The galaxy ancestor problem

Abstract

The Hubble Space Telescope (HST) findsgalaxies whose Tolman dimming exceeds 10 mag. Could evolution alone explain these as our ancestor galaxies or could they be representatives of quite a different dynasty whose descendants are no longer prominent today? We explore the latter hypothesis and argue that surface brightness selection effects naturally bring into focus quite different dynasties from different redshifts. Thus, the HSTz = 7 galaxies could be examples of galaxies whose descendants are both too small and too choked with dust to be recognizable in our neighbourhood easily today. Conversely, the ancestors of the Milky Way and its obvious neighbours would have completely sunk below the sky at z > 1.2, unless they were more luminous in the past, although their diffused light could account for the missing re-ionization flux. This Succeeding Prominent Dynasties Hypothesis (SPDH) fits the existing observations both naturally and well even without evolution, including the bizarre distributions of galaxy surface brightness found in deep fields, the angular size ∼(1 + z)⁻¹ law, ‘downsizing’ which turns out to be an ‘illusion’ in the sense that it does not imply evolution, ‘infant mortality’, that is, the discrepancy between stars born and stars seen, the existence of ‘red nuggets’, and finally the recently discovered and unexpected excess of quasar absorption line damped Lyα systems at high redshift. If galaxies were not significantly brighter in the past and the SPDH were true, then a large proportion of galaxies could remain sunk from sight, possibly at all redshifts, and these sunken galaxies could supply the missing re-ionization flux. We show that fishing these sunken galaxies out of the sky by their optical emissions alone is practically impossible, even when they are nearby. More ingenious methods are needed to detect them. It follows that disentangling galaxy evolution through studying ever higher redshift galaxies may be a forlorn hope because one could be comparing young oranges with old apples, not ancestors with their true descendants.

galaxies: evolution, galaxies: high-redshift, galaxies: ISM, quasars: absorption lines, galaxies: statistics, dark ages, reionization, first stars

1 Introduction

Attempts to decipher the evolution of the cosmos through studying high-redshift galaxies rely on the implicit assumption that those galaxies are, in some sense, the ancestors of the galaxies around us today. But what if they are not? We would not be comparing like with like and so be completely misled.

Tolman (1930) long ago argued that the surface brightness (SB) of galaxies dims with redshift z as (1 + z)⁻⁴, indeed proposed it as a test for expansion. Now that the new wide-field camera WFC3 on board the Hubble Space Telescope (HST; O'Connell et al. 2010; Mackenty et al. 2010) can routinely find galaxies at redshifts of 7 or more, it raises serious questions as to their nature. Their SBs as measured in our frame are similar to those of galaxies nearby, such as the Milky Way (MW). Thus, to be the ancestors of the local population they must have undergone enormous and implausible evolution (dimming by ∼9 mag), in lockstep with redshift. This might seem a fortuitous coincidence, particularly when the star formation histories of local galaxies show few signs of such dramatic evolution, testifying more to fairly constant rates of star formation throughout cosmic time (see e.g. Tosi 2008).

Here we explore an alternative hypothesis that the populations of galaxies which show up at different redshifts are different from one another. They are not ancestors and descendants, but members of quite distinct families. For instance, galaxies prominent at high redshift may form a physically compact, very high SB family which can take a lot of Tolman dimming (≥10 mag) without disappearing from our sight at redshift 7 or more. The problem then becomes explaining where their descendants are today. The HST observations show that they are very small (sub-kpc), dense, rather rare in comoving density terms, and have no dust absorption. Taking into account their small sizes, and self-absorption by dust, which would naturally be high in such systems today, their contemporary descendants might be inconspicuous amongst the population of currently prominent galaxies.

Conversely, as we will show, lacking significant evolution, more than half the light from the MW will appear to have sunk beneath the sky at redshift 0.5, and every last photon by redshift 1.2. Our predecessor galaxies might therefore be totally invisible as individuals at higher redshifts, although their integrated light could very well swamp the output of those few compact high-z galaxies we can still detect out there. It hardly needs to be said that such a population of sunken galaxies could dramatically impact our ideas of cosmic evolution. For instance, they could supply the presently missing ultraviolet (UV) photons needed to re-ionize the Universe. They could also explain the excess of quasar absorption line systems (QSOALs), while Lilly–Madau plots showing the combined star formation rates in the cosmos as a function of redshift would have to be seriously modified.

This Succeeding Prominent Dynasties Hypothesis (SPDH), as opposed to the current notion of the Evolving Single Dynasty Hypothesis (ESDH), has its roots in a number of older ideas. It is forgotten today, but before HST was launched it was anticipated that Tolman dimming would rob the sky of almost all high-z galaxies, and it should have come as more of a surprise to find that this was not the case. Local galaxies tend to have a rather tight distribution of SBs, the explanation for which is still controversial (e.g. Davies, Impey & Phillips 1999). However, if it is a selection effect, then the families of the wrong SB at any redshift will appear inconspicuous by comparison with other families of the ‘right’ SB. An observer looking back through redshift space would thus expect to see, thanks to Tolman, different prominent families at different epochs. In particular he or she would expect to see the more compact objects at higher redshifts, and would find angular diameters ∝(1 + z)⁻¹, which is exactly observed to be the case (Section 6).

Some of these ideas were explored in Phillipps, Davies & Disney (1990) which built on the earlier papers Disney & Phillipps (1983) and Disney (1976). However, the highest redshift being considered there was 0.3! The situation has certainly moved on in a number of respects: the observations, of course, the supersession of photography by linear electronic detectors, which makes the analysis markedly simpler, and the most fashionable cosmological model in which to set the calculations. Most importantly though, those earlier papers were missing a vital argument about the way to normalize Visibility, an argument that is supplied here in Section 4, and which makes a significant difference to the main inferences.

To what extent could pure luminosity evolution offset Tolman dimming? If we examine Bruzual–Charlot models (Charlot & Bruzual 1991; Bruzual & Charlot 1993, 2003), then the most extreme single-burst models fade by 8 mag over 10 Gyr. Almost all that fading occurs immediately after the burst. However, for the last 10 Gyr of such a model's life, which is a long period appropriate for this discussion, the fading is a mere 2–3 mag (e.g. Bruzual & Charlot 2003, fig. 4) which is comparable with Tolman dimming, which is 3 mag at redshift 1, 5 mag at redshift 2, 7 mag at redshift 4 and 9 mag at redshift 7. Models with steady star formation rates, which match the stellar populations of nearby galaxies (e.g. Tosi 2008), actually brighten with time, that is, fade with redshift, exacerbating the situation. It is hard to see how the high SBs seen in redshift 7 galaxies can be accounted for stellar evolution, not as we understand it now.

The purpose of this paper is to push the SPDH to the highest redshifts currently accessible to observations (∼10). If it can be tested to destruction so much the better because, if it is true, then deciphering galaxy evolution will be very much harder, and perhaps impossible for generations to come.

The rest of this paper is organized as follows:

• Section 2 It gives a schematic outline of how the hypothesis works, and some of the conclusions it leads to.

• Section 3 It demonstrates by calculation the non-intuitive but dramatic nature of SB selection effects, that is, how two plunging curves mean that only galaxies huddled perilously close to the sky are seen to any great distance (see Fig. 9).

• Section 4 It introduces a vital new argument to normalize galaxy visibility. It leads to the daunting conclusion that low-SB galaxies too dim to turn up in the Schmidt photographic surveys will never be detectable in the optical, at least not for generations to come. Thus, whole dynasties of sunken galaxies could exist, lurking just beneath the sky.

• Section 5 It incorporates Tolman dimming and cosmology into visibility theory to show how quickly redshift can drag galaxies below the sky. Thus, MWs would appear half sunk by redshift 0.5 and wholly sunk by z = 1.2.

• Section 6 It argues that a combination of high intrinsic SB and aberration will, at high z, bring to the surface an extremely compact dynasty of galaxies that are relatively inconspicuous nearby. Their apparent angular sizes will obey the angular diameter ∼(1 + z)⁻¹ law as observed.

• Section 7 It explains why the aforementioned z ∼ 7 galaxies can leave descendants in our neighbourhood which we would not find without a dedicated search partly because they would have choked with their own dust.

• Section 8 It explains why galaxies containing normal amounts of gas and dust can never reach very high SBs because dust absorption intervenes first. Even faint galaxy catalogues, which can dispense with angular size criteria (there being no very faint Galactic stars), will only find compact galaxies if they are dust-free.

• Section 9 It repeats the Visibility theory of Section 3 but for giant ellipticals which have a different light distribution. They should sink more slowly with redshift, leading to the illusion that they formed earlier than spirals. Fig. 9 demonstrates how perilously close all visibly prominent galaxies must huddle to the sky. The SPDH explains the recently discovered ‘red nuggets’ without recourse to either luminosity or size evolution, as a selection effect. They are extremely compact elliptical or spheroidal objects too small to be found nearby.

• Section 10 It argues that because low-SB galaxies sink at lower redshifts, there will be a downsizing illusion which has nothing to do with evolution but reflects a correlation between intrinsic SB and luminosity in the sense that intrinsically less luminous galaxies generally have dimmer intrinsic SBs and hence sink at lower redshifts. In this section, we briefly speculate about the so-called missing dwarfs predicted by cold dark matter.

• Section 11 It first argues that luminosity evolution is quite unable to explain the bizarre properties of redshift 4–7 galaxies and goes on to discuss several phenomena predicted by the SPDH including: (i) ‘infant mortality’ – the mismatch between the number of galaxies seen forming and the number seen later on. This is rather direct evidence that most high-redshift galaxies have indeed sunk; (ii) ‘unexpected QSOALs’ – the surprising number of damped Lyα absorbers (DLAs) recently found at high redshift, more evidence of a sunken population; and (iii) ‘re-ionization’ which can be explained by the diffused light of all the sunken dynasties. We conclude that the SPDH fits the high redshift galaxy observations in a natural and parsimonious way that the ESDH cannot. It remains to be tested by looking for the sunken and choked galaxies predicted to lie in large numbers, both near and far.

2 The Narrow Window

As anyone who has looked for M31 can testify, the problem of detecting galaxies in the optical is not so much lack of light as lack of contrast against the foreground sky. (M31 has a V mag of 3.4 which spread over its size of roughly 3 × 1 deg amounts to a SB = 21.2 V mag arcsec⁻², where the sky is about 21.5 at a fair site.) This can be quantified by calculating the Visibility V of a galaxy as a function of both its luminosity L and its SB contrast with the sky (in mag arcsec⁻² or μ), where the Visibility V is the relative volume in which it could be detected. The result is shown schematically in Fig. 1. Irrespective of luminosity there is a very narrow window in SB contrast in which it is easy to see galaxies. Calculations show that the full width at half-maximum (FWHM) for the Visibility of spirals and other exponential galaxies is less than 3 mag, and as is well known (Disney & Phillipps 1983; Davies et al. 1994) catalogues of galaxies appear to conform rather well to this theoretically predicted selection effect, though how many hidden galaxies lie undetected outside this narrow visibility window is still a large and open question (Impey & Bothun 1997; Heller et al. 1999). Some certainly exist: on the low-SB side lie Local Group galaxies discovered as a result of enhanced star counts, objects like Segue 1 with SBs ∼ 6 mag dimmer than the peak in Fig. 1 (Belokorov et al. 2007); and on the high-SB side lie ultracompact dwarfs distinguished from stars by their spectroscopic signatures with SBs ∼ 7 mag higher (Phillipps et al. 1998; Jones et al. 2006). Astronomers are surprised to find how narrow the SB window is. In Section 3, we justify it by calculation. Here we attempt a schematic explanation.

