Astronomy

Relationship between photometric colour and redshift

Relationship between photometric colour and redshift


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In this paper, the authors describe the theoretical relationship between photometric colours, particularly in Figures 1 and 6:

Since the colour is the difference between two magnitudes, and the redshift is defined as $z=frac{lambda'-lambda_0}{lambda_0}$

I would have thought that the colour-z relationships would simply be a downward curve, since the light gets redshifted (goes down in frequency), and the ratio of two magnitudes is constant with redshift.

Why do the colour-z relationships in this paper (and in others I've seen) show such a complex relationship?


The main reason is that the intrinsic spectra of galaxies are complex and therefore a redshift of their spectrum, whilst leading to a redder spectrum overall, does not necessarily lead to reddening in all colours.

For instance if there is apeak in the intrinsic spectrum, then as that peak moves redward, then colours formed from bands on the same side or straddling the peak in wavelength would behave differently.

The dramatic redward turn up at high redshifts (note, a large colour is redder) is caused by the "Lyman break" moving through the colour bands. Basically, the intrinsic spectrum is self-absorbed at wavelengths shortward of 91.2 nm by neutral hydrogen. This absorption edge is redshifts into the visible region at redshifts bigger than 3, causing the bluer bands to essentially disappear and any colour formed with them to become very red.


Relationship between photometric colour and redshift - Astronomy

A simple estimate of the photometric redshift would prove invaluable to forthcoming continuum surveys on the next generation of large radio telescopes, as well as mitigating the existing bias towards the most optically bright sources. While there is a well-known correlation between the near-infrared K-band magnitude and redshift for galaxies, we find the K - z relation to break down for samples dominated by quasi-stellar objects. We hypothesise that this is due to the additional contribution to the near-infrared flux by the active galactic nucleus, and, as such, the K-band magnitude can only provide a lower limit to the redshift in the case of active galactic nuclei, which will dominate the radio surveys. From a large optical dataset, we find a tight relationship between the rest-frame (U - K)/(W2 - FUV) colour ratio and spectroscopic redshift over a sample of 17 000 sources, spanning z ≈ 0.1-5. Using the observed-frame ratios of (U - K)/(W2 - FUV) for redshifts of z ≲ 1, (I - W2)/(W3 - U) for 1 ≲ z ≲ 3, and (I - W2.5)/(W4 - R) for z ≳ 3, where W2.5 is the λ = 8.0 μm magnitude and the appropriate redshift ranges are estimated from the W2 (4.5 μm) magnitude, we find this to be a robust photometric redshift estimator for quasars. We suggest that the rest-frame U - K colour traces the excess flux from the AGN over this wide range of redshifts, although the W2 - FUV colour is required to break the degeneracy.


History

The history of the subject begins with the development in the nineteenth century of wave mechanics and the exploration of phenomena associated with the Doppler effect. The effect is named after Christian Andreas Doppler who offered the first known physical explanation for the phenomenon in 1842. The hypothesis was tested and confirmed for sound waves by the Dutch scientist Christoph Hendrik Diederik Buys Ballot in 1845. Doppler correctly predicted that the phenomenon should apply to all waves, and in particular suggested that the varying colors of stars could be attributed to their motion with respect to the Earth. While this attribution turned out to be incorrect (stellar colors are indicators of a star's temperature, not motion), Doppler would later be vindicated by verified redshift observations.

The first Doppler redshift was described by French physicist Armand-Hippolyte-Louis Fizeau in 1848 who pointed to the shift in spectral lines seen in stars as being due to the Doppler effect. The effect is sometimes called the "Doppler-Fizeau effect". In 1868, British astronomer William Huggins was the first to determine the velocity of a star moving away from the Earth by this method.

In 1871, optical redshift is confirmed when the phenomenon is observed in Fraunhofer lines using solar rotation, about 0.1 Å in the red. In 1901 Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors.

The earliest occurrence of the term "red-shift" in print (in this hyphenated form), appears to be by American astronomer Walter S. Adams in 1908, where he mentions "Two methods of investigating that nature of the nebular red-shift". The word doesn't appear unhyphenated, perhaps indicating a more common usage of its German equivalent, Rotverschiebung, until about 1934 by Willem de Sitter.

Beginning with observations in 1912, Vesto Slipher discovered that most spiral nebulae had considerable redshifts. Subsequently, Edwin Hubble discovered an approximate relationship between the redshift of such "nebulae" (now known to be galaxies in their own right) and the distance to them with the formulation of his eponymous Hubble's law. These observations are today considered strong evidence for an expanding universe and the Big Bang theory.


Relationship between photometric colour and redshift - Astronomy

Using ˜5000 spectroscopically confirmed galaxies drawn from the Observations of Redshift Evolution in Large Scale Environments (ORELSE) survey we investigate the relationship between colour and galaxy density for galaxy populations of various stellar masses in the redshift range 0.55 ≤ z ≤ 1.4. The fraction of galaxies with colours consistent with no ongoing star formation (f q ) is broadly observed to increase with increasing stellar mass, increasing galaxy density, and decreasing redshift, with clear differences observed in f q between field and group/cluster galaxies at the highest redshifts studied. We use a semi-empirical model to generate a suite of mock group/cluster galaxies unaffected by environmentally specific processes and compare these galaxies at fixed stellar mass and redshift to observed populations to constrain the efficiency of environmentally driven quenching (Ψ convert ). High-density environments from 0.55 ≤ z ≤ 1.4 appear capable of efficiently quenching galaxies with log (M_/M_<⊙ >)> 10.45. Lower stellar mass galaxies also appear efficiently quenched at the lowest redshifts studied here, but this quenching efficiency is seen to drop precipitously with increasing redshift. Quenching efficiencies, combined with simulated group/cluster accretion histories and results on the star formation rate-density relation from a companion ORELSE study, are used to constrain the average time from group/cluster accretion to quiescence and the elapsed time between accretion and the inception of the quenching event. These time-scales were constrained to be <t convert > = 2.4 ± 0.3 and <t delay > = 1.3 ± 0.4 Gyr, respectively, for galaxies with log (M_/M_<⊙ >)> 10.45 and <t convert > = 3.3 ± 0.3 and <t delay > = 2.2 ± 0.4 Gyr for lower stellar mass galaxies. These quenching efficiencies and associated time-scales are used to rule out certain environmental mechanisms as being the primary processes responsible for transforming the star formation properties of galaxies over this 4 Gyr window in cosmic time.


