ dφ)reiφ. (3) In practice, one does not directly observe the source, but only its images. It is therefore convenient to replace, in the left-hand side of the above relation, the source dependent term rseiφs by its expression in terms of the image coordinates: D(drs/rs+idφs)=(D+r∂D/∂r) dr/r+(iD+ ∂D/∂φ)dφ. (4) Relation 3 gives the coordinates of an image point as a function of those of the corresponding source point. Indeed, drs and rsdφs are Cartesian coordinates having their origin at the centre of the source and the axis of abscissas radially outwards; similarly, dr and rdφ are Cartesian coordinates having their origin at the centre of the image and the axis of abscissas radially outwards (Figure 2.1). For Relation 3 to apply, drs and rsdφs must be small enough for the corresponding source points to stay away from the lens inner caustic. In practical cases, with the centre of the source located near the caustic, this will often not be the case if the source extension is such that it overlaps the caustic. The linear approximation of Relation 3 needs therefore to be used with care. Yet, it usefully serves several purposes, such as providing explicit expressions for the magnifications which may be calculated 35 using arbitrarily small values of drs and rsdφs or for giving a qualitative illustration of the main features in simple terms as is done below. With Relation 3 being linear, it is straightforward to express (dr/r, dφ)