Figure 1.

Schematic Wigwam diagram of the Visibility (i.e. relative volume in which it can be detected) of an exponential galaxy as a function of its surface brightness contrast Δμ (mag arcsec⁻²) to μ_c the lowest SB isophote that can be detected in the particular survey. In the usual convention, lower SBs are to the right-hand side, while the contrasts Δμ increase to the left-hand side. The diagram is the same for all luminosities which only effect the vertical scale, so regard L as fixed. The dashed line is the upper limit to the visibility set by the apparent magnitude limit m_c of the survey, and so is called V^m in the text. The smooth line is the upper limit to the visibility set by θ_c, the angular size limit of the survey defined at μ_c, and so is called V^θ in the text and labelled so in Fig. 3. To be visible any galaxy must lie beneath both lines, and so must lie in the shaded region A. Those at the left-hand side in region B are high-SB objects that appear too small. Those in region C are low-SB objects that appear too faint. Those in region D have no part of their images showing above the sky; they are entirely sunk beneath it. In practice, the FWHM of the visible window A is only 2.5 mag. This should be compared with Tolman dimming of 3 mag at a redshift of 1, and of 9 mag at a redshift of 7. As one looks to higher redshifts, Tolman dimming causes galaxies to march from the left-hand to right-hand side across the diagram, passing through the visibility window A, the wigwam, which is anchored in local coordinates by the brightness of the local sky (to which μ_c is related). In this rendering, the maximum heights of the two curves, dashed and smooth, have been arbitrarily set to be roughly equal; in practice, they can be altered by the survey parameters m_c, θ_c and μ_c (see Section 4 and Fig. 3 for an exact rendition with all the numbers put in).

To get into a given galaxy catalogue an object must obey two independent criteria. It must be bright enough to be detected, that is, exceed some limiting catalogue apparent magnitude m_c, yet large enough in angular size to be detected as an extended object. That is to say its apparent angular diameter θ, measured at some specified isophote μ_c, must exceed the minimum catalogue limit θ_c. (NB: Faint galaxy catalogues may not have an angular size limit because Galactic stars run out beyond m ∼ 25, but it turns out, by coincidence, that the results of this section apply to them too; see Section 8.)

If each galaxy is characterized by two parameters, absolute luminosity L and intrinsic SB (say, central SB μ₀ in mag arcsec⁻² or effective SB at half-light μ_1/2), then one can calculate the maximum distance d^m at which it can lie and still obey the magnitude criterion, and d^θ, the maximum distance at which it can lie and yet obey the angular criterion. Both distances scale as L^1/2 so we can set aside luminosity as a simple scaling parameter and investigate the more interesting dependencies of d^m and d^θ on the SB contrast Δμ = μ_c − μ₀ between the galaxy and the sky.

Fig. 1 illustrates what happens for objects with an exponentially declining light distribution (most galaxies bar giant ellipticals; see later). The dashed line shows d^m cubed (volume not distance is the important measure) and the smooth line shows d^θ cubed, both as a function of Δμ, the SB contrast. High-SB galaxies with large Δμ lie to the left-hand side and low-SB ones with small Δμ to the right-hand side. (NB: L is kept fixed!)

What is going on? Consider the dashed line. At high enough SB virtually all of a galaxy's light lies above μ_c, and V^m ≡ (d^m)³ does not vary with the contrast, so the line is flat. However, as the galaxy's SB is lowered (i.e. moves towards the right-hand side) so the contrast Δμ = μ_c − μ₀ drops, and more and more of its light falls below the limiting isophote μ_c until, when μ₀ = μ_c (i.e. Δμ = 0), it vanishes altogether (i.e. d^m and hence V^m → 0).

The smooth line corresponding to V^θ ≡ (d^θ)³ is more interesting. It has a fairly narrow peak because at high SB (towards the left-hand side) the galaxy must be physically small, while towards the right-hand side most of its light is dimmed below the limiting isophote μ_c, and what is left to measure above has a smaller and smaller apparent angular size until it vanishes altogether when μ₀ → μ_c and Δμ → 0.

Every galaxy in the catalogue must obey both criteria. Thus, it must lie in the shaded, wigwam-shaped area A beneath both the smooth line and the dashed line. Both lines plunge steeply, resulting in a narrow FWHM with a peak at P corresponding to an optimum contrast Δμ(P). Higher SB galaxies in region B lie above the smooth line, and will be too small in diameter to be seen as galaxies at any significant distance, while lower SB galaxies towards the right-hand side in region C lie above the dashed (green) line and will be too faint to be seen above the sky at any greater distance. Galaxies in region D are completely submerged below the sky, even their cores being dimmer than the limiting isophote μ_c.

Fig. 1, the Visibility or Wigwam diagram, is central to our hypothesis, and fundamental to galaxy research, and as such deserves careful study. Note first that is fixed in the observer's coordinate system and is independent of redshift. Any galaxy that is redshifted, and consequently dimmed by Tolman effects, is moved rightwards to lower SB. A prominent or high-visibility galaxy near the peak at P will slide rapidly down the dashed line to the right-hand side of region A until it is only visible nearby. (Actually, it will slide much faster because its apparent luminosity, which normalizes the height of the curves, is also falling at the same time due to Tolman effects.) Note, secondly, that the diagram applies to all (exponential) galaxies, irrespective of luminosity, which only changes the vertical scale. Note, thirdly, that μ_c, the outer isophotal level, is related to the sky brightness (at an appropriate wavelength) but is generally deeper, thanks to the accumulation of photons per detector pixel (see Section 4). Fourthly, the half width at half-maximum of the visibility window A is generally less than 2 mag. However, redshift dimming by (1 + z)⁻⁴ corresponds in magnitude to +10 log (1 + z); thus, 2 mag corresponds to a redshift of less than 0.6. This implies that even at redshifts of a half, ancestor galaxies are severely dimmed, and in many cases are sunk out of sight entirely. So even at moderate redshifts (0.5 ≤ z ≤ 1) the argument has to be made that the galaxies we do detect out there are really the ancestors of the MW and its catalogued neighbours.

If they are not our ancestors, then what else could they be? To answer that, it is necessary to discuss Tolman dimming. One factor of (1 + z) arises from relative time-dilatation in the source and one from photon weakening, that is, photons shifting to lower energy along their line of flight. The other two arise from simple aberration, that is to say that the source was closer to the observer and therefore looked bigger by a factor of (1 + z) in each dimension than it would do today (i.e. the convergence angle of its light was set at emission not detection).

Returning to Fig. 1 aberration means that a source that is in region B, and is therefore too compact to have much visibility nearby, can be apparently expanded by aberration and so appears relatively prominent at higher redshifts. To understand this, note that Fig. 1 has no vertical scale marked in; it shows the relative visibilities of galaxies with different SB contrasts. Remove the high-visibility galaxies (e.g. MWs) by redshift dimming, then other intrinsically higher SB objects fill the peak of the visibility window instead. It is always the galaxies whose apparent SBs match at the peak (approximately at Δμ = 3–4 mag) which at any redshift appear most prominent, that is, those for which

(1)

The narrowness of the visibility window (FWHM ∼ 2.5 mag, as we shall prove in Section 3), by comparison with Tolman dimming, can lead to some very surprising phenomena and illusions. For instance:

• (i)

  The apparent distribution of SBs among galaxies cannot change with redshift, for it is a consequence of the local window. This surprising prediction is observed (e.g. Jones & Disney 1997, see fig. 2). Tolman dimming is 3 mag by redshift 1 and 9 mag by redshift 7; thus, the observed constancy in Fig. 2 is most unlikely to be a consequence of dramatic stellar evolution which is nowhere apparent in the archaeology of our own and neighbouring galaxies (e.g. Tosi 2008, Tolstoy, Hill & Tosi 2009). See Section 11 for a discussion on evolution.

• (ii)

  Galaxies at redshifts >1 will sink below the sky, but their diffused radiation could still dominate the universe and lead to phenomena such as re-ionization.

• (iii)

  To be detected above the sky, high-z galaxies must have very high intrinsic SBs, and thus be very small for their luminosities. Unless galaxies are also undergoing dramatic size evolution, we must therefore see out there a new and distinctly different dynasty.

• (iv)

  If less luminous galaxies also have dimmer intrinsic SBs, as evidence suggests, then that alone would lead to the illusion of downsizing, that is, dwarf galaxies will apparently only lift themselves above the sky at recent epochs (Section 10).

• (v)

  There will be another illusion which we dub infant mortality. Infant galaxies may briefly lift themselves above the sky while undergoing vigorous star formation associated with their birth and then sink from sight leaving a shortage of older children (Section 11).

• (vi)

  Those disappeared children should nevertheless turn up in absorption as an excess of QSOALs at high z (Section 11).

Figure 2.

The distribution of central SBs of exponential galaxies in a typical HDF, in this case the WFPC2 I band. A connected pixel algorithm was used to identify images having ≥8 contiguous pixels (equivalent radius = 0.064 arcsec) above a detection threshold μ_L = 25.22 (Vega system) in the F814 filter. Visual morphological classification was performed on all images brighter than 28.0 mag. The galaxies classified as exponentials, based on the presence of discs and/or their light profiles on visual inspection, were fitted with exponential profiles and hence central SBs μ₀ (taken from Jones & Disney 1997). They are fairly sharply peaked at a SB, μ₀, 1–1.5 mag dimmer than the sky, exactly as predicted by visibility theory (see Section 6, equation 51). Since such frames contain galaxies from a wide range of redshifts, and thus Tolman dimming, it is very hard to understand such a sharp peak as anything but a profound selection effect operating in the observer's frame of reference.

Once one knows what to look for, phenomena (i) to (vi) are all plain to see in the observational literature.

Like anthropologists galaxy astronomers certainly have an ancestor problem. However, its solution may be naturally found within the SPDH scenario.

3 An Outline of Visibility Theory

The Visibility of galaxies is a subtle matter with a tangled history which, in the past, was complicated by the need to take into account photographic saturation, no longer generally necessary. Some of the papers are incomplete (Disney 1976; Disney & Phillipps 1983; van der Kruit 1987), some are misleading (McGaugh 1996) and some are wrong (Allen & Shu 1979).

All we attempt to do in this section, and in the simplest possible way, is justify the narrowness of the Visibility window A illustrated in Fig. 1 because it is so crucial to the main argument and because it comes as a surprise to most astronomers. To keep things simple, we consider only exponential galaxies and ignore absorption, Tolman dimming and cosmology for now. If we adopt de Vaucouleurs (1959) two-parameter intensity I(r) profiles for galaxies, that is,

(2)

(β = 1 for pure exponentials, β = 4 for giant ellipticals, with hybrid galaxies in between), then we can reach the main results analytically. It is easily shown that the apparent luminosity, integrated over the image out to angle Θ, is (β = 1 henceforth, until we reach Section 9)

(3)

so that as Θ → ∞ the total apparent luminosity

(4)

where I₀ is the central SB and α is the scalelength. Thus, from equation (4) we have

(5)

If the angular radius out to the outermost detectable isophote I_c is Θ_out ≡ Nα, which defines N, then the perceived angular diameter θ = 2Θ_out = 2Nα.