Spectroscopic failures in photometric redshift calibration: cosmological biases and survey requirements

15 widely-separated regions, each at least

20 arcmin in diameter, and reaching the faintest objects used in a given experiment, will likely be necessary if photometric redshifts are to be trained and calibrated with conventional techniques. Larger, more complete samples (i.e., with longer exposure times) can improve photo-z algorithms and reduce scatter further, enhancing the science return from planned experiments greatly (increasing the Dark Energy Task Force figure of merit by up to

50%) Options: This spectroscopy will most efficiently be done by covering as much of the optical and near-infrared spectrum as possible at modestly high spectral resolution (λ/Δλ >

3000), while maximizing the telescope collecting area, field of view on the sky, and multiplexing of simultaneous spectra. The most efficient instrument for this would likely be either the proposed GMACS/MANIFEST spectrograph for the Giant Magellan Telescope or the OPTIMOS spectrograph for the European Extremely Large Telescope, depending on actual properties when built. The PFS spectrograph at Subaru would be next best and available considerably earlier, c. 2018 the proposed ngCFHT and SSST telescopes would have similar capabilities but start later. Other key options, in order of increasing total time required, are the WFOS spectrograph at TMT, MOONS at the VLT, and DESI at the Mayall 4 m telescope (or the similar 4MOST and WEAVE projects) of these, only DESI, MOONS, and PFS are expected to be available before 2020. Table 2-3 of this white paper summarizes the observation time required at each facility for strawman training samples. To attain secure redshift measurements for a high fraction of targeted objects and cover the full redshift span of future experiments, additional near-infrared spectroscopy will also be required this is best done from space, particularly with WFIRST-2.4 and JWST Calibration: The first several moments of redshift distributions (the mean, RMS redshift dispersion, etc.), must be known to high accuracy for cosmological constraints not to be systematics-dominated (equivalently, the moments of the distribution of differences between photometric and true redshifts could be determined instead). The ultimate goal of calibration is to characterize these moments for every subsample used in analyses - i.e., to minimize the uncertainty in their mean redshift, RMS dispersion, etc. – rather than to make the moments themselves small. Calibration may be done with the same spectroscopic dataset used for training if that dataset is extremely high in redshift completeness (i.e., no populations of galaxies to be used in analyses are systematically missed). Accurate photo-z calibration is necessary for all imaging experiments Requirements: If extremely low levels of systematic incompleteness (<

0.1%) are attained in training samples, the same datasets described above should be sufficient for calibration. However, existing deep spectroscopic surveys have failed to yield secure redshifts for 30–60% of targets, so that would require very large improvements over past experience. This incompleteness would be a limiting factor for training, but catastrophic for calibration. If <

0.1% incompleteness is not attainable, the best known option for calibration of photometric redshifts is to utilize cross-correlation statistics in some form. The most direct method for this uses cross-correlations between positions on the sky of bright objects of known spectroscopic redshift with the sample of objects that we wish to calibrate the redshift distribution for, measured as a function of spectroscopic z. For such a calibration, redshifts of

100,000 objects over at least several hundred square degrees, spanning the full redshift range of the samples used for dark energy, would be necessary and Options: The proposed BAO experiment eBOSS would provide sufficient spectroscopy for basic calibrations, particularly for ongoing and near-future imaging experiments. The planned DESI experiment would provide excellent calibration with redundant cross-checks, but will start after the conclusion of some imaging projects. An extension of DESI to the Southern hemisphere would provide the best possible calibration from cross-correlation methods for DES and LSST. We thus anticipate that our two primary needs for spectroscopy – training and calibration of photometric redshifts – will require two separate solutions. For ongoing and future projects to reach their full potential, new spectroscopic samples of faint objects will be needed for training those new samples may be suitable for calibration, but the latter possibility is uncertain. In contrast, wide-area samples of bright objects are poorly suited for training, but can provide high-precision calibrations via cross-correlation techniques. Additional training/calibration redshifts and/or host galaxy spectroscopy would enhance the use of supernovae and galaxy clusters for cosmology. We also summarize additional work on photometric redshift techniques that will be needed to prepare for data from ongoing and future dark energy experiments. « less


Contents

Photometers employ the use of specialised standard passband filters across the ultraviolet, visible, and infrared wavelengths of the electromagnetic spectrum. [4] Any adopted set of filters with known light transmission properties is called a photometric system, and allows the establishment of particular properties about stars and other types of astronomical objects. [10] Several important systems are regularly used, such as the UBV system [11] (or the extended UBVRI system [12] ), near infrared JHK [13] or the Strömgren uvbyβ system. [10]

Historically, photometry in the near-infrared through short-wavelength ultra-violet was done with a photoelectric photometer, an instrument that measured the light intensity of a single object by directing its light onto a photosensitive cell like a photomultiplier tube. [4] These have largely been replaced with CCD cameras that can simultaneously image multiple objects, although photoelectric photometers are still used in special situations, [14] such as where fine time resolution is required. [15]

Modern photometric methods define magnitudes and colours of astronomical objects using electronic photometers viewed through standard coloured bandpass filters. This differs from other expressions of apparent visual magnitude [7] observed by the human eye or obtained by photography: [4] that usually appear in older astronomical texts and catalogues.

Magnitudes measured by photometers in some commonplace photometric systems (UBV, UBVRI or JHK) are expressed with a capital letter. e.g. 'V" (mV), "B" (mB), etc. Other magnitudes estimated by the human eye are expressed using lower case letters. e.g. "v", "b" or "p", etc. [16] e.g. Visual magnitudes as mv, [17] while photographic magnitudes are mph / mp or photovisual magnitudes mp or mpv. [17] [4] Hence, a 6th magnitude star might be stated as 6.0V, 6.0B, 6.0v or 6.0p. Because starlight is measured over a different range of wavelengths across the electromagnetic spectrum and are affected by different instrumental photometric sensitivities to light, they are not necessarily equivalent in numerical value. [16] For example, apparent magnitude in the UBV system for the solar-like star 51 Pegasi [18] is 5.46V, 6.16B or 6.39U, [19] corresponding to magnitudes observed through each of the visual 'V', blue 'B' or ultraviolet 'U' filters.