Thus,

(6)

From equation (2)

(7)

where μ_c and μ₀ are I_c and I₀ in magnitude, respectively. So, defining the vital SB contrast

(8)

(9)

Combining equation (4), (5) and (7) and recalling that l = dex(−0.4m) and I₀ = dex(−0.4μ₀)

(10)

or using

(11)

and replacing μ₀ by Δμ using equation (8)

(12)

It shows that angular size is a separable function of the absolute magnitude M and the SB contrast Δμ, as one might have expected.

To get into a sample or catalogue with a minimum angular size θ_c a galaxy must then be at a distance d^θ(in pc) such that

(13)

which could also be written as

(14)

which neatly separates the contrast, inside the curly brackets, the catalogue, inside the square brackets, and the luminosity factors in the expression for V^θ ∝ (d^θ)³. Note that the contrast dependence inside the curly brackets clearly has a maximum, which explains the shape of the smooth curve in Fig. 1.

Likewise, to find d^m and V^m we can calculate the apparent magnitude of the fraction ƒ of the galaxy light lying inside the outermost detectable isophote.ƒ is obtained simply by integrating equation (3) only to Θ_c, corresponding to I_c, in which case equation (6) becomes

(15)

where N = 0.92Δμ as always. So,

(16)

and

(17)

Thus, the maximum distance d^m to which the galaxy can be detected, without exceeding the catalogue limit m_c, is

(18)

where l_c is the apparent luminosity corresponding to m_c. In its cubed form equation (18) yields V^m the dashed line in Fig. 1 and 3 which reflects the monotonically falling nature of ƒ (see equation (15)) as the contrast Δμ, and hence N = 0.92Δμ, vanishes.

Figure 3.

The calculated Visibility window for exponential galaxies. The vertical scale shows the relative volumes within which galaxies with different surface brightness contrasts to the background (plotted horizontally) can be detected. Following the usual convention, this contrast Δμ, in mag, is plotted from the right-hand to left-hand side with high SB, that is, high-contrast galaxies to the left-hand side and low-SB galaxies to the right-hand side. The maximum heights of the two curves V^m (dashed) and V^θ (smooth) assume a sample for which , typical of all exponential galaxies, save those hundreds of pixels across. This is a typical Wigwam diagram for the visibility of galaxies of all kinds (see later). Since the vertical scale is arbitrary, the Wigwam diagram is valid irrespective of absolute luminosity, just as it is valid irrespective of the absolute survey depth (deepest isophotal level μ_c) because the horizontal axis is given only in contrast Δμ ≡ (μ_c − μ₀) where the latter is the central SB, measured in mag. To be detected, galaxies must lie inside the wigwam, the shaded area marked A, which we call the visibility window. Note how narrow it is, with a FWHM of 2.5 mag with a peak P at a contrast of 3.5 mag. Because the window is so narrow, redshift dimming will quickly move galaxies rightwards and out of sight into region C and even D. For future reference, note that even the V^θ (smooth) curve, by itself, has a FWHM of only 3 mag.

Having established the general shape of the smooth and dashed lines, V^θ and V^m in Fig. 1 and 3, respectively, what about their intersection point P which will vary with their relative heights? Dividing equation (14) by equation (18), we get

(19)

where we recall that N = 0.92Δμ. Fig. 1 and 3 show a plot of (d^θ)³ and (d^m)³ as a function of the SB contrast Δμ, that is, of N.

It is evident from the above equation that the relative heights of the two Visibilities can only be adjusted through the pure number

(20)

determined by the catalogue parameters m_c μ_c and θ_c. Now it turns out (next section) that m_c and μ_c are closely linked to one another by photon statistics, while θ_c is generally set by the telescope resolution. Thus, in practice Γ_c has a narrow range. Hence, the relative heights of our smooth and dashed lines, which define the Visibility window, cannot sensibly vary by much, and in particular its narrow aperture in contrast (<±1.5 mag) and its wigwam shape are more or less unavoidable as we shall see in the next section.

The net result of all the algebra is Fig. 3 which looks very much like the schematic Fig. 1 but now is anchored in numbers, in particular the very narrow FWHM (2.5 mag), and the position of the Visibility peak P 3.5 mag above contrast zero. The actual curves and consequent wigwam-shaped Visibility window were calculated from equation (14) and (18) assuming a value for Γ_c (equation (20)) of π typical of virtually all CCD surveys both in space and from the ground (Fig. 4; Section 4). By looking at Fig. 3 it is worth anticipating two points: (i) the contrast zero-point is locked to the absolute sky brightness being, for distant galaxies, about 5 mag dimmer than the sky in space and 6 mag dimmer than the (brighter) sky on the ground; and (ii) the 3.5 mag contrast at the peak refers only to the central brightest point of an exponential galaxy. Most of it huddles much closer to the sky (Fig. 9). [A more detailed description of Visibility theory can be found in Disney & Phillipps (1983), though the study lacks the vital arguments of the next section; also see Ellis, Perry & Sievers 1984).]

Figure 4.

The SB contrast Δμ(P), in mag, at the peak of the visibility wigwam (see Fig. 1 and 3) for exponential galaxies in surveys with different values of the pure number (see equation 20). Values were calculated numerically using the procedure discussed below equation (27). For the smooth and dashed curves to cross Δμ(P) must be >2.2 (see equation 27) and Γ_c must be <5 for the smallest galaxies. Thus, the practical range in Δμ(P) for nearly all surveys is narrow (3–4 mag). Thus, the visibility curve shown in Fig. 3 with and Δμ(P) = 3.5 is very typical for exponentials. Only very large galaxies with hundreds of pixels per diameter have Γ_c less than 2.2. For them the smooth V^θ visibility curve is all that applies. [NB: The algebra for ellipticals is slightly different but otherwise an identical procedure leads to a crossover at Δμ(P) = 10.4 mag for a Γ_c of 5 (see Fig. 7 and 9).]

4 Imprisoned by Light

The precise shape and location of the Visibility window for a given galaxy survey will depend on the relative heights of the smooth and dashed lines in Fig. 3 which in turn depend on the number n_p of photons gathered by the detector per arcsec². For instance, if the smooth line V^θ is lower than the dashed line V^m at all contrasts Δμ, then it is only the smooth line V^θ, with its FWHM and peak, which defines the Visibility window which then might be quite different from the Wigwam calculated in Fig. 3. It turned out that the relative heights were determined by the pure number Γ_c, but what determines Γ_c? This is an important step in the argument missing in Disney & Phillipps (1983).

Imagine a roughly circular source Θ arcsec in diameter where the detector has collected n_p photons arcsec⁻², mostly from the sky. The signal from the source

(21)

where = a level of signal from the source in photons collected per arcsec² averaged over the whole source area. I_sky is the foreground sky level and n_p is the total accumulated signal in photons arcsec⁻². Then from equation (20)

(22)

where I_c is the average level of signal within the outermost detectable contour in photons arcsec⁻². Thus, for the limiting case of the smallest sources detected in the catalogue, Θ → θ_c and

(23)

What are and I_c? They will be set by signal-to-noise ratio (S/N) considerations. For the whole source the signal is given by equation (21), while photon noise from the sky (assumed brighter than the source) is .

So, the S/N of the whole source or

(24)

I_c will likewise be set by the S/N in the outermost isophote which we will assume is qΘ wide (defining q). The signal in that outer isophote .

The noise in it , and so .

Thus,

(25)

Substituting equation (24) and (25) into equation (23) we have

(26)

where q, defined to be the width of the outer isophote as a fraction of the diameter, depends only on the sizes of the sources at the limit of detection.

From equation (26) it becomes clear that estimating Γ_c, and hence the location of the Visibility window, relies on picking appropriate values for the two limiting S/Ns, σ_m referring to the whole image and σ_θ to its outer isophote, which define the catalogue. If the noise is dominated by photon statistics, that is, it is binomial in nature, there is a rational way to select those ratios. They must be just high enough to avoid a significant number of false positives. As is well known in the binomial situation, the probability of a single false positive (i.e. single tail) is given in Table 1.

Table 1.

Probabilities of a false positive (single tail)

Thus, in a survey of a single CCD frame (∼10⁷ pixels) a formal choice of a discriminating σ_m = 5 should eliminate all but a handful of false positives. For a survey consisting of a fair number of CCD frames, S/N of 6–7 would be safer.

The case for σ_θ is different. The source has been selected; one needs only to be reasonably certain that the apparent outer isophote is real, that is, its probability as a false positive is less than, say, 5 or 10 per cent, in which case σ_θ ∼ 1.5 should suffice.

All that remains uncertain in equation (26) is q. Now for small extended sources containing only 10–20 optical pixels altogether, that is, galaxies at the limit of detectability as such in HDFs, q ≤ 1/3, that is, the diameter must be ≥3 times the outermost isophote width.

Thus, for HDFs . However, if we had the luxury of a catalogue comprised of large galaxies >100 pixels across, then q → 10⁻² and hence Γ_c → 0.5.

We can summarize the bounds on Γ_c as follows:

(27)

where very large galaxies having hundreds of resolution elements per diameter are on the left-hand side, and extremely small galaxies having three to five resolution elements per diameter are on the right-hand side.

Γ_c is so important because it determines the crossover point P, that is, Δμ(P) and thus the nature of the Visibility windows in diagrams such as Fig. 1 and 3. One can find Δμ(P) for a given Γ_c by equating V^θ to V^m and solving equation (19) for Δμ. There are no solutions for N < 2 (i.e. Δμ < 2/0.92 = 2.2) because then the dashed line always passes above the smooth line, and none for N > 4(Δμ > 4.5) because Γ_c must be <5 (see equation 27). In between there is a rather smooth, almost linear transition which passes through the (Γ_c, Δμ(P)) points (2.5, 3.0), (3.2, 3.5) and (4.9, 4.4) (see Fig. 4). Thus, Fig. 3 (π, 3.5) is completely typical of all but the largest galaxies with hundreds of detector pixels per diameter.

We can summarize a situation, which is very much simpler than it might have been, as follows. A search for exponential galaxies with a CCD detector will have a Visibility diagram much like Fig. 3, that is, a Wigwam diagram. Such galaxies will only be found in a narrow Visibility window centred at a contrast of between 3 and 4.5 mag, and the FWHM of that window will be 1 mag to the high-SB side, 1.5 mag to the low-SB side, making a total FWHM of only 2.5 mag in all. For very large (>100 detector pixels in diameter) galaxies, the situation is qualitatively different. Only the smooth curve in Fig. 3 then matters, in which case the Visibility window will be centred at a contrast of 2.2 mag with a FWHM of 3 mag, 2 mag on the high-SB side and 1 mag on the low-SB side (as assumed by Disney 1976, and reviewed by Impey & Bothun 1997).

Escaping. To emphasize how implacably we are imprisoned in our local cell of light, let us try to calculate a way out of it.

The number of galaxies of comoving density ϕ we will detect in a survey covering solid angle Ω will be

(28)

To move rightwards in Fig. 3, that is, to lower SB, it is the dashed line V^m which matters, that is, the galaxy's apparent magnitude must exceed the level of sky noise by some discriminating S/N factor σ.

For a circular source of diameter Θ arcsec in an observation containing n_p photons arcsec⁻² (equation 24)

Now Θ ∝ R/d (R = radius, d = distance). Thus,

(29)

or

(30)

Now

(31)

where T is the total survey time, t is the dwell time per frame and W is the solid angular area of the field of view of a single frame. This last equation is obvious but crucial because it argues that increasing the dwell time t in order to search for lower SB galaxies will not be so productive because it will, at the same time, reduce Ω and hence the volume that can be explored.

Thus,

(32)

However,

(33)

where D = telescope diameter, t = dwell time, Q = quantum efficiency of the system and Δλ = the bandwidth of the detector in Å, say, and we have assumed, for low-SB galaxies, that most of the collected photons come from the sky.

So, putting all together

(34)

a very important relation which neatly separates the galaxy properties (), the survey properties [] and the detector power {}.

From equation (34) we infer the following:

• (i)
  To acquire a certain number of galaxies of SB :

  

  (35)

  that is, for a drop in SB of 1 mag the dwell time must be increased by 3 mag, or bya factor of 16. Thus, to escape entirely out of our visibility window (FWHM = 2.5 mag) on the low-SB side, we would need to increase the dwell time by 2.5 × 3 mag or a factor of a thousand! We truly are imprisoned in our lighted cell. [Indeed the situation may be even worse than we have supposed. Thus far, we have assumed that the two galaxy parameters L and are independent, which may not be true. In so far as we can disentangle the two, which requires a sample selected by non-optical (e.g. 21 cm) means, the suggestion is that (Garcia-Appadoo et al. 2009; Chang et al. 2011). If that is true, then to see dim objects the dwell time t must increase not as but as ! (see also Section 10).
• (ii)
  The detector figure of merit {} is higher for CCDs than for Schmidt photographic plates {36 deg², Q ∼ 0.01}, provided the CCDs (Q ∼ 0.5) have >2000 pixels a side. The grasp of any survey, by equation (34), is

  

  (36)

  which means that 1-month-long CCD surveys with 4-m-class telescopes will be an order of magnitude less effective for finding low-SB galaxies than the combined Schmidt surveys covering the whole sky. However, if photon counting were the whole story, then SDSS ought to beat the Schmidt surveys by a factor of between 5 and 10, despite its very short dwell time ∼100 s. Unfortunately, very low SB galaxies can only be detected if they look apparently large (equation 24) when the unevenness of the sky background, not its photon statistics, becomes the predominant source of noise (Sabatini, Roberts & Davies 2003).
• (iii)

  One ray of hope is D³ in equation (36). Alas, large telescopes produce larger images which overfill the CCD detector pixels for diameters >2–3 m on the ground because optics cannot be made arbitrarily fast. In that case, W ∼ D⁻² requiring for a given . Telescope diameters of 100 m would be needed (see Section 6) to move one window width dimmer than we can see now. Only low-noise, high-quantum-efficiency detectors of far larger physical size than CCDs offer any prospect of escape.

• (iv)
  Thus far we have estimated everything in terms of Δμ ≡ μ_c − μ₀, where μ_c is so far numerically unspecified SB, presumably connected to the sky brightness μ_sky by S/N considerations. Recall that it is the SB of the outermost detectable contour in a galaxy where that contour width is, by definition, a fraction q of the galaxy's total angular diameter Θ. It follows from equation (25) that

  

  (37)
  For the smallest galaxies detectable in a survey, , so

  

  (38)

  where N′ ≈ Θ²n_p is the total number of photons collected, largely from the sky, from an area equivalent to the area of the whole source.

So far as space is concerned the high resolution of HST means that Θ is very small for faint galaxies (≤10⁻¹ arcsec) so that extremely long integrations (tens of orbits) are needed to achieve values of N′ as high as 10⁴ photons. It follows from equation (38) that

(39)

Thus in space μ_c is locked to the sky brightness. The position of the visibility window up there is not merely defined in terms of contrast but is in practice locked in absolute surface brightness terms too.

The same is true on the ground though the argument is slightly more subtle. According to equation (34), for a given

(40)

from which it might seem that a sufficiently long dwell time t might lead to the detection of arbitrarily dim galaxies. Not so because equation (40) ignores Tolman dimming. It is easy to show that such dimming modifies equation (40) to

If, because of the (1 + z)² term, you cannot afford z to rise above 0.2, say, then to find sufficient galaxies you must increase the area coverage Ω in equation (28) by taking a number of frames Ω/W = T/t. Putting in reasonable values for W (one CCD) and ϕ, it then transpires that to find a handful of low-SB L_* galaxies within z ≈ 0.2 would require

(41)

Now a pixel-matching (i.e. 2–3 M) telescope collects n_p ∼ 10 sky photons s⁻¹ arcsec⁻², so in an area of a 3 × 3 arcsec² galaxy n_p ∼ 10² sky photons s⁻¹ arcsec⁻² so that in a long campaign lasting T = 10⁷ s (i.e. t ≈ 10³ s) N′ = 10⁵ photons per galaxy area. So from equation (38)

(42)

which again is locked in absolute terms to the SB of the terrestrial sky (which at most wavelengths is at least 1 mag brighter than it is at HST).

The fundamental point is that the Visibility Wigwam diagrams are fixed not only in contrast terms but also in absolute surface brightness terms.

An alternative way to look at the matter is to investigate how the dimmest galaxy (SB ∼I_min) one can detect improves with the telescope diameter D. On the ground, because of the pixel-matching problem, I_min ∝ (DA)^−2/3, where A is the physical area of the detector. In space pixel matching is not an issue because the diffraction-limited angular resolution δθ ∝ D⁻¹. But then, for a fixed number P of pixels, the survey area Ω ∝ P(δθ)² ∝ PD⁻² and so again I_min ∝ D^−2/3P⁻¹. In other words, the telescope costs of escaping from the visibility window, be it in space or on the ground, become exorbitant. A factor of 10 improvement in I_min would imply an increase in the telescope diameter of 10^3/2 and hence in costs C of 10^3γ/2 if C ∝ D^γ. Since γ is usually reckoned to lie between 2.5 and 3, and certainly above 2, vast sums would be needed.

For all practical purposes then we are implacably imprisoned in our cell of light. Classes of low-SB galaxies unresolved into stars, which cannot already be seen in Schmidt surveys, are beyond the hope of discovery by optical means alone. It follows that large hidden populations of low-SB galaxies, both near and far, cannot be ruled out by optical observations alone. This is a much stronger statement than could have been made before and it relies on the arguments which led to equation (26).

5 How Galaxies Sink from Sight

The Visibility window depicted in Fig. 3 is immutable, mathematical and pinned in local coordinates because it shows the contrast to one's local sky, be it on the ground or in space. What we need to calculate next are the properties, in particular the sizes and intrinsic SBs, of the kinds of galaxies, seen at different redshifts, which will make it through that narrow window, particularly near its peak, taking into account the Tolman effects described above, which both dim a galaxy and increase its apparent size.

The (1 + z)⁻⁴ factor rapidly becomes very significant by comparison with the narrow FWHM (2.5 mag) of the visibility window. Even at z = 0.5 many of the most visible galaxies that are in region A (Fig. 1) at low redshift would be translated into region C and be far too dim to see. They have sunk. Their SB contrast now becomes

(43)

which implies that even galaxies at the peak of the visibility window at low redshift (where Δμ ∼ 3.5) will have zero contrast Δμ′, that is, will cross the dashed line (Fig. 1) and vanish entirely by a redshift of 1.2. To delineate that dashed line recall that the fraction of light detected above the outermost isophote μ_c is given by equation (15). Fig. 5 depicts ƒ(Δμ). More than 50 per cent of the light from a galaxy that would be at the peak nearby has already been lost at redshift 0.5, 82 per cent at redshift 1, and all by 1.2. These figures alone are enough to query the feasibility of trying to study galaxy evolution by using deep fields.

Figure 5.

The curve shows ƒ(Δμ), the fraction of an exponential galaxy's light seen above the outermost detectable isophote μ_c, plotted against the galaxy's contrast Δμ ≡ (μ_c − μ₀) in mag. It is calculated from equation (15) with Δμ modified using equation (43). Thus, a galaxy like the MW with an optimal contrast Δμ(P) ≈ 3.5 mag at z = 0 has an ƒ(Δμ) = 0.83 there, as shown by the tick mark. By the time it is removed to z = 0.5, ƒ(Δμ) has dropped to 0.4, by z = 1.0 to.06, and finally disappears altogether at z = 1.2 due to Tolman dimming. By contrast, the tick marks to the right-hand side of the line show an L^⋆ galaxy 9 mag higher in SB, a so-called ‘masquerade’. Even at z = 2 90 per cent of its light is still visible, and by z = 5 nearly 30 per cent is still left. It only sinks completely at a redshift of 7.

The galaxies that will appear instead at the peak of the window will be, as always, those with an apparent contrast Δμ′ of ∼ 3.5 mag. In other words, their intrinsic SBs will be given by (see equation 1)

(44)

or, at z = 0.5, 1.8 mag more brilliant than optimally visible galaxies nearby to us today at low redshift (μ₀ ∼ 21.5Vμ) and 3.0 mag more brilliant at z = 1. Indeed if one examines the Visibility window (Fig. 3), one sees, down at the FWHM, that the z = 1 galaxies now in the window must have emerged or surfaced from region B where they would be practically invisible at redshift zero.

How can redshifting, and hence dimming, a galaxy render it more visible? What the Visibility window illustrates are the relative Visibilities of galaxies with different SBs. Rare but high-visibility galaxies can be seen at great distances; common but low-visibility galaxies may rarely turn up close enough to us to be noticeable in surveys. If now we remove the local population to redshift 1, virtually all the previously prominent galaxies will sink below the sky, thanks to equation (43). Our high-SB specimen therefore has much less competition, and is correspondingly more prominent. In addition, it has gained through aberration, whereas removing it to z = 1 would normally render it too small to be seen as a galaxy (i.e. θ < θ_c) aberration may return it from the invisible region B into the visible window A.

Then in qualitative terms, removing any population of galaxies to higher redshifts will drastically alter their relative Visibilities, so that the previously prominent specimens sink partially, or wholly, out of sight, to be replaced there at the peak of the window by intrinsically more brilliant galaxies that were relatively inconspicuous at low z because of their small apparent sizes. It is time to make things quantitative.

Begin by calculating the apparent magnitude m(z) of galaxies that have peak Visibility (i.e. Δμ′ ≡ μ_c − μ₀) ≈ 3.5 at redshift z taking into account both Tolman dimming and cosmology. Apparent luminosity

(45)

where ƒ is the fraction of the light seen above the sky (equation (15)) and Δμ′ has been adjusted for redshift according to equation (43). Converting into magnitudes, with distances in Mpc,

(46)

d(z) is the proper comoving radial distance defined such that the comoving volume element out at z is corresponding to solid angle ΔΩ. [We do not need to employ the concepts of luminosity distance or angular size distance because we incorporate the (1 + z) factors directly into equations such as equation (45) and (46).]

Cosmology now enters only through the functional dependence of the comoving distance d(z) on z. It can be a complicated function depending, as it may, on the various model parameters , and so on. Here we use the empty-universe approximation

(47)

because it is simple and closely approximates the currently fashionable ΛCDM model. Between 0.1 < z < 10 the discrepancy is a maximum of 12 per cent (at z = 1) and for most of the range it is much less [as can easily be checked using Ned Wright's very useful online cosmology calculator (Wright 2006)]. Given uncertainties as to which is the correct model, and K corrections, dust and evolution, this approximation is more than satisfactory.