Magnitude differences between filters indicate colour differences and are related to temperature. [20] Using B and V filters in the UBV system produces the B–V colour index. [20] For 51 Pegasi, the B–V = 6.16 – 5.46 = +0.70, suggesting a yellow coloured star that agrees with its G2IV spectral type. [21] [19] Knowing the B–V results determines the star's surface temperature, [22] finding an effective surface temperature of 5768±8 K. [23]

Another important application of colour indices is graphically plotting star's apparent magnitude against the B–V colour index. This forms the important relationships found between sets of stars in colour–magnitude diagrams, which for stars is the observed version of the Hertzsprung-Russell diagram. Typically photometric measurements of multiple objects obtained through two filters will show, for example in an open cluster, [24] the comparative stellar evolution between the component stars or to determine the cluster's relative age. [25]

Due to the large number of different photometric systems adopted by astronomers, there are many expressions of magnitudes and their indices. [10] Each of these newer photometric systems, excluding UBV, UBVRI or JHK systems, assigns an upper or lower case letter to the filter used. e.g. Magnitudes used by Gaia are 'G' [26] (with the blue and red photometric filters, GBP and GRP [27] ) or the Strömgren photometric system having lower case letters of 'u', 'v', 'b', 'y', and two narrow and wide 'β' (Hydrogen-beta) filters. [10] Some photometric systems also have certain advantages. e,g. Strömgren photometry can be used to measure the effects of reddening and interstellar extinction. [28] Strömgren allows calculation of parameters from the b and y filters (colour index of by) without the effects of reddening, as the indices m 1 and c 1. [28]

There are many astronomical applications used with photometric systems. Photometric measurements can be combined with the inverse-square law to determine the luminosity of an object if its distance can be determined, or its distance if its luminosity is known. Other physical properties of an object, such as its temperature or chemical composition, may also be determined via broad or narrow-band spectrophotometry.

Photometry is also used to study the light variations of objects such as variable stars, minor planets, active galactic nuclei and supernovae, [7] or to detect transiting extrasolar planets. Measurements of these variations can be used, for example, to determine the orbital period and the radii of the members of an eclipsing binary star system, the rotation period of a minor planet or a star, or the total energy output of supernovae. [7]

A CCD camera is essentially a grid of photometers, simultaneously measuring and recording the photons coming from all the sources in the field of view. Because each CCD image records the photometry of multiple objects at once, various forms of photometric extraction can be performed on the recorded data typically relative, absolute, and differential. All three will require the extraction of the raw image magnitude of the target object, and a known comparison object. The observed signal from an object will typically cover many pixels according to the point spread function (PSF) of the system. This broadening is due to both the optics in the telescope and the astronomical seeing. When obtaining photometry from a point source, the flux is measured by summing all the light recorded from the object and subtracting the light due to the sky. [29] The simplest technique, known as aperture photometry, consists of summing the pixel counts within an aperture centered on the object and subtracting the product of the nearby average sky count per pixel and the number of pixels within the aperture. [29] [30] This will result in the raw flux value of the target object. When doing photometry in a very crowded field, such as a globular cluster, where the profiles of stars overlap significantly, one must use de-blending techniques, such as PSF fitting to determine the individual flux values of the overlapping sources. [31]

Calibrations Edit

After determining the flux of an object in counts, the flux is normally converted into instrumental magnitude. Then, the measurement is calibrated in some way. Which calibrations are used will depend in part on what type of photometry is being done. Typically, observations are processed for relative or differential photometry. [32] Relative photometry is the measurement of the apparent brightness of multiple objects relative to each other. Absolute photometry is the measurement of the apparent brightness of an object on a standard photometric system these measurements can be compared with other absolute photometric measurements obtained with different telescopes or instruments. Differential photometry is the measurement of the difference in brightness of two objects. In most cases, differential photometry can be done with the highest precision, while absolute photometry is the most difficult to do with high precision. Also, accurate photometry is usually more difficult when the apparent brightness of the object is fainter.

Absolute photometry Edit

To perform absolute photometry one must correct for differences between the effective passband through which an object is observed and the passband used to define the standard photometric system. This is often in addition to all of the other corrections discussed above. Typically this correction is done by observing the object(s) of interest through multiple filters and also observing a number of photometric standard stars. If the standard stars cannot be observed simultaneously with the target(s), this correction must be done under photometric conditions, when the sky is cloudless and the extinction is a simple function of the airmass.

Relative photometry Edit

To perform relative photometry, one compares the instrument magnitude of the object to a known comparison object, and then corrects the measurements for spatial variations in the sensitivity of the instrument and the atmospheric extinction. This is often in addition to correcting for their temporal variations, particularly when the objects being compared are too far apart on the sky to be observed simultaneously. [6] When doing the calibration from an image that contains both the target and comparison objects in close proximity, and using a photometric filter that matches the catalog magnitude of the comparison object most of the measurement variations decrease to null.

Differential photometry Edit

Differential photometry is the simplest of the calibrations and most useful for time series observations. [5] When using CCD photometry, both the target and comparison objects are observed at the same time, with the same filters, using the same instrument, and viewed through the same optical path. Most of the observational variables drop out and the differential magnitude is simply the difference between the instrument magnitude of the target object and the comparison object (∆Mag = C Mag – T Mag). This is very useful when plotting the change in magnitude over time of a target object, and is usually compiled into a light curve. [5]

Surface photometry Edit

For spatially extended objects such as galaxies, it is often of interest to measure the spatial distribution of brightness within the galaxy rather than simply measuring the galaxy's total brightness. An object's surface brightness is its brightness per unit solid angle as seen in projection on the sky, and measurement of surface brightness is known as surface photometry. [9] A common application would be measurement of a galaxy's surface brightness profile, meaning its surface brightness as a function of distance from the galaxy's center. For small solid angles, a useful unit of solid angle is the square arcsecond, and surface brightness is often expressed in magnitudes per square arcsecond.

A number of free computer programs are available for synthetic aperture photometry and PSF-fitting photometry.

SExtractor [33] and Aperture Photometry Tool [34] are popular examples for aperture photometry. The former is geared towards reduction of large scale galaxy-survey data, and the latter has a graphical user interface (GUI) suitable for studying individual images. DAOPHOT is recognized as the best software for PSF-fitting photometry. [31]

There are a number of organizations, from professional to amateur, that gather and share photometric data and make it available on-line. Some sites gather the data primarily as a resource for other researchers (ex. AAVSO) and some solicit contributions of data for their own research (ex. CBA):


5 Results

5.1 ImpZ redshifts

First, it is of interest to see how successful the photometric redshifts are when there is no AV freedom - that is, AV= 0 in the solutions. The results of this are plotted in the left-hand panel of Fig. 7, for ‘measure’ cases (black squares) and ‘limit’ cases (red crosses). It is immediately clear that not allowing AV freedom has caused many of the ImpZ solutions to be incorrect: the code has been forced to substitute (incorrectly) additional redshift in place of the reddening action of dust. Indeed, 11 of the 42 ‘measure’ and 70 of the 153 ‘limit’ sources (48 per cent of the sample) have photometric redshifts that lie outside the log(1 +zspec) ± 0.1 boundaries.

Photometric redshifts from ImpZ . Comparison of spectroscopic and ImpZ -derived redshifts for ‘measure’ cases (black squares) and ‘limit’ cases (red crosses). The left-hand panel shows results when AV freedom is not considered the middle panel shows results for unconstrained redshift and AV space and the right-hand panel shows results for redshift space constrained to be within 0.05 in log(1 +zphot) of zspec, but AV unconstrained. Dot-dashed lines denote an accuracy of 0.1 in log(1 +z), a typical photometric redshift accuracy.

Photometric redshifts from ImpZ . Comparison of spectroscopic and ImpZ -derived redshifts for ‘measure’ cases (black squares) and ‘limit’ cases (red crosses). The left-hand panel shows results when AV freedom is not considered the middle panel shows results for unconstrained redshift and AV space and the right-hand panel shows results for redshift space constrained to be within 0.05 in log(1 +zphot) of zspec, but AV unconstrained. Dot-dashed lines denote an accuracy of 0.1 in log(1 +z), a typical photometric redshift accuracy.

ImpZ is therefore successful at returning accurate redshifts. However, can it also provide a measure of the extinction compatible with that implied by the Balmer decrement? Since two of the 42 ‘measure’ sources obtain an incorrect photometric redshift this means that their extinction values are also likely to be incorrect. In order to remove this (small) source of error in the following comparisons with the Balmer-derived extinction, we now constrain the redshift range explored by ImpZ to lie within 0.05 in log(1 +zphot) of zspec (plotted in the right-hand panel of Fig. 7). Now, good solutions are found for all 42 ‘measure’ cases and 153 ‘limit’ cases, with σz= 0.06. It is the resulting ImpZ AV values from this setup that are considered from now on in the investigation.

5.2 Range of ImpZ AV allowed values

Although we wish to explore the accuracy of the extinction output from redshift codes by comparing to a sample with Balmer-derived measurements, we can also obtain an internal estimate of how well constrained the [z, template, AV]-solution is from the reduced χ 2 distribution. For the solution with the minimum χ 2 , χ 2 min, the following question can be asked: what range of AV produces a fit at or near the correct redshift (within 0.05 of log [1 +zspec]), with a reduced χ 2 within χ 2 min+1?

The results of asking this question of each source are illustrated in Fig. 8. It can be seen that, for the majority of ‘measure’ cases (black), the AV is quite well constrained, in most cases to within 0.3 or so in AV. Note that the lines in this plot can be discontinuous for example, object 150 in the plot has a best-fitting AV of 1.9 but has reasonable solutions in the AV ranges −0.3 to 0.4 and 0.8 to 1.1 which arise from fitting two other templates to the source. It is of interest to note that, whereas 57 per cent of ‘measure’ cases do not have discontinuous solutions, this is only true for 43 per cent of the ‘limit’ (cyan) cases. Thus, determining the AV via photometry for these sources is more problematic, just as it is via the Balmer ratio method. Considering the ‘measure’ cases in more detail, there are two objects for which the AV is poorly constrained (objects 83 and 159 in the figure, evidenced by their long black lines). Their best-fitting AV values are, respectively, 2.3 and 1.7, making them the most heavily extincted of the ImpZ ‘measure’ cases. Comparison with their Balmer [AV] (3 ± 1 and −1.6 ± 0.5 respectively) would support the result for the first source but the Balmer decrement for the second source would imply negative, or zero extinction, arguing against the ImpZ best-fitting result, or resulting in the interpretation that this source is problematic.

ImpZ AV allowed values. The range of AV parameter space for each source that provides a solution with a reduced χ 2 within χ 2 min+1 and that is at or near the correct redshift (within 0.05 of log [1 +zspec]). ‘Measure’ sources are shown as black lines, ‘limit’ sources as cyan lines. The best AV solution value is indicated as a black (‘measure’ case) or cyan (‘limit’ case) square.

ImpZ AV allowed values. The range of AV parameter space for each source that provides a solution with a reduced χ 2 within χ 2 min+1 and that is at or near the correct redshift (within 0.05 of log [1 +zspec]). ‘Measure’ sources are shown as black lines, ‘limit’ sources as cyan lines. The best AV solution value is indicated as a black (‘measure’ case) or cyan (‘limit’ case) square.

Fig. 9 illustrates the width of AV parameter space that lies within +1 of χ 2 min by plotting the distribution of this ‘width’ value (here the ‘width’ is simply defined as the maximum allowed AV minus the minimum allowed AV). For ‘measure’ cases (black) the distribution drops quite steeply with width such that more than half of the sources have a width of 0.4 or less. There is then a slight tail made up primarily of sources that had discontinuous solutions (such as one template with low AV and another template with higher AV), whilst the two sources with poorly constrained AV can be seen as a spike towards the maximum width of 3.3 (i.e the full −0.3 to 3 AV range). The distribution for the ‘limit’ cases is more clearly bimodal, with a similar set of reasonably well-constrained sources with an AV-width of 0.4 or less, but a much larger set of sources with poorly constrained AV. Again, this is likely to be a result of the nature of these ‘limit’ sources for which the dust extinction is hard to determine (via either the Balmer lines or photometry).

Histogram of the width of ImpZ AV allowed values. Distribution of the width of AV parameter space (defined by the minimum and maximum allowed AV) that lies within +1 of χ 2 min. ‘Measure’ sources are shown as a black line, ‘limit’ sources as a cyan dotted line.