Substituting equation (47) into equation (46) we have

(48)

Likewise to find θ′′(z) we use equation (47) for d(Mpc) and equation (12) becomes

(49)

where the (1 + z) term incorporates the aberration.

Fig. 6 employs the last two equations to investigate the appearances of two galaxies at different redshifts. The first galaxy is a MW and the second a hypothetical galaxy of the same intrinsic luminosity but with a SB no less than 9 mag (4000 times) higher. Notice first how quickly the MW sinks below the sky. By redshift one-half 56 per cent of its light has gone. By z = 0.9 the aberration cannot compensate for the sinking of its outer isophotes, and by z = 1.2 it has sunk completely. One cannot expect to see healthy, that is, more or less complete MWs much beyond a redshift of 0.5.

Figure 6.

The appearance of exponential galaxies as a function of their redshift z. The apparent magnitudes m(z) (left-hand panel) and angular sizes θ(z) (diameter in arcsec) (right-hand panel) are shown for two objects of very different intrinsic SBs: a MW labelled Mw and a masquerade labelled Mq which is a hypothetical galaxy of the same luminosity but which is 9 mag (4000 times) higher in SB, that is, 9 mag more brilliant. The abscissa is redshift (plotted logarithmically, as are the other quantities). Follow first the magnitude m(z) for the MW in the left-hand panel. Because of redshift dimming, and shrinkage of its outer detectable isophotes against the sky, it rises more and more rapidly until, by z = 1.2, it vanishes even from the deepest HDFs [magnitude limits ≈30 depending on colour]. Follow secondly the angular size θ(z) (right-hand panel) of the MW. It falls rapidly at first then slows as a number of factors including aberration kick-in, then falls again catastrophically as z → 1 when it loses contrast with the sky. It sinks completely out of sight when z → 1.2 (see equation 43). Now look at the masquerade. Outer-isophote loss is negligible; thus, its m(z) (left-hand panel) increases more gradually with z so that its m(z) ∼ 29 at a redshift ∼7. It is still visible to HST out there. Its angular size θ(z) (right-hand panel), which is barely an arcsec at low z, hardly changes with redshift, due to the (1 + z) aberration term so it is distinguishable to HST as a galaxy even at z ∼ 7.

Now look at the hypothetical masquerade which would be only 330 pc in diameter. Being (4000)^−1/2 smaller than the MW, its angular diameter at z ∼ 0.1 would only be 0.2 arcsec, so unless it was close (<50 Mpc) it would, from the ground, masquerade as a star, hence its name. However, by a redshift of 4 aberration is kicking in, while all the lower SB galaxies would have sunk, or be sinking out of sight, so that by z ∼ 7 it would be the most visible L_* galaxy in sight because its apparent SB would be ∼21.5μ, that is, 3.5 mag brighter than the SB limit μ_c (see Section 6). Its angular size would be ∼0.6 arcsec making it distinguishable to HST as non-stellar, while its magnitude would be ∼29.5 (Vega), and if z is increased above 7 so would its angular size, which would now be dominated by aberration. As we shall see later, it looks very like the z ∼ 7 galaxies being found with WFC3 on board HST. All this supposes it is largely transparent. See Section 8 for opacity.

We can summarize this section as follows. The sheer size of Tolman dimming at the high redshifts accessible with HST makes it almost certain that the population of galaxies we see out there is very different from, and may not even be related to, our conspicuous neighbours today. The narrowness of the Visibility window (Fig. 3) compared to Tolman dimming is such that, without dramatic and fortuitous amounts of evolution (up to and beyond 9 mag), our neighbours will fade dramatically beyond redshift 0.5 and sink altogether below our local sky at z ∼ 1.2. Whatever be the case nearby, the distant (z > 1) universe is almost certainly dominated by sunken galaxies that are invisible to us, sunken galaxies that would surely alter our ideas on the star formation history of the cosmos and its re-ionization, could we but detect them. Those who aim to decode these matters by looking at the high-redshift galaxies now visible with HST, even to decode galaxy evolution beyond redshift one-half, must first convince themselves that they are looking at our ancestors and not at a very different, higher SB population, the one that is most visible to us at that redshift, but which is inconspicuous nearby.

6 Why High-Redshift Galaxies Look Small

Technical developments, and in particular the fitting of the new WFC3 camera to HST, make it almost trivial to find galaxies out to redshift 7, and perhaps higher. Its near-infrared (near-IR) sensitivity out to 1.7 μm, its resolution there (∼0.1 arcsec) and its field of view (4.8 arcmin²) conspire to make it ∼30 times faster for finding such objects than previous space cameras like NICMOS. Such galaxies are observed in their rest-frame UV (0.1–0.2 μm) where prominent breaks in their spectra at Lyα and at the Lyman limit make for fairly unambiguous selection and photometric redshift measurements (e.g. Bouwens et al. 2010; Bunker et al. 2010; McLure et al. 2010; Oesch et al. 2010).

If, as we are supposing, SB selection through our narrow Visibility window dominates the appearance of galaxies out there, one can make several strong predictions:

• (i)

  All such high-z galaxies (indeed all exponential galaxies in the deep frames) should have a narrow range of apparent SB (∼3–4 mag).

• (ii)

  That range should be centred 3–4 mag higher than the limiting isophotal value for the observational data in question.

• (iii)

  For the high-z galaxies, Tolman dimming then implies that their intrinsic SBs must be very high, ∼9 mag higher than prominent galaxies nearby. This in turn implies that they must be physically very small, otherwise they would be superluminous.

• (iv)
  The apparent scalelength for such exponential galaxies should appear to decrease with redshift in a well-determined way, that is,

  

  (50)
• (iv)

  Either such supercompact galaxies have detectable descendants nearby, or there must be some plausible mechanism for explaining their absence (see the next section).

Let us now compare these predictions with observations:

• (i)

  The predicted constancy and scatter in SB is a direct consequence of the previous three sections and hardly needs further discussion.

• (ii)
  Where do we expect the central peak of the distribution of the SBs of galaxies in a Hubble Deep Field (HDF) window to lie? At peak we know (Section 3) Δμ(P) = μ_c − μ₀ = 3.5. For small galaxies in HDFs, it follows from equation (39)

  

  (51)

  Fig. 2 shows the distribution of SBs in one of the HDFs. As can been seen, it fits the prediction of visibility theory very well because the sky brightness in the I band at HST is 22.5μ (Vega).

  Given Tolman dimming, evolution and dust absorption, all of which could be very large in these circumstances, especially in the rest-frame UV, it is very hard to understand Fig. 2 as other than some kind of profound selection effect operating in the observer's frame of reference, as the SPDH suggests it is.

  Oesch et al. (2010) studied the structure of 16 z ∼ 7 galaxies in this sample and reported ‘with an average intrinsic size 0.7 ± 0.3 kpc these galaxies are found to be extremely compact, having an observed SB μ_J ≈ 26 mag arcsec⁻²’. Their fig. 2 shows the half-light radii tracking absolute magnitude so as to maintain that SB constancy, and in their fig. 5 they extend the sample to objects in the range z = 2 to 8, finding that the measured (as opposed to the corrected) UV SB which they interpret as a star formation rate remains relatively constant for the whole redshift range from z ∼ 7 to z ∼ 4 for galaxies with luminosities in the range 0.3–1L_⋆.

• (iii)
  Size evolution. For galaxies to be seen in the visibility window (equation 44) demands that their SB

  

  (52)

Now the physical scalelength . Thus, for a given L and I_c

(53)

therefore,

(54)

and

(55)

Thus, the apparent scalelength will be, thanks to aberration, a factor of (1 + z) larger, in which case we predict

(56)

Note that the above argument is morphology-independent, and applies as well to ellipticals as to exponentials.

Oesch et al. (2010) compare their measured r_1/2 values with (1 + z)^−m over the range z = 2 to 8 and report m = 1.2 ± 0.17 for luminous, 0.3–1L_{⋆, z = 3}, and m = 1.32 ± 0.52 for less luminous, 0.12–0.3L_{⋆, z = 3}, galaxies. ‘This is in agreement with previous estimates where the sizes were found to scale roughly according to (1 + z)⁻¹’ (Bouwens et al. 2004, 1996).

Earlier Buitrago et al. (2008) measured 80 giant galaxies (M > 10¹¹ M_⊙) in the range 1.7 < z < 3 using NICMOS and split the sample into discs and spheroids. Discs are 2.6 ± 0.3 smaller than today and spheroids 4.3 ± 0.7 smaller. The implied stellar densities in the past at ≈2 × 10¹⁰ M_⊙ kpc⁻³ ‘are very high and as high as globular clusters today’. The disc measurements too are obviously consistent with r_1/2 ≈ (1 + z)⁻¹.

The same fall-off in physical size with redshift proceeds all the way from z = 0 to 7 with R_1/2 ≈ (1 + z)⁻¹. For instance, Ryan et al. (2012) have recently used WFC3 in a 15-colour search to isolate a sample of early-type galaxies, this time in the interval z = 1.6 ± 0.6, and compared their sizes with a very large sample of equivalent SDSS galaxies at z ∼ 0.2. Again they parametrize the size decline as R_eff ∝ (1 + z)^−m and find m is mass-dependent this time, and

(57)

yielding m ∼ 1 for massive galaxies, and a statistical decline over all objects of a factor of 4 between redshifts 0 and 1.6.

The decline in galaxy size with redshift is the most remarkable and consistent result in all the HDF observations. It is predicted, indeed demanded by the SPD hypothesis in which Tolman dimming brings successively more compact galaxies to light at higher redshifts, while sinking entirely out of sight their less compact companions. [NB: As an aside, the confirmation of the prediction that angular size ∝ (1 + z)⁻¹ be taken, a la Tolman, as rather direct evidence, so far lacking, that the Universe is expanding. It would rely on the assumption that intrinsic SB does not change much with redshift, as suggested by the archaeology of nearby galaxies.]

7 Where have the Descendants Gone?

What happened to the spectacularly high SB galaxies we see back at redshift 7? Have they evolved away either by mergers or passive dimming, or are their descendants lurking around us today? We shall argue that their direct descendants could well be present in our neighbourhood but would have passed unnoticed because they are extremely inconspicuous, and for other three different reasons. First, their compact physical sizes translate into angular sizes so small that their angular size visibility V^θ will be down on normal galaxies by a factor of 60 cubed. Secondly, the dust grains in such compact objects would be on average 60 times closer to neighbouring stars than they would be in a MW galaxy today, and therefore be 3600 times more effective as absorbers. Very little of their optical light would therefore escape making even the nearest of them exceedingly faint. Finally, in comoving terms, they appear to be pretty rare which implies that the nearest of them would be far enough away to make them, in terms of angular size, barely distinguishable from stars. Take these arguments one by one:

• (i)

  Visibility. For compact objects it is angular size visibility V^θ which counts. According to equation (25) the most visible objects at redshift 7 must have a SB of 10 log (1 + 7) = 9 mag higher, and therefore a diameter 4.5 mag, or 60 times smaller than the galaxies in our vicinity. Thus, for a given luminosity their angular sizes would be 60 times smaller, and their Visibilities 60⁻³ ∼ 10⁻⁵ less. They will be extremely inconspicuous.