Histogram of the width of ImpZ AV allowed values. Distribution of the width of AV parameter space (defined by the minimum and maximum allowed AV) that lies within +1 of χ 2 min. ‘Measure’ sources are shown as a black line, ‘limit’ sources as a cyan dotted line.

This analysis suggests that the photometric redshift solution has an inherently low extinction precision (at least with five-band photometry), such that AV is precise to perhaps only 0.3 for most sources, and is poorly constrained for a small subset. Rather than defining this internally estimated width value as the error in the Phot [AV] value, we choose instead to take the opposite approach for the comparison with the Balmer [AV]. We will take the Phot [AV] at face value and use the supposed relation with the Balmer [AV] to provide an external estimate of the precision of the Phot [AV] measurements.

5.3 ImpZ - comparison with Balmer

We can test how well the Calzetti ratio holds by comparing the Phot [AV] with 0.44×Balmer [AV]. The distribution of AV residuals for ImpZ is plotted in Fig. 10. It can be seen that there is quite a spread to the distribution, although it is broadly centred on zero. Based on the findings in Section 5.2 on the precision of the Phot [AV] solutions, some of this spread can be expected to arise from this low precision. Some of it can also be attributed to the accuracy of the Balmer [AV], which is typically accurate to around 30 per cent.

AV residuals for hyperz (left) and ImpZ (right). The residual is 0.44×Balmer[AV]-Phot[AV].

AV residuals for hyperz (left) and ImpZ (right). The residual is 0.44×Balmer[AV]-Phot[AV].

Fig. 11 plots (purple squares) the ImpZ AV values and residuals as a function of Balmer [AV] (multiplied by the Calzetti 2001b factor of 0.44). Note that the ImpZ AV values have been taken at face-value and have not had an error assigned to them, since we wish to derive an error based on the comparison with the Balmer [AV] values.

Left: AV results for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of the Balmer[AV] (multiplied by the Calzetti (2001) factor of 0.44) versus the Phot [AV]. Dot-dashed lines denote residuals of 0.3, 0.5 and 0.7 in AV. Errors are not defined for the Phot [AV] values. Right: AV residuals for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of Balmer [AV] (multiplied by the Calzetti 2001 factor of 0.44) versus the residual 0.44×Balmer[AV]-Phot[AV].

Left: AV results for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of the Balmer[AV] (multiplied by the Calzetti (2001) factor of 0.44) versus the Phot [AV]. Dot-dashed lines denote residuals of 0.3, 0.5 and 0.7 in AV. Errors are not defined for the Phot [AV] values. Right: AV residuals for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of Balmer [AV] (multiplied by the Calzetti 2001 factor of 0.44) versus the residual 0.44×Balmer[AV]-Phot[AV].

It can be seen that the residuals are smallest for Phot [AV] values of around 0.5 to 1, and that the residuals increase as we move away from this region (to either higher AV or lower/negative AV). Hence, the correlation to Balmer [AV] appears to be best for sources of intermediate extinction. It is also clear that none of the sources that were calculated as having negative AV based on their Balmer lines obtain similar Phot [AV] values, lending weight to the supposition that the Balmer method has fallen down for these sources as a result of limitations in the technique. The source with the largest Balmer [AV], of 5.15±0.5, is also the source with the largest residual (of sources with non-negative AV). Being so extincted, it is likely that this object is quite extreme, so disagreement between the star- and gas-derived extinction measures is to be expected.

gives and an outlier fraction, η, of 50 per cent. These results are better than one would infer from the large extinction degeneracies seen in the SSP fitting that were discussed in Section 4.1. Thus, a relationship between Balmer-derived and photometry-derived extinction measures is obtainable, although not a strong one.

5.4 hyperz - comparison with Balmer

Applying hyperz in a ‘best-case’ configuration (constraining the hyperz redshift solutions to the spectroscopic values and excluding the elliptical template) gives similar statistics of and an outlier fraction, η, of 50 per cent. The distribution of AV residuals for hyperz is plotted in Fig. 10. As with ImpZ , the distribution is broad but reasonably well centred on zero.

Fig. 11 plots (cyan triangles) the hyperz AV values and residuals as a function of Balmer [AV] (multiplied by the Calzetti 2001b factor of 0.44). The residuals are again correlated with the Balmer [AV], being smallest for Phot [AV] values of around 0.5 to 1, and the negative Balmer [AV] sources are again in poor agreement.

Thus, hyperz and ImpZ portray a similar correlation to the Balmer [AV], although the agreement is noisy.

5.5 Comparison of ImpZ and hyperz

As well as comparing the extinction outputs of the two photometric redshift codes with the Balmer-derived values, it is instructive to compare them with one another to see if they tend to agree on a similar extinction value for a given source. A plot of ImpZ -AV versus hyperz -AV is given in Fig. 12.

Comparison between ImpZ and hyperz : AV results for ImpZ versus those from hyperz . The solid line is exact agreement, and dot-dashed lines are residuals of 0.3 in AV. Note that for plotting purposes the values have been randomly altered by up to 0.02 in the x and y directions in order to separate points with the same/very similar values.

Comparison between ImpZ and hyperz : AV results for ImpZ versus those from hyperz . The solid line is exact agreement, and dot-dashed lines are residuals of 0.3 in AV. Note that for plotting purposes the values have been randomly altered by up to 0.02 in the x and y directions in order to separate points with the same/very similar values.

This shows that the two codes are in reasonable agreement about the extinction of a given source. 25 of the 42 sources (60 per cent) agree within <0.4 in AV, and 35 sources (83 per cent) agree within <0.5. The main difference appears to be for five sources for which ImpZ gives a high value of AV > 0.8 whilst hyperz tends to return a smaller AV estimate. Two of these sources (this includes object 159 mentioned in Section 5.2) have Balmer decrements that imply negative, or zero, extinction, favouring the hyperz result or the interpretation that the sources are problematic. Two others have intermediate Balmer [AV], consistent with the results of either code, and one has a larger Balmer [AV] (this is object 83 mentioned in Section 5.2) thus favouring the ImpZ result.

Calculating similar statistics to when comparing with the Balmer-AV, comparison between the AV values of the two codes gives and an outlier fraction, η, of 17 per cent. This internal consistency check between the two codes gives increased confidence in the photometric redshift template-fitting method as a technique to obtain extinction.

5.6 The Calzetti ratio

χ 2 analysis. The reduced χ 2 for ImpZ (solid line with crosses) and hyperz (dot-dashed line, diamonds) as a function of γ, the chosen ratio between photometrically derived and Balmer-ratio-derived extinction measures. The χ 2 values that are 1 above the minimum in the two distributions are indicated by horizontal lines ( ImpZ , dotted hyperz , dashed).

χ 2 analysis. The reduced χ 2 for ImpZ (solid line with crosses) and hyperz (dot-dashed line, diamonds) as a function of γ, the chosen ratio between photometrically derived and Balmer-ratio-derived extinction measures. The χ 2 values that are 1 above the minimum in the two distributions are indicated by horizontal lines ( ImpZ , dotted hyperz , dashed).

The resulting statistical measures, , and the outlier fraction, η, are also plotted as a function of γ for the ImpZ and hyperz results ( Fig. 14). The left-hand panel shows how the rms in the residual, , varies with γ. A clear minimum is seen at γ∼ 0.15 to 0.35 for ImpZ results, and at around 0.2 to 0.4 for hyperz . A similar minimum is seen in the range γ∼ 0.2 to 0.35 for ImpZ results when the outlier fraction, η, is plotted against γ in the right-hand panel. For hyperz , the minimum is at around γ∼ 0.3 to 0.45.

Left: for ImpZ (solid line with crosses) and hyperz (dot-dashed blue line, diamonds) as a function of γ, the chosen ratio between photometrically derived and Balmer-ratio-derived extinction measures. The Calzetti value of γ= 0.44 is indicated as a long-dashed line. Right: percentage outliers for ImpZ (solid line with crosses) and hyperz (dot-dashed blue line, diamonds) as a function of γ. The Calzetti value of γ= 0.44 is indicated as a long-dashed line.

Left: for ImpZ (solid line with crosses) and hyperz (dot-dashed blue line, diamonds) as a function of γ, the chosen ratio between photometrically derived and Balmer-ratio-derived extinction measures. The Calzetti value of γ= 0.44 is indicated as a long-dashed line. Right: percentage outliers for ImpZ (solid line with crosses) and hyperz (dot-dashed blue line, diamonds) as a function of γ. The Calzetti value of γ= 0.44 is indicated as a long-dashed line.

This analysis suggests that, for this sample, the Calzetti ratio of 0.44 is a reasonable choice for γ within the accuracy of the method, although a value of ∼0.25 is preferred. A choice of γ= 0.25 gives the following statistics.

For ImpZ : and an outlier fraction, η, of 38 per cent.

For hyperz : and an outlier fraction, η, of 57 per cent.

As before, the distribution of AV residuals is plotted ( Fig. 15). It can be seen that the distribution is more peaked, although offset from zero.

AV residuals for hyperz (left) and ImpZ (right) with γ= 0.25. The residual is now 0.25×Balmer[AV]-Phot[AV].

AV residuals for hyperz (left) and ImpZ (right) with γ= 0.25. The residual is now 0.25×Balmer[AV]-Phot[AV].

Fig. 16 plots the Phot [AV] values and residuals as a function of Balmer [AV] (multiplied by γ= 0.25). It can be seen that, for lower Balmer [AV], the Balmer [AV] tends to underestimate the extinction compared with the Phot [AV] value. If the negative Balmer [AV] values are excluded, the remaining sources with positive Balmer [AV] do tend to follow the line denoting agreement, albeit with large scatter.

Left: AV results for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of the Balmer[AV] (multiplied by γ= 0.25) versus the photometrically derived AV. Dot-dashed lines denote residuals of 0.3, 0.5 and 0.7 in AV. Errors are not defined for the Phot [AV] values. Right: AV residuals for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of Balmer [AV] (multiplied by γ= 0.25) versus the residual 0.25×Balmer[AV]-Phot[AV].

Left: AV results for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of the Balmer[AV] (multiplied by γ= 0.25) versus the photometrically derived AV. Dot-dashed lines denote residuals of 0.3, 0.5 and 0.7 in AV. Errors are not defined for the Phot [AV] values. Right: AV residuals for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of Balmer [AV] (multiplied by γ= 0.25) versus the residual 0.25×Balmer[AV]-Phot[AV].

In Fig. 17, the sources with only a lower limit on their Balmer-derived extinction (that is, a minimum 3σ Hα detection but a limit only on the Hβ line flux) are plotted in comparison with the extinction as derived from the photometric redshift codes. Here, no ratio γ is applied to the Balmer AV. Instead, straight lines indicating different ratios are overplotted. Since these are lower limits, sources need to lie on, or to the left of, a line to imply consistency with that chosen ratio. It can be seen that these lower-limit sources are more consistent with lower values of γ. Of the 153 such sources, 146 (95 per cent) are consistent with the γ= 0.25 line when considering ImpZ solutions (purple squares), but only 130 (85 per cent) are consistent when considering the hyperz solutions (cyan triangles).

‘Limit’ cases: AV results for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of the Balmer [AV]‘limits’ versus the photometrically derived AV. Solid black lines denote γ values of 0.25, 0.44, 0.6 and 1.0 for the conversion factor relating Balmer [AV] and Phot [AV]. Sources need to lie on, or to the left of, a line to imply consistency with that chosen ratio.

‘Limit’ cases: AV results for hyperz (cyan triangles) and ImpZ (purple squares). The plot is of the Balmer [AV]‘limits’ versus the photometrically derived AV. Solid black lines denote γ values of 0.25, 0.44, 0.6 and 1.0 for the conversion factor relating Balmer [AV] and Phot [AV]. Sources need to lie on, or to the left of, a line to imply consistency with that chosen ratio.


6 Discussions and Conclusions

A number of redshift codes routinely output an AV value in addition to the best-fitting redshift, but little has been done to investigate the reliability and/or accuracy of such extinction measures. The main reason for this lies in the aim of such codes - they have been developed and optimized in order to derive redshifts. However, as the field of photometric redshift derivation matures, it is useful to consider some of the other parameters that redshift solutions produce.

In this paper we have asked whether a photometric redshift code can reliably determine dust extinction. The short answer would be: ‘not to a great accuracy’.

Using a sample with extinctions derived from Balmer flux ratios, the AV values produced by two photometric redshift codes, ImpZ and hyperz , have been compared with the Balmer [AV] values.