• (ii)

  Internal absorption. Large disc galaxies typically lose half their light to internal dust absorption (Disney, Davies & Phillipps 1989; Soifer, Helou & Werner 2008), but compact galaxies ought to lose vastly more. One will see into a disc galaxy ∼one mean-free-path λ, where λ = 1/nσ, where n is the particle density and σ is the particle cross-section for absorption. Shrinking the disc radially by a factor of 60 will increase n by ∼60², so the physical depth from which one could detect light would, crudely speaking, decrease by the same factor, leading to a loss of apparent luminosity ∼60² ∼ 9 mag. In other words, once a disc becomes optically thick, compacting it further cannot increase the apparent SB, and its apparent optical luminosity will decrease with its area.

• (iii)

  Rarity and apparent angular size. Mclure et al. (2010) fit a Schecter luminosity function to the faint end of the high-z sample (where the statistics are ‘better’) and arrive at a comoving density ϕ^⋆ = 7 × 10⁻⁴ Mpc⁻³ mag⁻¹ which is more than an order of magnitude below the local value. Ignoring clustering, the expected distance to the nearest one from us ought to be Mpc (distance modulus ∼ 29) and as the physical size ∼ 20 kpc/60 ∼ 300 pc, the angular size of the nearest one would be ∼10 arcsec, whilst most would look stellar. They would also be very faint, even the nearest z ∼ 7 descendant to us would have a B magnitude of [M_* + (m − M) + 9 mag (dust)] ∼ −20 + 29 + 9 ∼ 18 mag.

Attempts have been made to find ultracompact galaxies by setting spectroscopic fibres on bright star-like objects superposed on clusters (Drinkwater & Gregg 1998; Phillipps et al. 2005). There was some limited success with the discovery of ultracompact dwarf galaxies. However, we would expect that most, and certainly most of the bolometrically luminous ones, will be choked with their own smoke (dust). They might, however, turn up in dedicated searches in the far-IR.

The above discussion is highly simplistic, but the conclusions are so strong that one hardly needs to qualify them further. Even if they survive intact around us today, the descendants of redshift 7 exponential galaxies would pass unnoticed without a dedicated and extensive search in the far-IR.

8 Absorption and Visibility

The last section suggests that dust can have a dramatic effect on the Visibility of high-SB galaxies. There turns out to be a new Visibility V^τ due to such opacity but happily, and by coincidence, it lies exactly on top of V^θ for dust-to-gas ratios typical of local disc galaxies, and is independent of luminosity, that is, high SB and opacity go together. This coincidence has an important practical consequence. Faint galaxy catalogues take advantage of the rapid decline in Galactic stars beyond magnitude ∼25, so that fainter objects can be assumed to be virtually all galaxies, and the need for an angular size limit θ_c can be dropped. Thus, V^θ, the left-hand margin of the Visibility Wigwam diagram, becomes irrelevant, and so such catalogues could in principle detect extremely high SB objects. However, it turns out that they cannot because such high-SB discs will be opaque, and V^τ overlies θ_c almost exactly, supplying a left-hand margin for the wigwam instead. (This will not be true for dustless galaxies; see Section 9.)

Consider the intensity I of light emerging vertically from a mixed slab of stars and interstellar medium (ISM), a toy model of a galactic disc. In an obvious notation:

where ɛ_* and κ are the stellar emissivity and absorption coefficient per unit mass.

Then I(τ) = (ɛ_*/κ) [1 − exp (−τ)].

For an optically-thick slab, I ≈ (ɛ_*/κ), that is, the maximum SB depends not on the size but only on the intrinsic properties of the slab, that is, its opacity and emissivity per unit mass. Thus, dwarfs and giants will go opaque at the same SB. It is well known that the ISM goes optically thick at an H i column density ∼10²¹ H i atoms cm⁻², which corresponds to a disc surface density of ∼10 M_⊙ pc⁻² of gas and dust. If the embedded mass of stars is (1/ƒ_ISM) greater, then the SB of such an opaque slab will be ≈10/ƒ_ISMQ L_⊙ pc⁻², where Q is the typical mass-to-light ratio for disc stars in solar units. Since 1 L_⊙ pc⁻² in B corresponds to 27Bμ, then low-gas discs (ƒ_sim ≈ 0.1 and Q ∼ 2) will go optically thick at a disc surface density ∼ 50 M_⊙ pc⁻² and a local SB of 23–24Bμ; gassier discs (ƒ_ISM ≈ 1) at SBs 2–3 mag dimmer.

How does this compare with galaxies within the Visibility Wigwam diagram? According to equation (42):

μ(P) = μ_c − 3.5 ≈ μ_sky + 5.8 − 3.5 ≈ μ_sky + 2.3 ∼ 22.5 + 2.3 ∼ 24.8Bμ.

Thus, the inner parts of discs within the Wigwam diagram are optically thick already. This checks with far-IR observations which reveal that half of all the starlight in such galaxies is reprocessed on dust (e.g. Soifer et al. 2008).

This is a deliberately crude calculation but it demonstrates that, irrespective of mass, exponential galaxies containing normal amounts of gas and dust will not be found at SBs very much higher than already existing inside the Visibility Wigwam diagram because, no matter what their surface densities, they will be optically thick. Much more sophisticated discussions of disc opacity and SB can be found in Disney et al. (1989) and Davies & Burstein (1995). Note that at high redshifts we observe the rest-frame UV where dust absorption may be significantly higher.

How does opacity affect Visibility? If a disc becomes optically thick at some intensity, say, I(crit), then its optical luminosity L(opt) will decline as I(crit)R² as the radius is compressed further. Thus, the maximum distance d(max) to which it can be seen will decline as √L(opt) ∼ R and thus its Visibility V^τ ∼ R³. Thus, on a Visibility diagram, where (by definition) we hold L(bol) fixed, and , which is an identical scaling to V^θ. Both Visibilities decline as angular size cubed. Thus, the V^τ and V^θ lines for dusty galaxies lie more or less on top of one another. Even faint object catalogues will have Wigwam diagrams for dusty galaxies.

In summary, the Visibility of galaxies can be heavily affected by their dust contents, which is hardly surprising. It is not so obvious that high SB and high opacity generally go together, but they do, so the inner parts of discs become opaque before the outer, and galaxies with ‘normal’, that is, local amounts of dust and gas, no matter how compressed they are, can never attain SBs much higher than the MW or the Andromeda Galaxy. Since the V^θ and V^τ lines coincide, we thus cannot imagine that region B on any visibility diagram will be heavily populated, except by dust-free galaxies. Increasing redshift will thus not drag into the visibility window many dusty exponentials to replace those dimming into obscurity towards region C (see Fig. 3). A strong prediction of the SPDH is that nearly all very high z galaxies will appear relatively dust-free. That does not mean that they are actually dust-free because such light as does emerge mostly comes from the dust-free regions within them such as their upper surfaces (e.g. Disney et al. 1989).

9 How Elliptical Galaxies Sink

For simplicity, we have so far concentrated exclusively on exponentials. We now turn to giant ellipticals which have a softer light distribution:

(58)

though that also implies a small amount of luminosity in a sharp pip in the core. At first sight, it will look as if ellipticals have very different visibility functions from exponentials, reaching their peak angular size visibility V^θ at a central SB no less than 7 mag brighter than exponentials (Disney 1976). However, that turns out to be an artefact of the parametrization, and if a more physical SB measure μ_1/2 (the SB at half-light radius) is introduced, then one finds (e.g. Davies 1993) that the elliptical and exponential visibilities lie almost on top of one another, but with the FWHM of the ellipticals being somewhat broader (4.2 mag as opposed to 2.5).

The algebra is much the same with the following modifications:

Equation (3):

(59)

In equation 8: {} → {(0.92)⁴exp (−0.92Δμ/2)}, where the maximum of {} occurs at Δμ = 4/0.46 = 8.7 mag.

Also in equation (8):

(60)

d^m is identical but with ƒ(Δμ) (equation 15) replaced by ƒ_E(Δμ) where

(61)

with

(62)

To plot the two Visibilities V^θ and V^m together, we need first to adopt a value for Γ_c, and as we shall be most interested in apparently small distant galaxies we adopt a value at the upper limit of the Γ_c range of Γ_c = 5 (equation 21). That leads to a crossover point P at a contrast Δμ(P) = 10.4 mag and thus to the Visibility diagram shown in Fig. 7.

Figure 7.

Visibility as a function of surface brightness contrast Δμ ≡ (μ_c − μ₀) for giant elliptical galaxies. The magnitude-limited Visibility V^m is normalized to 1 at very high contrast, that is, high SB towards the left-hand side because all of the galaxy's light shows above the sky then. The angular size Visibility V^θ is the humped function. To the left-hand side its apparent size shrinks as, for a given luminosity, a galaxy must physically shrink as its SB increases. To the right-hand side, it shrinks as more and more of its outer light is lost below the sky. The relative heights of V^m and V^θ are determined by the pure number which must have a value close to 5 for all but very nearby galaxies hundreds of pixels in diameter (Section 4). The actual Visibility is the lower envelope of both curves, that is, the shaded area marked A with a peak at P where the two Visibilities intersect to make a wigwam. Only galaxies within region A are detected in a survey with limits (l_c, μ_c, θ_c) in the combination Γ_c as above. Galaxies in region B appear too small to be distinguishable as such, those in region C too faint, and those in region D both too small and too faint. The FWHM of the Visibility window A is 4.2 mag, as opposed to the narrower (2.5 mag) window for exponentials (see Fig. 3). The location of the elliptical peak P is at a contrast of Δμ(P) = 10.4 mag, far higher than the exponential peak at Δμ(P) = 3.5 because, by comparison, elliptical light distributions rise towards a sharper peak towards the core. However, that peak contains very little light and so a fairer measure of the SB of a galaxy is the SB at half-light = μ_1/2, and a fairer comparison of the Visibilities of the different kinds of galaxies is made using their μ_1/2 to measure a contrast Δμ_1/2 ≡ (μ_c − μ_1/2) (see Fig. 9). [NB: Very large galaxies, hundreds of pixels across, have lower Γ_c, and so their V^θ falls below their V_m, and hence their Visibility is defined entirely by their V^θ (see Disney 1976), and they do not have a Wigwam diagram.]

Once again notice the two Visibilities intersect (at P) and the result can only be another discontinuous, sharply peaked Wigwam Visibility function limited on the right-hand side by V^m and on the left-hand side by V^θ. The FWHM of the combined visibility curve enclosing the Visibility window A is 4.2 mag, 1.7 mag on the low-SB side, and 2.5 mag on the high-SB side. The normalization shown is such as to make V^m → 1 as the contrast Δμ → ∞. The Visibility shape, and some of the subsequent consequences, is different from Phillipps et al. (1990) because there no cognizance was taken of Γ_c, and so there was no unambiguous way to adjust the relative heights of V^θ and V^m.

Exactly as for exponentials, Tolman dimming and cosmology can be added to yield the apparent magnitude m_E(z) and angular size (diameter) as (see equation 44):

(63)

where Δμ′ = Δμ − 10 log (1 + z), while (equation 47)

(64)

[The dex(−5) scalefactor accounts for the difference between the 10 pc in (m − M) and the Mpc used in H₀.]