First, it was demonstrated that the inclusion of AV was crucial in order to obtain photometric redshifts of high accuracy and reliability, such that 95 per cent of the ImpZ results agreed with the spectroscopic redshifts to better than 0.1 in log(1 +z). Without the inclusion of AV freedom, there was a systematic and incorrect offset to higher photometric redshifts, with many more incorrect redshift solutions. The existence of some negative AV solutions may be indicative of the need for a bluer template in the template set, or for the inclusion of some additional free parameter in the fits. As the most important feature for template-fitted photometric redshifts is the location and identification of ‘breaks’ in the SED, the inclusion of AV freedom can be seen, at first-order, as a modifier of the template SED's slope, but it does not have a strong effect on the breaks themselves. Hence a similar improvement may be achievable via a ‘tilting’ parameter, or similar, which would act to alter the slope of the template SEDs. Since the addition of dust extinction has a physical basis, however, this is a preferable parameter, as long as we can demonstrate that there is some correlation between the best-fitting phot [AV] and the actual (or in this case, that measured via the Balmer ratio) dust extinction of the source. Thus, once the ability to derive good redshifts for the sample had been demonstrated a comparison between the Phot [AV] and Balmer [AV] was carried out.

The correlation between the Phot [AV] and the Balmer [AV] was similar for both codes, but was noisy and not particularly strong. Based on direct comparison between the two codes, and investigations into the χ 2 solution space, it was found that a good part of this noise is derived from the inherent lack of precision that the Phot [AV] solution has (perhaps 0.3 in AV say), no doubt since it is based on only five photometric measurements. Additional noise arises from the precision of the Balmer [AV], typically accurate to perhaps 30 per cent, as a result of the resolution of the spectrographic data. Given these errors, the correlation seen was, in fact, quite good.

The correlation was improved somewhat when the empirical value of γ= 0.44, the ratio between gas- and star- derived extinction, as determined by Calzetti (2001b), was allowed to vary. From least-squares fitting, the minimum in the reduced χ 2 distribution was found for γ∼ 0.25 ± 0.2.

The Calzetti ratio of 0.44 means that there is a factor of about 2 difference in reddening, such that the ionized gas (as measured by the Balmer decrement) is twice as reddened as the stellar continua (as measured by the photometry) (e.g. Fanelli, O'Connell & Thuan 1988 Calzetti et al. 1994). This implies that the covering factor of the dust is larger for the gas than for the stars, which can be explained by the fact that the ionizing stars are short-lived and so for their lifetime remain relatively close to their (dusty) birthplace, whilst the majority of stars contributing to the galaxy's overall optical luminosity are longer-lived and can migrate away from their dusty origins.

For the sample of galaxies in this paper, this factor of 2 difference in covering factor implied by the Calzetti ratio is found to be plausible, given the errors of the method. The sample has some preference for an increased covering factor, which implies that these galaxies are undergoing more rapid, ‘bursty’ star formation than the galaxies Calzetti used in her derivation. Perhaps more importantly, the results demonstrate the pitfalls of assuming that star- and gas-based extinction measures will give the same dust extinction given some conversion factor. Thus, correlation to Balmer-derived values are modulo the uncertainty in comparing star- and gas-based extinction measures.

However, the results presented here show that, given certain considerations, there is potential in the application of photometric codes to derive a reliable extinction measure, although the precision is currently low. It is expected that the ability of photometric redshift codes to determine extinction will improve with the availability of more photometric bands (here, there are five wide-band filters between 3000 and 9000 Å). A sample with a combination of wide- and narrow-band filters, with good wavelength coverage and range (in particular, extension to near-IR) will break many of the degeneracies and allow the codes to differentiate accurately between different possible fits.

The results also show that it is important to note that this will be a measure of the star-based extinction, and will not necessarily be well correlated with the extinction to the ionized regions of a galaxy.

Acknowledgments

We would like to thank Michael Rowan-Robinson for discussions on the nature of dust extinction and SED templates. The referee provided astute suggestions and comments on this work and we extend our thanks. We also thank those responsible for the CNOC2 survey whose data we have used here. The Canada-France-Hawaii Telescope is operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique de France, and the University of Hawaii.


Relationship between photometric colour and redshift - Astronomy

PhD Committee members: Scott Dodelson, LianTao Wang, Steve Meyer

Thesis Abstract: The existence of a quasi-deSitter expansion in the early universe, known as inflation, generates the seeds of large-scale structures and is one of the foundations of the standard cosmological model. The main observational predictions of inflation include the existence of a nearly scale-invariant primordial power spectrum that is imprinted on the cosmic microwave background (CMB), that has been corroborated with remarkable precision in recent years. Generalizations of the vanilla single-field slow-roll inflation provide a wealth of observational signatures in the power spectrum and the non-Gaussianity of fluctuations of the CMB, and this motivates a technique that can evaluate predictions of inflation beyond the slow-roll approximation called the generalized slow-roll (GSR). I will describe the latest searches for signatures of slow-roll violations in the Planck data using the GSR formalism, which is an ideal framework to probe inflationary models in this regime.

Ph.D. Committee members: Scott Dodelson, Stephan Meyer, Craig Hogan

"To constrain cosmology, and in particular to probe dark energy, from deep optical imaging surveys such as the Dark Energy Survey (DES), requires precise estimates of the redshifts of the distant galaxies they observe. Traditionally, these redshift estimates are made using galaxy colors, but this technique has known limitations and biases. Jennifer's thesis work involved the testing and implementation of a novel technique for estimating redshifts of galaxies, using the fact that they cluster in space with galaxies for which the redshifts may be known from spectroscopic measurements. Using simulations, Jen found that this "clustering redshift" technique accurately reconstructs the galaxy redshift distribution for a survey such as DES. She then applied this technique to determine the redshift distribution for several million galaxies in the first year of DES data, an important result that should prove extremely valuable for the cosmological analysis of these data."
- Joshua A. Frieman, PhD advisor

Thesis Abstract: Accurate determination of photometric redshifts and their errors is critical for large scale structure and weak lensing studies for constraining cosmology from deep, wide imaging surveys. Current photometric redshift methods suffer from bias and scatter due to incomplete training sets. Exploiting the clustering between a sample of galaxies for which we have spectroscopic redshifts and a sample of galaxies for which the redshifts are unknown can allow us to reconstruct the true redshift distribution of the unknown sample. Here we use this method in both simulations and early data from the Dark Energy Survey (DES) to determine the true redshift distributions of galaxies in photometric redshift bins. We find that cross-correlating with the spectroscopic samples currently used for training provides reliable estimates of the true redshift distribution in a photometric redshift bin. We discuss the use of the cross-correlation method in validating template- or learning-based approaches to redshift estimation and its future use in Stage IV surveys.

Ph.D. Committee members: Scott Dodelson, Joshua A. Frieman, Donald Q. Lamb

"In his PhD thesis Benedikt Diemer has shown that radial density profiles of dark matter halos cannot be characterized only as a function of halo mass, as was thought previously, but also depend on the mass accretion rate of halos. The work has resulted in a new model that accurately describes halo profiles in simulations from small radii out to 10 virial radii. Likewise, Benedikt has shown that halo concentrations depend not only on the halo mass (or more precisely on halo peak height), but also on the local slope of the power spectrum. Overall, this thesis showed that previously believed "universality" of the halo profiles is limited. Beyond just criticizing previous models, new models were developed that take into account the extra dependencies of halo profile parameters on the mass accretion rate and power law slope."
- Andrey V. Kravtsov, Ph.D. advisor

Thesis Abstract: We present a systematic study of the density profiles of dark matter halos in LCDM cosmologies, focusing on the question whether these profiles are "universal", i.e., whether they follow the same functional form regardless of halo mass, redshift, cosmology, and other parameters. The inner profile can be described as a function of mass and concentration, and we thus begin by investigating the universality of the concentration-mass relation. We propose a universal model in which concentration is a function only of a halo's peak height and the local slope of the matter power spectrum. This model matches the concentrations in LCDM and scale-free simulations, correctly extrapolates over 16 orders of magnitude in halo mass, and differs significantly from all previously proposed models at high masses and redshifts. Testing the universality of the outer regions, we find that the profiles are remarkably universal across redshift when radii are rescaled by R200m, whereas the inner profiles are most universal in units of R200c, highlighting that universality may depend upon the definition of the halo boundary. Furthermore, we discover that the profiles exhibit significant deviations from the supposedly universal analytic formulae previously suggested in the literature, such as the NFW and Einasto forms. In particular, the logarithmic slope of the profiles of massive or rapidly accreting halos steepens more sharply than predicted around

R200m, where the steepness increases with increasing peak height or mass accretion rate. We propose a new, accurate fitting formula that takes these dependencies into account. Finally, we demonstrate that the profile steepening corresponds to the caustic at the apocenter of infalling matter on its first orbit. We call the location of the caustic the splashback radius, Rsp, and propose this radius as a new, physically motivated definition of the halo boundary. We discuss potential observational signatures of Rsp that would allow us to estimate the mass accretion rate of halos.

Ph.D. Committee members: Paolo Privitera, Scott Wakely, Scott Dodelson.

Thesis Abstract: The workings of the most energetic astrophysical accelerators in the Universe are encoded in the origin of ultrahigh energy cosmic rays (UHECRs). Current observations by the Auger Observatory, the largest UHECR observatory, show a spectrum that agrees with an extragalactic origin, as well as an interesting transition in chemical composition from light element to heavier element as energy increases. Candidate sources range from young neutron stars to gamma-ray bursts and events in active galaxies. In this talk, I will discuss newborn pulsars as the sources of ultrahigh energy cosmic rays. I will show that a newborn pulsar model naturally injects heavier elements and can fit the observed spectrum once propagation in the supernova remnant is taken into account. With the proper injection abundances, integrated cosmic rays from the extragalactic pulsar population can match observation in all aspects - energy spectrum, chemical composition, and anisotropy. I will then examine the fingerprints of their Galactic counterparts on cosmic ray spectrum. Furthermore, I will consider the multi-messenger smoking gun of this scenario - the detectability of high energy neutrinos from pulsars in the Local Universe.

Ph.D. Committee members: Hsiao-Wen Chen, Andrey Kravtsov, Rich Kron

"Dr. Louis Abramson is an expert on the observation and phenomenological modeling of galaxy evolution, with a particular focus on the relationship between bulk statistical observables of galaxies, such as the distributions of star-formation-rate and mass over cosmic time, and the star formation histories of galaxies. His work during his Ph.D. has led to several new insights into the relationship between the passive (i.e., bulges) and actively star-forming components of galaxies, and led to a clear understanding that the scatter of galaxies across the so-called 'star forming main-sequence' is a critical observable to consider in further analyses, which he will continue as a postdoc at UCLA."
- Michael D. Gladders, Ph.D. advisor

Thesis Abstract: Galaxy star formation histories (SFHs) form a central thread of the cosmological narrative. Understanding them is therefore a central mission of the study of galaxy evolution. Although an ever-better picture is emerging of the build-up of the stellar mass of the *average* galaxy over time, the relevance of this track to the growth of *individual* galaxies is unclear. Largely, this ambiguity is due to the availability of only loose, ensemble-level constraints at any redshift appreciably greater than zero. In this talk, I outline how one of these constraints -- the the star formation rate/stellar mass relation -- shapes empirically based SFH models, especially in terms of the *diversity* of paths leading to a given end-state. I show that two models propose very different answers to this question -- galaxies grow *together* vs. galaxies grow *apart* -- corresponding (largely) to two different interpretations of the scatter in instantaneous galaxy growth rates at fixed stellar mass -- unimportant vs. essential. I describe how these interpretations affect one's stance on the profundity of galaxy "bimodality," the role of quenching mechanisms, and the influence of environment on galaxy evolution. Finally, after endorsing one of the models, I present some predictions that --- given upcoming observations --- should have the power to prove me right or wrong.


Title: QSO redshift estimates from optical, near-infrared and ultraviolet colours

0.1 - 5. Using the observed-frame ratios of (U K)/(W2-FUV) for redshifts of z > 1, (I-W2)/(W3-U) for 1 < z < 3 and (I-W2.5)/(W4-R) for z > 3, where W2.5 is the 8.0 micron magnitude and the appropriate redshift ranges are estimated from the W2 (4.5 micron) magnitude, we find this to be a robust photometric redshift estimator for quasars. We suggest that the rest-frame U-K colour traces the excess flux from the AGN over this wide range of redshifts, although the W2-FUV colour is required to break the degeneracy.