Fig. 8 shows how the apparent magnitudes and sizes of giant ellipticals fade with redshift. It is an analogue to Fig. 6 but for ellipticals, and like that diagram it too compares a galaxy of ‘normal’, that is, ‘local’ SB with a masquerade, that is to say one which has a SB 9 mag more brilliant so as to give a maximum visibility at z = 7. It is interesting to compare Fig. 8 and 6.

Figure 8.

The apparent magnitude m(z) (left-hand panel) and angular size (right-hand panel) (diameter in arcsec) of an L* elliptical with normal SB [μ₀ ∼ 15μ, μ_1/2 ∼ 23μ in the V band] and a masquerade L* elliptical with a SB 9 mag more brilliant, as a function of redshift z. The normal giant elliptical crosses the HDF line (∼30 mag) at z ∼ 2 but the masquerade one reaches z ∼ 5–6 before it is extinguished, because it loses very little outer light below the sky. For angular sizes, aberration kicks in so the masquerade reaches a minimum angular size of ∼2 arcsec at z ∼ 1 and it apparently grows gradually thereafter. [The odd aberration effect seen on θ(z) for the normal giant elliptical is of no consequence, because by then its m(z) has long since fallen below the sky.] Compare with Fig. 6 for exponentials (see text). For a given luminosity ellipticals can be seen significantly farther away than exponentials, which might give the false impression that they formed earlier.

Figure 9.

The visibilities of both extreme morphologies of galaxies as a function of their contrast Δμ with the sky, this time expressed in terms of their half-light SBs μ_1/2 (exponentials: dotted; ellipticals: solid). Now Δμ → Δμ_1/2 ≡ (μ_c − μ_1/2)). In this more physically representative measure, the two visibility wigwams fall almost on top of one another. They continue to the right-hand side of Δμ_1/2 = 0 simply because half of their light lies below their μ_1/2. What is most remarkable is how perilously close both wigwams huddle to the sky. Galaxies that are marginally dimmer, for instance, dwarfs, or higher z objects, quickly disappear altogether. That suggests a natural explanation for downsizing as a selection effect which has nothing to do with evolution. [And what about the so-called missing dwarfs in the CDM paradigm?]

First notice the gentler slopes of m(z) for the ellipticals at high z. They do not fade away so quickly with z because they have a light distribution with a steep core, and this reflects in a difference between ƒ(Δμ) (equation 15) and ƒ_E(Δμ) (equation 61). So while a MW dies completely, both in size and magnitude, at z = 1.2, m(z) for a giant elliptical does not reach the typical HDF limit of ∼29 until z ∼ 2. This extra Visibility range in V^m allows the aberration to kick in more decisively ensuring the much more upturned θ_E(z) curves at high z. Thus, a normal giant elliptical would appear large enough to be seen as a galaxy with HST out to a redshift of 4–5 if, by z = 2, it had not already faded in magnitude below 29.

Comparing Fig. 8(a) with Fig. 8(b), the masquerade elliptical (i.e. the one with a SB 9 mag more brilliant than a normal giant elliptical) is too compact to lose much of its outer skirt of light below the sky so that it is only dimmed by distance, cosmology and the Tolman (1 + z)⁻² effect. The magnitude difference between normal and masquerade ellipticals is entirely due to the aforesaid skirt effect.

In summary, normal giant ellipticals ought to be seen out to higher z (2 with HST) than discs of the same luminosity (z = 1.2). This is the opposite conclusion to that reached by Phillipps et al. (1990) and is accounted for purely by Γ_c (Section 4). This could give the misleading impression that giant ellipticals formed before other galaxies.

The one dramatic difference between giant ellipticals and exponentials is the position of Δμ(P) at the point where the Visibility reaches a maximum, that is, at the centre of the Visibility window A; Δμ(P) = 10.4 for ellipticals there, whereas for exponentials it is equal to 3.5. But this is an artefact of the parametrization. Ellipticals have a pip of light in their core which yields a correspondingly bright μ₀ which, however, is representative of very little luminosity in total. Better therefore to use Δμ_1/2 where

(65)

and μ_1/2 is the SB the half-light radius.

It is trivial to show

(66)

Thus,

(67)

for exponentials, while for ellipticals

(68)

Thus,

(69)

So in terms of a more representative measure of SB, the half-light SB μ_1/2, the visibilities of both breeds of galaxies lie almost exactly on top of one another; see Fig. 9. It is a remarkable diagram that ought to give galaxy astronomers food for thought and we discuss it further in Section 10. For now, at least, it argues that both extreme light distributions lie so close to one another, as far as Visibility is concerned, that so should all the intermediate types.

What about faint galaxy catalogues that search for red galaxies using near-IR arrays? They can dispense with an angular size limitation because there will be no Galactic stars at faint enough magnitudes. Thus, they will have no angular size Visibility, and for early-type galaxies with no dust, no absorption Visibility either. Thus, the high-SB (left-hand panel) envelope to the Visibility Wigwam diagram will be missing and faint near-IR catalogues should be capable of finding very high SB, compact, early-type galaxies which will have no detectable counterparts nearby. Why not? Because catalogues of nearby galaxies will have an angular size criterion d^θ, hence a V^θ, and hence a Visibility Wigwam as shown in Fig. 8 and 9.

Such hypothetical galaxies, which are a direct prediction of visibility theory and the SPDH, have recently been found and labelled ‘red nuggets’ (Cimatti et al. 2004; Damjanova et al. 2009). They have elliptical profiles, are found in the redshift range 1 ≤ z ≤ 2 (beyond that their rest-frame UV would be too faint to detect), and are luminous (∼L*) but have radial sizes two to four times smaller than giant ellipticals nearby (median R_E ∼ 4 kpc). The median density of red nuggets in 1.1 ≤ z ≤ 2 is 2 × 10⁻⁵ Mpc⁻³, whereas in the relatively nearby SDSS sample it is ≤3 × 10⁻⁸ Mpc⁻³. When observed they have normal, that is, nearby SBs μ_1/2 of 19–20 mag in Gunn r, but when corrected for Tolman dimming these are raised 2–4 mag higher. As we shall argue (Section 11) such a large departure cannot be explained by luminosity evolution, nor can anyone explain their growth in size by a factor of 3 in radius between z = 1.5 and today (e.g. Damjanova 2009). Whether they are pure ellipticals, or in some cases the spheroidal components of larger discs (van der Wel et al. 2011), they have exactly the properties predicted by the SPDH which requires neither growth in size nor excessive luminosity evolution. As such they ought to have counterparts relatively nearby but finding them will be difficult because of their small angular sizes. Ignoring clustering the nearest objects ought to lie at a distance of ∼ Mpc, have an angular size of ∼15 arcsec and an apparent magnitude ∼−20 + 1.5 (evol) + 32 ∼ 14 mag. However, if they are heavily clustered like ellipticals and extremely red objects (McCarthy 2004), the nearest ones could be five times farther away, making them challenging objects to find nearby from the ground. Their colours should be much bluer than background early types because one will be looking at a flatter part of the spectral energy distribution. The recent detection of red nuggets shows that galaxies do indeed exist which are extremely difficult or impossible to find nearby, the key assumption of the SPDH.

10 Downsizing: a Different Explanation

Another remarkable phenomenon among apparently faint galaxies is downsizing (Cowie, Songhaila & Hu 1996; Heavens et al. 2004; Thomas et al. 2005; Perez-Gonzalez et al. 2007a). In purely observational terms, it is the appearance of lower luminosity and dwarf galaxies in apparent magnitude selected samples only at lower redshifts (z ≤ 0.5), and when they do appear they have comparatively blue colours and strong emission lines. If interpreted in terms of the ESD hypothesis, it requires giant galaxies to form their stars first and dwarfs last, the very reverse of expectations based on hierarchical galaxy formation, the fashionable cosmogonic hypothesis.

However, downsizing, as we shall next argue, can also be an entirely natural outcome of the alternative SPD scenario, where it has no implications for the ordering of galaxy evolution. The only assumption required is that lower luminosity galaxies have, in a statistical sense, dimmer intrinsic SBs. Because of obvious selection effects such an assumption is not easy to demonstrate unequivocally, but most observations, as well as common sense, speak in its favour. For instance, observations of H i-selected samples, which are unaffected by optical selection effects, certainly show such a correlation (Garcia-Appadoo et al. 2009; Chang et al. 2012) with

(70)

implying global stellar densities that are independent of luminosity L. The 1/3 then arises because the path-length through a more luminous galaxy scales in that proportion.

The rationale for downsizing under the SPDH is immediately apparent in Fig. 9. What we see there is that visible galaxies of all types ought to huddle perilously close to the limit set by the local sky, and observations going back to Holmberg (1965), Freeman (1970), Disney (1976) and Davies et al. (1994) testify that this is indeed the case observationally. If now dwarf galaxies carry the further handicap of a lower intrinsic SB, then they will naturally be the first to sink below the sky. A smaller amount of redshift dimming should suffice to sink dwarfs entirely out of sight, whereas giants will still be visible farther out; observationally speaking, that is downsizing.

To see how potent this kind of downsizing is, we calculate how rapidly the Visibility of an exponential galaxy (most dwarfs are exponentials) will fall if we lower its SB according to equation (69) and then redshift it. If we start with an exponential of optimum SB (i.e. at the peak of the Visibility window with Δμ = 3.5 mag), lowering its SB will run it towards the right-hand side in Fig. 1, 3 and 9, that is, towards the boundary determined by d^m and hence V^m. In those V^m expressions, the intrinsic SB enters explicitly only through ƒ(Δμ), or if redshift dimming is allowed for in addition then through ƒ(Δμ′) where as usual Δμ′ = Δμ − 10 log (1 + z). We can thus estimate what happens to the visibility of a lower luminosity galaxy purely as a result of its extra SB dimming, by comparison with an L_* galaxy of normal, that is, local SB. Both galaxies will of course fade with redshift but the dimmer dwarf will fade by more as it is quickly swallowed up by the sky. The situation is best summarized in the following Table 2 for three galaxies, an L_*, a 0.1L_* and a 0.01L_*, all obeying equation (69) and with the L_* having optimum SB at the apex of the Visibility window at z = 0.

Table 2.

Dimming and downsizing

Recall that ƒ is the fraction of a galaxy's light still visible above the sky, and that ƒ^3/2 is proportional to the Visibility. The giant elliptical does not sink completely until z ≈ 1.2, the low-luminosity galaxy has virtually gone by z = 0.7 and the dwarf by 0.4.

The other important aspect of downsizing as observed is that the low-luminosity and dwarf galaxies are bluer and have stronger emission lines at a given redshift. However, that could be explained by a natural SB selection effect. Galaxies approaching total immersion would be far more prominent if they were undergoing temporary bursts of star formation, with a consequent increase in SB. Take the dwarf in the table at z = 0.4. If its SB were to increase by 0.35 mag, we can see that its Visibility would increase by a factor of (0.04/0.005) or 8, and if by 0.7 mag then by over 20. (The ‘half-baked’ appearances of many HDF galaxies could be due to the extra star formation required to lift such galaxies above the sky.)

In summary, the SPD hypothesis has a natural explanation for the downsizing observations in terms of Visibility. It assumes only that intrinsic SBs generally fall with luminosity, and implies nothing about evolution.

[PS: One wonders if a similar effect could explain the so-called missing dwarfs problem which afflicts CDM (e.g. Klypin et al 1999). If the correlation between SB and luminosity in equation (69) holds all the way down to very faint objects, then their SBs and Visibilities would render them exceedingly hard to find, even nearby. Equation (69) implies that Δμ_1/2 = −1/3 ΔM. Thus, a.001L* exponential dwarf would have a SB = −1/3 × 7.5 = 2.5 mag dimmer than a giant at the peak of the window, and its Visibility would consequently be very small, and if the correlation were slightly steeper, for example, Δμ_1/2 = −1/2 ΔM, which is probably not ruled out by the observations, then Δμ(P) would be 3.5 − (7.5/2), that is, would be negative and such a peak dwarf would be totally and irretrievably sunk below the sky at any redshift.]

11 Discussion

If the universe is expanding, then the associated Tolman dimming should render conventional galaxies undetectable at high z. Most should be heavily affected by the sky at redshift 0.5, all totally submerged by redshift 2. The fact that we can easily see galaxies out to redshift 7 means either that conventional galaxies have undergone the most dramatic evolution (ESDH) or that the galaxies out there belong to different populations, different dynasties, whose descendants have not so far been identified nearby (SPDH).

We have argued (Section 1) that pure luminosity evolution (<3 mag) will have a hard job compensating for Tolman dimming at a redshift of 2 (5 mag); at a redshift of 7 (9 mag) it seems to be out of the question, especially as models with steady star formation rates, which match the stellar populations of nearby galaxies (e.g. Tosi 2008), actually brighten with time, that is, fade with redshift. Whether or not the SPDH is true it seems that the conventional ESDH cannot explain the latest HDFs.

If our neighbourhood galaxies are role models, then high-redshift galaxies are truly bizarre. They are an order of magnitude smaller in physical size, while their intrinsic SBs must be 9 mag or 4000 times higher. Moreover, and this is even more extraordinary, they must have systematically adjusted their sizes and their SBs over cosmic time so as to squeeze themselves through the narrow visibility window at all the various intermediate redshifts where they can be seen. Such dramatic evolution is hardly consistent with the archaeology of nearby galaxies whose star formation histories seem rather steady and quiescent over time. Furthermore, the high-redshift galaxies are, in comoving terms, rather rare (e.g. McLure et al. 2010), too few in number to provide the UV radiation needed to re-ionize the IGM at redshifts between 6 and 11 (Robertson et al. 2010), and this difficulty is compounded if downsizing is a physical, as opposed to an illusory, phenomenon, for the lower luminosity galaxies form too late to contribute to the UV budget when it would be necessary.

The SPDH requires no such dramatic evolution, and explains both downsizing (Section 9) and galaxy expansion (Section 6) as illusory phenomena, the side effects of Visibility theory, that is, SB selection. It also leads, through its postulation of large numbers of sunken galaxies, particularly at high redshift, to a natural solution for the re-ionization problem, and it is interesting that there are two other strong hints in the recent literature at the presence of such a sunken high-redshift population: one due to what we call infant mortality and the other due to an excess of high-z QSOALs. We discuss these next.

Infant mortality. In the ESD hypothesis, the stellar mass density at any epoch ought to equal the accumulated rate of star formation over all preceding epochs:

In the SPD scenario, this, however, will no longer apparently be the case because in moving from z to z + dz some lower SB galaxies will appear to sink beneath the sky due to Tolman dimming [i.e. ], while other higher SB objects will apparently surface (thanks to aberration) to partially replace them. Thus, the integral above should be replaced by

where the net {} could be either positive or negative, depending on the distribution of galaxy numbers as a function of intrinsic SB. All one can say for sure is that there is no reason to expect a good match between observed stellar densities and accumulated past star formation, and such a mismatch has been noted by many observers. For instance, Perez-Gonzalez et al. (2008) remark: ‘We find that the cosmic SFR densities estimated by differentiating the evolution of cosmic stellar mass density do not match the observations based on direct SFR tracers as also noted by Rudnick et al. (2006), Hopkins & Beacom (2006) and Burch et al. (2006). The mismatch up to z ∼ 2 (a factor of 1.7) could be explained by changing the IMF. But as z rises from 2 to 4.5 the discrepancy is larger (a factor 4 to 5 …)’. This highly significant mismatch is in the sense that less galaxies are observed in each redshift bin than the total of previous star formation would lead one to expect. Under the SPD hypothesis this would be naturally explained if most high-redshift galaxies sink below the sky once their most vigorous period of star formation comes to an end. This is rather direct evidence of the SPDH, though perhaps not conclusive.

Sunken DLAs. Even where they cannot be seen in emission, sunken galaxies should still show up in absorption, in particular as QSOALs, probably of DLAs, that is, of high column density variety. The number detected in the redshift range

where g(z) dz is the physical path-length derived from some cosmological model, ϕ(z, L) is the comoving density of galaxies of luminosity L, and A(z, L) is their effective cross-section per galaxy. As is well known (e.g. Wolfe, Gawiser & Prochaska 2005) if ϕ(z, L) corresponds to the local value for L* galaxies, A(L, z) ∼ observed optical area of L*, and a reasonable extra boost (by a factor of 2–15) is allowed for the dwarf contribution associated with the L*, then there is a fair correspondence with the observed N(z), that is, between 5 and 10 per cent of high-redshift quasars (z ≤ 6) have DLAs in their spectra. Unfortunately, that sets no absolute limit on ϕ, and hence on a population of sunken galaxies, without some independent knowledge of the cross-section A(L, z) which is not available. One can always increase ϕ by decreasing A. Nevertheless, if the SPD hypothesis is true, one might expect a mismatch between the z dependence of N(z) and the number density inferred from the rate of galaxy formation, in the sense that that there will probably be more DLAs in the distance, corresponding to the extra galaxies out there which have sunk from sight, and in a qualitative sense at least that is what seems to be observed in the large SDSS sample of quasars recently analysed by Prochaska & Wolfe (2009). Instead of a decline in cross-section [our N(z), their l(z)] and in comoving H i density with rising redshift, caused by the decline in the number of already formed disc galaxies with z, they find an increase by a factor of 2 between z = 2 and 4.5 (only 2 Gyr) which they find ‘a profound and surprising result’. There may be other explanations (which they mention) but it seems qualitatively consistent with the idea of a larger proportion of sunken galaxy absorbers at higher redshifts.

So there is significant indirect evidence in favour of the SPD hypothesis and its implication that the universe is stuffed with hidden galaxies. Moreover, the SPDH is an almost inevitable consequence of Visibility theory, which is hardly radical, but usually neglected. If hidden galaxies are not ubiquitous, it will take a great number of fortuitous coincidences to explain why all the detected galaxies in the universe have arranged themselves, at all redshifts, so as to squeeze through our narrow, parochial, visibility window (see e.g. Fig. 2).

Indirect evidence is all very well but direct evidence of hidden galaxies, particularly of such a rich population as the SPD hypothesis requires, would be far more persuasive. If hidden galaxies are so common why have not they turned up in dedicated searches with large telescopes, and why have not far more of them appeared in the blind H i surveys (that are of course free of SB selection) that we and others have recently been carrying out?

With the benefit of hindsight, we can answer both questions. The argument at the end of Section 4 is new. We and others understandably supposed that a large enough optical telescope, fitted with CCDs and dedicated to the search, would turn up low-SB galaxies, if they exist. But, alas, that is simply not true, and equation (34) reveals why. When it comes to searching for low-SB galaxies, Tolman dimming, together with the small sizes of CCDs, more than cancels out their high quantum efficiencies and infinite dynamic ranges. The Schmidt photographic surveys, completed in the 1980s, represent the best that can be done. That is a depressing admission, but there seems to be no practical way around equation (34) and (41) in Section 4.

The blind 21-cm surveys which we and others have carried out such as HIPASS (Meyer et al. 2003), HIJASS (Lang et al. 2003), AGES (Cortese et al. 2008) and ALFALFA (Giovanelli et al. 2005) have turned up only one truly invisible giant galaxy, Virgo HI21 (Davies et al. 2004; Minchin et al. 2005, 2007), and there is even some dispute about that (Haynes, Giovanelli & Kent 2007). In fact HIPASS’ failure to turn up a single invisible galaxy among its 4000 detections in the Southern hemisphere has been used to claim (Doyle et al. 2005; Wong et al. 2006; Wong et al. 2009) that hidden galaxies do not exist, or are very rare (and we were party to that claim). Our re-analysis (Disney 2008; Disney & Lang, in preparation) shows, however, that the claim was based on a grossly optimistic estimate for the reliability of the optical identifications involved. When clustering is allowed for, there could well be 100 dark galaxies in the sample, that is, H i sources that have been misidentified with optically-bright objects that are fortuitously close by in both angular and redshift space, and blind surveys with a larger dish will not improve the situation, because their extra resolution is exactly counterbalanced by the extra distance at which the typical sources will be found (Disney 2008). Anyway when objects such as Malin 1 certainly exist, a giant low-SB galaxy 200 kpc across, containing >10¹¹ M_⊙ of H i (Bothun & Impey 1989), one has to be cautious about blind-scanning techniques in general. Such objects nearby will be much larger than the scanning radio beams, and so tend to be lost in the process of noise subtraction. None of the existing blind H i surveys, in our opinion, sets strong constraints on the presence of H i-rich hidden galaxies nearby. In such surveys, absence of evidence is not strong evidence of absence.

Nevertheless it remains vital to pin down some of the hypothetical hidden galaxies. At low redshifts in the H i suspicious optical identifications should be vigorously pursued with interferometers, and the compact descendants of the z = 7 galaxies, now small, faint and choked with dust (Section 7), might still be locatable in the far-IR relatively nearby. At high redshift, sunken MWs beyond a z of 1.2 may have regions, if seen in emission lines, that will still rise above the sky in objective prism surveys. Moreover, they, and their elliptical counterparts (z > 2), should still give rise to faint supernovae which may turn up in what otherwise appears to be intergalactic space, and a start has been made in that direction (Hayward, Irwin & Bregman 2005).

In one sense, one must hope that the SPD hypothesis is wrong, for if it is right then extragalactic research is going to be so much harder. The obvious programme of decoding galaxy formation and evolution simply by building larger instruments such as JWST or ELT to look at fainter, more distant objects will not work because a given dynasty of galaxies will remain visible through our visibility window for only a limited range of redshifts, that is, for only a restricted portion of its life. We might see the infants of one dynasty, the children of another, the adults of a third, and the grizzled elders of a fourth only among our neighbours.

On the other hand, the SPD hypothesis has strong epistemic advocates. It is extremely parsimonious (Gauch 2005) relying as it does only on Tolman dimming and visibility theory, the last of which we have been at some pains to explain and defend. Neither is it the least radical in the sense that it employs assumptions outside very ordinary physics, and it is vulnerable in that it predicts the existence of whole dynasties of galaxies which are presently undetected, but whose existence may eventually be possible to affirm or deny.

Acknowledgments

We thank Mathias Disney (UCL Geography) for the diagrams, Nino Disney, Richard Elliott and Joe Romano (University of Texas, Brownsville) for indispensable help with word processing, and Peter Coles for drawing our attention to Ned Wright's very useful online cosmology calculator. We would also like to thank an anonymous referee who made some wise and challenging comments which led to some significant changes and improvements in the initial draft. MJD especially wants to thank fellow members of the HST FOC and WFC3 camera teams from whom he learned so much between 1976 and 2011, and the European Space Agency (via the European Space Telescope Coordinating Facility) for travel support over the period.